Where. T b b wvtv ]. [)(= comprises of linear and angular velocities. B. Dynamic Model. The dynamic model of WMR is designed by Euler-. Lagrange equations.
Adaptive Trajectory Control of Wheeled Mobile Robot (WMR) Kanwal Naveed, Zeashan Khan, M. Salman, M. Bilal malik, M. Usman Ali Department of Electrical Engineering College of Electrical and Mechanical Engineering, NUST Rawalpindi, Pakistan Abstract— A wheeled mobile robot belongs to the class of Nonholonomic systems with highly nonlinear dynamics. In order to achieve the task of trajectory tracking, a direct model reference adaptive control (D-MRAC) scheme is established which overcomes the parametric uncertainties by making use of an adaptive law for control parameters. The control scheme employs Lyapunov stability principle to derive the adaptation law. This criterion guarantees asymptotic stability. A comparison between the Model Reference Adaptive Control and an optimal robust linear quadratic integrator is also presented, which demonstrate the better performance of proposed controller in presence of parametric uncertainties. Keywords—mobile robots; dynamic model; adaptive control; trajectory tracking;
In the second part of this paper, a linear quadratic integrator (LQI) is developed for trajectory control. It is compared with MRAC through simulations. It is shown that even a very little model uncertainty leads to a degraded performance of linear controller thus proving the supremacy of MRAC. The paper constitutes of following sections: Section 2 develops complete dynamic and kinematic model of WMR; Section 3, which discuss the adaptive controller; Section 4, which develops LQI control scheme for WMR; Section 5, which presents simulation results and comparison between the two control schemes and finally section 6, which incorporates conclusions and remarks II.
I.
KINEMATIC AND DYNAMIC MODEL
INTRODUCTION
Wheel is one of the prime scientific inventions of all times. Wheeled mobile robots belong to the class of unmanned ground vehicles (UGVs). Among many different configurations of WMR, the most renowned is a differential drive robot. These robots incorporate fast maneuvering and energy saving characteristic, thereby backing their potential application in areas requiring task performance through an automated system. However, it is necessary for a mobile robot to follow some predefined track regardless of the nature of task. The predefined trajectory helps mobile robot follow the correct direction in a precise manner. Therefore, it is very important to control the trajectory of a mobile robot, not only to keep it on a particular track but also to reach the target within the specified time limit For this purpose, a direct model reference adaptive control (MRAC) strategy is introduced for trajectory control of WMR in existence of model uncertainty. This strategy incorporates kinematic and dynamic controller in contrast to [1]-[5], which incorporated only either kinematic or dynamic controller. It is being assumed that kinematic model is well established and model uncertainties are present only in dynamics of WMR. Therefore, adaptive part of the controller is developed using dynamic model of WMR. The kinematic controller is responsible for providing the desired velocity profiles to the dynamic controller [1], which in turn provides torques to the actuators of WMR for producing the necessary velocities. Adaptive law for MRAC is based on Lyapunov stability approach [6], which guarantees asymptotic stability.
A. Kinematic Model The Kinematic model of Wheeled Mobile Robot defines the present state (position and rotation) of the robot under the condition that initial states are known [5]. WMR possesses three degree of freedom therefore; its position is well defined with the help of three variables. For this purpose, generalized coordinates are assumed for WMR q [ x y ]T configuration.
Fig. 1 Configuration of Wheeled Mobile Robot
TABLE I.
