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Tracking control for multiple-integrator systems is regarded as a fundamental problem associated with nonlinear dynamic systems in the physical and mathematical sciences, with many applications in engineering fields. In this paper, we adopt the Zhang neural network method to solve this nonlinear dynamic problem. In addition, in order to adapt to the requirements of real-world hardware implementations with higher-order precision for this problem, the multiple-order derivatives in the Zhang neural network method are estimated using backward finite-divided difference formulas with quadratic-order precision, thus producing time delays. As such, we name the proposed method the Zhang neural network method with time delay. Moreover, we present five theorems to describe the convergence property of the Zhang neural network method without time delay and the quadratic-order error pattern of the Zhang neural network method with time delay derived from the backward finite-divided difference formulas with quadratic-order precision, which specifically demonstrate the effect of the time delay. Finally, tracking controllers with quadratic-order precision for multiple-integrator systems are constructed using the Zhang neural network method with time delay, and two numerical experiments are presented to substantiate the theoretical results for the Zhang neural network methods with and without time delay.

In the engineering field, a complex system or problem can be transformed into a relatively simple system or problem through equivalency. In this paper, the repetitive motion planning scheme of redundant robot manipulators with time‐varying physical limits are to be optimized via Zhang equivalency (ZE) approach, which includes Zhang equality equivalency (ZEE) approach and Zhang inequality equivalency (ZIE) approach. The ZE approach is presented, theoretically proved, and applied to optimize the control scheme of redundant robot manipulators at the velocity layer. After that, an enhanced repetitive motion planning scheme via ZE (ERMPS‐ZE) on the basis of quadratic programming is finally built up, which is subject to a ZEE constraint and a time‐varying unified ZIE constraint for the solution of RMP in redundant robot manipulator systems. The simulation results on UR10 robot manipulator substantiate superiorities of the proposed scheme with regard to practicality, validity, and completeness over the traditional scheme. In addition, the results of simulation experiments under different limit situations show the adjusting manners and characteristics of the redundant robot manipulator adopting ERMPS‐ZE when the manipulator encounters physical limits, which validate the correctness and efficacy of the ERMPS‐ZE. Finally, the physical experiment also substantiates the practicability of the ERMPS‐ZE.

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