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。

A recurrent neural network called the dual neural network is proposed in this Letter for solving the strictly convex quadratic programming problems. Compared to other recurrent neural networks, the proposed dual network with fewer neurons can solve quadratic programming problems subject to equality, inequality, and bound constraints. The dual neural network is shown to be globally exponentially convergent to optimal solutions of quadratic programming problems. In addition, compared to neural networks containing high-order nonlinear terms, the dynamic equation of the proposed dual neural network is piecewise linear, and the network architecture is thus much simpler. The global convergence behavior of the dual neural network is demonstrated by an illustrative numerical example. Ù 2002 Elsevier Science B.V. All rights reserved.

以平方可积空间上的复指数Fourier级数作为激励函数构造了新型Fourier神经元网络，并推导出采用加号逆表示的网络权值直接确定公式，克服了传统BP神经网络收敛速度慢、易陷于局部极小点、迭代学习易发生振荡等缺陷。并在此基础上构造了隐神经元衍生算法，克服了传统BP神经网络难以确定最优网络拓扑结构的缺点。理论分析及仿真实验表明，该复指数Fourier神经元网络能够一步计算网络最优权值且能自适应调整网络结构，对随机加性噪声具有抑制作用，并能高精度逼近非连续函数。

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.

本文利用优化方法及递归神经网络实时求解器消除冗余机器手臂在运动过程中出现的角偏差问题。鉴于机器手臂都存在着关节物理约束，我们的优化方案也因此考虑关节极限和关节速度极限的躲避。更重要的是，本文详细分析了该成功解决关节角偏差问题的二次型性能指标的设计原理。仿真结果证实了该方法的可行性与有效性。