PARAMETERS OF WHEELED MOBILE ROBOT
Parameters
Description Left and right wheel radius
r 2b d mc mw Iz Iw Pc
m M (q) 0 mc d sin
Distance between left and right wheels Distance between
Po
and
Pc
Platform mass of WMR without wheels Each driving wheel plus motor rotor mass Platform moment of inertia without wheels Each wheel and the motor rotor moment of inertia about the wheel axis The center of mass of WMR
The kinematic model is based on the assumption that WMR moves on a flat surface with no side slip [8]. Linear velocity vb and angular velocity wb of WMR is responsible for controlling the motion of WMR. With the help of these velocities, the kinematics of WMR can be defined by following relation: cos 0 (1) q Gv (t ) sin 0 0 1 Where v(t ) [vb velocities.
wb ]T comprises of linear and angular
B. Dynamic Model The dynamic model of WMR is designed by EulerLagrange equations. These equations are considered over Newton mechanics to avoid any additional complexity. WMR possesses zero potential energy because it travels on a plane surface. Consequently, the Lagrange equation reduces to kinetic energy only: L
1 2
nj
v
T
jm jv j
wT j I j w j
(2)
j 1
After constructing the Lagrange following Euler–Lagrange equation is used: L L d (3) FG dt q j q j Where, FG denotes generalized forces. Solving (3), the dynamic model takes the following shape: M (q)q F (q, q )q B(q) E(q) T
(4)
Where M(q) is an nn inertia matrix, F(q) is a nn matrix which represents the centripetal and Coriolis terms, B(q) is an n(n-m) input matrix; τ is the (n-m)1 vector of torque produced by the actuators of wheels; and λ is the m1 vector representing limitation forces.
0 m mc d cos
mc d sin mc d cos I
0 0 mc d cos F (q, q ) 0 0 mc d sin 0 0 0 cos cos 1 B(q) sin sin r b b
(5)
Where m mc 2mw and I I c 2I m mc d 2 2mw b 2 are defined in table I. The dynamic model is simplified by replacing torque with actuator dynamics. Therefore, it is assumed that each wheel is driven by a brushed DC motor incorporating mechanical gear system. The equation of motor armature is given as: u c Lc
dic ic Rc K e m dt
(6)
Where K e is defined as the back electromotive force constant. Keeping in view the relation between torque and armature current ic that is provided in [8], the torque delivered to right and left wheels is written as:
1 r r ucr vb K1 u K 2W w , W 1 l cl b r
b r b r
(7)
This will result into the following form of dynamic model of WMR: M (q)q F (q, q )q B(q)( K1uc K2Wv) E(q)T
(8)
The state space representation of dynamic model is obtained by taking derivate of (1) and replacing it in (8):
Mv(t ) F (q ).v(t ) K1 B .u c
(9)
Where M G T MG, F G T MG G T FG K 2 BW B GT B
(10) The complete state space representation incorporating the kinematic and dynamic model of WMR can be represented by the combination (9) and (1) in following form:
0 Gv z 1 .u c 1 M F v K 1 M B
z qT
III.
vT
Where A M u1 Fu and B M u1 .The reference model with the desired stated and the reference input can be written as: (11)
Controller design problem for trajectory tracking of WMR is divided in two stages: A. Dynamic Controller The Dynamics of wheeled mobile robot are nonlinear in nature including time varying and uncertain terms: M (q)v(t ) F (q ).v(t ) u b
(12)
Where, u b K1 B .u c is the controlled input. To control such a system, a direct model reference adaptive control strategy is devised. The convergence of this scheme requires that the plant is slowly time varying and is close to the desired model. In this paper, the second condition is eased by taking advantage of any known dynamics [8]. The dynamics of WMR can be separated in terms of „known‟ and „unknown‟ dynamics in the following form: M M k M u* (M k I ) ( I M u* ) (M k I ) M u
(13) The subscript „k‟ and „u‟ denotes known and unknown parameters. The known dynamics are used to calculate the linear constant controller gain with the help of feedback linearization. Note that, if parametric uncertainty is absent; then, M u I (M u* 0) and Fu 0 . Substituting, (13) in (12) we get: M u (q)v(t ) Fu (q ).v(t ) ub'
If
(18)
u b (M k I )v(t ) Fk (q )v(t )
