Neural-network control of mobile manipulators

Neural-Network Control of Mobile Manipulators

Sheng Lin and A. A. Goldenberg, Fellow, IEEE

Abstract

In this paper, a neural network (NN)-based methodology is developed for the motion control of mobile manipulators subject to kinematic constraints. The dynamics of the mobile manipulator is assumed to be completely unknown, and is identified on-line by the NN estimators. No preliminary learning stage of NN weights is required. The controller is capable of disturbance-rejection in the presence of unmodeled bounded disturbances. The tracking stability of the closed-loop system, the convergence of the NN weight-updating process and boundedness of NN weight estimation errors are all guaranteed. Experimental tests on a four-DOF manipulator arm illustrate that the proposed controller significantly improves the performance in comparison with conventional robust control.

Index Terms-Closed-loop stability, disturbance rejection, dynamics interaction, mobile manipulator, neural networks (NNs), parameter uncertainty.

I. INTRODUCTION

A MOBILE manipulator is a robotic arm mounted on a moving base. The base mobility substantially increases the size of the robot workspace, and enables better positioning of the manipulator for efficient task execution. Mobile manipulator systems have been suggested for various applications, e.g., tasks involving hazardous environments, such as explosive handling, waste management, outdoor exploration and space operations. The motion of the mobile base is subject to holonomic or nonholonomic kinematics constraints, which renders the control of mobile manipulator very challenging. Furthermore, the complex physical structure, the highly coupled dynamics between the mobile base and the mounted manipulator arm, and the mobility of the wheeled mobile base are some of the features that substantially increase the difficulty of system design and control.

In recent years, there has been a growing interest in the motion control of mobile manipulators. Hootsmans and Dubowsky [1] developed a control method based on an extended Jacobian to compensate for dynamic interactions between the manipulator and base. Yamamoto et al. [2] addressed the coordination of locomotion and manipulator motion between the base and the arm, and the problem of following a moving surface. Liu and Lewis [3] developed a decentralized robust controller for trajectory tracking of the mobile manipulator end-effector. Khatib [4] addressed payload modeling and control issue. Tahboub [5]

Manuscript received April 11, 2000; revised December 7, 2000.
S. Lin is with the Mechanical and Industrial Engineering Department, University of Toronto, Toronto, ON M5S 3G8, Canada (e-mail: slin@mie.utoronto.ca).
A. A. Goldenberg is with the Mechanical and Industrial Engineering Department, University of Toronto, Toronto, ON M5S 3G8, Canada. He is also with Engineering Services, Inc. (ESI), Toronto, ON, Canada (e-mail: golden@esit.com).

Publisher Item Identifier S 1045-9227(01)05537-0.
developed a robust adaptive controller for trajectory tracking of a mobile manipulator when it moves on uneven terrain. Chung et al. [6] proposed an interaction controller for the control of mobile manipulator, consisting of a robust adaptive manipulator controller and an input-output linearizing controller. Dixon et al. [18] developed a robust tracking and regulation controller for mobile robots.

Most approaches require the precise knowledge of dynamics of the mobile manipulator, or, they simplify the dynamics model by ignoring dynamics issues, such as vehicle dynamics, payload dynamics, dynamics interactions between the base and the arm, and unknown disturbances such as the dynamic effect caused by terrain irregularity. These issues make the published schemes inappropriate for realistic applications for the following reasons. Due to its complexity, the precise dynamics model of a mobile manipulator is normally unobtainable. When the mobile manipulator moves at a relatively high speed, ignoring the vehicle dynamics and the dynamics interactions between the arm and the base may cause unbearable vibrations of the system [1]. Finally, when the robot moves on uneven terrain (e.g., for outdoor exploration), ignoring disturbances generated by terrain irregularities may lead to tip-over.

In the recent years, neural networks (NNs), with their strong learning capability, have proven to be a suitable tool for controlling complex nonlinear dynamic systems. The basic idea behind NN-based control is to use an NN estimator to identify the unknown nonlinear dynamics and compensate for it. Also, the NN-based approach can deal with the control of nonlinear systems that may not be linearly parameterizable, as required in the adaptive approach.

With regard to neural networks, they have been widely adopted in the modeling and control of robotic manipulators [7], [19]. Polycarpou [8] provided a systematic methodology for the identification of nonlinear systems using neural networks. Jin et al. [9] developed a strict theoretical basis for the stable design of NN-based manipulator control. Gorinevsky et al. [10] applied neural networks to path-planning and tracking control problems of mobile robots. Lewis et al. [7] proposed an NN control scheme that guarantees closed-loop performance in terms of small tracking errors and bounded controls. Fierro and Lewis [11] further applied this methodology to the control of a nonholonomic mobile robot where unmodeled bounded disturbances and/or unstructured unmodeled dynamics of the vehicle are considered.

One main issue of all the contributions mentioned above is that the effectiveness was illustrated only through simulations. Very few results were reported to have been verified experimentally. Moreover, to the best of our knowledge, no satisfactory experimental results have been reported in the literature on NN-based stable closed-loop robotic control. Also, it is well

known that good simulation performance of a control law normally does not guarantee its applicability in real systems due to unmodeled dynamics, difficulty in parameter adjustment or other limitations.

