International Journal of Control, Automation, and Systems (2011) 9(3):534-541 DOI 10.1007/s12555-011-0313-1

http://www.springer.com/12555

Adaptive Observer-based Trajectory Tracking Control of Nonholonomic Mobile Robots Bong Seok Park, Jin Bae Park*, and Yoon Ho Choi Abstract: In this paper, an adaptive observer-based trajectory tracking problem is solved for nonholonomic mobile robots with uncertainties. An adaptive observer is first developed to estimate the unmeasured velocities of a mobile robot with model uncertainties. Using the designed observer and the backstepping technique, a trajectory tracking controller is designed to generate the torque as an input. Using Lyapunov stability analysis, we prove that the closed-loop system is asymptotically stable with respect to the estimation errors and tracking errors. Finally, the simulation results are presented to validate the performance and robustness of the proposed control system against uncertainties. Keywords: Adaptive observer, backstepping, nonholonomic mobile robots, robot dynamics, robot kinematics.

1. INTRODUCTION The control of nonholonomic mobile robots has many difficulties due to the nonholonomic constraints. In addition, it is well known that any continuous time invariant feedback control law cannot make the wheeled mobile robots asymptotically stable [1]. Therefore, many efforts have been devoted to the tracking control of nonholonomic mobile robots [2-4]. All these controllers consider only the kinematic model of a mobile robot, which implies that “perfect velocity” tracking is assumed to generate the actual control inputs. But it is not easy to design a dynamic controller for the realization of perfect velocity tracking. Hence, some results have been proposed to design the controller at the torque level. In [5], the kinematic and torque controllers were integrated by using the backstepping technique. In [6] and [7], the sliding-mode control method was proposed for trajectory tracking of nonholonomic mobile robots. A waveletnetwork-based controller was designed for mobile robots with unstructured dynamics and disturbances in [8]. An adaptive torque control input for the mobile robot was proposed to solve the unified tracking and regulation control problems in [9]. It should be noted that all these schemes are concerned __________ Manuscript received March 14, 2009; revised April 9, 2010 and August 19, 2010; accepted January 3, 2011. Recommended by Editor Jae-Bok Song. This work was supported in part by the Brain Korea 21 in 2010 and by the Human Resources Development Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea Government Ministry of Knowledge Economy. (No. 2007-P-EP-HM-E08-0000). Bong Seok Park and Jin Bae Park are with the Department of Electrical and Electronic Engineering, Yonsei University, Shinchon-dong, Seodaemum-gu, Seoul 120-749, Korea (e-mails: [email protected], [email protected]). Yoon Ho Choi is with the Department of Electronic Engineering, Kyonggi University, Suwon-si, Kyonggi-do 443-760, Korea (e-mail: [email protected]). * Corresponding author. © ICROS, KIEE and Springer 2011

with state-feedback control. State feedbacks have been available when the velocity is measured, but unfortunately in practice, the use of velocity measurement is not generally desirable because velocity measurement devices are often contaminated by a considerable amount of noise. From this observation, many results on the output feedback control of a robot manipulator were addressed in [10-14]. However, many solutions proposed for the robot manipulators cannot directly be applied to the mobile robot due to the quadratic cross terms of unmeasured velocities and the nonholonomic constraints of the mobile robot. For solving this problem, the output feedback stabilization of nonholonomic systems in chained form with parameter uncertainties was considered in [15]. Besancon [16] proposed an output feedback controller using the solution of a differential equation. Do et. al [17] derived a coordinate transformation to cancel the velocity cross terms in the mobile robot dynamics. However, to the best of our knowledge, there is still no result available on the adaptive output feedback control problem for the mobile robot at the torque level. Accordingly, we propose an adaptive output feedback controller for the trajectory tracking of uncertain nonholonomic mobile robots. For the design of an adaptive output feedback control system, we first develop an adaptive observer, which can deal with parametric uncertainties. To solve the main difficulty caused by the quadratic velocity terms depending on the system parameters, we introduce a transformation matrix which makes the Coriolis-centripetal matrix to keep the skew symmetric property after transformation. Then, the backstepping technique is applied to design the controller for the mobile robot at the torque level. From Lyapunov stability analysis, we prove the closed-loop stability and derive the adaptation law. This paper is organized as follows: Section 2 introduces the kinematics and dynamics of the mobile robot. In Section 3, we propose an adaptive observer-

