Adaptive feedback linearizing control of nonholonomic wheeled mobile robots in presence of parametric and nonparametric uncertainties

Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

Adaptive feedback linearizing control of nonholonomic wheeled mobile robots in presence of parametric and nonparametric uncertainties Khoshnam Shojaei, Alireza Mohammad Shahri n, Ahmadreza Tarakameh Mechatronics and Robotics Research Laboratory, Electronic Research Center, Electrical Engineering Department, Iran University of Science and Technology, Tehran, Iran

a r t i c l e in f o

a b s t r a c t

Article history: Received 3 August 2009 Received in revised form 3 July 2010 Accepted 13 July 2010

In this paper, the integrated kinematic and dynamic trajectory tracking control problem of wheeled mobile robots (WMRs) is addressed. An adaptive robust tracking controller for WMRs is proposed to cope with both parametric and nonparametric uncertainties in the robot model. At first, an adaptive nonlinear control law is designed based on input–output feedback linearization technique to get asymptotically exact cancellation of the parametric uncertainty in the WMR parameters. The designed adaptive feedback linearizing controller is modified by two methods to increase the robustness of the controller: (1) a leakage modification is applied to modify the integral action of the adaptation law and (2) the second modification is an adaptive robust controller, which is included to the linear control law in the outer loop of the adaptive feedback linearizing controller. The adaptive robust controller is designed such that it estimates the unknown constants of an upper bounding function of the uncertainty due to friction, disturbances and unmodeled dynamics. Finally, the proposed controller is developed for a type (2, 0) WMR and simulations are carried out to illustrate the robustness and tracking performance of the controller. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Adaptive robust Feedback linearization Parametric uncertainty Trajectory tracking WMR

1. Introduction The problem of motion control of Wheeled Mobile Robots (WMRs) is extensively studied in past few decades [1,4,5,8,10]. An important motion control problem is the trajectory tracking which is concerned with the design of a controller to force a WMR to track a geometric path with an associated timing law [7]. A variety of control algorithms for trajectory tracking problem are developed in the literature [14,15,17,18,20,21]. Because of the challenging nonlinear model of WMRs, the feedback linearization technique is one of the successful design approaches to solve this problem. There are many works that propose tracking controllers based on feedback linearization for WMRs [2,3,6,11,14,16,27]. Campion et al. [27] investigated the controllability and feedback linearizability of the nonholonomic systems. Andrea-Novel et al. [13] applied the linearization technique to achieve tracking control of mobile robots. In [16], a tracking controller is proposed based on input–output feedback linearization for a nonsquare WMR system. Oriolo et al. [14] presented a design and experimental validation of dynamic feedback linearization to solve the trajectory tracking problem. However, most of them ignore the WMR dynamics in the design of controllers, which are inefficient n

Corresponding author. Tel.: +98 21 77240492 3; fax: +98 21 77240490. E-mail addresses: [email protected], [email protected], [email protected] (K. Shojaei), [email protected] (A. Mohammad Shahri), [email protected] (A. Tarakameh). 0736-5845/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2010.07.007

for high speed massive WMRs. In addition, the works which propose the feedback linearization controllers for both kinematic and dynamic models of the WMRs mostly apply exact models and ignore their parametric uncertainties (for example, see [2]). This problem may cause not to achieve the exact cancellation of nonlinearities in the WMR model by input–output feedback linearization technique. Fortunately, adaptive control strategies present a reasonable solution to overcome parametric uncertainties. There are many key works to address the problem of tracking control of feedback linearizable systems [9]. However, the authors believe that the adaptive version of input–output feedback linearization control, as a powerful technique, has not been paid enough attention to solve the trajectory tracking problem of WMRs. Note that the main drawback of this control strategy is that the inversion of the estimate of decoupling matrix may not exist when the parameter estimates tend to zero. Therefore, this may lead to the divergence of tracking errors. This problem was solved by a technique, which restricts the parameters estimate to lie within some prior bounds [23,26]. Another problem is that the integral-type of adaptive laws may lose their stability in presence of nonparametric uncertainties such as disturbances. Robust control strategies can modify the adaptive control law to overcome nonparametric uncertainties. This modification may be carried out in the design stage of the adaptation or control law. The main contribution and novelty of the present work lies in designing an adaptive input–output feedback linearizing controller to solve the integrated kinematic and dynamic

K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

trajectory tracking problem of WMRs. The proposed controller is modified by: (1) a leakage modification on the parameter update rule to avoid parameters drift due to the nonparametric uncertainties, (2) a robust controller with an adaptive upper bounding function to compensate for the nonparametric uncertainties, which are motivated from the textbook of Lewis et al. [19] on the robot manipulators. Consequently, the formulation of the adaptive control law is also developed for the type (2, 0) WMR. Furthermore, in contrast to previous works, our proposed controller provides an actuator-level control signal from a practical viewpoint. The rest of the paper is structured as follows. After a review of the kinematic and dynamic model of WMRs in Section 2, the tracking controller is proposed based on SPR-Lyapunov design approach in Section 3. The adaptive tracking controller is modified to be robust against nonparametric uncertainties in Section 4. Simulation results are presented for type (2, 0) WMR to illustrate the robustness and tracking performance of the proposed controllers in Section 5. Finally, conclusion and future works are presented in Section 6.

2. Kinematic and dynamic model of a nonholonomic WMR In this section, we review a mathematical formulation of wheeled mobile robots with nonholonomic constraints which are moving on a planar surface. It is assumed that the configuration of the WMR is described by n generalized coordinates, q, subject to m constraints (mon) as follows: _ ¼ Ck,j ðq, qÞ

n X

ck,ji ðqÞq_ i ¼ 0,

j ¼ 1,:::,m

ð1Þ

i¼1

where it includes k holonomic and m k nonholonomic constraints, which all of them may be written in the form of Ak ðqÞq_ ¼ 0