In this paper, an NN-based control methodology is proposed for the joint-space position control of a mobile manipulator. Two NN controllers are developed to control the arm and the base independently. Each NN controller output comprises a linear control term (classical PID) and an NN compensation term. The NN compensation term is used for on-line estimation of unknown nonlinear dynamics caused by parameter uncertainty and disturbances. No preliminary learning stage is required for the NN weights. The control scheme is capable of disturbance-rejection in the presence of unknown bounded disturbances. The NN on-line learning laws are original. The tracking stability of the closed-loop system, the convergence of the NN learning process and the boundedness of NN weight estimation errors are all rigorously proven. In order to validate the proposed NN-controllers and NN learning laws, extensive experiments have been conducted on a real robotic manipulator. The experimental results are also presented in this paper.

This paper is organized as follows. Section II provides some necessary mathematical preliminaries. The dynamic model of mobile manipulators subject to kinematic constraints is derived in Section III. The proposed theory is developed in Section IV. Section V presents the experimental results to illustrate the effectiveness of the proposed theory. Some remarks are made in the final Section VI.

II. MATHEMATICAL PRELIMINARIES

Definition 1 (Vector and Matrix Norms): Let $\Re^{n}$ denote the space of real $n$-dimensional vector. The norm of a vector $x \in$ $\Re^{n}$ is defined as

$$\|x\|=\sqrt{x^{T} x}$$

The norm of a matrix is defined as

$$\|A\|=\sqrt{\lambda_{\max }\left(A^{T} A\right)}$$

where $\lambda_{\max }(\circ)$ is the maximum eigenvalue of $A^{T} A$.
The Frobenius norm of a matrix is defined as the root of the sum of the squares of all elements

$$\|A\|_{F}^{2}=\operatorname{tr}\left(A^{T} A\right)$$

where the trace $\operatorname{tr}(A)$ satisfies $\operatorname{tr}(A)=\operatorname{tr}\left(A^{T}\right)$ for any $A \in \Re^{n \times n}$. For any $B \in \Re^{m \times n}$ and $C \in \Re^{n \times m}$, $\operatorname{tr}(B C)=\operatorname{tr}(C B)$. Suppose that $A$ is positive definite, then for any $B \in \Re^{m \times n}$

$$\operatorname{tr}\left(B A B^{T}\right) \geq 0$$

with equality if $B$ is the $m \times n$ zero matrix, and

$$\frac{d}{d t}\{\operatorname{tr}(A(t))\}=\operatorname{tr}\left(\frac{d A(t)}{d t}\right)$$

Definition 2 (Uniform Ultimate Boundedness, UUB) [7]: Consider the dynamic system $\dot{x}=f(x, t)$ with $x \in \Re^{n}$. Let the initial time be $t_{0}$, and the initial condition be $x_{0} \equiv x\left(t_{0}\right)$. The equilibrium point $x_{e}$ is said to be uniformly ultimately bounded if there exists a compact set $S \subset \Re^{n}$ so that for all
$x_{0} \in S$ there exists a bound $B$ and a time $T\left(B, x_{0}\right)$, such that $\left\|x(t)-x_{e}\right\| \leq B$ for all $t \geq t_{0}+T$.

III. DYNAMICS MODELING OF A MOBILE MANIPULATOR

A. General

The dynamics of a mobile manipulator subject to kinematic constraints can be obtained using the Lagrangian approach in the form [7]

$$M(q) \ddot{q}+C(q, \dot{q}) \dot{q}+F(q, \dot{q})+A^{T}(q) \lambda+\tau_{d}=E(q) \tau$$

where $r$ kinematic constraints are described by

$$A(q) \dot{q}=0$$

and $q \in \Re^{p}$ denotes the $p$ generalized coordinates, $M(q) \in \Re^{p \times p}$ is the symmetric and positive definite inertia matrix, $C(q, \dot{q}) \in \Re^{p \times p}$ represents the centripetal and Coriolis matrix, $F(q, \dot{q}) \in \Re^{p}$ represents the friction and gravitational vector, $A(q) \in \Re^{r \times p}$ is the constraint matrix, $\lambda \in \Re^{r}$ is the Lagrange multiplier which denotes the vector of constraint forces, $\tau_{d} \in \Re^{p}$ denotes bounded unknown disturbances including unstructured dynamics, $E(q) \in \Re^{p \times(p-r)}$ is the input transformation matrix, and $\tau \in \Re^{p-r}$ is the torque input vector.

In (6), the following properties hold [7].
Property 1 (Parameter Boundedness):

$$\begin{aligned} & M_{\min } I_{n} \leq M(q) \leq M_{\max } I_{n} \\ & C(q, \dot{q}) \leq C_{b}(q)\|\dot{q}\| \end{aligned}$$

where

$M_{\min }, M_{\max }$	positive scalar constants which are dependant on the mass properties and constraint matrix;
$I_{n}$	$p \times p$ identity matrix;
$C_{b}(q)$	positive definite function of $q$.

Property 2 (Skew Symmetricity):

$$\begin{aligned} \dot{M}-2 C & =-\left(\dot{M}-2 C\right)^{T} \\ \dot{M} & =C+C^{T} \end{aligned}$$

