Abstract— This paper addresses the trajectory tracking control problem of a nonholonomic wheeled mobile robot in presence of modeling uncertainties. A trajectory tracking controller is proposed by combination of an adaptive feedback linearizing controller and an adaptive robust PID control law. A SPR- Lypunov stability analysis demonstrates that tracking errors and parameters estimation errors are globally uniformly ultimately bounded (GUUB) and exponentially converge to a small ball containing the origin. The proposed adaptive-robust tracking controller is successfully applied to a nonholonomic wheeled mobile robot and experimental results are presented to validate the effectiveness of the proposed controller.
Index Terms – Adaptive Control, Feedback linearization, Trajectory tracking, Uncertainty, Wheeled mobile robots.
I. INTRODUCTION
HE motion control problem of nonholonomic mechanical systems has attracted a remarkable attention during past years [9]. According to the basic theorem of Brockett [1], such systems can not be stabilized at any equilibrium configuration by smooth static state feedback.
From a review of the literature [9], this well-known theorem and challenging nonlinear nature of nonholonomic systems motivate many researchers to be focused on motion control problem of these systems. In spite of much effort on design of stabilizing controllers for nonholonomic systems, limited types of feedback controllers have been developed. Some fundamental results on modeling, control and stabilization of nonholonomic systems have been reported in [3] and [4].
Some researchers proposed stabilizing controllers for nonholonomic systems by converting them into the chained form [10], [18], [20]. Among various motion control problems of nonholonomic systems, most of researches have been concentrated on the tracking of a geometric path with an associated timing law so-called trajectory tracking. A variety of control algorithms for trajectory tracking problem is developed in the literature [8], [19], [22].
Differential geometric control theory may be traditionally utilized to design feedback linearizing controllers to solve trajectory tracking problem of nonholonomic systems [14].
From a review of the literature, following results are summarized with regard to application of this technique for nonholonomic robotic systems: 1. a nonholonomic robot is
Kh. Shojaei and A. M. Shahri are with the Electrical Engineering Department, Iran University of Science and Technology, Tehran, Iran. (e- mail: [email protected], [email protected]).
B. Tabibian is with the Computer Engineering Department, Iran University of Science and Technology, Tehran, Iran. (e-mail:
controllable and its equilibrium point can be made Lyapunov stable, but can not be made asymptotically stable by a smooth static state feedback [4]. 2. It has been shown that nonholonomic systems are not input-state linearizable.
However, if we choose a proper set of output equations, it may be input-output linearizable [6], [8]. 3. The dimension of the largest feedback linearizable subsystem of the kinematic and dynamic model of a nonholonomic system is two times of the number of its actuators and the controllability index of each subsystem is equal to 2 [3]. 4.
The internal dynamics of these systems are stable [13].
There exists invaluable works that propose trajectory tracking controllers based on feedback linearization for nonholonomic systems which most of them are developed for nonholonomic wheeled mobile robots [3], [8], [15], [19].
However, they use exact kinematic and dynamic model of nonholonomic systems. Feedback linearization is based on cancellation of nonlinear terms. Therefore, this cancellation may not be achieved perfectly in presence of uncertainties in nonholonomic robotic systems.
Based on the current knowledge of the authors, there are few works to propose robust feedback linearizing controllers for trajectory tracking of nonholonomic robotic systems in presence of both parametric and nonparametric uncertainties. This paper will be focused on the tracking problem of a nonholonomic wheeled mobile robot (WMR) and its main contributions lie in the following points: 1. an adaptive-robust controller is designed to solve the trajectory tracking problem of an electrically driven nonholonomic WMR in presence of parametric and nonparametric uncertainties. The proposed tracking controller comprises an adaptive feedback linearizing control law in the inner loop and an adaptive robust PID control law in the outer loop. 2.
The SPR-Lyapunov stability analysis is applied to demonstrate that the tracking errors are globally uniformly ultimately bounded (GUUB) and they exponentially converge to a small ball containing the origin. 3.