ð2Þ

195

_ A Rnn a matrix where M1(q)ARn  n is the inertia matrix, C1 ðq, qÞ _ A RðnmÞ1 which denotes the coriolis and centripetal forces. FðqÞ denotes the friction vector. B1(q)ARn  (n  m) is the input transformation matrix, tAR(n  m)  1 the torque vector which is generated by wheels actuators, tdAR(n  m)  1 denotes the bounded unknown disturbances, and lARm  1 the vector of constraint forces. Property 1. M1(q) is a symmetric and positive-definite matrix which is upper and lower bounded, that is, m1 r JM1 ðqÞJ r m2 , where m1and m2 are positive scalar constants. _ A RðnmÞ1 , includes Remark 1. The friction term in Eq. (7),FðqÞ viscous and dynamic frictions in concern with the robot wheels _ rf1 :q:þ _ such that :FðqÞ: f2 , where f1 and f2 are positive constants. The disturbance vector, tdAR(n  m)  1, may include unstructured unmodeled dynamics, for example, dynamics of the castor wheel, power amplifiers for actuators and sensor dynamics such that Jtd J r t1 where t1AR is the upper bound of td. To include actuator dynamics in (7), it is assumed that the robot wheels are driven by n m similar brush DC motors with mechanical gears. Fig. 1 shows the simplified drive system. The electrical equation of the motor armature is written as follows: ua ¼ La

dia þRa ia þ Kb y_ M dt

ð8Þ

where Kb is the back EMF constant. The parameters La and Ra denote the inductance and impedance of the armature circuit, respectively. By ignoring the armature inductance, and considering the relation between torque and armature current (i.e. tM ¼Ktia) and relations between torque and velocity before and after gears (i.e. t ¼n tM and y_ M ¼ n y_ ), the delivered torque to WMR wheels by actuators is given by

t ¼ K1 ua K2 y_

ð9Þ

where Ak ðqÞ A Rmn is a full-rank matrix. Assume that S(q)¼ [s1(q),...,sn  m(q)]T is also a full-rank matrix that is made up of a set of smooth and linearly independent vector fields, si(q)ARn, i¼1,..., n  m, in the null space of Ak(q).(See [2] for more details), i.e. Ak ðqÞSðqÞ ¼ 0

ð3Þ

According to (2) and (3), it is possible to write the kinematic equation of WMR motion in terms of an auxiliary vector time function v(t)ARn  m which is called pseudo-velocity vector as q_ ¼ SðqÞvðtÞ

DC motor

ð4Þ T

where v(t)¼[v1(t),y,vn  m(t)] . The WMR dynamic model is derived by Lagrangian mechanics. First, the Lagrangian L which is the difference between kinetic and potential energy of the system, must be calculated. Because of the planar motion, the potential energy of the WMR is zero. Therefore, the Lagrangian is only equal to kinetic energy

θM

Robot Wheel

τM θ

n

L¼

i 1X ½vT m v þ oTi Ii oi  2i¼1 i i i

ð5Þ

Then, one may use Euler–Lagrange equation incorporating velocity constraints in the following form:   d @L @L ¼ FG ð6Þ  dt @q_ i @qi where FG denotes the generalized forces. After calculating (6), the dynamic model of WMR may be written as follows: _ q_ þB1 ðqÞFðqÞ _ þ B1 ðqÞtd ¼ B1 ðqÞtAk ðqÞT l M1 ðqÞq€ þC1 ðq, qÞ

ð7Þ

τ

nM :nw

direction of movement Fig. 1. Drive system for each wheel.

196

K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

where K1 ¼(nKt/Ra), K2 ¼nKbK1, n ¼nw/nM are the gear ratios and Kt the torque constant of the motor. Eq. (9) may be rewritten as

where xARn and f(x), q(x, y), gi(x, y) and x(x,t) are smooth vector fields on Rn with g(0, y) a0.

t ¼ K1 ua K2 Xv,

Remark 4. Based on the study of WMRs dynamics shown in (15), the following results might be summarized:

ð10Þ

(n  m)  (n  m)

where XAR is a transformation matrix which transforms wheel velocities to pseudo-velocities vector. Substituting (10) in (7) results in T

_ q_ þB1 ðqÞFðqÞ _ þ B1 ðqÞtd ¼ B1 ðqÞðK1 ua K2 XvÞAk ðqÞ l M1 ðqÞq€ þ C1 ðq, qÞ ð11Þ For controller design purposes, the state space representation can be derived by taking time derivative of the kinematic model (4) _ q€ ¼ SðqÞv þSðqÞv_

ð12Þ

Next, by replacing (4) and (12) in (11) and multiplying the result by ST and considering (3), one obtains _ _ þ td ¼ K1 B1 ua _ þ C 1 ðqÞvðtÞ M 1 vðtÞ þFðqÞ

ð13Þ

1. The system is controllable and its equilibrium point xe ¼0 can be made Lyapunov stable, but cannot be made asymptotically stable by a smooth state feedback [24]. 2. The internal dynamics of a WMR is stable, when the mobile robot moves forward but is unstable when it moves backward [25]. 3. If at least one constraint is nonholonomic, it has been shown that the WMR system is not input-state linearizable. But if we choose a proper set of output equations, it may be input– output linearizable [2,12]. 4. The largest feedback linearizable subsystem of system (15) has dimension 2(n  m) and the relative degree of the system with respect to each output is 2 [27].

where M 1 ¼ ST M1 S, T

B1 ¼ S B1 ,

_ ¼ ST M1 S_ þ ST CS þK2 B1 X, C 1 ðqÞ

td ¼ B1 td

F ¼ B1 F,

ð14Þ

The kinematic model (4) and dynamic equation shown by (13) can be integrated into the following state space representation in companion form: # # " " #   " 0 0 Sv q_ 1 1 1 x_ ¼ ua þ þ ¼ _ þ td Þ K1 M 1 B 1 M 1 ðFðqÞ M 1 C 1 v v_ ð15Þ (2n  m)

where xAR is the state vector. This representation allows us to apply the differential geometric control theory to solve the trajectory tracking problem. Property 2. M 1 ðqÞ is a symmetric and positive-definite matrix, which is upper and lower bounded, that is, m1 r:M 1 ðqÞ: rm2 , where m1 and m2 are positive scalar constants. Remark 2. In the derived dynamic model, it is assumed the dynamics of the un-powered castor wheels are ignored to reduce the complexity of the model. However, this assumption imposes some source of uncertainty to the WMR system.

The output equations are functions of position state variables q. Since the number of degrees of freedom of the WMR system is n m, we have n  m independent position output equations y ¼ hðxÞ ¼ ½h1 ðqÞ,. . .,hnm ðqÞT

ð17Þ

Definition 1. Given a smooth bounded reference trajectory yr(t) ¼h(qr(t)), which is generated by a reference mobile robot, and supposing that qr satisfies the velocity constraintsAðqr Þ q_ r ¼ 0, then the integrated kinematic and dynamic tracking control problem is to design a feedback control for the system (15) and (17) such that it satisfies lim ðyðtÞyr ðtÞÞ ¼ 0

