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Trajectory Tracking Control of the Nonholonomic Mobile Robot using Torque Method and Neural Network

Controlling a mobile robot to follow a predefined trajectory is a challenging task due to the nonlinear
chacteristic and nonholonomic constraint of the robot. According to Brocket theory, a nonholonomic system is not able to be asymptotically stable using the smooth and time invariant control laws. Some methods to stablize the nonholonomic system through feedback control have been proposed. They however often assume ideal conditions. Others focus on determining uncertainties in measurements and model parameters and try to fix them by using hybrid feedback control or velocity chart control. These methods are usually complex and difficult to implement. Our approach is the use of Lyapunov function technique to design a stable controller for nonholonomic systems [1],[2]. The goal is the optimization in motion of the robot during the path following process. From the robot kinematics, the uncertainties in system parameters are determined and compenstated by the implementation of an extended Kalman filter. But this stage only focuses on the kinematics while the dynamics parameters such as the robot’s load which plays an important role in the stable of the robot are not concerned. In addition, non-parameter uncertainties such as high-frequency unmodeled dynamics, actuator dynamics, structural vibrations, measurement noises, computstion roundoff error, and sampling delay also need be considered. Thus, the problem of kinematics and dynamics control of nonholonomic system is challenging. A number of approaches to control the system with nonholonomic constraint have been introduced [3],[4],[6]. In [9-16], authors were combined the dynamics model of the mobile robot to the kinematics controller with nonholonomic constraint. However, we find it difficult to implement closely the dynamics model of the mobile robot due to the non-quantified parameters of dynamics as well as the usually-variable robot’s load. These errors are mainly caused by the uncertainties of such a model and the non-parameters, which is composed of: 1-high frequency unmodeled dynamics, such as actuator dynamics or structral vibrations; 2 measurement noise; 3-computation roundoff error and sampling delay. Thus, the problem of kinematics and dynamics control of non-holonomic system is absolutely challenging. The typical method is to solve this is considered as “adaptive control”. For example, the backstepping method of Wang et al. [17] and R.Fierro et al. [18], the sliding-mode techniques in [19-20], were applied to reduce sway for an offshore container crane. These methods also implemented neural network to compensate the uncertainties such as the combination between the backstepping method and the neural network in [9,11]. In [21], a combination between the RBFNN controller and the sliding-mode techniques for the path following task of an omnidirectional wheeled mobile manipulators was applied.
The asymptotically stabilization were proven theoretically and experimentally. In [9], the author presented a control method using neural network in which online learning law of weight factors is used to compensate the uncertainties caused by error in dynamics model. If the dynamics model contains non-parameter uncertainties, the asymptotically stabilization is then not assured. In this paper, the tracking control algorithm based on the torque method is presented. The controller is of feedforward being in combination with proportional type. The non-parameter uncertainties and dynamics model errors are compensated by using the RBFNN. The asymptotic stabilization of the whole system is proven by the Lyapunov theory. Our main job is that the splitting of the path following tasks into two independent control loops. The outer loop is employed to control the kinematics such as the determination of tangent and angular velocities so that the errors in position
and direction go toward zero (globally asymptotically stabilization according to the Lyapunov theory); output of this controller is sent to the inner control loop. The inner control loop is used to control the dynamics. output of this controller is sent to the inner control loop. The inner control loop is used to control the dynamics. In this control loop, This controller is designed by the combination between the
feedforward, the scale techniques, and compensated-non parameter-RBFNN type. The paper is organized as follows. Section 2 briefly is introduced the kinematics and dynamics of the mobile robot.
Section 3 is described the process to design the controller. Section 4 is presented the simulation results and finally is section 5. 
A typical example of a nonholonomic mobile robot is shown in Fig. 1 [9-11]. 
Figure 1. A non holonomic mobile robot platform 
The target of the control problem is to design an adaptive control law in which the position vector q and the velocity q to hold the position vector r and the demanded velocity in terms of unknown dynamics parameters of the mobile robot. The desired position and velocity vectors are represented. 
The position tracking error between the reference and the actual robot could be expressed in the mobile robot’s coordinator as follows. 
The neural network is approximated by Gaussian RBF network consisting of three layers: input layer, hidden layer with n nodes that contains the Gaussian function, and the output layer with linear function of n neurons. The RBF network structure satisfies the conditions of the Stone-Weierstrass theorem. Hidden layer neuron is the Gaussian function with the form: 
Figure 2. The approximate neural network of Wσ function 
This paper proposes a control method for trajectory tracking of nonholonomic mobile robot. The main
contribution is the proposal of a controller with two control loops. One is for the kinematics. The other is for the dynamics. In addition, a neural network is introduced to deal with uncertainties of the dynamics model. The global stability of the system is proven. The simulation results confirmed the effectiveness of the method. 
                                                                              TS. Trần Thuận Hoàng


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