Research ArticleNonholonomic Motion Planning Strategy forUnderactuated Manipulator
Liang Li Yuegang Tan and Zhang Li
School of Mechanical and Electronic Engineering Wuhan University of Technology Wuhan 430070 China
Correspondence should be addressed to Liang Li leeliang126com
Received 28 June 2014 Accepted 25 November 2014 Published 18 December 2014
Academic Editor Tarek M Sobh
Copyright copy 2014 Liang Li et alThis 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 develops nonholonomic motion planning strategy for three-joint underactuated manipulator which uses only twoactuators and can be converted into chained form Since the manipulator was designed focusing on the control simplicity there areseveral issues for motion planning mainly including transformation singularity path estimation and trajectory robustness in thepresence of initial errors which need to be considered Although many existing motion planning control laws for chained formsystem can be directly applied to themanipulator and steer it to desired configuration coordinate transformation singularities oftenhappen We propose two mathematical techniques to avoid the transformation singularities Then two evaluation indicators aredefined and used to estimate control precision and linear approximation capability In the end the initial error sensitivity matrix isintroduced to describe the interference sensitivity which is called robustness The simulation and experimental results show thatan efficient and robust resultant path of three-joint underactuated manipulator can be successfully obtained by use of the motionplanning strategy we presented
1 Introduction
In the past few years nonholonomic motion planning hasbecome an attractive research field Much theoretical devel-opment and application have been exploited This paper canbe viewed as the further study on the basis of [1] which pro-posed 119899-joint underactuated manipulator [2] And inspiredby [3ndash5] its physical design is carried out mainly focusingon the kinematic model with triangular structure whichcan be converted into chained form and achieve the controlsimplicity
Although motion planning control law can steer thechained form system to desired state there exists transfor-mation singularity while chained form path is transformedback into actual coordinates Even in the absence of singu-larity planning nonholonomic motion is not an easy taskTransformation singularities add a second level of diffi-culty we must take into account both transformation sin-gularities and nonholonomic constraint In order to solvetransformation singularity problem the path in chainedform space should remain in the singularity-free regions Inother words when a path does not belong to those regions
in chained form coordinate transformation singularity willappear in actual coordinate Therefore transformation sin-gularity avoidance is equivalent to obstacle avoidance inchained form space Singularity regions can be checked bydiffeomorphism of chained form conversion but their shapeand location depend on the mechanical structure and cannotbe generically described for an arbitrary mechanism [6ndash8] Most of the techniques developed for path planning areclassified as geometric path planners [9ndash11] An alternativemethodology of obstacle avoidance is artificial potential field(APF) which considers the robot as a particle or rigid bodywithout constraints [12] and significant effort has been de-voted to elimination of local minima [13 14] References [15ndash17] indicate that a control law with topological propertyhas better obstacle avoidance capability by setting a serialof intermediate points and applies the sinusoidal inputs lawto steer the tractor-trailer system However some motionplanning laws which can achieve a better resultant path thansinusoidal does for the underactuated manipulator have notopological property such as time polynomial inputs
To decrease the trajectory oscillations and increase pre-cision for motion planning in practical experiments two
Hindawi Publishing CorporationJournal of RoboticsVolume 2014 Article ID 743857 10 pageshttpdxdoiorg1011552014743857
2 Journal of Robotics
Angular displacement sensorJoint 1Joint 2 Joint 3
Timing beltCylindrical gearBevel gear
Actuator
Actuator
Disk A
Disk B
Figure 1 Structural diagram of three-joint underactuated manipu-lator
indicators are defined to evaluate the planning path perfor-mance of approximating linear path and control precisionThen we propose the notion of initial state errors robustnesswhich is used to evaluate sensitivity in the presence of initialerrors
This paper is organized as follows Section 2 gives a briefintroduction for underactuated manipulator Section 3 illus-trates the methods for transformation singularity avoidanceof the three-joint prototype Sections 4 and 5 introduce thepath estimation and robustness respectively Conclusionsand future works are given in Section 6
2 An Introduction toUnderactuated Manipulator
Figures 1 and 2 show the structural diagram and photo ofthe prototype three-joint underactuated manipulator withtwo actuators Obviously the configuration of the 119899-jointunderactuated manipulator has 119899 + 1 dimensions whichare determined by angular displacement 120579
119894(119894 = 1 2 119899)
of each joint and the angle 1205931of a horizontally mounted
actuator And another vertically mounted actuator connectswith joint 1
Figure 3 shows the detailed structure of joint 119894 of theunderactuatedmanipulator Each joint except the last one hastwo sets of same size friction disk transmission mechanisms[18] P1 P2 and P3 represent the rolling without slippingcontact points between disk119860 and disk 119861 Disk 119861
119894with radius
1199031rotates around the fixed axis with a given angular velocity
119894which drives disk 119860
119894to rotate The angular velocity
119894of
disk 119860119894is divided into two parts one is transmitted into disk
119861119894+1
as angular velocity input 119894+1
through a set of bevel gearsand spur gears the other drives the next joint 119894 + 1 to rotatethrough a set of friction diskmechanisms and timing belts
120601119894
and 120573119894represent the angular velocity of disk 119861
1015840
119894with radius 119903
2
and disk 1198601015840
119894 respectively 119877 and 119903
3express the distance from
rotation axle of disk119860 to rolling contact points P1 P2 and P3The underactuated manipulatorrsquos nonholonomic con-
straints are due to the rolling contact between disk119860 and disk119861 By setting configuration variables [120593
119899minus1 1205791 1205792 120579
119899]119879
Figure 2 Prototype of three joints and two actuators
120579i+1120579i
Oi+1Oi
Oi+1Oi
OD
OB
Disk A998400i
Disk Ai
Disk A998400i+1
Disk Ai+1
r3120573i
Ri
r2 120601i
r1i
Time belt
Disk
B998400 i
Disk
Bi
Disk
B998400 i+1
Disk
Bi+1
Joint i + 1Joint i
r1i+1
Spur gearBevel gear
P1
P2
P3
Figure 3 Structure diagram of joint 119894 mechanical system
the kinematic model can be described as follows (for detailedmathematical derivation see [2])
This formulation of the kinematic model has the trian-gular structure which can be converted into chained formEquations (1) can be expressed as follows
119902 = 1198921(119902) sdot 119906
1+ 1198922(119902) sdot 119906
2 (2)
Journal of Robotics 3
where 119902 ≜ [120593119899minus1
1205791 120579
119899]119879
1198921(119902) = [1 0 119896
1198991198883
119899minus1119875119899minus2
1 119896119892119875119899minus2
1]
119879
1198922(119902) = [0 1 0 0]
119879
(3)
1198921(119902) and 119892
2(119902) are vector fields This system is said to be
driftless that is to say the state of the system does not driftwhen the controls are set to zero
3 Transformation Singularity Avoidance
31 Conversion into Chained Form Inspired by [3] Sordalenshows conversions of the kinematic model of a car with119899 trailers The underactuated manipulator is designed toconvert into chained form and achieve control simplicityalthough a little complication should be added to mechanicalstructures
Nonlinear coordinate transformation and input feedbacktransformation of kinematics model of three-joint underac-tuated manipulator can be expressed as follows
1199114= 1205793
1199113=
1205971199114
1205971199024
1198914(1199023) = (
1198963
119896119892
) 1198883
2
1199112=
1205971199113
1205971199023
1198913(1199022) =
minus311989631198962
1198962
119892
1198882
11198882
21199041
2
1199111= 1205932
(4)
V1= 1199061= 1205932
V2=
31198962
21198963
1198963
119892
1198884
1(211988821199042
2minus 1198883
2) 1199061+
611989621198963
1198962
119892
119888111990411198882
211990421199062
(5)
where 119904119895
119894= sin119895120579
119894and 119888119895
119894= cos119895120579
119894
Reference [2] has proven nonholonomy and controllabil-ity of the 119899-joint underactuated manipulator Then we willdiscuss that conversions are diffeomorphic Inspired by [3]there is Theorem 1 as follows
Theorem 1 The chained form conversions of underactuatedmanipulator are diffeomorphic if and only if the Jacobian119891119894+1
(119902119894) of kinematic model in (1) is nonsingular
Equation (7) shows that the coordinated conversions for119899-joint manipulator are diffeomorphic except at 120579
119894= 0
Therefore the kinematic model can be diffeomorphicallyconvertible into chained form in the subspace 120579
119894isin (minus1205872 0)cup
(0 1205872)
32 Three-Joint Underactuated Manipulator Motion PlanningThere are two major control schemes for chained form Oneis open loop control and the other is feedback control Amajor advantage of open loop control is that solutions forpractical applications with low computational cost can beprovided Furthermore chained form of feedback control hastwo drawbacks one is that stabilizing chained system to thenonzero configuration is extremely difficult in practice theother is that obstacle or singularity avoidance problem ishardly solved by some form of feedback control laws becauseof no specified extent of overshoot
There are many existing open loop controllers for thechained form such as sinusoidal inputs time polynomialsinputs and piecewise constant inputs Any one of the controllaws can be applied to the underactuated manipulator Sincetime polynomials inputs law is easily obtained by solvingsimple algebraic equations and generates a better resultantpath for three-joint underactuated manipulator it is given asfollows
V1= 1198870
V2= 1198880+ 1198881119905 + 11988821199052
(8)
Although the control law polynomial inputs can steerchained form state 119911 to their desired configuration there is noguarantee that this path when mapped back into the actualcoordinate will avoid the transformation singularity Thatmeans we must check every path and ensure the existenceof every state variable in configuration space If a singularitydoes really happen some measures should be taken to find avalid path
The transformation singularity happens at 120579119894
= 0 andtrajectories in the chained form space should satisfy theconditions which can be directly checked from (4) and (7)Actually the primary cause of transformation singularity ofthree-joint manipulator is due to 119911
2gt 0 because expression
of 1199112not only is more complex but also is affected by 120579
2
The two mathematical techniques avoiding transformationsingularities are proposed in the case of 119911
2gt 0 as in the
following section
33 Constraint Point Method In order to avoid transforma-tion singularity the chained form state existing singularityregion is modified by setting some points which are calledconstraint points The constraint point method is just usedto avoid the transformation singularity in the path Thepolynomial inputs law with 119899 states and 119898 constraint pointsfor chained form is established as follows
V1= 1198870
V2=
119899minus2
sum
119894=0
119888119894119905119894+
119898
sum
119895=1
119889119895119905119899+119895minus2
(9)
4 Journal of Robotics
where 119894 isin 0 1 119899 minus 2 119895 isin 1 2 119898 and 119889119895is coef-
ficient Constraint points (119905119895 119911119894(119905119895)) are applied to chained
form state 119911119894(119905) which can be expressed as follows
1199111 (
119905) = 1198870119905 + 1199111 (
0)
1199112 (
119905) =
119899minus2
sum
119894=0
1
119894 + 1
119888119894119905119894+1
+
119898
sum
119895=1
1
119899 + 119895 minus 1
119889119895119905119899+119895minus1
+ 1199112 (
0)
1199113 (
119905) = 1198870(
119899minus2
sum
119894=0
1
(119894 + 2) (119894 + 1)
119888119894119905119894+2
+
119898
sum
119895=1
1
(119899 + 119895) (119899 + 119895 minus 1)
119889119895119905119899+119895
)
+ 11988701199112 (
0) sdot 119905 + 1199113 (
0)
119911119899 (
119905) = 119887119899minus2
0(
119899minus2
sum
119894=0
119894
(119894 + 119899 minus 1)
119888119894119905119894+119899minus1
+
119898
sum
119895=1
(119899 + 119895 minus 2)
(2119899 + 119895 minus 3)
1198891198951199052119899+119895minus3
)
+
119899
sum
119896=2
119887119899minus119896
0sdot 119905119899minus119896
sdot 119911119896 (
0)
(119899 minus 119896)
(10)
Obviously the state 119911(119905) is function of initial conditions119911(0) as well as 119899 + 119898 undetermined coefficients (119887
0 1198880
119888119899minus2
1198891 119889
119898) Equations (10) inputs will steer the
