Research Article Extreme Learning Machine Assisted Adaptive...

Research ArticleExtreme Learning Machine Assisted Adaptive Control ofa Quadrotor Helicopter

Yu Zhang1 Zheng Fang2 and Hongbo Li3

1School of Aeronautics and Astronautics Zhejiang University Zhejiang Hangzhou 310027 China2State Key Laboratory of Synthetical Automation for Process Industries Northeastern University Liaoning Shenyang 110189 China3Department of Computer Science and Technology Tsinghua University Beijing 100084 China

Correspondence should be addressed to Zheng Fang fangzhengmailneueducn

Received 21 August 2014 Revised 10 December 2014 Accepted 14 December 2014

Academic Editor Yi Jin

Copyright copy 2015 Yu Zhang et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Control of quadrotor helicopters is difficult because the problem is naturally nonlinear The problem becomes more challengingfor common model based controllers when unpredictable uncertainties and disturbances in physical control system are taken intoaccount This paper proposes a novel intelligent controller design based on a fast online learning method called extreme learningmachine (ELM) Our neural controller does not require precise system modeling or prior knowledge of disturbances and wellapproximates the dynamics of the quadrotor at a fast speed The proposed method also incorporates a sliding mode controller forfurther elimination of external disturbances Simulation results demonstrate that the proposed controller can reliably stabilize aquadrotor helicopter in both agitated attitude and position control tasks

1 Introduction

Unmanned aerial vehicles (UAVs) have received considerableattention in recent years due to their wide military and civil-ian applications Their typical applications include collectingdata monitoring surveillance investigation and inspection[1] which can be used in scenarios such as environmentalmonitoring resource exploration agriculture surveying traf-fic control weather forecasting aerial photography disasterssearch and rescue Particularly unmanned rotorcrafts play animportant role in these applications because of their flexibilitysuch as hovering and vertical take-off and landing (VTOL)However conventional rotorcraft with a main rotor and a tailrotor has extremely complex dynamics Its maneuverabilityis greatly limited since it is very difficult to design a con-troller with high performance Meanwhile a special kind ofrotorcrafts called quadrotor helicopter which has a compactform is becoming more and more popular than conventionalrotorcrafts as they are mechanically and dynamically simplerand easier to control In spite of this the quadrotor is still adynamically unstable system and its controller design is alsochallenging because of the inherent system characteristics

such as nonlinearities cross couplings due to the gyroscopiceffects and underactuation [2] Besides all the applicationsmentioned above require a vehicle with stable and accurateperformance of motion control Therefore how to design ahigh quality controller for quadrotor is an important andmeaningful problem

Manydifferent control theories andmethods are employedto design the attitude stabilizer or motion controller forquadrotors [3] In the early stage Bouabdallah et al first applytwo different control techniques PID and linear quadratic(LQ) techniques to a microquadrotor called OS4 [2] Sub-sequently other research works using PID [4 5] or LQ [6]methods are also reported These two kinds of controllersare easy to design and implement However they cannothandle the unmolded dynamics and external disturbancesBecause of the nonlinearity feedback linearization technol-ogy is adopted to design the controller for quadrotors [78] However precise model of quadrotors which is requiredfor linearization is difficult to obtain Backstepping designmethod [9ndash11] is another choice to deal with the nonlinearmodels of quadrotors [12 13] but the controller is usually vul-nerable to parameters uncertainty To deal with the dynamic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 905184 12 pageshttpdxdoiorg1011552015905184

2 Mathematical Problems in Engineering

uncertainty and external disturbances sliding mode control[14] and 119867 infinite control [15 16] algorithms are used toimprove the robustness of the system However slidingmodecontroller is likely to have chattering phenomena in bothsides of sliding mode surface due to the delay of sensorsor actuators and 119867 infinite control method which requiresapproximate linearization near the equilibrium point of thesystem is not suitable for aggressive control

The performance of model based controllers abovedegrades significantly in case of model uncertainties andunknown disturbances One potential way to solve thisproblem is intelligent control methods such as fuzzy logiccontrol [17 18] neural networks control [19 20] and learningbased control [21] Onemain challenge of utilizing these tech-niques is the convergence performance of controllers Slowconvergence may cause failure in real time control system

In this paper a novel computational intelligence tech-nique called extreme learning machine (ELM) [22] is intro-duced to control the quadrotors by compensating thedynamic uncertainties and the external disturbances ELMtheories have been successfully improved recently by Caoet al [23] and widely used in several control systems [24]Essentially it is a learning policy for generalized single hiddenlayer feedforward networks (SLFNs) whose input weightand hidden layer do not need to be tuned Compared withbackpropagation (BP) method and support vector machines(SVMs) ELM provides better generalization performanceat a much faster learning speed and with least humanintervention [25] Thus ELM can be used to estimate andcompensate the uncertainties and disturbances of the systemssimultaneously in real time

There are three main contributions of this paper Firstwe employ Lyapunov second method to minimize the costfunction of ELM and satisfy the quadrotor control systemstability simultaneously under the framework of ELM So itis a development of ELM theory Second traditional neuralnetwork based controller is facing two problems One is thattoo many parameters need to be initialized and tuned Theother is slow convergence Since ELM can converge very fastwith its input weight and hidden nodes parameters fixed thetwo problems above are significantly alleviated when ELMis employed to the quadrotor control Third the quadrotorcontrol system is a complicated dynamic systemThe stabilityof the ELM based control system is proved

This paper is organized as follows In Section 2 themathematical model of ELM is presented Kinematics anddynamics models of quadrotors are described in Section 3 InSection 4 the details of designing an ELM-assisted quadrotorcontroller are presented The stability of the proposed con-troller is also proved in this section Simulation results aregiven in Section 5 to demonstrate the performance of the pro-posed controller Finally the paper is concluded in Section 6

2 Preliminary on Extreme Learning Machine

In this section the basic idea of ELM is briefly reviewedto provide a background for designing controller for thequadrotor ELM is a special SLFN whose learning speed

can be much faster than conventional feedforward networklearning algorithm such as BP algorithm while obtainingbetter generalization performance [26]The essence of ELM isthat the input weights and the parameters of the hidden layerdo not need to adjust during the learning procedure We takea SLFNwith 119871 hidden nodes as an exampleThe output of theSLFN can be modeled as

119891119871(x) =

119871

sum

119894=1

120573119894119866(x c

119894 119886119894) 119909 isin R119899 c

119894isin R119899 (1)

where 120573119894is the output weight connecting the 119894th hidden node

to the output node 119866(x c119894 119886119894) is the activation function of

the 119894th hidden node and c119894and 119886

119894are the parameters of

the activation function which are randomly generated andthen fixed afterwards Furthermore there are two kinds ofhidden nodes Usually additive hidden nodes use Sigmoid orthreshold activation function as follows

119866(x c119894 119886119894) =

1 minus 119890minus(c119894x+119886119894)

1 + 119890minus(c119894x+119886119894) (2)

where c119894is the input weight vector for the 119894th hidden node

and 119886119894is the bias of the 119894th hidden node For RBF hidden

nodes Gaussian or triangular activation function is used foractivation which can be given by

119866(x c119894 119886119894) = exp(minus

1003817100381710038171003817x minus c119894

1003817100381710038171003817

2

21198862

119894

) (3)

where c119894and 119886119894are the center and impact factor of the 119894th RBF

node respectivelyThen 119873 sample pairs (x

119896 y119896) isin R119899 times R119898 (119896 = 1 119873)

are used to train the SLFN If this network can approximate119873 samples with zero error there must exist 120573lowast

119894 c119894 and 119886

119894such

that119871

sum

119894=1

120573lowast

119894119866(x119896 c119894 119886119894) = y119896 119896 = 1 119873 (4)

The previous equation can be rewritten compactly as

H120573lowast = Y (5)

where

H =[[

[

119866(x1 c1 1198861) sdot sdot sdot 119866(x

1 c119871 119886119871)

d


119873 c119871 119886119871)

]]

]119873times119871

120573lowast= [120573lowast1198791

sdot sdot sdot 120573lowast119879119871]119879

Y = [Y1198791sdot sdot sdot Y119879

119871]119879

(6)

ELM aims to minimize not only the training error but alsothe norm of output weights which would yield a bettergeneralization performance [25] In other words ELM try tominimize the training error as well as the norm of the outputweights So the objective function can be expressed as

min1003817100381710038171003817H120573 minus Y100381710038171003817100381710038171003817100381710038171205731003817100381710038171003817 (7)

