Research ArticleAn Improved ZMP-Based CPG Model of BipedalRobot Walking Searched by SaDE
H F Yu E H K Fung and X J Jing
Department of Mechanical Engineering Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong
Correspondence should be addressed to H F Yu fai5121988yahoocomhk
Received 31 October 2013 Accepted 23 December 2013 Published 4 March 2014
Academic Editors D K Pratihar and K Terashima
Copyright copy 2014 H F Yu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper proposed amethod to improve the walking behavior of bipedal robot with adjustable step length Objectives of this paperare threefold (1) Genetic Algorithm Optimized Fourier Series Formulation (GAOFSF) is modified to improve its performance(2) Self-adaptive Differential Evolutionary Algorithm (SaDE) is applied to search feasible walking gait (3) An efficient method isproposed for adjusting step length based on themodified central pattern generator (CPG)modelTheGAOFSF ismodified to ensurethat trajectories generated are continuous in angular position velocity and acceleration After formulation of the modified CPGmodel SaDE is chosen to optimize walking gait (CPGmodel) due to its superior performanceThrough simulation results dynamicbalance of the robot with modified CPG model is better than the original one In this paper four adjustable factors (119877hssupport119877hsswing119877kssupport and119877ksswing) are added to the joint trajectoriesThrough adjusting these four factors joint trajectories are changedand hence the step length achieved by the robot Finally the relationship between (1) the desired step length and (2) an appropriateset of 119877hssupport 119877hsswing 119877kssupport and 119877ksswing searched by SaDE is learnt by Fuzzy Inference System (FIS) Desired joint angles canbe found without the aid of inverse kinematic model
1 Introduction
Recently many approaches have been adopted for generationof bipedal walking gait Some researches [1ndash3] adopted asimplified dynamicmodel to generate walking gait calculatedthrough inverse kinematic model which is complex andhence the computation load is high
Inspired by neural science some researchers investigatedcentral pattern generator (CPG) The prime reason forarousing their interest is that CPG models provide severalparameters for modulation of locomotion such as step strideand rhythm and are suitable to integrate feedback sensorsHence a good interaction between the robot and the envi-ronment can be achieved [4] According to Ijspeert [4] CPGbecomesmore andmore popular in robot community Taga etal [5] integrated feedbacks with neural oscillators for unpre-dicted environment Yang et al [6] Shafii et al [7] and Yazdiet al [8] utilized TFS to formulate ZMP-based CPG modelas the basic walking pattern of bipedal robot Or [9] pre-sented a hybrid CPG-ZMP control system for flexible spine
humanoid robot Aoi and Tsuchiya [10] proposed a loco-motion control system based on CPG model for straightand curved walking Farzenah et al [11] noted that manyresearches on CPG model are designed for specific motiononly and thus cannot generate arbitrary walking gait such aschanging step length and proposed 31 TS-Fuzzy systems foradjusting speed and step length
Ijspeert [4] and Gong et al [12] stated that stochasticpopulation-based optimization algorithms have been chosento optimize parameters of CPG model in many studiesGenetic Algorithm (GA) [6 13] Genetic Programming (GP)[14] Particle Swarm Optimization (PSO) [7] and Bee Algo-rithm [8] are adopted in searching the parameters of CPGmodel Besides the above-mentioned techniques there arestill other gradient-free optimization techniques Storn andPrice [15] proposed Differential Evolution (DE) and con-ducted comparisons with some prominent algorithms suchas Adaptive Simulated Annealing (ASA) and the BreederGenetic Algorithm (BGA) DE outperforms the above-men-tioned prominent algorithms in terms of least number of
Hindawi Publishing CorporationISRN RoboticsVolume 2014 Article ID 241767 16 pageshttpdxdoiorg1011552014241767
2 ISRN Robotics
generations for finding global minimum [15] Similar resultsare reported in the following studies [16ndash18] Hegery et al [16]carried out comparisons between DE and GA on N-Queenand travelling salesman problem and concluded that theperformance of DE is better Tusar and Filipic [17] carried outcomparisons between DE-based variants DEMO and basicGA on multiobjective optimization problem and their resultshowed that DEMO outperforms basic GA Vesterstroslashm andThomsen [18] noted that DE outperforms PSO and Evolu-tionary Algorithms (EAs) on majority of numerical bench-mark problems DE consists of population size (NP) scalingfactor (119865) and crossover rate (CR) which significantly affectthe performance of DE [19ndash22] Different problems requiredifferent parameters and strategies for effective optimizationEven in the same problem different regions of search spacemay require different strategies and parameters for betterperformance [20] It is time consuming to search the mostappropriate strategy and parameters by trial and error HenceOmran et al [21] and Brest et al [22] have proposed differentmethods to adjust CR and F However appropriate mecha-nism for choosing suitable strategies is not considered in [2122] Qin et al [20] proposed Self-adaptive Differential Evo-lution Algorithm (SaDE) which can adjust CR 119865 and choosestrategy automatically during optimization SaDE outper-forms conventional DE variants and the other adaptive DEsuch as SDE [21] and jDE [22] in terms of higher successfulrate
Based on the above-mentioned findings this paperfocuses on (1) CPG model for trajectory generation and (2)providing an efficient method to adjust step length In thispaper original CPG model proposed by Yang et al [6] isadopted since it provides a good foundation for the goal statedin this paperThe angular velocity of trajectories generated byGAOFSF [6] is usually discontinuous which has an adverseeffect on ZMP As a result GAOFSF ismodified to ensure thatthe trajectory generated is continuous in angular position andits first and second derivatives After formulation of modifiedCPGmodel parameters of CPGmodel are searched based onkinematic and dynamic constraints It shows that the problemcan be formulated as a multiobjectives and multiconstraintsoptimizationGradient-free optimization technique is chosensince a set of parameters is searched in a highly irregular andmultidimensional space which cannot be handled by stan-dard gradient-based search method [12] SaDE is chosen asthe method for optimizing the walking gait of robot in thispaper because (1) its performance is superior and (2) appro-priate strategies and parameters are not chosen manuallyBased on [6 23] step length can be varied by simply changingseveral adjustable factors of GAOFSF Look-up table pro-posed by Yang et al [6] is not adopted in this paper because(1) a lot of memory is occupied if tremendous data is storedand (2) arbitrary step length within specific range cannot becommanded to the robot To deal with this problem fourparameters (119877hssupport 119877hsswing 119877kssupport and 119877ksswing) areadded to themodified CPGmodel and searched by SaD FourFIS systems are used to learn the relationship between (1)desired step lengths and 119877hssupport 119877hsswing 119877kssupport and119877ksswing Then desired joint angles can be found without theaid of inverse kinematics
Table 1 Joint number of support leg and swing leg corresponds todifferent joints
Joints of leg Support leg Swing legAnkle joints 1-2 5-6Knee joint 3 4Hip joints 4ndash6 1ndash3
Table 2 Height of each link in support leg and swing leg
Link number Support leg Swing leg0 0026m NA1 004m 0028m2 00645m 00385m3 00645m 00645m4 00385m 0064m5 0028m 004m6 0096m 0026m
2 Kinematic Model and DynamicModel of Bipedal Robot
In Figure 1 it shows that the bipedal robot consists of 12DoF Each leg has 6 DoF RL and LL represent right and leftlegs respectively while 1 2 3 6 represent joint numberFigure 1 assumes that right leg is the support leg while the leftleg is the swing leg Joint number 119899 of support leg and swingleg corresponds to different joint shown in Table 1 Also 119911-axis of local coordinates attached on different joints acts asrotation axis and the direction of rotation is determined byright-hand rule
Information of physical dimension of bipedal robot ismeasured and simplified based on a modified Kondo-3HVThe total mass of the bipedal robot is about 14 kg The massof upper trunk and lower body is about 04 kg and 1 kgrespectively Since the mass of each link in lower body isalmost the same then their masses are simply obtained by1 kg12 = 0083 kg The height of each link is shown inTable 2
Since Denavit-Hartenberg notation [24 25] and iterativeNewton-Euler dynamic algorithm [26 27] are maturelydeveloped and commonly used in many studies of bipedalrobot these two methods are adopted to formulate forwardkinematic model and inverse dynamic model respectivelyin this paper Since this paper focuses on bipedal walkingon horizontal flat plane in sagittal plane (parallel to the119884119866
minus 119885119866
plane of global coordinate) only ZMP119910
shown in(1) is considered and acts as an indicator to evaluate thedynamic equilibrium of the bipedal robot In Figure 2 a localcoordinate is attached to the center of support foot to observethe variation of ZMP
119910
with time
ISRN Robotics 3
LinkRL6
LinkRL5
LinkRL4
LinkRL3
LinkRL2
LinkRL1
LinkRL0 LinkLL6
LinkLL5
LinkLL4
LinkLL3
LinkLL2
LinkLL1
XRL6
XRL5
XRL4
XRL3
XRL2
XRL1
YRL6
YRL5
YRL4
YRL3
YRL2
YRL1
ZRL6
ZRL5
ZRL4
ZRL3
ZRL2
ZRL1
XLL6
XLL5
XLL4
XLL3
XLL2
XLL1
YLL6
YLL5
YLL4
YLL3
YLL2
YLL1
ZLL6
ZLL5
ZLL4
ZLL3
ZLL2
ZLL1
ZG
XG YG
Figure 1 Schematic diagram of bipedal robot
Consider
ZMP119910
= (
6
sum
119894=1
119898119894119877119871
(119894119877119871
+ 119892) 119910119894119877119871
+
6
sum
119894=1
119898119894119871119871
(119894119871119871
+119892) 119910119894119871119871
minus
6
sum
119894=1
119898119894119877119871
119910119911119894119877119871
minus
6
sum
119894=1
119898119894119871119871
119910119911119894119871119871
)
times (
6
sum
119894=1
(119894119877119871
+ 119892)119898119894119877119871
+
6
sum
119894=1
(119894119871119871
+ 119892)119898119894119871119871
)
minus1
(1)
3 Formulation of ModifiedZMP-Based CPG Model
Hip and knee trajectories of GAOFSF [6] consist of twodifferent sections For hip joint two different TruncatedFourier Series (TFS) are used to formulate the upper portion(120579ℎ
+) and lower portion (120579ℎ
minus) of trajectory For knee jointTFS and lock phase are joined together to formulate thewholetrajectory Based on Figure 3 the angular position is observedto be continuous However angular velocity of searchedtrajectory generated by GAOFSF is usually discontinuous (1)at the transition between 120579
ℎ
+ and 120579ℎ
minus (1199053
and 1199056
or 1199050
) and
ZZMPXZMP
YZMP
Figure 2 Local coordinate system attached on the foot sole (topview)
(2) at the beginning (1199051
or 1199054
) and the end of lock phase(1199052
or 1199055
) Abrupt change in angular velocity has an adverseeffect on ZMP and hence the dynamic equilibrium of bipedalrobot In this paper the GAOFSF is modified to ensure thatthe trajectories generated are continuous in angular positionand its first derivative and second derivative [119905
5
1199052
] and[1199052
1199055
] are regarded as the period of left support phase andright support phase respectively Then 119905
2
and 1199055
are the
4 ISRN Robotics
Right hipLef hip
Right kneeLef knee
Low
er p
ortio
nU
pper
por
tion
ck
cht
120579
t0 t1 t2 t3 t4 t5 t6
120587120596i
120587120596k
Figure 3 General shape of GAOFSF [6]
landing time of left support phase and right support phaserespectively [6] The time duration (119905
5
minus 1199052
) of one step isset as 1 s to ensure a reasonable walking speed 119905
2
and 1199055
areset as 07 s and 17 s while 119905
3
and 1199056
are set as 1 s and 2 sFour different parameters (119877hssupport 119877hsswing 119877kssupport and119877ksswing) are added for adjustment of step length Details offinding the values of these four parameters are discussed inSection 5 In this section investigation is mainly focused onthe formulation of modified CPG model
31 Hip Trajectory in Sagittal Plane Since the peaks of 120579ℎ
+
and 120579ℎ
minus are different two different TFS are required toformulate the hip trajectory Yang et al [6] have proposed5th order TFS and showed that the amplitudes of 4th and 5thorders are too small which can be neglected 3rd order TFS isadopted In (2)-(3) angular velocity of 120579
minus
ℎ
and 120579+
ℎ
at 1199053
and 1199056
is set as equal to ensure angular velocity is continuous Thustwomore orders are added to 120579
ℎ
minus to satisfy this constraint andare calculated by (4)-(5)
Consider
120579minus
ℎ
(1199056
) = 120579+
ℎ
(1199056
) (2)
120579minus
ℎ
(1199053
) = 120579+
ℎ
(1199053
) (3)
1198615
=minus (2119860
1
+ 61198603
+ 21198611
+ 61198613
)
10 (4)
1198614
=(minus1198601
+ 21198602
minus 31198603
minus 1198611
minus 21198612
minus 31198613
minus 51198615
)
4 (5)
119905 isin [1199050
1199053
] 120579rhs = 120579ℎ
minus
(119905) 120579rhs = 120579ℎ
+
(119905)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ119905) + 119888ℎ
(6)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ119905) + 119888ℎ
119905 isin [1199053
1199056
] 120579rhs = 120579ℎ
+
(119905 minus 1199053
) 120579lhs = 120579ℎ
minus
(119905 minus 1199053
)
(7)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (8)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (9)
120596ℎ
=120587
1199053
minus 1199050
(10)
32 Knee Trajectory in Sagittal Plane In the knee trajectoryYang et al [6] proposed a lock phase ([119905
1
1199052
] and [1199054
1199055
])in knee trajectory which assumes constant joint angle andzero angular velocity and accelerationThis introduces abruptchange in angular velocity The main goals to be achievedin the modified knee trajectory are (1) continuity in angularposition velocity and acceleration and (2) the advantageproposed in GAOFSF that can be maintained Then a newformulation is proposed in the following equation
119905 isin [1199050
1199052
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199052
1199056
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199052
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199050
1199055
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199055
1199056
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199055
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
120596119896
=2120587
1199056
minus 1199050
(11)
ISRN Robotics 5
In (11) lock phase is canceled to ensure continuous angularvelocity By adjusting 119888119896 [6] GAOFSF can achieve energyefficient and stable ldquobent kneerdquo walking gait This advantageis still remained in the proposedmodifiedmodelThe generalshape of the hip and knee trajectories generated by modifiedCPG model is shown in Section 6
33 Ankle Trajectory in Sagittal Plane Right and left anklejoint trajectories are simply formulated as (12) to ensure thatthe trunk is upright and swing foot is parallel to the horizontalflat plane
