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Mnif, F. & Touati, F. / An Adaptive Control Scheme for Nonholonomic Mobile Robot with Parametric Uncertanity, pp. 059 - 063, International Journal of Advanced Robotic Systems, Volume 2, Number1 (2005), ISSN 1729-8806

An Adaptive Control Scheme for Nonholonomic Mobile Robot with Parametric Uncertainty

1

Mnif, F. 1,2 & Touati, F.1 Sultan Qaboos University, Department of Electrical and Computer Engineering, Oman 2 Institut National des Sciences Appliquées et de Technologie, Intelligent Control, Design and Optimization of Complex Systems Research Unit, Tunisia

Abstract: This paper addresses the problem of stabilizing the dynamic model of a nonholonomic mobile robot. A discontinuous adaptive state feedback controller is derived to achieve global stability and convergence of the trajectories of the of the closed loop system in the presence of parameter modeling uncertainty. This task is achieved by a non smooth transformation in the original system followed by the derivation of a smooth time invariant control in the new coordinates. The stability and convergence analysis is built on Lyapunov stability theory. Keywords: nonholonomic system, adaptive control, car-like vehicle.

1. Introduction In the past few years, a considerable interest has motivated researchers in the analysis and control design of underactuated and nonholonomic systems. From a theoretical point of view, this interest was sparked by the fact that such system present challenges problem in control theory. At same time, these systems do arise in a number of very important practical applications in the area of robotic vehicle control, namely in land, marine and aerospace vehicles, for which the number of actuators is smaller than the number of degrees of freedom. Wheeled mobile robots (WMRs) are increasingly present in industrial and service robotics, particularly when flexible motion capabilities are required on reasonably smooth ground and surfaces [1]. Several mobility configurations can be found in these applications. The most common are the tricycle and the car-like drive. Kinematics study of several configurations of WMRs can be found in [2]. Beside the relevance in industrial applications, the problem of autonomous motion planning and control of WMRs has attracted the interest of researchers in view of its theoretical challenges. In particular, these systems are typically examples of nonholonomic mechanical systems [3]. In the absence of workspace obstacles, the basic motion tasks assigned to a WMR may be reduced to moving

between two postures and following a given trajectory. From a control viewpoint, the peculiar nature of nonholonomic kinematics makes the control problem easier than the first; in fact, it is known [4], that feedback stabilization at a given posture cannot be achieved via smooth time invariant control. This indicates that the problem is really nonlinear; linear control is ineffective, even locally, and innovative design is required. The trajectory tracking problem of WMRs was globally solved in [5] by using nonlinear feedback control, and independently in [6] and [7] through the use of dynamic feedback linearization. Recursive backstepping control schemes for chained forms of WMRs have been also addressed by several authors [8]. Despite the substantial research effort, some fundamental problems still remain open. In particular, all the control laws presented in the above mentioned papers are nonrobust against parameter uncertainties. In this paper, a new solution to the problem of regulating the dynamic of a nonholonomic wheeled robot of the WMR. A simple discontinuous adaptive state feedback controller that yields global convergence of the closed loop system in the presence of parametric modeling uncertainty is derived. This is achieved by resorting to a polar representation of the kinematic model of the mobile robot in the original state space followed by the derivation of a smooth time invariant control law in the new coordinates. 059

2. Dynamic model of the WMR and Problem formulation The WMR is a unicycle type as shown in Figure 1. It has two identical parallel rear wheels controlled by two independent actuators and a steering front wheel. It is assumed that the plane of each wheel is perpendicular to the ground and the contact between the wheels and the ground in non-slipping, i.e. the velocity of the center of mass of the robot is orthogonal to the rear wheels axis. By assuming this, a nonholonomic constraint on the motion of the mobile robot of the form

x sin θ − y cos θ = 0

(1)

is imposed Fig. 1. The WMR It is also assumed that the masses of the wheels are negligible and that the center of mass of the mobile robot is located in the middle of the rear wheels axis. The torques developed by each actuator on the rear wheels are τ 1, 2 i=1,2.

Polar Transformation Consider the following coordinate transformation

e = x2 + y2

The kinematics and dynamics of the mobile robot are modeled by the equations:

x = −e cos(θ + β )

x = v cos θ

y = −e sin(θ + β )

y = v sin θ

θ = ω

(2)

mv = F

x y

β

where v and ω denote the linear and angular velocity of the body reference frame {B} with respect to the inertial

reference frame {U }. the control inputs are the force F

along the vehicle axis x B and the torque N about its vertical axes z B . It is easy to see that

1 F = (τ 1 + τ 2 ) R

(3)

L (τ 1 + τ 2 ) R

(4)

where R is the radius of the rear wheels and 2L is the length of the axis between them. m and I are the mass and moment of inertia of the robot respectively. With the above notation, the control problem is to derive the controls for τ 1 and τ 2 to regulate {B} to is the goal inertial frame {G} = {U } , in the presence of uncertainty in parameters m, I, R, and L.