Experimental results on a commercial nonholonomic WMR are presented to evaluate the effectiveness of the proposed controller.
The rest of the paper is organized as follows. In section II, the kinematic and dynamic models of a nonholonomic WMR are briefly reviewed. Section III proposes the trajectory tracking controller and the stability analysis of the closed-loop system. Section IV presents experimental results on robuLAB 10 WMR to verify the effectiveness of the proposed control law. Finally, section V concludes the paper.
Adaptive-robust feedback linearizing control of a nonholonomic wheeled mobile robot
Khoshnam Shojaei, Alireza Mohammad Shahri, Member, IEEE, Behzad Tabibian
T
Advanced Intelligent Mechatronics Montréal, Canada, July 6-9, 2010
II. PROBLEM STATEMENT
In this section, a mathematical formulation of a non- holonomic differential drive is reviewed. It is assumed that the WMR has two motorized wheels on an axis that independently drive the robot. The Center of mass of the robot is located in PC. The point PO is the origin of the local coordinate frame that is attached to the WMR body. The point PL is a virtual reference point on x axis of the local frame at a distance L (look-ahead distance) of PO (see Fig.
1). The WMR parameters are listed as follows:
bis the distance between each driving wheel and the axis of symmetry; r is the radius of each driving wheel;
mC denotes 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;
C 2 w
m=m + m is the total mass of the robot; IC is the moment of inertia of the robot without the driving wheels and the rotors of the motors about a vertical axis through PC ; Iw denotes the moment of inertia of each wheel and the motor rotor about the wheel axis; Im is the moment of inertia of each wheel and the motor rotor about a wheel diameter; adenotes the length of the platform in the direction perpendicular to the driving wheel axis; dis the distance from PO to PC along the positive x-axis.
By considering assumptions in [2], [3], three generalized coordinates, q, are assumed to describe the WMR model. It is possible to write the kinematic equation of WMR motion under some velocity constraints as:
cos 0
( ). ( ), ( ) sin 0
0 1
q S q v t S q
ϕ ϕ
⎡ ⎤
⎢ ⎥
= = ⎢ ⎥
⎢ ⎥
⎣ ⎦
(1) where v is made up of linear and angular velocities. The WMR dynamic model is derived by Lagrangian mechanics.
After calculation of Lagrange equation, the dynamic model of WMR may be written as follows:
( ) ( , ) ( , ) ( ). ( )T
M q q+C q q q U q q+ =B q τ−A q λ (2) where q = [xO, yO, φ]T (see Fig. 1), M(q) is the inertia matrix, C is a matrix which denotes the Coriolis and centripetal forces. B(q) is the input transformation matrix. τ is the torque vector which is generated by two wheels actuators.