Mathematical Problems in Engineering 3

1

2

3

4

(a)

1

2

3

4

(b)

1

2

3

4

(c)

1

2

3

4

(d)

Figure 1 Quadrotor schematic The white arrow width is proportional to rotor rotational speed and the black arrows show the movingdirection of the quadrotor

Finally the minimal norm least-square method insteadof the standard optimization method was adopted in theoriginal implementation of ELM [25] and the closed formsolution is obtained

120573lowast= HdaggerY (8)

whereHdagger is theMoore-Penrose generalized inverse of matrixH

3 Quadrotor Helicopters Model

The quadrotor helicopter has four rotors in cross config-uration As we can see from Figure 1 the two pairs ofrotors (1 3) and (2 4) always turn in opposite directionsBy changing the rotor speed we can move the vehicles indifferent directions in 3D space Initially suppose all therotors have the same speed as shown in Figure 1(c) increasingthe four rotors speeds together generates upward movementThen increasing or decreasing 2 and 4 rotors speed inversely

will change the attitude of the quadrotor It generates rollrotation as well as lateral motion Changing 1 and 3 rotorsspeed in the same way produces the pitch rotation as well asthe longitudinal movements (see Figure 1(d)) Finally if thecounter-torque resulting from rotor (1 3) is different fromthat of rotor (2 4) the yaw rotations of the quadrotor aregenerated as shown in Figures 1(a) and 1(b)

31 Kinematic Model To facilitate the model descriptionquadrotor reference frames are defined first We considerearth fixed frame 119864-119883119884119885 as an inertia frame while frame119861-119909119910119911 is a body fixed frame as shown in Figure 2

To transform an attitude from the body frame (119909 119910and 119911) to the inertia frame (119883 119884 and 119885) the coordinatetransformation matrix [1]

R = [[

119888120579119888120595 119904120601119904120579119888120595 minus 119888120601119904120595 119888120601119904120579119888120595 + 119904120601119904120595

119888120579119904120595 119904120601119904120579119904120595 + 119888120601119888120595 119888120601119904120579119904120595 minus 119904120601119888120595

minus119904120579 119904120601119888120579 119888120601119888120579

]

]

(9)


x

y

z

Z

XY

E

B

Figure 2 Quadrotor reference frames

where 119888 and 119904 denote cosine and sine functions Then therelationship between the euler rates 120578 = [ 120601 120579 ]

119879

and angu-lar body rates 120596 = [119901 119902 119903]

119879 can be represented by

120596 = R119903120578 (10)

where

R119903= [

[

1 0 minus sin(120579)0 cos(120601) sin(120601) cos(120579)0 minus sin(120601) cos(120601) cos(120579)

]

]

(11)

Particularly when the quadrotor works around the hoverposition the following condition can be approximately sat-isfied

R119903asymp I3times3 (12)

where I is the identity matrix

32 DynamicModel Since the quadrotor helicopter can fly in3D space both rotational and translationalmotions should beconsidered in system modeling The rotational equations ofmotion can be derived in the body frame using the Newton-Euler method while the translational equations of motion arederived in the inertia navigation frameusingNewtonrsquos secondlaw [1] Many works about quadrotor modeling have beenreported so here we directly give the full state space model[27] of a quadrotor for controller design

The state vector is defined as X =

[120601 120601 120579 120579 120595 119911 119909 119910 119910]119879

where 120601 120579 and 120595

are the attitude angles which represent roll pitch and yaw(119909 119910 119911) is the quadrotorrsquos position in the inertia frameIf we define the speeds of the four rotors as Ω

1 Ω2 Ω3

and Ω4 the control input vector can be further defined as

U = [1198801 1198802 1198803 1198804]119879 which is mapped by

1198801= 119896119865(Ω2

1+ Ω2

2+ Ω2

3+ Ω2

4)

1198802= 119896119865(minusΩ2

2+ Ω2

4)

1198803= 119896119865(Ω2

1minus Ω2

3)

1198804= 119896119872(Ω2

1minus Ω2

2+ Ω2

3minus Ω2

4)

(13)where 119896

119865and 119896

119872are the aerodynamic force and moment

constants respectively In this case 1198801is the total thrust

generated from the four rotors 1198802 1198803 and 119880

4are the

equivalent of the torques around 119909- 119910- and 119911-axis in bodyframe

Then the state space model can be given by the followingcompact form

X = f(XU) (14)where

f(XU) =

((((((((((((((((((((((

(

120601

1205791198861+ 1205791198862Ω119903+ 11988711198802

120579

1206011198863minus 1206011198864Ω119903+ 11988721198803

120579 1206011198865+ 11988731198804

119892 minus(cos120601 cos 120579)119880

1

119898

(cos120601 sin 120579 cos120595 + sin120601 sin120595)1198801

119898

119910

(cos120601 sin 120579 sin120595 minus sin120601 cos120595)1198801

119898

))))))))))))))))))))))

)

(15)

1198861=

(119868119910119910minus 119868119911119911)

119868119909119909

1198862=119869119903

119868119909119909

1198863=(119868119911119911minus 119868119909119909)

119868119910119910

1198864=119869119903

119868119910119910

1198865=

(119868119909119909minus 119868119910119910)

119868119911119911

1198871=

119897

119868119909119909

1198872=

119897

119868119910119910

1198873=1

119868119911119911

(16)

and 119868119909119909 119868119910119910 and 119868

119911119911are the moment of inertia around 119909- 119910-

and 119911-axis respectively 119869119903is the propeller inertia coefficient

119897 is the arm length of the quadrotor


Positioncontroller

Desiredtrajectory

Attitudecontroller

Rotors

speed

generation

Quadrotordynamics

Ω1

Ω2

Ω3

Ω4

x x y y z z

U1

U2 U3 U4

120601d 120579d 120595d

120601 120579 120595 120601 120579 120595

xd xdyd yd

zd zd120601d120579d 120595d

Figure 3 The structure of the quadrotor control system

4 Controller Design Using ELM

By investigating the relationship between the state variablesin (15) the attitude of the quadrotor does not depend on thetranslational motion while the translation of the quadrotordepends on the attitude angles Thus the whole systemcan be decoupled into two subsystems inner loop attitudesubsystem and outer loop position subsystem respectivelyThe structure of the whole control system is described inFigure 3 where (119909

119889 119889 119910119889 119910119889 119911119889 119889) is the coordinate of the

desired trajectory and (120601119889 120579119889 120595119889 120601119889 120579119889 119889) is the desired

attitude angles and their rates generated by the positioncontroller 119880

1is produced by the position controller related

to the altitude of the vehicle while the 1198802 1198803 and 119880

4are

calculated by the attitude controller

41 Position Control Loop The outer loop is a positioncontrol system which is much slower compared to the innerloop attitude control system In addition the desired rollpitch and yaw angles are normally small that makes theposition subsystem an approximately linear system near theequilibrium points A simple PD controller is sufficient tocontrol the quadrotorrsquos position

For horizontal movement according to the desired tra-jectory the expected accelerations in 119883 and 119884 direction canbe calculated by the PD controller

119889= 119896119901119909(119909119889minus 119909) + 119896

119889119909(119889minus )

119910119889= 119896119901119910(119910119889minus 119910) + 119896

119889119910( 119910119889minus 119910)

(17)

where (119896119901119909 119896119889119909) and (119896

119901119910 119896119889119910) are the proportional and

differential control gains in119883 and119884 directions Control gainsare acquired using pole placement designmethod tomake theposition system stable When the yaw angle of the quadrotorkeeps fixed we can substitute

119889and 119910

119889into (15) and obtain

(cos120601119889sin 120579119889cos120595 + sin120601

119889sin120595)119880

1

119898= 119889

(cos120601119889sin 120579119889sin120595 minus sin120601

119889cos120595)119880

1

119898= 119910119889

(18)

Using the small angle assumption around the equilibriumposition (18) can be simplified as

(120579119889cos120595 + 120601

119889sin120595)119880

1

119898= 119889

(120579119889sin120595 minus 120601

119889cos120595)119880

1

119898= 119910119889

(19)

Then the reference roll and pitch angles can be solved by

[120601119889

120579119889

] =119898

1198801

[sin120595 cos120595minus cos120595 sin120595]

minus1

[119889

119910119889

] (20)

For vertical movements the altitude control input can becalculated by

1198801=

1

cos(120601) cos(120579)

sdot ((1 + 11989611198962)(119911119889minus 119911) + (119896

1+ 1198962)(119889minus ) + 119898119892)