B. Dynamics Modeling of the Mobile Manipulator

The generalized coordinates $q$ may be separated into two sets $q=\left(q_{v}^{T}, q_{r}^{T}\right)^{T}$, where $q_{v} \in \Re^{m}$ describes the generalized coordinates appearing in the constraint equations (7), and $q_{r} \in \Re^{n}$ are the free generalized coordinates; $p=m+n$. Therefore, (7) can be simplified to

$$A_{v}\left(q_{v}\right) \dot{q}_{v}=0$$

with $A_{v}\left(q_{v}\right) \in \Re^{r \times m}$. Assume that the robot is fully actuated, then (6) can be further rewritten as

$$\begin{aligned} & \binom{M_{11} \quad M_{12}}{M_{21} \quad M_{22}}\binom{\ddot{q}_{v}}{\ddot{q}_{r}}+\left(\begin{array}{cc} C_{11} & C_{12} \\ C_{21} & C_{22} \end{array}\right)\binom{\dot{q}_{v}}{\dot{q}_{r}} \\ & +\binom{F_{1}}{F_{2}}+\binom{A_{v}^{T}\left(q_{v}\right) \lambda}{0}+\binom{\tau_{d 1}}{\tau_{d 2}} \\ & =\binom{E_{v} \tau_{v}}{\tau_{r}} \end{aligned}$$

where $\tau_{v} \in \Re^{m-r}$ represents the actuated torque vector of the constrained coordinates, those related to the constrained motion of the wheels, the joints, and the end-effector. For simplicity in the theoretical derivation, hereinafter we consider only the case where the vehicle motion is constrained. However, the proposed theory can be easily extended to include joint and/or end-effector constraints. $E_{v} \in \Re^{m \times(m-r)}$ represents the input transformation matrix; $\tau_{r} \in \Re^{n}$ the actuating torque vector of the free coordinates; $\tau_{d 1}$ and $\tau_{d 2}$ are disturbance torques bounded by $\left\|\tau_{d 1}\right\| \leq \tau_{1 N}$ and $\left\|\tau_{d 2}\right\| \leq \tau_{2 N}$, with $\tau_{1 N}$ and $\tau_{2 N}$ some positive constants. The constraint equation (10) and the first row of (11) form the vehicle subdynamics, and the second row of (11) represents the arm subdynamics.

It is straightforward to show that the following properties hold.

Property 3: $M_{11}$ and $M_{22}$ are both symmetric and positive definite, and

$$\begin{aligned} & \dot{M}_{11}-2 C_{11}=-\left(\dot{M}_{11}-2 C_{11}\right)^{T} \\ & \dot{M}_{22}-2 C_{22}=-\left(\dot{M}_{22}-2 C_{22}\right)^{T} \end{aligned}$$

Property 4:

$$\begin{aligned} & \dot{M}_{21}=C_{21}+C_{12}^{T} \\ & M_{12}=M_{21}^{T} \end{aligned}$$

Let $S_{v}\left(q_{v}\right) \in \Re^{m \times(m-r)}$ denote a full rank matrix formed by $(m-r)$ columns that span the null space of $A_{v}\left(q_{v}\right)$ defined in (10), i.e.,

$$S^{T}\left(q_{v}\right) A_{v}^{T}\left(q_{v}\right)=0$$

From (14), we can find an auxiliary vector $v(t) \in \Re^{m-r}$ such that for all $t$

$$\dot{q}_{v}=S\left(q_{v}\right) v(t)$$

and its derivative is

$$\ddot{q}_{v}=S\left(q_{v}\right) \dot{v}+\dot{S}\left(q_{v}\right) v$$

Equation (15) is called the steering system. $v(t)$ can be regarded as a velocity input vector steering the state vector $q$ in state space.

Equations (11) and (15) describe the dynamics equations of mobile manipulators subject to kinematic constraints. The constraints may consist of kinematic constraints on the vehicle base and task-space constraints when the end-effector is required to follow a certain surface [2], etc. Without loss of generality, in this paper we only consider the case that the mobile base is subject to holonomic or nonholonomic kinematic constraints.

C. Dynamic Modeling of the Vehicle Base

Let us consider the first $m$-equations of (11)

$$\begin{aligned} & M_{11} \ddot{q}_{v}+M_{12} \ddot{q}_{r}+C_{11} \dot{q}_{v}+C_{12} \dot{q}_{r} \\ & \quad+F_{1}+A_{v}^{T} \lambda+\tau_{d 1}=E_{v} \tau_{v} \end{aligned}$$

Multiplying both sides of (17) by $S^{T}$ and using (14) to eliminate the constraint force we obtain

$$\begin{aligned} & S^{T} M_{11} \ddot{q}_{v}+S^{T} M_{12} \ddot{q}_{r}+S^{T} C_{11} \dot{q}_{v} \\ & \quad+S^{T} C_{12} \dot{q}_{r}+S^{T} F_{1}+S^{T} \tau_{d 1}=S^{T} E_{v} \tau_{v} \end{aligned}$$

Substituting (15) and (16) into (18) yields

$$\begin{aligned} & S^{T} M_{11} S \dot{v}+S^{T} M_{11} \dot{S} v+S^{T} M_{12} \ddot{q}_{r}+S^{T} C_{11} S v \\ & \quad+S^{T} C_{12} \dot{q}_{r}+S^{T} F_{1}+S^{T} \tau_{d 1}=S^{T} E_{v} \tau_{v} \end{aligned}$$

Let us rewrite (19) in a compact form as

$$\bar{M}_{11} \dot{v}+\bar{C}_{11} v+f_{1}+\bar{\tau}_{d 1}=\bar{\tau}_{v}$$

where $\bar{M}_{11}=S^{T} M_{11} S, \bar{C}_{11}=S^{T} C_{11} S+S^{T} M_{11} \dot{S}, \bar{\tau}_{d 1}=$ $S^{T} \tau_{d 1} ;\left\|\bar{\tau}_{d 1}\right\| \leq \bar{\tau}_{1 N}$ with $\bar{\tau}_{1 N}$ some positive constant (notice that both $\tau_{d 1}$ and $S$ are bounded), and

$$\begin{aligned} \bar{\tau}_{v} & =S^{T} E_{v} \tau_{v} \\ f_{1} & =S^{T}\left(M_{12} \ddot{q}_{r}+C_{12} \dot{q}_{r}+F_{1}\right) \end{aligned}$$

$f_{1}$ consists of the gravitational and friction force vector $F_{1}$, the disturbances on the vehicle base (e.g., terrain disturbance force), and the dynamic interaction with the mounted manipulator arm (i.e., $M_{12} \ddot{q}_{r}+C_{12} \dot{q}_{r}$ ) which has been shown to have significant effect on the base motion, thus it needs to be compensated for [2].

