Motivated by handling joint physical limits, environmental obstacles and various performance indices, researchers have developed a general quadratic-programming (QP) formulation for the redundancy resolution of robot manipulators. Such a general QP formulation is subject to equality constraint, inequality constraint and bound constraint, simultaneously. Each of the constraints has interpretably physical meaning and utility. Motivated by the real-time solution to the robotic problems, dynamic system solvers in the form of recurrent neural networks (RNN) have been developed and employed. This is in light of their parallel-computing nature and hardware implementability. In this paper, we have reviewed five RNN models, which include state-of-the-art dual neural networks (DNN) and LVI-based primal-dual neural networks (LVI-PDNN). Based on the review of the design experience, this paper proposes the concept, requirement and possibility of developing a future recurrent neural network model for solving online QP problems in redundant robotics; i.e., a piecewiselinear primal neural network.

为解决在速度层上无穷范数最小化模型中可能出现的不连续点问题，提出一种基于双判据方法的二次型优化模型。冗余机器人运动规划与控制模型可以统一各种关节物理极限，如关节变量极限与关节速度极限。同时该模型又可以最终转化为一个标准的二次规划问题。为了实时求解该二次规划问题，提出一种基于线性变分不等式（LVI）的原对偶神经网络。该神经网络作为实时求解器具有简单的分段线性结构和较高的计算效率。计算机对PUMA560机器手臂的模拟仿真表明，该方案具有灵活性和有效性。

In this paper, a primal-dual neural network based on linear variational inequalities (LVI) is presented for the online repetitive motion planning of PA10 robot arm, a kinematically redundant manipulator. To do this, a drift-free criterion is exploited. In addition, the physical constraints such as joint limits and joint velocity limits are incorporated into the problem formulation of such a redundancy-resolution scheme. The scheme is finally reformulated as a quadratic-programming (QP) problem. As a QP real-time solver, the LVI-based primal-dual neural network is designed based on the QP-LVI conversion and Karush-Kuhn-Tucker (KKT) condition. With simple piecewise-linear dynamics and global exponential convergence to optimal solutions, it can handle general QP and linear programming (LP) problems in the same inverse-free manner. The repetitive motion planning scheme and the LVI-based primal-dual neural network are simulated successfully based on PA10 robot arm, with effectiveness demonstrated.

为避免权值反复迭代修正的冗长BP训练过程，避免传统方法陷入局部极小点，根据多项式理论，构造了一种新型前向神经网络模型，推导了基于最速下降法的误差反传算法和基于伪逆的直接确定法。仿真结果显示，迭代方法和伪逆直接确定法都能达到比较高的工作精度（10-6）。

基于函数逼近理论，构造一种Chebyshev基函数神经网络模型。推导出该网络模型的权值直接确定方法，可一步计算出权值，克服了传统BP神经网络学习率选取困难、学习过程冗长和易陷入局部极小等缺点。在此基础上，设计了基于二分搜索的结构自适应算法，根据精度要求自动确定网络最优结构。理论分析及仿真验证均表明，该网络不仅能够快速地完成网络权值确定和结构自适应，且具有优异的学习与逼近能力，而且对随机加性噪声也具有较好的抑制作用。