An Adaptive Control Scheme for Nonholonomic Mobile Robot with Parametric Uncertainty

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.

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

\dot{x} sin θ - \dot{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.

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

\begin{array}{l} \dot{x} = v cos θ \\ \dot{y} = v sin θ \\ \dot{θ} = ω \\ m \dot{v} = F \\ I \dot{ω} = N \end{array}

(2)

where v and

\underset{.}{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

F = \frac{1}{R} (τ_{1} + τ_{2})

(3)

N = \frac{L}{R} (τ_{1} + τ_{2})

(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 τ₁ and τ₂ to regulate {B} to is the goal inertial frame {G}={U}, in the presence of uncertainty in parameters m, I, R, and L.

Polar Transformation

Consider the following coordinate transformation

\begin{array}{l} e = \sqrt{x^{2} + y^{2}} \\ x = - e cos (θ + β) \\ y = - e sin (θ + β) \\ θ + β {= tan}^{- 1} \frac{x}{y} \end{array}

(5)

where e is the length of the vector

\vec{O_{B} O_{u}}

and β 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

\begin{array}{l} \dot{e} = - v cos β \\ \dot{β} = \frac{sin β}{e} v - ω \\ \dot{θ} = ω \\ m \dot{v} = F \\ I \dot{ω} = N \end{array}

(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 = \frac{v}{e}, σ = β + θ

and rewrite the equations of motion (6)

\begin{array}{l} \dot{e} = ρ sin β \\ \dot{β} = ρ sin β - ω \\ \dot{ω} = \frac{N}{I} \end{array}

(7)

and

\begin{array}{l} \dot{e} = - ρ cos β e \\ \dot{ρ} = \frac{F}{m e} + ρ^{2} cos β \end{array}

(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

V_{1} = \frac{1}{2} k_{σ} σ^{2} + \frac{1}{2} β^{2}

(9)

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

{\dot{V}}_{1} = β [k_{1} k_{σ} σ \frac{sin β}{β} + k_{1} sin β - ω]

(10)

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

\begin{array}{l} α_{1} (σ, β) = k_{1} k_{σ} σ \frac{sin β}{β} + k_{1} sin β + k_{2} β \\ k_{2} > 0 \end{array}

(11)

virtual control law.

Introduce the error variable

z_{1} = ω - α_{1}

(12)

and compute V̇₁ to obtain

{\dot{V}}_{1} = - k_{2} β^{2} - β z_{1}

Step2: (Backstepping) The function V₁ is now augmented with a quadratic term in z₁ to obtain the new candidate Lyapunov function

V_{2} = V_{1} + \frac{1}{2} z_{1}^{2}

The time derivative of V₂ can be written as

{\dot{V}}_{2} = - k_{2} β^{2} + z_{1} [\frac{N}{I} - f (σ, β, z_{1}, ρ)]

where

f (σ, β, z_{1}, ρ) = \frac{\partial α}{\partial σ} \dot{σ} + \frac{\partial α}{\partial β} β - β

\begin{array}{l} = k_{1} k_{σ} \dot{σ} \frac{sin β}{β} + k_{1} k_{σ} σ \dot{β} \frac{β cos β - sin β}{β^{2}} \\ + k_{1} \dot{β} cos β + k_{2} \dot{β} + β \end{array}

Note that the terms $\frac{sin β}{β}$ and $\frac{β cos β - sin β}{β^{2}}$ are well defined according to the l'Hôpital rule. Let the control for N as

N = I f (σ, β, z_{1}, ρ) - k_{3} z_{1}, k_{3} > 0

Then ${\dot{V}}_{2} = - k_{2} β^{2} - \frac{k_{3}}{I} z_{1}^{2} \leq 0$

which negative semi-definite.

Step 3: Distance regulation

Consider the distance subsystem (8) and define a new error variable z₂ = ρ – k₁ and the Lyapunov function candidate as

V_{3} = V_{2} + \frac{1}{2} z_{2}^{2}

Differentiating V₃ yields to

{\dot{V}}_{3} = - k_{2} β^{2} - \frac{k_{3}}{I} z_{1}^{2} + z_{2} [\frac{F}{m e} + g (σ, β, ρ)]

The last two terms of g are due to the fact that ρ in not constant, but ρ = k₁ + z₂ instead. They are computed by replacing ρ by k₁ + z₁ in the expression for V₁ and propagating the corresponding terms down to V̇₃. Choosing the control input as

F = - m g (σ, β, ρ) e - k_{4} z_{2} e

then the time derivative of V₃ becomes

{\dot{V}}_{3} = - k_{2} β^{2} - \frac{k_{3}}{I} z_{1}^{2} - \frac{k_{4}}{m} z_{2}^{2}

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 = τ₁ – τ₂ and u₂ = τ₁ + τ₂. From (3) and (4) the dynamic equations of the mobile robot become

\dot{ω} = \frac{u_{1}}{c_{1}}, \dot{v} = \frac{u_{2}}{c_{2}}

where

c_{1} = \frac{I R}{L}

and c₂ = mR are the unknown parameters of the system. Let the augmented candidate Lyapunov function

V_{4} = V_{3} + \frac{1}{2 c_{1} γ_{1}} {\tilde{c}}_{1}^{2} + \frac{1}{2 c_{2} γ_{2}} {\tilde{c}}_{2}^{2}

where c˜_i, i = 1,2 are parameter estimation error such that c˜_i = c_i – ĉ_i where ĉ_i, i = 1,2 are the nominal values of c_i.