Property 5: $\dot{M}_{11}-2 \bar{C}_{11}$ is skew-symmetric. Proof:

$$\begin{aligned} & \dot{M}_{11}-2 \bar{C}_{11} \\ & \quad=2 S^{T} M_{11} \dot{S}+S^{T} \dot{M}_{11} S-2 S^{T} M_{11} \dot{S}-2 S^{T} C_{11} S \\ & \quad=S^{T}\left(\dot{M}_{11}-2 C_{11}\right) S \end{aligned}$$

Since $\dot{M}_{11}-2 C_{11}$ is skew-symmetric, therefore, $\dot{\bar{M}}_{11}-2 \bar{C}_{11}$ is also skew-symmetric.

D. Dynamics Modeling of the Mounted Manipulator Arm

Consider the last $n$-equations of (11)

$$M_{21} \ddot{q}_{v}+M_{22} \ddot{q}_{r}+C_{21} \dot{q}_{v}+C_{22} \dot{q}_{r}+F_{2}+\tau_{d 2}=\tau_{r}$$

Rearrange (23) as follows:

$$M_{22} \ddot{q}_{r}+C_{22} \dot{q}_{r}+\left(M_{21} \ddot{q}_{v}+C_{21} \dot{q}_{v}+F_{2}\right)+\tau_{d 2}=\tau_{r}$$

Equation (24) represents the dynamic equation of the mounted manipulator arm. The terms in the brackets consist of the dynamic interaction term (i.e., $M_{21} \ddot{q}_{v}+C_{21} \dot{q}_{v}$ ), the gravitational and friction force vector $F_{2}$, and the disturbance on the manipulator.

Equations (15), (20), and (24) form the complete dynamic model of the mobile manipulator subject to kinematics constraints. In the following sections, the control methodology will be developed based on this model.

IV. THEORY

A. General

In most applications mobile manipulators are required to perform tasks (defined in the task-space) which are depicted as certain desired trajectories $x_{v}^{d}(t)$. In order to achieve this objective, joint-space reference trajectories $q_{d}(t)=$ $\left(q_{v d}^{T}(t), q_{v d}^{T}(t)\right)^{T}$ must first be derived, where $q_{r d}(t)$ denotes the reference arm trajectory, and $q_{v d}(t)$ the reference vehicle trajectory. Joint torque commands are then generated by the controller to make the mobile manipulator track the reference joint-space trajectories, and consequently, the desired task-space trajectories.

In this paper, it is assumed that reference joint-space trajectories are available (i.e., they have already been derived based on desired task-space trajectories). The main concern is to provide proper joint torque input that can guarantee stable joint space tracking in the presence of parameter uncertainty and unknown disturbances. The dynamics interaction between the base and the arm is also taken into account in this consideration.

In this section, NN-based control laws and NN weight updating laws will be derived for the stable joint-space tracking of a mobile manipulator described by (15), (20), and (24). The design procedure is as follows: 1) the mobile manipulator dynamics is redefined as an error dynamics based on a set of carefully chosen Lyapunov subfunctions; 2) NN on-line estimators are constructed and new NN learning laws are proposed; 3) new control laws for the manipulator arm and vehicle are derived by taking into account the dynamic couplings between the two; and 4) a proof on the tracking stability of the overall closed-loop system and the boundedness on NN weight estimation errors is provided.

B. Feedback Control of a Mobile Robot With Independently Actuated Wheels

Consider the vehicle subdynamics represented by (15) and (20). Tracking control of the steering system (15) has been extensively addressed in the literature [13], [17], [18]. For example, for a wheeled mobile robot with two independently actuated wheels, the kinematics parameters in (15) are defined as

$$S\left(q_{v}\right)=\left[\begin{array}{cc} \cos \theta & 0 \\ \sin \theta & 0 \\ 0 & 1 \end{array}\right], \quad v=\binom{v}{\omega} \quad \text { and } \quad q_{v}=\binom{x}{y}$$

where $(x, y)$ represent the Cartesian coordinates of the cart, $\theta$ its orientation, $v$ and $\omega$ its linear and angular velocities, respectively. Let the reference motion of the cart be prescribed as

$$\left(\begin{array}{c} \dot{x}_{r} \\ \dot{y}_{r} \\ \dot{\theta}_{r} \end{array}\right)=\left[\begin{array}{cc} \cos \theta_{r} & 0 \\ \sin \theta_{r} & 0 \\ 0 & 1 \end{array}\right]\binom{v_{r}}{\omega_{r}}$$

where $v_{r}>0$ and $\omega_{r}$ are reference linear and angular velocities, respectively. Stable linear and nonlinear velocity feedback laws for (25) can be found in [14] to achieve the asymptotic tracking. For instance, the following feedback velocity input guarantees that the position tracking of (26) is asymptotically stable [14]:

$$v_{c}=\binom{v_{c}}{\omega_{c}}=\left[\begin{array}{cc} v_{r} \cos e_{3}+k_{1} e_{1} \\ \omega_{r}+k_{2} v_{r} e_{2}+k_{3} v_{r} \sin e_{3} \end{array}\right]$$

where positive constants $k_{1}, k_{2}$ and $k_{3}$ are control gains, and the tracking errors are defined as

$$\left(\begin{array}{c} e_{1} \\ e_{2} \\ e_{3} \end{array}\right)=\left[\begin{array}{ccc} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array}\right]\left(\begin{array}{c} x_{r}-x \\ y_{r}-y \\ \theta_{r}-\theta \end{array}\right)$$

The stability of the tracking system can be proven by choosing the following Lyapunov function [11]:

$$L=k_{1}\left(e_{1}^{2}+e_{2}^{2}\right)+2 k_{3} v_{r}\left(1-\cos e_{3}\right)$$

The derivative of $L$ along the system trajectories is guaranteed to be negative definite [14].