The vector U denotes model uncertainties. A(q) is a full-rank matrix which is obtained from motion constraints (see [2]), and λ is the vector of constraint forces. These matrices are expressed as follows:
0 sin
( ) 0 cos
sin cos
C C
C C
m m d
M q m m d
m d m d I
ϕ ϕ
ϕ ϕ
⎡ ⎤
⎢ ⎥
=⎢ − ⎥
⎢ − ⎥
⎣ ⎦
(3)
Fig. 1. Configuration of a non-holonomic differential drive WMR
0 0 cos cos cos
( , ) 0 0 sin , ( ) 1 sin sin ,
0 0 0
C C
m d
C q q m d B q
r b b
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
=⎢ ⎥ = ⎢ ⎥
⎢ ⎥ ⎢ − ⎥
⎣ ⎦ ⎣ ⎦
(4) where
2 2
2 , 2 2
C w C m C w
m=m + m I=I + I +m d + m b (5) To include actuator dynamic in (2), it is assumed that the robot wheels are driven by two similar brush DC motors with mechanical gears. The electrical equation of the motor armature is written as follows:
a a a/ a a b
u =L di dt+R i +K θ (6) where ua denotes the vector of the motor input voltage, Kb is the back EMF constant, La and Ra are the armature inductance and resistance, respectively. By ignoring the inductance of armature circuit, and considering the relation between torque and armature current and relations between torque and velocity before and after gears,
. , . , . ,
M K iτ a n M M n
τ = τ = τ θ = θ (7) the delivered torque vector to the right and left wheels by actuators is given by:
1 ar 2 ,
r r
al
l l
k u k
u
τ θ
τ θ
⎡ ⎤ ⎡ ⎤
⎡ ⎤= ⎢ ⎥− ⎢ ⎥
⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (8) where
1 ( / a) , 2 . b 1
k = nKτ R k =n K k (9) where n is the gear ratio and Kτ is torque constant of the motor. Considering the relation between angular velocities of wheels and pseudo-velocities, we have
1 2
1
, 1
ar
r r
al
l r
b
u v r r
k k X X
u b
r r
τ
τ ω
⎡ ⎤
⎢ ⎥
⎡ ⎤
⎡ ⎤ ⎡ ⎤
= ⎢ ⎥− = ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎢⎣ − ⎥⎦
(10)
1 2
( ) ( , ) ( ).( . a ) ( )T
M q q+C q q q U+ =B q k u −k X v −A q λ (11) For controller design purposes, the state space representation can be derived by taking time derivative of the kinematic model (1):
( ). ( ).
q=S q v +S q v (12) Next, by replacing (1) and (12) in (11) and multiplying the result by ST and considering that A(q).S(q)=0 (see [2], [3]), we obtain
( ) ( ). ( ) 1 . a
Mv t +C q v t +U =k B u (13) where
, ( ) 2 ,
, ( , ) ( , )
T T T
T T
M S MS C q S MS S CS k BX B S B U q q S U q q
= = + +
= = (14)
The kinematic model (1) and dynamic equation shown by (13) can be integrated into the following state space representation in companion form:
1 1
( ) 1
( ) ( , ) ( )
0
( ) , ( , ) ,
. 0
0
, ( ( ) ( ) ( , ))
a
x
x f x g x u x Sv
f x g x
k M B
M C q v t U q q ξ
θ ξ
θ −
= −
= + +
= =
− +
⎡ ⎤
⎡ ⎤ ⎢ ⎥
⎢ ⎥ ⎣ ⎦ ⎣ ⎦
⎡ ⎤
⎢ ⎥
⎣ ⎦
(15)
where x = [qT,vT]T is the state vector. This representation allows us to apply the differential geometric control theory for trajectory tracking problem.
III. CONTROLLER DESIGN
Since the number of degrees of freedom of the nonholonomic WMR is instantaneously two, we choose two independent position output equations as follows:
( ) [ O cos , O sin ]T
y =h x = x +L ϕ y +L ϕ (16) 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, we obtain:
2 ( ) ( ) ( )
h , y L h xf L L h xf D x ua
y =J Sv = + ξ + (17)
where Jh is the Jacobian matrix, L denotes the Lie derivative, D(x)=LfLgh(x) is a decoupling matrix. Assuming that det(D(x))≠0, the system (17) is input-output linearizable.
Assumption 1: Measurements of all states, i.e.
[ T, T]T
x = q v , are available in real-time.
Assumption 2: Pseudo-velocities, i.e. ( )v t =[vr( ),t ωr( )]t T, are bounded for all time t >0.