ð18Þ

t-1

The basic approach to obtain a linear input–output relation is to repeatedly differentiate the outputs so that they are explicitly related to inputs. After differentiating, one obtains y_ j ¼ Lf hj þ Lq hj þLx hj þ

n m X

ðLgi hj Þuai ¼ Jhj Sv,

j ¼ 1,2,:::,nm

ð19Þ

i¼1

Remark 3. In this paper, it is assumed that parametric uncertainties are caused by inaccurate measurement of the WMR parameters such as mass, moment of inertia and actuators parameters. Moreover, some parameters may be time-varying. For example, mass and moment of inertia may vary because of loading or unloading of some objects on the WMR chassis. Nonparametric uncertainties may be caused by unmodeled dynamics of the system such as castor wheels of the WMR, nonidealities of the mechanical system such as backlash, viscous and dynamic frictions which are related to the robot wheels and wheel slippage, etc. However, the kinematic perturbation [30] due to slippage of wheels is not considered as nonparametric uncertainties in designing of the controller. 3. Controller design A trajectory tracking control law can be designed based on adaptive feedback linearization technique for the WMR system as given in (15). The presented system in (15) might be summarized as the following affine MIMO nonlinear model: x_ ¼ f ðxÞ þqðx, yÞ þ

n m X i¼1

gi ðx, yÞuai þ xðx,tÞ

ð16Þ

which it is not related to the actuators input. By differentiating again, it yields y€ j ¼ L2f hj þ Lf Lq hj þ Lf Lx hj þ Lq Lf hj þ L2q hj þ Lq Lx hj þ Lx Lf hj þ Lx Lq hj þL2x hj þ

n m X

Lgi ðLf hj Þuai þ

i¼1

n m X

Lgi ðLq hj Þuai þ

i¼1

n m X

!

Lgi ðLx hj Þuai

i¼1

ð20Þ where it is obvious that Lgi(Lfhj)a0. After some simplifications, we may rewrite (20) for the entire system as y€ ¼ L2f hðxÞ þ Lq Lf hðxÞ þLx Lf hðxÞ þ Lg Lf hðxÞ:ua where Lg Lf hðxÞ : ¼ DðxÞ 2 ... Lg1 Lf h1 6 & DðxÞ ¼ 4 ^ Lg1 Lf hnm . . .

is defined as the decoupling matrix 3 Lgnm Lf h1 7 ^ 5 Lgnm Lf hnm

ð21Þ

ð22Þ

Assuming the condition det(D(x)a0) is satisfied, the system (16) is input–output linearizable. The following nonlinear feedback: ua ¼ D1 ðxÞðZL2f hðxÞLq Lf hðxÞÞ,

ð23Þ

K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

linearizes and decouples the WMR system into n  m double integrators as follows: 2 3 2 3 2 3 Lx Lf h1 Z1 y€ 1 6€ 7 6 7 6L L h 7 6 y 2 7 6 Z2 7 6 x f 2 7 6 7¼6 7þ6 7 ð24Þ 6^ 7 6^ 7 6^ 7 4 5 4 5 4 5 Lx Lf hnm Znm y€ nm where Zj, j ¼1,2,y,n m represents the new external inputs. Now, it is assumed that there are p uncertain parameters in the WMR model and there is no nonparametric uncertainty, i.e. LxLfh(x) ¼0. Theorem 1 is presented to solve the integrated kinematic and dynamic tracking control problem of WMRs in presence of parametric uncertainty based on the following Lemma and assumptions; Assumption 1. Measurements of all states, i.e. x ¼[qT,vT]T, are available in real-time. Assumption 2. Pseudo-velocities, i.e. v(t)¼[v1(t),y,vn  m(t)]T, are bounded for all time t40. +

Lemma 1. Given a differentiable function f(t):R -R, if f(t)AL2 and f_ ðtÞ A L1 , then f(t) tends to zero as t-N, where LN denotes bounded function set and L2 denotes square integrable function set [28]. Theorem 1. Provided that the reference trajectory yr(t) is selected to be bounded for all times t40, and under Assumptions 1 and 2, the following adaptive tracking controller guarantees that all signals in the closed-loop system are bounded and the tracking error e(t) ¼y(t)  yr(t) converges to zero as t-N ^ 1 ðxÞðZL2 hLq L^ f hÞ, ua ¼ D f _

y^ ¼ GW T E1

ð25Þ

where WAR(n  m)  p is the regression matrix, E1ARn  m a vector of filtered error signals and GAR   p a symmetric and positive definite matrix as the adaptive gain. b1AR(n  m)  (n  m) and b2AR(n  m)  (n  m) are the diagonal matrices, which denote derivative and proportional gains of the linear control law for the entire system, respectively. Proof. According to Certainty Equivalence principle, we need to replace D(x) and LqLfh(x) by their estimates in decoupling control law (23)& ^ ua ¼ D

1

ðxÞðZL2f hLq L^ f hÞ,

ð26Þ

Zj ¼ y€ jr þ b1j ðy_ jr y_ j Þ þ b2j ðyjr yj Þ, j ¼ 1,2,. . .nm

^ DðxÞ ¼ Lg^ Lf hðxÞ,Lq L^ f h ¼ Lq^ Lf h

ð27Þ

By substituting (26) in (21), we have ^ 1 ðxÞðZL2 hLq L^ f hÞ y€ ¼ L2f hðxÞ þ Lq Lf hðxÞ þ DðxÞD f

ð28Þ

After some manipulation, Eq. (28) may easily be written in the following form: ~ D ^ 1 ðxÞðZL2 hLq L^ f hÞ y€ ¼ Z þ Lq L~ f hðxÞ þ DðxÞ f

ð29Þ

e€ j þ b1j e_ j þ b2j ej ¼ Wj y~

ð33Þ

where Wj ¼[wj1, wj2,y,wjp] is the jth row of regression matrix. For the purpose of adaptation, one may use the following filtered error signal for jth output:

ej ¼ e_ j þ aj ej

ð34Þ

Since e_ j ¼ y_ j y_ jr is known as a function of measured states by considering (19), it is obvious that ej is available. The parameter aj is chosen such that the following transfer function is strictly positive real (SPR): Tj ðsÞ ¼

s þ aj s2 þ b1j s þ b2j

ð35Þ

This means that Tj(s) is analytic in the closed right-half plane and Re(Tj(jo))40. Accordingly, by positive real lemma [23], there exist the positive definite matrices Pj and Qj such that ATj Pj þ Pj Aj ¼ Qj , ð36Þ

where matrices Aj, Bj and Cj are defined by minimal state space realization of (33) and (34) in the following form: X_ j ¼ Aj Xj þBj Wj y~

ej ¼ Cj Xj

ð37Þ

where Xj ¼ ½ej , e_ j T is the state variable and " #   h i 0 1 0 Aj ¼ , Bj ¼ , Cj ¼ aj 1 b2j b1j 1

ð38Þ

As a result, the entire system error equation may be written as X_ ¼ AX þ BW y~ ð39Þ

where AAR2(n m)  2(n m), BAR2(n m)  (n m) and CAR(n m)  2(n m) are block diagonal matrices A ¼ diagðA1 ,A2 ,. . .,Anm Þ, C ¼ diagðC1 ,C2 ,. . .,Cnm Þ,

