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。

本论文揭示，作为两种并行的神经计算模型，BP和Hopfield类型神经网络都可以有效地对二次型V（x）=T/2+TVxxPxqx实现最小化求解。而且，尽管BP和Hopfield类型神经网络在网络设计思想和网络结构上呈现出很大的差异，但是它们在二次型函数最小化问题上都表现出了相同的学习能力，这说明两者具有本质的联系。

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。

In this Letter, by following Zhang et al. is method, a recurrent neural network (termed as Zhang neural network, ZNN) is developed and analyzed for solving online the time-varying convex quadraticprogramming problem subject to time-varying linear-equality constraints. Different from conventional gradient-based neural networks (GNN), such a ZNN model makes full use of the time-derivative information of time-varying coefficient. The resultant ZNN model is theoretically proved to have global exponential convergence to the time-varying theoretical optimal solution of the investigated time-varying convex quadratic program. Computer-simulation results further substantiate the effectiveness, efficiency and novelty of such ZNN model and method.

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.