(21)

using backstepping design method where 1198961and 1198962are pos-

itive real numbers and 119898119892 is the term to balance the gravityPlease note that all the inputs have their own saturationfunctions

42 Attitude Control Loop The quadrotorrsquos attitude systemis sensitive to the disturbances It is also a nonlinear systemwith unmodeled dynamic uncertainties compared to thereal quadrotor system Hence an ELM based SLFN withvery fast learning speed is employed to adaptively learn andestimate the nonlinear model including the uncertaintiesof the quadrotorrsquos attitude system in real time Meanwhilethe proposed neural controller incorporates a sliding modecontroller to deal with the external disturbancesThe stabilityof the attitude control system is proved using the Lyapunovtheory

Since the roll pitch and yaw subsystems have the sameform of expression without loss of generality we take rollsubsystem as an example to introduce the design procedureAlthough three attitude subsystems are coupled with eachother the coupled items can be considered the unknownnon-linear functions of dynamic system or internal disturbancesand thus be adaptively estimated by the SLFN with ELM


0 1 2

0

0 1 2

0

ReferencePDSMC

SMC with NNSMC with ELM

ReferencePDSMC


0505 1515

minus02

minus01

minus02

minus01

01

02

03

01

02

03

Time (s) Time (s)

120579 (r

ad)

120601(r

ad)

Figure 4 Roll and pitch angle regulation

421 Model Based Control Law According to (15) the rollsubsystem is expressed in the following form

1= 1199092

2= 119891(x) + 119892(x)119906 + 119889

119910 = 1199091

(22)

where 119891(x) is assumed to be unknown but bounded and 119892(x)is the input gain Particularly in roll subsystem 119892(x) = 119887

1=

119897119868119909119909

which is known 119889 is the unknown disturbances It isbounded and satisfies |119889| le 119889max where 119889max is the upperbound of the disturbances Then the output tracking errorE = [119890 119890]119879 where 119890 = 119910

119889minus 1199091and 119890 = 119910

119889minus 1199092

Suppose all the functions and parameters in (22) areprecisely known the perfect control law 119906lowast can be obtainedusing feedback linearization method as follows

119906lowast=119868119909119909

119897( 119910119889minus 119891(x) minus 119889 + K119879E) (23)

where K = [1198701 1198702]119879 is a real number vector Substitute (23)

into (22) the following error equation is obtained

119890 + 1198702119890 + 1198701= 0 (24)

1198701and 119870

2can be determined when all the roots of the

polynomial 1199042 + 1198702119904 + 1198701= 0 are in the open left half plane

which indicates that the tracking error will converge to zero

422 Neural Controller Structure Since 119891(x) is coupledwith other variables and cannot be accurately modeled 119889

is also unknown disturbances control law of (23) cannotbe directly implemented Here we present an ELM-assistedneural controller to solve the problem

The inner loop controller is mainly composed of twoparts One is the neural controller 119906

119899and the other is the

sliding mode controller 119906119904 So the overall control law

119906 = 119906119899+ 119906119904 (25)

The neural controller is used to approximate the unknownfunction 119891(x) while the sliding mode controller is employedto eliminate the external disturbances 119889

For the neural controller according to (23) the optimalneural control law is expected as

119906lowast

119899=119868119909119909

119897( 119910119889minus 119891(x) + K119879E) (26)

Here a SLFN whose parameters are determined based on theELM is employed to approximate the above desired neuralcontrol lawThen the actual neural control law 119906

119899is given by

119906119899= H(r c a)120573 (27)

where r = [ 119910119889 xE]119879 is the input vector As we mentioned

in Section 2 c and a are hidden node parameters which aregenerated randomly and then fixed Training a SLFN withELM is the equivalent of finding a least-square solution of theoutput weights 120573lowast such that

lowast

119899= H(r c a)120573lowast + 120576(r) (28)

where 120576(r) is the approximation error This error arises if thenumber of hidden nodes is much less than the number of


training samples [24] but it is bounded with the constant 120576119873

that is |120576(r)| le 120576119873

For the sliding mode controller standard sliding modecontrol law has the form

119906119904= 119866119904(x) sgn(119891(E)) (29)

where 119866119904(x) is the sliding mode gain and 119891(E) is the sliding

mode surface functionSince we have the forms of the neural controller and the

slidingmode controller the next step is to derive the adaptivelaw so that120573 converges to120573lowastThe parameters of the slidingmode controller are determined The details are given in thenext subsection

43 Adaptive Control Law and Stability Analysis The stableadaptive law for 120573 and the parameters of the sliding modecontroller are derived using Lyapunov second method

Based on (22) (25) and (26) the tracking error equationcan be obtained

E = ΛE + B[119906lowast119899minus 119906119899minus 119906119904minus119889119868119909119909

119897] (30)

where

Λ = [0 1

minus1198701minus1198702

]

B = [[

0

119897

119868119909119909

]

]

(31)

Substituting (27) (28) into (30) yields

E = ΛE + B[H(r c a) + 120576(r) minus 119906119904 minus119889119868119909119909

119897] (32)

where = 120573lowast minus 120573 Then the following Lyapunov function isconsidered to derive the stable tuning law for 120573

119881 =1

2E119879PE + 1

2120578119879

(33)

where P is a symmetric and positive definite matrix and 120578 isa positive constant which is referred to as the learning rate ofthe SLFN

According to (32) the derivative of the Lyapunov func-tion is acquired as

=1

2[E119879PE + E119879PE] + 1

120578

120573119879

= minus1

2E119879QE + (E119879PBH +

1

120578

120573119879

)

minus E119879PB(119889119868119909119909119897minus 120576(r)) minus E119879PB119906

119904

(34)

where

Q = minus(Λ119879P + PΛ) (35)

SupposeQ is selected as a symmetric definite matrix the firstitem of the Lyapunov function will converge to zero ThenP can be calculated by solving (35) According to the seconditem of the Lyapunov function can be eliminated if we let

120573119879

= minus120578E119879PBH (36)

Consider 120573 = minus based on the definition of the Thus the

tuning rule for the output weight is acquired

119879

= 120578E119879PBH (37)

After that (34) becomes

= minus1

2E119879QE minus E119879PB(119889119868119909119909

119897minus 120576(r)) minus E119879PB119906

119904 (38)

To make (38) less than or equal to zero the sliding modecontroller 119906

119904can be determined as

119906119904= (

119889max119868119909119909119897

+ 120576119873)sgn(E119879PB) (39)

Therefore the following relationship can be derived from(39)

le minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus E119879PB((119889max119868119909119909119897

+ 120576119873) sgn(E119879PB))

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(

119889max119868119909119909119897

+ 120576119873)

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889| minus 119889max)119868119909119909

119897

+10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(|120576(r)| minus 120576119873)

le 0

(40)

Equation (40) indicates that is negative semidefinite andthe control system is guaranteed to be stable Please note thattraditional ELM trains their output weights using the least-square error method but in this paper we derived the tuningrate of the output weights using Lyapunov second methodIt is an online sequential learning process that makes thecontroller applicable in real time The input weight and theparameters of the hidden nodes are randomly assigned as thenormal ELM algorithm


0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8

Time (s)Time (s)0505 1515

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2

Figure 5 Control inputs 1198802and 119880

3

Finally the overall control laws for quadrotorrsquos attitudesystem are given by

1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)

where the subscripts 120601 120579 and 120595 represent the roll pitch andyaw subsystem respectively To avoid chattering problem thesgn(sdot) function can be replaced by the saturation functionsat(sdot)

5 Simulations

In this section attitude and position control simulationsare both implemented on a nonlinear quadrotor model toevaluate the performance of the proposed controller Wealso compare our method to some other controllers todemonstrate the effectiveness of ELMThe parameters of thequadrotor for simulation are measured from a real platformas listed in Table 1

0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573

Figure 6 Learning curve of 120573 in attitude regulation control

Table 1 Parameters of the quadrotor

Name Parameter Value UnitMass 119898 22 kgInertia around 119909 axis 119868

1199091199091676119890 minus 2 kgsdotm2

Inertia around 119910 axis 119868119910119910

1676119890 minus 2 kgsdotm2



Rotor inertia 119869119903

01 kgsdotm2

Arm length 119897 018 m

51 Simulation Setup Two cases are tested for attitude controlsimulations The first case is to regulate the quadrotorhelicopter from an initial attitude (120601

