Tracking control of mobile robots using saturation ... - IEEE Xplore

Proceedings of the 1999 IEEE International Conference on Robotics & Automation Detroit, Michigan May 1999

Tracking Control of Mobile Robots Using Saturation Feedback Controller Ti-Chung Lee

Kai-Tai Song

Department of Electrical Engineering Ming Hsin Institute of Technology Hsinchu 304, TAIWAN

Ching-Hung Lee

Ching-Cheng Teng

Department of Electrical and Control Engineering National Chiao Tung University Hsinchu, TAIWAN

feedback whereas the second one uses time-varying continuous feedback, which was first investigated by Samson in [21]. Subsequent to these investigations, Pomet [20] then proposed the “smooth” feedback control laws. But they were to solve the regulation problem found to yield a slow asymptotic convergence. In order to obtain faster convergence (e.g., exponential convergence), an alternative approach was proposed by M’Closkey and Murray in [15] initially and taken up in several studies [ 15,161 subsequently. The tracking problem for mobile robots has also attracted as much attention among researches [6,10,17,19, 211. Using Barbalet lemma or the backstepping method, some controllers have been proposed such that the mobile robots could globally follow special paths such as circles and straight lines. Despite the apparent advancement of the above methods, there remains however, several main Keywords: Mobile robots, tracking, time-varying systems, restrictions on their applications: (a) In the tracking problem, nonlinear systems, backstepping. only some special cases (straight lines or circles) are solved. (b) In other cases, for example, the tracking problem with 1. Introduction the linear and the angular velocities approaching to zero The control problem of nonholonomic mechanical remains unsolved. systems (e. g., mobile robots) has attracted considerable In applications, it is preferred to solve the tracking attention among the control community recently. Attempts problem and the regulation problem simultaneously using a to control such systems are, however, deceptively simple. single controller, otherwise, there will be a need to switch The challenge of this problem is reflected in the fact that a between two controllers of different types. In this paper, mobile robot in the plane possesses three degrees of both the tracking problem and the regulation problem of freedom of motion, which have to be controlled by only mobile robots will be solved simultaneously without any two control inputs and under the nonholonomic constraint. further assumptions except for some regular assumptions. Several researchers have shown, based on Brockett’s Moreover, for the saturation constraint in control inputs (the theorem [5], that such a system is open-loop controllable, linear and angular velocities) are also taken into but not stabilizable by pure smooth time-invariant feedback, consideration in the tracking problem. For practical see [2,17]. That is, there does not exist a smooth, or applications, the bounds on the velocities of wheels must be continuous, feedback. Recently, methods for solving this attended to avoid the high-gain control signal. problem can be put into two categories. The first one is The organization of this paper is as follows. Some based on the motion planning by using the vision guided, notations and the problem formulation are presented in environment prediction, artificial neural network, or other Section 2. Main results including the control law and the sensory information, [7,8,13,23,24]. stability analysis are given in Section 3. In Section 4, some The second category is using the nonlinear system results from computer simulation and practical experiment theorems for solving this problem, e.g., the hybrid control, for mobile robots are given to verify the proposed tracking the backstepping technique, the sliding model control, the controller. Section 5 presents the discussion for fuzzy logic control, and the adaptive control, etc., experimental environment, results, extension of the [ 1,3,4,9,11,20]. In this paper, our approach categorizes the proposed approach, and future research. The conclusion second one. In this category, two main research directions remark is given in Section 6. have been adopted for mobile robots. The frst direction, started from Bloch et al. in [3], uses the discontinuous

Abstract: A general tracking control problem with saturation constraint for nonholonomic mobile robots is proposed and solved using the backstepping technique. A global result is given in which some art$cial assumptions about the linear and the angular velocities of mobile robots from recent literature are dropped The proposed controller can simultaneously solve both the tracking problem and the regulation problem of mobile robots. With the proposed control laws, mobile robots can now globally follow any path such as a straight line, a circle and the path approaching to the origin using a single controller. Computer simulations are presented, which confirm the effectiveness of the tracking control laws. Moreover, practical experimental results concerning the tracking control are reported with saturation constraint for mobile robots.