Here Ar K and Br I . The system matrix of reference model is a diagonal matrix in which the terms K11 , K 22 are constant and equal to i which is inversely proportional to the time constant of desired transient behavior. The direct model reference adaptive control strategy makes an effort to derive the error between the reference and actual states to zero asymptotically. In this way, the adaptive scheme will force the WMR to follow desired trajectory. Defining xd=vd, the error between the reference and actual states takes the following form: e xd x
Dynamics of error are obtained by taking derivative of above equation and making use of equations (17) and (18): e Ar e ( Ar A) x Bu b' Br u ref
u ref v d Kvd
(21)
The control law can be written in the following form: u b' u l u a
(22)
This controller has two parts. The linear feedback controller derived from the „known dynamics‟ is given as follows: u l Kx u ref
(23)
(15)
u is known, then from equation (16) ub can be recovered
as:
The adaptive controller devised for the „unknown part‟ is given as: u a x r u ref
(16)
After the removal of known dynamics by transformation of (15), the control problem for (14) can simply be stated as that of designing u b' to obtain ub such that the actual velocities of WMR follow the desired velocities. In this way, WMR achieves the task of trajectory tracking. Defining x v we get: x (t ) Ax(t ) Bu b'
(20)
(14)
' b
ub ub' (M k I )v(t ) Fk (q )v(t )
(19)
The reference input is given as:
F Fk Fu
u b'
x d Ar x d Br u ref
CONTROLLER DESIGN
(17)
(24)
Equation (24) includes two gains, which play important role in the design of adaptive controller. These gains are not constant but are incessantly updated such that the error approaches zero asymptotically. The update law is provided by the Lyapunov stability criterion. Replacing (22) in (20), the error dynamics are obtained in following form: e Ar e Ce De u ref
(25)
Ce M u1 ( Fu K ) K
(26)
De I M u1 ( I r )
(27)
Therefore, the equation (32) reduces to 1 L e T Qe 2
Theorem 1 Let P be a positive matrix obtained from Lyapunov equation with Q 0 : ArT P PAr Q
(28)
If the filtered error is defined as: we Pe
(29)
Moreover, if the adaptive gains in (24) are updated such as:
a wev T , r b weu ref
(30)
Where a and b are positive scalars. Then by utilizing (30) in the control rule demonstrated from (22) to (24) asymptotic stability of closed-loop error (20) is guaranteed. Proof: Choose the following Lyapunov candidate function using P 0 :
L
From (37) it is easily deduced that L is decreasing constantly. This proves that e approaches zero asymptotically. Equation (35) and (36) provides with the update rule for the adaptive part of the controller. However if the system is completely known ( M u I , Fu 0, 0, r 0 ) then (20) is reduced to the following form: (38) e Ar e Consequently, the only task is to choose a positive diagonal matrix Ar , so that the asymptotic stability of error is guaranteed. B. Kinematic Controller: The basic kinematic model of WMR is exploited to develop the kinematic controller. The sole purpose of this controller is to provide desired velocity profiles to the dynamic controller. Kinematic controller is established using error between the desired and the actual trajectory and is given by [1].
1 T 1 1 e Pe tr C eT YCe DeT De tr DeT YDe DeT De 2 a b (31)
v d cos w d sin
where, Y M u . After differentiating L and using some common mathematics properties we obtain: YC 1 L e T ArT P PAr e tr C eT Pex T e 2 a
(32)
In order to make L negative definite the last two terms can be equated to zero. This is possible if „P‟ is chosen from (28) and by choosing adaptation laws as in (30). Taking derivative of (26) and (27) while assuming that plant parameters are changing slowly we obtain:
Where,
mx x d n x tanh n sin x m cos y y d n y tanh n y
~ x ~ y
(39)
Here ~ x x d x, ~ y y d y are the errors between the actual and the desired position. Moreover, mx and my > 0 are the controller gains and nx, ny > 0 are saturation constants.
YD T tr DeT Peu ref e b
(37)
C e M u1
(33)
D e M u1 r
(34)
a wev T
(35)
r b weu ref
(36)
IV.