According to certainty equivalence principle, following decoupling control law is proposed:
1 2
( ) (. ),
a ˆ f
u =D− x η−L h (18) where
D ˆ
is the estimate of the decoupling matrix D which is defined as follows:( ) ˆ ( )
ˆ g f ,
D x =L L h x (19) By substituting (18) in (17), one gets
2 ( ) ( ). ˆ 1( ).( 2 ) ( )
f f f
y =L h x +D x D− x η−L h +L L h xξ (20)
After some manipulation, equation (20) may easily be re- written in the following form
1 2
( )x ˆ ( ).(x f ) f ( ),
y = +η D D− η−L h +L L h xξ (21) where
( )x ( )x ˆ( )x
D =D −D (22) By considering the presented model of the WMR, the term
( )x ˆ 1( )x
D D− in equation (21) is computed in terms of parameters estimation error and the following parametric model may be readily derived:
. f ( )
y =η+W θ +L L h xξ (23) where θ is the vector of parameters estimation errors and the matrix W is a 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 [2], [12]. Assume that the external control input ηj for j-th subsystem of (24) is chosen such that j-th output, yj(t), tracks the desired output, yjr(t), in the outer loop:
0 ( ) ,
t
j yjr kpj ej k evj j ka j ej d vR j
η = − − −
∫
τ τ + (24)where kpj, kvj and kaj denote gains of the outer loop controller, vRj is a robust control term which must be designed to compensate for nonparametric uncertainty LξLfh(x). Applying the robustified PID control law given in (24) to j-th subsystem of (23) leads to the following error equation:
0 ( )
( ( )) , 1, 2
t
j vj j pj j a j j j
j R j
f
e k e k e k e d W
v j
L L h xξ
τ τ θ
+ + + =
+ + =
∫
(25) where Wj is the j-th row of regression matrix. For the purpose of adaptation, one may use the following filtered error signal for j-th output:1 2 0t ( )
j ej jej j ej d
ε = +β +β
∫
τ τ (26) Since ej =yj −yjr is known as a function of measured states by considering (17), it is obvious that εj is available.The parameters β1j and β2j are chosen such that the transfer function of the closed-loop error system (25) and (26) will be strictly positive real (SPR) [12]:
2
1 2
3 2
( ) j j
j
vj pj a j
s s
H s
s k s k s k
β β
+ +
= + + + (27) The transfer function (27) is SPR if and only if Hj(s) is analytic in Re[s]>0 [12] and
2
| |
Re[ ( )] 0, ( , ),
lim [ ( )] 0,
j
j
H j
Re H j
ω
ω ω
ω ω
→∞
> ∀ ∈ −∞ ∞
> (28) By applying these conditions to (27), one may show that Hj(s) is SPR if following conditions hold:
2 2
1 2
, , 2
/ , /
pj vj pj vj a j pj a j vj
j pj vj j a j vj
k k k k k k k k
k k k k
β β
< > >
= = (29) Accordingly, by positive real lemma (see [12]), there exists positive definite matrices Pj and Qj such that
T , T
j j j j j j j j
A P +P A = −Q P B =C (30) where matrices Aj, Bj and Cj are defined by minimal state space realization of (25) and (26) in the following form:
( ( )
,
j j j j j ( ))j R j
j j j
X A X B W f v
C X
L L h xξ θ
ε
= + + +
= (31)
where
0
2 1
( )
0 1 0 0
0 0 1 , 0 ,
1 1
t T
j j j j
T j
j j j j
a j pj vj
X e d e e
A B C
k k k
τ τ
β β
⎡ ⎤
= ⎢⎣ ⎥⎦
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
=⎢ ⎥ =⎢ ⎥ =⎢ ⎥
⎢− − − ⎥ ⎢ ⎥⎣ ⎦ ⎢⎣ ⎥⎦
⎣ ⎦
∫
(32)
As a result, the error equation for the entire system is written as:
( f ( ) R), 1
X =A X +B Wθ+L L h xξ +v E =CX (33) where A, B and C are block diagonal matrices:
1 2 1 2 1 2
1 2
[ , ]
( , ), ( , ), ( , ),
, ,
T T T T
X X X Xn m
A diag A A B diag B B C diag C C
= −
= = =
… (34)
The Lyapunov equation (30) is also written for entire system as follows:
T , T
A P+PA= −Q PB =C (35) where
1 2 1 2
( , ), ( , ),
P=diag P P Q =diag Q Q (36) In order to design the robust control vR, we assume that
||LξLfh(x)|| < ρ(v) where ρ(v) is an upper bounding function.
