Simultaneous Stabilization and Tracking of Wheeled Mobile Robotsvia Chained Form*

Yaonan Wang, Zhiqiang Miao, Yanjie Chen

Abstract—A time-varying controller is proposed to simulta-neously solve the stabilization and the tracking problem ofwheeled mobile robots via their equivalent chained form. Thecontrol is designed based on Lyapunov’s direct method andthe backstepping technique. The outstanding feature of theproposed controller is computationally simple due to its full useof the existing results on stabilization and tracking control forthe equivalent chained form systems of wheeled mobile robots.Simulation results for a wheeled mobile robot are provided toillustrate the effectiveness of the proposed controller.

I. INTRODUCTION

In the past decade, the control of mechanical systems withnonholonomic constraints has attracted considerable attentionin the control community. Mobile robots with two indepen-dent drive wheels, or their feedback equivalent chained formsystems, have served as benchmark mechanical example withsubstantial engineering interest. The challenge of controllingsuch systems comes from the fact that the motion of a wheeledmobile robot in a plane possesses three degrees of freedom(DOF); while there are only two available control inputsunder the nonholonomic constraint. By applying Brockett’stheorem [1], it is shown that a nonholonomic system cannot beasymptotically stabilized by smooth or even continuous, pure-state feedback. Consequently, a good number of novel ideasand feedback design strategies have been proposed for thestabilization problem, including: discontinuous time-invariantfeedback [2]– [4], smooth time-varying feedback [5]– [9],and hybrid feedback [10], [11]. See the survey paper [12]for more details. Specifically, the first time-varying controlmethod was proposed by Samson in [5] for the stabilization ofcart. This approach was further developed for the stabilizationof a car-like mobile robot with a steering wheel [6]. Theessential idea of Samson’s result is using a heat functionto provide ”persistent excitation” and a feedback term tointroduce ”dissipation”.

In parallel, the tracking control problem of nonholonomicsystems has also received a great deal of attention because ofits practical importance. In the case of mobile robot, solutionsto trajectory tracking were introduced in [13]– [16], and [17]–[22] to cover a broader class of systems in the chained form. Toensure asymptotic tracking, the controllers in these referencesrequire that the reference velocities satisfy some kind ofpersistent excitation (PE) condition. For example, in [13], the

* This work was supported by the National Natural Science Foundation ofChina (grant 61175075), and National High Technology Research and Devel-opment Program of China (863 Program: 2012AA111004, 2012AA112312).

The authors are with the College of Electrical and Information Engi-neering, Hunan University, Changsha, 410082, China. miaozhiqiang@hnu.edu.cn

reference linear velocity must nonzero; in [14], the referenceangular velocity is assumed to satisfy the PE condition; and in[15] and [16], the reference linear velocity or angular velocitymust not converge to zero. Roughly speaking, the fulfillmentof a PE condition implies that the desired reference trajectoryis ”a moving trajectory”, instead of fixed set-point. Theseassumptions make it impossible for a single controller to solveboth the stabilization problem and the tracking problem.

The stabilization and tracking problems of nonholonomicsystems are studied separately. However, in practical applica-tions, it is preferable to solve the stabilization problem and thetracking problem simultaneously using a single controller. Thisproblem was solved in [23] for the first time by introducinga time-varying velocity feedback controller to achieve bothstabilization and tracking of unicycle-modeled mobile robotsat the kinematics level. Recently, Do et al. [24], [25] focusedon the dynamic model using the backstepping technique,where the torque and force were taken as the control inputs.However, their controller is quite complicated and computa-tionally demanding, and the control gains must be selectedvery carefully to satisfy some constraints.

Inspired by Samson’s time-varying feedback method, thispaper presents a novel controller at the torque level to si-multaneously solve the stabilization and the tracking problemof mobile robot. The outstanding feature of our controller iscomputationally simple due to its fully use of the existingresults on stabilization [6] and tracking control [20], [21] forthe equivalent chained form systems of wheeled mobile robots.The controller is developed based on backstepping techniqueand Lyapunov’s direct method, and guarantees the globalasymptotic convergence of the regulation and tracking errorto the origin. Simulation results for a wheeled mobile robotare presented to illustrate the effectiveness of the proposedcontroller.