chained form system from 119911119894(0) to 119911
119894(119879) in finite time 119879
and pass through constraint state 119911119894(119905119895) at time 119905
119894 It can be
seen clearly that the existence and uniqueness of solutionsto (10) can guarantee the chained form state 119911(119905) satisfyingnonholonomy at constraint points
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
119879)
1199113 (
119879)
119911119899 (
119879)
1199111198941(1199051)
119911119894119898
(119905119898)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
= 119872(119879)
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1198880
1198881
119888119899
1198891
119889119898
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
+ 119863
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
0)
1199113 (
0)
119911119899 (
0)
1199111198941 (
0)
119911119894119898 (
0)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(11)
where 119911119894119898(119905119898) is point on the 119911
119894(119905) curve at time 119905
119898isin (0 119879)
and the matrix119872(119879) has the form
119872(119879) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879
1198792
2
sdot sdot sdot
119879119899minus1
119899 minus 1
sdot sdot sdot
119879119899+119898minus1
119899 + 119898 minus 1
11988701198792
2
11988701198793
3
sdot sdot sdot
(119899 minus 2)1198870119879119899
119899
sdot sdot sdot
(119899 + 119898 minus 2)1198870119879119899+119898
(119899 + 119898)
119887119899minus2
0119879119899minus1
(119899 minus 1)
119887119899minus2
0119879119899
119899
sdot sdot sdot
(119899 minus 2)119887119899minus2
01198792119899minus3
(2119899 minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
01198792119899+119898minus3
(2119899 + 119898 minus 3)
1198871198941minus2
01199051198941minus1
1
(1198941minus 1)
1198871198941minus2
01199051198941
1
1198941
sdot sdot sdot
(119899 minus 2)1198871198941minus2
0119905(119899+1198941minus3)
1
(119899 + 1198941minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
0119905119899+119898+119894119898minus3
1
(119899 + 119898 + 1198941minus 3)
119887119894119898minus2
0119905119894119898minus1
119898
(119894119898
minus 1)
119887119894119898minus2
0119905119894119898119898
119894119898
sdot sdot sdot
(119899 minus 2)119887119894119898minus2
0119905119899+119894119898minus3
119898
(119899 + 119894119898
minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119894119898minus2
0119905119899+119898+119894119898minus3
119898
(119899 + 119898 + 119894119898
minus 3)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(12)
Cramer law guarantees existence and uniqueness ofsolutions to (11) if and only if matrix119872(119879) is nonsingular for1198870
= 0
Remark 2 When 1199111(0) = 119911
1(119879) 119872(119879) becomes singular
This case can be dealt with by the intermediate point methodmentioned in [19]
Journal of Robotics 5
0 5 10 15 20 25 30Time (s)
minus08
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Figure 4 Chained form state 119911
Remark 3 This major advantage of the constraint pointmethod is an effective and simple way Of course othermeth-ods such as penalty function method [20] can be applied toavoid transformation singularity
Initial and desired configuration are given as 119902(0) = [1∘
1∘ 1∘]119879 and 119902(119879) = [30
∘ 30∘ 30∘]119879 in the configuration space
which correspond to 119885(0) = [minus0032 0711 0018]119879 and
119885(119879) = [minus0532 0462 0524]119879 in chained form space respec-
tively Figure 4 indicates the trajectory of 119885 without anyconstraint points Although the state 119885 can move to desiredconfiguration 119911(119879) 119911
2(119905) path has two zero-crossing points
against the conditions of keeping negative The inversechained form transformation singularity happens in intervalbetween two zero-crossing points as shown in Figure 4cyan marker of 119911
2(119905) trajectory Figure 5 indicates that a
constraint point (5 minus02) is chosen in the coordinate systemThe initial and desired configuration of control parameter119885 are redefined as 119885(0) = [minus0032 0711 0018 minus0032]
119879
and 119885(119879) = [minus0532 0462 0524 minus02]119879 It can be seen that
1199112(119905) can still move to target values 119911(119879) smoothly and avoid
placement versus time It is clear that the joint angles canreach the desired configuration smoothly without any trans-formation singularities
34 Input Control Parameter Adjustment Method Althoughconstraint point method has low numerical computationalcost the choice of constraint points sometimes needs a lotof trials for transformation singularity avoidance Inspiredby overparameterization of sinusoidal inputs in [19] theinput control parameter adjustment method is presentedThe actual coordinate of underactuated manipulator has 4
0 5 10 15 20 25 30Time (s)
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Constraint point(5 minus02)
Figure 5 Chained form state 119911 with one constraint point
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 6 Joints angular displacement with constraint point (simu-lation 1)
dimensions [120593119899minus1
1205791 1205792 1205793]119879 and our goal is to steer 3-
dimensional state variable 120579119894(119905) to 120579
119894(119879) 119894 isin 1 2 3 Since 119887
0
can be expressed as 1198870= (1205932(119879) minus 120593
2(0))119879 we can deal with
the polynomials input law by choosing an appropriate 1198870in
singularity-free regions to achieve transformation singularityavoidance This method is applied to three-joint underactu-ated manipulator
As mentioned the main condition of avoidance transfor-mation singularity is
1199112 (
119905) lt 0 119905 isin [0 119879] (13)
6 Journal of Robotics
Equation (13) can be satisfied if maximum value of 1199112(119905)
keeps negative The time corresponding to stagnation pointof 1199112(119905) can be expressed as follows
1199051=
minus1198881+ radic1198882
1minus 411988801198882
21198882
1199052=
minus1198881minus radic1198882
1minus 411988801198882
21198882
1198892(1199112(1199052))
1198891199052
= minusradic1198882
1minus 411988801198882lt 0
(14)
From (11) 1199112(1199052) can be represented as function of control
parameter 1198870
119888119894= 119888119894(1198870)
1199112(1199052) =
1198882
3
1199053
2+
1198881
2
1199052
2+ 11988801199052+ 1199112 (
0)
119894 = 0 1 2
(15)
Consider (15) as a mapΦ1198870
119868119877 997904rArr 119868119877 (16)
which maps input control parameter 1198870into the maximum
1199112(1199052) Obviously the mapping Φ
1198870is surjection (many-to-
one) since the choice of 1198870for solving 119911
2(1199052) is not unique
The range of value 1198870can be determined by solving roots of
(15)In Figure 7 simulation 2 shows joint angles versus time
for three-joint underactuated manipulator The joint anglesof initial configuration and terminal configuration are 120579(0) =
[5∘ 5∘ 5∘]119879 and 120579(119879) = [30
∘ 30∘ 30∘]119879 respectively Input
control parameter 1205932= 34∘ since there is no transformation
singularity if and only if 1205932is always chosen in the range of
3235∘ to 3544∘ It can be seen that the joints angles reach thedesired configuration after the setting time
A major advantage of this approach is that analyticalsolution of control parameter 119887
0can be calculated in transfor-
mation singularity-free region And the order of polynomialsdoes not increase in comparison to constraint point methodsince a valid path without singularity depends on choosingsuitable value of 119887
0 Comparing overparameterization meth-
ods the choice of input control parameter 1198870is not arbitrary
but definiteThis method leaves one parameter free and searches for a
value in this parameter space which guarantees that the pathbelongs to transformation singularity-free region Howeveraccording to Galois theory closed-form solution does notexist for polynomial order higher than 4 In other wordsthere is no analytical solution but numerical one with highcomputational cost for 119887
0when underactuated manipulator
has 5 or more joints
4 Nonholonomic Path Estimation
Although a valid path without transformation singularitycan be generated in the previous sections actual shape of
0 5 10 15 20 25 300
10
20
30
50
60
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 7 Joints angular displacement at 1205932= 34∘ (simulation 2)
path shows a detour to desired configuration Meanderingtrajectory with high amplitude causes the control losing inpractice
41 Nonholonomic Path Evaluation Scheme A resultant pathof underactuated manipulator must satisfy nonholonomicconstraint which is essentially nonlinear A reference pathis established to be a straight line connecting initial con-figuration with desired configuration in the configurationspace We define two indicators approximation distance 119889
119886
and extreme point number 119905119901 The indicator 119889
119886 which is the
maximum value of vertical distance between the referencepath and the point on the nonholonomic path can evaluateapproaching the reference path capability 119905
119901 which is the
number of extreme points on the path estimates the numberof actuators rotation direction changes
The nonholonomic path evaluation method in [5] esti-mates the distance between linear path and planned pathand does not precisely describe the extent of meanderingof the path An evaluation scheme is proposed to decreasethe numbers of changes of actuator rotational directionand errors from transmission of manipulator such as back-lash
Let us consider a configuration space CS with 119897 CS times
CS rarr CS[01] being a local planner for two configurations119886 119887 isin CS and 119897(119886 119887) is a nonholonomic path 119897
119886119887(119905) 119905 isin [0 1]
such that 119897119886119887
(0) = 119886 119897119886119887
(1) = 119887 The reference path 119875(119886 119887) isstraight line 119905 isin [0 1]
Definition 4 119889119886is the maximum value of vertical distance
between the reference path and the point on the nonholo-nomic path and let forall119899 ℎ isin CS as shown in Figure 8 be
Journal of Robotics 7
Desired configuration
Nonholonomic path
(deg
)
Reference path
Initial configuration
Time
a
m
h
b
n
120572
l1
l2
la
Figure 8 Illustration of the indicators of evaluation
119897119886119887
(119905) = 119899 119901119886119887
(1199051015840) = ℎ 119899119898 isin CS and la sdot l1 = 0 119901
119886119887(119905) =
((119905 minus 1)(0 minus 1)) sdot 119886 + ((119905 minus 0)(1 minus 0)) sdot 119887
Definition 5 119905119901is the number of extreme points of the actual
trajectory The actuator rotational direction changes at time119905119894if 119889(119905119894minus 120578) sdot 119889(119905
119894+ 120578) lt 0 where 120578 gt 0 and 119889(119905
119894plusmn 120578) is the
vertical distance at 119905119894plusmn 120578 between the actual trajectory and
reference path
We make further explanation about two indicators forsearching optimal pathThe values of 119889
119886and 119905119901are calculated
within control parameter 1198870range which guarantees that the
path belongs to transformation singularity-free region Inpractice the performance for motion planning of three-jointmanipulator is mainly decided by 119905
119901 With slight increment
of 119905119901 the errors of transmission parts of underactuated
manipulator will be increased significantly Therefore 119905119901is
more significant than 119889119886for performance of nonholonomic
motion planning In conclusion a search in parameter space1198870is carried out to calculate the above two indicators with
priority to 119905119901 In other words the minimum value 119889
119886is
searched in the min(119905119901) region
To verify the usefulness of the evaluation scheme sim-ulation 3 has been carried out for three-joint underactuatedmanipulator Let 120593
2be free and initial configuration and
desired configuration are 120579(0) = [10∘ 10∘ 10∘]119879 and 120579(119879) =
In simulation 3 we just estimate the first joint angletrajectory instead of all of them because the underactuatedmanipulator is an open-chain mechanism and the actuator isconnected with joint 1 as shown in Figures 1 and 2 Equations(4) reveal that inverse mapping from 119911
2to 1205791not only is
more complicated than others but also depends on 1205792 The
simulations (1 2 and 3) show that the 1205791trajectory has more
oscillation in comparison with 1205792and 1205793
Figure 10 illustrates the experimental result In additionthere is existence of 5∘ maximum error on the 120579
3curve
8 Journal of Robotics
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
Figure 1 Structural diagram of three-joint underactuated manipu-lator
indicators are defined to evaluate the planning path perfor-mance of approximating linear path and control precisionThen we propose the notion of initial state errors robustnesswhich is used to evaluate sensitivity in the presence of initialerrors
This paper is organized as follows Section 2 gives a briefintroduction for underactuated manipulator Section 3 illus-trates the methods for transformation singularity avoidanceof the three-joint prototype Sections 4 and 5 introduce thepath estimation and robustness respectively Conclusionsand future works are given in Section 6