119894= 1205878 120579

119894= 1205878 120595

119894=

0) to a target attitude (120601119905= minus12058716 120579

119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)

Figure 7 Roll and pitch angle tracking

The attitude response speed and accuracy can be evaluatedin this case The second case is an attitude tracking missionThe quadrotorrsquos attitude is set to follow a trajectory (120601

119889=

(1205878) sin(2120587 times 025119905) 120579119889= (1205878) sin(2120587 times 025119905)) while the

yaw angle keeps fixedTo further demonstrate the controller performance posi-

tion tracking simulations are also implemented where thequadrotor is expected to follow a trajectory such as straightline and circle

In all the cases above the model uncertainty 119889 in (22)is given by 05 lowast sin(120587119905) and 119889max = 05 The controllerparameters are chosen as follows 119870

1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873

= 14 The number ofhidden nodes 119871 = 6 RBF nodes are selected as the hiddennodeswhose parameter c is generated in the interval [minus512058716512058716] and 119886 = 04

To show the feasibility and advantage of the proposedcontroller we compare our method (SMC with ELM) to aproportion-differentiation controller (PD) a standard slidingmode controller (SMC) and a SMC with back propagationbased neural network (SMCwith NN)The parameters of thePD controller are chosen as follows 119896

119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896

119889120595= 1 Slidingmode control

gain 119866 = 145 Sliding mode surface parameters 1198881= 8 and

1198882= 1 The initial parameters of the SMC with NN are the

same as the SMC with ELM

52 Results The results of roll (120601) and pitch (120579) angles reg-ulation using different controllers are illustrated in Figure 4As we can see the proposed controller makes the attitudeangles converge faster to the reference values than the other

controllers PD controller has attitude overshoot problemwhich may result in vehiclersquos vibration This problem can beavoided by decreasing PD control gains but the responsespeed is sacrificed The result from SMC with NN has largeerrors because of the slow convergence problem The controlinputs related to roll and pitch are shown in Figure 5 Allof them are bounded and applicable Inputs in our methodare large at the beginning but decrease quickly This is thereason why the response speed and accuracy are obtainedsimultaneously In Figure 6 it indicates that the outputweights 120573 trained by ELM can converge faster than BP basedneural networksThus the contribution of ELM in quadrotorcontroller design is further confirmed

Attitude tracking results are shown in Figure 7 ELM-assisted controller makes the attitude follow the referencetrajectory pretty well while PD and SMC have a delay intracking process The result from SMC with NN also cannotfollow the reference input well It shows that the proposedcontroller has the best tracking performance among the fourcontrollers Tracking errors of this case are shown in Figure 8We can see that our proposed method has the smallest errorFigure 9 shows the better convergency of output weights 120573using ELM than traditional neural networks

The results of straight line tracking are shown in Figure 10which demonstrate better tracking performance of proposedcontroller As we can see from Figure 11 SMC with ELM hasthe smallest error compared to the others In addition the PDand SMC with NN cannot damp the errors quickly and havesome oscillations The learning curve of ELM also convergesfast as shown in Figure 12

Position tracking on a circle trajectory is shown inFigure 13 It seems that all methods can give a stable tracking


0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)

Figure 8 Attitude tracking errors

0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573

Figure 9 Learning curve of 120573 in attitude tracking control

0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)

Figure 10 Position tracking on a straight line

performance However as we can see the SMC with ELMhas the best performance in terms of stability and trackingaccuracyThe training process of 120573 in ELM is also better thanBPbased neural networks as given in Figure 14 ELMcan con-verge quickly and remains unchanged afterwards in this test

6 Conclusions

In this paper an ELM-assisted adaptive controller combinedwith a sliding mode controller is designed for control ofa quadrotor helicopter A single hidden layer feedforwardnetwork whose input weights and hidden node parametersare generated randomly and fixed afterwards is used toapproximate the unknown dynamic model and internaluncertainties Different from the standard ELM algorithmthe output weights of this neural network are updated basedon the Lyapunov second method to guarantee the stabilityof the attitude control system A sliding mode controlleris employed to compensate the approximation error of theSLFN and eliminate the external disturbances Plenty ofsimulations on quadrotorrsquos attitude and position control areimplemented to validate the effectiveness of the proposedcontrol scheme The simulation results show that the pro-posed controller has better performance on response speedand control accuracy than simple PD controller Further-more the comparison with the standard SMC and SMC withNN indicates that the extreme learningmachine has potentialability to handle the unmodeled uncertainty problem in thecontrol domain because of its fast convergence capability andgood approximation ability


0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion

Figure 11 Tracking errors in119883 and 119884 direction

0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573

Figure 12 Learning curve of 120573 in position tracking control(straight line)

0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)

Figure 13 Position tracking on a circle trajectory

0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573

Figure 14 Learning curve of 120573 in position tracking control(circle)

For future work the idea of composite design andreinforcement learning [28] could be borrowed to developnew ELM algorithm to achieve better results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61005085) and FundamentalResearch Funds for the Central Universities (2012QNA4024N120408002)


References

[1] A Nagaty S Saeedi C Thibault M Seto and H Li ldquoControland navigation framework for quadrotor helicoptersrdquo Journal ofIntelligent and Robotic Systems vol 70 no 1ndash4 pp 1ndash12 2013

[2] S Bouabdallah A Noth and R Siegwart ldquoPID vs LQ controltechniques applied to an indoor micro Quadrotorrdquo in Pro-ceedings of the IEEERSJ International Conference on IntelligentRobots and Systems (IROS rsquo04) pp 2451ndash2456 IEEE October2004

[3] Y Li and S Song ldquoA survey of control algorithms for quadrotorunmanned helicopterrdquo in Proceedings of the IEEE 5th Inter-national Conference on Advanced Computational Intelligence(ICACI rsquo12) pp 365ndash369 October 2012

[4] S Gonzalez-Vazquez and J Moreno-Valenzuela ldquoA new non-linear PIPID controller for quadrotor posture regulationrdquo inProceedings of the Electronics Robotics and AutomotiveMechan-ics Conference (CERMA rsquo10) pp 642ndash647 Morelos MexicoSeptemberndashOctober 2010

[5] A L SalihMMoghavvemiHA FMohamed andK SGaeidldquoFlight PID controller design for a UAV quadrotorrdquo ScientificResearch and Essays vol 5 no 23 pp 3660ndash3667 2010

[6] L Minh and C Ha ldquoModeling and control of quadrotorMAV using vision-based measurementrdquo in Proceedings of theInternational Forumon Strategic Technology (IFOST rsquo10) pp 70ndash75 IEEE October 2010

[7] N G Shakev A V Topalov O Kaynak and K B Shiev ldquoCom-parative results on stabilization of the quad-rotor rotorcraftusing bounded feedback controllersrdquo Journal of Intelligent andRobotic Systems vol 65 no 1ndash4 pp 389ndash408 2012

[8] H Voos ldquoNonlinear control of a quadrotor micro-uav usingfeedback-linearizationrdquo in Proceedings of the IEEE InternationalConference on Mechatronics (ICM rsquo09) pp 1ndash6 IEEE April2009

[9] B Xu F Sun H Liu and J Ren ldquoAdaptive kriging controllerdesign for hypersonic flight vehicle via back-steppingrdquo IETControl Theory amp Applications vol 6 no 4 pp 487ndash497 2012

[10] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014

[11] B Xu F Sun C Yang D Gao and J Ren ldquoAdaptive discrete-time controller designwith neural network for hypersonic flightvehicle via back-steppingrdquo International Journal of Control vol84 no 9 pp 1543ndash1552 2011

[12] A Das F Lewis and K Subbarao ldquoBackstepping approach forcontrolling a quadrotor using lagrange form dynamicsrdquo Journalof Intelligent and Robotic Systems vol 56 no 1-2 pp 127ndash1512009

[13] A Honglei L Jie W Jian W Jianwen and M HongxuldquoBackstepping-based inverse optimal attitude control ofquadrotorrdquo International Journal of Advanced Robotic Systemsvol 10 article no 223 2013

[14] A R Patel M A Patel and D R Vyas ldquoModeling and analysisof quadrotor using sliding mode controlrdquo in Proceedings of the44th Southeastern Symposium on System Theory (SSST rsquo12) pp111ndash114 IEEE March 2012

[15] J Gadewadikar F L Lewis K Subbarao K Peng and B MChen ldquoH-infinity static output-feedback control for rotorcraftrdquoJournal of Intelligent and Robotic Systems Theory and Applica-tions vol 54 no 4 pp 629ndash646 2009