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−1

where e is the length of the vector O B Ou and

Iω = N

N=

θ + β = tan

(5)

denotes the angle measured from x B to d. Differentiating (5) with respect to time, the dynamics of the WMR in the new coordinate system can be written as

e = −v cos β sin β β = v −ω e θ = ω mv = F Iω = N

(6)

Note that the coordinate transformation is valid only for non zero value of e. This will introduce a discontinuity in the control law that will be derived later and which will obviate the basic limitations imposed by Brockett. 3 Nonlinear Controller Design Step 1: Define the variables

ρ=

v , σ = β +θ e

and rewrite the equations of motion (6)

= k1 k σ σ

e = ρ sin β

β = ρ sin β − ω

(7)

N ω = I and

(8)

where the system has been divided into two subsystems that will henceforth be referred to as the heading and distance subsystems (7) and (8). Consider the heading system (7) and suppose at this stage that ρ = k1 > 0 . Define the Lyapunov function

V1 = 12 kσ σ 2 + 12 β 2

(9)

Differentiating (9) along the trajectories of (7), we obtain

⎡ ⎤ sin β + k1 sin β − ω ⎥ V1 = β ⎢k1 kσ σ β ⎣ ⎦

β

(10)

+ k1 sin β + k 2 β

k2 > 0

(11)

virtual control law. Introduce the error variable

z1 = ω − α 1

(12)

and compute V1 to obtain

(Backstepping)

+ k1 β cos β + k 2 β + β sin β

β

and

β cos β − sin β are β2

well defined according to the l’Hôpital rule. Let the control for N as

Then

N = If (σ , β , z1 , ρ ) − k 3 z1 , k 3 > 0 k V2 = − k 2 β 2 − 3 z12 ≤ 0 I

which negative semi-definite. Step 3: Distance regulation Consider the distance subsystem (8) and define a new error variable z 2 = ρ − k1 and the Lyapunov function candidate as Differentiating V3 yields to

k ⎡F ⎤ + g (σ , β , ρ )⎥ V3 = − k 2 β 2 − 3 z12 + z 2 ⎢ I ⎣ me ⎦ The last two terms of g are due to the fact that ρ in not constant, but ρ = k1 + z 2 instead. They are computed by replacing ρ by k1 + z1 in the expression for V1 and

propagating the corresponding terms down to V3 . Choosing the control input as

The

function

then the time derivative of V3 becomes

V1 is

now

augmented with a quadratic term in z1 to obtain the new candidate Lyapunov function

V2 = V1 + 12 z12 The time derivative of V2 can be written as

⎡N ⎤ V2 = − k 2 β + z1 ⎢ − f (σ , β , z1 , ρ )⎥ ⎣I ⎦ 2

where

β

F = − mg (σ , β , ρ )e − k 4 z 2 e

V1 = −k 2 β 2 − βz1 Step2:

β cos β − sin β + k1 k σ σβ 2

V3 = V2 + 12 z 22

Following the nomenclature in [8] and let w be a virtual control input and

sin β

β

Note that the terms

e = − ρ cos βe F ρ = + ρ 2 cos β me

α 1 (σ , β ) = k1 kσ σ

sin β

f (σ , β , z1 , ρ ) =

∂α ∂α σ + β −β ∂σ ∂β

k k V3 = − k 2 β 2 − 3 z12 − 4 z 22 I m which is negative semi-definite. Step 4: Parameter adaptation Suppose that the values of the system parameters m, I, L, and R are not known precisely. Define the control inputs u i , i =1, 2 as u1 = τ 1 − τ 2 and u 2 = τ 1 + τ 2 . From (3) and (4) the dynamic equations of the mobile robot become

ω =

u1 u , v = 2 c1 c2

061

where

c1 =

IR L

and

c 2 = mR are

the

unknown

The complete control law is thus given by

parameters of the system. Let the augmented candidate Lyapunov function

V 4 = V3 +

⎡ u1 ⎤ ⎧ cˆ1 f (.) − k 3 z1 ⎢ ⎥ ⎪ − k θ − k θ ⎪ θ θ u=⎢ ⎥=⎨ ⎢u 2 ⎥ ⎪− cˆ 2 g (.) e − k 4 z 2 e ⎢ ⎥ ⎪ 0 ⎣ ⎦ ⎩

1 ~2 1 ~2 c1 + c2 2c1γ 1 2c 2 γ 2

where c~i , i =1,2 are parameter estimation error such

that c~i = ci − cˆi where cˆi , i =1,2 are the nominal values

with the adaptation laws

cˆ1 = −γ 1 z1 f (.)