B ¼ diagðB1 ,B2 ,. . .,Bnm Þ, ð40Þ

T T . The Lyapunov Eq. (36) is also written for and X ¼ ½X1T ,X2T ,. . .,Xnm entire system as follows:

AT P þ PA ¼ Q ,

where

PB ¼ C T ~ ^ DðxÞ ¼ DðxÞDðxÞ

ð32Þ

This yields the following error equation:

E1 ¼ CX

where

Lq L~ f hðxÞ ¼ Lq Lf hðxÞLq L^ f hðxÞ,

where y~ ¼ ½y~ 1 , y~ 2 ,. . ., y~ p T is the vector of parameters’ estimates errors and the matrix WAR(n  m)  p is the regression matrix, which is made up of time known functions which are assumed to be bounded. Now, the adaptive law may be derived by SPR-Lyapunov design approach which is motivated from Sastry and Bodson [22], Ioannou [23] and the work of Craig [26]. Assume that the external control input Zj for jth subsystem of (31) is chosen so that jth output, yj(t), tracks the desired output, yjr(t), in the outer loop

Pj Bj ¼ CjT

_ þ b2 ðyr yÞ, Z ¼ y€ r þ b1 ðy_ r yÞ

197

ð41Þ

ð30Þ

Then, one may readily derive the following parametric model from (29):

where,

y€ ¼ Z þ W y~

P ¼ diagðP1 ,P2 ,. . .,Pnm Þ,

ð31Þ

Q ¼ diagðQ1 ,Q2 ,. . .,Qnm Þ,

ð42Þ

198

K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

Now, one may define the following Lyapunov function to derive the adaptive law: T VðX, y~ Þ ¼ X T PX þ y~ G1 y~

ð43Þ

By taking time derivative of the proposed function and applying (39) and (41), we may write T _ V_ ðX, y~ Þ ¼ X T QX þ 2y~ ðW T E1 þ G1 y~ Þ

ð44Þ

One may choose _ y~ ¼ GW T E

ð45Þ

1

to be assured that the derivative of Lyapunov function is negative _ _ definite. Since y is a constant parameter, then y~ ¼ y^ and the

To achieve robustness to uncertainty d, following modifications may be applied to the proposed controller. 4.1. Modification of the adaptation law This modification seems to be necessary to avoid parameter drift due to the uncertainty d. Following theorem is presented to increase the robustness of the adaptation law in (25). Theorem 2. The following adaptive law in the proposed controller of Theorem 1 guarantees that the tracking errors and parameter estimation errors are uniformly ultimately bounded. _ _ y^ ¼ y~ ¼ GW T E1 GSy^ , ð51Þ

adaptive law in (25) is easily obtained. As a result, we have V_ ðX, y~ Þ ¼ X T QX r lmin ðQ ÞJXJ2

ð46Þ

This means that XAL2 and thanks to the Lyapunov theory, we have X, y~ A L1 . Therefore, E1 ¼CXAL2 and E1ALN. By considering _ the adaptive law, we have Jy^ J rJGJ JW T J JE1 J which together _ with E1AL2 and WALN implies that y^ A L2 . Since X, y~ A L1 , then it follows that X_ ¼ AX þBW y~ A L1 and E_ 1 ¼ C X_ A L1 . Finally, since E1 , E_ 1 A L1 and E1AL2, by Lemma 1, we conclude that E1-0 as _ t-N, which, in turn, implies that y^ -0 as t-N. This result shows that the tracking errors ej and e_ j are asymptotically stable. However, from this analysis, we only conclude that the parameter estimation error remains bounded.

_ Proof. After substituting y~ from (51) in (50) and using y^ ¼ yy~ , it yields& T T T V_ ðX, y~ Þ ¼ X T QX þ 2y~ Sy2y~ Sy~ þ 2d E1

Considering the minimum singular values of matrices Q and S, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lmin ðQ T Q Þ and mS ¼ lmin ðST SÞ, and the fact that

i.e. mQ ¼

JdJ r d, we have T V_ ðX, y~ Þ rmQ JXJ2 þ2mS 99y~ 99 99y992mS Jy~ J2 þ 2d JE1 J

ð53Þ

Considering that 1 1 2 2 1 1 99y~ 99 þ k2 99y99  ð 99y~ 99k99y99Þ2 , 2 2 k 2k2 one may write

Remark 5. Note that if WeLN, a similar adaptive scheme may be derived by using normalization techniques to cope with unbounded regression matrix.

99y~ 99 99y99 ¼

Remark 6. As it is standard in the literature, in order that the parameter error converges to zero exponentially fast, the following Persistency of Excitation (PE) condition must be satisfied over any interval of time of length T0 Z t þ T0 aI r WðtÞT WðtÞdt r bI ð47Þ

99y~ 99 99y99 r

ð54Þ

1 1 2 2 99y~ 99 þ k2 99y99 2 2k2

ð55Þ

where kAR + . We may also write T

T

mQ JXJ2 þ 2d JE1 J r mQ JXJ2 þ 2d JCJ JXJ

!2 2 1 1 pffiffiffiffiffiffiffi 2 2 r  mQ JXJ2  mQ JXJ pffiffiffiffiffiffiffi JCJd þ JCJ2 d mQ mQ 2 2

t

where a is the level of excitation and b 40 a constant parameter. Remark 7. The determinant of decoupling matrix in the control law in Eq. (25) may include some of estimated parameters y^ . Hence, prior bounds on these parameters are sufficient to guarantee nonsingularity of the decoupling matrix. As implied in Sastry’s work [9], several techniques exist in the literature for this purpose [22,23]. This remark and Assumptions 1 and 2 imply that WALN.