More generally, consider the steering system (15). Define the tracking error as $\tilde{q}_{v}=q_{v d}-q_{v}$. It is assumed that there exist a Lyapunov function $V_{1}\left(\tilde{q}_{v}, t\right)$, a positive continuous function $W_{1}(t)>0$, and a reference smooth feedback velocity $\alpha(\cdot)$, such that

$$\left.\frac{\partial V_{1}}{\partial t}+\frac{\partial V_{1}}{\partial \tilde{q}_{v}} \dot{\tilde{q}}_{v}\right|_{\tilde{q}_{v}=S\left(q_{v}\right) \alpha} \leq-W_{1}(t) \quad \text { when } \tilde{q}_{v} \neq 0$$

The objective hereinafter is to derive proper torque input $\bar{\tau}_{v}(t)$ in (20), such that the velocity $v(t)$ [defined in (15)] tracks the reference velocity $\alpha(\cdot)$.

The following assumption is made in the sequel.
Assumption 1 (Bounded Reference Trajectory): The desired trajectories of both vehicle base and manipulator arm are bounded by

$$\left\|\begin{array}{c} q_{v d} \\ \alpha \\ \dot{\alpha} \\ q_{r d} \\ \tilde{q}_{r d} \\ \tilde{q}_{r d} \end{array}\right\| \leq q_{B}$$

with $q_{B}$ some known positive constant.

C. Lyapunov Function for the Vehicle Base

Define the vehicle velocity tracking error $z$ as

$$z=v-\alpha$$

Differentiating (32), multiplying both sides by $\bar{M}_{11}$ and substituting (20) into it yields

$$\bar{M}_{11} \dot{z}+\bar{C}_{11} z+f_{1}+\bar{\tau}_{d 1}+\bar{M}_{11} \dot{\alpha}+\bar{C}_{11} \alpha=\bar{\tau}_{v}$$

Equation (33) represents the vehicle dynamics in terms of tracking errors.

Let us choose the Lyapunov function $V_{2}$ as

$$V_{2}=\frac{1}{2} z^{T} \bar{M}_{11} z$$

Based on Property 3, $\bar{M}_{11}$ is a symmetric positive definite matrix.

Differentiating (34) along the system trajectories yields

$$\dot{V}_{2}=z^{T} \bar{M}_{11} \dot{z}+\frac{1}{2} z^{T} \dot{\bar{M}}_{11} z$$

Substituting (33) into (35) we have

$$\begin{aligned} \dot{V}_{2}= & z^{T}\left\{\bar{\tau}_{v}-f_{1}-\bar{\tau}_{d 1}-\bar{M}_{11} \dot{\alpha}-\bar{C}_{11} \alpha\right\} \\ & +\frac{1}{2} z^{T}\left(\dot{\bar{M}}_{11}-2 \bar{C}_{11}\right) z \\ = & z^{T}\left\{\bar{\tau}_{v}-f_{1}-\bar{\tau}_{d 1}-\bar{M}_{11} \dot{\alpha}-\bar{C}_{11} \alpha\right\} \end{aligned}$$

D. Lyapunov Function for the Manipulator Arm

Now consider the arm subdynamics (24). Let us define the arm tracking error and its derivative as

$$e=q_{r d}-q_{r}, \quad \dot{e}=\dot{q}_{r d}-\dot{q}_{r}$$

and the filtered tracking error as

$$r=\dot{c}+k e$$

where $k=k^{T}>0$. In (38), $r$ can be regarded as an input to a linear dynamic system with state variable $e$. Therefore, when $r \rightarrow 0$, it can guarantee that $e \rightarrow 0$ through linear control theory [7].

Let us differentiate (38) to obtain

$$\dot{r}=\ddot{c}+k \dot{c}=\ddot{q}_{r d}-\ddot{q}_{r}+k \dot{c}$$

From (37)-(39) we have

$$\begin{aligned} \dot{q}_{r} & =\dot{q}_{r d}-(r-k e) \\ \ddot{q}_{r} & =\ddot{q}_{r d}-\dot{r}+k(r-k e) \end{aligned}$$

The manipulator dynamics (24) can then be formulated in terms of the filtered error as

$$\begin{aligned} M_{22} \dot{r}= & M_{22} \ddot{c}+M_{22} k \dot{c} \\ = & M_{22} \ddot{q}_{r d}-M_{22} \ddot{q}_{r}+M_{22} k(r-k e) \\ = & M_{22} \ddot{q}_{r d}+C_{22} \dot{q}_{r}+\left(M_{21} \ddot{q}_{v}+C_{21} \dot{q}_{v}+F_{2}+\tau_{d 2}\right) \\ & -\tau_{r}+M_{22} k(r-k e) \\ = & M_{22} \ddot{q}_{r d}+C_{22} \dot{q}_{r d}-C_{22}(r-k e) \\ & +\left(M_{21} \ddot{q}_{v}+C_{21} \dot{q}_{v}+F_{2}+\tau_{d 2}\right) \\ & -\tau_{r}+M_{22} k(r-k e) \\ = & \left(M_{22} \ddot{q}_{r d}+C_{22} \dot{q}_{r d}+F_{2}+\tau_{d 2}\right)+M_{21} \ddot{q}_{v} \\ & +C_{21} \dot{q}_{v}-\tau_{r}+\left(M_{22} k-C_{22}\right)(r-k e) \\ = & -\tau_{r}+\left(M_{22} k-C_{22}\right)(r-k e)+f_{2}+\tau_{d 2} \end{aligned}$$

with

$$f_{2}=M_{21} \ddot{q}_{v}+C_{21} \dot{q}_{v}+M_{22} \ddot{q}_{r d}+C_{22} \dot{q}_{r d}+F_{2}$$