Adaptive Observer-based Trajectory Tracking Control of Nonholonomic Mobile Robots

based trajectory tracking controller for the mobile robot with uncertainties, and analyze the stability of the proposed control system. Section 4 presents some simulation results and Section 5 gives some conclusions. 2. PROBLEM STATEMENT We consider the mobile robot with two degrees of freedom as shown in Fig. 1. The kinematics and dynamics of nonholonomic mobile robots are described by the following differential equations [18]:  cos φ  q
= J ( q ) z = 0.5r  sin φ  R −1  Mz
+ C (q
) z + Dz = τ ,

cos φ  z  sin φ   1  ,  z2  R −1 

(1)

where q = [ x y θ ]T ∈ »3 ; x, y are the coordinates of P0, and φ is the heading angle of the mobile robot, z = [z1 z2]T ∈ » 2 ; z1 and z2 represent the angular velocities of right and left wheels, respectively. R is the half of the width of the mobile robot and r is the radius of the wheel, m12   0 φ
 m M =  11 , C (q
) = 0.5R −1r 2 mc d  , 
 m12 m11   −φ 0  0  d 2 −2 2 D =  11  , m11 = 0.25R r (mR + I ) + Iω ,  0 d 22  m12 = 0.25R −2 r 2 (mR 2 − I ), m = mc + 2mω , I = mc d 2 + 2mω R 2 + I c + 2 I m , τ = [τ1 τ 2 ]T .

In these expressions, d is the distance from the center of mass Pc of the mobile robot to the middle point P0 between the right and left driving wheels. mc and mω are the mass of the body and the wheel with a motor, respectively. Ic, Iω, and Im are the moment of inertia of the body about the vertical axis through Pc, the wheel with a motor about the wheel axis, and the wheel with a motor about the wheel diameter, respectively. The positive terms dii, i =1, 2, are the damping coefficients. τ is the control torque applied to the wheels of the mobile robot.

535

In this paper, it is assumed that the position measurements q are available but that the velocities are not. The problem to be solved is the asymptotic tracking problem of a reference trajectory without velocity measurement. In other words, the purpose is to design an adaptive output feedback control law u which allows the mobile robot to track the desired trajectory generated by the following reference robot:  x
r = vr cos φr ,   y
r = vr sin φr , 
φr = ω r ,

(2)

where xr, yr, and φr are the position and orientation of the reference robot. vr and ωr are the linear and angular velocities of the reference robot, respectively. Assumption 1: The reference signal η r = [vr ω r ]T and its first derivative are available, and bounded. In addition, vr > 0. Remark 1: In Assumption 1, vr > 0 means that this paper is only focused on a controller design for the trajectory tracking problem of mobile robots. That is, the case of vr = 0 is not considered. 3. MAIN RESULTS 3.1. Adaptive observer design To design the adaptive observer, we transform the system (1) into a more appropriate representation using the following relationship: η = B −1 z,

(3)

where v 1 1 R  η =  , B =  . r 1 − R  ω 

Hence, (1) can be written as cos φ q
= J (q)η =  sin φ  0

0 v 0    , ω 1    Qη
= −Cq (ω )η − Dqη + Bqτ ,

(4)

where 0 (m + m12 ) / R  Q =  11 , 0 R (m11 − m12 )    0 −ω  Cq (ω ) = 0.5R −1r 2 mc d  , ω 0   d11 + d 22  Dq =  2 R  d11 − d 22  2

Fig. 1. Mobile robot with two actuated wheels.

d11 − d 22   2 −1  , Bq = Q( MB) . R(d11 + d 22 )   2

Several properties of the dynamic equation (4) are as follows: Property 1: The matrix Q is symmetric and positive