II. PROBLEM FORMULATION

As shown in Fig. 1, the configuration of the differential-drive wheeled mobile robot (WMR) can be described by q =[x, y, θ]T , where (x, y) are the coordinates of the midpointQ between the two driving wheels, and θ is the orientationangle of the mobile robot. The mobile base satisfies the ”purerolling without side slipping” condition, which is described bythe following nonholonomic constraint:

x sin θ − y cos θ = 0 (1)

Define the constraint matrix J(q) as

J(q) = [sin θ,− cos θ, 0] (2)

2014 11th IEEE InternationalConference on Control & Automation (ICCA)June 18-20, 2014. Taichung, Taiwan

978-1-4799-2837-8/14/$31.00 ©2014 IEEE 643

Fig. 1. Wheeled mobile robot configuration.

The constraint (1) can be rewritten as

J(q)q = 0 (3)

According to the Euler-Lagrangian formulation, the dynam-ics of the mobile robot can be described as

M(q)q + C(q, q)q +G(q) = JT (q)λ+B(q)τ (4)

where M(q) ∈ R3×3 is the symmetric, positive definite inertiamatrix, C(q, q) ∈ R3×3 is the centripetal and coriolis matrix,G(q) ∈ R3 is the gravitational vector, J(q) ∈ R1×3 isthe constrained matrix, λ ∈ R is the associated Lagrangemultiplier, τ ∈ R2 is the vector of control inputs force,B(q) ∈ R3×2 is the input transformation matrix.

Let S(q) ∈ R3×2 be a full rank matrix defined by

S(q) =

cos θ 0

sin θ 0

0 1

(5)

It is easy to verify that ST (q)JT (q) = 0, then there exists anauxiliary vector of independent generalized velocities v ∈ R2

that make the whole system (3) and (4) be transformed into amore appropriate representation for control purposes:

q = S(q)v (6)M1(q)v + C1(q, q)v +G1(q) = B1(q)τ (7)

where M1(q) = STM(q)S,C1(q, q) = ST (M(q)S +C(q, q)S), G1(q) = STG(q), B1(q) = STB(q).

For ease of controller design in this article, the existingresults for the control of nonholonomic canonical forms inthe literature are exploited [26]. Considering the coordinatetransformationz = T1(q) and state feedback v = T2(q)uwhich are defined asz1z2

z3

=

0 0 1

sin θ − cos θ 0

cos θ sin θ 0

xyθ

(8)

[v1

v2

]=

[z2 1

1 0

][u1

u2

](9)

Using the above coordinate transformation, the kinematicmodel of WMR given by Eq. (6) can be converted to thechained form

z1 = u1, z2 = u1z3, z3 = u2 (10)

where z = [z1, z2, z3]T ∈ R3 is the state, u = [u1, u2]

T ∈ R2

is the generalized velocity vector. Based on the above trans-formation, the dynamic model is converted as

M2(z)u+ C2(z, z)u+G2(z) = B2(z)τ (11)

where M2(z) = TT2 (q)M1(q)T2(q)|q=T−1

1 (z), C2(z, z) =

TT2 (q)(M1(q)T2(q) + C1(q, q)T2(q))|q=T−1

1 (z), G2(z) =

TT2 (q)G1(q)|q=T−1

1 (z), B2(z) = TT2 (q)B1(q)|q=T−1

1 (z). Thefollowing properties of the model (11) can be easily proved:Property 1: Matrix M2(z) is symmetric positive definite;M2(z), C2(z, z) and G2(z) are bounded.Property 2: M2 − 2C2 is a skew-symmetric matrix.

We assume that the reference trajectory is generated by thevirtual robot:

xd = v1d cos(θd)

yd = v1d sin(θd)

θd = v2d

(12)

where qd = [xd, yd, θd]T are the position and orientation of the

virtual robot. vd = [v1d, v2d]T are the velocities of the virtual

robot. Applying the coordinate transformation zd = T1(qd)and the state feedback vd = T2(qd)ud, so that zd and ud

satisfy the equation(10). It is assumed that ud = [u1d, u2d]T

satisfy the following assumption:Assumption 1: The reference signals u1d, u2d, u1d and u2d arebounded. In addition, one of the following conditions holds:

C 1) There exist T , µ1 > 0 such that∫ t+T

t

(u21d(s) + u2

2d(s))ds ≥ µ1, ∀t ≥ 0 (13)

C 2) There exists µ2 > 0 such that∫ ∞

0

(|u1d(s)|+ |u2d(s)|)ds ≤ µ2 (14)

In this paper, our purpose is to design a (single) continuousfeedback control law τ for the system (10)-(11) that simultane-ously solves stabilization and tracking for a desirable referencetrajectory zd(t), in particular,

limt→∞

(z(t)− zd(t)) = 0 (15)

Remark 1: It should be noted that asymptotic convergence ofthe regulation and tracking error to the origin is achieved inpur paper. Based on dynamic oscillators [27] and transversefunctions [28], unified frameworks were provided for both thetracking and the regulation problem, however, only uniformlyultimately bounded rather than asymptotic convergence of thetracking error was achieved.

III. CONTROL DESIGN

In this section, the controller at torque level is designed us-ing Lyapunov-based approach and the backstepping technique.The structure of the control system is illustrated in Fig. 2.

644

Fig. 2. Structure of the proposed control system.

A. Kinematic Control

As a precursor to the full dynamic system study, let us firstneglect the dynamics of system (10)-(11) and consider thekinematic tracking problem only. Now u is assumed to be thevirtual control, the object is to design the control input u suchthat (15) holds. To facilitate controller design, an appropriatetracking error is introduced as follows:

ze =

z1 − z1d

z2 − z2d − z3d(z1 − z1d)

z3 − z3d

(16)

Then, the following error model can be attained:

˙z1e = u1 − u1d

˙z2e = u1z3e − u2dz1e

˙z3e = u2 − u2d

(17)

Considering the following Lyapunov function candidate

V1 =1

2(z21e + k2z

22e + z23e) (18)

where k2 is a positive constant. Along the trajectories of (17),the time derivative of is given as

V1 = z1e(u1−u1d−k2u2dz2e)+z3e(u2−u2d+k2u1z2e) (19)

Choosing the control u as

u1 = u1d − k1z1e + k2u2dz2e + ϑ1

u2 = u2d − k3z3e + k2u1z2e(20)

where k1, k3 are positive constants. Then substituting theexpression of u into (19), we have

V1 = −k1z21e − k3z

23e + ϑ1z1e (21)

Moreover, the time-varying signal ϑ1 is defined by

ϑ1 = ρ(t)h(t, ze) (22)

withρ = −(|u1d(t)|+ |u2d(t)|)ρ, ρ(0) = 1 (23)

and ze = [z2e, z3e]T , h(t, ze) is a function of class Cp+1,

with all successive partial derivatives uniformly bounded withrespect to t, and is required to have the following twoproperties:

P 1) h(t, ze) is uniformly bounded with respect to t and ze,i.e., there exists a constant h0 > 0, such that

|h(t, ze)| ≤ h0, ∀t > 0, ze ∈ R2 (24)

P 2) h(t, 0) = 0, and there is a time-diverging sequence{ti}i∈N , and a positive continuous function α(·), suchthat

∥ze∥ ≥ l > 0 =⇒j=p∑j=1

(∂jh

∂tj(ti, ze)

)2

≥ α(l) > 0, ∀i

(25)where N denotes the set of natural numbers.

Remark 2: The conditions imposed by P 1 and P 2 uponh(t, ze) are not severe and can be easily met. For example,the following three functions all satisfy the properties

h(t, ze) = h0 tanh(a∥ze∥b) sin(ct)

h(t, ze) =2h0∥ze∥b

1 + ∥ze∥2bsin(ct)

h(t, ze) =2h0

πarctan(a∥ze∥b) sin(ct)

with a = 0, b > 0, c = 0.Based on the above analysis, the original system (17) is

transformed intoz1e = −k1z1e + k2u2dz2e + ϑ1

z2e = u1z3e − u2dz1e

z3e = −k2u1z2e − k3z3e

(26)

Before giving the stability analysis of closed-loop system (26),we first present a technical lemma.