Considering (15), LξLfh(x) is calculated as follows:
1
2
( ) ( ) (( ( )
) ( ) ( ) )
( ) h
d
f q S q M C q
k BX v t F q L L h xξ J
τ
−
+ + +
= − (37)
where
1 2
( ) [ T ( ), T ( )]T
h h h
J q = J q J q (38) is the Jacobian matrix which is made up of Jacobians of all output equations with respect to q. By considering the model properties of the WMR, one may conclude that
[ ]
2
1 2 3
2
1 2 3
( ) ,
( ) 1 , T
v v v
Y v v v
ρ α α α
α α α α
= + +
=⎡⎣ ⎤⎦ =
(39) and α is a vector of unknown constants of the bounding function. Then, the following theorem is presented to solve the integrated kinematic and dynamic trajectory tracking control problem of the nonholonomic WMR in presence of both parametric and nonparametric uncertainties.
Theorem 1: Under assumptions 1 and 2, the following adaptive-robust tracking controller guarantees that the position and velocity tracking errors,
e
j, e
j and parameter estimation errors are globally uniformly ultimately bounded (GUUB) and exponentially converge to a small ball containing the origin:1
2 1 0
1
1 1 1 1 0
2 1 2 2 0
( ).( 2 ),
ˆ ( )
( ) ,
ˆ ( )
ˆ (ˆ ),
ˆ (ˆ ), ˆ( ) ( ).ˆ
ˆ
a
t
r p v a
T
T
u x f
E v
y K e K e K e d
v E W E
Y E v Y v
D η L h
η τ τ ρ
ρ γ
θ σ θ θ
α σ α α ρ α
= − −
= − − − −
+
= Γ − Γ −
= Γ − Γ − =
∫
(40)where D xˆ ( ) denotes the estimation of decoupling matrix, W is the regression matrix, E1 is a vector of filtered error signals , Γ1 and Γ2 are symmetric and positive definite matrices as the adaptive gains, Kp , Kv and Ka are diagonal matrices which denote proportional, derivative and integral gains of the PID control law for the entire system, respectively. Parameters σ1, σ2 and γ are also small positive constants.
Proof: Let us consider the following Lyapunov function:
1 1
1 2
( , , ) 1/ 2 T 1/ 2 T 1/ 2 T
V X θ α = X PX + θ Γ−θ+ α Γ−α (41) where α α α= − ˆ and θ θ θ= − ˆ. By differentiating (41) and applying (33) and (35), we have
1
1 1
1
1 1 2
( , , ) ( )
( ( ))
1/ 2 T T T
T T T
R f
V X X QX W E
v E L L h xξ E
θ α θ θ
α α
−
−
= − + + Γ
+ + + Γ (42)
Considering the adaptation laws in (40), we write (42) as
1 0 1
1 1 2 0
(ˆ )
( ( )) (ˆ )
1/ 2 T T TR
T T T T
f
V X QX v E
E Y E
L L h xξ
θ σ θ θ
α α σ α α
= − + − +
+ − + − (43)
By using the parametric form of the upper bound of LξLfh(x), i.e. ρ =Yα, we have
1 1
1 1 0 2 0
ˆ ˆ
( ) ( )
1/ 2 T TR T T
T T T T
V X QX v E Y E
Y E
α
α θ σ θ θ α σ α α
≤ − + +
− + − + − (44)
Then, by choosing the robust control law as
2
1ˆ /(ˆ 1 )
vR = −E ρ ρ E +γ (45) and substituting it in (44), one obtains
2
1 1
1 1
1 0 2 0
1 1
1 0 2 0
( ˆ) ( , , ) ˆ
ˆ
ˆ ˆ
( ) ( )
ˆ ˆ
1 / 2 /( )
ˆ ˆ
( ) ( ).