Bong Seok Park, Jin Bae Park, and Yoon Ho Choi

536

definite. Let Qm denotes the minimum eigenvalue of Q. Then, Qm ≤ Q. Property 2: The Coriolis-centripetal matrix Cq(ω) is skew symmetric and upper bounded, that is ||CQ(ω)||2 ≤ CM ||ω||, where CM is the known positive constant. Property 3: The matrix Dq is symmetric and positive definite. Assumption 2: All parameters of robot dynamics (4) are constants but unknown, and lie in a compact set. Assumption 3: The velocity η of the mobile robot is bounded, that is ||η|| ≤VM, where VM is the known positive constant. Remark 2: The difficulty in designing an observerbased output feedback for the mobile robot without measuring velocities arises from the quadratic dependence on the velocities. To solve this problem, Do et al. [17] use the coordinate transformation which cancels the quadratic dependence on velocities. However, this method cannot be applied to the mobile robot with parameter uncertainties because the quadratic velocity terms depend on the system parameters. Thus, we introduce a transformation matrix Q to design the adaptive observer without using coordinate transformation. Multiplying both sides of (1) by Q, we can make the Coriolis-centripetal matrix to keep the skew symmetric property after the transformation using (3). The skew symmetric property is used to eliminate the Corioliscentripetal matrix in designing the daptive observer. We now design the adaptive observer to estimate the velocities of the mobile robot. Define the following coordinate transformation. x1 = qˆ, x2 = ηˆ − l1 H1 (q − x1 ),

(5)

0 0 H2 =  . 0 1  Remark 3: The dynamics (7) cannot be directly implementable because of the presence of the unknown signal η. Therefore, we use the dynamics (6) to implement the adaptive observer for the mobile robot, and the dynamics (7) is used to construct the observer error dynamics. Remark 4: In order to implement the adaptive observer proposed in (6), Qˆ −1 must be exist. To guarantee the existence of Qˆ −1 , we choose the proper bounds on the estimate of Q. The stability can be proved by using the following Lemma [19]. Lemma 1: Consider the scalar function α = (θ −

θˆ)( ρ − θˆ), with θ , θˆ, ρ ∈ » n and ai ≤ θ i ≤ bi . If θˆ = κ (a, b, ρ ) ρ , where κ (a, b, ρ ) is the diagonal matrix with entries 0,if θˆi ≤ ai , ρ i ≤ 0  κ i (a, b, ρ ) = 0,if θˆi ≥ bi , ρ i ≥ 0  1, otherwise 

then α ≤ 0. The proof of Lemma 1 can be obtained by substituting (8) into the scalar function α. From (4) and (7), we can obtain the following observer error dynamics: q
 = J (q )η − L1q , Qη
 = −Y1 (ζ 1 ,ηˆ ,τ )θ1 − Cq (ω )η − Cq (ω )ηˆ − Dqη

(9)

− L2 q − l1QH 2η ,

where where

0 0 0  H1 =  , 0 0 1 

Y1 (ζ 1 ,ηˆ ,τ )θ
1 = Q
ζ 1 + C
q (ωˆ )ηˆ + D
qηˆ − B
qτ ,

qˆ and ηˆ are the estimates of q and η, respectively. l1 is a design parameter. Under the new x-coordinates, we propose the following adaptive observer for the system (4): x
1 = J ( q )( x2 + l1 H1 (q − x1 )) + L1 (q − x1 ), x
2 = Qˆ −1 (−Cˆ q (ωˆ )ηˆ − Dˆ qηˆ + Bˆ qτ + L2 (q − x1 ))

(6)

where L1 and L2 are the observer gain matrices. Qˆ , Cˆ q , Dˆ q , and Bˆ q are the estimated matrices. Substituting

(5) into (6) yields

qˆ
= J (q)ηˆ + L1 (q − qˆ ), Qˆηˆ
= −Cˆ q (ωˆ )ηˆ − Dˆ qηˆ + Bˆ qτ + L2 (q − qˆ ) ˆ (η − ηˆ ), +l1QH 2

ζ 1 = [ζ 11 ζ 12 ]T = Qˆ −1 (−Cˆ q (ωˆ )ηˆ − Dˆ qηˆ + Bˆ qτ + L2 q
),

θ
1 = θ1 − θˆ1 , Q
= Q − Qˆ , C
q = Cq − Cˆ q , D
q = Dq − Dˆ q ,

B
q = Bq − Bˆ q , q
= q − qˆ, η
= η − ηˆ, and θˆ1 is the estimated value of unknown parameter