2 An Introduction toUnderactuated Manipulator
Figures 1 and 2 show the structural diagram and photo ofthe prototype three-joint underactuated manipulator withtwo actuators Obviously the configuration of the 119899-jointunderactuated manipulator has 119899 + 1 dimensions whichare determined by angular displacement 120579
119894(119894 = 1 2 119899)
of each joint and the angle 1205931of a horizontally mounted
actuator And another vertically mounted actuator connectswith joint 1
Figure 3 shows the detailed structure of joint 119894 of theunderactuatedmanipulator Each joint except the last one hastwo sets of same size friction disk transmission mechanisms[18] P1 P2 and P3 represent the rolling without slippingcontact points between disk119860 and disk 119861 Disk 119861
119894with radius
1199031rotates around the fixed axis with a given angular velocity
119894which drives disk 119860
119894to rotate The angular velocity
119894of
disk 119860119894is divided into two parts one is transmitted into disk
119861119894+1
as angular velocity input 119894+1
through a set of bevel gearsand spur gears the other drives the next joint 119894 + 1 to rotatethrough a set of friction diskmechanisms and timing belts
120601119894
and 120573119894represent the angular velocity of disk 119861
1015840
119894with radius 119903
2
and disk 1198601015840
119894 respectively 119877 and 119903
3express the distance from
rotation axle of disk119860 to rolling contact points P1 P2 and P3The underactuated manipulatorrsquos nonholonomic con-
straints are due to the rolling contact between disk119860 and disk119861 By setting configuration variables [120593
119899minus1 1205791 1205792 120579
119899]119879
Figure 2 Prototype of three joints and two actuators
120579i+1120579i
Oi+1Oi
Oi+1Oi
OD
OB
Disk A998400i
Disk Ai
Disk A998400i+1
Disk Ai+1
r3120573i
Ri
r2 120601i
r1i
Time belt
Disk
B998400 i
Disk
Bi
Disk
B998400 i+1
Disk
Bi+1
Joint i + 1Joint i
r1i+1
Spur gearBevel gear
P1
P2
P3
Figure 3 Structure diagram of joint 119894 mechanical system
the kinematic model can be described as follows (for detailedmathematical derivation see [2])
This formulation of the kinematic model has the trian-gular structure which can be converted into chained formEquations (1) can be expressed as follows
119902 = 1198921(119902) sdot 119906
1+ 1198922(119902) sdot 119906
2 (2)
Journal of Robotics 3
where 119902 ≜ [120593119899minus1
1205791 120579
119899]119879
1198921(119902) = [1 0 119896
1198991198883
119899minus1119875119899minus2
1 119896119892119875119899minus2
1]
119879
1198922(119902) = [0 1 0 0]
119879
(3)
1198921(119902) and 119892
2(119902) are vector fields This system is said to be
driftless that is to say the state of the system does not driftwhen the controls are set to zero
3 Transformation Singularity Avoidance
31 Conversion into Chained Form Inspired by [3] Sordalenshows conversions of the kinematic model of a car with119899 trailers The underactuated manipulator is designed toconvert into chained form and achieve control simplicityalthough a little complication should be added to mechanicalstructures
Nonlinear coordinate transformation and input feedbacktransformation of kinematics model of three-joint underac-tuated manipulator can be expressed as follows
1199114= 1205793
1199113=
1205971199114
1205971199024
1198914(1199023) = (
1198963
119896119892
) 1198883
2
1199112=
1205971199113
1205971199023
1198913(1199022) =
minus311989631198962
1198962
119892
1198882
11198882
21199041
2
1199111= 1205932
(4)
V1= 1199061= 1205932
V2=
31198962
21198963
1198963
119892
1198884
1(211988821199042
2minus 1198883
2) 1199061+
611989621198963
1198962
119892
119888111990411198882
211990421199062
(5)
where 119904119895
119894= sin119895120579
119894and 119888119895
119894= cos119895120579
119894
Reference [2] has proven nonholonomy and controllabil-ity of the 119899-joint underactuated manipulator Then we willdiscuss that conversions are diffeomorphic Inspired by [3]there is Theorem 1 as follows
Theorem 1 The chained form conversions of underactuatedmanipulator are diffeomorphic if and only if the Jacobian119891119894+1
(119902119894) of kinematic model in (1) is nonsingular
Equation (7) shows that the coordinated conversions for119899-joint manipulator are diffeomorphic except at 120579
119894= 0
Therefore the kinematic model can be diffeomorphicallyconvertible into chained form in the subspace 120579
119894isin (minus1205872 0)cup
(0 1205872)
32 Three-Joint Underactuated Manipulator Motion PlanningThere are two major control schemes for chained form Oneis open loop control and the other is feedback control Amajor advantage of open loop control is that solutions forpractical applications with low computational cost can beprovided Furthermore chained form of feedback control hastwo drawbacks one is that stabilizing chained system to thenonzero configuration is extremely difficult in practice theother is that obstacle or singularity avoidance problem ishardly solved by some form of feedback control laws becauseof no specified extent of overshoot
There are many existing open loop controllers for thechained form such as sinusoidal inputs time polynomialsinputs and piecewise constant inputs Any one of the controllaws can be applied to the underactuated manipulator Sincetime polynomials inputs law is easily obtained by solvingsimple algebraic equations and generates a better resultantpath for three-joint underactuated manipulator it is given asfollows
V1= 1198870
V2= 1198880+ 1198881119905 + 11988821199052
(8)
Although the control law polynomial inputs can steerchained form state 119911 to their desired configuration there is noguarantee that this path when mapped back into the actualcoordinate will avoid the transformation singularity Thatmeans we must check every path and ensure the existenceof every state variable in configuration space If a singularitydoes really happen some measures should be taken to find avalid path
The transformation singularity happens at 120579119894
= 0 andtrajectories in the chained form space should satisfy theconditions which can be directly checked from (4) and (7)Actually the primary cause of transformation singularity ofthree-joint manipulator is due to 119911
2gt 0 because expression
of 1199112not only is more complex but also is affected by 120579
2
The two mathematical techniques avoiding transformationsingularities are proposed in the case of 119911
2gt 0 as in the
following section
33 Constraint Point Method In order to avoid transforma-tion singularity the chained form state existing singularityregion is modified by setting some points which are calledconstraint points The constraint point method is just usedto avoid the transformation singularity in the path Thepolynomial inputs law with 119899 states and 119898 constraint pointsfor chained form is established as follows
V1= 1198870
V2=
119899minus2
sum
119894=0
119888119894119905119894+
119898
sum
119895=1
119889119895119905119899+119895minus2
(9)
4 Journal of Robotics
where 119894 isin 0 1 119899 minus 2 119895 isin 1 2 119898 and 119889119895is coef-
ficient Constraint points (119905119895 119911119894(119905119895)) are applied to chained
form state 119911119894(119905) which can be expressed as follows
1199111 (
119905) = 1198870119905 + 1199111 (
0)
1199112 (
119905) =
119899minus2
sum
119894=0
1
119894 + 1
119888119894119905119894+1
+
119898
sum
119895=1
1
119899 + 119895 minus 1
119889119895119905119899+119895minus1
+ 1199112 (
0)
1199113 (
119905) = 1198870(
119899minus2
sum
119894=0
1
(119894 + 2) (119894 + 1)
119888119894119905119894+2
+
119898
sum
119895=1
1
(119899 + 119895) (119899 + 119895 minus 1)
119889119895119905119899+119895
)
+ 11988701199112 (
0) sdot 119905 + 1199113 (
0)
119911119899 (
119905) = 119887119899minus2
0(
119899minus2
sum
119894=0
119894
(119894 + 119899 minus 1)
119888119894119905119894+119899minus1
+
119898
sum
119895=1
(119899 + 119895 minus 2)
(2119899 + 119895 minus 3)
1198891198951199052119899+119895minus3
)
+
119899
sum
119896=2
119887119899minus119896
0sdot 119905119899minus119896
sdot 119911119896 (
0)
(119899 minus 119896)
(10)
Obviously the state 119911(119905) is function of initial conditions119911(0) as well as 119899 + 119898 undetermined coefficients (119887
0 1198880
119888119899minus2
1198891 119889
119898) Equations (10) inputs will steer the
chained form system from 119911119894(0) to 119911
119894(119879) in finite time 119879
and pass through constraint state 119911119894(119905119895) at time 119905
119894 It can be
seen clearly that the existence and uniqueness of solutionsto (10) can guarantee the chained form state 119911(119905) satisfyingnonholonomy at constraint points
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
119879)
1199113 (
119879)
119911119899 (
119879)
1199111198941(1199051)
119911119894119898
(119905119898)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
= 119872(119879)
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1198880
1198881
119888119899
1198891
119889119898
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
+ 119863
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
0)
1199113 (
0)
119911119899 (
0)
1199111198941 (
0)
119911119894119898 (
0)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(11)
where 119911119894119898(119905119898) is point on the 119911
119894(119905) curve at time 119905
119898isin (0 119879)
and the matrix119872(119879) has the form
119872(119879) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879
1198792
2
sdot sdot sdot
119879119899minus1
119899 minus 1
sdot sdot sdot
119879119899+119898minus1
119899 + 119898 minus 1
11988701198792
2
11988701198793
3
sdot sdot sdot
(119899 minus 2)1198870119879119899
119899
sdot sdot sdot
(119899 + 119898 minus 2)1198870119879119899+119898
(119899 + 119898)
119887119899minus2
0119879119899minus1
(119899 minus 1)
119887119899minus2
0119879119899
119899
sdot sdot sdot
(119899 minus 2)119887119899minus2
01198792119899minus3
(2119899 minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
01198792119899+119898minus3
(2119899 + 119898 minus 3)
1198871198941minus2
01199051198941minus1
1
(1198941minus 1)
1198871198941minus2
01199051198941
1
1198941
sdot sdot sdot
(119899 minus 2)1198871198941minus2
0119905(119899+1198941minus3)
1
(119899 + 1198941minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
0119905119899+119898+119894119898minus3
1
(119899 + 119898 + 1198941minus 3)
119887119894119898minus2
0119905119894119898minus1
119898
(119894119898
minus 1)
119887119894119898minus2
0119905119894119898119898
119894119898
sdot sdot sdot
(119899 minus 2)119887119894119898minus2
0119905119899+119894119898minus3
119898
(119899 + 119894119898
minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119894119898minus2
0119905119899+119898+119894119898minus3
119898
(119899 + 119898 + 119894119898
minus 3)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(12)
Cramer law guarantees existence and uniqueness ofsolutions to (11) if and only if matrix119872(119879) is nonsingular for1198870
= 0
Remark 2 When 1199111(0) = 119911
1(119879) 119872(119879) becomes singular
This case can be dealt with by the intermediate point methodmentioned in [19]
Journal of Robotics 5
0 5 10 15 20 25 30Time (s)
minus08
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Figure 4 Chained form state 119911
Remark 3 This major advantage of the constraint pointmethod is an effective and simple way Of course othermeth-ods such as penalty function method [20] can be applied toavoid transformation singularity
Initial and desired configuration are given as 119902(0) = [1∘
1∘ 1∘]119879 and 119902(119879) = [30
∘ 30∘ 30∘]119879 in the configuration space
which correspond to 119885(0) = [minus0032 0711 0018]119879 and
119885(119879) = [minus0532 0462 0524]119879 in chained form space respec-
tively Figure 4 indicates the trajectory of 119885 without anyconstraint points Although the state 119885 can move to desiredconfiguration 119911(119879) 119911
2(119905) path has two zero-crossing points
against the conditions of keeping negative The inversechained form transformation singularity happens in intervalbetween two zero-crossing points as shown in Figure 4cyan marker of 119911
2(119905) trajectory Figure 5 indicates that a
constraint point (5 minus02) is chosen in the coordinate systemThe initial and desired configuration of control parameter119885 are redefined as 119885(0) = [minus0032 0711 0018 minus0032]
119879
and 119885(119879) = [minus0532 0462 0524 minus02]119879 It can be seen that
1199112(119905) can still move to target values 119911(119879) smoothly and avoid
placement versus time It is clear that the joint angles canreach the desired configuration smoothly without any trans-formation singularities
34 Input Control Parameter Adjustment Method Althoughconstraint point method has low numerical computationalcost the choice of constraint points sometimes needs a lotof trials for transformation singularity avoidance Inspiredby overparameterization of sinusoidal inputs in [19] theinput control parameter adjustment method is presentedThe actual coordinate of underactuated manipulator has 4
0 5 10 15 20 25 30Time (s)
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Constraint point(5 minus02)