1/ 2
T T
T T
T
T T
Y E E
V X X QX Y E
Y E
X QX Y E Y E
θ α α α
α γ
θ σ θ θ α σ α α
α γ α γ
θ σ θ θ α σ α α
≤ − − +
+
+ − + −
≤ − + +
+ − + −
(46)
Since
1 1
ˆ ˆ
0≤Yα E γ/(Yα E +γ)≤γ (47) and considering that θ θ θˆ= − and α α αˆ = − , one may write (46) as follows:
1 0
1 2 0 2
( , , ) ( )
( )
1/ 2 T T
T T T
V X θ α X QX γ θ σ θ θ θ σ θ α σ α α α σ α
≤ − + + −
− + − − (48)
Now, by referring to [12] and considering the minimum singular value of a matrix A, i.e. µA = λmin(A AT ), we have
2 2
1 2
2 2 2
1 0 2 2
2 2
2 0
1 1
( , , ) (1 )
2 2
1 1
|| || (1 )
2 2
1 || || .
2
V X θ α µQ X µ θ
κ
µ κ θ θ µ α
κ
µ κ α α γ
≤ − − −
+ − − −
+ − +
(49)
By defining the following parameters,
1 2 1 2
3 2 2
2 2 2 2
1 0 2 0
1 1
0, (1 ) 0,
2 2
(1 1 ) 0, 2
1 1
|| || || || ,
2 2
2 / 2
ν µQ ν µ
κ ν µ
κ
ε µ κ θ θ µ κ α α γ
κ
= > = − >
= − >
= − + − +
>
(50)
Inequality (49) is re-written as follows:
2 2 2
1 2 3
( , , )
V X θ α ≤ −ν X −ν θ −ν α +ε (51) On the other hand, the Lyapunov function (41) may be stated as:
1 1
max max 1 max 2
2 2 2
( , , ) ( ) ( ) ( )
V X θ α ≤λ P X +λ Γ− θ +λ Γ− α
(52) where λmax(.) is used to denote the maximum eigenvalue of a matrix. Thus, it follows that for
{
1 max 2 max 11 3 max 21}
min / ( ), / ( ), / ( ) ,
cv = ν λ P ν λ Γ− ν λ Γ−
(53)
inequality (51) becomes ( , , ) v ( , , )
V X θ α ≤ −c V X θ α +ε
(54) After solving the differential inequality (54), we have
( ) (0) c tv / v(1 c tv ), [0, )
V t ≤V e− +ε c −e− ∀ ∈t ∞
(55)
This implies that tracking errors are GUUB and exponentially converge to a small ball containing the origin.
□
Remark 1: The stability of the internal and zero dynamics of a WMR in motion was analyzed by some researchers. The interested reader is referred to the work of Yun et al. [13].