+ l1 H1 L1 ( q − x1 ),

where

(8)

(7)

vector θ1 ∈ »8 . The regressor matrix Y1 (ζ 1 ,ηˆ,τ ) is defined as

Y1 (ζ 1 ,ηˆ,τ ) ζ 0 ωˆ 2 =  11  0 ζ 12 vˆωˆ

vˆ ωˆ 0 0 vˆ ωˆ

−(τ1 + τ 2 ) 0

 0 . −(τ1 − τ 2 )  (10)

Then, we have the following result. Proposition 1: Suppose that the dynamics (4) is observed by the adaptive observer (6) under Assumption 1-3. If the adaptation law is chosen as follows:
θˆ1 = −κ1Γ1Y1T (ζ 1 ,ηˆ ,τ )η ,

(11)

Adaptive Observer-based Trajectory Tracking Control of Nonholonomic Mobile Robots

where Γ1 > 0 and κ1 is the diagonal matrix defined in (8), then the observer error dynamics (9) is asymptotically stable at the origin, and the estimated parameters are bounded. Proof: See Appendix A. Remark 5: The adaptation law (11) cannot be directly implementable due to the presence of the unknown signal η. Thus, we use the following adaptation law q
(t ) t θˆ1 (t ) = − ∫ f (ζ )d ζ − κ1Γ1 ∫ (Y1T J + L1q
)d λ , q
(0)

0

(17)

where α
v = k1 (ωˆ + ω ) ye − k1 (vˆ + v ) + k1vr cos φe + v
r cos φe − vr (ωr − ωˆ − ω ) sin φe , α
ω = ω
r + γ [ ye v
r − vr (ωˆ + ω ) xe 1

(12)

+ vr2 sin φe ]∫ cos(λφe )d λ − γ ye vr (ωr − ωˆ 0

1

− ω ) ∫ sin(λφe )λ d λ + γ k2 (ωr − ωˆ − ω ).

where f (ζ )

v
e = u1 − α
v , ω
e = u2 + l1ω − α
ω ,

537

0

= κ1Γ1Y1T

 cos φ J+ =  0

+

(ζ ,ηˆ,τ ) J ,

sin φ 0

We choose the actual control input u as follows:

0 . 1 

u1 = xe − k3ve + k1 (ωˆ ye − vˆ + vr cos φe ) + v
r cos φe − vr (ω r − ωˆ ) sin φe − k1ve ye2 − ve vr2 ,

In this paper, we use (12) to design the observer and (11) is only used for the proof of Theorem 1. 3.2. Controller design In this subsection, we design an adaptive output feedback controller using the backstepping technique. To prepare for the controller design, we write (4) and (7) in conjunction with the control torque τ = Bˆ −q1 (Cˆ q (ωˆ )ηˆ + ˆ ) as follows: Dˆ qηˆ − L2 q
+ Qu x
= vˆ cos φ + v cos φ , y
= vˆ sin φ + v sin φ , φ
= ωˆ + ω ,

(13)

v
ˆ = u1 , ω
ˆ = u + l ω , 2

1

1 u2 = φe − k4ωe + ω
r + γ ( ye v
r − vrωˆ xe γ 1

+ vr2 sin φe )∫ cos(λφe )d λ − γ ye vr (ωr 0

1

− ωˆ )∫ sin(λφe )λ d λ + γ k2 (ωr − ωˆ ) − γωe vr2 ( xe2 + ye2 ), 0

where k3 and k4 are positive constants. The stability of the control system is provided in the following therorem whose proof is given in Appendix B. Theorem 1: Consider the uncertain mobile robot system described by (4). Under the Assumptions 1-3, if the control input (18) together with the adaptive observer (6) is applied to the mobile robot, then the position and velocity tracking errors as well as the velocity estimation errors converge to zero. 4. SIMULATIONS

T

where u = [u1 u2] is the new control input to be designed. Step 1: Define the error for x, y, and φ as follows: xe = ( xr − x) cos φ + ( yr − y ) sin φ , ye = −( xr − x) sin φ + ( yr − y ) cos φ ,

(14)

φe = φr − φ .

