Original articles
Robust zeroing neural network for fixed-time kinematic control of wheeled mobile robot in noise-polluted environment

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Abstract

Based on a new robust zeroing neural network (RZNN) model, the trajectory tracking control of a wheeled mobile robot (WMR) within fixed-time in noise-polluted environment is presented in this paper. Unlike most of the previous reported works, the RZNN model approach for trajectory tracking control of the WMR reaches fixed-time convergence and noise suppression simultaneously. Besides, detailed theoretical analysis of its convergence and robustness are provided. Numerical simulation verification is also provided to demonstrate the superior robustness and accurateness of the RZNN model approach for trajectory tracking control of the WMR in noise-polluted environment. Both of the theoretical analysis and numerical simulation results verify the effectiveness and robustness of the RZNN model approach.

Introduction

As the wheeled mobile robots (WMR) have been gradually applied in the fields of industry, transportation service, safety, and other fields, the motion control problem of WMR has drawn considerable attention in recent decades [24]. The motion control of WMR can be divided into two main categories: trajectory tracking and path following. As a significant motion control problem, the trajectory tracking of WMR is deeply investigated in recent years [2], [6], [18], [19], [28]. The realization of trajectory tracking of WMR is related to the design of a controller to enforce the WMR to track a user-defined trajectory, which usually can be treated as an inverse kinematic problem (IKP) [21], [31]. A lot of control algorithms for the effective realization of trajectory tracking of WMR have been reported [3], [4], [8], [15], [16], [22], [23], [32].

With the rapid development of artificial intelligence, the intelligent methods such as fuzzy logic and neural networks are widely adopted to design the controllers for the controlling of the WMR. In recent years, many researchers use the fuzzy logic methods to overcome dynamic uncertainties of WMR. In Ref. [5], Davood Nazari et al. proposed an optimal Mamdani-type fuzzy logic controller to cope with both parametric and non-parametric uncertainties in robot model; In [40], adaptive cerebellar model articulation controller network and fuzzy logic controller were used to control an autonomous mobile robot by human gestures; In [17], a fuzzy control system with probabilistic roadmap is proposed for path planning and target seeking behaviors of mobile robot. It is well-known​ that if the robustness properties are fully considered in the control procedure, the fuzzy logic method is effective in uncertain environment for the control of WMR [13]. However, lack of systematic definition of fuzzy rules and fuzzy membership functions is the significant drawback of the Fuzzy logic method. Most of the existing fuzzy rules are defined by users according to their personal information, although system performance is the same, the fuzzy rules will be different, for which it is difficult to obtain an optimal solution. The neural network approaches are also applied for the control of the WMR. By using a set of boundary functions and applying them to controller design, an adaptive neural control for dual-arm robot system with prescribed tracking performance and guaranteed global stability is realized in [34] by Yang et al. Adaptive neural network control of an uncertain full-state constraints n-link robot was introduced in [7], and it was effective to deal with uncertainties and disturbances of the system. In [36], adaptive neural control has been used for robotic manipulators with output constraints and uncertainties, and its feasibility and superiority were verified by simulation results.

As a powerful approach for solving time-varying problems, the zeroing neural network (ZNN) proposed in [37] was deeply investigated in recent years [9], [11], [12], [27], [29], [30]. In spite of its great success in solving time-varying problems, most of the existing ZNN applications focus on the free of noise environment, and noises are inevitable, such as unexpected impulsive, electromagnetic interferences and random noise interacted with environment. All the above unexpected interferences can be considered as dynamic noises for the corresponding dynamic system. We all know that dynamic noise is very harmful to dynamic systems, and dynamic systems are very vulnerable to noises and interference. Sometimes, the accuracy and stability of the dynamic system will be severely deteriorated by dynamic noises, and even fail to complete the assigned tasks. In other words, most of the existing ZNN are noise-free models, and its noise compatibility is not considered. For example, a finite-time convergent ZNN was presented in [25], which has faster convergence ability. In addition, a noise-free ZNN model for solving time-varying linear equation was proposed in [33], and it was also effectively used to the control of robot manipulators. In order to improve its noise compatibility, a nonlinear neural dynamics and a noise-tolerant ZNN model are presented in [10], [26], the nonlinear neural dynamics and noise-tolerant ZNN models work properly in noise-polluted environment, but they only achieve exponential convergence, not finite-time convergence or fixed-time convergence. Inspired by the above mentioned issues, a new robust zeroing neural network (RZNN) model for the fixed-time trajectory tracking control of WMR in time-varying noise-polluted environment is presented in this work. By using a novel activation function, the proposed RZNN model possesses the abilities of fixed-time convergence and additional time-varying noise suppression simultaneously, which are significant improvements of the existing ZNN models in robustness and effectiveness.