$f_{2}$ consists of the manipulator dynamics $\left(M_{22} \ddot{q}_{r d}+C_{22} \dot{q}_{r d}+\right.$ $F_{2}$ ), and the dynamics of interaction with the vehicle base $\left(M_{21} \ddot{q}_{v}+C_{21} \dot{q}_{v}\right)$.

To design the manipulator torque input, we choose the Lyapunov function as

$$V_{3}=\frac{1}{2} r^{T} M_{22} r$$

Notice that $M_{22}$ is a symmetric positive definite matrix (Property 3). Differentiating (44) along the system trajectories we obtain

$$\begin{aligned} \dot{V}_{3}= & r^{T} M_{22} \dot{r}+\frac{1}{2} r^{T} \dot{M}_{22} r \\ = & r^{T}\left\{-\tau_{r}+\left(M_{22} k-C_{22}\right)(r-k e)+f_{2}+\tau_{d 2}\right\} \\ & +\frac{1}{2} r^{T} \dot{M}_{22} r \\ = & r^{T}\left\{-\tau_{r}+M_{22} k r-\left(M_{22} k-C_{22}\right) k e+f_{2}+\tau_{d 2}\right\} \\ & +\frac{1}{2} r^{T}\left(\dot{M}_{22}-2 C_{22}\right) r \\ = & r^{T}\left\{-\tau_{r}+M_{22} k r+\left(C_{22}-M_{22} k\right) k e+f_{2}+\tau_{d 2}\right\} \end{aligned}$$

E. Lyapunov Function for the Mobile Manipulator

Let us consider the overall dynamics (11) that combines both the arm and vehicle dynamics. Let us choose the Lyapunov function as

$$V_{4}=V_{1}+\frac{1}{2}\binom{S z}{-r}^{T} M\binom{S z}{-r}$$

In the proposed Lyapunov function $V_{4}, V_{1}$ [see (30)] is used to account for the nonholonomic steering system (15), and the second term accounts for the vehicle base and manipulator arm dynamics, as well as the dynamic couplings between the two.

From (46) we have

$$\begin{aligned} V_{4}= & V_{1}+\frac{1}{2}\binom{S z}{-r}^{T}\left(\begin{array}{cc} M_{11} & M_{12} \\ M_{12}^{T} & M_{22} \end{array}\right)\binom{S z}{-r} \\ = & V_{1}+\frac{1}{2}(S z)^{T} M_{11}(S z)-\frac{1}{2} r^{T} M_{12}^{T}(S z) \\ & -\frac{1}{2}(S z)^{T} M_{12} r+\frac{1}{2} r^{T} M_{22} r \\ = & V_{1}+\frac{1}{2} z^{T}\left(S^{T} M_{11} S\right) z-r^{T} M_{12}^{T}(S z)+\frac{1}{2} r^{T} M_{22} r \\ = & V_{1}+\frac{1}{2} z^{T} \bar{M}_{11} z-r^{T} M_{12}^{T}(S z)+\frac{1}{2} r^{T} M_{22} r \\ = & V_{1}+V_{2}+V_{3}-r^{T} M_{12}^{T}(S z) \end{aligned}$$

Differentiating (47) yields

$$\dot{V}_{4}=\dot{V}_{1}+\dot{V}_{2}+\dot{V}_{3}-\frac{d}{d t}\left\{r^{T} M_{22}(S z)\right\}$$

Substituting (30), (36), and (45) into (48) yields

$$\begin{aligned} \dot{V}_{4} \leq & -W_{1}(t)+z^{T}\left\{\bar{\tau}_{v}-f_{1}-\bar{M}_{11} \dot{\alpha}-\bar{C}_{11} \alpha-\bar{\tau}_{d 1}\right\} \\ & +r^{T}\left\{-\tau_{r}+M_{22} k r-\left(M_{22} k-C_{22}\right) k e+f_{2}+\tau_{d 2}\right\} \\ & -\frac{d}{d t}\left\{r^{T} M_{22}(S z)\right\} \end{aligned}$$

From the definition of $f_{1}$ in (22) and (40), (41) we have

$$\begin{aligned} f_{1}= & S^{T}\left(M_{12} \ddot{q}_{r}+C_{12} \dot{q}_{r}+F_{1}\right) \\ = & S^{T}\left\{M_{12}\left(\ddot{q}_{r d}-\dot{r}+k(r-k e)\right)\right. \\ & \left.+C_{12}\left(\dot{q}_{r d}-(r-k e)\right)+F_{1}\right\} \\ = & -S^{T}\left(M_{12} \dot{r}+\left(C_{12}-M_{12} k\right)(r-k e)\right) \\ & +\underbrace{S^{T}\left\{M_{12} \ddot{q}_{r d}+C_{12} \dot{q}_{r d}+F_{1}\right\}}_{\bar{f}_{1}} \\ = & -S^{T}\left(M_{12} \dot{r}+\left(C_{12}-M_{12} k\right)(r-k e)\right)+\bar{f}_{1} \end{aligned}$$