[16] G V Raffo M G Ortega and F R Rubio ldquoAn integralpredictivenonlinear 119867

infincontrol structure for a quadrotor

helicopterrdquo Automatica vol 46 no 1 pp 29ndash39 2010[17] C Coza C Nicol C J B Macnab and A Ramirez-Serrano

ldquoAdaptive fuzzy control for a quadrotor helicopter robust towind buffetingrdquo Journal of Intelligent and Fuzzy Systems vol 22no 5-6 pp 267ndash283 2011

[18] D Gautam and C Ha ldquoControl of a quadrotor using asmart self-tuning fuzzy PID controllerrdquo International Journal ofAdvanced Robotic Systems vol 10 article 380 2013

[19] T Dierks and S Jagannathan ldquoOutput feedback control of aquadrotor UAV using neural networksrdquo IEEE Transactions onNeural Networks vol 21 no 1 pp 50ndash66 2010

[20] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[21] P Bouffard A Aswani and C Tomlin ldquoLearning-based modelpredictive control on a quadrotor onboard implementation andexperimental resultsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 279ndash284 IEEEMay 2012

[22] G-B Huang D H Wang and Y Lan ldquoExtreme learningmachines a surveyrdquo International Journal of Machine Learningand Cybernetics vol 2 no 2 pp 107ndash122 2011

[23] J Cao Z Lin G-B Huang and N Liu ldquoVoting based extremelearning machinerdquo Information Sciences vol 185 pp 66ndash772012

[24] H-J Rong and G-S Zhao ldquoDirect adaptive neural controlof nonlinear systems with extreme learning machinerdquo NeuralComputing and Applications vol 22 no 3-4 pp 577ndash586 2013

[25] G-B Huang Q-Y Zhu and C-K Siew ldquoReal-time learningcapability of neural networksrdquo IEEE Transactions on NeuralNetworks vol 17 no 4 pp 863ndash878 2006

[26] G-B Huang Q-Y Zhu and C-K Siew ldquoExtreme learningmachine theory and applicationsrdquoNeurocomputing vol 70 no1ndash3 pp 489ndash501 2006

[27] S Bouabdallah and R Siegwart ldquoFull control of a quadrotorrdquoin Proceedings of the IEEERSJ International Conference onIntelligent Robots and Systems (IROS rsquo07) pp 153ndash158 IEEEOctober 2007

[28] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


uncertainty and external disturbances sliding mode control[14] and 119867 infinite control [15 16] algorithms are used toimprove the robustness of the system However slidingmodecontroller is likely to have chattering phenomena in bothsides of sliding mode surface due to the delay of sensorsor actuators and 119867 infinite control method which requiresapproximate linearization near the equilibrium point of thesystem is not suitable for aggressive control

The performance of model based controllers abovedegrades significantly in case of model uncertainties andunknown disturbances One potential way to solve thisproblem is intelligent control methods such as fuzzy logiccontrol [17 18] neural networks control [19 20] and learningbased control [21] Onemain challenge of utilizing these tech-niques is the convergence performance of controllers Slowconvergence may cause failure in real time control system

In this paper a novel computational intelligence tech-nique called extreme learning machine (ELM) [22] is intro-duced to control the quadrotors by compensating thedynamic uncertainties and the external disturbances ELMtheories have been successfully improved recently by Caoet al [23] and widely used in several control systems [24]Essentially it is a learning policy for generalized single hiddenlayer feedforward networks (SLFNs) whose input weightand hidden layer do not need to be tuned Compared withbackpropagation (BP) method and support vector machines(SVMs) ELM provides better generalization performanceat a much faster learning speed and with least humanintervention [25] Thus ELM can be used to estimate andcompensate the uncertainties and disturbances of the systemssimultaneously in real time

There are three main contributions of this paper Firstwe employ Lyapunov second method to minimize the costfunction of ELM and satisfy the quadrotor control systemstability simultaneously under the framework of ELM So itis a development of ELM theory Second traditional neuralnetwork based controller is facing two problems One is thattoo many parameters need to be initialized and tuned Theother is slow convergence Since ELM can converge very fastwith its input weight and hidden nodes parameters fixed thetwo problems above are significantly alleviated when ELMis employed to the quadrotor control Third the quadrotorcontrol system is a complicated dynamic systemThe stabilityof the ELM based control system is proved

This paper is organized as follows In Section 2 themathematical model of ELM is presented Kinematics anddynamics models of quadrotors are described in Section 3 InSection 4 the details of designing an ELM-assisted quadrotorcontroller are presented The stability of the proposed con-troller is also proved in this section Simulation results aregiven in Section 5 to demonstrate the performance of the pro-posed controller Finally the paper is concluded in Section 6

2 Preliminary on Extreme Learning Machine

In this section the basic idea of ELM is briefly reviewedto provide a background for designing controller for thequadrotor ELM is a special SLFN whose learning speed

can be much faster than conventional feedforward networklearning algorithm such as BP algorithm while obtainingbetter generalization performance [26]The essence of ELM isthat the input weights and the parameters of the hidden layerdo not need to adjust during the learning procedure We takea SLFNwith 119871 hidden nodes as an exampleThe output of theSLFN can be modeled as

119891119871(x) =

119871

sum

119894=1

120573119894119866(x c

119894 119886119894) 119909 isin R119899 c

119894isin R119899 (1)

where 120573119894is the output weight connecting the 119894th hidden node

to the output node 119866(x c119894 119886119894) is the activation function of

the 119894th hidden node and c119894and 119886

119894are the parameters of

the activation function which are randomly generated andthen fixed afterwards Furthermore there are two kinds ofhidden nodes Usually additive hidden nodes use Sigmoid orthreshold activation function as follows

119866(x c119894 119886119894) =

1 minus 119890minus(c119894x+119886119894)

1 + 119890minus(c119894x+119886119894) (2)

where c119894is the input weight vector for the 119894th hidden node

and 119886119894is the bias of the 119894th hidden node For RBF hidden

nodes Gaussian or triangular activation function is used foractivation which can be given by

119866(x c119894 119886119894) = exp(minus

1003817100381710038171003817x minus c119894

1003817100381710038171003817

2

21198862

119894

) (3)

where c119894and 119886119894are the center and impact factor of the 119894th RBF

node respectivelyThen 119873 sample pairs (x

119896 y119896) isin R119899 times R119898 (119896 = 1 119873)

are used to train the SLFN If this network can approximate119873 samples with zero error there must exist 120573lowast

119894 c119894 and 119886

119894such

that119871

sum

119894=1

120573lowast

119894119866(x119896 c119894 119886119894) = y119896 119896 = 1 119873 (4)

The previous equation can be rewritten compactly as

H120573lowast = Y (5)

where

H =[[

[


1 c119871 119886119871)

d


119873 c119871 119886119871)

]]

]119873times119871

120573lowast= [120573lowast1198791

sdot sdot sdot 120573lowast119879119871]119879

Y = [Y1198791sdot sdot sdot Y119879

119871]119879

(6)

ELM aims to minimize not only the training error but alsothe norm of output weights which would yield a bettergeneralization performance [25] In other words ELM try tominimize the training error as well as the norm of the outputweights So the objective function can be expressed as

min1003817100381710038171003817H120573 minus Y100381710038171003817100381710038171003817100381710038171205731003817100381710038171003817 (7)


1

2

3

4

(a)

1

2

3

4

(b)

1

2

3

4

(c)

1

2

3

4

(d)










R = [[

119888120579119888120595 119904120601119904120579119888120595 minus 119888120601119904120595 119888120601119904120579119888120595 + 119904120601119904120595

119888120579119904120595 119904120601119904120579119904120595 + 119888120601119888120595 119888120601119904120579119904120595 minus 119904120601119888120595

minus119904120579 119904120601119888120579 119888120601119888120579

]

]

(9)


x

y

z

Z

XY

E

B



119879



120596 = R119903120578 (10)

where

R119903= [

[


]

]

(11)






[120601 120601 120579 120579 120595 119911 119909 119910 119910]119879

where 120601 120579 and 120595


1 Ω2 Ω3



1198801= 119896119865(Ω2

1+ Ω2

2+ Ω2

3+ Ω2

4)

1198802= 119896119865(minusΩ2

2+ Ω2

4)

1198803= 119896119865(Ω2

1minus Ω2

3)

1198804= 119896119872(Ω2

1minus Ω2

2+ Ω2

3minus Ω2

4)