Consider
120579as = minus 120579hs minus 120579ks (12)
4 Optimization of BasicWalking Pattern by SaDE
This section focuses on how to search the basic walkingpattern of bipedal robot by SaDE [19 20] To simplify thewhole process in this section 119877hssupport 119877hsswing 119877kssupportand 119877ksswing are set as 1The following are the main objectivesof basic walking pattern to be achieved through using SaDE
(1) The desired step length is set as 005m(2) Upper bound and lower bound of swing height are set
as 002m and 001m respectively(3) Premature landing should not occur throughout the
walking cycle(4) ZMP is within the area of support polygon through-
out the walking cycle
The procedure of SaDE takes the following steps Also aflow chart of SaDE is shown in Figure 30 of the Appendixto facilitate the understanding of readers Details of theprocedure are discussed in Sections 41ndash47
(1) Select fitness functions (119891119894
) and constraints (119878119894
)(2) Code the parameters to be searched to form target
vector (119883119894119866
)(3) Initialize the first generation (119866
1
) of population (119894)(4) Initialize crossover rate (CR
119906119894119895) scaling factor (119865
119906119894119895)
and probability (119875119895
) for each mutation strategies (119895)based on CR
119895
and 119865119895
(5) Select mutation strategies (119895) based on probability
(119875119895
) through MATLAB function rand() and performmutation operation to generate mutant vectors (V
119894
)(6) Perform crossover operation to generate trial vectors
(119906119894
)(7) Perform selection operation to select next generation
of target vectors (119883119894119866+1
)(8) During learning period (LP) record the successful
time (ST119895
) and failure time (FT119895
) of each strategy (119895)(9) During learning period (LP) record values of CR
119906119894119895
and119865119906119894119895
for each strategy (119895) that successfully help thetrial vectors (119906
119894
) enter the next generation 119866119894+1
(10) Upon completion of LP probability (119875119895
) for choosingsuitable strategies (119895) is adjusted based on successfulrate of strategy 119895 (SR
119895
) calculated through ST119895
andFT119895
(11) Upon completion of LP CR119895
and 119865119895
are adjustedbased on the mean values of stored CR
119906119894119895and 119865
119906119894119895
respectively
(12) Repeat procedures (4)ndash(11) until maximum genera-tion is completed
41 Fitness Functions and Constraints (Step 1) Fitness func-tions (119891
119899
) and constraints (119878119899
) are designed ormodified basedon [6] Since trunk occupies a large proportion of the wholebodymass abrupt change in trunk velocity can lead to abruptchange in ZMP
119910
1198911
is formulated as follows
119881trunk =119873
sum
119899=1
119881trunk119899
119873
1198911
= radic
119873
sum
119899=1
(119881trunk119899 minus 119881trunk)2
119873
(13)
where119873 is number of dataStrike velocity during landing should be as small as
possible since impact between landing foot and the groundcan cause mechanical wear of parts and unstable walking 119891
2
is stated as follows [6]
1198912
= radic119881strike1199092
+ 119881strike1199102
+ 119881strike1199112
(14)
If ZMP119910
is within the area of support polygon[0061 minus0061] throughout the walking cycle dynamicequilibrium of the robot is satisfactory To deal with thediscrepancies between simulation model and physicaltest bed a safety factor (001m) is added to ensure goodperformance during experiment 119878
1
is formulated as follows
1198781
=
119873
sum
119899=1
max (10038161003816100381610038161003816ZMP119910119899
The total scores used for evaluating each target vector(119883119894119866
) are formulated as follows
scores = minus1200 +
119873
sum
119894=1
120596119900119899
119891119899
+
119873
sum
119894=1
120596119901119899
119878119899
(23)
These weightings are set by trial and error to ensuretheir importance is almost the same and walking gait withsatisfactory performance can be obtained
120596119900
= [1000 500]
120596119901
= [2500 285 265 100 2200 12000 300 960 25000]
(24)
where 120596119900119899
is weighting of fitness functions and 120596119901119899
isweighting of constraints
42 Initialization of Parameters of Target Vectors (119883119894119866
)(Steps 2 and 3) The target vector (119883
119894119866
) is formulated asfollows
119883119894119866
= [1198601
1198602
1198603
1198611
1198612
1198613
1198621
1198622
1198623
119888ℎ 119888119896] (25)
Then the parameters of population (119894) are initialized byMATLAB function rand() which generates [0 1] randomly
43 Initialization of Parameters of SaDE (Step 4) Since therange [minus1 1] of parameters to be searched is small thenpopulation (119894) of one generation (119866) is set as 30 to balancebetween good diversity of population and low computationload Four mutation strategies (119895 = 1 2 3 4) are adoptedFor strategy 119895 MATLAB function 119899119900119903119898119903119899119889 (mean value120590) is used to generate CR
119906119894119895and 119865
119906119894119895for each trial vector
(119906119894
) In the initial phase the crossover rate (CR119906119894119895) should
not be too high to prevent premature convergence while thescaling factor (119865
119906119894119895) should not be too small to affect the
exploration ability Hence a moderate value (05) is assignedto CR
119895
and 119865119895
(mean value) and 120590 is set as 01 This functioncan generate random numbers from the normal distributionthrough mean value (CR
119895
119865119895
) and standard deviation (120590) 119875119895
is set as 025 for strategy 119895 so that each strategy has the equalchance to be chosen during the first learning period
44 Mutation Operation of SaDE (Step 5) DErand1DErand2 and DEcurrent-to-rand2 provide goodexploration ability while DEcurrent-to-best2 demonstratesgood convergence speed To balance between explorationability and convergence speed these four strategies areadopted These four mutation strategies are stated in(27)ndash(30) as follows
(1) DErand1
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (27)
(2) DErand2
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) + 119865119894119895
(1198831199034119866
minus 1198831199035119866
)
(28)
ISRN Robotics 7
(3) DErand-to-best2
V119894119866
= 119883119894119866
+ 119865119894119895
(119883best119866 minus 119883119894119866
) + 119865119894119895
(1198831199031119866
minus 1199091199032119866
)
+ 119865119894119895
(1198831199033119866
minus 1198831199034119866
)
(29)
(4) DEcurrent-to-rand2
V119894119866
= 119883119894119866
+ 119865119894119895
(1198831199031119866
minus 119883119894119866
) + 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (30)
45 Crossover Operation of SaDE (Step 6) Because of itspopularity binomial crossover operator [20] is utilized in allmutation strategies
If (rand () le CR119894119895
) || (119902 == 119902rand)
119906119902119894119866
= V119902119894119866
Else if (rand () gt CR119894119895
)
119906119902119894119866
= 119909119902119894119866
End
(31)
where 119902 is the parameter number (1 2 3 4 11) in thetarget (119883
119894119866
) and mutant (V119894119866
) vector Arbitrary parameternumber is assigned to 119902rand to ensure that trial vector (119906
119894119866
)is different from target vector (119883
119894119866
) If values of 119906119902119894119866
exceedthe desired range [minus1 1] of parameters to be searched 119906
119902119894119866
is reset by MATLAB function rand ()
46 Selection Operation of SaDE (Step 7) If score of trialvector (119906
119894119866
) is lower than that of target vector (119909119894119866
) then trialvector enters the next generation Otherwise target vectorenters the next generation
If scores119906119894119866
le scores119909119894119866
119909119894119866+1
= 119906119894119866
Else
119909119894119866+1
= 119909119894119866
End
(32)
47 Adjustment of 119875119895
119862119877119895
and 119865119895
(Steps 8ndash11) Initially alearning period (LP = 5 119892119890119899119890119903119886119905119894119900119899119904) is assigned to balancebetween good sample size of data and update frequencyDuring learning period successful times (ST
119895
) and failuretimes (FT
119895
) of each mutation strategy (119895 = 1 2 3 4) in eachgeneration are recorded For example if strategy 119895 is chosenand helps one trial vector (119906
119894119866
) to enter next generationthen ST
119895
is added by 1 Otherwise FT119895
is added by 1 Oncelearning period is completed 119875
Step length (m) Lower bound of maximum desired swingheight (m)
005m 001m004m 0009m003m 0008m002m 0007m001m 0006m
rate (SR119895
) Then ST119895
and FT119895
are reset to zero for the nextlearning period to eliminate effect of the past data
SR119895
=
ST119895
FT119895119866
+ ST119895119866
119875119895
=
SR119895
sum119873=4
119895=1
SR119895
(33)
Similar approach is applied to adjust CR119895
and 119865119895
Duringlearning period (LP = 5 generations) for strategy 119895 CR
119906119894119895and
119865119906119894119895
corresponding to the trial vectors (119906119894119866
) that successfullyenters next generation are stored in database Once learningperiod is completed mean value (CR
119895
and 119865119895
) of the storedvalue corresponding to strategy 119895 is calculated and is used togenerate a new set of CR
119906119894 119895and 119865
119906119894119895 Then storage data of
CR119906119894119895
and 119865119906119894119895
are removed for the next learning period toeliminate the effects of past data
5 Optimization of Factors (119877hssupport 119877hsswing119877kssupport and 119877ksswing) for Step LengthAdjustment
51 Objective Functions and Constraints of SaDE The objec-tive functions and constraints mentioned in Section 4 areadopted to search appropriate value of 119877hssupport 119877hsswing119877kssupport and119877ksswingfinal corresponding to different desiredstep length (004m 003m 002m and 001m)
52 Lower Bound of Maximum Desired Swing Height It isnatural that the maximum swing height becomes lower toconsume less energy if step length is smaller Lower bound ofswing height is reset as lower value for different step lengths(Table 3) while the upper bound (002m) remains the same
53 Smooth Transition of 119877jointsupportswing Values of119877jointsupportswing (joint = rhs lhs rks and lks) of support legand swing leg are different At the moment of landing time119877jointsupport is changed to 119877jointswing since the support leg ischanged to swing leg in the next step In order to have asmooth transition fifth order polynomial shown in equation(34) is utilized The period of transition time (119905
119891
minus 1199050
) is setas 04 s which is 40 of time duration (1 s) of one step tobalance between rapid transition time and good dynamicequilibrium
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
generations for finding global minimum [15] Similar resultsare reported in the following studies [16ndash18] Hegery et al [16]carried out comparisons between DE and GA on N-Queenand travelling salesman problem and concluded that theperformance of DE is better Tusar and Filipic [17] carried outcomparisons between DE-based variants DEMO and basicGA on multiobjective optimization problem and their resultshowed that DEMO outperforms basic GA Vesterstroslashm andThomsen [18] noted that DE outperforms PSO and Evolu-tionary Algorithms (EAs) on majority of numerical bench-mark problems DE consists of population size (NP) scalingfactor (119865) and crossover rate (CR) which significantly affectthe performance of DE [19ndash22] Different problems requiredifferent parameters and strategies for effective optimizationEven in the same problem different regions of search spacemay require different strategies and parameters for betterperformance [20] It is time consuming to search the mostappropriate strategy and parameters by trial and error HenceOmran et al [21] and Brest et al [22] have proposed differentmethods to adjust CR and F However appropriate mecha-nism for choosing suitable strategies is not considered in [2122] Qin et al [20] proposed Self-adaptive Differential Evo-lution Algorithm (SaDE) which can adjust CR 119865 and choosestrategy automatically during optimization SaDE outper-forms conventional DE variants and the other adaptive DEsuch as SDE [21] and jDE [22] in terms of higher successfulrate
Based on the above-mentioned findings this paperfocuses on (1) CPG model for trajectory generation and (2)providing an efficient method to adjust step length In thispaper original CPG model proposed by Yang et al [6] isadopted since it provides a good foundation for the goal statedin this paperThe angular velocity of trajectories generated byGAOFSF [6] is usually discontinuous which has an adverseeffect on ZMP As a result GAOFSF ismodified to ensure thatthe trajectory generated is continuous in angular position andits first and second derivatives After formulation of modifiedCPGmodel parameters of CPGmodel are searched based onkinematic and dynamic constraints It shows that the problemcan be formulated as a multiobjectives and multiconstraintsoptimizationGradient-free optimization technique is chosensince a set of parameters is searched in a highly irregular andmultidimensional space which cannot be handled by stan-dard gradient-based search method [12] SaDE is chosen asthe method for optimizing the walking gait of robot in thispaper because (1) its performance is superior and (2) appro-priate strategies and parameters are not chosen manuallyBased on [6 23] step length can be varied by simply changingseveral adjustable factors of GAOFSF Look-up table pro-posed by Yang et al [6] is not adopted in this paper because(1) a lot of memory is occupied if tremendous data is storedand (2) arbitrary step length within specific range cannot becommanded to the robot To deal with this problem fourparameters (119877hssupport 119877hsswing 119877kssupport and 119877ksswing) areadded to themodified CPGmodel and searched by SaD FourFIS systems are used to learn the relationship between (1)desired step lengths and 119877hssupport 119877hsswing 119877kssupport and119877ksswing Then desired joint angles can be found without theaid of inverse kinematics
Table 1 Joint number of support leg and swing leg corresponds todifferent joints
Joints of leg Support leg Swing legAnkle joints 1-2 5-6Knee joint 3 4Hip joints 4ndash6 1ndash3
Table 2 Height of each link in support leg and swing leg
Link number Support leg Swing leg0 0026m NA1 004m 0028m2 00645m 00385m3 00645m 00645m4 00385m 0064m5 0028m 004m6 0096m 0026m
2 Kinematic Model and DynamicModel of Bipedal Robot
In Figure 1 it shows that the bipedal robot consists of 12DoF Each leg has 6 DoF RL and LL represent right and leftlegs respectively while 1 2 3 6 represent joint numberFigure 1 assumes that right leg is the support leg while the leftleg is the swing leg Joint number 119899 of support leg and swingleg corresponds to different joint shown in Table 1 Also 119911-axis of local coordinates attached on different joints acts asrotation axis and the direction of rotation is determined byright-hand rule