of ci . The time derivative of V 4 can be computed to give ⎤ c~ ⎡u ⎡u ⎤ c~ V4 = −k2 β 2 + z1 ⎢ 1 − f (.)⎥ + z1 ⎢ 2 − g (.)⎥ − 1 c1 − 2 c2 c2γ 2 ⎣ c1 ⎦ ⎦ c1γ 1 ⎣ c2e and choose the control laws as

u1 = cˆ1 f (.) − k 3 z1 u 2 = −cˆ 2 g (.)e − k 4 z 2 e to obtain

k k c~ cˆ V4 = −k 2 β 2 − 3 z12 − 4 z 22 − 1 [ z1 f (.) + 1 ] c1 c2 c1 γ1 c~ cˆ + 2 [ z 2 g (.) − 2 ] c2 γ2 Choose the adaptation laws as

cˆ1 = −γ 1 z1 f (.) cˆ 2 = γ 2 z 2 g (.)

to yield

k k V4 = − k 2 β 2 − 3 z12 − 4 z 22 c2 c1

cˆ2 = γ 2 z2 g (.)

4. Simulation Results To illustrate the performance of the proposed control scheme, consider a WMR with the following parameters: m = 12kg , I = 1.5kgm 2 , L = 0.5m , R = 0.05m . The control parameters are chosen as k1 = 0.7 , k 2 = 0.1 , k 2 = 0.1 , k 4 = 0.7 , γ 1 = 0.1 , γ 2 = 6 , kσ = 3 , kθ = 0.9 and kθ = 0.4 . The

initial

estimates

for

the

vehicle

ˆ = 20kg , Iˆ = 2kgm 2 , Lˆ = 0.6m and Rˆ = 0.1m . were m Figure 2 shows the time responses of the relevant WMR variables for the initial conditions ( x 0 , y 0 , θ 0 , v 0 , ω 0 ) = (−4,2,0.8,0,0) . 5. Conclusions This paper proposed a new adaptive control law for an example of uncertain nonholonomic systems, the wheeled mobile robot. A discontinuous time invariant nonlinear adaptive control was derived to yield convergence of the states of the system under parameter uncertainties. The Lyapunov theory was used to derive the control approach and simulation results were presented to illustrate the approach.

which is negative semi-definite.

0

Step 5: Switching control law

-0.5 -1 -1.5 x (m)

It has been assumed that the WMR will never start or reach the position x = y = 0 because of the nondefinition of (6) at e = 0. To avoid this problem we introduce a switching control law. Define the following control law for e = 0 as

e≠0 e=0 e≠0 e=0

for for for for

-2 -2.5 -3

u1 = − kθθ − kθ θ u2 = 0 where kθ and kθ are positive constants.

062

-3.5 -4

0

2

4

6

8

10 t (sec)

(a)

12

14

16

18

20

2

0.8

1.8

0.6

1.6

0.4

1.4

0.2 c 2 hat

y (m )

1.2 1 0.8

0 -0.2

0.6

-0.4

0.4

-0.6

0.2 0

0

2

4

6

8

10 t (sec)

12

14

16

18

-0.8

20

0

2

(b)

4

6

8

10 12 time (sec)

14

16

18

20

(b) Figure 3: Time evolution of parameter estimation a: cˆ1 , b: cˆ 2

1.4

1.2

6. References

theta (rad)

1

0.8

0.6

0.4

0.2

0

0

2

4

6

8

10 t (sec)

12

14

16

18

20

(c) Figure 2: Time responses of a: x(t), b:y(t) and c: θ (t ) 2

1.5

c 1 hat

1

0.5

0

-0.5

-1

0

2

4

6

8

10 12 time (sec)

(a)

14

16

18

20

Schraft, R.D. and Schmierer, G. Serviceroboter. Springer Verlag, 1998. Alexander, J.C., and Maddocks, J.H.,On the Kinematics of wheeled mobile robot, International Journal of Robotics Research. 8, 5, pp. 15-27. 1989. Neimark,, J.I., and Fufaev, F.A., Dynamics of Nonholonomic Systems. American mathematical Society, Providence, RI. 1972 Campion, G., d’Andrea-Novel, B, and Bastin, G. Modeling and state feedback control of nonholonomic mechanical systems. Proc. of the 30th IEEE Conf. on Decision and Control, Brighton, UK, 1184-1189. 1991 Samson, C., and Ait-Abderrahim K,. Feedback control of a nonholonomic wheeled cart in Cartesian space, Proc. of the IEEE International Conference on Robotics and Automation, Sacramento, CA, pp. 1136-1141. 1991 De Luca, A. and Di Benedetto, M.D. Control of nonholonomic systems via dynamic compensation. Kybernetica, 29, 6. 1993 d’Andrea-Novel, B., Bastin, Campion, G.: Control of nonholonomic wheeled mobile robots by state feedback linearization. International Journal of Robotics Research, 14, 6, pp. 543-559, 1995. Jiang, Z-P, Nijimer, H.,: A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Transactions on Automatic Control, 44, 2, pp. 265-279, 1999. Brockett, R.W., Asymptotic stability and feedback stabilization. Progress in Math., vol. 27, Birkhauser, pp. 181-208, 1993.

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