Figure 5 Chained form state 119911 with one constraint point
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 6 Joints angular displacement with constraint point (simu-lation 1)
dimensions [120593119899minus1
1205791 1205792 1205793]119879 and our goal is to steer 3-
dimensional state variable 120579119894(119905) to 120579
119894(119879) 119894 isin 1 2 3 Since 119887
0
can be expressed as 1198870= (1205932(119879) minus 120593
2(0))119879 we can deal with
the polynomials input law by choosing an appropriate 1198870in
singularity-free regions to achieve transformation singularityavoidance This method is applied to three-joint underactu-ated manipulator
As mentioned the main condition of avoidance transfor-mation singularity is
1199112 (
119905) lt 0 119905 isin [0 119879] (13)
6 Journal of Robotics
Equation (13) can be satisfied if maximum value of 1199112(119905)
keeps negative The time corresponding to stagnation pointof 1199112(119905) can be expressed as follows
1199051=
minus1198881+ radic1198882
1minus 411988801198882
21198882
1199052=
minus1198881minus radic1198882
1minus 411988801198882
21198882
1198892(1199112(1199052))
1198891199052
= minusradic1198882
1minus 411988801198882lt 0
(14)
From (11) 1199112(1199052) can be represented as function of control
parameter 1198870
119888119894= 119888119894(1198870)
1199112(1199052) =
1198882
3
1199053
2+
1198881
2
1199052
2+ 11988801199052+ 1199112 (
0)
119894 = 0 1 2
(15)
Consider (15) as a mapΦ1198870
119868119877 997904rArr 119868119877 (16)
which maps input control parameter 1198870into the maximum
1199112(1199052) Obviously the mapping Φ
1198870is surjection (many-to-
one) since the choice of 1198870for solving 119911
2(1199052) is not unique
The range of value 1198870can be determined by solving roots of
(15)In Figure 7 simulation 2 shows joint angles versus time
for three-joint underactuated manipulator The joint anglesof initial configuration and terminal configuration are 120579(0) =
[5∘ 5∘ 5∘]119879 and 120579(119879) = [30
∘ 30∘ 30∘]119879 respectively Input
control parameter 1205932= 34∘ since there is no transformation
singularity if and only if 1205932is always chosen in the range of
3235∘ to 3544∘ It can be seen that the joints angles reach thedesired configuration after the setting time
A major advantage of this approach is that analyticalsolution of control parameter 119887
0can be calculated in transfor-
mation singularity-free region And the order of polynomialsdoes not increase in comparison to constraint point methodsince a valid path without singularity depends on choosingsuitable value of 119887
0 Comparing overparameterization meth-
ods the choice of input control parameter 1198870is not arbitrary
but definiteThis method leaves one parameter free and searches for a
value in this parameter space which guarantees that the pathbelongs to transformation singularity-free region Howeveraccording to Galois theory closed-form solution does notexist for polynomial order higher than 4 In other wordsthere is no analytical solution but numerical one with highcomputational cost for 119887
0when underactuated manipulator
has 5 or more joints
4 Nonholonomic Path Estimation
Although a valid path without transformation singularitycan be generated in the previous sections actual shape of
0 5 10 15 20 25 300
10
20
30
50
60
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 7 Joints angular displacement at 1205932= 34∘ (simulation 2)
path shows a detour to desired configuration Meanderingtrajectory with high amplitude causes the control losing inpractice
41 Nonholonomic Path Evaluation Scheme A resultant pathof underactuated manipulator must satisfy nonholonomicconstraint which is essentially nonlinear A reference pathis established to be a straight line connecting initial con-figuration with desired configuration in the configurationspace We define two indicators approximation distance 119889
119886
and extreme point number 119905119901 The indicator 119889
119886 which is the
maximum value of vertical distance between the referencepath and the point on the nonholonomic path can evaluateapproaching the reference path capability 119905
119901 which is the
number of extreme points on the path estimates the numberof actuators rotation direction changes
The nonholonomic path evaluation method in [5] esti-mates the distance between linear path and planned pathand does not precisely describe the extent of meanderingof the path An evaluation scheme is proposed to decreasethe numbers of changes of actuator rotational directionand errors from transmission of manipulator such as back-lash
Let us consider a configuration space CS with 119897 CS times
CS rarr CS[01] being a local planner for two configurations119886 119887 isin CS and 119897(119886 119887) is a nonholonomic path 119897
119886119887(119905) 119905 isin [0 1]
such that 119897119886119887
(0) = 119886 119897119886119887
(1) = 119887 The reference path 119875(119886 119887) isstraight line 119905 isin [0 1]
Definition 4 119889119886is the maximum value of vertical distance
between the reference path and the point on the nonholo-nomic path and let forall119899 ℎ isin CS as shown in Figure 8 be
Journal of Robotics 7
Desired configuration
Nonholonomic path
(deg
)
Reference path
Initial configuration
Time
a
m
h
b
n
120572
l1
l2
la
Figure 8 Illustration of the indicators of evaluation
119897119886119887
(119905) = 119899 119901119886119887
(1199051015840) = ℎ 119899119898 isin CS and la sdot l1 = 0 119901
119886119887(119905) =
((119905 minus 1)(0 minus 1)) sdot 119886 + ((119905 minus 0)(1 minus 0)) sdot 119887
Definition 5 119905119901is the number of extreme points of the actual
trajectory The actuator rotational direction changes at time119905119894if 119889(119905119894minus 120578) sdot 119889(119905
119894+ 120578) lt 0 where 120578 gt 0 and 119889(119905
119894plusmn 120578) is the
vertical distance at 119905119894plusmn 120578 between the actual trajectory and
reference path
We make further explanation about two indicators forsearching optimal pathThe values of 119889
119886and 119905119901are calculated
within control parameter 1198870range which guarantees that the
path belongs to transformation singularity-free region Inpractice the performance for motion planning of three-jointmanipulator is mainly decided by 119905
119901 With slight increment
of 119905119901 the errors of transmission parts of underactuated
manipulator will be increased significantly Therefore 119905119901is
more significant than 119889119886for performance of nonholonomic
motion planning In conclusion a search in parameter space1198870is carried out to calculate the above two indicators with
priority to 119905119901 In other words the minimum value 119889
119886is
searched in the min(119905119901) region
To verify the usefulness of the evaluation scheme sim-ulation 3 has been carried out for three-joint underactuatedmanipulator Let 120593
2be free and initial configuration and
desired configuration are 120579(0) = [10∘ 10∘ 10∘]119879 and 120579(119879) =
In simulation 3 we just estimate the first joint angletrajectory instead of all of them because the underactuatedmanipulator is an open-chain mechanism and the actuator isconnected with joint 1 as shown in Figures 1 and 2 Equations(4) reveal that inverse mapping from 119911
2to 1205791not only is
more complicated than others but also depends on 1205792 The
simulations (1 2 and 3) show that the 1205791trajectory has more
oscillation in comparison with 1205792and 1205793
Figure 10 illustrates the experimental result In additionthere is existence of 5∘ maximum error on the 120579
3curve
8 Journal of Robotics
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
2(119902) are vector fields This system is said to be
driftless that is to say the state of the system does not driftwhen the controls are set to zero
3 Transformation Singularity Avoidance
31 Conversion into Chained Form Inspired by [3] Sordalenshows conversions of the kinematic model of a car with119899 trailers The underactuated manipulator is designed toconvert into chained form and achieve control simplicityalthough a little complication should be added to mechanicalstructures
Nonlinear coordinate transformation and input feedbacktransformation of kinematics model of three-joint underac-tuated manipulator can be expressed as follows
1199114= 1205793
1199113=
1205971199114
1205971199024
1198914(1199023) = (
1198963
119896119892
) 1198883
2
1199112=
1205971199113
1205971199023
1198913(1199022) =
minus311989631198962
1198962
119892
1198882
11198882
21199041
2
1199111= 1205932
(4)
V1= 1199061= 1205932
V2=
31198962
21198963
1198963
119892
1198884
1(211988821199042
2minus 1198883
2) 1199061+
611989621198963
1198962
119892
119888111990411198882
211990421199062
(5)
where 119904119895
119894= sin119895120579
119894and 119888119895
119894= cos119895120579
119894
Reference [2] has proven nonholonomy and controllabil-ity of the 119899-joint underactuated manipulator Then we willdiscuss that conversions are diffeomorphic Inspired by [3]there is Theorem 1 as follows
Theorem 1 The chained form conversions of underactuatedmanipulator are diffeomorphic if and only if the Jacobian119891119894+1
(119902119894) of kinematic model in (1) is nonsingular
Equation (7) shows that the coordinated conversions for119899-joint manipulator are diffeomorphic except at 120579
119894= 0
Therefore the kinematic model can be diffeomorphicallyconvertible into chained form in the subspace 120579
119894isin (minus1205872 0)cup
(0 1205872)
32 Three-Joint Underactuated Manipulator Motion PlanningThere are two major control schemes for chained form Oneis open loop control and the other is feedback control Amajor advantage of open loop control is that solutions forpractical applications with low computational cost can beprovided Furthermore chained form of feedback control hastwo drawbacks one is that stabilizing chained system to thenonzero configuration is extremely difficult in practice theother is that obstacle or singularity avoidance problem ishardly solved by some form of feedback control laws becauseof no specified extent of overshoot
There are many existing open loop controllers for thechained form such as sinusoidal inputs time polynomialsinputs and piecewise constant inputs Any one of the controllaws can be applied to the underactuated manipulator Sincetime polynomials inputs law is easily obtained by solvingsimple algebraic equations and generates a better resultantpath for three-joint underactuated manipulator it is given asfollows
V1= 1198870
V2= 1198880+ 1198881119905 + 11988821199052
(8)
Although the control law polynomial inputs can steerchained form state 119911 to their desired configuration there is noguarantee that this path when mapped back into the actualcoordinate will avoid the transformation singularity Thatmeans we must check every path and ensure the existenceof every state variable in configuration space If a singularitydoes really happen some measures should be taken to find avalid path
The transformation singularity happens at 120579119894
= 0 andtrajectories in the chained form space should satisfy theconditions which can be directly checked from (4) and (7)Actually the primary cause of transformation singularity ofthree-joint manipulator is due to 119911
2gt 0 because expression
of 1199112not only is more complex but also is affected by 120579
2
The two mathematical techniques avoiding transformationsingularities are proposed in the case of 119911
2gt 0 as in the
following section
33 Constraint Point Method In order to avoid transforma-tion singularity the chained form state existing singularityregion is modified by setting some points which are calledconstraint points The constraint point method is just usedto avoid the transformation singularity in the path Thepolynomial inputs law with 119899 states and 119898 constraint pointsfor chained form is established as follows
V1= 1198870
V2=
119899minus2
sum
119894=0
119888119894119905119894+
119898
sum
119895=1
119889119895119905119899+119895minus2
(9)
4 Journal of Robotics
where 119894 isin 0 1 119899 minus 2 119895 isin 1 2 119898 and 119889119895is coef-
ficient Constraint points (119905119895 119911119894(119905119895)) are applied to chained
form state 119911119894(119905) which can be expressed as follows
1199111 (
119905) = 1198870119905 + 1199111 (
0)
1199112 (
119905) =
119899minus2
sum
119894=0
1
119894 + 1
119888119894119905119894+1
+
119898
sum
119895=1
1
119899 + 119895 minus 1
119889119895119905119899+119895minus1
+ 1199112 (
0)
1199113 (
119905) = 1198870(
119899minus2
sum
119894=0
1
(119894 + 2) (119894 + 1)
119888119894119905119894+2
+
119898
sum
119895=1
1
(119899 + 119895) (119899 + 119895 minus 1)
119889119895119905119899+119895
)
+ 11988701199112 (
0) sdot 119905 + 1199113 (
0)
119911119899 (
119905) = 119887119899minus2
0(
119899minus2
sum
119894=0
119894
(119894 + 119899 minus 1)
119888119894119905119894+119899minus1
+
119898
sum
119895=1
(119899 + 119895 minus 2)
(2119899 + 119895 minus 3)
1198891198951199052119899+119895minus3
)
+
119899
sum
119896=2
119887119899minus119896
0sdot 119905119899minus119896
sdot 119911119896 (
0)
(119899 minus 119896)
(10)