Information of physical dimension of bipedal robot ismeasured and simplified based on a modified Kondo-3HVThe total mass of the bipedal robot is about 14 kg The massof upper trunk and lower body is about 04 kg and 1 kgrespectively Since the mass of each link in lower body isalmost the same then their masses are simply obtained by1 kg12 = 0083 kg The height of each link is shown inTable 2
Since Denavit-Hartenberg notation [24 25] and iterativeNewton-Euler dynamic algorithm [26 27] are maturelydeveloped and commonly used in many studies of bipedalrobot these two methods are adopted to formulate forwardkinematic model and inverse dynamic model respectivelyin this paper Since this paper focuses on bipedal walkingon horizontal flat plane in sagittal plane (parallel to the119884119866
minus 119885119866
plane of global coordinate) only ZMP119910
shown in(1) is considered and acts as an indicator to evaluate thedynamic equilibrium of the bipedal robot In Figure 2 a localcoordinate is attached to the center of support foot to observethe variation of ZMP
119910
with time
ISRN Robotics 3
LinkRL6
LinkRL5
LinkRL4
LinkRL3
LinkRL2
LinkRL1
LinkRL0 LinkLL6
LinkLL5
LinkLL4
LinkLL3
LinkLL2
LinkLL1
XRL6
XRL5
XRL4
XRL3
XRL2
XRL1
YRL6
YRL5
YRL4
YRL3
YRL2
YRL1
ZRL6
ZRL5
ZRL4
ZRL3
ZRL2
ZRL1
XLL6
XLL5
XLL4
XLL3
XLL2
XLL1
YLL6
YLL5
YLL4
YLL3
YLL2
YLL1
ZLL6
ZLL5
ZLL4
ZLL3
ZLL2
ZLL1
ZG
XG YG
Figure 1 Schematic diagram of bipedal robot
Consider
ZMP119910
= (
6
sum
119894=1
119898119894119877119871
(119894119877119871
+ 119892) 119910119894119877119871
+
6
sum
119894=1
119898119894119871119871
(119894119871119871
+119892) 119910119894119871119871
minus
6
sum
119894=1
119898119894119877119871
119910119911119894119877119871
minus
6
sum
119894=1
119898119894119871119871
119910119911119894119871119871
)
times (
6
sum
119894=1
(119894119877119871
+ 119892)119898119894119877119871
+
6
sum
119894=1
(119894119871119871
+ 119892)119898119894119871119871
)
minus1
(1)
3 Formulation of ModifiedZMP-Based CPG Model
Hip and knee trajectories of GAOFSF [6] consist of twodifferent sections For hip joint two different TruncatedFourier Series (TFS) are used to formulate the upper portion(120579ℎ
+) and lower portion (120579ℎ
minus) of trajectory For knee jointTFS and lock phase are joined together to formulate thewholetrajectory Based on Figure 3 the angular position is observedto be continuous However angular velocity of searchedtrajectory generated by GAOFSF is usually discontinuous (1)at the transition between 120579
ℎ
+ and 120579ℎ
minus (1199053
and 1199056
or 1199050
) and
ZZMPXZMP
YZMP
Figure 2 Local coordinate system attached on the foot sole (topview)
(2) at the beginning (1199051
or 1199054
) and the end of lock phase(1199052
or 1199055
) Abrupt change in angular velocity has an adverseeffect on ZMP and hence the dynamic equilibrium of bipedalrobot In this paper the GAOFSF is modified to ensure thatthe trajectories generated are continuous in angular positionand its first derivative and second derivative [119905
5
1199052
] and[1199052
1199055
] are regarded as the period of left support phase andright support phase respectively Then 119905
2
and 1199055
are the
4 ISRN Robotics
Right hipLef hip
Right kneeLef knee
Low
er p
ortio
nU
pper
por
tion
ck
cht
120579
t0 t1 t2 t3 t4 t5 t6
120587120596i
120587120596k
Figure 3 General shape of GAOFSF [6]
landing time of left support phase and right support phaserespectively [6] The time duration (119905
5
minus 1199052
) of one step isset as 1 s to ensure a reasonable walking speed 119905
2
and 1199055
areset as 07 s and 17 s while 119905
3
and 1199056
are set as 1 s and 2 sFour different parameters (119877hssupport 119877hsswing 119877kssupport and119877ksswing) are added for adjustment of step length Details offinding the values of these four parameters are discussed inSection 5 In this section investigation is mainly focused onthe formulation of modified CPG model
31 Hip Trajectory in Sagittal Plane Since the peaks of 120579ℎ
+
and 120579ℎ
minus are different two different TFS are required toformulate the hip trajectory Yang et al [6] have proposed5th order TFS and showed that the amplitudes of 4th and 5thorders are too small which can be neglected 3rd order TFS isadopted In (2)-(3) angular velocity of 120579
minus
ℎ
and 120579+
ℎ
at 1199053
and 1199056
is set as equal to ensure angular velocity is continuous Thustwomore orders are added to 120579
ℎ
minus to satisfy this constraint andare calculated by (4)-(5)
Consider
120579minus
ℎ
(1199056
) = 120579+
ℎ
(1199056
) (2)
120579minus
ℎ
(1199053
) = 120579+
ℎ
(1199053
) (3)
1198615
=minus (2119860
1
+ 61198603
+ 21198611
+ 61198613
)
10 (4)
1198614
=(minus1198601
+ 21198602
minus 31198603
minus 1198611
minus 21198612
minus 31198613
minus 51198615
)
4 (5)
119905 isin [1199050
1199053
] 120579rhs = 120579ℎ
minus
(119905) 120579rhs = 120579ℎ
+
(119905)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ119905) + 119888ℎ
(6)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ119905) + 119888ℎ
119905 isin [1199053
1199056
] 120579rhs = 120579ℎ
+
(119905 minus 1199053
) 120579lhs = 120579ℎ
minus
(119905 minus 1199053
)
(7)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (8)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (9)
120596ℎ
=120587
1199053
minus 1199050
(10)
32 Knee Trajectory in Sagittal Plane In the knee trajectoryYang et al [6] proposed a lock phase ([119905
1
1199052
] and [1199054
1199055
])in knee trajectory which assumes constant joint angle andzero angular velocity and accelerationThis introduces abruptchange in angular velocity The main goals to be achievedin the modified knee trajectory are (1) continuity in angularposition velocity and acceleration and (2) the advantageproposed in GAOFSF that can be maintained Then a newformulation is proposed in the following equation
119905 isin [1199050
1199052
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199052
1199056
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199052
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199050
1199055
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199055
1199056
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199055
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
120596119896
=2120587
1199056
minus 1199050
(11)
ISRN Robotics 5
In (11) lock phase is canceled to ensure continuous angularvelocity By adjusting 119888119896 [6] GAOFSF can achieve energyefficient and stable ldquobent kneerdquo walking gait This advantageis still remained in the proposedmodifiedmodelThe generalshape of the hip and knee trajectories generated by modifiedCPG model is shown in Section 6
33 Ankle Trajectory in Sagittal Plane Right and left anklejoint trajectories are simply formulated as (12) to ensure thatthe trunk is upright and swing foot is parallel to the horizontalflat plane
Consider
120579as = minus 120579hs minus 120579ks (12)
4 Optimization of BasicWalking Pattern by SaDE
This section focuses on how to search the basic walkingpattern of bipedal robot by SaDE [19 20] To simplify thewhole process in this section 119877hssupport 119877hsswing 119877kssupportand 119877ksswing are set as 1The following are the main objectivesof basic walking pattern to be achieved through using SaDE
(1) The desired step length is set as 005m(2) Upper bound and lower bound of swing height are set
as 002m and 001m respectively(3) Premature landing should not occur throughout the
walking cycle(4) ZMP is within the area of support polygon through-
out the walking cycle
The procedure of SaDE takes the following steps Also aflow chart of SaDE is shown in Figure 30 of the Appendixto facilitate the understanding of readers Details of theprocedure are discussed in Sections 41ndash47
(1) Select fitness functions (119891119894
) and constraints (119878119894
)(2) Code the parameters to be searched to form target
vector (119883119894119866
)(3) Initialize the first generation (119866
1
) of population (119894)(4) Initialize crossover rate (CR
119906119894119895) scaling factor (119865
119906119894119895)
and probability (119875119895
) for each mutation strategies (119895)based on CR
119895
and 119865119895
(5) Select mutation strategies (119895) based on probability
(119875119895
) through MATLAB function rand() and performmutation operation to generate mutant vectors (V
119894
)(6) Perform crossover operation to generate trial vectors
(119906119894
)(7) Perform selection operation to select next generation
of target vectors (119883119894119866+1
)(8) During learning period (LP) record the successful
time (ST119895
) and failure time (FT119895
) of each strategy (119895)(9) During learning period (LP) record values of CR
119906119894119895
and119865119906119894119895
for each strategy (119895) that successfully help thetrial vectors (119906
119894
) enter the next generation 119866119894+1
(10) Upon completion of LP probability (119875119895
) for choosingsuitable strategies (119895) is adjusted based on successfulrate of strategy 119895 (SR
119895
) calculated through ST119895
andFT119895
(11) Upon completion of LP CR119895
and 119865119895
are adjustedbased on the mean values of stored CR
119906119894119895and 119865
119906119894119895
respectively
(12) Repeat procedures (4)ndash(11) until maximum genera-tion is completed
41 Fitness Functions and Constraints (Step 1) Fitness func-tions (119891
119899
) and constraints (119878119899
) are designed ormodified basedon [6] Since trunk occupies a large proportion of the wholebodymass abrupt change in trunk velocity can lead to abruptchange in ZMP
119910
1198911
is formulated as follows
119881trunk =119873
sum
119899=1
119881trunk119899
119873
1198911
= radic
119873
sum
119899=1
(119881trunk119899 minus 119881trunk)2
119873
(13)
where119873 is number of dataStrike velocity during landing should be as small as
possible since impact between landing foot and the groundcan cause mechanical wear of parts and unstable walking 119891
2
is stated as follows [6]
1198912
= radic119881strike1199092
+ 119881strike1199102
+ 119881strike1199112
(14)
If ZMP119910
is within the area of support polygon[0061 minus0061] throughout the walking cycle dynamicequilibrium of the robot is satisfactory To deal with thediscrepancies between simulation model and physicaltest bed a safety factor (001m) is added to ensure goodperformance during experiment 119878
1
is formulated as follows
1198781
=
119873
sum
119899=1
max (10038161003816100381610038161003816ZMP119910119899
The total scores used for evaluating each target vector(119883119894119866
) are formulated as follows
scores = minus1200 +
119873
sum
119894=1
120596119900119899
119891119899
+
119873
sum
119894=1
120596119901119899
119878119899
(23)
These weightings are set by trial and error to ensuretheir importance is almost the same and walking gait withsatisfactory performance can be obtained
120596119900
= [1000 500]
120596119901
= [2500 285 265 100 2200 12000 300 960 25000]
(24)
where 120596119900119899
is weighting of fitness functions and 120596119901119899
isweighting of constraints
42 Initialization of Parameters of Target Vectors (119883119894119866
)(Steps 2 and 3) The target vector (119883
119894119866
) is formulated asfollows
119883119894119866
= [1198601
1198602
1198603
1198611
1198612
1198613
1198621
1198622
1198623
119888ℎ 119888119896] (25)
Then the parameters of population (119894) are initialized byMATLAB function rand() which generates [0 1] randomly
43 Initialization of Parameters of SaDE (Step 4) Since therange [minus1 1] of parameters to be searched is small thenpopulation (119894) of one generation (119866) is set as 30 to balancebetween good diversity of population and low computationload Four mutation strategies (119895 = 1 2 3 4) are adoptedFor strategy 119895 MATLAB function 119899119900119903119898119903119899119889 (mean value120590) is used to generate CR
119906119894119895and 119865
119906119894119895for each trial vector
(119906119894
) In the initial phase the crossover rate (CR119906119894119895) should
not be too high to prevent premature convergence while thescaling factor (119865
119906119894119895) should not be too small to affect the
exploration ability Hence a moderate value (05) is assignedto CR
119895
and 119865119895
(mean value) and 120590 is set as 01 This functioncan generate random numbers from the normal distributionthrough mean value (CR
119895
119865119895
) and standard deviation (120590) 119875119895
is set as 025 for strategy 119895 so that each strategy has the equalchance to be chosen during the first learning period
44 Mutation Operation of SaDE (Step 5) DErand1DErand2 and DEcurrent-to-rand2 provide goodexploration ability while DEcurrent-to-best2 demonstratesgood convergence speed To balance between explorationability and convergence speed these four strategies areadopted These four mutation strategies are stated in(27)ndash(30) as follows
(1) DErand1
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (27)
(2) DErand2
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) + 119865119894119895
(1198831199034119866
minus 1198831199035119866
)
(28)
ISRN Robotics 7
(3) DErand-to-best2
V119894119866
= 119883119894119866
+ 119865119894119895
(119883best119866 minus 119883119894119866
) + 119865119894119895
(1198831199031119866
minus 1199091199032119866
)
+ 119865119894119895
(1198831199033119866
minus 1198831199034119866
)
(29)
(4) DEcurrent-to-rand2
V119894119866
= 119883119894119866
+ 119865119894119895
(1198831199031119866
minus 119883119894119866
) + 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (30)
45 Crossover Operation of SaDE (Step 6) Because of itspopularity binomial crossover operator [20] is utilized in allmutation strategies
If (rand () le CR119894119895
) || (119902 == 119902rand)
119906119902119894119866
= V119902119894119866
Else if (rand () gt CR119894119895
)
119906119902119894119866
= 119909119902119894119866
End
(31)
where 119902 is the parameter number (1 2 3 4 11) in thetarget (119883
119894119866
) and mutant (V119894119866
) vector Arbitrary parameternumber is assigned to 119902rand to ensure that trial vector (119906
119894119866
)is different from target vector (119883
119894119866
) If values of 119906119902119894119866
exceedthe desired range [minus1 1] of parameters to be searched 119906
119902119894119866
is reset by MATLAB function rand ()
46 Selection Operation of SaDE (Step 7) If score of trialvector (119906
119894119866
) is lower than that of target vector (119909119894119866
) then trialvector enters the next generation Otherwise target vectorenters the next generation
If scores119906119894119866
le scores119909119894119866
119909119894119866+1
= 119906119894119866
Else
119909119894119866+1
= 119909119894119866
End
(32)
47 Adjustment of 119875119895
119862119877119895
and 119865119895
(Steps 8ndash11) Initially alearning period (LP = 5 119892119890119899119890119903119886119905119894119900119899119904) is assigned to balancebetween good sample size of data and update frequencyDuring learning period successful times (ST
119895
) and failuretimes (FT
119895
