Terminal Recurrent Neural Networks for Time-Varying Reciprocal Solving With Application to Trajectory Planning of Redundant Manipulators

Time-varying matrix reciprocal problems are widely appeared in different matrix computations and engineering fields. Neural networks as a powerful tool have been developed to solve the time-varying problems. Recurrent neural networks (RNNs) are designed considering mainly for two aspects: 1) convergent precision and 2) convergent time. The core part of the existed neural methods is to design various kinds of activation function for time-varying matrix solving. However, most of the activation functions of neural networks are with infinite value, which demands long convergent time and are not applicable in practical engineering fields. This note proposes theoretical analyses and simulation results on the performance of terminal RNN (TRNN) and accelerated TRNN (ATRNN) with finite-time convergence, which is not only designed for constant matrix inversions but also for time-varying reciprocal matrix. Compared to the traditional RNNs, TRNNs are of limit-valued activation function and possess a finite time convergence property. The simulation results for time-varying reciprocal solving validate the perfect performance solved by TRNN and ATRNN. In addition, a quadratic program (QP) of velocity minimization based on TRNN is proposed to solve the trajectory tracking problems without considering the initial position error of the redundant manipulators. Finally, practical experiments of the redundant manipulators based on PUMA560 show the effectiveness and accuracy of the proposed approaches.

I. Introduction

The solutions for time-varying reciprocal problems of matrix are one of the most important subjects in a variety of applications, such as statistics [1], signal processing [2]–[4], and robot control [5], [6]. The main solutions for solving time-varying reciprocal problem are digital calculation methods. However, for many large-scale problems, the burdens of the computing are large and the inverse of matrices takes a lot of time. Furthermore, the rank of the parameter matrices is to be full, which may demand higher requirements for calculations. From the prospective of the fundamental mathematical operations, most of the related algorithms are not efficient or only for static problems [7]–[9]. When applied to time-varying situations, such as reciprocal problems, it is insufficient to deal the situation that the minimum arithmetic operations of underlying time-varying problems are needed to be taken cube of the matrix dimension.