ð52Þ

2 1 2 r  mQ JXJ2 þ JCJ2 d mQ 2

ð56Þ

Inequalities (55) and (56) may help write (53) as follows: 2 1 1 1 2 2 2 V_ ðX, y~ Þr  mQ JXJ2 2mS Jy~ J2 þ 2mS ð 2 99y~ 99 þ k2 99y99 Þþ JCJ2 d 2 2 mQ 2k

2 1 1 2 2 r mQ JXJ2 2mS ð1 2 ÞJy~ J2 þ mS k2 99y99 þ JCJ2 d 2 mQ 2k

4. Modification for robustness

ð57Þ

In practice, the WMR model is also subjected to nonparametric uncertainties which are described by Remark 3. Thus, we suppose that LxLfh(x)a0 in (21). By following (26) through (31), one may rewrite (31) as y€ ¼ Z þ W y~ þ d

ð48Þ

where d ¼LxLfh(x) denotes the input–output map of the uncertainty x(x,t) in the system (16). Then, by considering (32)–(39), the entire system error equation might be re-written as X_ ¼ AX þBðW y~ þ dÞ E1 ¼ CX

ð49Þ

By defining the following parameters: 1 2

s1 ¼ mQ 4 0, s2 ¼ 2mS ð1

1 Þ 4 0 and 2k2

r ¼ mS k2 99y992 þ

2

mQ

JCJ2 d

2

ð58Þ Inequality (57) can be rewritten as V_ ðX, y~ Þ rs1 JXJ2 s2 Jy~ J2 þ r

ð59Þ

On the other hand, the Lyapunov function in (43) may be stated as VðX, y~ Þ r lmax ðPÞJXJ2 þ lmax ðG1 ÞJy~ J2

ð60Þ

It follows that for By differentiating (43) and substituting (49) in the result, we have ( _ T V_ ðX, y~ Þ ¼ X T QX þ 2y~ ðW T E1 þ G1 y~ Þ þ 2d E1 T

k ¼ min ð50Þ

s1

,

s2

lmax ðPÞ lmax ðG1 Þ

) ,

ð61Þ

K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

Eq. (59) becomes V_ ðX, y~ Þ r kVðX, y~ Þ þ r

ð62Þ

After solving the differential inequality (62), we have VðtÞ r Vð0Þe

kt

þ

r

kt

ð1e

Þ,

8t A ½0,1Þ

k We can utilize (43) to write

VðtÞ Z lmin ðPÞJXJ2 ,

VðtÞ Z lmin ðG1 ÞJy~ J2

V

lmin ðPÞ

2

Jy~ J:: r

,

V

lmin ðG1 Þ

:

ð73Þ

The following theorem is presented to increase the robustness of the proposed adaptive trajectory tracking controller in Theorem 1 against nonparametric uncertainties.

ð64Þ

Noting (63), V is upper bounded which together with (64) results in JXJ2 r

^ ¼ Y a^ the where km is a positive scalar control constant and r estimate of the bounding function. To achieve a^ , an adaptive law is defined as

a_^ ¼ gY T JE1 J, ð63Þ

199

ð65Þ

Since r is bounded, Eq. (62) implies that X and y~ are uniformly ultimately bounded and robustness of the adaptive law is achieved against uncertainty d. This result also shows that the tracking errors ej and e_ j , j ¼1,2,y, n  m, are uniformly ultimately bounded. & Remark 8. The presented modification on adaptive law (51) is called sigma-modification, which is introduced by Ioannou and Sun [23]. The main drawback of such modification is that the size of the ultimate bound of tracking errors depends on external disturbances and it cannot be freely adjusted by control parameters.

Theorem 3. Under Assumptions 1 and 2, the following control law guarantees the asymptotic stability of the tracking errors (e _ in the outer loop of the proposed feedback linearization and e) controller in Theorem 1 _ þ b2 ðyr yÞ þvR , ZR ¼ y€ r þ b1 ðy_ r yÞ ð74Þ where vR is the adaptive robust control law which is defined by (71)–(73). Proof. Let us consider the following Lyapunov function:& V1 ðX, y~ , a~ , mÞ ¼ VðX, y~ Þ þ a~ T g1 a~ þ k1 m m

T _ V_ 1 ðX, y~ , a~ , mÞ ¼ X T QX þ2vTR E1 þ 2d E1 þ 2a~ T g1 a_~ þ k1 m m

ð76Þ

Using (72), the upper bound of d and considering r ¼Ya and

a_~ ¼ gY T :E1 :, V_ 1 ðX, y~ , a~ , mÞ r X T QX þ 2vTR E1 þ 2Y aJE1 J2Y a~ JE1 Jm

4.2. Modification of the control law

ð75Þ

where a~ ¼ aa^ and VðX, y~ Þ are defined in (43). By differentiating (75) and applying (67) and (45), we have

ð77Þ

By substituting (71), we have The linear control law in the outer loop of the feedback linearization controller, i.e. Z, may be robustified to compensate for the uncertainty d as follows. One may write (48) as y€ ¼ ZR þW y~ þ d

ð66Þ

where ZR ¼ Z +vR and vR is a robust control term which is proposed here. Again, by considering (32)–(39) and Eq. (66), the entire system error equation might be re-written as X_ ¼ AX þBðW y~ þ d þ vR Þ E1 ¼ CX

ð67Þ

In order to design vR, we assume that JdJ r rðq,vÞ and r(q,v) is an upper bounding function. Considering (15) and (19), we have 1

_ þ td Þ Lx Lf hðxÞ ¼ Jh ðqÞSðqÞM1 ðFðqÞ

ð68Þ

T T T Jh ðqÞ ¼ ½Jh1 ðqÞ, Jh2 ðqÞ,. . ., Jhnm ðqÞT

where is the Jacobian matrix which is made up of Jacobians with respect to outputs. By considering the mentioned WMR properties and Remark 1, one may conclude that 1

_ þ Jtd JÞ r a1 JvJ þ a2 JdJ rJJh ðqÞSðqÞJM 1 JJðJFðqÞJ

ð69Þ

Therefore, the bounding function is written as rðq,vÞ ¼ a1 JvJ þ a2 , which is defined in the parametric form, r ¼Ya, where  T   ð70Þ Y ¼ JvJ 1 , a ¼ a1 a2 and a is a vector of unknown constants of the bounding function. Then, a robust control vR is proposed as follows: 2

vR ¼ 

^ 2E1 r , ^ JE1 J þ mðtÞ 2r

ð71Þ

where m(t)¼ m0mf(t) is a strictly positive time function, which R1 satisfies the condition t0 mf ðsÞds ¼ Cg o1,8t0 ; for all t Zt0 [29]. One possible choice is

m_ ðtÞ ¼ km mðtÞ, mð0Þ 40,

ð72Þ

4ðY a^ Þ2 ET1 E1 þ 2Y a^ JE1 Jm V_ 1 ðX, y~ , a~ , mÞ rX T QX 2Y a^ JE1 J þ m r X T QX þ

2Y a^ JE1 Jm m 2Y a^ JE1 J þ m

ð78Þ

and thus

2Y a^ JE1 J 1 V_ 1 ðX, y~ , a~ , mÞ r X T QX þ m 2Y a^ JE1 J þ m

ð79Þ

Since the last term is always less than zero, we have V_ 1 ðX, y~ , a~ , mÞ r X T QX

ð80Þ

This means that X, y~ A L1 and a similar argument to Theorem 1 proves that tracking errors are asymptotically stable. Remark 9. When m(t) is selected such that it does not tend to zero in the limit, a sigma-modification of the adaptation law for the estimated parameter a^ in (73) seems to be necessary for robustness enhancement. As a result, the uniform ultimate boundedness of the tracking errors may be achieved. The interested reader is referred to [29] for more details. 5. A design example 5.1. Application to type (2, 0) WMR The configuration of type (2, 0) WMR is shown in Fig. 2. The WMR has two conventional fixed wheels on a single common axle and a castor wheel to maintain the equilibrium of the robot. The centre of mass of the robot is located in PC ¼(xC,yC). The point P0 ¼(xO,yO) is the origin of the local coordinate frame that is attached to the WMR body and is located at a distance d from PC. The point PL ¼(xL,yL) is a virtual reference point on x axis of the local frame at a distance L (look-ahead distance) of PO. The parameter 2b is the distance between two fixed wheels. The radius of each wheel is denoted by r.