) of each mutation strategy (119895 = 1 2 3 4) in eachgeneration are recorded For example if strategy 119895 is chosenand helps one trial vector (119906
119894119866
) to enter next generationthen ST
119895
is added by 1 Otherwise FT119895
is added by 1 Oncelearning period is completed 119875
Step length (m) Lower bound of maximum desired swingheight (m)
005m 001m004m 0009m003m 0008m002m 0007m001m 0006m
rate (SR119895
) Then ST119895
and FT119895
are reset to zero for the nextlearning period to eliminate effect of the past data
SR119895
=
ST119895
FT119895119866
+ ST119895119866
119875119895
=
SR119895
sum119873=4
119895=1
SR119895
(33)
Similar approach is applied to adjust CR119895
and 119865119895
Duringlearning period (LP = 5 generations) for strategy 119895 CR
119906119894119895and
119865119906119894119895
corresponding to the trial vectors (119906119894119866
) that successfullyenters next generation are stored in database Once learningperiod is completed mean value (CR
119895
and 119865119895
) of the storedvalue corresponding to strategy 119895 is calculated and is used togenerate a new set of CR
119906119894 119895and 119865
119906119894119895 Then storage data of
CR119906119894119895
and 119865119906119894119895
are removed for the next learning period toeliminate the effects of past data
5 Optimization of Factors (119877hssupport 119877hsswing119877kssupport and 119877ksswing) for Step LengthAdjustment
51 Objective Functions and Constraints of SaDE The objec-tive functions and constraints mentioned in Section 4 areadopted to search appropriate value of 119877hssupport 119877hsswing119877kssupport and119877ksswingfinal corresponding to different desiredstep length (004m 003m 002m and 001m)
52 Lower Bound of Maximum Desired Swing Height It isnatural that the maximum swing height becomes lower toconsume less energy if step length is smaller Lower bound ofswing height is reset as lower value for different step lengths(Table 3) while the upper bound (002m) remains the same
53 Smooth Transition of 119877jointsupportswing Values of119877jointsupportswing (joint = rhs lhs rks and lks) of support legand swing leg are different At the moment of landing time119877jointsupport is changed to 119877jointswing since the support leg ischanged to swing leg in the next step In order to have asmooth transition fifth order polynomial shown in equation(34) is utilized The period of transition time (119905
119891
minus 1199050
) is setas 04 s which is 40 of time duration (1 s) of one step tobalance between rapid transition time and good dynamicequilibrium
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
Hip and knee trajectories of GAOFSF [6] consist of twodifferent sections For hip joint two different TruncatedFourier Series (TFS) are used to formulate the upper portion(120579ℎ
+) and lower portion (120579ℎ
minus) of trajectory For knee jointTFS and lock phase are joined together to formulate thewholetrajectory Based on Figure 3 the angular position is observedto be continuous However angular velocity of searchedtrajectory generated by GAOFSF is usually discontinuous (1)at the transition between 120579
ℎ
+ and 120579ℎ
minus (1199053
and 1199056
or 1199050
) and
ZZMPXZMP
YZMP
Figure 2 Local coordinate system attached on the foot sole (topview)
(2) at the beginning (1199051
or 1199054
) and the end of lock phase(1199052
or 1199055
) Abrupt change in angular velocity has an adverseeffect on ZMP and hence the dynamic equilibrium of bipedalrobot In this paper the GAOFSF is modified to ensure thatthe trajectories generated are continuous in angular positionand its first derivative and second derivative [119905
5
1199052
] and[1199052
1199055
] are regarded as the period of left support phase andright support phase respectively Then 119905
2
and 1199055
are the
4 ISRN Robotics
Right hipLef hip
Right kneeLef knee
Low
er p
ortio
nU
pper
por
tion
ck
cht
120579
t0 t1 t2 t3 t4 t5 t6
120587120596i
120587120596k
Figure 3 General shape of GAOFSF [6]
landing time of left support phase and right support phaserespectively [6] The time duration (119905
5
minus 1199052
) of one step isset as 1 s to ensure a reasonable walking speed 119905
2
and 1199055
areset as 07 s and 17 s while 119905
3
and 1199056
are set as 1 s and 2 sFour different parameters (119877hssupport 119877hsswing 119877kssupport and119877ksswing) are added for adjustment of step length Details offinding the values of these four parameters are discussed inSection 5 In this section investigation is mainly focused onthe formulation of modified CPG model
31 Hip Trajectory in Sagittal Plane Since the peaks of 120579ℎ
+
and 120579ℎ
minus are different two different TFS are required toformulate the hip trajectory Yang et al [6] have proposed5th order TFS and showed that the amplitudes of 4th and 5thorders are too small which can be neglected 3rd order TFS isadopted In (2)-(3) angular velocity of 120579
minus
ℎ
and 120579+
ℎ
at 1199053
and 1199056
is set as equal to ensure angular velocity is continuous Thustwomore orders are added to 120579
ℎ
minus to satisfy this constraint andare calculated by (4)-(5)
Consider
120579minus
ℎ
(1199056
) = 120579+
ℎ
(1199056
) (2)
120579minus
ℎ
(1199053
) = 120579+
ℎ
(1199053
) (3)
1198615
=minus (2119860
1
+ 61198603
+ 21198611
+ 61198613
)
10 (4)
1198614
=(minus1198601
+ 21198602
minus 31198603
minus 1198611
minus 21198612
minus 31198613
minus 51198615
)
4 (5)
119905 isin [1199050
1199053
] 120579rhs = 120579ℎ
minus
(119905) 120579rhs = 120579ℎ
+
(119905)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ119905) + 119888ℎ
(6)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ119905) + 119888ℎ
119905 isin [1199053
1199056
] 120579rhs = 120579ℎ
+
(119905 minus 1199053
) 120579lhs = 120579ℎ
minus
(119905 minus 1199053
)
(7)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (8)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (9)
120596ℎ
=120587
1199053
minus 1199050
(10)
32 Knee Trajectory in Sagittal Plane In the knee trajectoryYang et al [6] proposed a lock phase ([119905
1
1199052
] and [1199054
1199055
])in knee trajectory which assumes constant joint angle andzero angular velocity and accelerationThis introduces abruptchange in angular velocity The main goals to be achievedin the modified knee trajectory are (1) continuity in angularposition velocity and acceleration and (2) the advantageproposed in GAOFSF that can be maintained Then a newformulation is proposed in the following equation
119905 isin [1199050
1199052
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199052
1199056
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199052
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199050
1199055
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199055
1199056
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199055
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
120596119896
=2120587
1199056
minus 1199050
(11)
ISRN Robotics 5
In (11) lock phase is canceled to ensure continuous angularvelocity By adjusting 119888119896 [6] GAOFSF can achieve energyefficient and stable ldquobent kneerdquo walking gait This advantageis still remained in the proposedmodifiedmodelThe generalshape of the hip and knee trajectories generated by modifiedCPG model is shown in Section 6
33 Ankle Trajectory in Sagittal Plane Right and left anklejoint trajectories are simply formulated as (12) to ensure thatthe trunk is upright and swing foot is parallel to the horizontalflat plane
Consider
120579as = minus 120579hs minus 120579ks (12)
4 Optimization of BasicWalking Pattern by SaDE
This section focuses on how to search the basic walkingpattern of bipedal robot by SaDE [19 20] To simplify thewhole process in this section 119877hssupport 119877hsswing 119877kssupportand 119877ksswing are set as 1The following are the main objectivesof basic walking pattern to be achieved through using SaDE
(1) The desired step length is set as 005m(2) Upper bound and lower bound of swing height are set
as 002m and 001m respectively(3) Premature landing should not occur throughout the
walking cycle(4) ZMP is within the area of support polygon through-
out the walking cycle
The procedure of SaDE takes the following steps Also aflow chart of SaDE is shown in Figure 30 of the Appendixto facilitate the understanding of readers Details of theprocedure are discussed in Sections 41ndash47
(1) Select fitness functions (119891119894
) and constraints (119878119894
)(2) Code the parameters to be searched to form target
vector (119883119894119866
)(3) Initialize the first generation (119866
1
) of population (119894)(4) Initialize crossover rate (CR
119906119894119895) scaling factor (119865
119906119894119895)
and probability (119875119895
) for each mutation strategies (119895)based on CR
119895
and 119865119895
(5) Select mutation strategies (119895) based on probability
(119875119895
) through MATLAB function rand() and performmutation operation to generate mutant vectors (V
119894
)(6) Perform crossover operation to generate trial vectors
(119906119894
)(7) Perform selection operation to select next generation
of target vectors (119883119894119866+1
)(8) During learning period (LP) record the successful
time (ST119895
) and failure time (FT119895
) of each strategy (119895)(9) During learning period (LP) record values of CR
119906119894119895
and119865119906119894119895
for each strategy (119895) that successfully help thetrial vectors (119906
119894
) enter the next generation 119866119894+1
(10) Upon completion of LP probability (119875119895
) for choosingsuitable strategies (119895) is adjusted based on successfulrate of strategy 119895 (SR
119895
) calculated through ST119895
andFT119895
(11) Upon completion of LP CR119895
and 119865119895
are adjustedbased on the mean values of stored CR
119906119894119895and 119865
119906119894119895
respectively
(12) Repeat procedures (4)ndash(11) until maximum genera-tion is completed
41 Fitness Functions and Constraints (Step 1) Fitness func-tions (119891
119899
) and constraints (119878119899
) are designed ormodified basedon [6] Since trunk occupies a large proportion of the wholebodymass abrupt change in trunk velocity can lead to abruptchange in ZMP
119910
1198911
is formulated as follows
119881trunk =119873
sum
119899=1
119881trunk119899
119873
1198911
= radic
119873
sum
119899=1
(119881trunk119899 minus 119881trunk)2
119873
(13)
where119873 is number of dataStrike velocity during landing should be as small as
possible since impact between landing foot and the groundcan cause mechanical wear of parts and unstable walking 119891
2
is stated as follows [6]
1198912
= radic119881strike1199092
+ 119881strike1199102
+ 119881strike1199112
(14)
If ZMP119910
is within the area of support polygon[0061 minus0061] throughout the walking cycle dynamicequilibrium of the robot is satisfactory To deal with thediscrepancies between simulation model and physicaltest bed a safety factor (001m) is added to ensure goodperformance during experiment 119878
1
is formulated as follows
1198781
=
119873
sum
119899=1
max (10038161003816100381610038161003816ZMP119910119899
The total scores used for evaluating each target vector(119883119894119866
) are formulated as follows
scores = minus1200 +
119873
sum
119894=1
120596119900119899
119891119899
+
119873
sum
119894=1
120596119901119899
119878119899
(23)
These weightings are set by trial and error to ensuretheir importance is almost the same and walking gait withsatisfactory performance can be obtained
120596119900
= [1000 500]
120596119901
= [2500 285 265 100 2200 12000 300 960 25000]
(24)
where 120596119900119899
is weighting of fitness functions and 120596119901119899
isweighting of constraints
42 Initialization of Parameters of Target Vectors (119883119894119866
)(Steps 2 and 3) The target vector (119883
119894119866
) is formulated asfollows
119883119894119866
= [1198601
1198602
1198603
1198611
1198612
1198613
1198621
1198622
1198623
119888ℎ 119888119896] (25)
Then the parameters of population (119894) are initialized byMATLAB function rand() which generates [0 1] randomly
43 Initialization of Parameters of SaDE (Step 4) Since therange [minus1 1] of parameters to be searched is small thenpopulation (119894) of one generation (119866) is set as 30 to balancebetween good diversity of population and low computationload Four mutation strategies (119895 = 1 2 3 4) are adoptedFor strategy 119895 MATLAB function 119899119900119903119898119903119899119889 (mean value120590) is used to generate CR
119906119894119895and 119865
119906119894119895for each trial vector
(119906119894
) In the initial phase the crossover rate (CR119906119894119895) should
not be too high to prevent premature convergence while thescaling factor (119865
119906119894119895) should not be too small to affect the
exploration ability Hence a moderate value (05) is assignedto CR
119895
and 119865119895
(mean value) and 120590 is set as 01 This functioncan generate random numbers from the normal distributionthrough mean value (CR
119895
119865119895
) and standard deviation (120590) 119875119895
is set as 025 for strategy 119895 so that each strategy has the equalchance to be chosen during the first learning period
44 Mutation Operation of SaDE (Step 5) DErand1DErand2 and DEcurrent-to-rand2 provide goodexploration ability while DEcurrent-to-best2 demonstratesgood convergence speed To balance between explorationability and convergence speed these four strategies areadopted These four mutation strategies are stated in(27)ndash(30) as follows
(1) DErand1
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (27)
(2) DErand2
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) + 119865119894119895
(1198831199034119866
minus 1198831199035119866
)
(28)
ISRN Robotics 7
(3) DErand-to-best2
V119894119866
= 119883119894119866
+ 119865119894119895
(119883best119866 minus 119883119894119866
) + 119865119894119895
(1198831199031119866
minus 1199091199032119866
)
+ 119865119894119895
(1198831199033119866
minus 1198831199034119866
)
(29)
(4) DEcurrent-to-rand2
V119894119866
= 119883119894119866
+ 119865119894119895
(1198831199031119866
minus 119883119894119866
) + 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (30)
45 Crossover Operation of SaDE (Step 6) Because of itspopularity binomial crossover operator [20] is utilized in allmutation strategies
If (rand () le CR119894119895
) || (119902 == 119902rand)
119906119902119894119866
= V119902119894119866
Else if (rand () gt CR119894119895
)
119906119902119894119866
= 119909119902119894119866
End
(31)
where 119902 is the parameter number (1 2 3 4 11) in thetarget (119883
119894119866
) and mutant (V119894119866
) vector Arbitrary parameternumber is assigned to 119902rand to ensure that trial vector (119906
119894119866
)is different from target vector (119883
119894119866
) If values of 119906119902119894119866
exceedthe desired range [minus1 1] of parameters to be searched 119906
119902119894119866
is reset by MATLAB function rand ()
46 Selection Operation of SaDE (Step 7) If score of trialvector (119906
119894119866
) is lower than that of target vector (119909119894119866
) then trialvector enters the next generation Otherwise target vectorenters the next generation
