Equivalency is a powerful approach that can transform an original problem into another problem that is relatively more ready to be resolved. In recent years, Zhang neurodynamics equivalency (ZNE), in the form of neurodynamics or recurrent neural networks (RNNs), has been investigated, abstracted, and proposed as a process that can equivalently solve equations at different levels. After long-term research, we have noticed that the ZNE can not only work with equations, but also inequations. Thus, the ZNE of inequation type is proposed, proved, and applied in this study. The ZNE of inequation type can transform different-level bound constraints into unified-level bound constraints. Applications of the jerk-level ZNE of bound constraints, equation constraints, and objective indices ultimately build up effective time-varying quadratic-programming schemes for cyclic motion planning and control (CMPC) of single and dual robot-arm systems. In addition, as an effective time-varying quadratic-programming solver, a projection neural network (PNN) is introduced. Experimental results with single and dual robot-arm systems substantiate the correctness and efficacy of ZNE and especially the ZNE of inequation type. Comparisons with conventional methods also exhibit the superiorities of ZNE.

中文翻译：

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机械臂系统循环运动的约束约束、方程约束和目标指标的急动级张神经动力学等效性


等价是一种强大的方法，可以将一个原始问题转化为另一个相对更容易解决的问题。近年来，张氏神经动力学等价（ZNE）以神经动力学或循环神经网络（RNN）的形式被研究、抽象并提出为可以等价求解不同层次方程的过程。经过长期的研究，我们发现ZNE不仅可以处理方程，还可以处理不等式。由此，本研究提出、证明并应用了不等式型ZNE。不等式类型的ZNE可以将不同层次的边界约束转化为统一层次的边界约束。边界约束、方程约束和目标指数的急动级 ZNE 的应用最终为单和双机器人手臂系统的循环运动规划和控制（CMPC）建立了有效的时变二次规划方案。此外，作为一种有效的时变二次规划求解器，引入了投影神经网络（PNN）。单臂和双臂系统的实验结果证实了ZNE，特别是不等式型ZNE的正确性和有效性。与传统方法的比较也显示了ZNE的优越性。