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(n m ¼2):

YG y

vlx

a

x L

d yO

h2 ðqÞT ¼ ½xO þ Lcos j, yO þL sin jT ð85Þ ~ D ^ 1 ðxÞ in (29), After the calculation of L2f hðxÞ, Lq L~ f hðxÞ and DðxÞ one gets 2 3 @ 6 @q ðJh1 ðqÞSðqÞvÞSðqÞv 7 6 7 L2f hðxÞ ¼ 6 @ ð86Þ 7, 4 5 ðJh2 ðqÞSðqÞvÞSðqÞv @q

y ¼ hðxÞ ¼ ½h1 ðqÞ,

θl

b

ϕ

PL

PO

PC

2

vr

~ D ^ 1 ðxÞ ¼ 4 DðxÞ

vrx Castor wheel

1

1 1 ðy~ 5 y^ 5 y~ 6 y^ 6 Þcos j sin j

2r

O

6 SðqÞ ¼ 4 sin j 0

7 0 5, 1



m M1 ¼ 0

 0 , I

2

2K2 6 r2 6 C1 ¼ 4 _ mC dj

3

2 1 7 6r 7 6 and X ¼ 4 1 2b2 K2 5 r r2 _ mC dj

b r 

2K2 mC d mC d , y3 ¼ , , y2 ¼ m I mr2 K K b y5 ¼ 1 and y6 ¼ 1 , Ir mr

y4 ¼

3 7 7, b5 r

Sv 0 "

Q ðx, yÞ ¼

 ,

" qðx, yÞ ¼

ð82Þ

#

0 Q ðx, yÞ

# y1 vr y2 o2r , y3 or vr y4 or

" and "

GðyÞ ¼

gðx, yÞ ¼

y5 y5 y6 y6

0 GðyÞ

r

3

4

ð89Þ

where w11 ¼ vr cos j, w12 ¼ o2r cos j, w13 ¼ or vr L sin j, w14 ¼ or Lsin j 1 w15 ¼ y^ 5 ðF1 cos2 j þ F2 sin j cos jÞ, 1 w16 ¼ y^ 6 ðF1 sin2 jF2 sin j cos jÞ, w21 ¼ vr sin j,

w22 ¼ o2r sin j, w23 ¼ or vr L cos j, w24 ¼ or Lcos j, 1 w25 ¼ y^ 5 ðF1 sin j cos j þ F2 sin2 jÞ, 1 w26 ¼ y^ 6 ðF1 sin j cos j þ F2 cos2 jÞ:

ð90Þ

where @ ðJ SvÞSvw11 y^ 1 w12 y^ 2 w13 y^ 3 w14 y^ 4 , @q h1 @ F2 ¼ Z2  ðJh2 SvÞSvw21 y^ 1 w22 y^ 2 w23 y^ 3 w24 y^ 4 : @q

F1 ¼ Z1 

ð91Þ

The block diagram of the adaptive feedback linearizing controller is shown in Fig. 3. 5.2. Simulation results

2b2 K2 , Ir2

we have 

2

the regression matrix in (31) is obtained as " # w11 w12 w13 w14 w15 w16 , W¼ w21 w22 w23 w24 w25 w26

where m¼mC + 2mw, I ¼ IC þ 2Im þ mC d2 þ 2mw b2 and B1 ¼ X T . The parameter mC is the mass of the platform without the driving wheels and the rotors of the DC motors, mw denotes the mass of each driving wheel plus the rotor of its motor, IC denotes the moment of inertia of the platform without the driving wheels and the rotors of the motors about a vertical axis through PC and Im denotes the moment of inertia of each wheel and the motor rotor about a wheel diameter. The parameters mass (m), moment of inertia (I), wheel radius (r), distance between two wheels (2b) and actuator parameters (K1 and K2) are supposed to be uncertain. Then, by substituting (81) in (14) and (15), and defining the following new uncertain parameters:

y1 ¼

5

ð88Þ

XG

ð81Þ

f ðxÞ ¼

1

3 y~ 1 vr cos jy~ 2 o2r cos jy~ 3 or vr Lsin j þ y~ 4 or Lsin j 5, Lq L~ f hðxÞ ¼ 4 ~ y vr sin jy~ o2 sin j þ y~ or vr L cos jy~ or Lcos j

If the generalized coordinates vector is selected to be q¼ [xO,yO,j]T, one velocity constraint is obtained as y_ O cos j x_ O sin j ¼ 0. Thus, we define pseudo-velocities as v(t)¼[v(t), o(t)]T which are linear and angular velocities of the WMR body. According to the notation introduced before, the following kinematic and dynamic matrices are obtained: 3

1

y~ 5 y^ 5 sin2 j þ y~ 6 y^ 6 cos2 j

3

2

Fig. 2. Configuration of type (2, 0) wheeled mobile robot.

0

1

ð87Þ

θr

xO

cos j

1

b

1

2

1

y~ 5 y^ 5 cos2 j þ y~ 6 y^ 6 sin2 j ðy~ 5 y^ 5 y~ 6 y^ 6 Þcos j sin j

# ð83Þ

# ð84Þ

The following output variables are chosen to track a desired trajectory based on look-ahead control method

In this section, some computer simulations were performed in order to show the tracking performance and robustness of the proposed controllers under parametric and nonparametric uncertainties. The WMR parameters are chosen to match with a real world mobile robot, and Gaussian white noise is also added to the states to simulate a localization system. All simulations are carried out based on Euler approximation with a time step of 20 ms. Real physical parameters of WMR and control parameters are listed as follows: r ¼0.15 m, b¼0.75 m, d¼0.3 m, L¼0.1 m, mC ¼36 kg, mw ¼1 kg, Im ¼0.0025 kg m2, IC ¼15.625 kg m2, sampling time dt¼0.02 s, K1 ¼7.2 and K2 ¼2.59. Thus, the real values of new parameters in (82) are computed as y ¼[6.06, 0.284, 0.54, 6.48, 1.26, 1.8]T. In order to provide a smooth navigation, a qffiffiffiffiffiffiffi critically damped system is chosen by setting b1j ¼ 2 b2j with

K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

201

Disturbance, sensors error source Actuators input saturation η

y (t)

β

2

β

Nonlinear Feedback

Inner loop feedback

vR

1

_e +

xO , yO , ϕ , v , ω

Outer loop feedback _ e+

yr (t)

Output y Equation

Adaptive Law

Robust term yr (t)

Wheeled mobile robot

ua

y

Lf h

Fig. 3. Block diagram of the proposed controller.