If scores119906119894119866
le scores119909119894119866
119909119894119866+1
= 119906119894119866
Else
119909119894119866+1
= 119909119894119866
End
(32)
47 Adjustment of 119875119895
119862119877119895
and 119865119895
(Steps 8ndash11) Initially alearning period (LP = 5 119892119890119899119890119903119886119905119894119900119899119904) is assigned to balancebetween good sample size of data and update frequencyDuring learning period successful times (ST
119895
) and failuretimes (FT
119895
) of each mutation strategy (119895 = 1 2 3 4) in eachgeneration are recorded For example if strategy 119895 is chosenand helps one trial vector (119906
119894119866
) to enter next generationthen ST
119895
is added by 1 Otherwise FT119895
is added by 1 Oncelearning period is completed 119875
Step length (m) Lower bound of maximum desired swingheight (m)
005m 001m004m 0009m003m 0008m002m 0007m001m 0006m
rate (SR119895
) Then ST119895
and FT119895
are reset to zero for the nextlearning period to eliminate effect of the past data
SR119895
=
ST119895
FT119895119866
+ ST119895119866
119875119895
=
SR119895
sum119873=4
119895=1
SR119895
(33)
Similar approach is applied to adjust CR119895
and 119865119895
Duringlearning period (LP = 5 generations) for strategy 119895 CR
119906119894119895and
119865119906119894119895
corresponding to the trial vectors (119906119894119866
) that successfullyenters next generation are stored in database Once learningperiod is completed mean value (CR
119895
and 119865119895
) of the storedvalue corresponding to strategy 119895 is calculated and is used togenerate a new set of CR
119906119894 119895and 119865
119906119894119895 Then storage data of
CR119906119894119895
and 119865119906119894119895
are removed for the next learning period toeliminate the effects of past data
5 Optimization of Factors (119877hssupport 119877hsswing119877kssupport and 119877ksswing) for Step LengthAdjustment
51 Objective Functions and Constraints of SaDE The objec-tive functions and constraints mentioned in Section 4 areadopted to search appropriate value of 119877hssupport 119877hsswing119877kssupport and119877ksswingfinal corresponding to different desiredstep length (004m 003m 002m and 001m)
52 Lower Bound of Maximum Desired Swing Height It isnatural that the maximum swing height becomes lower toconsume less energy if step length is smaller Lower bound ofswing height is reset as lower value for different step lengths(Table 3) while the upper bound (002m) remains the same
53 Smooth Transition of 119877jointsupportswing Values of119877jointsupportswing (joint = rhs lhs rks and lks) of support legand swing leg are different At the moment of landing time119877jointsupport is changed to 119877jointswing since the support leg ischanged to swing leg in the next step In order to have asmooth transition fifth order polynomial shown in equation(34) is utilized The period of transition time (119905
119891
minus 1199050
) is setas 04 s which is 40 of time duration (1 s) of one step tobalance between rapid transition time and good dynamicequilibrium
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
landing time of left support phase and right support phaserespectively [6] The time duration (119905
5
minus 1199052
) of one step isset as 1 s to ensure a reasonable walking speed 119905
2
and 1199055
areset as 07 s and 17 s while 119905
3
and 1199056
are set as 1 s and 2 sFour different parameters (119877hssupport 119877hsswing 119877kssupport and119877ksswing) are added for adjustment of step length Details offinding the values of these four parameters are discussed inSection 5 In this section investigation is mainly focused onthe formulation of modified CPG model
31 Hip Trajectory in Sagittal Plane Since the peaks of 120579ℎ
+
and 120579ℎ
minus are different two different TFS are required toformulate the hip trajectory Yang et al [6] have proposed5th order TFS and showed that the amplitudes of 4th and 5thorders are too small which can be neglected 3rd order TFS isadopted In (2)-(3) angular velocity of 120579
minus
ℎ
and 120579+
ℎ
at 1199053
and 1199056
is set as equal to ensure angular velocity is continuous Thustwomore orders are added to 120579
ℎ
minus to satisfy this constraint andare calculated by (4)-(5)
Consider
120579minus
ℎ
(1199056
) = 120579+
ℎ
(1199056
) (2)
120579minus
ℎ
(1199053
) = 120579+
ℎ
(1199053
) (3)
1198615
=minus (2119860
1
+ 61198603
+ 21198611
+ 61198613
)
10 (4)
1198614
=(minus1198601
+ 21198602
minus 31198603
minus 1198611
minus 21198612
minus 31198613
minus 51198615
)
4 (5)
119905 isin [1199050
1199053
] 120579rhs = 120579ℎ
minus
(119905) 120579rhs = 120579ℎ
+
(119905)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ119905) + 119888ℎ
(6)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ119905) + 119888ℎ
119905 isin [1199053
1199056
] 120579rhs = 120579ℎ
+
(119905 minus 1199053
) 120579lhs = 120579ℎ
minus
(119905 minus 1199053
)
(7)
120579ℎ
minus
=
5
sum
119899=1
119877hsswingsupport119861119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (8)
120579ℎ
+
=
3
sum
119899=1
119877hsswingsupport119860119899 sin (119899120596ℎ (119905 minus 1199053
)) + 119888ℎ (9)
120596ℎ
=120587
1199053
minus 1199050
(10)
32 Knee Trajectory in Sagittal Plane In the knee trajectoryYang et al [6] proposed a lock phase ([119905
1
1199052
] and [1199054
1199055
])in knee trajectory which assumes constant joint angle andzero angular velocity and accelerationThis introduces abruptchange in angular velocity The main goals to be achievedin the modified knee trajectory are (1) continuity in angularposition velocity and acceleration and (2) the advantageproposed in GAOFSF that can be maintained Then a newformulation is proposed in the following equation
119905 isin [1199050
1199052
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199052
1199056
]
120579lks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199052
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199050
1199055
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 + (1199056
minus 1199055
)))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
119905 isin [1199055
1199056
]
120579rks =3
sum
119899=1
119877ksswingsupport119862119899 sin (119899120596119896 (119905 minus 1199055
))
+
119873=3
sum
119899=1
119877ksswingsupport119862119899 + 119888119896
120596119896
=2120587
1199056
minus 1199050
(11)
ISRN Robotics 5
In (11) lock phase is canceled to ensure continuous angularvelocity By adjusting 119888119896 [6] GAOFSF can achieve energyefficient and stable ldquobent kneerdquo walking gait This advantageis still remained in the proposedmodifiedmodelThe generalshape of the hip and knee trajectories generated by modifiedCPG model is shown in Section 6
33 Ankle Trajectory in Sagittal Plane Right and left anklejoint trajectories are simply formulated as (12) to ensure thatthe trunk is upright and swing foot is parallel to the horizontalflat plane
Consider
120579as = minus 120579hs minus 120579ks (12)
4 Optimization of BasicWalking Pattern by SaDE
This section focuses on how to search the basic walkingpattern of bipedal robot by SaDE [19 20] To simplify thewhole process in this section 119877hssupport 119877hsswing 119877kssupportand 119877ksswing are set as 1The following are the main objectivesof basic walking pattern to be achieved through using SaDE
(1) The desired step length is set as 005m(2) Upper bound and lower bound of swing height are set
as 002m and 001m respectively(3) Premature landing should not occur throughout the
walking cycle(4) ZMP is within the area of support polygon through-
out the walking cycle
The procedure of SaDE takes the following steps Also aflow chart of SaDE is shown in Figure 30 of the Appendixto facilitate the understanding of readers Details of theprocedure are discussed in Sections 41ndash47
(1) Select fitness functions (119891119894
) and constraints (119878119894
)(2) Code the parameters to be searched to form target
vector (119883119894119866
)(3) Initialize the first generation (119866
1
) of population (119894)(4) Initialize crossover rate (CR
119906119894119895) scaling factor (119865
119906119894119895)
and probability (119875119895
) for each mutation strategies (119895)based on CR
119895
and 119865119895
(5) Select mutation strategies (119895) based on probability
(119875119895
) through MATLAB function rand() and performmutation operation to generate mutant vectors (V
119894
)(6) Perform crossover operation to generate trial vectors
(119906119894
)(7) Perform selection operation to select next generation
of target vectors (119883119894119866+1
)(8) During learning period (LP) record the successful
time (ST119895
) and failure time (FT119895
) of each strategy (119895)(9) During learning period (LP) record values of CR
119906119894119895
and119865119906119894119895
for each strategy (119895) that successfully help thetrial vectors (119906
119894
) enter the next generation 119866119894+1
(10) Upon completion of LP probability (119875119895
) for choosingsuitable strategies (119895) is adjusted based on successfulrate of strategy 119895 (SR
119895
) calculated through ST119895
andFT119895
(11) Upon completion of LP CR119895
and 119865119895
are adjustedbased on the mean values of stored CR
119906119894119895and 119865
119906119894119895
respectively
(12) Repeat procedures (4)ndash(11) until maximum genera-tion is completed
41 Fitness Functions and Constraints (Step 1) Fitness func-tions (119891
119899
) and constraints (119878119899
) are designed ormodified basedon [6] Since trunk occupies a large proportion of the wholebodymass abrupt change in trunk velocity can lead to abruptchange in ZMP
119910
1198911
is formulated as follows
119881trunk =119873
sum
119899=1
119881trunk119899
119873
1198911
= radic
119873
sum
119899=1
(119881trunk119899 minus 119881trunk)2
119873
(13)
where119873 is number of dataStrike velocity during landing should be as small as
possible since impact between landing foot and the groundcan cause mechanical wear of parts and unstable walking 119891
2
is stated as follows [6]
1198912
= radic119881strike1199092
+ 119881strike1199102
+ 119881strike1199112
(14)
If ZMP119910
is within the area of support polygon[0061 minus0061] throughout the walking cycle dynamicequilibrium of the robot is satisfactory To deal with thediscrepancies between simulation model and physicaltest bed a safety factor (001m) is added to ensure goodperformance during experiment 119878
1
is formulated as follows
1198781
=
119873
sum
119899=1
max (10038161003816100381610038161003816ZMP119910119899
The total scores used for evaluating each target vector(119883119894119866
) are formulated as follows
scores = minus1200 +
119873
sum
119894=1
120596119900119899
119891119899
+
119873
sum
119894=1
120596119901119899
119878119899
(23)
These weightings are set by trial and error to ensuretheir importance is almost the same and walking gait withsatisfactory performance can be obtained
120596119900
= [1000 500]
120596119901
= [2500 285 265 100 2200 12000 300 960 25000]
(24)
where 120596119900119899
is weighting of fitness functions and 120596119901119899
isweighting of constraints
42 Initialization of Parameters of Target Vectors (119883119894119866
)(Steps 2 and 3) The target vector (119883
119894119866
) is formulated asfollows
119883119894119866
= [1198601
1198602
1198603
1198611
1198612
1198613
1198621
1198622
1198623
119888ℎ 119888119896] (25)
Then the parameters of population (119894) are initialized byMATLAB function rand() which generates [0 1] randomly
43 Initialization of Parameters of SaDE (Step 4) Since therange [minus1 1] of parameters to be searched is small thenpopulation (119894) of one generation (119866) is set as 30 to balancebetween good diversity of population and low computationload Four mutation strategies (119895 = 1 2 3 4) are adoptedFor strategy 119895 MATLAB function 119899119900119903119898119903119899119889 (mean value120590) is used to generate CR
119906119894119895and 119865
119906119894119895for each trial vector
(119906119894
) In the initial phase the crossover rate (CR119906119894119895) should
not be too high to prevent premature convergence while thescaling factor (119865
119906119894119895) should not be too small to affect the
exploration ability Hence a moderate value (05) is assignedto CR
119895
and 119865119895
(mean value) and 120590 is set as 01 This functioncan generate random numbers from the normal distributionthrough mean value (CR
119895
119865119895
) and standard deviation (120590) 119875119895
is set as 025 for strategy 119895 so that each strategy has the equalchance to be chosen during the first learning period
44 Mutation Operation of SaDE (Step 5) DErand1DErand2 and DEcurrent-to-rand2 provide goodexploration ability while DEcurrent-to-best2 demonstratesgood convergence speed To balance between explorationability and convergence speed these four strategies areadopted These four mutation strategies are stated in(27)ndash(30) as follows
(1) DErand1
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (27)
(2) DErand2
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) + 119865119894119895
(1198831199034119866
minus 1198831199035119866
)
(28)
ISRN Robotics 7
(3) DErand-to-best2
V119894119866
= 119883119894119866
+ 119865119894119895
(119883best119866 minus 119883119894119866
) + 119865119894119895
(1198831199031119866
minus 1199091199032119866
)
+ 119865119894119895
(1198831199033119866
minus 1198831199034119866
)
(29)
(4) DEcurrent-to-rand2
V119894119866
= 119883119894119866
+ 119865119894119895
(1198831199031119866
minus 119883119894119866
) + 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (30)
45 Crossover Operation of SaDE (Step 6) Because of itspopularity binomial crossover operator [20] is utilized in allmutation strategies
If (rand () le CR119894119895
) || (119902 == 119902rand)
119906119902119894119866
= V119902119894119866
Else if (rand () gt CR119894119895
)
119906119902119894119866
= 119909119902119894119866
End
(31)
where 119902 is the parameter number (1 2 3 4 11) in thetarget (119883
119894119866
) and mutant (V119894119866
) vector Arbitrary parameternumber is assigned to 119902rand to ensure that trial vector (119906
119894119866
)is different from target vector (119883
119894119866
) If values of 119906119902119894119866
exceedthe desired range [minus1 1] of parameters to be searched 119906
119902119894119866
is reset by MATLAB function rand ()
46 Selection Operation of SaDE (Step 7) If score of trialvector (119906
119894119866
) is lower than that of target vector (119909119894119866
) then trialvector enters the next generation Otherwise target vectorenters the next generation
If scores119906119894119866
le scores119909119894119866
119909119894119866+1
= 119906119894119866
Else
119909119894119866+1
= 119909119894119866
End
(32)
47 Adjustment of 119875119895
119862119877119895
and 119865119895
(Steps 8ndash11) Initially alearning period (LP = 5 119892119890119899119890119903119886119905119894119900119899119904) is assigned to balancebetween good sample size of data and update frequencyDuring learning period successful times (ST
119895
