Time-varying matrix reciprocal problems are widely appeared in different matrix computations and engineering fields. Neural networks as a powerful tool have been developed to solve the time-varying problems. Recurrent neural networks (RNNs) are designed considering mainly for two aspects: 1) convergent precision and 2) convergent time. The core part of the existed neural methods is to design various kinds of activation function for time-varying matrix solving. However, most of the activation functions of neural networks are with infinite value, which demands long convergent time and are not applicable in practical engineering fields. This note proposes theoretical analyses and simulation results on the performance of terminal RNN (TRNN) and accelerated TRNN (ATRNN) with finite-time convergence, which is not only designed for constant matrix inversions but also for time-varying reciprocal matrix. Compared to the traditional RNNs, TRNNs are of limit-valued activation function and possess a finite time convergence property. The simulation results for time-varying reciprocal solving validate the perfect performance solved by TRNN and ATRNN. In addition, a quadratic program (QP) of velocity minimization based on TRNN is proposed to solve the trajectory tracking problems without considering the initial position error of the redundant manipulators. Finally, practical experiments of the redundant manipulators based on PUMA560 show the effectiveness and accuracy of the proposed approaches.

Zeroing neural network (ZNN) with different activation functions (AFs) for finding zero-result of time-varying quadratic programming (TVQP) with no noises are revisited. To improve the convergent speed of the ZNN and resist various noises occurred in the real application, two robust terminal recurrent neural network (RTRNN) models by adding two different activation functions (AFs) are presented for the exact solution of the TVQP problem facing various noises. The appearing advantage of the pre-specified time of the RTRNN model is independent of the initial status of a generated system and the convergent time can be accelerated in advance, which is much superior than the finite time performance with regard to the initial status. In addition, the pre-specified convergent time of the RTRNN is mathematically discussed in detail under external noises. Simulated comparisons between the proposed RTRNN and the state-of-the-art neural networks substantiate the predefined time performance and strong robustness.

In this article, we derive the predictor–corrector (PC) methods with three-order convergent precision, together with a class of specific general linear three-step (GLTS) rules provided. Afterward, a time-varying optimization (TVO) problem, which is deemed as a discrete TVO has been formulated and studied. The classical discrete zeroing neural network via Zhang et al. discretization (ZD-DZNN) is often utilized to obtain the solution. Actually, the stepsize domain of the DZNN model is a great factor for the dynamical stability. To enlarge the stepsize domain of the DZNN model, specific GLTS-type PC-DZNN models are applied to solve the TVO problem. Theoretical analyses show that better stability of the DZNN can be achieved by PC methods. Numerical simulative comparisons between the proposed PC-DZNN models and the ZD-DZNN in terms of stability are provided for further illustrations. In addition, motion planning of a PA10 manipulator and physical kinematics on UR5 formed as a TVO problem has been solved efficiently by applying the specific GLTS-type PC-DZNN models.

To amplify the achievements on Zeroing neural network (ZNN) and widen the application of fuzzy logic system (FLS), a complex fuzzy parameter zeroing neural network (CFPZNN) model is established to address the time-variant complex Sylvester equation problem. Varying from the fixed parameters in conventional ZNN (CZNN) or time-varying parameters in ZNN (TVP-ZNN), the fuzzy parameter generated by the FLS fluctuates according with convergent error and adjusts the convergent rate adaptively. Three different activated functions (AFs) equipped with the CFP-ZNN model are analyzed and discussed. Finite convergence characteristic of the CFP-ZNN model with Signbi-power (SBP) is testified. Furthermore, various membership functions (MFs) and various fuzzy control output values are studied and compared to exhibit the performance of the CFPZNN model. Theoretical analyses and comparable simulation results among different ZNN-based neural network models in dealing with time-variant complex Sylvester equations are welly coincided.

To assist redundant manipulator to complete complex repetitive trajectory in a preseted time, this paper provides a solution to the kinematic assignment and presents a predefined time fuzzy zeroing neural network with event-triggered mechanism (ETM-PTFZNN). The repetitive kinematics of the redundant manipulator is originally formulated as a time-varying quadratic programming (TVQP) problem, and the ETM-PTFZNN is engaged to solve the corresponding TVQP, where the fuzzy predefined time (PT) convergence is obtained by the fuzzy system and PT activation function (AF) simultaneously. Furthermore, event-triggered mechanism (ETM) is introduced to update the fuzzy parameters of the ETM-PTFZNN orderly, which greatly alleviates the calculation burden of the ETM-PTFZNN. Theoretical analyses and simulative profiles reveal that the ETM-PTFZNN model can realize predefined time characteristic, robustness, adaptive stability and repetitive trajectory for the TVQP problem of kinematic planning for manipulators.

For repeatable motion of redundant mobile manipulators, the flexible base platform and the redundant manipulator have to be returned to the desired initial position simultaneously after completing the given tasks. To remedy deviations between initial position and desired position of each kinematic joint angle, a special kind of repeatable optimization for kinematic energy minimization based on terminal‐time Zhang neural network (TTZNN) with finite‐time convergence is proposed for inverse kinematics of mobile manipulators. It takes the advantages that each joint of the manipulator is required to return to the desired initial position not considering the initial orientation of itself for realizing repeatable kinematics control. Unlike the existed training methods, such an optimization of kinematic energy scheme based on TTZNN can not only reduce the convergent position error of each joint to zero in finite time, but also improve the convergent precision. Theoretical analysis and verifications show that the proposed optimal kinematic energy scheme accelerates the convergent rate, which is tended to be applied in practical robot kinematics. Simulation results on the manipulator with three mobile wheels substantiate the timeliness and repetitiveness of the proposed optimization scheme.