Before the end of the introduction section, the key innovations of this work are summarized below.

(1) Different from the existing noise-free ZNN models, by using a novel activation function, a new RZNN model for the fixed-time trajectory tracking control of WMR in time-varying noise-polluted environment is proposed in this paper, and its fixed-time convergence and additional time-varying noise suppression abilities significantly strengthen the robustness and effectiveness of the existing ZNN models.

(2) The robustness and effectiveness of the proposed RZNN model in noise-polluted environment are verified by rigorous mathematical analysis.

(3) Numerical simulations and comprehensive comparisons with the existing ZNN models for the trajectory tracking control of WMR further demonstrate the robustness and effectiveness of the proposed RZNN model.

Section snippets

Problem statement

In this section, we first present the three-dimensional structure and geometrical model of a WMR, and then its corresponding inverse kinematic model is introduced.

For the purpose of better understanding, the three-dimensional structure and geometrical model of a WMR is presented in Fig. 1. As seen from Fig. 1(a), a WMR consists of a movable base platform and a six-joint manipulator, and its geometrical model is presented in Fig. 1(b).

The notations in Fig. 1(b) are presented below.

  • (1)

    Po: center

Robust zeroing neural network (RZNN) model

In this section, we propose a new RZNN model for solving the IKP of the WMR in Fig. 1, and the proposed RZNN model for solving the IKP of the WMR reaches fixed-time convergence noise suppression simultaneously.

First, we define a vector-valued error function to monitor and control the process of the IKP of the WMR. E(t)=rmd(t)rm(t)

Here, rmd(t)Rm is the desired path to be tracked, and rm(t) is the actual path of the end-effector of the WMR, and ei(t) stands for the ith element of E(t). As seen

RZNN model analysis

As the basis of discussing and analyzing the novel time-fixed convergence ZNN model, the following Definition and Lemma should be presented in advance.

Definition 1

[1]

The ZNN model is deemed as fixed-time stability if its convergence time t is 0 < t < Tmax and the upper bound value Tmax is not dependent on initial conditions of the model.

Lemma 1

[1], [20]

If a radially unbounded function F is continuous and positive definite, and the solutions x(t) of the ZNN model fulfill the following time-varying inequality: Fx(t)aFpx(t)+

Simulations and comparisons

In this section, simulation results of a WMR with its three-dimensional model presented in Fig. 1 are investigated to demonstrate the robustness and effectiveness of the proposed RZNN model (9) in dynamic noise-polluted environment for solving the IKP of the WMR. Part A is the simulation and analysis of the RZNN model. For comparison, part B is simulation and analysis of the original ZNN model.

Conclusion

In this work, a new RZNN model for solving inverse kinematic problem (IKP) of wheeled mobile robot (WMR) is proposed and investigated. Unlike most of the existing neural network models in ideal environment without considering internal noise and outside disturbance, during the execution of solving IKP of WMR with the proposed RZNN model, four classic dynamic noises are considered and investigated. Rigorous theoretical analysis and verification on the noise and disturbance canceling ability of

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 61875054), Natural Science Foundation of Hunan Province (Grant No. 2020JJ4315, No. 2020JJ5199), Scientific Research Fund of Hunan Provincial Education Department (Grant No. 20B216, No. 20C0786, No. 18C0296).

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