Obviously the state 119911(119905) is function of initial conditions119911(0) as well as 119899 + 119898 undetermined coefficients (119887
0 1198880
119888119899minus2
1198891 119889
119898) Equations (10) inputs will steer the
chained form system from 119911119894(0) to 119911
119894(119879) in finite time 119879
and pass through constraint state 119911119894(119905119895) at time 119905
119894 It can be
seen clearly that the existence and uniqueness of solutionsto (10) can guarantee the chained form state 119911(119905) satisfyingnonholonomy at constraint points
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
119879)
1199113 (
119879)
119911119899 (
119879)
1199111198941(1199051)
119911119894119898
(119905119898)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
= 119872(119879)
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1198880
1198881
119888119899
1198891
119889119898
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
+ 119863
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
0)
1199113 (
0)
119911119899 (
0)
1199111198941 (
0)
119911119894119898 (
0)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(11)
where 119911119894119898(119905119898) is point on the 119911
119894(119905) curve at time 119905
119898isin (0 119879)
and the matrix119872(119879) has the form
119872(119879) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879
1198792
2
sdot sdot sdot
119879119899minus1
119899 minus 1
sdot sdot sdot
119879119899+119898minus1
119899 + 119898 minus 1
11988701198792
2
11988701198793
3
sdot sdot sdot
(119899 minus 2)1198870119879119899
119899
sdot sdot sdot
(119899 + 119898 minus 2)1198870119879119899+119898
(119899 + 119898)
119887119899minus2
0119879119899minus1
(119899 minus 1)
119887119899minus2
0119879119899
119899
sdot sdot sdot
(119899 minus 2)119887119899minus2
01198792119899minus3
(2119899 minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
01198792119899+119898minus3
(2119899 + 119898 minus 3)
1198871198941minus2
01199051198941minus1
1
(1198941minus 1)
1198871198941minus2
01199051198941
1
1198941
sdot sdot sdot
(119899 minus 2)1198871198941minus2
0119905(119899+1198941minus3)
1
(119899 + 1198941minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
0119905119899+119898+119894119898minus3
1
(119899 + 119898 + 1198941minus 3)
119887119894119898minus2
0119905119894119898minus1
119898
(119894119898
minus 1)
119887119894119898minus2
0119905119894119898119898
119894119898
sdot sdot sdot
(119899 minus 2)119887119894119898minus2
0119905119899+119894119898minus3
119898
(119899 + 119894119898
minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119894119898minus2
0119905119899+119898+119894119898minus3
119898
(119899 + 119898 + 119894119898
minus 3)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(12)
Cramer law guarantees existence and uniqueness ofsolutions to (11) if and only if matrix119872(119879) is nonsingular for1198870
= 0
Remark 2 When 1199111(0) = 119911
1(119879) 119872(119879) becomes singular
This case can be dealt with by the intermediate point methodmentioned in [19]
Journal of Robotics 5
0 5 10 15 20 25 30Time (s)
minus08
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Figure 4 Chained form state 119911
Remark 3 This major advantage of the constraint pointmethod is an effective and simple way Of course othermeth-ods such as penalty function method [20] can be applied toavoid transformation singularity
Initial and desired configuration are given as 119902(0) = [1∘
1∘ 1∘]119879 and 119902(119879) = [30
∘ 30∘ 30∘]119879 in the configuration space
which correspond to 119885(0) = [minus0032 0711 0018]119879 and
119885(119879) = [minus0532 0462 0524]119879 in chained form space respec-
tively Figure 4 indicates the trajectory of 119885 without anyconstraint points Although the state 119885 can move to desiredconfiguration 119911(119879) 119911
2(119905) path has two zero-crossing points
against the conditions of keeping negative The inversechained form transformation singularity happens in intervalbetween two zero-crossing points as shown in Figure 4cyan marker of 119911
2(119905) trajectory Figure 5 indicates that a
constraint point (5 minus02) is chosen in the coordinate systemThe initial and desired configuration of control parameter119885 are redefined as 119885(0) = [minus0032 0711 0018 minus0032]
119879
and 119885(119879) = [minus0532 0462 0524 minus02]119879 It can be seen that
1199112(119905) can still move to target values 119911(119879) smoothly and avoid
placement versus time It is clear that the joint angles canreach the desired configuration smoothly without any trans-formation singularities
34 Input Control Parameter Adjustment Method Althoughconstraint point method has low numerical computationalcost the choice of constraint points sometimes needs a lotof trials for transformation singularity avoidance Inspiredby overparameterization of sinusoidal inputs in [19] theinput control parameter adjustment method is presentedThe actual coordinate of underactuated manipulator has 4
0 5 10 15 20 25 30Time (s)
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Constraint point(5 minus02)
Figure 5 Chained form state 119911 with one constraint point
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 6 Joints angular displacement with constraint point (simu-lation 1)
dimensions [120593119899minus1
1205791 1205792 1205793]119879 and our goal is to steer 3-
dimensional state variable 120579119894(119905) to 120579
119894(119879) 119894 isin 1 2 3 Since 119887
0
can be expressed as 1198870= (1205932(119879) minus 120593
2(0))119879 we can deal with
the polynomials input law by choosing an appropriate 1198870in
singularity-free regions to achieve transformation singularityavoidance This method is applied to three-joint underactu-ated manipulator
As mentioned the main condition of avoidance transfor-mation singularity is
1199112 (
119905) lt 0 119905 isin [0 119879] (13)
6 Journal of Robotics
Equation (13) can be satisfied if maximum value of 1199112(119905)
keeps negative The time corresponding to stagnation pointof 1199112(119905) can be expressed as follows
1199051=
minus1198881+ radic1198882
1minus 411988801198882
21198882
1199052=
minus1198881minus radic1198882
1minus 411988801198882
21198882
1198892(1199112(1199052))
1198891199052
= minusradic1198882
1minus 411988801198882lt 0
(14)
From (11) 1199112(1199052) can be represented as function of control
parameter 1198870
119888119894= 119888119894(1198870)
1199112(1199052) =
1198882
3
1199053
2+
1198881
2
1199052
2+ 11988801199052+ 1199112 (
0)
119894 = 0 1 2
(15)
Consider (15) as a mapΦ1198870
119868119877 997904rArr 119868119877 (16)
which maps input control parameter 1198870into the maximum
1199112(1199052) Obviously the mapping Φ
1198870is surjection (many-to-
one) since the choice of 1198870for solving 119911
2(1199052) is not unique
The range of value 1198870can be determined by solving roots of
(15)In Figure 7 simulation 2 shows joint angles versus time
for three-joint underactuated manipulator The joint anglesof initial configuration and terminal configuration are 120579(0) =
[5∘ 5∘ 5∘]119879 and 120579(119879) = [30
∘ 30∘ 30∘]119879 respectively Input
control parameter 1205932= 34∘ since there is no transformation
singularity if and only if 1205932is always chosen in the range of
3235∘ to 3544∘ It can be seen that the joints angles reach thedesired configuration after the setting time
A major advantage of this approach is that analyticalsolution of control parameter 119887
0can be calculated in transfor-
mation singularity-free region And the order of polynomialsdoes not increase in comparison to constraint point methodsince a valid path without singularity depends on choosingsuitable value of 119887
0 Comparing overparameterization meth-
ods the choice of input control parameter 1198870is not arbitrary
but definiteThis method leaves one parameter free and searches for a
value in this parameter space which guarantees that the pathbelongs to transformation singularity-free region Howeveraccording to Galois theory closed-form solution does notexist for polynomial order higher than 4 In other wordsthere is no analytical solution but numerical one with highcomputational cost for 119887
0when underactuated manipulator
has 5 or more joints
4 Nonholonomic Path Estimation
Although a valid path without transformation singularitycan be generated in the previous sections actual shape of
0 5 10 15 20 25 300
10
20
30
50
60
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 7 Joints angular displacement at 1205932= 34∘ (simulation 2)
path shows a detour to desired configuration Meanderingtrajectory with high amplitude causes the control losing inpractice
41 Nonholonomic Path Evaluation Scheme A resultant pathof underactuated manipulator must satisfy nonholonomicconstraint which is essentially nonlinear A reference pathis established to be a straight line connecting initial con-figuration with desired configuration in the configurationspace We define two indicators approximation distance 119889
119886
and extreme point number 119905119901 The indicator 119889
119886 which is the
maximum value of vertical distance between the referencepath and the point on the nonholonomic path can evaluateapproaching the reference path capability 119905
119901 which is the
number of extreme points on the path estimates the numberof actuators rotation direction changes
The nonholonomic path evaluation method in [5] esti-mates the distance between linear path and planned pathand does not precisely describe the extent of meanderingof the path An evaluation scheme is proposed to decreasethe numbers of changes of actuator rotational directionand errors from transmission of manipulator such as back-lash
Let us consider a configuration space CS with 119897 CS times
CS rarr CS[01] being a local planner for two configurations119886 119887 isin CS and 119897(119886 119887) is a nonholonomic path 119897
119886119887(119905) 119905 isin [0 1]
such that 119897119886119887
(0) = 119886 119897119886119887
(1) = 119887 The reference path 119875(119886 119887) isstraight line 119905 isin [0 1]
Definition 4 119889119886is the maximum value of vertical distance
between the reference path and the point on the nonholo-nomic path and let forall119899 ℎ isin CS as shown in Figure 8 be
Journal of Robotics 7
Desired configuration
Nonholonomic path
(deg
)
Reference path
Initial configuration
Time
a
m
h
b
n
120572
l1
l2
la
Figure 8 Illustration of the indicators of evaluation
119897119886119887
(119905) = 119899 119901119886119887
(1199051015840) = ℎ 119899119898 isin CS and la sdot l1 = 0 119901
119886119887(119905) =
((119905 minus 1)(0 minus 1)) sdot 119886 + ((119905 minus 0)(1 minus 0)) sdot 119887
Definition 5 119905119901is the number of extreme points of the actual
trajectory The actuator rotational direction changes at time119905119894if 119889(119905119894minus 120578) sdot 119889(119905
119894+ 120578) lt 0 where 120578 gt 0 and 119889(119905
119894plusmn 120578) is the
vertical distance at 119905119894plusmn 120578 between the actual trajectory and
reference path
We make further explanation about two indicators forsearching optimal pathThe values of 119889
119886and 119905119901are calculated
within control parameter 1198870range which guarantees that the
path belongs to transformation singularity-free region Inpractice the performance for motion planning of three-jointmanipulator is mainly decided by 119905
119901 With slight increment
of 119905119901 the errors of transmission parts of underactuated
manipulator will be increased significantly Therefore 119905119901is
more significant than 119889119886for performance of nonholonomic
motion planning In conclusion a search in parameter space1198870is carried out to calculate the above two indicators with
priority to 119905119901 In other words the minimum value 119889
119886is
searched in the min(119905119901) region
To verify the usefulness of the evaluation scheme sim-ulation 3 has been carried out for three-joint underactuatedmanipulator Let 120593
2be free and initial configuration and
desired configuration are 120579(0) = [10∘ 10∘ 10∘]119879 and 120579(119879) =
In simulation 3 we just estimate the first joint angletrajectory instead of all of them because the underactuatedmanipulator is an open-chain mechanism and the actuator isconnected with joint 1 as shown in Figures 1 and 2 Equations(4) reveal that inverse mapping from 119911
2to 1205791not only is
more complicated than others but also depends on 1205792 The
simulations (1 2 and 3) show that the 1205791trajectory has more
oscillation in comparison with 1205792and 1205793
Figure 10 illustrates the experimental result In additionthere is existence of 5∘ maximum error on the 120579
3curve
8 Journal of Robotics
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
where 119894 isin 0 1 119899 minus 2 119895 isin 1 2 119898 and 119889119895is coef-
ficient Constraint points (119905119895 119911119894(119905119895)) are applied to chained
form state 119911119894(119905) which can be expressed as follows
1199111 (
119905) = 1198870119905 + 1199111 (
0)
1199112 (
119905) =
119899minus2
sum
119894=0
1
119894 + 1
119888119894119905119894+1
+
119898
sum
119895=1
1
119899 + 119895 minus 1
