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Eleven-Point Gradient-Zhang Dynamics Algorithm for Time-Dependent Nonlinear Equality-Constraint Programming and Manipulator Application
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DOI：10.1109/TASE.2023.3340291.png)
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Abstract
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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.
Keywords:
Time-dependent nonlinear equality-constraint programming
gradient-Zhang dynamics
Zhang time discretization
numerical algorithm
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