Differentiating (14) along the solution of (13) yields xe = (ω
+ ωˆ ) ye − v
− vˆ + vr cos φe , y e = −(ω
+ ωˆ ) xe + vr sin φe , φ = ω − (ω
+ ωˆ ). e

(15)

r

Choose a virtual control α= [α v αω ]T of ηˆ as follows: α v = k1 xe + vr cos φe , 1

αω = ω r + γ vr ye ∫ cos(λφe )d λ + γ k2φe ,

(16)

0

where k1, k2, and γ are the positive constants. Step 2: Define the error ηe = [ve ωe ]T for ηˆ such as ηe = ηˆ − α . Then, differentiating it along the solutions of (13) and (16) yields

In this section, we perform the simulation for the tracking control of the nonholonomic mobile robot to demonstrate the validity of the proposed adaptive observer-based control method. The physical parameters for the mobile robot are chosen as R = 0.75m, d = 0.3m, r = 0.15m, mc= 28kg, mω=1kg, Ic=15.625kg, Iω =0.005 kg · m2, Im=0.0025kg · m2, and d11 = d22 = 5m. In this simulation, we assume that all of these parameters are unknown. The design parameters are chosen as k1 = k2 = 1, k3 =1.1, k4 =26.6, γ =2, l1=7, L1= diag(2,2,2), and Γ = diag(0.01, 0.01, 0.01, 2, 0.01,1, 0.01, 0.01) where diag(·) denotes the diagonal matrix. The reference velocities vr, ωr for generating the reference trajectory are chosen as follows: πt 0 ≤ t < 10 : vr = 0.15(1 − cos( 10 )) m/s, ω r = 0 rad/s, πt 10 ≤ t < 30 : vr = 0.3m/s, ωr = 0.15(1 + cos( 10 )) rad/s, πt 30 ≤ t < 50 : vr = 0.3m/s, ωr = −0.15(1 + cos( 10 )) rad/s,

50 ≤ t < 60 : vr = 0.3m/s, ω r = 0 rad/s.

The initial postures for the reference robot and the actual robot are ( xr yr φr ) = (0 m 0 m 0 rad), ( x y φ ) =(−0.5m

538

Bong Seok Park, Jin Bae Park, and Yoon Ho Choi

−0.5m 0.5rad), respectively. In order to demonstrate the effectiveness of the proposed approach, the noise for the position and orientation measurements is generated by a normal distribution random noise. The variance of the noise is chosen to represent a 10% error in terms of the position and orientation measurements. For the practical applicability of the controller, bounds on the torque is imposed: τ1 = τ 2 ≤ 30kgf ⋅ cm. The simulation results are shown in Figs. 2 and 3. Fig. 2 shows the tracking result and the tracking errors of the nominal control without the adaptive observer. Fig. 3 shows the simulation results for the proposed scheme. In Fig. 2, one can see that the measurement errors degrade the performance. On the other hand, the tracking and observer errors tend to zero in Fig. 3. In other words, the performance of the system has been improved with respect to the nominal control without the adaptive observer. Therefore, the simulation results demonstrate the effectiveness of the proposed adaptive observerbased controller. Moreover, to show the response of different observer gain, we simulate with the observer gain L1= diag(0.01, 0.01, 0.01). Fig. 4 shows the simulation results and Table 1 shows the average errors. As shown in Table 1, the high observer gain has the smaller tracking errors. Thus, one can see that the performance of the system will improve with the higher gain.

(a) Tracking performance.

(b) Tracking errors.

(a) Tracking performance.

(c) Control inputs.

(b) Tracking errors.

(d) Observer errors.

Fig. 2. Simulation results of the nominal control without the adaptive observer.

Fig. 3. Simulation results of the proposed scheme with L1=diag(2.5,2.5,2.5).

Adaptive Observer-based Trajectory Tracking Control of Nonholonomic Mobile Robots

539


− Dqη − L2 q − l1QH 2η ) − θ1Γ1−1θˆ1 = −q T L1q − ηT ( Dq + Cq (ω ))η + ηT ( J T (q) − L2 )q
− θ1T (Y1Tη + Γ1−1θˆ1 ) − ηT l1QH 2η − ηT Cq (ω )ηˆ.