) and failuretimes (FT
119895
) of each mutation strategy (119895 = 1 2 3 4) in eachgeneration are recorded For example if strategy 119895 is chosenand helps one trial vector (119906
119894119866
) to enter next generationthen ST
119895
is added by 1 Otherwise FT119895
is added by 1 Oncelearning period is completed 119875
Step length (m) Lower bound of maximum desired swingheight (m)
005m 001m004m 0009m003m 0008m002m 0007m001m 0006m
rate (SR119895
) Then ST119895
and FT119895
are reset to zero for the nextlearning period to eliminate effect of the past data
SR119895
=
ST119895
FT119895119866
+ ST119895119866
119875119895
=
SR119895
sum119873=4
119895=1
SR119895
(33)
Similar approach is applied to adjust CR119895
and 119865119895
Duringlearning period (LP = 5 generations) for strategy 119895 CR
119906119894119895and
119865119906119894119895
corresponding to the trial vectors (119906119894119866
) that successfullyenters next generation are stored in database Once learningperiod is completed mean value (CR
119895
and 119865119895
) of the storedvalue corresponding to strategy 119895 is calculated and is used togenerate a new set of CR
119906119894 119895and 119865
119906119894119895 Then storage data of
CR119906119894119895
and 119865119906119894119895
are removed for the next learning period toeliminate the effects of past data
5 Optimization of Factors (119877hssupport 119877hsswing119877kssupport and 119877ksswing) for Step LengthAdjustment
51 Objective Functions and Constraints of SaDE The objec-tive functions and constraints mentioned in Section 4 areadopted to search appropriate value of 119877hssupport 119877hsswing119877kssupport and119877ksswingfinal corresponding to different desiredstep length (004m 003m 002m and 001m)
52 Lower Bound of Maximum Desired Swing Height It isnatural that the maximum swing height becomes lower toconsume less energy if step length is smaller Lower bound ofswing height is reset as lower value for different step lengths(Table 3) while the upper bound (002m) remains the same
53 Smooth Transition of 119877jointsupportswing Values of119877jointsupportswing (joint = rhs lhs rks and lks) of support legand swing leg are different At the moment of landing time119877jointsupport is changed to 119877jointswing since the support leg ischanged to swing leg in the next step In order to have asmooth transition fifth order polynomial shown in equation(34) is utilized The period of transition time (119905
119891
minus 1199050
) is setas 04 s which is 40 of time duration (1 s) of one step tobalance between rapid transition time and good dynamicequilibrium
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
In (11) lock phase is canceled to ensure continuous angularvelocity By adjusting 119888119896 [6] GAOFSF can achieve energyefficient and stable ldquobent kneerdquo walking gait This advantageis still remained in the proposedmodifiedmodelThe generalshape of the hip and knee trajectories generated by modifiedCPG model is shown in Section 6
33 Ankle Trajectory in Sagittal Plane Right and left anklejoint trajectories are simply formulated as (12) to ensure thatthe trunk is upright and swing foot is parallel to the horizontalflat plane
Consider
120579as = minus 120579hs minus 120579ks (12)
4 Optimization of BasicWalking Pattern by SaDE
This section focuses on how to search the basic walkingpattern of bipedal robot by SaDE [19 20] To simplify thewhole process in this section 119877hssupport 119877hsswing 119877kssupportand 119877ksswing are set as 1The following are the main objectivesof basic walking pattern to be achieved through using SaDE
(1) The desired step length is set as 005m(2) Upper bound and lower bound of swing height are set
as 002m and 001m respectively(3) Premature landing should not occur throughout the
walking cycle(4) ZMP is within the area of support polygon through-
out the walking cycle
The procedure of SaDE takes the following steps Also aflow chart of SaDE is shown in Figure 30 of the Appendixto facilitate the understanding of readers Details of theprocedure are discussed in Sections 41ndash47
(1) Select fitness functions (119891119894
) and constraints (119878119894
)(2) Code the parameters to be searched to form target
vector (119883119894119866
)(3) Initialize the first generation (119866
1
) of population (119894)(4) Initialize crossover rate (CR
119906119894119895) scaling factor (119865
119906119894119895)
and probability (119875119895
) for each mutation strategies (119895)based on CR
119895
and 119865119895
(5) Select mutation strategies (119895) based on probability
(119875119895
) through MATLAB function rand() and performmutation operation to generate mutant vectors (V
119894
)(6) Perform crossover operation to generate trial vectors
(119906119894
)(7) Perform selection operation to select next generation
of target vectors (119883119894119866+1
)(8) During learning period (LP) record the successful
time (ST119895
) and failure time (FT119895
) of each strategy (119895)(9) During learning period (LP) record values of CR
119906119894119895
and119865119906119894119895
for each strategy (119895) that successfully help thetrial vectors (119906
119894
) enter the next generation 119866119894+1
(10) Upon completion of LP probability (119875119895
) for choosingsuitable strategies (119895) is adjusted based on successfulrate of strategy 119895 (SR
119895
) calculated through ST119895
andFT119895
(11) Upon completion of LP CR119895
and 119865119895
are adjustedbased on the mean values of stored CR
119906119894119895and 119865
119906119894119895
respectively
(12) Repeat procedures (4)ndash(11) until maximum genera-tion is completed
41 Fitness Functions and Constraints (Step 1) Fitness func-tions (119891
119899
) and constraints (119878119899
) are designed ormodified basedon [6] Since trunk occupies a large proportion of the wholebodymass abrupt change in trunk velocity can lead to abruptchange in ZMP
119910
1198911
is formulated as follows
119881trunk =119873
sum
119899=1
119881trunk119899
119873
1198911
= radic
119873
sum
119899=1
(119881trunk119899 minus 119881trunk)2
119873
(13)
where119873 is number of dataStrike velocity during landing should be as small as
possible since impact between landing foot and the groundcan cause mechanical wear of parts and unstable walking 119891
2
is stated as follows [6]
1198912
= radic119881strike1199092
+ 119881strike1199102
+ 119881strike1199112
(14)
If ZMP119910
is within the area of support polygon[0061 minus0061] throughout the walking cycle dynamicequilibrium of the robot is satisfactory To deal with thediscrepancies between simulation model and physicaltest bed a safety factor (001m) is added to ensure goodperformance during experiment 119878
1
is formulated as follows
1198781
=
119873
sum
119899=1
max (10038161003816100381610038161003816ZMP119910119899
The total scores used for evaluating each target vector(119883119894119866
) are formulated as follows
scores = minus1200 +
119873
sum
119894=1
120596119900119899
119891119899
+
119873
sum
119894=1
120596119901119899
119878119899
(23)
These weightings are set by trial and error to ensuretheir importance is almost the same and walking gait withsatisfactory performance can be obtained
120596119900
= [1000 500]
120596119901
= [2500 285 265 100 2200 12000 300 960 25000]
(24)
where 120596119900119899
is weighting of fitness functions and 120596119901119899
isweighting of constraints
42 Initialization of Parameters of Target Vectors (119883119894119866
)(Steps 2 and 3) The target vector (119883
119894119866
) is formulated asfollows
119883119894119866
= [1198601
1198602
1198603
1198611
1198612
1198613
1198621
1198622
1198623
119888ℎ 119888119896] (25)
Then the parameters of population (119894) are initialized byMATLAB function rand() which generates [0 1] randomly
43 Initialization of Parameters of SaDE (Step 4) Since therange [minus1 1] of parameters to be searched is small thenpopulation (119894) of one generation (119866) is set as 30 to balancebetween good diversity of population and low computationload Four mutation strategies (119895 = 1 2 3 4) are adoptedFor strategy 119895 MATLAB function 119899119900119903119898119903119899119889 (mean value120590) is used to generate CR
119906119894119895and 119865
119906119894119895for each trial vector
(119906119894
) In the initial phase the crossover rate (CR119906119894119895) should
not be too high to prevent premature convergence while thescaling factor (119865
119906119894119895) should not be too small to affect the
exploration ability Hence a moderate value (05) is assignedto CR
119895
and 119865119895
(mean value) and 120590 is set as 01 This functioncan generate random numbers from the normal distributionthrough mean value (CR
119895
119865119895
) and standard deviation (120590) 119875119895
is set as 025 for strategy 119895 so that each strategy has the equalchance to be chosen during the first learning period
44 Mutation Operation of SaDE (Step 5) DErand1DErand2 and DEcurrent-to-rand2 provide goodexploration ability while DEcurrent-to-best2 demonstratesgood convergence speed To balance between explorationability and convergence speed these four strategies areadopted These four mutation strategies are stated in(27)ndash(30) as follows
(1) DErand1
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (27)
(2) DErand2
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) + 119865119894119895
(1198831199034119866
minus 1198831199035119866
)
(28)
ISRN Robotics 7
(3) DErand-to-best2
V119894119866
= 119883119894119866
+ 119865119894119895
(119883best119866 minus 119883119894119866
) + 119865119894119895
(1198831199031119866
minus 1199091199032119866
)
+ 119865119894119895
(1198831199033119866
minus 1198831199034119866
)
(29)
(4) DEcurrent-to-rand2
V119894119866
= 119883119894119866
+ 119865119894119895
(1198831199031119866
minus 119883119894119866
) + 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (30)
45 Crossover Operation of SaDE (Step 6) Because of itspopularity binomial crossover operator [20] is utilized in allmutation strategies
If (rand () le CR119894119895
) || (119902 == 119902rand)
119906119902119894119866
= V119902119894119866
Else if (rand () gt CR119894119895
)
119906119902119894119866
= 119909119902119894119866
End
(31)
where 119902 is the parameter number (1 2 3 4 11) in thetarget (119883
119894119866
) and mutant (V119894119866
) vector Arbitrary parameternumber is assigned to 119902rand to ensure that trial vector (119906
119894119866
)is different from target vector (119883
119894119866
) If values of 119906119902119894119866
exceedthe desired range [minus1 1] of parameters to be searched 119906
119902119894119866
is reset by MATLAB function rand ()
46 Selection Operation of SaDE (Step 7) If score of trialvector (119906
119894119866
) is lower than that of target vector (119909119894119866
) then trialvector enters the next generation Otherwise target vectorenters the next generation
If scores119906119894119866
le scores119909119894119866
119909119894119866+1
= 119906119894119866
Else
119909119894119866+1
= 119909119894119866
End
(32)
47 Adjustment of 119875119895
119862119877119895
and 119865119895
(Steps 8ndash11) Initially alearning period (LP = 5 119892119890119899119890119903119886119905119894119900119899119904) is assigned to balancebetween good sample size of data and update frequencyDuring learning period successful times (ST
119895
) and failuretimes (FT
119895
) of each mutation strategy (119895 = 1 2 3 4) in eachgeneration are recorded For example if strategy 119895 is chosenand helps one trial vector (119906
119894119866
) to enter next generationthen ST
119895
is added by 1 Otherwise FT119895
is added by 1 Oncelearning period is completed 119875
Step length (m) Lower bound of maximum desired swingheight (m)
005m 001m004m 0009m003m 0008m002m 0007m001m 0006m
rate (SR119895
) Then ST119895
and FT119895
are reset to zero for the nextlearning period to eliminate effect of the past data
SR119895
=
ST119895
FT119895119866
+ ST119895119866
119875119895
=
SR119895
sum119873=4
119895=1
SR119895
(33)
Similar approach is applied to adjust CR119895
and 119865119895
Duringlearning period (LP = 5 generations) for strategy 119895 CR
119906119894119895and
119865119906119894119895
corresponding to the trial vectors (119906119894119866
) that successfullyenters next generation are stored in database Once learningperiod is completed mean value (CR
119895
and 119865119895
) of the storedvalue corresponding to strategy 119895 is calculated and is used togenerate a new set of CR
119906119894 119895and 119865
119906119894119895 Then storage data of
CR119906119894119895
and 119865119906119894119895
are removed for the next learning period toeliminate the effects of past data
5 Optimization of Factors (119877hssupport 119877hsswing119877kssupport and 119877ksswing) for Step LengthAdjustment
51 Objective Functions and Constraints of SaDE The objec-tive functions and constraints mentioned in Section 4 areadopted to search appropriate value of 119877hssupport 119877hsswing119877kssupport and119877ksswingfinal corresponding to different desiredstep length (004m 003m 002m and 001m)
52 Lower Bound of Maximum Desired Swing Height It isnatural that the maximum swing height becomes lower toconsume less energy if step length is smaller Lower bound ofswing height is reset as lower value for different step lengths(Table 3) while the upper bound (002m) remains the same
53 Smooth Transition of 119877jointsupportswing Values of119877jointsupportswing (joint = rhs lhs rks and lks) of support legand swing leg are different At the moment of landing time119877jointsupport is changed to 119877jointswing since the support leg ischanged to swing leg in the next step In order to have asmooth transition fifth order polynomial shown in equation(34) is utilized The period of transition time (119905
119891
minus 1199050
) is setas 04 s which is 40 of time duration (1 s) of one step tobalance between rapid transition time and good dynamicequilibrium
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
The total scores used for evaluating each target vector(119883119894119866
) are formulated as follows
scores = minus1200 +
119873
sum
119894=1
120596119900119899
119891119899
+
119873
sum
119894=1
120596119901119899
119878119899
(23)
These weightings are set by trial and error to ensuretheir importance is almost the same and walking gait withsatisfactory performance can be obtained
120596119900
= [1000 500]
120596119901
= [2500 285 265 100 2200 12000 300 960 25000]
(24)
where 120596119900119899
is weighting of fitness functions and 120596119901119899
isweighting of constraints
42 Initialization of Parameters of Target Vectors (119883119894119866
)(Steps 2 and 3) The target vector (119883
119894119866
) is formulated asfollows
119883119894119866
= [1198601
1198602
1198603
1198611
1198612
1198613
1198621
1198622
1198623
119888ℎ 119888119896] (25)
Then the parameters of population (119894) are initialized byMATLAB function rand() which generates [0 1] randomly