(13)where 119896

119865and 119896




4are the



X = f(XU) (14)where

f(XU) =

((((((((((((((((((((((

(

120601

1205791198861+ 1205791198862Ω119903+ 11988711198802

120579

1206011198863minus 1206011198864Ω119903+ 11988721198803

120579 1206011198865+ 11988731198804

119892 minus(cos120601 cos 120579)119880

1

119898


119898

119910


119898

))))))))))))))))))))))

)

(15)

1198861=

(119868119910119910minus 119868119911119911)

119868119909119909

1198862=119869119903

119868119909119909

1198863=(119868119911119911minus 119868119909119909)

119868119910119910

1198864=119869119903

119868119910119910

1198865=

(119868119909119909minus 119868119910119910)

119868119911119911

1198871=

119897

119868119909119909

1198872=

119897

119868119910119910

1198873=1

119868119911119911

(16)

and 119868119909119909 119868119910119910 and 119868





Positioncontroller

Desiredtrajectory

Attitudecontroller

Rotors

speed

generation

Quadrotordynamics

Ω1

Ω2

Ω3

Ω4

x x y y z z

U1

U2 U3 U4

120601d 120579d 120595d

120601 120579 120595 120601 120579 120595

xd xdyd yd

zd zd120601d120579d 120595d









4are




119889= 119896119901119909(119909119889minus 119909) + 119896

119889119909(119889minus )

119910119889= 119896119901119910(119910119889minus 119910) + 119896

119889119910( 119910119889minus 119910)

(17)

where (119896119901119909 119896119889119909) and (119896



119889and 119910


(cos120601119889sin 120579119889cos120595 + sin120601

119889sin120595)119880

1

119898= 119889


119889cos120595)119880

1

119898= 119910119889

(18)


(120579119889cos120595 + 120601

119889sin120595)119880

1

119898= 119889

(120579119889sin120595 minus 120601

119889cos120595)119880

1

119898= 119910119889

(19)


[120601119889

120579119889

] =119898

1198801


minus1

[119889

119910119889

] (20)


1198801=

1

cos(120601) cos(120579)

sdot ((1 + 11989611198962)(119911119889minus 119911) + (119896

1+ 1198962)(119889minus ) + 119898119892)

(21)






0 1 2

0

0 1 2

0

ReferencePDSMC


ReferencePDSMC


0505 1515

minus02

minus01

minus02

minus01

01

02

03

01

02

03

Time (s) Time (s)

120579 (r

ad)

120601(r

ad)



1= 1199092

2= 119891(x) + 119892(x)119906 + 119889

119910 = 1199091

(22)


1=

119897119868119909119909


119889minus 1199091and 119890 = 119910

119889minus 1199092


119906lowast=119868119909119909

119897( 119910119889minus 119891(x) minus 119889 + K119879E) (23)



119890 + 1198702119890 + 1198701= 0 (24)

1198701and 119870









119906 = 119906119899+ 119906119904 (25)



119906lowast

119899=119868119909119909

119897( 119910119889minus 119891(x) + K119879E) (26)


119899is given by

119906119899= H(r c a)120573 (27)



lowast

119899= H(r c a)120573lowast + 120576(r) (28)




that is |120576(r)| le 120576119873


119906119904= 119866119904(x) sgn(119891(E)) (29)







119897] (30)

where

Λ = [0 1


]

B = [[

0

119897

119868119909119909

]

]

(31)



119897] (32)


119881 =1

2E119879PE + 1

2120578119879

(33)



=1

2[E119879PE + E119879PE] + 1

120578

120573119879

= minus1

2E119879QE + (E119879PBH +

1

120578

120573119879

)


119904

(34)

where

Q = minus(Λ119879P + PΛ) (35)


120573119879

= minus120578E119879PBH (36)



119879

= 120578E119879PBH (37)


= minus1

2E119879QE minus E119879PB(119889119868119909119909

119897minus 120576(r)) minus E119879PB119906

119904 (38)



119906119904= (

119889max119868119909119909119897

+ 120576119873)sgn(E119879PB) (39)


le minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus E119879PB((119889max119868119909119909119897

+ 120576119873) sgn(E119879PB))

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(

119889max119868119909119909119897

+ 120576119873)

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889| minus 119889max)119868119909119909

119897

+10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(|120576(r)| minus 120576119873)

le 0

(40)



0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8


minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2


3


1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)


5 Simulations


0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573




1199091199091676119890 minus 2 kgsdotm2






01 kgsdotm2



119894= 1205878 120579

119894= 1205878 120595

119894=


119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

4

(a)

1

2

3

4

(b)

1

2

3

4

(c)

1

2

3

4

(d)










R = [[

119888120579119888120595 119904120601119904120579119888120595 minus 119888120601119904120595 119888120601119904120579119888120595 + 119904120601119904120595

119888120579119904120595 119904120601119904120579119904120595 + 119888120601119888120595 119888120601119904120579119904120595 minus 119904120601119888120595

minus119904120579 119904120601119888120579 119888120601119888120579

]

]

(9)


x

y

z

Z

XY

E

B



119879



120596 = R119903120578 (10)

where

R119903= [

[


]

]

(11)






[120601 120601 120579 120579 120595 119911 119909 119910 119910]119879

where 120601 120579 and 120595


1 Ω2 Ω3



1198801= 119896119865(Ω2

1+ Ω2

2+ Ω2

3+ Ω2

4)

1198802= 119896119865(minusΩ2

2+ Ω2

4)

1198803= 119896119865(Ω2

1minus Ω2

3)

1198804= 119896119872(Ω2

1minus Ω2

2+ Ω2

3minus Ω2

4)

(13)where 119896

119865and 119896




4are the



X = f(XU) (14)where

f(XU) =

((((((((((((((((((((((

(

120601

1205791198861+ 1205791198862Ω119903+ 11988711198802

120579

1206011198863minus 1206011198864Ω119903+ 11988721198803

120579 1206011198865+ 11988731198804

119892 minus(cos120601 cos 120579)119880

1

119898


119898

119910


119898

))))))))))))))))))))))

)

(15)

1198861=

(119868119910119910minus 119868119911119911)

119868119909119909

1198862=119869119903

119868119909119909

1198863=(119868119911119911minus 119868119909119909)

119868119910119910

1198864=119869119903

119868119910119910

1198865=

(119868119909119909minus 119868119910119910)

119868119911119911

1198871=

119897

119868119909119909

1198872=

119897

119868119910119910

1198873=1

119868119911119911

(16)

and 119868119909119909 119868119910119910 and 119868





Positioncontroller

Desiredtrajectory

Attitudecontroller

Rotors

speed

generation

Quadrotordynamics

Ω1

Ω2

Ω3

Ω4

x x y y z z

U1

U2 U3 U4

120601d 120579d 120595d

120601 120579 120595 120601 120579 120595

xd xdyd yd

zd zd120601d120579d 120595d









4are




119889= 119896119901119909(119909119889minus 119909) + 119896

119889119909(119889minus )

119910119889= 119896119901119910(119910119889minus 119910) + 119896

119889119910( 119910119889minus 119910)

(17)

where (119896119901119909 119896119889119909) and (119896



119889and 119910


(cos120601119889sin 120579119889cos120595 + sin120601

119889sin120595)119880

1

119898= 119889


119889cos120595)119880

1

119898= 119910119889

(18)


(120579119889cos120595 + 120601

119889sin120595)119880

1

119898= 119889

(120579119889sin120595 minus 120601

119889cos120595)119880

1

119898= 119910119889

(19)


[120601119889

120579119889

] =119898

1198801


minus1

[119889

119910119889

] (20)


1198801=

1

cos(120601) cos(120579)

sdot ((1 + 11989611198962)(119911119889minus 119911) + (119896

1+ 1198962)(119889minus ) + 119898119892)

(21)






0 1 2

0

0 1 2

0

ReferencePDSMC


ReferencePDSMC


0505 1515

minus02

minus01

minus02

minus01

01

02

03

01

02

03

Time (s) Time (s)

120579 (r

ad)

120601(r

ad)



1= 1199092

2= 119891(x) + 119892(x)119906 + 119889

119910 = 1199091

(22)


1=

119897119868119909119909


119889minus 1199091and 119890 = 119910

119889minus 1199092


119906lowast=119868119909119909

119897( 119910119889minus 119891(x) minus 119889 + K119879E) (23)



119890 + 1198702119890 + 1198701= 0 (24)