(A.2) Substituting (11) into (A.2), and considering Lemma 1, we obtain

V
o = −qT L1q − ηT Dqη + ηT ( J T (q) − L2 )q − ηT l1QH 2η − ηT H 3η,

(A.3)

where (a) Tracking performance.

0 −ω  H 3 = 0.5R −1r 2 mc d  . 0 v 

From Properties 1-2 and Assumption 3, η
T (l1QH 2 + H 3 )η
can be represented as follows: η
T (l1QH 2 + H 3 )η
≥ (l1Qm − CM VM )η
T η
.

Then, (A.3) can be rewritten as V
o ≤ −qT L1q − ηT Dqη + ηT ( J T (q) − L2 )q − (l1Qm − CM VM )ηTη.

Therefore, by choosing L2 = J T (q), we obtain

(b) Tracking errors. Fig. 4. Simulation results of the proposed scheme with L1=diag(0.01,0.01,0.01). Table 1. Average tracking errors. xe

ye

φe

L1 = diag(2.5, 2.5, 2.5)

0.0122

0.0447

0.0069

L2 = diag(0.01,0.01,0.01)

0.0170

0.0377

0.0056

In this paper, an adaptive observer-based controller for nonholonomic mobile robots with parametric uncertainties has been proposed. First, the adaptive observer has been designed to deal with parametric uncertainties. To solve the main difficulty caused by the quadratic velocity terms depend on the system parameters, we transform the robot dynamics. Second, the backstepping technique is applied to design the controller for mobile robots at the torque level. The closed-loop stability is proved by using Lyapunov stability theory. APPENDIX A Consider the Lyapunov function 1 T (q
q
+ η
T Qη
+ θ
1T Γ1−1θ
1 ). 2

(A.4)

Since Qm, CM, and VM are bounded, we can choose l1 such that

V
o ≤ −qT L1q − ηT Dqη,

(A.5)

which means E1 = (q
,η
) and θ
1 are bounded. This implies that θˆ is bounded. Moreover, E1(t) is 1

5. CONCLUSIONS

Vo =

V
o ≤ −qT L1q − ηT Dqη − (l1Qm − CM VM )ηTη.

uniformly continuous because its time derivative along the solution of the closed loop system is bounded. Hence, it satisfies the conditions of Barbalat’s lemma, which then guarantees that E1(t)→0 as t→∞. APPENDIX B Consider the Lyapunov function candidate V = Vc + Vo ,

(B.1)

where Vc = 12 ( xe2 + ye2 + γ1 φe2 + ve2 + ωe2 ), and Vo is given by (A.1). The time derivative of Vc along (15) and (17) is V
c = xe [(ω + ωe + αω ) ye − (ve + α v ) − v + vr cos φe ] ye [−(ω + ωe + αω ) xe + vr sin φe ]

(A.1)

The time derivative of Vo along (9) yields Vo = q
T ( J (q)η
− L1q
) + η
T (−Y1θ
1 − Cq (ω )η
− Cq (ω
)ηˆ

1 + φe [ω r − ω − ωe − αω ]ve (u1 − α
v ) γ + ωe (u2 + l1ω − α
ω ).

Substituting (18) into (B.2) yields

(B.2)

Bong Seok Park, Jin Bae Park, and Yoon Ho Choi

540

φ
e = −γ k2φe + p (t ),

1  e − k2φe2 − φeω − k3ve2 − k1ve2 ye2 V
c = −k1 xe2 − vx γ

1

where p(t ) = −γ vr ye ∫ cos(λφe )d λ − ωe − ω
. A direct

 e − vr ω ve sin φe − k4ωe2 − ve2 vr2 − k1ω ye ve + k1vv

0

1

− γωe2 vr2 ( xe2 + ye2 ) + l1ωeω + γ vr ω xeωe ∫ cos(λφe )d λ 0

1

 e. − γ vr ω yeωe ∫ sin(λφe )d λ + γ k2ωω 0

application of Lemma 2 in [4] gives that p(t) converges to zero. Therefore, ye(t) must converge to zero because (φe , ωe , ω
) tends to zero and vr > 0 by Assumption 1.