IV. EXPERIMENTAL RESULTS
The proposed controller has been successfully implemented on robuLAB 10 WMR which is manufactured by robosoft Inc. The robuLAB 10 is a differentially-driven WMR which is equipped with sonar, a laser range finder, a wireless LAN for the communication, 12V batteries, two DC motors to drive wheels that each wheel is equipped with an incremental encoder for the localization system which updates the relative pose of the WMR every 200 ms. Two passive castor wheels are placed in the rear and front of the WMR to preserve its equilibrium. The WMR does not accept the motors voltage as the input. It is only commanded by linear and angular velocities which are denoted by vref,1
and vref,2, respectively. Therefore, in order to test the
proposed control law (40) experimentally, the following model is used which is presented by De La Cruz et al. [21]:
[ ]
[ ]
[ ]
[ ]
1 2
1 1 2
1 1
2 2
2
1 1 2 2 1 2 3 1 2 4 2
1 ,1 2 , 2
1 2
( ) cos sin 0 0 ,
( , ) 0 0 0 0 ,
( , ) 0 0 0 0 ,
( ) 0 0 0 ,
, ,
( ) ( , ) ( , ) ( ),
T
T
T
T
ref ref
v v
f x v v v
g x g x x
p v p v p v v p v
x f x g x v g x v x
ϕ ϕ
θ θ
θ θ
ξ ξ ξ
ξ δ ξ δ
θ θ ξ
=
=
=
=
= − + = − − +
= + + +
(56)
11 12
21 22
1 2
11 1 1 2
1 2
12 2 1 2
1 2
21 1 1 2
1 2
22 2 1 2
ˆ ( cos sin .cos ),
ˆ ( sin sin .cos ),
ˆ ( sin .cos sin ),
ˆ ( sin .cos cos ),
w w
W w w
w w w w
θ ϕ ϕ ϕ
θ ϕ ϕ ϕ
θ ϕ ϕ ϕ
θ ϕ ϕ ϕ
−
−
−
−
⎡ ⎤
⎢ ⎥
⎣ ⎦
=
= Φ + Φ
= Φ − Φ
= Φ + Φ
= −Φ + Φ
(57)
1 1 (Jh1.Sv Sv). , 2 2 (Jh2.Sv Sv). .
q q
η ∂ η ∂
Φ = − Φ = −
∂ ∂ (58)
and x = [x, y, ϕ , v1 , v2] denotes the state vector, , 1, , 4
i i
p = … are unknown constant parameters, the terms δv1 and δv2 denote norm-bounded unstructured uncertainties due to frictions and unmodeled dynamics. The interested reader is referred to [21] and [22] for more details about the above presented model. In this experiment, Microsoft Robotics Studio (MSRS) is used to implement the proposed controller. The MSRS execute the controller program code and generates the linear and angular velocities to be commanded to RobuLAB 10 through a wireless LAN. It is assumed that there is no knowledge about the parameters of the WMR. Moreover, in order to test the robustness of the controller, the robot is loaded and unloaded by some objects during the motion. The controller parameters for the experiment are set to kp = 1, kv = 2, ka = 0.2, β1 = 0.5 and β2
= 0.1 to guarantee the SPR-Lyapunov stability. This time, a circle-shaped trajectory is considered as the reference input which is specified by y1r( )t =0.5+cos(ωrt),
2r( ) 0.5 sin( r )
y t = + ωt . Note that ωr must be chosen small because the controller performance degrades when ωr is far from zero (here, ωr = 0.05). The initial values of WMR motion are set to x(0)=0.75 m; y(0)=0.25 m; φ(0)=00; v1(0)
= 0; v2(0) = 0. In order to show the tracking performance and robustness of the proposed controller, a feedback linearizing controller and an adaptive feedback linearizing controller (see [23]) are also tested on the WMR. Fig. 3 illustrates one of experimental results which shows the desired trajectory (dashed line) and trajectories of robuLAB 10 which are the results of three controllers. Further results such as time evolution of the states are omitted here due to the limited space. More evaluation of the proposed controller will be presented in the future works.
V. CONCLUSION
In this paper, the trajectory tracking problem of the
-0.5 0 0.5 1 1.5 -0.5
0 0.5 1 1.5
x (m)
y (m)
Feedback linearizing controller Desired trajectory
Proposed controller Adaptive feedback linearizing controller
nonholonomic robotic systems has been addressed. An adaptive-robust tracking control law has been proposed based on feedback linearization technique and the PID controller. The global uniform ultimate boundedness stability of the position and velocity tracking errors was proved by SPR-Lyapunov stability analysis. Experimental results have been carried out to verify the effectiveness of the proposed controller.
Fig. 2. RobuLAB 10 wheeled mobile platform.
Fig. 3. Desired trajectory (dashed line), WMR trajectory by the feedback linearizing controller (dashed-dotted line), adaptive feedback linearizing controller (dotted line), the proposed controller (bold solid line) in x-y plane.
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