LQI CONTROLLER
Conventional linear quadratic regulator (LQR) can be applied to nonlinear systems using feedback linearization. However, it does not include an integrator. This infers that the steady state of LQR is not offset error free when the system is subjected to model uncertainty [9]. Therefore, the standard LQ problem must be reformed to include an integrator term in it for robustness. This modification can be explained with the help of a general linear system in [9] and can be extended to nonlinear systems by using feedback or global linearization [7]. Consider an „m’ input „m’ output „n’ dimensional plant: x(t ) Ax(t ) Bu(t ), y(t ) Cx(t )
(40)
It is assumed that the above plant is controllable and observable. The modified cost function for integral LQ control is written as: J y(t ) r Qy y(t ) r zT (t ) Rz (t )dt
0
T
(41)
Q y , R are symmetric and positive definite, r is reference
35
input to be followed with z being the derivative of actual input.
30
25
A new (m+n) state equation is defined with v(t ) being the derivative of actual input as in (42): (42)
20
Y(m)
e 0 C e 0 d~ x (t ) ~~ ~ Ax (t ) Bv(t ) x 0 A x Bv(t ) dt
LQI MRAC Reference trajectory
15
10
5
Therefore, the new cost function is given as:
Q J ~ x T (t ) y 0 0
t
0 ~ x (t ) z T (t ) Rz (t )dt 0
0
(43)
-5 -5
Fig. 2
r y( )d K
25
30
35
Desired and actual trajectories (Square shaped) using LQI and MRAC for nominal plant model
LQI MRAC Reference trajectory
25
20
Y(m) 15
x x(t )
(46)
10
This LQI controller can be employed to nonlinear systems by feedback linearization of (12). V.
20
30
t
0
15
35
(45)
The input is obtained after integrating (44) as: u (t ) K r
10
40
~ With P being the solution of Riccati equation given below:
0 0
5
X(m)
Thus, the problem is now formulated to standard LQ problem. The feedback control law is given by: ~ ~ (44) v(t ) R 1 B T P~ x (t ) K r e(t ) K x x(t )
~ ~ ~~ ~ ~ ~ ~ Q A T P PA PBR 1 B T P y 0
0
5
0
SIMULATION RESULTS
In order to evaluate and compare the performance of both control techniques, simulations are performed using MATLAB/SIMULINK. Comparison between the two techniques is performed based on performance indices such as Mean square error (MSE), Integral square error (ISE), Integral absolute error (IAE) and Integral Time weighted absolute error (ITAE). The physical and design parameters for computer simulations are obtained from [8]: r 0.15m, b 0.75m, d 0.3m, mc 36kg , m w 1kg , I m 0.0025kg .m 2 , I c 15,625kg .m 2 , K 1 7.2, K 2 2.59, m x m y 10, n x n y 4, a b 100, K diag (1000,1000), K x 9, K r 40
The results of Matlab/Simulink simulations are shown in Fig. 2-5 which shows the tracking performance for a square trajectory. Simulations are performed for the nominal plant as well as in the presence of 30% parameter perturbation for both types of controllers. Performance indices are presented in Table-II.
-5 -10
-5
0
5
10
15
20
25
30
35
40
X(m)
Fig. 3
Desired and actual trajectories (Square shaped) using LQI and MRAC for 30% perturbed plant model
From the simulations and indices, it is evident that the adaptive controller exhibits improved performance as compared to LQI technique in the presence of plant perturbations. Table II.