Non-adaptive Controller AFL Controller Desired Path

15 10 5 0

0

20

40

60 80 Time (sec)

100

120

140

0 15

20

40

60 80 Time (sec)

100

120

140

25 35 y2 (m)

Y (meter)

30

20 y1 (m)

35

20

15

0

5

10 X (meter)

15

30 25 20

20

Fig. 4. WMRs trajectories for the adaptive controller (bold solid line) and nonadaptive controller (dashed–dotted line) in presence of parametric uncertainty.

b2j ¼1 in proportional-derivative (PD) controller (32), which together with aj ¼ b2j/b1j provides the presented SPR conditions for the transfer function (35). Moreover, the look-ahead distance L must be chosen appropriately. Since the parameter L appears in the determinant of the decoupling matrix, small values of L may result in large control signals. Large values of L may also lead to poor tracking performance. In the first simulation, the adaptive tracking controller is tested only for uncertain parameters. It is assumed that there is no knowledge about the WMR parameters. For simulation purposes, a smooth desired trajectory is chosen as follows: xr ðtÞ ¼ xg þ R cosðor tÞ, yr ðtÞ ¼ yg þ R sinðor tÞ,

ð92Þ

where (xg,yg) ¼(10 m, 25 m) and R ¼7.5 m are the centre and radius of the circular trajectory, respectively, and or ¼0.05. Note that or must be chosen small because the controller performance degrades when or is far from zero. The initial position and orientation of WMR, initial values of the estimated parameters of WMR and the adaptation gain are selected as: T

xð0Þ ¼ ½15,25,0,0,0 ,

y^ ð0Þ ¼ ½1,1,1,1,1,1T , G ¼ diagð½10,5,10,10,1,10Þ:

ð93Þ

Fig. 4 shows the desired and WMR trajectories for the adaptive controller and a nonadaptive feedback linearizing controller. As one can see, parametric uncertainties have undesirable effects on the tracking performance of the nonadaptive feedback linearizing controller because of the lack of exact cancellation for nonlinear dynamics. But the position tracking errors asymptotically converge to zero for the adaptive controller. The estimated parameters are also bounded which are shown in Fig. 5. In the second simulation, the modified adaptive tracking controllers are tested for both parametric and nonparametric uncertainties. Real physical parameters of WMR are listed as follows: r¼0.05 m, b¼0.3 m, d ¼0.15 m, L¼0.2 m, mC ¼ 10 kg, mw ¼0.2 kg, Im ¼ 0.006 kg m2, IC ¼3 kg m2, sampling time dt ¼0.02 s, K1 ¼0.261 and K2 ¼0.267. Thus, the real values of new parameters in (82) are computed as y ¼[20.52, 0.144, 0.46, 5.87,0.5, 0.48]T. A smooth desired eight-shaped trajectory is chosen as follows: y1d ðtÞ ¼ xg þ Rsinð2or tÞ, y2d ðtÞ ¼ yg þR sinðor tÞ,

ð94Þ

where (xg, yg)¼(2.5 m, 5.5 m) and R¼2 m are parameters of the trajectory and or ¼0.05. To illustrate the robustness of the proposed controller, the uncertain values of parameters are set to y^ ð0Þ ¼ ½1, 1, 1, 1, 1, 1T and the following models are chosen _ to simulate the nonparametric uncertainties such as friction FðqÞ

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K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

and disturbance td: _ ¼ Fv q_ þ Kd sgnðqÞ, _ FðqÞ td ¼ ½Ad sinðod tÞ,Ad sinðod tÞT

controller without modification. Parameters of the adaptive robust control law are chosen as m(0) ¼1, km ¼1. As illustrated by the figure, the position tracking errors are asymptotically stable for the modified tracking controller. In most of the simulation results, the adaptive tracking controller without modification loses its stability in the presence of large nonparametric uncertainties. However, the modified adaptive controllers maintain their stability and show remarkable tracking performance such that the root mean squared error values reduce less than 0.1 m in x and y directions. Figs. 7 and 9 illustrate the generated control signals for all controllers. The high-frequency oscillations in the control signals of Figs. 7 and 9 are caused by the high-frequency sinusoidal disturbance in (95). Moreover, the saturation-type controller (71) is a continuous approximation of the sliding mode control which also generates chattering in the control signal of Fig. 9a. The chattering is undesirable in practice since it may excite unmodeled high-frequency dynamics. One may compromise between the tracking accuracy and smoothness of the control signal by well tuning of the function m(t) in (71). In real robotic applications, high bandwidth actuators are required to cope with such oscillations. Otherwise, the controller performance may be degraded in real world experiments. As one may infer from (71) and (72), the robust control is discontinuous in the limit and the chattering of the control signal in Fig. 9a is increasing as time goes to infinity. To alleviate the effect of chattering for all t Z0, one may choose m(t) ¼ m0, where m0 is a constant value. However, as stated in Remark 9, this choice of m(t) may result in uniform ultimate boundedness of the tracking errors and the asymptotic stability may be lost. As stated by Remark 3, our proposed controllers cannot overcome the kinematic perturbation due to wheels slippage. The slippage causes that the kinematic constraints (2) and kinematic Eq. (4) do not hold true anymore. Another simulation is carried out to illustrate the performance of the proposed adaptive robust controller in presence of kinematic perturbation due to lateral wheel slippage. As reported by [30], one may use the following perturbed kinematic model instead of (4):

ð95Þ

where Fv ¼ 0.9, Kd ¼0.75, Ad ¼10 and od ¼3 rad/s. The first term in the right-hand side of the first Eq. in (95) denotes the viscous friction. The term sgn(.) denotes the signum function, which simply model the dynamic friction. In addition, the control signals for actuators are saturated to lie within 9ua9 r24 V to simulate a real world WMR system. Note that the controller gains must be chosen such that the control signals do not exceed beyond this bound. Otherwise, the saturation nonlinearity of the actuators must be taken into account in the controller design process. The mobile robot starts from x(0)¼[2.5,6,  p/6,0,0]T to track the generated trajectory by (94). Fig. 6 shows the robustness and performance of the robust adaptive tracking controller and the adaptive controller without modification. The adaptive tracking controller with s-modification on adaptation law clearly shows more robustness to disturbance and friction. Fig. 8 illustrates the tracking performance and robustness of the adaptive robust feedback linearizing controller in comparison with the adaptive

Estimated parameters

2.5

θ1 θ2

2

θ3 θ4 θ5

1.5

θ6 1 0.5 0

0

20

40

60 80 Time (sec)

100

120

q_ ¼ SðqÞv þ rU ðtÞ½sin j,cos j,0T

140

where rU(t)AR denotes a bounded disturbance. In this simulation, it is assumed that rU(t)¼0.1[H(t 50) H(t 70)], where H(.) denotes the standard Heaviside step function. Fig. 10 shows that the tracking

Fig. 5. Estimated WMR parameters.