0-7803-5180-0-5/99 $10.00 0 1999 IEEE

2639

2. Preliminaries 2.1 Notation

where

1. % denotes the set of all real numbers and %+ denotes the set of all nonnegative real numbers. 2. A function f :8++ 8 is uniformly continuous if for any E > O , there exists a 6 ( ~ ) > 0such that if IxI-x,I 0 as

the

desired

linear

velocity

satisfying

0 Iv,(v, - Jx,T+y, 7) c ,v and the desired angular velocity satisfying suplw, @)I 0 and 0 < a < vmX-sup v, . Note that

(8) (9)

/v~I/ul/+~v,cosx,~~v,, (10) satisfying the saturation constraint of the line velocity. Using the saturation control (9), we have V, = -2sat,(k,,x,)x, +2x,vpsinxo. (1 1) Next, introduce a new variable &h(t)x i,) = x, + I 1+v,':

fi/+y,where

t, will be specified in the following theorem. Note that if ( Y o , x , , x , ) converges to zero, then the stability of

will be guaranteed. With (12), system (6a) is transformed into = +I x, + P ( x 0 , x,, x , , t ) (13) where (x,,,x,,x,)

9

a ( x , , x , , t ) = 1-

Eh,

1+y': '

vi,€%.

This means that lim ?,(t) = 0 (by the Lyapunov stability 1-

theorem, see [21]). Now, we are in a position to present main results. Theorem 1 Consider a simpl$ed model (3) of mobile robots. Then the tracking problem with saturation constraint can be solved using control laws (9) and (14) with h(t) = 1 + y cos(t - t o ) , where to will be defined in the proof oftheorem.

Proof: Due to limited space, the proof is omitted here. A similar proof can be found in the result of [141.

120

with h(t) = l+ycos(t-t,),O < y < 1 , and 0 < E
0 and b>O. Then, application, the above condition can be chosen by = -a(x,,x,,t)sat,(k,x,) . (15) simulation results. In general, the constraint for v, and Note that lc i4 m 3 = 0 . So, it is possible to choose a small w,, are larger than the condition (18). The proposed algorithms have been verified by using ~ wrnx - sup(w,(t)l. Then E > 0 and b>O such that S U P ~ U , ,< computer simulation and practical experiments on an I20 tm (w(t)lI luol+(wrl< wmy satisfying the saturation constraint experimental mobile robot. The robot's hardware structure has two independent drive wheels and a free caster for of the angular velocity. Define V,(X,) = . We then balance. The motion of the robot is controlled by changing obtain the velocities of the left and right wheels. Two dedicated motion control chips HCTL-1100 from HP were employed .I ,I

x,

'x

264 1

for servo control of the two drive wheels. The integral velocity control mode was used in the experiments. The control computer only needs to send commands to the chips, which take care of the motor servo control. The position estimation of the robot is conducted using an odometer, which samples the left and right wheel velocities to calculate the current posture of the robot. The formula of this estimator is presented in (1 9) (23).

-

xnm = x,, + A T . v .cos(eo,d (21) (22) Y”ew= Yo, + A T . v .Sin(O0,d1 e, =e,, + A T . @ (23) where E is the distance between two drive wheels and T is the sampling period. The desired velocity calculated in the tracking algorithm is transferred into the left and right wheel velocities using v, = v , +

2 L.w,,

(24)

Two trajectories have been used to show the performance of the control law. Computer simulations were first conducted for these trajectories and the parameters (h, k,, y , E ) are determined by examining the desirable tracking performance. In the following the simulation and experimental results of tracking as well as docking performance are presented respectively for circular and parallel parking trajectories. The model of these trajectory generators are illustrated below: W Circular trajectory: X, = X, + R.sin(c.t) y , = y , -R.cos(c.t)

v?=c.R w, = c

where ( x,, yc ) is the center coordinate R is the radius 9

Parallel parking

2c~a’sin2(c(t+;5))+b2cos’(2c(t+$))0 1 t