From the definition of $f_{2}$ in (16) and (43) we have

$$\begin{aligned} f_{2}= & M_{21}(\dot{S} v+S \dot{v})+C_{21} S v+\underbrace{\left(M_{22} \ddot{q}_{r d}+C_{22} \dot{q}_{r d}+F_{2}\right)}_{\bar{f}_{2}} \\ = & \bar{f}_{2}+\left(M_{21} S\right) \dot{v}+\left(M_{21} \dot{S}+C_{21} S\right) v \\ = & \bar{f}_{2}+\left(M_{21} S\right)(\dot{z}+\dot{\alpha})+\left(M_{21} \dot{S}+C_{21} S\right)(z+\alpha) \end{aligned}$$

Substituting (50) and (51) into (49) and after some collections of them we have

$$\begin{aligned} \dot{V}_{4} \leq & -W_{1}(t)+z^{T}\left\{\bar{\tau}_{v}-\bar{M}_{11} \dot{\alpha}-\bar{C}_{11} \alpha\right\} \\ & +r^{T}\left\{-\tau_{r}+M_{22} k r-\left(M_{22} k-C_{22}\right) k e\right\} \\ & -z^{T} \bar{f}_{1}+r^{T} \bar{f}_{2}-\frac{d}{d t}\left\{r^{T} M_{22}(S z)\right\} \\ & -z^{T} \bar{\tau}_{d 1}+r^{T} \tau_{d 2} \end{aligned}$$

First of all, we carry out the following derivations:

$$\begin{aligned} & -z^{T} \bar{f}_{1}+r^{T} \bar{f}_{2}-\frac{d}{d t}\left\{r^{T} M_{22}(S z)\right\} \\ & =-z^{T} \bar{f}_{1}+r^{T} \bar{f}_{2}+(S z)^{T}\left\{M_{12} \dot{r}+\left(C_{12}-M_{12} k\right)(r-k e)\right\} \\ & +r^{T}\left\{M_{21} S \dot{z}+M_{21} S \dot{\alpha}+M_{21} \dot{S} z+M_{21} \dot{S} \alpha+C_{21} S z\right. \\ & \left.+C_{21} S \alpha\right\}-\dot{r}^{T} M_{21} S z-r^{T} \dot{M}_{21} S z \\ & -r^{T} M_{21} \dot{S} z-r^{T} M_{21} S \dot{z} \\ & =-z^{T} \bar{f}_{1}+r^{T} \bar{f}_{2}+(S z)^{T}\left\{\left(C_{12}-M_{12} k\right)(r-k e)\right\} \\ & +r^{T}\left\{M_{21} S \dot{\alpha}+M_{21} \dot{S} \alpha+C_{21} S \alpha-C_{21} S z\right\} \\ & =-z^{T} \bar{f}_{1}+r^{T} \bar{f}_{2}+(S z)^{T}\left\{-C_{12} k e-M_{12} k(r-k e)\right\} \\ & +r^{T}\left\{M_{21} S \dot{\alpha}+M_{21} \dot{S} \alpha+C_{21} S \alpha\right\} \end{aligned}$$

where Properties 4 and 5 have been used in the previous derivations.

Substituting (53) into (52) we obtain

$$\begin{aligned} \dot{V}_{4} \leq & -W_{1}(t)+z^{T}\left\{\bar{\tau}_{v}-\bar{M}_{11} \dot{\alpha}-\bar{C}_{11} \alpha\right\} \\ & +r^{T}\left\{-\tau_{r}+M_{22} k r-\left(M_{22} k-C_{22}\right) k e\right\} \\ & -z^{T} \bar{f}_{1}+r^{T} \bar{f}_{2}+(S z)^{T}\left\{-C_{12} k e-M_{12} k(r-k e)\right\} \\ & +r^{T}\left\{M_{21} S \dot{\alpha}+M_{21} \dot{S} \alpha+C_{21} S \alpha\right\}-z^{T} \bar{\tau}_{d 1}+r^{T} \tau_{d 2} \\ = & -W_{1}(t)+z^{T}\left\{\bar{\tau}_{v}-\bar{M}_{11} \dot{\alpha}-\bar{C}_{11} \alpha-\bar{f}_{1}-S^{T}\right. \\ & \left.\cdot\left\{C_{12} k e+M_{12} k(r-k e)\right\}\right\} \\ & +r^{T}\left\{-\tau_{r}+M_{22} k r+\left(C_{22}-M_{22} k\right) k e+\bar{f}_{2}\right. \\ & \left.+M_{21} S \dot{\alpha}+M_{21} \dot{S} \alpha+C_{21} S \alpha\right\}-z^{T} \bar{\tau}_{d 1}+r^{T} \tau_{d 2} \end{aligned}$$

Therefore

$$\begin{aligned} \dot{V}_{4} \leq & -W_{1}(t)+z^{T}\left\{\bar{\tau}_{v}-\Psi_{1}\right\}+r^{T}\left\{-\tau_{r}+\Psi_{2}\right\} \\ & -z^{T} \bar{\tau}_{d 1}+r^{T} \tau_{d 2} \end{aligned}$$

with the unknown nonlinear terms

$$\begin{aligned} \Psi_{1}= & \bar{M}_{11} \dot{\alpha}+\bar{C}_{11} \alpha+\bar{f}_{1}+S^{T}\left\{C_{12} k e+M_{12} k(r-k e)\right\} \\ \Psi_{2}= & M_{22} k r+\left(C_{22}-M_{22} k\right) k e+\bar{f}_{2}+M_{21} S \dot{\alpha} \\ & +M_{21} \dot{S} \alpha+C_{21} S \alpha \end{aligned}$$

where

$$\begin{aligned} & \bar{f}_{1}=S^{T}\left\{M_{12} \ddot{q}_{r d}+C_{12} \dot{q}_{r d}+F_{1}\right\} \\ & \bar{f}_{2}=M_{22} \ddot{q}_{r d}+C_{22} \dot{q}_{r d}+F_{2} \end{aligned}$$