Lemma 1 Let V : R+ → R+ be continuously differentiableand W : R+ → R+ uniformly continuous satisfying that, foreach t ≥ 0,

V (t) ≤ −W (t) + p1(t)V (t) + p2(t)√V (t) (27)

with both p1(t) and p2(t) are non-negative and belong to L1-space. Then, there exists a constant c, such that W (t) → 0and V (t) → c as t → ∞.

Proof: First, we prove that V (t) is bounded. Accordingto (27), we have

V (t) ≤ p1(t)V (t) + p2(t)√V (t) (28)

645

which implies the following inequality when V = 0 :

d(√

V (t))

dt≤ p1(t)

2

√V (t) +

p2(t)

2(29)

When V = 0, it can be verified that the upper right-hand derivative D+(

√V (t)) ≤ p2(t)/2, Hence, D+(

√V (t))

satisfies (29) for all values of V . By the comparison lemma,√V (t) satisfies the inequality√V (t) ≤

(√V (0) +

∫ t

0

exp(−∫ s

0

p1(τ)

2dτ)

p2(s)

2ds

)× exp

(∫ t

0

p1(s)

2ds

)(30)

Since p1(t) and p2(t) are non-negative, andexp(−

∫ s

0p1(τ)

2 dτ) ≤ 1,∀s ≥ 0, equation (30) impliesthat√

V (t) ≤(√

V (0) +

∫ t

0

p2(s)

2ds

)exp

(∫ t

0

p1(s)

2ds

)(31)

Since both p1(t) and p2(t) belong to L1 space, V (t) isbounded. That is, for any r > 0 and for any initial condition√V (0) ≤ r, there exists a positive constant δ such that√

V (t) ≤ δ, ∀t ≥ 0. (32)

Then from (27), we have

V (t) ≤ −W (t) + δ2p1(t) + δp2(t) (33)

which implies

d

dt

(V (t)− δ2

∫ t

0

p1(s)ds− δ

∫ t

0

p2(s)ds

)≤ 0 (34)

It follows that (V (t) − δ2∫ t

0p1(s)ds − δ

∫ t

0p2(s)ds) is non-

increasing. Since V (t) is bounded from below by zero, V (t)tends to a finite nonnegative constant.

On the other hand, it follows from (27) that

V (t) +

∫ t

0

W (s)ds ≤ V (0) + δ2∫ t

0

p1(s)ds

+ δ

∫ t

0

p2(s)ds < ∞, ∀√V (0) ≤ r,∀t ≥ 0. (35)

The above inequality implies that W (t) belongs to L1-space.Thus, by Barbalat’s lemma, W (t) asymptotically converges tozero. This completes the proof.

We now use Lemma 1 to investigate the stability propertyof closed-loop system (26), the result is stated in the followingtheorem:

Theorem 1 Consider the kinematic simultaneous stabilizationand tracking problem of subsystem (10). Under Asumption 1,the closed-loop system (26) is globally asymptotically stable.Thus, the time-varying control law (20) makes (15) holds.

Proof: Solving the differential equation (23), we have

ρ(t) = exp

(−∫ t

0

(|u1d(s)|+ |u2d(s)|)ds)

(36)

which means 0 ≤ ρ(t) ≤ 1. If C 1 holds, then based onthe stability results on linear time-varying systems, ρ(t) isexponentially convergent, and ρ(t) ∈ L1. If C 2 holds, then0 < exp (−µ2) < ρ(t) ≤ 1.

We first consider the case of C 1. Since |h(t, ze)| ≤ h0, and|z1e| ≤

√2V1, from (21), we have

V1 ≤ −k1z21e − k3z

23e + ρ(t)|h(t, ze)||z1e|

≤ −k1z21e − k3z

23e + h0ρ(t)

√2V1 (37)

Because C 1 holds, ρ(t) ∈ L1. By means of Lemma 1, we havez1e, z3e tend to zero, and z2e tends to a constant as t → ∞.Since limt→∞ z1e(t) = 0, applying the extended version ofBarbalat’s lemma [6] to the first equation of (26), yields

limt→∞

u2d(t)z2e(t) = 0 (38)

Similarly, because limt→∞ z3e(t) = 0, applying the extendedversion of Barbalat’s lemma to the last equation of (26), yields

limt→∞

u1d(t)z2e(t) = 0 (39)

Combining (38) and (39), we have

limt→∞

(u21d(t) + u2

2d(t))z2e(t) = 0 (40)

Following (40), it can be easily proved that limt→∞ z2e(t) = 0by contradiction. Thus, ∥ze∥ is asymptotically convergent.