43 Initialization of Parameters of SaDE (Step 4) Since therange [minus1 1] of parameters to be searched is small thenpopulation (119894) of one generation (119866) is set as 30 to balancebetween good diversity of population and low computationload Four mutation strategies (119895 = 1 2 3 4) are adoptedFor strategy 119895 MATLAB function 119899119900119903119898119903119899119889 (mean value120590) is used to generate CR
119906119894119895and 119865
119906119894119895for each trial vector
(119906119894
) In the initial phase the crossover rate (CR119906119894119895) should
not be too high to prevent premature convergence while thescaling factor (119865
119906119894119895) should not be too small to affect the
exploration ability Hence a moderate value (05) is assignedto CR
119895
and 119865119895
(mean value) and 120590 is set as 01 This functioncan generate random numbers from the normal distributionthrough mean value (CR
119895
119865119895
) and standard deviation (120590) 119875119895
is set as 025 for strategy 119895 so that each strategy has the equalchance to be chosen during the first learning period
44 Mutation Operation of SaDE (Step 5) DErand1DErand2 and DEcurrent-to-rand2 provide goodexploration ability while DEcurrent-to-best2 demonstratesgood convergence speed To balance between explorationability and convergence speed these four strategies areadopted These four mutation strategies are stated in(27)ndash(30) as follows
(1) DErand1
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (27)
(2) DErand2
V119894119866
= 1198831199031119866
+ 119865119894119895
(1198831199032119866
minus 1198831199033119866
) + 119865119894119895
(1198831199034119866
minus 1198831199035119866
)
(28)
ISRN Robotics 7
(3) DErand-to-best2
V119894119866
= 119883119894119866
+ 119865119894119895
(119883best119866 minus 119883119894119866
) + 119865119894119895
(1198831199031119866
minus 1199091199032119866
)
+ 119865119894119895
(1198831199033119866
minus 1198831199034119866
)
(29)
(4) DEcurrent-to-rand2
V119894119866
= 119883119894119866
+ 119865119894119895
(1198831199031119866
minus 119883119894119866
) + 119865119894119895
(1198831199032119866
minus 1198831199033119866
) (30)
45 Crossover Operation of SaDE (Step 6) Because of itspopularity binomial crossover operator [20] is utilized in allmutation strategies
If (rand () le CR119894119895
) || (119902 == 119902rand)
119906119902119894119866
= V119902119894119866
Else if (rand () gt CR119894119895
)
119906119902119894119866
= 119909119902119894119866
End
(31)
where 119902 is the parameter number (1 2 3 4 11) in thetarget (119883
119894119866
) and mutant (V119894119866
) vector Arbitrary parameternumber is assigned to 119902rand to ensure that trial vector (119906
119894119866
)is different from target vector (119883
119894119866
) If values of 119906119902119894119866
exceedthe desired range [minus1 1] of parameters to be searched 119906
119902119894119866
is reset by MATLAB function rand ()
46 Selection Operation of SaDE (Step 7) If score of trialvector (119906
119894119866
) is lower than that of target vector (119909119894119866
) then trialvector enters the next generation Otherwise target vectorenters the next generation
If scores119906119894119866
le scores119909119894119866
119909119894119866+1
= 119906119894119866
Else
119909119894119866+1
= 119909119894119866
End
(32)
47 Adjustment of 119875119895
119862119877119895
and 119865119895
(Steps 8ndash11) Initially alearning period (LP = 5 119892119890119899119890119903119886119905119894119900119899119904) is assigned to balancebetween good sample size of data and update frequencyDuring learning period successful times (ST
119895
) and failuretimes (FT
119895
) of each mutation strategy (119895 = 1 2 3 4) in eachgeneration are recorded For example if strategy 119895 is chosenand helps one trial vector (119906
119894119866
) to enter next generationthen ST
119895
is added by 1 Otherwise FT119895
is added by 1 Oncelearning period is completed 119875
Step length (m) Lower bound of maximum desired swingheight (m)
005m 001m004m 0009m003m 0008m002m 0007m001m 0006m
rate (SR119895
) Then ST119895
and FT119895
are reset to zero for the nextlearning period to eliminate effect of the past data
SR119895
=
ST119895
FT119895119866
+ ST119895119866
119875119895
=
SR119895
sum119873=4
119895=1
SR119895
(33)
Similar approach is applied to adjust CR119895
and 119865119895
Duringlearning period (LP = 5 generations) for strategy 119895 CR
119906119894119895and
119865119906119894119895
corresponding to the trial vectors (119906119894119866
) that successfullyenters next generation are stored in database Once learningperiod is completed mean value (CR
119895
and 119865119895
) of the storedvalue corresponding to strategy 119895 is calculated and is used togenerate a new set of CR
119906119894 119895and 119865
119906119894119895 Then storage data of
CR119906119894119895
and 119865119906119894119895
are removed for the next learning period toeliminate the effects of past data
5 Optimization of Factors (119877hssupport 119877hsswing119877kssupport and 119877ksswing) for Step LengthAdjustment
51 Objective Functions and Constraints of SaDE The objec-tive functions and constraints mentioned in Section 4 areadopted to search appropriate value of 119877hssupport 119877hsswing119877kssupport and119877ksswingfinal corresponding to different desiredstep length (004m 003m 002m and 001m)
52 Lower Bound of Maximum Desired Swing Height It isnatural that the maximum swing height becomes lower toconsume less energy if step length is smaller Lower bound ofswing height is reset as lower value for different step lengths(Table 3) while the upper bound (002m) remains the same
53 Smooth Transition of 119877jointsupportswing Values of119877jointsupportswing (joint = rhs lhs rks and lks) of support legand swing leg are different At the moment of landing time119877jointsupport is changed to 119877jointswing since the support leg ischanged to swing leg in the next step In order to have asmooth transition fifth order polynomial shown in equation(34) is utilized The period of transition time (119905
119891
minus 1199050
) is setas 04 s which is 40 of time duration (1 s) of one step tobalance between rapid transition time and good dynamicequilibrium
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
45 Crossover Operation of SaDE (Step 6) Because of itspopularity binomial crossover operator [20] is utilized in allmutation strategies
If (rand () le CR119894119895
) || (119902 == 119902rand)
119906119902119894119866
= V119902119894119866
Else if (rand () gt CR119894119895
)
119906119902119894119866
= 119909119902119894119866
End
(31)
where 119902 is the parameter number (1 2 3 4 11) in thetarget (119883
119894119866
) and mutant (V119894119866
) vector Arbitrary parameternumber is assigned to 119902rand to ensure that trial vector (119906
119894119866
)is different from target vector (119883
119894119866
) If values of 119906119902119894119866
exceedthe desired range [minus1 1] of parameters to be searched 119906
119902119894119866
is reset by MATLAB function rand ()
46 Selection Operation of SaDE (Step 7) If score of trialvector (119906
119894119866
) is lower than that of target vector (119909119894119866
) then trialvector enters the next generation Otherwise target vectorenters the next generation
If scores119906119894119866
le scores119909119894119866
119909119894119866+1
= 119906119894119866
Else
119909119894119866+1
= 119909119894119866
End
(32)
47 Adjustment of 119875119895
119862119877119895
and 119865119895
(Steps 8ndash11) Initially alearning period (LP = 5 119892119890119899119890119903119886119905119894119900119899119904) is assigned to balancebetween good sample size of data and update frequencyDuring learning period successful times (ST
119895
) and failuretimes (FT
119895
) of each mutation strategy (119895 = 1 2 3 4) in eachgeneration are recorded For example if strategy 119895 is chosenand helps one trial vector (119906
119894119866
) to enter next generationthen ST
119895
is added by 1 Otherwise FT119895
is added by 1 Oncelearning period is completed 119875
Step length (m) Lower bound of maximum desired swingheight (m)
005m 001m004m 0009m003m 0008m002m 0007m001m 0006m
rate (SR119895
) Then ST119895
and FT119895
are reset to zero for the nextlearning period to eliminate effect of the past data
SR119895
=
ST119895
FT119895119866
+ ST119895119866
119875119895
=
SR119895
sum119873=4
119895=1
SR119895
(33)
Similar approach is applied to adjust CR119895
and 119865119895
Duringlearning period (LP = 5 generations) for strategy 119895 CR
119906119894119895and
119865119906119894119895
corresponding to the trial vectors (119906119894119866
) that successfullyenters next generation are stored in database Once learningperiod is completed mean value (CR
119895
and 119865119895
) of the storedvalue corresponding to strategy 119895 is calculated and is used togenerate a new set of CR
119906119894 119895and 119865
119906119894119895 Then storage data of
CR119906119894119895
and 119865119906119894119895
are removed for the next learning period toeliminate the effects of past data
5 Optimization of Factors (119877hssupport 119877hsswing119877kssupport and 119877ksswing) for Step LengthAdjustment
51 Objective Functions and Constraints of SaDE The objec-tive functions and constraints mentioned in Section 4 areadopted to search appropriate value of 119877hssupport 119877hsswing119877kssupport and119877ksswingfinal corresponding to different desiredstep length (004m 003m 002m and 001m)
52 Lower Bound of Maximum Desired Swing Height It isnatural that the maximum swing height becomes lower toconsume less energy if step length is smaller Lower bound ofswing height is reset as lower value for different step lengths(Table 3) while the upper bound (002m) remains the same
53 Smooth Transition of 119877jointsupportswing Values of119877jointsupportswing (joint = rhs lhs rks and lks) of support legand swing leg are different At the moment of landing time119877jointsupport is changed to 119877jointswing since the support leg ischanged to swing leg in the next step In order to have asmooth transition fifth order polynomial shown in equation(34) is utilized The period of transition time (119905
119891
minus 1199050
) is setas 04 s which is 40 of time duration (1 s) of one step tobalance between rapid transition time and good dynamicequilibrium
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
54 Fuzzy Inference System (FIS) Four FISs are adoptedto learn the relationship between (1) 119877hssupport 119877hsswing119877kssupport and119877ksswing and (2) desired step length (5 cm 4 cm
3 cm 2 cm 1 cm and 0 cm)The architecture of FIS proposedby [28] is shown in Figure 4 The desired step length (119863(119899))is the input of FIS while 119877jointsupportswing is the output of FISwhich is obtained through the summation of119891
119898
(119863(119899)) Basedon [24] 119891
119898
(119863(119899)) is a function of desired step and can berepresented as a first order polynomial stated in (40)
In this section the membership function (119872119898
isparameter that affects the width of Gaussian function of119872
119898
119888119898
is parameter that affects the center of Gaussian functionof119872119898
120596119898
is firing strength of rule119898 and 120596119898
is normalizedfiring strength
This section focuses on adjusting the consequent param-eters (119896
1198981
1198961198982
) of 119891119898
(119883) by least square estimation (LSE)[28] stated in (41) Gradient descent algorithm is not adoptedto adjust premise parameters since the size of training data(119873 = 6) is relatively small Hence premise parameters arefixed during the training process In this paper 119888
1minus5
are setas 02 04 06 08 and 1 respectively while 120590
1minus5
are set as03 After 50 training cycles an appropriate set of consequentparameters is found Details of result in this section are statedin Section 6
Consider
119870lowast
= (119880119879
119880)minus1
119880119879
119884 (41)
119870lowast
= [11989611
11989612
11989621
11989622
11989651
11989652] (42)
119880 =[[
[
1205961
(1) 1205961
(1)119863 (1) 1205962
(1) 1205962
(1)119863 (1) 1205965
(1) 1205965
(1)119863 (1)
1205961
(119873) 1205961
(119873)119863 (119873) 1205962
(119873) 1205962
(119873)119863 (119873) 1205965
(119873) 1205965
(119873)119863 (119873)
]]
]
(43)
where 119863 is vector of desired step length and 119884 is vector ofdesired output
6 Results and Discussions
61 Scores and Searched Parameters of Modified CPG ModelAccording to Figure 5 the maximum generation is set as200 The scores of the best target vector is minus11635 Thevalues of fitness functions and constraints are [0018msminus1
00369msminus1] and [0m 0msminus10msminus1 0 rad 0 rad 0m 0m0m and 0m] respectively The searched parameters are[02290 00257 00016 minus03161 minus00199 minus00001 02892 0086100004 minus01786 00619] The results show that the modifiedCPG model can achieve a satisfactory performance
62 Kinematic Aspects of Modified CPG Model The walkinggait searched by SaDE is visualized in Figure 6 Figure 7
ISRN Robotics 11
0 20 40 60 80 100 120 140 160 180 200
0200400
Generations
Scor
es
ScoresavgScoresbest
minus1200
minus1000
minus800
minus600
minus400
minus200
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
Figure 15 Average scores and scores of best target vector of originalCPG model
0 005 01 015 020
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus005
Figure 16 Walking gait of original CPG model searched by SaDE
shows that step achieved by bipedal robot is the same as thedesired step length (005m) In Figure 8 maximum heightof swing foot is satisfactory based on Table 3 Swing footlands on the ground at desired landing time (119905
2
= 07 s and1199055
= 17 s) The joint trajectories of modified CPG modelare shown in Figure 9 Based on Figures 10 and 11 angularposition and velocity of each joint trajectory is observedto be continuous and smooth Because of the heavy trunkmass abrupt change in 119881trunk119910 should be prevented FromFigure 12 it shows that 119881trunk119910 is continuous and smoothexcept the landing time
63 Dynamic Aspects of Modified CPG Model Figure 13shows that dynamic equilibrium of the robot is satisfac-tory since ZMP
119910
is within the area of support polygon[minus0061 0061]m The length of foot sole is 0122m whileZMP119910
margin is defined to be (length of foot sole2) minus
max(|ZMP119910
|) Based on Figure 13 ZMP119910
margin is observedto be at least 003mwhich is large enough to allow larger stepZMP119910
is smooth and the only abrupt change happens at thelanding time (119905
2
= 07 s and 1199055
= 17 s) since ZMP119910
shifts tothe support foot of the next step at these two moments
64 Observations on Mutation Strategies of SaDE Duringsearching parameters of modified CPG model it shows
001 003 005 007 0090
00501
01502
02503
03504
Hei
ght (
m)
Distance (m)minus005 minus003 minus001
Figure 17 The landing instant of right swing leg (original CPGmodel)