119889119895119905119899+119895minus1
+ 1199112 (
0)
1199113 (
119905) = 1198870(
119899minus2
sum
119894=0
1
(119894 + 2) (119894 + 1)
119888119894119905119894+2
+
119898
sum
119895=1
1
(119899 + 119895) (119899 + 119895 minus 1)
119889119895119905119899+119895
)
+ 11988701199112 (
0) sdot 119905 + 1199113 (
0)
119911119899 (
119905) = 119887119899minus2
0(
119899minus2
sum
119894=0
119894
(119894 + 119899 minus 1)
119888119894119905119894+119899minus1
+
119898
sum
119895=1
(119899 + 119895 minus 2)
(2119899 + 119895 minus 3)
1198891198951199052119899+119895minus3
)
+
119899
sum
119896=2
119887119899minus119896
0sdot 119905119899minus119896
sdot 119911119896 (
0)
(119899 minus 119896)
(10)
Obviously the state 119911(119905) is function of initial conditions119911(0) as well as 119899 + 119898 undetermined coefficients (119887
0 1198880
119888119899minus2
1198891 119889
119898) Equations (10) inputs will steer the
chained form system from 119911119894(0) to 119911
119894(119879) in finite time 119879
and pass through constraint state 119911119894(119905119895) at time 119905
119894 It can be
seen clearly that the existence and uniqueness of solutionsto (10) can guarantee the chained form state 119911(119905) satisfyingnonholonomy at constraint points
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
119879)
1199113 (
119879)
119911119899 (
119879)
1199111198941(1199051)
119911119894119898
(119905119898)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
= 119872(119879)
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1198880
1198881
119888119899
1198891
119889119898
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
+ 119863
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1199112 (
0)
1199113 (
0)
119911119899 (
0)
1199111198941 (
0)
119911119894119898 (
0)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(11)
where 119911119894119898(119905119898) is point on the 119911
119894(119905) curve at time 119905
119898isin (0 119879)
and the matrix119872(119879) has the form
119872(119879) =
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
119879
1198792
2
sdot sdot sdot
119879119899minus1
119899 minus 1
sdot sdot sdot
119879119899+119898minus1
119899 + 119898 minus 1
11988701198792
2
11988701198793
3
sdot sdot sdot
(119899 minus 2)1198870119879119899
119899
sdot sdot sdot
(119899 + 119898 minus 2)1198870119879119899+119898
(119899 + 119898)
119887119899minus2
0119879119899minus1
(119899 minus 1)
119887119899minus2
0119879119899
119899
sdot sdot sdot
(119899 minus 2)119887119899minus2
01198792119899minus3
(2119899 minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
01198792119899+119898minus3
(2119899 + 119898 minus 3)
1198871198941minus2
01199051198941minus1
1
(1198941minus 1)
1198871198941minus2
01199051198941
1
1198941
sdot sdot sdot
(119899 minus 2)1198871198941minus2
0119905(119899+1198941minus3)
1
(119899 + 1198941minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119899minus2
0119905119899+119898+119894119898minus3
1
(119899 + 119898 + 1198941minus 3)
119887119894119898minus2
0119905119894119898minus1
119898
(119894119898
minus 1)
119887119894119898minus2
0119905119894119898119898
119894119898
sdot sdot sdot
(119899 minus 2)119887119894119898minus2
0119905119899+119894119898minus3
119898
(119899 + 119894119898
minus 3)
sdot sdot sdot
(119899 + 119898 minus 2)119887119894119898minus2
0119905119899+119898+119894119898minus3
119898
(119899 + 119898 + 119894119898
minus 3)
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
(12)
Cramer law guarantees existence and uniqueness ofsolutions to (11) if and only if matrix119872(119879) is nonsingular for1198870
= 0
Remark 2 When 1199111(0) = 119911
1(119879) 119872(119879) becomes singular
This case can be dealt with by the intermediate point methodmentioned in [19]
Journal of Robotics 5
0 5 10 15 20 25 30Time (s)
minus08
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Figure 4 Chained form state 119911
Remark 3 This major advantage of the constraint pointmethod is an effective and simple way Of course othermeth-ods such as penalty function method [20] can be applied toavoid transformation singularity
Initial and desired configuration are given as 119902(0) = [1∘
1∘ 1∘]119879 and 119902(119879) = [30
∘ 30∘ 30∘]119879 in the configuration space
which correspond to 119885(0) = [minus0032 0711 0018]119879 and
119885(119879) = [minus0532 0462 0524]119879 in chained form space respec-
tively Figure 4 indicates the trajectory of 119885 without anyconstraint points Although the state 119885 can move to desiredconfiguration 119911(119879) 119911
2(119905) path has two zero-crossing points
against the conditions of keeping negative The inversechained form transformation singularity happens in intervalbetween two zero-crossing points as shown in Figure 4cyan marker of 119911
2(119905) trajectory Figure 5 indicates that a
constraint point (5 minus02) is chosen in the coordinate systemThe initial and desired configuration of control parameter119885 are redefined as 119885(0) = [minus0032 0711 0018 minus0032]
119879
and 119885(119879) = [minus0532 0462 0524 minus02]119879 It can be seen that
1199112(119905) can still move to target values 119911(119879) smoothly and avoid
placement versus time It is clear that the joint angles canreach the desired configuration smoothly without any trans-formation singularities
34 Input Control Parameter Adjustment Method Althoughconstraint point method has low numerical computationalcost the choice of constraint points sometimes needs a lotof trials for transformation singularity avoidance Inspiredby overparameterization of sinusoidal inputs in [19] theinput control parameter adjustment method is presentedThe actual coordinate of underactuated manipulator has 4
0 5 10 15 20 25 30Time (s)
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Constraint point(5 minus02)
Figure 5 Chained form state 119911 with one constraint point
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 6 Joints angular displacement with constraint point (simu-lation 1)
dimensions [120593119899minus1
1205791 1205792 1205793]119879 and our goal is to steer 3-
dimensional state variable 120579119894(119905) to 120579
119894(119879) 119894 isin 1 2 3 Since 119887
0
can be expressed as 1198870= (1205932(119879) minus 120593
2(0))119879 we can deal with
the polynomials input law by choosing an appropriate 1198870in
singularity-free regions to achieve transformation singularityavoidance This method is applied to three-joint underactu-ated manipulator
As mentioned the main condition of avoidance transfor-mation singularity is
1199112 (
119905) lt 0 119905 isin [0 119879] (13)
6 Journal of Robotics
Equation (13) can be satisfied if maximum value of 1199112(119905)
keeps negative The time corresponding to stagnation pointof 1199112(119905) can be expressed as follows
1199051=
minus1198881+ radic1198882
1minus 411988801198882
21198882
1199052=
minus1198881minus radic1198882
1minus 411988801198882
21198882
1198892(1199112(1199052))
1198891199052
= minusradic1198882
1minus 411988801198882lt 0
(14)
From (11) 1199112(1199052) can be represented as function of control
parameter 1198870
119888119894= 119888119894(1198870)
1199112(1199052) =
1198882
3
1199053
2+
1198881
2
1199052
2+ 11988801199052+ 1199112 (
0)
119894 = 0 1 2
(15)
Consider (15) as a mapΦ1198870
119868119877 997904rArr 119868119877 (16)
which maps input control parameter 1198870into the maximum
1199112(1199052) Obviously the mapping Φ
1198870is surjection (many-to-
one) since the choice of 1198870for solving 119911
2(1199052) is not unique
The range of value 1198870can be determined by solving roots of
(15)In Figure 7 simulation 2 shows joint angles versus time
for three-joint underactuated manipulator The joint anglesof initial configuration and terminal configuration are 120579(0) =
[5∘ 5∘ 5∘]119879 and 120579(119879) = [30
∘ 30∘ 30∘]119879 respectively Input
control parameter 1205932= 34∘ since there is no transformation
singularity if and only if 1205932is always chosen in the range of
3235∘ to 3544∘ It can be seen that the joints angles reach thedesired configuration after the setting time
A major advantage of this approach is that analyticalsolution of control parameter 119887
0can be calculated in transfor-
mation singularity-free region And the order of polynomialsdoes not increase in comparison to constraint point methodsince a valid path without singularity depends on choosingsuitable value of 119887
0 Comparing overparameterization meth-
ods the choice of input control parameter 1198870is not arbitrary
but definiteThis method leaves one parameter free and searches for a
value in this parameter space which guarantees that the pathbelongs to transformation singularity-free region Howeveraccording to Galois theory closed-form solution does notexist for polynomial order higher than 4 In other wordsthere is no analytical solution but numerical one with highcomputational cost for 119887
0when underactuated manipulator
has 5 or more joints
4 Nonholonomic Path Estimation
Although a valid path without transformation singularitycan be generated in the previous sections actual shape of
0 5 10 15 20 25 300
10
20
30
50
60
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 7 Joints angular displacement at 1205932= 34∘ (simulation 2)
path shows a detour to desired configuration Meanderingtrajectory with high amplitude causes the control losing inpractice
41 Nonholonomic Path Evaluation Scheme A resultant pathof underactuated manipulator must satisfy nonholonomicconstraint which is essentially nonlinear A reference pathis established to be a straight line connecting initial con-figuration with desired configuration in the configurationspace We define two indicators approximation distance 119889
119886
and extreme point number 119905119901 The indicator 119889
119886 which is the
maximum value of vertical distance between the referencepath and the point on the nonholonomic path can evaluateapproaching the reference path capability 119905
119901 which is the
number of extreme points on the path estimates the numberof actuators rotation direction changes
The nonholonomic path evaluation method in [5] esti-mates the distance between linear path and planned pathand does not precisely describe the extent of meanderingof the path An evaluation scheme is proposed to decreasethe numbers of changes of actuator rotational directionand errors from transmission of manipulator such as back-lash
Let us consider a configuration space CS with 119897 CS times
CS rarr CS[01] being a local planner for two configurations119886 119887 isin CS and 119897(119886 119887) is a nonholonomic path 119897
119886119887(119905) 119905 isin [0 1]
such that 119897119886119887
(0) = 119886 119897119886119887
(1) = 119887 The reference path 119875(119886 119887) isstraight line 119905 isin [0 1]
Definition 4 119889119886is the maximum value of vertical distance
between the reference path and the point on the nonholo-nomic path and let forall119899 ℎ isin CS as shown in Figure 8 be
Journal of Robotics 7
Desired configuration
Nonholonomic path
(deg
)
Reference path
Initial configuration
Time
a
m
h
b
n
120572
l1
l2
la
Figure 8 Illustration of the indicators of evaluation
119897119886119887
(119905) = 119899 119901119886119887
(1199051015840) = ℎ 119899119898 isin CS and la sdot l1 = 0 119901
119886119887(119905) =
((119905 minus 1)(0 minus 1)) sdot 119886 + ((119905 minus 0)(1 minus 0)) sdot 119887
Definition 5 119905119901is the number of extreme points of the actual
trajectory The actuator rotational direction changes at time119905119894if 119889(119905119894minus 120578) sdot 119889(119905
119894+ 120578) lt 0 where 120578 gt 0 and 119889(119905
119894plusmn 120578) is the
vertical distance at 119905119894plusmn 120578 between the actual trajectory and
reference path
We make further explanation about two indicators forsearching optimal pathThe values of 119889
119886and 119905119901are calculated
within control parameter 1198870range which guarantees that the
path belongs to transformation singularity-free region Inpractice the performance for motion planning of three-jointmanipulator is mainly decided by 119905
119901 With slight increment
of 119905119901 the errors of transmission parts of underactuated
manipulator will be increased significantly Therefore 119905119901is
more significant than 119889119886for performance of nonholonomic
motion planning In conclusion a search in parameter space1198870is carried out to calculate the above two indicators with
priority to 119905119901 In other words the minimum value 119889
119886is
searched in the min(119905119901) region
To verify the usefulness of the evaluation scheme sim-ulation 3 has been carried out for three-joint underactuatedmanipulator Let 120593
2be free and initial configuration and
desired configuration are 120579(0) = [10∘ 10∘ 10∘]119879 and 120579(119879) =
In simulation 3 we just estimate the first joint angletrajectory instead of all of them because the underactuatedmanipulator is an open-chain mechanism and the actuator isconnected with joint 1 as shown in Figures 1 and 2 Equations(4) reveal that inverse mapping from 119911
2to 1205791not only is
more complicated than others but also depends on 1205792 The