The time derivative of V₄ can be computed to give

{\dot{V}}_{4} = - k_{2} β^{2} + z_{1} [\frac{u_{1}}{c_{1}} - f (.)] + z_{1} [\frac{u_{2}}{c_{2 e}} - g (.)] - \frac{{\tilde{c}}_{1}}{c_{1} γ_{1}} {\dot{\bar{c}}}_{1} - \frac{{\tilde{c}}_{2}}{c_{2} γ_{2}} {\dot{\bar{c}}}_{2}

and choose the control laws as

\begin{array}{l} u_{1} = {\hat{c}}_{1} f (.) - k_{3} z_{1} \\ u_{2} = - {\hat{c}}_{2} g (.) e - k_{4} z_{2} e \end{array}

to obtain

\begin{array}{l} {\dot{V}}_{4} = - k_{2} β^{2} - \frac{k_{3}}{c_{1}} z_{1}^{2} - \frac{k_{4}}{c_{2}} z_{2}^{2} - \frac{{\tilde{c}}_{1}}{c_{1}} [z_{1} f (.) + \frac{{\dot{\hat{c}}}_{1}}{γ_{1}}] \\ + \frac{{\tilde{c}}_{2}}{c_{2}} [z_{2} g (.) - \frac{{\dot{\hat{c}}}_{2}}{γ_{2}}] \end{array}

Choose the adaptation laws as

\begin{array}{l} {\dot{\hat{c}}}_{1} = - γ_{1} z_{1} f (.) \\ {\dot{\hat{c}}}_{2} = γ_{2} z_{2} g (.) \end{array}

to yield

{\dot{V}}_{4} = - k_{2} β^{2} - \frac{k_{3}}{c_{1}} z_{1}^{2} - \frac{k_{4}}{c_{2}} z_{2}^{2}

which is negative semi-definite.

Step 5: Switching control law

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

\begin{array}{l} u_{1} = - k_{\overset{..}{θ}} \dot{θ} - k_{θ} θ \\ u_{2} = 0 \end{array}

where

k_{\dot{θ}}

and k_θ are positive constants.

The complete control law is thus given by

u = [\begin{array}{l} u_{1} \\ u_{2} \end{array}] = {\begin{array}{l} {\hat{c}}_{1} f (.) - k_{3} z_{1} & for & e \neq 0 \\ - k_{\overset{..}{θ}} \dot{θ} - k_{θ} θ & for & e = 0 \\ - {\hat{c}}_{2} g (.) e - k_{4} z_{2} e & for & e \neq 0 \\ 0 & for & e = 0 \end{array}

with the adaptation laws

\begin{array}{l} {\dot{\hat{c}}}_{1} = - γ_{1} z_{1} f (.) \\ {\dot{\hat{c}}}_{2} = γ_{2} z_{2} g (.) \end{array}

4. Simulation Results

To illustrate the performance of the proposed control scheme, consider a WMR with the following parameters: m = 12kg, I = 1.5kgm², L = 0.5m, R = 0.05m. The control parameters are chosen as k₁ = 0.7, k₂ = 0.1, k₂ = 0.1, k₄ = 0.7, γ₁ = 0.1, γ₂ = 6, k_σ = 3, $k_{\dot{θ}} = 0.9$ and k_θ = 0.4. The initial estimates for the vehicle were m̂ = 20kg, Î = 2kgm², L̂ = 0.6m and R̂ = 0.1m. Figure 2 shows the time responses of the relevant WMR variables for the initial conditions (x₀, y₀, θ₀, v₀, ω₀) = (–4,2,0.8,0,0).

Figure 2:

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

Figure 3:

Time evolution of parameter estimation a: ĉ₁, b: ĉ₂

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.

References

Schraft

R.D.

Schmierer

Serviceroboter. Springer Verlag, 1998.

Alexander

J.C.

Maddocks

J.H.

, On the Kinematics of wheeled mobile robot, International Journal of Robotics Research. 8, 5, pp. 15–27. 1989.

Neimark

J.I.

Fufaev

F.A.

, Dynamics of Nonholonomic Systems. American mathematical Society, Providence, RI. 1972.

Campion

d'Andrea-Novel

Bastin

Modeling and state feedback control of nonholonomic mechanical systems. Proc. of the 30^th IEEE Conf. on Decision and Control, Brighton, UK, 1184–1189. 1991.

Samson

Ait-Abderrahim

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

Di Benedetto

M.D.

Control of nonholonomic systems via dynamic compensation. Kybernetica, 29, 6. 1993.

d'Andrea-Novel

Bastin Campion

: 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

,: 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.