8

AFL Robust AFL

e1 (m)

1

7

ð96Þ

0

6

0

20

40

60 80 Time (sec.)

100

120

0

20

40

60 80 Time (sec.)

100

120

0.5

5 e2 (m)

Y (meter)

-1

4 3 0

1

2 3 X (meter)

4

5

0 -0.5

140

Fig. 6. WMR trajectory for both controllers in presence of parametric and nonparametric uncertainties: robust adaptive controller with s-modification (bold solid line) and adaptive feedback linearizing controller without modification (dashed–dotted line) (a). WMR tracking errors for both controllers: robust adaptive controller (bold solid line) and adaptive feedback linearizing controller without modification (dashed–dotted line) (b).

40 20 0 -20 -40

203

ua [1]

40 20 0 -20 -40

0

20

40

60 80 100 Time (sec.)

120

140

40 20 0 -20 -40

ua [2]

ua [2]

ua [1]

K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

0

20

40

60 80 Time (sec.)

100

120

140

40 20 0 -20 -40

0

20

40

60 80 Time (sec.)

100

120

140

0

20

40

60 80 Time (sec.)

100

120

140

Fig. 7. The generated control signals for adaptive feedback linearizing controller with s-modification (a) and adaptive controller without modification (b).

e1 (m)

7.5 7 Y (meter)

6.5

AFL Adptive Robust FL

1 0 -1 -2

6

0

20

40

60 80 Time (sec)

100

120

140

20

40

60 80 Time (sec)

100

120

140

5.5 1

5 e2 (m)

4.5 4 3.5 3

0 -1

0

1

2 3 X (meter)

4

0

Fig. 8. WMR trajectory for both the controllers in presence of parametric and nonparametric uncertainties: adaptive robust controller (bold solid line) and adaptive feedback linearizing controller without modification (dashed–dotted line) (a). WMR tracking errors for both controllers: adaptive robust controller (bold solid line) and adaptive feedback linearizing controller without modification (dashed–dotted line) (b).

ua [1]

ua [1]

50 0 -50

0

20

40

60 80 100 Time (sec)

120

140

ua [2]

ua [2]

50 0 -50

0

20

40

60 80 100 Time (sec)

120

140

40 20 0 -20 -40

40 20 0 -20 -40

0

20

40

60 80 100 Time (sec)

120

140

0

20

40

60 80 100 Time (sec)

120

140

Fig. 9. The generated control signals for the adaptive feedback linearizing controller with adaptive robust controller (a) and adaptive controller without modification (b).

performance and robustness of the proposed controller is highly degraded in presence of the kinematic perturbation. Improvement of the controller robustness demands more investigations which determine the direction of our future works.

6. Conclusion and future works A trajectory tracking controller has been designed based on feedback linearization technique for nonholonomic wheeled

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K. Shojaei et al. / Robotics and Computer-Integrated Manufacturing 27 (2011) 194–204

8 e1 (m)

7 6.5 Y (meter)

AFL Controller

0

7.5

Adaptive robust FL

-0.5 -1

6

0

20

40

60 80 Time (sec)

100

120

140

0

20

40

60 80 Time (sec)

100

120

140

5.5 5 e2 (m)

4.5 4 3.5 3

0

1

2 3 X (meter)

4

0.4 0.2 0 -0.2 -0.4

5

Fig. 10. The generated trajectories by the adaptive controller and proposed adaptive robust feedback linearizing controller in presence of lateral slippage of wheels (a) and the tracking errors for both the controllers (b).

mobile robots. The proposed controller can solve the integrated kinematic and dynamic tracking problem in presence of both parametric and nonparametric uncertainties. Following points were considered in the design process of the control law: (1) an adaptive law was designed based on SPR-Lyapunov approach to achieve robustness to parametric uncertainties. (2) It was shown that the adaptive tracking controller can be modified in order to compensate for nonparametric uncertainties due to friction, disturbances and unmodeled dynamics. (3) From a practical viewpoint, the control law was designed such that it can provide a control signal in the actuator level. Theoretical proofs have shown that the closed-loop system is stable under parametric and nonparametric uncertainties. Simulation results for a type (2, 0) WMR were also presented to demonstrate the robustness and tracking performance of the proposed controller. Since the measurements of all state variables are not available for feedback, the design of observer-based adaptive-robust control laws for the tracking problem of nonholonomic WMRs is the subject of our future works, which is not sufficiently addressed in the literature. Based on the simulated results, another subject of the future work is the modeling and compensation of the kinematic perturbations due to slippage of WMR wheels. References [1] Campion G, Bastin G, Andrea-Novel BD. Structural properties and classification of kinematic and dynamic models of wheeled mobile robots. IEEE Trans Robotics Autom 1996;12(1):47–62. [2] Sarkar N, Yun X, Kumar V. Control of mechanical systems with rolling constraint: application to dynamic control of mobile robots. Int J Robotics Res 1994;13(1):55–69. [3] Coelho P, Nunes U. Path-following control of mobile robots in presence of uncertainties. IEEE Trans Robotics 2005;21(2). [4] Samson C. Control of chained systems application to path following and timevarying point-stabilization of mobile robots. IEEE Trans Autom Control 1997;1:64–77. [5] McCloskey R, Murray R. Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. IEEE Trans Autom Control 1997. [6] Coelho P, Nunes U. Lie algebra application to mobile robot control: a tutorial. Robotica 2003;21(5):483–93. [7] Aguiar JPPedro. Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty. IEEE Trans Autom Control 2007;52(8):1362–79. [8] Kolmanovsky I, McClamroch H. Developments in nonholonomic control problems. IEEE Control Syst Mag 1995:20–36. [9] Sastry SS, Isidori A. Adaptive control of linearizable systems. IEEE Trans Autom Control 1989;AC-34:1123–31.

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