simulations (1 2 and 3) show that the 1205791trajectory has more
oscillation in comparison with 1205792and 1205793
Figure 10 illustrates the experimental result In additionthere is existence of 5∘ maximum error on the 120579
3curve
8 Journal of Robotics
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
Remark 3 This major advantage of the constraint pointmethod is an effective and simple way Of course othermeth-ods such as penalty function method [20] can be applied toavoid transformation singularity
Initial and desired configuration are given as 119902(0) = [1∘
1∘ 1∘]119879 and 119902(119879) = [30
∘ 30∘ 30∘]119879 in the configuration space
which correspond to 119885(0) = [minus0032 0711 0018]119879 and
119885(119879) = [minus0532 0462 0524]119879 in chained form space respec-
tively Figure 4 indicates the trajectory of 119885 without anyconstraint points Although the state 119885 can move to desiredconfiguration 119911(119879) 119911
2(119905) path has two zero-crossing points
against the conditions of keeping negative The inversechained form transformation singularity happens in intervalbetween two zero-crossing points as shown in Figure 4cyan marker of 119911
2(119905) trajectory Figure 5 indicates that a
constraint point (5 minus02) is chosen in the coordinate systemThe initial and desired configuration of control parameter119885 are redefined as 119885(0) = [minus0032 0711 0018 minus0032]
119879
and 119885(119879) = [minus0532 0462 0524 minus02]119879 It can be seen that
1199112(119905) can still move to target values 119911(119879) smoothly and avoid
placement versus time It is clear that the joint angles canreach the desired configuration smoothly without any trans-formation singularities
34 Input Control Parameter Adjustment Method Althoughconstraint point method has low numerical computationalcost the choice of constraint points sometimes needs a lotof trials for transformation singularity avoidance Inspiredby overparameterization of sinusoidal inputs in [19] theinput control parameter adjustment method is presentedThe actual coordinate of underactuated manipulator has 4
0 5 10 15 20 25 30Time (s)
minus06
minus04
minus02
0
02
04
06
08
Z
z4
z2z3
z4
z2
z3
Constraint point(5 minus02)
Figure 5 Chained form state 119911 with one constraint point
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 6 Joints angular displacement with constraint point (simu-lation 1)
dimensions [120593119899minus1
1205791 1205792 1205793]119879 and our goal is to steer 3-
dimensional state variable 120579119894(119905) to 120579
119894(119879) 119894 isin 1 2 3 Since 119887
0
can be expressed as 1198870= (1205932(119879) minus 120593
2(0))119879 we can deal with
the polynomials input law by choosing an appropriate 1198870in
singularity-free regions to achieve transformation singularityavoidance This method is applied to three-joint underactu-ated manipulator
As mentioned the main condition of avoidance transfor-mation singularity is
1199112 (
119905) lt 0 119905 isin [0 119879] (13)
6 Journal of Robotics
Equation (13) can be satisfied if maximum value of 1199112(119905)
keeps negative The time corresponding to stagnation pointof 1199112(119905) can be expressed as follows
1199051=
minus1198881+ radic1198882
1minus 411988801198882
21198882
1199052=
minus1198881minus radic1198882
1minus 411988801198882
21198882
1198892(1199112(1199052))
1198891199052
= minusradic1198882
1minus 411988801198882lt 0
(14)
From (11) 1199112(1199052) can be represented as function of control
parameter 1198870
119888119894= 119888119894(1198870)
1199112(1199052) =
1198882
3
1199053
2+
1198881
2
1199052
2+ 11988801199052+ 1199112 (
0)
119894 = 0 1 2
(15)
Consider (15) as a mapΦ1198870
119868119877 997904rArr 119868119877 (16)
which maps input control parameter 1198870into the maximum
1199112(1199052) Obviously the mapping Φ
1198870is surjection (many-to-
one) since the choice of 1198870for solving 119911
2(1199052) is not unique
The range of value 1198870can be determined by solving roots of
(15)In Figure 7 simulation 2 shows joint angles versus time
for three-joint underactuated manipulator The joint anglesof initial configuration and terminal configuration are 120579(0) =
[5∘ 5∘ 5∘]119879 and 120579(119879) = [30
∘ 30∘ 30∘]119879 respectively Input
control parameter 1205932= 34∘ since there is no transformation
singularity if and only if 1205932is always chosen in the range of
3235∘ to 3544∘ It can be seen that the joints angles reach thedesired configuration after the setting time
A major advantage of this approach is that analyticalsolution of control parameter 119887
0can be calculated in transfor-
mation singularity-free region And the order of polynomialsdoes not increase in comparison to constraint point methodsince a valid path without singularity depends on choosingsuitable value of 119887
0 Comparing overparameterization meth-
ods the choice of input control parameter 1198870is not arbitrary
but definiteThis method leaves one parameter free and searches for a
value in this parameter space which guarantees that the pathbelongs to transformation singularity-free region Howeveraccording to Galois theory closed-form solution does notexist for polynomial order higher than 4 In other wordsthere is no analytical solution but numerical one with highcomputational cost for 119887
0when underactuated manipulator
has 5 or more joints
4 Nonholonomic Path Estimation
Although a valid path without transformation singularitycan be generated in the previous sections actual shape of
0 5 10 15 20 25 300
10
20
30
50
60
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 7 Joints angular displacement at 1205932= 34∘ (simulation 2)
path shows a detour to desired configuration Meanderingtrajectory with high amplitude causes the control losing inpractice
41 Nonholonomic Path Evaluation Scheme A resultant pathof underactuated manipulator must satisfy nonholonomicconstraint which is essentially nonlinear A reference pathis established to be a straight line connecting initial con-figuration with desired configuration in the configurationspace We define two indicators approximation distance 119889
119886
and extreme point number 119905119901 The indicator 119889
119886 which is the
maximum value of vertical distance between the referencepath and the point on the nonholonomic path can evaluateapproaching the reference path capability 119905
119901 which is the
number of extreme points on the path estimates the numberof actuators rotation direction changes
The nonholonomic path evaluation method in [5] esti-mates the distance between linear path and planned pathand does not precisely describe the extent of meanderingof the path An evaluation scheme is proposed to decreasethe numbers of changes of actuator rotational directionand errors from transmission of manipulator such as back-lash
Let us consider a configuration space CS with 119897 CS times
CS rarr CS[01] being a local planner for two configurations119886 119887 isin CS and 119897(119886 119887) is a nonholonomic path 119897
119886119887(119905) 119905 isin [0 1]
such that 119897119886119887
(0) = 119886 119897119886119887
(1) = 119887 The reference path 119875(119886 119887) isstraight line 119905 isin [0 1]
Definition 4 119889119886is the maximum value of vertical distance
between the reference path and the point on the nonholo-nomic path and let forall119899 ℎ isin CS as shown in Figure 8 be
Journal of Robotics 7
Desired configuration
Nonholonomic path
(deg
)
Reference path
Initial configuration
Time
a
m
h
b
n
120572
l1
l2
la
Figure 8 Illustration of the indicators of evaluation
119897119886119887
(119905) = 119899 119901119886119887
(1199051015840) = ℎ 119899119898 isin CS and la sdot l1 = 0 119901
119886119887(119905) =
((119905 minus 1)(0 minus 1)) sdot 119886 + ((119905 minus 0)(1 minus 0)) sdot 119887
Definition 5 119905119901is the number of extreme points of the actual
trajectory The actuator rotational direction changes at time119905119894if 119889(119905119894minus 120578) sdot 119889(119905
119894+ 120578) lt 0 where 120578 gt 0 and 119889(119905
119894plusmn 120578) is the
vertical distance at 119905119894plusmn 120578 between the actual trajectory and
reference path
We make further explanation about two indicators forsearching optimal pathThe values of 119889
119886and 119905119901are calculated
within control parameter 1198870range which guarantees that the
path belongs to transformation singularity-free region Inpractice the performance for motion planning of three-jointmanipulator is mainly decided by 119905
119901 With slight increment
of 119905119901 the errors of transmission parts of underactuated
manipulator will be increased significantly Therefore 119905119901is
more significant than 119889119886for performance of nonholonomic
motion planning In conclusion a search in parameter space1198870is carried out to calculate the above two indicators with
priority to 119905119901 In other words the minimum value 119889
119886is
searched in the min(119905119901) region
To verify the usefulness of the evaluation scheme sim-ulation 3 has been carried out for three-joint underactuatedmanipulator Let 120593
2be free and initial configuration and
desired configuration are 120579(0) = [10∘ 10∘ 10∘]119879 and 120579(119879) =
In simulation 3 we just estimate the first joint angletrajectory instead of all of them because the underactuatedmanipulator is an open-chain mechanism and the actuator isconnected with joint 1 as shown in Figures 1 and 2 Equations(4) reveal that inverse mapping from 119911
2to 1205791not only is
more complicated than others but also depends on 1205792 The
simulations (1 2 and 3) show that the 1205791trajectory has more
oscillation in comparison with 1205792and 1205793
Figure 10 illustrates the experimental result In additionthere is existence of 5∘ maximum error on the 120579
3curve
8 Journal of Robotics
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
Equation (13) can be satisfied if maximum value of 1199112(119905)
keeps negative The time corresponding to stagnation pointof 1199112(119905) can be expressed as follows
1199051=
minus1198881+ radic1198882
1minus 411988801198882
21198882
1199052=
minus1198881minus radic1198882
1minus 411988801198882
21198882
1198892(1199112(1199052))
1198891199052
= minusradic1198882
1minus 411988801198882lt 0
(14)
From (11) 1199112(1199052) can be represented as function of control
parameter 1198870
119888119894= 119888119894(1198870)
1199112(1199052) =
1198882
3
1199053
2+
1198881
2
1199052
2+ 11988801199052+ 1199112 (
0)
119894 = 0 1 2
(15)
Consider (15) as a mapΦ1198870
119868119877 997904rArr 119868119877 (16)
which maps input control parameter 1198870into the maximum
1199112(1199052) Obviously the mapping Φ
1198870is surjection (many-to-
one) since the choice of 1198870for solving 119911
2(1199052) is not unique
The range of value 1198870can be determined by solving roots of
(15)In Figure 7 simulation 2 shows joint angles versus time
for three-joint underactuated manipulator The joint anglesof initial configuration and terminal configuration are 120579(0) =
[5∘ 5∘ 5∘]119879 and 120579(119879) = [30
∘ 30∘ 30∘]119879 respectively Input
control parameter 1205932= 34∘ since there is no transformation
singularity if and only if 1205932is always chosen in the range of
3235∘ to 3544∘ It can be seen that the joints angles reach thedesired configuration after the setting time
A major advantage of this approach is that analyticalsolution of control parameter 119887
0can be calculated in transfor-
mation singularity-free region And the order of polynomialsdoes not increase in comparison to constraint point methodsince a valid path without singularity depends on choosingsuitable value of 119887
0 Comparing overparameterization meth-
ods the choice of input control parameter 1198870is not arbitrary
but definiteThis method leaves one parameter free and searches for a
value in this parameter space which guarantees that the pathbelongs to transformation singularity-free region Howeveraccording to Galois theory closed-form solution does notexist for polynomial order higher than 4 In other wordsthere is no analytical solution but numerical one with highcomputational cost for 119887
0when underactuated manipulator
has 5 or more joints
4 Nonholonomic Path Estimation
Although a valid path without transformation singularitycan be generated in the previous sections actual shape of
0 5 10 15 20 25 300
10
20
30
50
60
40
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 7 Joints angular displacement at 1205932= 34∘ (simulation 2)
path shows a detour to desired configuration Meanderingtrajectory with high amplitude causes the control losing inpractice
41 Nonholonomic Path Evaluation Scheme A resultant pathof underactuated manipulator must satisfy nonholonomicconstraint which is essentially nonlinear A reference pathis established to be a straight line connecting initial con-figuration with desired configuration in the configurationspace We define two indicators approximation distance 119889
119886
and extreme point number 119905119901 The indicator 119889
119886 which is the
maximum value of vertical distance between the referencepath and the point on the nonholonomic path can evaluateapproaching the reference path capability 119905
