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标题“Global Tracking Control of a Wheeled Mobile Robot Using RBF Neural Networks”表明，本文研究的主题是利用径向基函数（Radial Basis Function, RBF）神经网络来实现轮式移动机器人的全局轨迹跟踪控制。这涉及到在移动机器人控制系统中采用神经网络的先进方法。研究论文中经常使用的关键概念如“全局跟踪”、“自适应神经网络控制”、“不确定性建模”等，均在这篇文章的标题中得到体现。描述部分指出，文章提出了一个新的自适应位置跟踪控制方案，通过使用RBF神经网络对不确定性进行建模。在反馈补偿方案中，机器人参考位置和实际位置的信息都被用作神经网络的输入。与现有结果相比，主要优势在于可以确保闭环系统的全局稳定性，并且可以根据参考信号预先确定神经网络的逼近域。文章最后通过一个模拟示例展示了所提出控制方案的有效性。在控制理论和应用领域，神经网络，特别是RBF网络，因其在紧凑域内的良好逼近能力和适应性，已经成为解决不确定非线性系统控制问题的重要工具。近几十年来，自适应神经网络控制（Adaptive Neural Network Control, ANNC）已经成为控制不确定非线性系统的强大方法论，并在理论和应用实践中取得了大量进展。然而，由于实际情况中移动机器人的不确定性和复杂性，控制这类机器人存在许多难题。因此，ANNC方案已经被广泛应用于移动机器人跟踪控制问题中，并取得了一系列有趣的结果。文章中提到的关键技术包括： 1. 径向基函数神经网络（RBF NNs）：这些网络在处理非线性建模和逼近方面表现出色，特别是在控制系统中对不确定性的适应和逼近。RBF NNs具有一个由中心点和宽度参数定义的隐藏层，能够逼近任意连续函数。 2. 自适应控制（Adaptive Control）：自适应控制系统能够自动调整控制参数来应对系统中的不确定性和变化。这对于处理动态变化的环境和内部参数未知或变化的系统尤为重要。 3. 全局稳定性（Global Stability）：在控制系统中，全局稳定性意味着系统对于所有的初始条件都将稳定于一个期望的状态。这是评估控制算法性能的一个重要指标。 4. 逼近域的确定（Determination of Approximation Domain）：在神经网络中，逼近域的确定意味着找到一个能够通过神经网络准确逼近真实系统行为的输入空间区域。 5. 反馈补偿（Feedback Compensation）：反馈补偿是指通过实时监测系统输出，并将其与期望输出进行比较，然后通过反馈环节调整控制输入，以减小二者之间的差异。 6. 轮式移动机器人（Wheeled Mobile Robots）：这类机器人通常使用轮子进行移动，其控制系统的设计和实现面临许多挑战，如运动学建模、路径规划和动态控制等。文章提出的控制方案不仅利用了RBF神经网络对移动机器人不确定性的建模能力，而且在控制策略设计上考虑了全局稳定性和逼近域的确定，这些特性共同为移动机器人的精确轨迹跟踪提供了新的可能。通过研究自适应神经网络控制和RBF神经网络在移动机器人领域的应用，可以提高机器人在复杂环境中执行任务的能力，使其在实际应用中更具鲁棒性和适应性。

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Global Tracking Control of a Wheeled Mobile

Robot Using RBF Neural Networks

,

Jian Wu, Dong Zhao, and Weisheng Chen



Department of Mathematics, Xidian University, Xi’an 710071, China

wshchen@126.com

Abstract. In this paper, the global tracking control problem for a class

of wheeled mobile robots is considered and a new adaptive position track-

ing control scheme is proposed where radial basis function (RBF) neural

network (NN) is utilized to model the uncertainty. The feedback com-

pensation scheme is obtained, where the information of reference posi-

tion and real position of robot are both used as the NN input. Compered

with the existing results, the main advantage is that the global stability

of the closed-loop system can be ensured and the NN approximation do-

main can be determined based on the reference signal a prior. Finally,

a simulation example is provided to demonstrate the eﬀectiveness of the

proposed control scheme.