We now move to the case of C2. We first show that z1e isbounded. From (21), we have

V1 ≤ −k1z21e − k3z

23e + ρ(t)|h(t, ze)||z1e|

≤ −k1z21e + h0|z1e| (41)

It follows that if |z1e| ≥ h0/k1, then V1 ≤ 0, thus z1e isbounded.

Then we will shown that z2e and z3e tend to zero as t → ∞.For the last two equations of (26), we define the Lyapunovfunction as

V2 =1

2(k2z

22e + z23e) (42)

the time derivative of V2 is given as

V2 = −k3z23e − k2u2dz1ez2e (43)

Since |z2e| ≤√2V2/k2, we have

V2 ≤ −k3z23e − k2|u2d||z1e|

√2k2V2 (44)

Because z1e is bounded and u2d ∈ L1, u2dz1e ∈ L1. By meansof Lemma 1, we have z3e tends to zero, and z2e tends to aconstant as t → ∞. Since limt→∞ z3e(t) = 0, applying theextended version of Barbalat’s lemma to the last equation of(26), yields

limt→∞

u1(t)z2e(t) = 0 (45)

We now proceed by contradiction. Assume that u1(t) doesnot tend to zero. Considering that z2e(t) tends to a constanttogether with equation (45), then z2e(t) tends to zero. Byuniformly continuity and since ϑ1(t, 0) = 0, ϑ1(t, ze) also

646

tends to zero. Considering that z2e is bounded and u2d ∈ L1,then u2dz2e ∈ L1, and the first equation of (26):

z1e = −k1z1e + k2u2dz2e + ϑ1 (46)

can be viewed as a stable linear system subjected to an additiveperturbation which asymptotically vanishes. As a consequence,z1e tends to zero. As z1e and ϑ1 tend to zero, u1d and u2dz2ebelong to L1, this in turn implies that u1(t) tends to zero,yielding a contradiction. Therefore, u1(t) must asymptoticallytend to zero.

Differentiating the expression of u1 with respect to time andusing the convergence of u1d, u2d, u1d, u2d and ∥ ˙ze∥ to zero,we get

u1(t) =∂ϑ1

∂t(t, ze) + o(t)

= ρ(t)∂h

∂t(t, ze) +

∂ρ

∂th(t, ze) + o(t)

= ρ(t)∂h

∂t(t, ze) + o′(t) (47)

where limt→∞ o(t) = 0, and

o′(t) = −(|u1d(t)|+ |u2d(t)|)ϑ1(t, ze) + o(t).

Because u1d(t) and u2d(t) belong to L1, ϑ1(t, ze) is uni-formly bounded, we conclude that o′(t) tends to zero. Since(∂h/∂t)(t, ze) is uniformly continuous and 0 < exp(−µ2) <ρ(t) ≤ 1, (∂h/∂t)(t, ze) tends to zero by application of theextended version of Barbalat’s lemma.

By repeating the same procedure as many times as neces-sary, we show that (∂jh/∂tj)(t, ze) tend to zero, (1 ≤ j ≤ p).Therefore

limt→∞

j=p∑j=1

(∂jh

∂tj(t, ze)

)2

= 0 (48)

Assume now that ∥ze(t)∥ remains larger than some positivenumber l. The previous convergence result is then not com-patible with the P 2 property imposed on the function h(t, ze).Therefore, ∥ze(t)∥ asymptotically converges to zero. Then byuniformly continuity and ϑ1(t, 0) = 0, ϑ1(t, ze) tends to zero.

In view of the expression of u1, asymptotical convergenceof z1e(t) to zero readily follows. Thus, ze(t) is asymptoticallyconvergent independent of initial state. This completes theproof of Theorem 1.

B. Dynamic Control

In this part, the control law τ is designed based on thekinematic controller using the bacstepping approach, so thatthe problem of simultaneous stabilization and tracking can alsobe solved using mechanical torques.

