Eleven-Point Gradient-Zhang Dynamics Algorithm for Time-Dependent Nonlinear Equality-Constraint Programming and Manipulator Application

The problem of time-dependent nonlinear equality-constraint programming (TDNECP) is a hot topic in various scientific and engineering fields. In this paper, the problem-solving of TDNECP is investigated thoroughly. Combining the gradient dynamics (GD) and the Zhang dynamics (ZD), this study proposes the continuous-time gradient-ZD (GZD) model for the problem-solving of TDNECP by constructing the integrated feedback term from GD and ZD. In addition, a novel eleven-point Zhang time discretization (ZTD-XI) formula is proposed. The convergenceness and order of truncation errors of the proposed ZTD-XI formula are rigorously proven through mathematical derivation. On the basis of the proposed ZTD-XI formula and the continuous-time GZD model, a novel eleven-point discrete-time GZD-based (DTGZD-XI) algorithm is proposed and generalized to obtain the optimal solution of TDNECP. Alongside this, the two-, five-, and eight-point discrete-time GZD-based and GD-based algorithms for the problem-solving of TDNECP are prepared for comparison purpose. Finally, challenging numerical experiments and UR5 manipulator-based computer simulations are carried out. The results further substantiate the effectiveness and superiority of the proposed DTGZD-XI algorithm. Note to Practitioners—This paper is motivated by the need for a high-precision and online dynamics-based algorithm to solve time-dependent nonlinear equality-constraint programming. While existing dynamics-based approaches can be applied to this problem, they often suffer from lagging error and accuracy issues. To address these problems, we propose an integrated dynamics approach, namely the gradient-Zhang dynamics (GZD), along with a novel eleven-point Zhang time discretization (ZTD-XI) formula that achieves order-6 precision. On the basis of the proposed ZTD-XI formula and the continuous-time GZD model, a novel eleven-point discrete-time GZD-based (DTGZD-XI) algorithm is proposed and generalized to obtain the optimal solution of time-dependent nonlinear equality-constraint programming. We also provide a block diagram and pseudocode to help practitioners apply the algorithm conveniently. The effectiveness, convergence, and superiority of the proposed DTGZD-XI algorithm are validated through numerical experiments and UR5 manipulator-based computer simulations. In future research, we will extend the proposed algorithm to plan the motion of multiple redundant manipulators with more complicated constraints, including obstacle avoidance and posture planning.