Keywords: Wheeled Mobile Robot, Global Tracking, Adaptive Neural

Network Control, Determination of Approximation Domain.

1 Introduction

Due to the well-known fact that neural networks (NNs), including radial basis

function (RBF) NNs [1], high-order NNs [2], multilayer NNs [3], ect., have good

approximation capabilities and adaptation abilities over a compact domain, they

paly an important role in the control community, especially in uncertain non-

linear system control. During the past several decades, adaptive neural network

control (ANNC) has evolved as a powerful methodology for uncertain nonlinear

systems. A large amount of progress in ANNC has been obtained in theory and

practical applications, see. e.g., [1,4,5] and the reference therein.

On the other hand, there exist many diﬃcult problems in controlling mobile

robots because of the inevitable uncertainties in practical situations. Therefore,

ANNC schemes have widely been employed to solving the tracking control prob-

lem of mobile robots in the past several years and lots of interesting results

have been obtained (e.g., see [6-13]). In [11], the authors investigated asymp-

totic adaptive NN tracking control of nonholonomic mobile robot formations



This work is supported by National Natural Science Foundation of China

(61174213,61203074), the Program for New Century Excellent Talents in Univer-

sity (NCET-10-0665).



Fundamental Research Funds for the Central Universities (K5051370014).



Corresponding author.

C. Guo, Z.-G. Hou, and Z. Zeng (Eds.): ISNN 2013, Part II, LNCS 7952, pp. 139–146, 2013.

 Springer-Verlag Berlin Heidelberg 2013

140 J. Wu, D. Zhao, and W. Chen

and proved the global asymptotic stability of the followers. Chaitanya [12] re-

solved the speed jump problem in the tracking control of a mobile robot using

a single layer neural networks (NNs) structure. Moreover, there are other ap-

proaches to study the control problem of robotic systems with uncertainties in

the existing literature (e.g., see [14],[15]).

However, the above adaptive control has inherent drawbacks, we know NN uni-

versal approximation property holds only over a compact set, so recently there are a

few results solving the global stability of closed-loop systems ( e.g., see [16]). Mean-

while, we can not ﬁnd appropriate methods to determine the RBF NN approxima-

tion domain since it relies on the uncertain position information. Motivated by the

aforementioned discusses, we will attempt to solve this problem. First, according to

the bound of reference signals, we determine the RBF NN approximation domain,

then designing a feedforward compensation scheme and a feedback compensation

scheme such that closed-loop systems reach to be globally stable, respectively.

2 Preliminaries and Problem Description

2.1 RBF NN Approximation

NNs have been found to be very useful for controlling nonlinear systems with

uncertainties. In particular, RBF NN is usually used as a tool for modeling non-

linear functions due to its simple structure and good approximation ability[14].

Therefore, in this paper, we employ the RBF NN to approximate an unknown

continuous function f(χ),χ ∈ Ω ⊆ R

,whereΩ is a compact set, i.e.,

f(χ)=Φ

(χ)W + (χ)(1)

where Φ(χ):Ω → R

is a vector-valued function and the neural node number

m>1. The components of Φ(χ), denoted by s

(χ), 1 ≤ k ≤ m, are called the

basis functions that are commonly chosen as Gaussian functions with the form

(χ)=exp



−

(χ − μ

)

(χ − μ

)



,k=1, 2, ...m (2)

where μ

∈ Ω is a constant vector which is called the center of s

(χ), and η

> 0

is a real number which is called the width of s

(χ). The optimal weight vector

W =[w

... w

]

is deﬁned as

W := arg min

W ∈R



sup

χ∈Ω

|f(χ) − Φ

(χ)

W |



(3)

and (χ) is the inherent neural networks approximation error, which can be

decreased by increasing the neural network number m.

2.2 Main Lemmas

The following four lemmas are critical and usually used to design ANNC scheme

and analyze the closed-loop system stability.

Lemma 1 [17]. Young inequality states that if a and b are nonnegative real

numbers and p and q are positive real numbers such that

=1,then

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