1198701and 119870









119906 = 119906119899+ 119906119904 (25)



119906lowast

119899=119868119909119909

119897( 119910119889minus 119891(x) + K119879E) (26)


119899is given by

119906119899= H(r c a)120573 (27)



lowast

119899= H(r c a)120573lowast + 120576(r) (28)




that is |120576(r)| le 120576119873


119906119904= 119866119904(x) sgn(119891(E)) (29)







119897] (30)

where

Λ = [0 1


]

B = [[

0

119897

119868119909119909

]

]

(31)



119897] (32)


119881 =1

2E119879PE + 1

2120578119879

(33)



=1

2[E119879PE + E119879PE] + 1

120578

120573119879

= minus1

2E119879QE + (E119879PBH +

1

120578

120573119879

)


119904

(34)

where

Q = minus(Λ119879P + PΛ) (35)


120573119879

= minus120578E119879PBH (36)



119879

= 120578E119879PBH (37)


= minus1

2E119879QE minus E119879PB(119889119868119909119909

119897minus 120576(r)) minus E119879PB119906

119904 (38)



119906119904= (

119889max119868119909119909119897

+ 120576119873)sgn(E119879PB) (39)


le minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus E119879PB((119889max119868119909119909119897

+ 120576119873) sgn(E119879PB))

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(

119889max119868119909119909119897

+ 120576119873)

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889| minus 119889max)119868119909119909

119897

+10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(|120576(r)| minus 120576119873)

le 0

(40)



0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8


minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2


3


1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)


5 Simulations


0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573




1199091199091676119890 minus 2 kgsdotm2






01 kgsdotm2



119894= 1205878 120579

119894= 1205878 120595

119894=


119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

y

z

Z

XY

E

B



119879



120596 = R119903120578 (10)

where

R119903= [

[


]

]

(11)






[120601 120601 120579 120579 120595 119911 119909 119910 119910]119879

where 120601 120579 and 120595


1 Ω2 Ω3



1198801= 119896119865(Ω2

1+ Ω2

2+ Ω2

3+ Ω2

4)

1198802= 119896119865(minusΩ2

2+ Ω2

4)

1198803= 119896119865(Ω2

1minus Ω2

3)

1198804= 119896119872(Ω2

1minus Ω2

2+ Ω2

3minus Ω2

4)

(13)where 119896

119865and 119896




4are the



X = f(XU) (14)where

f(XU) =

((((((((((((((((((((((

(

120601

1205791198861+ 1205791198862Ω119903+ 11988711198802

120579

1206011198863minus 1206011198864Ω119903+ 11988721198803

120579 1206011198865+ 11988731198804

119892 minus(cos120601 cos 120579)119880

1

119898


119898

119910


119898

))))))))))))))))))))))

)

(15)

1198861=

(119868119910119910minus 119868119911119911)

119868119909119909

1198862=119869119903

119868119909119909

1198863=(119868119911119911minus 119868119909119909)

119868119910119910

1198864=119869119903

119868119910119910

1198865=

(119868119909119909minus 119868119910119910)

119868119911119911

1198871=

119897

119868119909119909

1198872=

119897

119868119910119910

1198873=1

119868119911119911

(16)

and 119868119909119909 119868119910119910 and 119868





Positioncontroller

Desiredtrajectory

Attitudecontroller

Rotors

speed

generation

Quadrotordynamics

Ω1

Ω2

Ω3

Ω4

x x y y z z

U1

U2 U3 U4

120601d 120579d 120595d

120601 120579 120595 120601 120579 120595

xd xdyd yd

zd zd120601d120579d 120595d









4are




119889= 119896119901119909(119909119889minus 119909) + 119896

119889119909(119889minus )

119910119889= 119896119901119910(119910119889minus 119910) + 119896

119889119910( 119910119889minus 119910)

(17)

where (119896119901119909 119896119889119909) and (119896



119889and 119910


(cos120601119889sin 120579119889cos120595 + sin120601

119889sin120595)119880

1

119898= 119889


119889cos120595)119880

1

119898= 119910119889

(18)


(120579119889cos120595 + 120601

119889sin120595)119880

1

119898= 119889

(120579119889sin120595 minus 120601

119889cos120595)119880

1

119898= 119910119889

(19)


[120601119889

120579119889

] =119898

1198801


minus1

[119889

119910119889

] (20)


1198801=

1

cos(120601) cos(120579)

sdot ((1 + 11989611198962)(119911119889minus 119911) + (119896

1+ 1198962)(119889minus ) + 119898119892)

(21)






0 1 2

0

0 1 2

0

ReferencePDSMC


ReferencePDSMC


0505 1515

minus02

minus01

minus02

minus01

01

02

03

01

02

03

Time (s) Time (s)

120579 (r

ad)

120601(r

ad)



1= 1199092

2= 119891(x) + 119892(x)119906 + 119889

119910 = 1199091

(22)


1=

119897119868119909119909


119889minus 1199091and 119890 = 119910

119889minus 1199092


119906lowast=119868119909119909

119897( 119910119889minus 119891(x) minus 119889 + K119879E) (23)



119890 + 1198702119890 + 1198701= 0 (24)

1198701and 119870









119906 = 119906119899+ 119906119904 (25)



119906lowast

119899=119868119909119909

119897( 119910119889minus 119891(x) + K119879E) (26)


119899is given by

119906119899= H(r c a)120573 (27)



lowast

119899= H(r c a)120573lowast + 120576(r) (28)




that is |120576(r)| le 120576119873


119906119904= 119866119904(x) sgn(119891(E)) (29)







119897] (30)

where

Λ = [0 1


]

B = [[

0

119897

119868119909119909

]

]

(31)



119897] (32)


119881 =1

2E119879PE + 1

2120578119879

(33)



=1

2[E119879PE + E119879PE] + 1

120578

120573119879

= minus1

2E119879QE + (E119879PBH +

1

120578

120573119879

)


119904

(34)

where

Q = minus(Λ119879P + PΛ) (35)


120573119879

= minus120578E119879PBH (36)



119879

= 120578E119879PBH (37)


= minus1

2E119879QE minus E119879PB(119889119868119909119909

119897minus 120576(r)) minus E119879PB119906

119904 (38)



119906119904= (

119889max119868119909119909119897

+ 120576119873)sgn(E119879PB) (39)


le minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus E119879PB((119889max119868119909119909119897

+ 120576119873) sgn(E119879PB))

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(

119889max119868119909119909119897

+ 120576119873)

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889| minus 119889max)119868119909119909

119897

+10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(|120576(r)| minus 120576119873)

le 0

(40)



0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8


minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2


3


1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)


5 Simulations


0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573




1199091199091676119890 minus 2 kgsdotm2






01 kgsdotm2



119894= 1205878 120579

119894= 1205878 120595

119894=


119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Positioncontroller

Desiredtrajectory

Attitudecontroller

Rotors

speed

generation

Quadrotordynamics

Ω1

Ω2

Ω3

Ω4

x x y y z z

U1

U2 U3 U4

120601d 120579d 120595d

120601 120579 120595 120601 120579 120595

xd xdyd yd

zd zd120601d120579d 120595d









4are




119889= 119896119901119909(119909119889minus 119909) + 119896

119889119909(119889minus )

119910119889= 119896119901119910(119910119889minus 119910) + 119896

119889119910( 119910119889minus 119910)

(17)

where (119896119901119909 119896119889119909) and (119896



119889and 119910


(cos120601119889sin 120579119889cos120595 + sin120601

119889sin120595)119880

1

119898= 119889


119889cos120595)119880

1

119898= 119910119889

(18)


(120579119889cos120595 + 120601

119889sin120595)119880

1

119898= 119889

(120579119889sin120595 minus 120601

119889cos120595)119880

1

119898= 119910119889

(19)


[120601119889

120579119889

] =119898

1198801


minus1

[119889

119910119889

] (20)


1198801=

1

cos(120601) cos(120579)

sdot ((1 + 11989611198962)(119911119889minus 119911) + (119896

1+ 1198962)(119889minus ) + 119898119892)

(21)






0 1 2

0

0 1 2

0

ReferencePDSMC


ReferencePDSMC


0505 1515

minus02

minus01

minus02

minus01

01

02

03

01

02

03

Time (s) Time (s)

120579 (r

ad)

120601(r

ad)



1= 1199092

2= 119891(x) + 119892(x)119906 + 119889

119910 = 1199091

(22)