To this end, we reformulate the kinematic controller givenin (20) as a desired signal as follows

η1 = u1d − k1z1e + k2u2dz2e + ϑ1

η2 = u2d − k3z3e − k2η1z2e(49)

Denote u = u− η, the subsystem (11) can be rewritten as

M2(z) ˙u+ C2(z, z)u = B2(z)τ

− (M2(z)η + C2(z, z)η +G2(z)) (50)

If the control law is chosen as

τ = B−12 (z) [−Kpu+ (M2(z)η + C2(z, z)η +G2(z)] (51)

where Kp is a positive definite matrix, it can be proved thatze(t) asymptotically converge to zero along a similar reasoningprocess in the proof of Theorem 1. The completed stabilityanalysis is omitted due to limited space, however it will appearin our further work.

IV. SIMULATION RESULTSA simplified model of wheeled mobile robot moving on a

horizontal plane is used for simulation. The dynamic modelcan be expressed as

mx =1

P(τ1 + τ2) cos θ + λ sin θ

my =1

P(τ1 + τ2) sin θ − λ cos θ

I0θ =L

P(τ1 − τ2)

(52)

where m is the mass of the mobile robot, I0 being its inertiamoment around the vertical axis at point Q, P is the radiusof the wheels and 2L the length of axis of the front wheels,and τ1 and τ2 are the torques provided by motors.

The following four cases are simulated to illustrate theeffectiveness of the proposed controller:Case 1: Set-point stabilization: v1d = 0, v2d = 0;Case 2: Approach to a set-point: v1d = e−0.1t, v2d = e−t;Case 3: Tracking a line path: v1d = 1, v2d = 0;Case 4: Tracking a circle: v1d = 1, v2d = 1.The reference trajectory qd(t) = [xd(t), yd(t), θd(t)]

T isgenerated by the desired velocities v1d(t) and v2d(t) with theinitial conditions as qd(0) = [0, 0, 0]T .

In the simulation, the physical parameters of WMR aretaken as m = 12, I0 = 5, L = 0.2, P = 0.1. The initialpositions and velocities of the wheeled robot are picked asq(0) = [−0.5,−0.5, 0]T , v(0) = [0, 0]T . The control parame-ters are set as k1 = 3, k2 = 5, k3 = 5, Kp = diag[10, 10].The nonlinear time-varying function h(t, ze) is chosen ash(t, ze) = 15 tanh(z22e + z23e) sin 2t. Simulation results areshown in Figs. 3-6. It can be seen from the figures that thetracking errors all approach to zero and the mobile robotfollows the desired path with perfect performance.

V. CONCLUSION

A single controller is developed to simultaneously solvethe tracking and regulation problems of a mobile robot. Thecontroller fully exploit the existing results on stabilizationand tracking control for the equivalent chained form systemsof wheeled mobile robots. Simulation results confirm theeffectiveness of the proposed time-varying controller. Futureworks may to extend the proposed methodology to mobilerobots with dynamic uncertainties.

647

−1 −0.8 −0.6 −0.4 −0.2 0

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

x/m

y/m

Start point of WMR

(a) Robot position in (x, y) plane

0 5 10 15 20 25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t/s

Err

ors

x−xd

y−yd

θ−θd

(b) Tracking errors

Fig. 3. Simulation results of Case 1.

−2 −1 0 1 2 3 4 5 6 7−2

−1

0

1

2

3

4

5

6

7

8

9

x/m

y/m

ActualDesired

Start point of WMR

(a) Robot position in (x, y) plane

0 5 10 15 20 25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t/s

Err

ors

x−xd

y−yd

θ−θd

(b) Tracking errors

Fig. 4. Simulation results of Case 2.

0 5 10 15 20 25−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

x/m

y/m

ActualDesired

Strat point of WMR

(a) Robot position in (x, y) plane

0 5 10 15 20 25−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t/s

Err

ors

x−xd

y−yd

θ−θd

(b) Tracking errors

Fig. 5. Simulation results of Case 3.

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

1.5

2

2.5

x/m

y/m

ActualDesired

Start poiont of WMR

(a) Robot position in (x, y) plane

0 5 10 15 20 25−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t/s

Err

ors

x−xd

y−yd

θ−θd

(b) Tracking errors

Fig. 6. Simulation results of Case 4.

REFERENCES

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