The nonlinear terms $\Psi_{1}$ and $\Psi_{2}$ in (56) and (57) are to be identified on-line using NN estimators. In the development of the NN on-line estimators, radial basis function (RBF) network with fixed centers and widths is employed. It is not difficult to extend the results to the case where a multilayer perceptions (MLPs) network [3] is used.

F. Neural Network Controller Design for the Mobile Manipulator

The RBF network has been shown to have universal approximation ability to approximate any smooth function on a com-
pact set $S$, simply connected set of $\Re^{n}$ [7], [20]. Let $f(\cdot): S \rightarrow$ $\Re^{n}$ be a smooth function and $\{\phi(x)\}$ be a basis set, where $f(\cdot) \in C^{m}(S)$, the space of continuous functions. Then, for each $f(\cdot) \in C^{m}(S)$, there exist a weight matrix $W$ such that

$$f(x)=W^{T} \phi(x)+\varepsilon$$

with the estimation error bounded by

$$\|\varepsilon\|<\varepsilon_{N}$$

for a given constant $\varepsilon_{N}$. It was shown in [20] that the set of RBFs forms a basis.

In light of the universal approximation ability of the RBF network, $\Psi_{1}$ and $\Psi_{2}$ defined in (56) and (57), respectively, may be identified using RBF nets with sufficiently high number of hidden-layer neurons such that

$$\begin{aligned} & \Psi_{1}=W_{1}^{T} h_{1}(x)+\varepsilon_{1}(x) \\ & \Psi_{2}=W_{2}^{T} h_{2}(x)+\varepsilon_{2}(x) \end{aligned}$$

where $x$ is the input pattern to the neural networks defined as

$$\begin{aligned} x & \equiv\left[\begin{array}{llllll} q_{c d}^{T} & \alpha^{T} & \dot{\alpha}^{T} & \tilde{q}_{v}^{T} & z^{T} & c^{T} & q_{c d}^{T} & \tilde{q}_{c d}^{T} & \ddot{q}_{r d}^{T} \end{array}\right]^{T} \\ & \in \Re^{(5 m+5 n-3 r)} \end{aligned}$$

$W_{1} \in \Re^{n_{1} \times(m-r)}$ and $W_{2} \in \Re^{n_{2} \times n}$ are ideal and unknown weights, respectively, which are assumed to be constant and bounded by

$$\left\|W_{1}\right\|_{F} \leq W_{1 B}, \quad\left\|W_{2}\right\|_{F} \leq W_{2 B}$$

with $W_{1 B}$ and $W_{2 B}$ some known positive constants; $n_{1}$ and $n_{2}$ are the numbers of hidden-layer neurons of the two RBF nets, respectively; the approximation errors $\varepsilon_{1} \in \Re^{m-r}, \varepsilon_{2} \in \Re^{n}$ are bounded by $\left\|\varepsilon_{1}\right\| \leq \varepsilon_{1 N}$ and $\left\|\varepsilon_{2}\right\| \leq \varepsilon_{2 N}$, with $\varepsilon_{1 N}$ and $\varepsilon_{2 N}$ two positive constants; $h_{1}(x): \Re^{5 m+5 n-3 r} \rightarrow \Re^{n_{1}}$ and $h_{2}(x): \Re^{5 m+5 n-3 r} \rightarrow \Re^{n_{2}}$ are properly chosen RBFs for the hidden-layer neurons of the two nets, with $x$ as the input pattern of the input layers to the two nets. The basis functions can be chosen as the Gaussian functions defined as [7]

$$\begin{aligned} & h_{1 i}(x)=\exp \left(\frac{-\left\|x-c_{1 i}\right\|^{2}}{\sigma_{1 i}^{2}}\right), \quad i=1,2, \ldots, n_{1} \\ & h_{2 i}(x)=\exp \left(\frac{-\left\|x-c_{2 i}\right\|^{2}}{\sigma_{2 i}^{2}}\right), \quad i=1,2, \ldots, n_{2} \end{aligned}$$

where $c_{1 i}$ and $c_{2 i}$ are centers, and $\sigma_{1 i}$ and $\sigma_{2 i}$ are widths, which are all chosen a priori and kept fixed throughout for simplicity. Therefore, only the weights $W_{1}$ and $W_{2}$ are adjustable during the learning process.

The estimates of $\Psi_{1}$ and $\Psi_{2}$ are given by

$$\begin{aligned} & \hat{\Psi}_{1}=\hat{W}_{1}^{T} h_{1}(x) \\ & \hat{\Psi}_{2}=\hat{W}_{2}^{T} h_{2}(x) \end{aligned}$$

Thus, the main objective is to design proper control laws [i.e., torque inputs in (20) and (24)], and stable NN learning laws, such that the unknown robot dynamics [i.e., (56) and (57)] can be largely compensated for by the NN estimators, and the stability of the robot error dynamics [i.e., (33) and (42)] and the boundedness on the NN estimation weights can be guaranteed. This is achieved through the following theorem.