1=

119897119868119909119909


119889minus 1199091and 119890 = 119910

119889minus 1199092


119906lowast=119868119909119909

119897( 119910119889minus 119891(x) minus 119889 + K119879E) (23)



119890 + 1198702119890 + 1198701= 0 (24)

1198701and 119870









119906 = 119906119899+ 119906119904 (25)



119906lowast

119899=119868119909119909

119897( 119910119889minus 119891(x) + K119879E) (26)


119899is given by

119906119899= H(r c a)120573 (27)



lowast

119899= H(r c a)120573lowast + 120576(r) (28)




that is |120576(r)| le 120576119873


119906119904= 119866119904(x) sgn(119891(E)) (29)







119897] (30)

where

Λ = [0 1


]

B = [[

0

119897

119868119909119909

]

]

(31)



119897] (32)


119881 =1

2E119879PE + 1

2120578119879

(33)



=1

2[E119879PE + E119879PE] + 1

120578

120573119879

= minus1

2E119879QE + (E119879PBH +

1

120578

120573119879

)


119904

(34)

where

Q = minus(Λ119879P + PΛ) (35)


120573119879

= minus120578E119879PBH (36)



119879

= 120578E119879PBH (37)


= minus1

2E119879QE minus E119879PB(119889119868119909119909

119897minus 120576(r)) minus E119879PB119906

119904 (38)



119906119904= (

119889max119868119909119909119897

+ 120576119873)sgn(E119879PB) (39)


le minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus E119879PB((119889max119868119909119909119897

+ 120576119873) sgn(E119879PB))

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(

119889max119868119909119909119897

+ 120576119873)

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889| minus 119889max)119868119909119909

119897

+10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(|120576(r)| minus 120576119873)

le 0

(40)



0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8


minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2


3


1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)


5 Simulations


0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573




1199091199091676119890 minus 2 kgsdotm2






01 kgsdotm2



119894= 1205878 120579

119894= 1205878 120595

119894=


119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2

0

0 1 2

0

ReferencePDSMC


ReferencePDSMC


0505 1515

minus02

minus01

minus02

minus01

01

02

03

01

02

03

Time (s) Time (s)

120579 (r

ad)

120601(r

ad)



1= 1199092

2= 119891(x) + 119892(x)119906 + 119889

119910 = 1199091

(22)


1=

119897119868119909119909


119889minus 1199091and 119890 = 119910

119889minus 1199092


119906lowast=119868119909119909

119897( 119910119889minus 119891(x) minus 119889 + K119879E) (23)



119890 + 1198702119890 + 1198701= 0 (24)

1198701and 119870









119906 = 119906119899+ 119906119904 (25)



119906lowast

119899=119868119909119909

119897( 119910119889minus 119891(x) + K119879E) (26)


119899is given by

119906119899= H(r c a)120573 (27)



lowast

119899= H(r c a)120573lowast + 120576(r) (28)




that is |120576(r)| le 120576119873


119906119904= 119866119904(x) sgn(119891(E)) (29)







119897] (30)

where

Λ = [0 1


]

B = [[

0

119897

119868119909119909

]

]

(31)



119897] (32)


119881 =1

2E119879PE + 1

2120578119879

(33)



=1

2[E119879PE + E119879PE] + 1

120578

120573119879

= minus1

2E119879QE + (E119879PBH +

1

120578

120573119879

)


119904

(34)

where

Q = minus(Λ119879P + PΛ) (35)


120573119879

= minus120578E119879PBH (36)



119879

= 120578E119879PBH (37)


= minus1

2E119879QE minus E119879PB(119889119868119909119909

119897minus 120576(r)) minus E119879PB119906

119904 (38)



119906119904= (

119889max119868119909119909119897

+ 120576119873)sgn(E119879PB) (39)


le minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus E119879PB((119889max119868119909119909119897

+ 120576119873) sgn(E119879PB))

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(

119889max119868119909119909119897

+ 120576119873)

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889| minus 119889max)119868119909119909

119897

+10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(|120576(r)| minus 120576119873)

le 0

(40)



0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8


minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2


3


1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)


5 Simulations


0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573




1199091199091676119890 minus 2 kgsdotm2






01 kgsdotm2



119894= 1205878 120579

119894= 1205878 120595

119894=


119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











that is |120576(r)| le 120576119873


119906119904= 119866119904(x) sgn(119891(E)) (29)







119897] (30)

where

Λ = [0 1


]

B = [[

0

119897

119868119909119909

]

]

(31)



119897] (32)


119881 =1

2E119879PE + 1

2120578119879

(33)



=1

2[E119879PE + E119879PE] + 1

120578

120573119879

= minus1

2E119879QE + (E119879PBH +

1

120578

120573119879

)


119904

(34)

where

Q = minus(Λ119879P + PΛ) (35)


120573119879

= minus120578E119879PBH (36)



119879

= 120578E119879PBH (37)


= minus1

2E119879QE minus E119879PB(119889119868119909119909

119897minus 120576(r)) minus E119879PB119906

119904 (38)



119906119904= (

119889max119868119909119909119897

+ 120576119873)sgn(E119879PB) (39)


le minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus E119879PB((119889max119868119909119909119897

+ 120576119873) sgn(E119879PB))

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889|119868119909119909

119897+ |120576(r)|)

minus10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(

119889max119868119909119909119897

+ 120576119873)

= minus1

2E119879QE + 10038161003816100381610038161003816E

119879PB10038161003816100381610038161003816(|119889| minus 119889max)119868119909119909

119897

+10038161003816100381610038161003816E119879PB10038161003816100381610038161003816(|120576(r)| minus 120576119873)

le 0

(40)



0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8


minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2


3


1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)


5 Simulations


0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573




1199091199091676119890 minus 2 kgsdotm2






01 kgsdotm2



119894= 1205878 120579

119894= 1205878 120595

119894=


119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2

0

2

4

6

8

PDSMC


PDSMC


0 1 2

0

2

4

6

8


minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

U3

U2


3


1198802= H120601(r120601 c120601 a120601)120573120601

+ (119889max119868119909119909

119897+ 120576119873)sgn(E119879

120601P120601B120601)

1198803= H120579(r120579 c120579 a120579)120573120579

+ (

119889max119868119910119910

119897+ 120576119873)sgn(E119879

120579P120579B120579)

1198804= H120595(r120595 c120595 a120595)120573120595

+ (119889max119868119911119911 + 120576119873)sgn(E119879

120595P120595B120595)

(41)


5 Simulations


0 1 2

3


Time (s)02 04 06 08 12 14 16 18

26

265

27

275

28

285

29

295

120573




1199091199091676119890 minus 2 kgsdotm2






01 kgsdotm2



119894= 1205878 120579

119894= 1205878 120595

119894=


119905= minus12058716 120595

119905= 0)


0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

ReferencePDSMC


0 1 2 3 4 5

0

minus02

minus03

minus04

minus01

Time (s)Time (s)

03

02

01

04

0

minus02

minus03

minus04

minus01

03

02

01

04

120579 (r

ad)

ReferencePDSMC


120601(r

ad)



119889=





1= 8 119870

2= 128 Q =

diag(100 100) 120578 = 005 and 120576119873



119901120601= 5 119896

119889120601= 12

119896119901120579= 5 119896119889120579= 12 119896

119901120595= 5 and 119896











0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

0

005

015

0 1 2 3 4 5

0

005

015

PDSMC


PDSMC


Time (s) Time (s)

02

01

01

minus02

minus015

minus01

minus005

minus02

minus015

minus01

minus005

120579tr

acki

ng er

rors

(rad

)

120601 tr

acki

ng er

rors

(rad

)


0 1 2 3 4 505 15 25 35 4526

27

28

29

3

31

32

33


Time (s)

120573


0 1 2 3 4 5 60

1

2

3

4

5

6

ReferencePDSMC


x (m)

y(m

)



6 Conclusions



0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

0

0 1 2 3 4 5

0

minus02

minus02

minus03

minus04

minus05

minus01

minus01

01

01

02

02

03

04

05

Time (s)Time (s)

PDSMC


PDSMC


Erro

rs in

Ydi

rect

ion

Erro

rs in

Xdi

rect

ion


0 1 2 3 4 5

2

Time (s)05 15 25 35 45

19

195

205

23

215

22

225

21

235


120573


0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

ReferencePDSMC


X (m)

Y(m

)


0 2 4 6 8 10 12 14 16 18 20

2

3

4

Time (s)

15

25

35


120573





Acknowledgments



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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