that mutation strategy 2 and strategy 3 do not favor forsearching feasible walking gait since they are suppressedcompletely after 60 generations (Figure 14) Figure 5 showsthat scores of target vectors tend to converge in the range ofgeneration 100 to 200 Strategy 1 can help trial vector convergefaster since it involves the best trial vector in the mutationoperation Hence from generation 100 to 200 the probabilityfor choosing strategy 1 is always higher than that of strategy 4
65 Results of Original CPG Model Except constraint 1 theoriginal CPG model [6] is searched by SaDE under the samesetting Safety factor is set as 0m since it becomes difficult tosearch a feasible walking gait for original CPGmodel if safetyfactor is set as 001m Based on Figure 15 the scores of thebest trial vector is minus11059 Also the searched parameters are[00519 00012 00070 minus05237 minus00053 minus00068 03579 0145900025 00891 01736] The values of fitness functions andconstraints are [00694msminus1 00484msminus1] and [00002m0msminus1 0msminus1 0 rad 0 rad 0m 0m 0m 0m] respectivelyCompared with the performance of modified CPG modelthe original CPG model is less satisfactory The walking gaitof original CPG model searched by SaDE is visualized inFigure 16 Also in Figure 17 desired step length (005m) issuccessfully achieved In Figure 18maximumheight of swingfoot is satisfactory based on Table 3 Also swing foot lands onthe ground at desired landing time (119905
2
= 07 s and 1199055
= 17 s)The joint trajectories of modified CPG model are shown
in Figure 19 Based on Figures 20 and 21 angular velocityof each joint trajectory is observed to be discontinuousThe standard deviation (120590original = 00694msminus1) of 119881Trunk119910is larger than that (120590modified = 0018msminus1) of modifiedCPG model The prime reason is that in Figure 22 119881Trunk119910is observed to be discontinuous because of discontinuityin angular velocity In Figure 23 obvious abrupt changein ZMP
119910
is observed Also ZMP119910
exceeds the area ofsupport polygon (119878
1
= 00002m) This shows that dynamicequilibrium of bipedal robot is affected by discontinuousangular velocity Although the area of foot sole can be madelarger to tolerate the abrupt change of ZMP
119910
the agility of therobot is sacrificed For the original CPG model a larger stepis not allowed since ZMP
119910
has exceeded the support polygonwhen the robot walks with 5 cm step length
12 ISRN Robotics
0 02 04 06 08 1 12 14 16 18 20
0002
0004
0006
0008
001
0012
Time (s)
Swin
g he
ight
(m)
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
Figure 18 Variation of height of swing foot with time (original CPGmodel)
0 02 04 06 08 1 12 14 16 18 2
0
02
04
06
08
Time (s)
Join
t ang
le (r
ad)
minus02
minus04
minus06
LhsRhs
LksRks
Figure 19 Joint trajectories of original CPG model searched bySaDE
0 01 02
005
115
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus05 minus04 minus03 minus02 minus01minus2
minus15
minus1
minus05
Figure 20 120579hs versus 120579hs of original CPG model
66 Results of 119877hssupport 119877kssupport 119877hsswing and 119877ksswingSearched by SaDE 119877hssupport 119877kssupport 119877hsswing and 119877ksswingsearched by SaDE are shown in Table 4 In Figure 24 it showsthat the bipedal robot can achieve desired step length (4 cm3 cm 2 cm and 1 cm) Also in Figure 25 themaximum swingheight corresponding to different desired step length reachesreasonable swing height based on Table 3 and no prematurelanding occurs during walking
In Figure 26 ZMP119910
is always within the area of supportpolygon [0061 minus0061] Hence the dynamic equilibrium of
0 01 02 03 04 05 06 07
0
05
1
15
2
Angular position (rad)
Ang
ular
velo
city
(rad
s)
minus1
minus05
Figure 21 120579ks versus 120579ks of original CPG model
0 02 04 06 08 1 12 14 16 18 2
0
005
01
015
02
Time (s)
minus005
Vy
(ms
)
Figure 22 Variation of 119881Trunk119910 with time (original CPG model)
0 02 04 06 08 1 12 14 16 18 2
0001002003004005006007
Time (s)
minus001
minus002
minus003
ZMP y
(m)
Figure 23 Variation of ZMP119910
with time (original CPG model)
001 003 005 007 0090
00501
01502
02503
03504
Distance (m)
Hei
ght (
m)
minus001minus003minus005
Figure 24 Different step length achieved by bipedal robot
ISRN Robotics 13
0 02 04 06 08 1 12 14 16 18 20
000100020003000400050006000700080009
001
Time (s)
Hei
ght (
m)
4 cm3 cm
2 cm1 cm
Figure 25 Variation of height of swing foot with time (differentdesired step length)
0 02 04 06 08 1 12 14 16 18 2
00005
0010015
0020025
Time (s)
minus0005
minus001
minus0015
minus002
minus0025
ZMP y
(m)
4 cm3 cm
2 cm1 cm
Figure 26 Variation of ZMP119910
with time (different desired steplength)
0 02 04 06 08 1 12 14 16 18 2
0
001
002
003
Time (s)
minus001
minus002
minus003
ZMP y
(m)
0 cm 5 cm0 cm 4 cm0 cm 3 cm
0 cm 2 cm0 cm 1 cm
Figure 27 Variation of ZMP119910
corresponding to different desiredstep length change
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
Figure 29 Height of swing foot (arbitrary desired step lengthcommand)
the robot for different desired step length is satisfactoryThenan attempt is made to show that when desired step length isadjusted in the next step ZMP
119910
can still bemaintainedwithinthe area of support polygon [minus0061 0061] The bipedalrobot is at rest initially and is commanded to change its steplength at 119905
2
= 07 s In Figure 27 it shows that if change in steplength is larger then change in ZMP
119910
is larger It also showsthat in the case of the largest change in step length (0 cm to5 cm) ZMP
119910
is still within area of support polygon Hencerobot can keep its dynamic balance well when desired steplength is changed in the next step
67 Relationship between 119877hssupport 119877kssupport 119877hsswing and119877ksswing and Desired Step Length Learnt by FIS The param-eters of FLS tuned by LSE are shown in Table 5 Arbi-trary desired step lengths (0041 cm 0033 cm 0025 cmand 0017 cm) within a specific range (0 cmndash5 cm) are com-manded to the robot In Figure 28 it shows that the robotcan achieve arbitrary desired step length successfully InFigure 29 premature landing does not occur throughout thewhole walking cycle Hence the relationship between (1)119877hssupport119877kssupport119877hsswing and119877ksswing and (2) desired steplength is learnt successfully
14 ISRN Robotics
Yes
No
Is the maximum generation completed
End
No
Yes
Select fi and Si
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
(1) Initialize the first generation (G1) of population (i) of trial vectors (XiG)
(2) Assign initial mean value of crossover rate (CRj) scaling factor (Fj) and probability (Pj)
(3) Initialize crossover rate (CRu119894119895) and scaling factor (Fu119894119895 ) based on CRj and Fj
Code the parameters to be searched to form target vector (XiG)
One of four mutation strategies (j) is chosen for mutation operation through probability to
generate mutant factors (i)
Perform crossover operation to generate trial vectors (uiG)
Perform selection operation to select next generation of target vectors (XiG+1
by comparing the scores of trial vectors (uiG) and target vectors (XiG)
ui+1 wins XiG wins
)
(1) For strategyj values of CRu119894 jand Fu119894j
of corresponding successful ui+1 are recorded
(2) Successful time (STj) is added by 1
(3) Failure time (FTj) is added by 0
(1) Successful time (STj) is added by 0
(2) Failure time (FTj) is added by 1
Is learning period of CRj Fj and Pj completed
(1) CRj and Fj are updated based on the mean values of the stored data of CRu119894119895and Fu119894119895
(2) Probability (Pj) of choosing each mutation strategies (j) is adjusted based on SRj
calculated through STj and FTj
(3) Reset STj and FTj to zero
(4) Clear storage data of CRj and Fj
(5) Restart learning period of (1) Pj and (2) CRj and Fj
Figure 30 Flow chart of SaDE
7 Conclusions
In this paper GAOFSF [6] is modified to ensure that trajecto-ries generated are continuous in angular position and its firstand second derivatives Through simulations bipedal robotwith modified CPG model yields better dynamic balanceSaDE is firstly applied to search the parameters of CPGmodelof bipedal walking Performance of modified CPG model
searched by SaDE is found to be satisfactory Four adjustableparameters (119877hssupport 119877kssupport 119877hsswing and 119877ksswing) areadded to modified CPG model and searched by SaDE Therobot is able to adjust its step length by simply changingthese four factors Instead of using look-up table [6] therelationship between (1) 119877hssupport 119877kssupport 119877hsswing and119877ksswing and (2) desired step length is successfully learntSimulation results show that the robot is able to walk with
ISRN Robotics 15
arbitrary desired step length within specific range (0 cmndash5 cm) The desired joint angles can be obtained without theaid of inverse kinematics
Appendix
For more details see Figure 30
Symbols
119860 119861 119862 Coefficients of truncated Fourier seriesCR Crossover rateCR Mean crossover rate119891 Fitness functions119865 Mutation rate119865 Mean mutation rate119892 Gravity119867swing foot Height of swing foot119867desired Desired height of swing foot119867ground Height of ground119898119894
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
119878 Constraints119906 Trial vectorV Mutant vector119881trunk Velocity of trunk119881trunk Mean velocity of trunk119881strike Velocity of swing foot at landing time120596ℎ
120596119896
Natural frequency of hip and kneetrajectory
120596119900119901
Weighting factor of fitnessfunctionsconstraints
119883 Target vector119910 119910-Coordinate of CoM of link 119894
119911 119911-Coordinate of CoM of link 119894
119910 Linear acceleration in 119910-direction at CoMof link 119894
Linear acceleration in 119911-direction at CoMof link 119894
119877 Scaling factor of truncated Fourier series120579hsks Hipknee joint angle in sagittal plane120583119860119898
Degree of membership function120590 Standard deviation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the Department of Mechani-cal Engineering ofTheHong Kong Polytechnic University forproviding the research studentship
References
[1] S Kajita F Kanehiro K Kaneko et al ldquoBiped walking patterngeneration by using preview control of zero-moment pointrdquo inProceedings of the IEEE International Conference onRobotics andAutomation pp 1620ndash1626 September 2003
[2] J-Y Kim I-W Park and J-H Oh ldquoWalking control algorithmof biped humanoid robot on uneven and inclined floorrdquo Journalof Intelligent and Robotic Systems vol 48 no 4 pp 457ndash4842007
[3] K Erbatur O Koca E Taskiran M Yilmaz and U SevenldquoZMP based reference generation for biped walking robotsrdquoWorld Academy of Science Engineering and Technology vol 58pp 943ndash950 2009
[4] A J Ijspeert ldquoCentral pattern generators for locomotion controlin animals and robots a reviewrdquoNeural Networks vol 21 no 4pp 642ndash653 2008
[5] G Taga Y Yamaguchi and H Shimizu ldquoSelf-organized controlof bipedal locomotion by neural oscillators in unpredictableenvironmentrdquo Biological Cybernetics vol 65 no 3 pp 147ndash1591991
[6] L Yang C-M Chew T Zielinska and A-N Poo ldquoA uniformbiped gait generator with offline optimization and onlineadjustable parametersrdquo Robotica vol 25 no 5 pp 549ndash5652007
[7] N Shafii L P Reis and N Lau ldquoBiped walking using coronaland sagittal movements based on truncated Fourier seriesrdquo inProceedings of 14th Annual Robo Cup International Symposiumpp 324ndash335 2010
[8] E Yazdi V Azizi and A T Haghighat ldquoEvolution of bipedlocomotion using bees algorithm based on truncated Fourierseriesrdquo in Proceedings of the World Congress on Engineering andComputer Science pp 378ndash382 2010
[9] J Or ldquoA hybridCPG-ZMP control system for stable walking of asimulated flexible spine humanoid robotrdquoNeural Networks vol23 no 3 pp 452ndash460 2010
[10] S Aoi and K Tsuchiya ldquoAdaptive behavior in turning of anoscillator-driven biped robotrdquo Autonomous Robots vol 23 no1 pp 37ndash57 2007
[11] Y Farzaneh A Akbarzadeh and A A Akbaria ldquoOnline bio-inspired trajectory generation of seven-link biped robot basedon T-S fuzzy systemrdquo Applied Soft Computing 2013
[12] D Gong J Yan and G Zuo ldquoA review of gait optimizationbased on evolutionary computationrdquo Applied ComputationalIntelligence and Soft Computing vol 2010 Article ID 413179 12pages 2010
[13] H Inada and K Ishii ldquoBipedal walk using a central patterngeneratorrdquo in Proceedings of International Congress Series pp185ndash188 2004
[14] K Miyashita S Ok and K Hase ldquoEvolutionary generation ofhuman-like bipedal locomotionrdquoMechatronics vol 13 no 8-9pp 791ndash807 2003
[15] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[16] BHegery C CHung andK Kasprak ldquoA comparative study ondifferential evolution and genetic algorithms for some combi-natorial problemsrdquo in Proceedings of 8th Mexican InternationalConference on Artificial Intelligence 2009
[17] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithms in multi-objective optimizationrdquo in Proceedings of
16 ISRN Robotics
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
4th International Conference on Evolutionary Multi-CriterionOptimization 2007
[18] J Vesterstroslashm and R Thomsen ldquoA comparative study of differ-ential evolution particle swarm optimization and evolutionaryalgorithms on numerical benchmark problemsrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo04) pp1980ndash1987 June 2004
[19] A K Qin and P N Suganthan ldquoSelf-adaptive differentialevolution algorithm for numerical optimizationrdquo in Proceedingsof the IEEE Congress on Evolutionary Computation (IEEE CECrsquo05) pp 1785ndash1791 September 2005
[20] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009
[21] M G H Omran A Salman and A P Engelbrecht ldquoSelf-adaptive differential evolutionrdquo in Proceedings of InternationalConference on Computational Intelligence and Security pp 192ndash199 2005
[22] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[23] L Yang C-M Chew and A-N Poo ldquoReal-time bipedalwalking adjustment modes using truncated fourier series for-mulationrdquo in Proceedings of the 7th IEEE-RAS InternationalConference on Humanoid Robots (HUMANOIDS rsquo07) pp 379ndash384 December 2007
[24] C Hern ndez-Santos E Rodriguez-Leal R Soto and JL Gordillo ldquoKinematics and dynamics of a new 16DOFHumanoid biped robot with active toe jointrdquo InternationalJournal of Advanced Robotic Systems vol 9 no 190 2012
[25] K-C Choi H-J Lee and M C Lee ldquoFuzzy posture controlfor biped walking robot based on force sensor for ZMPrdquo inProceedings of the SICE-ICASE International Joint Conferencepp 1185ndash1189 October 2006
[26] D Tlalolini Y Aoustin and C Chevallereau ldquoDesign of awalking cyclic gait with single support phases and impacts forthe locomotor system of a thirteen-link 3D biped using theparametric optimizationrdquoMultibody System Dynamics vol 23no 1 pp 33ndash56 2010
[27] A P Sudheer R Vijayakumar and K P Mohanda ldquoStable gaitsynthesis and analysis of a 12-degree of freedom biped robotin sagittal and frontal planesrdquo Journal of Automation MobileRobotics and Intelligent System vol 6 no 4 pp 36ndash44 2012
[28] J-S R Jang ldquoANFIS adaptive-network-based fuzzy inferencesystemrdquo IEEE Transactions on Systems Man and Cyberneticsvol 23 no 3 pp 665ndash685 1993