(B.3) Using Young’s inequality, we obtain

[1]

φ 2 ω
2 x 2 v
2 − k3ve2 Vc ≤ −k1 xe2 + e + − k2φe2 + e + γ 4γ 2 2 k ω
2 ω
2 + 1 + vr2 ve2 + 4 4 2 2 2 l ω k v
+ k1ve2 + 1 − k4ωe2 − γωe2 vr2 ( xe2 + ye2 ) + 1 e 4 2 2
γ k ω
2 ω
2 γω + + γ vr2 xe2ωe2 + + γ vr2 ye2ωe2 + γ k2ωe2 + 2 2 2 4 1 2 1 2 ≤ −(k1 − ) xe − (k2 − )φe − (k3 − k1 )ve2 γ 2 − k1ve2 ye2

− ve2 vr2

[2]

+ k1ve2 ye2

[3]

[4]

l12 1 k 3 γ γk − γ k2 )ωe2 + ( + 1 + + + 2 )η
Tη
. 2 4γ 4 4 2 4 (B.4)

[5]

Then, from (A.4) and (B.4), we obtain the time derivative of (B.1) as follows:

[6]

− ( k4 −

V
= V
c + V
o 1 1 ≤ −(k1 − ) xe2 − (k2 − )φe2 − (k3 − k1 )ve2 γ 2 − ( k4 −

l12 − γ k2 )ωe2 2

− (l1Qm − CM VM −

(B.7)

[7] (B.5)

1 k1 3 γ γ k2 T − − − − )η η 4γ 4 4 2 4

− qT L1q − ηT Dqη.

By choosing k1 = 1 + k1* , k2 = 1 + k2* , k3 = k1 + k3* , k4 = 2 γ l12 k γ γk + γ k2 + k4* , and l1 = (CM VM + 3 + 1 + 1 + + 2 2 4 4 4γ 2 4 +l1* ) / Qm , we obtain

[8]

[9]

[10]

V
≤ −k1* xe2 − k2*φe2 − k3*ve2 − k4*ωe2 − l1*η Tη − q T L1q , (B.6)

where k1* , k2* , k3* , k4* , and l1* are the positive constants. From the definition of V and (B.6), we conclude that V(t) is bounded, which implies that E2 = ( xe , φe , ve , ωe ,η
, q
) are bounded for all t ≥ t0 ≥ 0. By integrating both sides of (B.6) and Barbalat’s lemma in [21], we have that E(t) → 0 as t → 0. To prove the convergence of ye, we substitute αω given in (16) into the φ
e -dynamics in (15) giving

[11]

[12]

[13]

REFERENCES R. W. Brockett, “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory, Birkhauser, Basel, pp. 181-191. 1983. Y. Kanayama, Y. Kimura, F. Miyazaki, and T. Noguchi, “A stable tracking control method for a nonholonomic mobile robot,” Proc. IEEE Int. Conf. on Intelligent Robotics and Automation, pp. 12361241, 1991. C. Samson and K. Ait-Abderrahim, “Feedback control of a nonholonomic wheeled cart in Cartesian space,” Proc. IEEE Int. Conf. Robot and Automation, pp. 1136-1141, 1991. Z. P. Jiang and H. Nijmeijer, “Tracking control of mobile robots: a case study in backstepping,” Automatica, vol. 33, no. 7, pp. 1393-1399, July 1997. R. Fierro and F. L. Lewis, “Control of a nonholonomic mobile robot: backstepping kinematics into dynamics,” Journal of Robotic Systems, vol. 14, no. 3, pp. 149-163, 1997. J. M. Yang and J. H. Kim, “Sliding mode control of trajectory tracking of nonholonomic wheeled mobile robots,” IEEE Trans. Robot. Automat., vol. 15, no. 3, pp. 578-587, June 1999. D. Chwa, “Sliding-mode tracking control of nonholonomic wheeled mobile robots in polar coordinates,” IEEE Trans. Control Systems Technol., vol. 12, no. 4, pp. 637-644, July 2004. C. D. Sousa, E. M. Hemerly, and R. K. H. Galvao, “Adaptive control for mobile robot using wavelet networks,” IEEE Trans. Sys., Man, and Cybern., vol. 32, no. 4, pp. 493-504, August 2002. W. E. Dixon, M. S. de Queiroz, D. M. Dawson, and T. J. Flynn, “Adaptive tracking and regulation of a wheeled mobile robot with controller/update law modularity,” IEEE Trans. Contr. Syst. Technol., vol. 12, no. 1, pp. 138-147, January 2004. E. Kim, “Output feedback tracking control of robot manipulators with model uncertainty via adaptive fuzzy logic,” IEEE Trans. Fuzzy Systems, vol. 12, no. 3, pp. 368-378, June 2004. F. Calugi, A. Robertsson, and R. Johansson, “Output feedback adaptive control of robot manipulators using observer backstepping,” Proc. IEEE Int. Conf. on Intel. Robots and Systems, pp. 2091-2096, 2002. S. J. Yoo, J. B. Park, and Y. H. Choi, “Adaptive output feedback control of flexible-joint robots using neural networks: dynamic surface design approach,” IEEE Trans. Neural Networks, vol. 19, no. 10, pp. 1712-1726, October 2008. S. J. Yoo, J. B. Park, and Y. H. Choi, “Output feed-