COMPARISON OF PERFORMANCE INDICES FOR 30% PERTURBATION CASE
Performance Indices
MSE
ISE
IAE
ITAE
Along x-axis
Using LQI
14.293
142.972
31.0076
150.564
Using MRAC
1.303x10-5
1.303x10-4
0.0284
0.1682
Along y-axis
Using LQI
4.779
142.972
31.0076
150.564
Using MRAC
1.79x10-5
1.79x10-4
0.0289
0.2247
1.5
1 LQI MRAC
LQI MRAC
1 0.5
Error along Y (m)
Error along X (m)
0.5
0
0
-0.5
-0.5
-1 -1
-1.5
0
1
2
3
4
5
6
7
8
9
10
-1.5
0
1
2
3
4
Time(sec)
5
6
7
8
9
10
Time(sec)
(a)
(b)
Fig. 4. Error between desired and actual trajectory using LQI and MRAC for nominal models (a) X-axis, (b) Y-axis
8
3 LQI MRAC
6
2
4
1
Error along Y (m)
Error along X (m)
LQI MRAC
2
0
-2
0
-1
-2
-4
-3
-6
-4
-8
0
1
2
3
4
5
6
7
8
9
10
Time(sec)
(a)
-5
0
1
2
3
4
5
6
7
8
9
10
Time(sec)
(b)
Fig. 5 Error between desired and actual trajectory using LQI and MRAC for 30% perturbed models (a) X-axis, (b) Y-axis
VI.
CONCLUSION
This paper presents two different controllers in order to achieve the task of WMR trajectory tracking. The first one is a direct model reference adaptive (D-MRAC) controller that consists of kinematic and dynamic controllers for trajectory tracking. The adaptive part of the controller is incorporated in the dynamic controller. Second one is the linear quadratic plus integral (LQI) controller which uses a performance index to achieve trajectory tracking as well as
an integral term for robustness. A comparison between the two controllers is performed based on the nominal and the perturbed plant models. Different performance indices are used to quantify the tracking quality. The simulation results have shown that despite the robustness of LQI, the controller performance is inferior to adaptive controller when it comes to plant parameter uncertainty as evident from Table II.
REFERENCES [1]
F.N. Martins, W.C. Celeste, R. Carelli, M. Sarcinelli-Filho, and T.F. Bastos-Filho, “An adaptive dynamic controller for autonomous mobile robot trajectory tracking,” Control Engineering Practice, vol.16, pp. 1354–1363, 2008. [2] P.A. Nino-Suarez, E. Aranda-Bricaire, and M. Velasco-Villa, “Discrete-time sliding mode trajectory-tracking control for a wheeled mobile Robot,” Proceedings of the 45th IEEE Conference on Decision and Control, December 2006. [3] Y. Kanayama, Y. Kimura, F. Miyazaki, and T. Noguchi, "A stable tracking control method for an autonomous mobile robot," IEEE International Conference on Robotics and Automation, vol.1, pp. 384389, May 1990. [4] G. Klancar, I. Skrjanc, “Tracking-error model-based predictive control for mobile robots in real time,” Robotics and Autonomous Systems, vol.6, no.55, 2007. [5] T. Fukao, H. Nakagawa, N. Adachi, “Adaptive tracking control of a nonholonomic mobile robot”, IEEE Transactions on Robotics and Information, vol.16, no.5, 2000 [6] G. Maliotis, “A hybrid model reference adaptive control/computed torque control scheme for robotic manipulators,” Proceedings of the Institution of Mechanical Engineers, vol. 205 no. 3, pp. 215-221, August 1991. [7] L. R. Hunt, R. Su, G. Meyer, “Global transformation of non- linear systems”. IEEE Transactions on Automattion Control, 1983 [8] E. Canigur and M. Ozkan. “Model Reference adaptive control of a nonholomonic wheeled mobile robot for trajectory tracking”. IEEE International Conferencce on INISTA, 2012 [9] J. Choi, LQI-note, Lecture notes in Linear Systems and Control, 2010 [10] J. J Craig, P.Hsu, S. Shankar, “Adaptive control of mechanical manipulators”. 1988 [11] K. Naveed and Z. H. Khan “Adaptive trajectory tracking of wheeled mobile robot with uncertain parameters”, Computational Intelligence for decision support in cyber physical systems, 2014, Springer (To appear)