119901 which is the
number of extreme points on the path estimates the numberof actuators rotation direction changes
The nonholonomic path evaluation method in [5] esti-mates the distance between linear path and planned pathand does not precisely describe the extent of meanderingof the path An evaluation scheme is proposed to decreasethe numbers of changes of actuator rotational directionand errors from transmission of manipulator such as back-lash
Let us consider a configuration space CS with 119897 CS times
CS rarr CS[01] being a local planner for two configurations119886 119887 isin CS and 119897(119886 119887) is a nonholonomic path 119897
119886119887(119905) 119905 isin [0 1]
such that 119897119886119887
(0) = 119886 119897119886119887
(1) = 119887 The reference path 119875(119886 119887) isstraight line 119905 isin [0 1]
Definition 4 119889119886is the maximum value of vertical distance
between the reference path and the point on the nonholo-nomic path and let forall119899 ℎ isin CS as shown in Figure 8 be
Journal of Robotics 7
Desired configuration
Nonholonomic path
(deg
)
Reference path
Initial configuration
Time
a
m
h
b
n
120572
l1
l2
la
Figure 8 Illustration of the indicators of evaluation
119897119886119887
(119905) = 119899 119901119886119887
(1199051015840) = ℎ 119899119898 isin CS and la sdot l1 = 0 119901
119886119887(119905) =
((119905 minus 1)(0 minus 1)) sdot 119886 + ((119905 minus 0)(1 minus 0)) sdot 119887
Definition 5 119905119901is the number of extreme points of the actual
trajectory The actuator rotational direction changes at time119905119894if 119889(119905119894minus 120578) sdot 119889(119905
119894+ 120578) lt 0 where 120578 gt 0 and 119889(119905
119894plusmn 120578) is the
vertical distance at 119905119894plusmn 120578 between the actual trajectory and
reference path
We make further explanation about two indicators forsearching optimal pathThe values of 119889
119886and 119905119901are calculated
within control parameter 1198870range which guarantees that the
path belongs to transformation singularity-free region Inpractice the performance for motion planning of three-jointmanipulator is mainly decided by 119905
119901 With slight increment
of 119905119901 the errors of transmission parts of underactuated
manipulator will be increased significantly Therefore 119905119901is
more significant than 119889119886for performance of nonholonomic
motion planning In conclusion a search in parameter space1198870is carried out to calculate the above two indicators with
priority to 119905119901 In other words the minimum value 119889
119886is
searched in the min(119905119901) region
To verify the usefulness of the evaluation scheme sim-ulation 3 has been carried out for three-joint underactuatedmanipulator Let 120593
2be free and initial configuration and
desired configuration are 120579(0) = [10∘ 10∘ 10∘]119879 and 120579(119879) =
In simulation 3 we just estimate the first joint angletrajectory instead of all of them because the underactuatedmanipulator is an open-chain mechanism and the actuator isconnected with joint 1 as shown in Figures 1 and 2 Equations(4) reveal that inverse mapping from 119911
2to 1205791not only is
more complicated than others but also depends on 1205792 The
simulations (1 2 and 3) show that the 1205791trajectory has more
oscillation in comparison with 1205792and 1205793
Figure 10 illustrates the experimental result In additionthere is existence of 5∘ maximum error on the 120579
3curve
8 Journal of Robotics
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
Definition 5 119905119901is the number of extreme points of the actual
trajectory The actuator rotational direction changes at time119905119894if 119889(119905119894minus 120578) sdot 119889(119905
119894+ 120578) lt 0 where 120578 gt 0 and 119889(119905
119894plusmn 120578) is the
vertical distance at 119905119894plusmn 120578 between the actual trajectory and
reference path
We make further explanation about two indicators forsearching optimal pathThe values of 119889
119886and 119905119901are calculated
within control parameter 1198870range which guarantees that the
path belongs to transformation singularity-free region Inpractice the performance for motion planning of three-jointmanipulator is mainly decided by 119905
119901 With slight increment
of 119905119901 the errors of transmission parts of underactuated
manipulator will be increased significantly Therefore 119905119901is
more significant than 119889119886for performance of nonholonomic
motion planning In conclusion a search in parameter space1198870is carried out to calculate the above two indicators with
priority to 119905119901 In other words the minimum value 119889
119886is
searched in the min(119905119901) region
To verify the usefulness of the evaluation scheme sim-ulation 3 has been carried out for three-joint underactuatedmanipulator Let 120593
2be free and initial configuration and
desired configuration are 120579(0) = [10∘ 10∘ 10∘]119879 and 120579(119879) =
In simulation 3 we just estimate the first joint angletrajectory instead of all of them because the underactuatedmanipulator is an open-chain mechanism and the actuator isconnected with joint 1 as shown in Figures 1 and 2 Equations(4) reveal that inverse mapping from 119911
2to 1205791not only is
more complicated than others but also depends on 1205792 The
simulations (1 2 and 3) show that the 1205791trajectory has more
oscillation in comparison with 1205792and 1205793
Figure 10 illustrates the experimental result In additionthere is existence of 5∘ maximum error on the 120579
3curve
8 Journal of Robotics
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
The backlash at the bevel gear the low stiffness of the longshaft transmission parts with low resolution and the lackof drive torque under guaranteeing rolling without slippingcondition would have caused these errors In addition theerror of 120579
3is obviously larger than the others because of
cumulative error of transmission system
5 Nonholonomic Motion Planning Robustness
Since motion planning is an open loop control a resultantpath will be affected by various errors Although the motionplanning using time polynomial inputs shows the satisfactoryresult in the previous section actual path needs to be checkedin the presence of errors
51 A Simulation with Initial Condition Error for OptimalPath Figure 11 illustrates that a simulation in the presenceof initial errors 120579(0) = [95
∘ 105∘ 95∘]119879 which implies each
joint angle has 05∘ error Other parameters are the same assimulation 3 As shown in Figure 11 simulation 4 result shows119889119886
= 5757∘ and 119905
119901= 2 of 120579
1 The maximum distance 119889
119886in
simulation 4 is 36 times higher than that in simulation 3 andthe rotation direction of joint 1 will change twice It is easilyconcluded that the actual performance of motion planning isgreatly affected by 05 degrees in the presence of initial errorsSimulation 4 results illustrate the lack of robustness and highsensitivity with initial condition errors
52 Initial Condition Error Sensitivity Analysis In order tospecify effect of initial errors the expression of 119911(119905) can beobtained while the time polynomial inputs law is applied tochained form system existing initial error
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
The usefulness of motion planning robustness is exper-imentally verified as shown in Figures 15 and 16 Althoughthere are little differences between simulations (5 6) andexperiments (2 3) Figures 15 and 16 are similar to Figures13 and 14 respectively Actually the experimental results arequite acceptable if the errors of mechanical structure areneglected
6 Conclusions and Further Works
For solving transformation singularity problem of three-joint underactuated manipulator we present two simpleand effective mathematical techniques to find a valid pathin actual coordinate Then a motion planning strategy isdeveloped to estimate efficiency and open loop robustness ofresultant pathThis strategy is dealt with by two steps the firststep is to estimate the efficiency and capability of linearization
0 5 10 15 20 25 30
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 14 Joint angles displacement with initial errors (simulation6)
approximation for a resultant path Although nonholonomicpath estimation scheme can generate a satisfactory resultantpath actual trajectory is greatly affected in the presence ofinitial errorThe second step is to generate a robustness trajec-tory based on initial condition error sensitivity analysis Thesimulation and experimental results show that the motionplanning strategy can improvemotion planning performancefor three-joint underactuated manipulator
With increase of number of joints the kinematic modelwill be complex and the conversion into the chained formwill become ill-conditionedThe next work is how to improve
10 Journal of Robotics
0 5 10 15 20 25 30 35 40 455
10
15
20
25
30
35
40
45
50
55
Time (s)
120579(d
eg)
1205791
1205792
1205793
120579112057921205793
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
Figure 16 Joints angular displacement with initial errors (experi-ment 3)
the mechanism structure and find valid path for high dimen-sion system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the Specialized Research Fund forthe Doctoral Program of Higher Education (20100143110012)China and Fundamental Research Funds for the CentralUniversities (145204002)
References
[1] Y Tan L LiM Liu andGChen ldquoDesign andpath planning forcontrollable underactuated manipulatorrdquo International Journalof Advancements in Computing Technology vol 4 no 2 pp 212ndash221 2012
[2] T G Tan Research on underactuated manipulator and motioncontrol based on nonholonomic constrains [PhD thesis] WuhanUniversity of Technology 2005
[3] O J Sordalen ldquoConversion of the kinematics of a car withn trailers into a chained formrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation pp 382ndash387 Atlanta Georgia USA May 1993
[4] R M Murray and S S Sastry ldquoSteering nonholonomic systemsin chained formrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 1121ndash1126 Brighton UK December1991
[5] Y Nakamura W Chung and O J Sordalen ldquoDesign andcontrol of the nonholonomic manipulatorrdquo IEEE Transactionson Robotics and Automation vol 17 no 1 pp 48ndash59 2001
[6] H G Tanner S G Loizou and K J Kyriakopoulos ldquoNonholo-nomic navigation and control of cooperating mobile manipu-latorsrdquo IEEE Transactions on Robotics and Automation vol 19no 1 pp 53ndash64 2003
[7] Y Q Ren and B L Ma ldquoGlobal stabilization of a class ofnonholonomic systemsrdquo Zidonghua XuebaoActa AutomaticaSinica vol 33 no 9 pp 979ndash984 2007
[8] B L Ma ldquoGlobal k-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 pp 194ndash201 2009
[9] A Krontiris S Louis and K E Bekris ldquoMulti-level formationroadmaps for collision-free dynamic shape changes with non-holonomic teamsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 1570ndash1575 2012
[10] JWit C D Crane III andD Armstrong ldquoAutonomous groundvehicle path trackingrdquo Journal of Robotic Systems vol 21 no 8pp 439ndash449 2004
[11] WD Esquivel and L E Chiang ldquoNonholonomic path planningamong obstacles subject to curvature restrictionsrdquoRobotica vol20 no 1 pp 49ndash58 2002
[12] Y Zhu T Zhang and J-Y Song ldquoPath planning for nonholo-nomic mobile robots using artificial potential field methodrdquoControlTheory and Applications vol 27 no 2 pp 152ndash158 2010
[13] J Barraquand B Langlois and J-C Latombe ldquoNumericalpotential field techniques for robot path planningrdquo IEEE Trans-actions on SystemsMan and Cybernetics vol 22 no 2 pp 224ndash241 1992
[14] S Garrido L Moreno D Blanco and M F Martın ldquoRoboticmotion using harmonic functions and finite elementsrdquo Journalof Intelligent and Robotic Systems Theory and Applications vol59 no 1 pp 57ndash73 2010
[15] S Sekhavat and J-P Laumond ldquoTopological property forcollision-free nonholonomic motion planning the case ofsinusoidal inputs for chained form systemsrdquo IEEE Transactionson Robotics and Automation vol 14 no 5 pp 671ndash680 1998
[16] E Papadopoulos I Papadimitriou and I Poulakakis ldquoPoly-nomial-based obstacle avoidance techniques for nonholonomicmobile manipulator systemsrdquo Robotics and Autonomous Sys-tems vol 51 no 4 pp 229ndash247 2005
[17] S Sekhavat and J P Laumond ldquoTopological property oftrajectories computed from sinusoidal inputs for nonholonomicchained form systemsrdquo in Proceedings of the 13th IEEE Interna-tional Conference on Robotics and Automation vol 4 pp 3383ndash3388 April 1996
[18] Y G Tan and Z D Zhou ldquoMotion Decomposition and Com-position Transmission Mechanism Based on Friction DiskrdquoPatent of the Peoplersquos Republic of China No ZL 02147771XSeptember 2004
[19] D Tilbury R M Murray and S S Sastry ldquoTrajectory genera-tion for theN-trailer problemusing goursat normal formrdquo IEEETransactions on Automatic Control vol 40 no 5 pp 802ndash8191995
[20] A W Divelbiss and J T Wen ldquoA path space approach tononholonomic motion planning in the presence of obstaclesrdquoIEEE Transactions on Robotics and Automation vol 13 no 3pp 443ndash451 1997