Adaptive Observer-based Trajectory Tracking Control of Nonholonomic Mobile Robots

[14]

[15]

[16]

[17]

[18]

[19] [20]

[21]

back dynamic surface control of flexible-joint robots,” Int. Journal of Control, Automation, and Systems, vol. 6, no. 2, pp. 223-233, April 2008. J. M. Valenzuela, V. Santibanez, and R. Campa, “On output feedback tracking control of robot manipulators with bounded torque input,” Int. Journal of Control, Automation, and Systems, vol. 6, no. 1, pp. 76-85, February 2008. S. S. Ge, Z. Wang, and T. H. Lee, “Adaptive stabilization of uncertain nonholonomic systems by state and output feedback,” Automatica, vol. 39, no. 8, pp. 1451-1460, August 2003. G. Besancon, “Global output feedback tracking control for a class of Lagrangian systems,” Automatica, vol. 36, no. 12, pp. 1915-1921, December 2000. K. D. Do and J. Pan, “Global output-feedback path tracking of unicycle-type mobile robots,” Robotics Comput-Integr Manuf, vol. 22, no. 2, pp. 166-179, April 2006. T. Fukao, H. Nakagawa, and N. Adachi, “Adaptive tracking control of a nonholonomic mobile robot,” IEEE Trans. Robot. Automat., vol. 16, no. 5, pp. 609-615, October 2000. M. R. Sirouspour and S. E. Salcudean, “Nonlinear control of hydraulic robots,” IEEE Trans. Robot. Automat., vol. 17, no. 2, pp. 173-182, April 2001. M. Erlic and W. S. Lu, “A reduced-order adaptive velocity observer for manipulator control,” IEEE Trans. Robot. Automat., vol. 11, no. 2, pp. 293-303, April 1995. J. J. Slotine and W. P. Li, Applied Nonlinear Control, Prentice-Hall, Eaglewood Cliffs, NJ, 1991.

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Bong Seok Park received his B.S. and M.S. degrees in Electrical and Electronic Engineering from Yonsei University, Seoul, Korea in 2005 and 2008, respectively, where he is currently working toward a Ph.D. degree. His research interests include nonlinear control, adaptive control, formation control, and the control of mobile robots. Jin Bae Park received his B.S. degree in Electrical Engineering from Yonsei University, Seoul, Korea, in 1977 and his M.S. and Ph.D. degrees in Electrical Engineering from Kansas State University, Manhattan, in 1985 and 1990, respectively. Since 1992 he has been with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, where he is currently a Professor. His research interests include robust control and filtering, nonlinear control, mobile robot, fuzzy logic control, neural networks, and genetic algorithms. He has served as vice-president for the Institute of Control, Robotics and Systems. He served as Editor-in-Chief for the International Journal of Control, Automation, and Systems. Yoon Ho Choi received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from Yonsei University, Seoul, in 1980, 1982 and 1991, respectively. Since 1993, he has been with School of Electronic Engineering at Kyonggi University, where he is currently a Professor. From 2000 to 2002, he was with the Department of Electrical Engineering, Ohio State University, where he was a Visiting Scholar. His research interests include nonlinear control theory, intelligent control, biped and mobile robots, web-based control system and wavelet transform. He had served as a director for the Institute of Control, Robotics and Systems.