Different from the common static and continuous-time dynamic problems of unconstrained/constrained nonlinear optimization, this article aims to investigate a discrete-time dynamic problem of nonlinear optimization with multiple types of constraints, which can be succinctly termed as future multiconstrained nonlinear optimization (FMCNO) problem because of the unknown future. Considering the unique advantages of neural networks with parallelism and fuzzy control systems (FCSs) with adaptivity, a fuzzy-enhanced robust discretized zeroing neural network (FER-DZNN) model is proposed to address the FMCNO problem. Specifically, by introducing a fuzzy factor outputted from an FCS with dual inputs, the FER-DZNN model is designed on the basis of an FER evolution rule and a five-step look-ahead discretization rule. Moreover, theoretical results are provided to indicate the convergence and robustness of the FER-DZNN model under various noises. Finally, two illustrative examples, including an application example to robotic manipulator control, are presented to substantiate the superior convergent and robust performance of the FER-DZNN model under various noises for addressing the FMCNO problem.

Parallel robots including Gough–Stewart platforms are widely applied in industrial factories and medical fields. This article investigates a kinematically redundant Gough–Stewart robot, designs, and compares three novel zeroing neural networks (ZNNs) to serve as the redundancy-resolution controllers. Unlike the existing ZNN controllers, the newly designed ZNN controllers are endowed with the capability to handle joint constraints by using a nonlinear complementarity problem function without introducing any extra hyperparameters, guaranteeing the safety of the robot. The proposed ZNN controllers are training-free, noniterative, and more accurate, as compared with other typical neural controllers for kinematic control of the Gough–Stewart robot. Theoretically, the convergence analyses of the ZNN controllers are rigorously carried out. Corresponding discrete neural controllers are established and then applied to the kinematically redundant Gough–Stewart robot with two path-tracking tasks exemplified. The path-tracking results comparatively substantiate the effectiveness and superiority of the ZNN controllers for redundancy resolution under joint constraints.

Heterogeneous continuum robots (HCRs) with different structures have been designed for different purposes, whereas the coordination of multiple HCRs has received little attention. On one hand, multiple HCRs coordination brings the possibility of performing complicated tasks. On the other hand, the structural diversity of HCRs poses great difficulties to their modeling and control. This article proposes a model-free scheme for the synchronous motion control of multiple HCRs. The control problem is formulated as two convex optimization problems in a model-free closed-loop framework, including control quantity estimation and Jacobian matrix estimation. The proposed approach aims at addressing the synchronous motion problem of multiple HCRs in a decentralized way. The design of model-free feedback control guarantees the high adaptability of the proposed method for a wide range of HCRs. Simulation studies are performed to verify the effectiveness and adaptability of the proposed scheme for multiple HCRs. Comparative studies verify that the tracking error synthesized by the proposed method is about two-thirds lower than that of the existing method while the computational cost is similar, which reveals the merit of the proposed method in terms of accuracy. Finally, the feasibility and effectiveness of the proposed method are also verified by hardware-in-loop simulations and physical experiments on the synchronous motion control of a cable-driven continuum robot and a concentric-tube robot.

To effectively diminish the kinetic energy dissipation, joint-angle drift, and joint-velocity discontinuity problems, and simultaneously achieve the end-effector position and direction control as well as the avoidance of joint-physical limits, a novel velocity layer weighted multicriteria optimization scheme is proposed, which outperforms the traditional schemes for the robotic manipulators with multiple constraints. Besides, considering that the existing joint-limit conversion strategies are not differentiable everywhere or with relatively complex formulation, a new exponential joint-limit conversion strategy is introduced to facilitate the dynamic quadratic programming reformulation of the proposed scheme. For easier numerical realization and real-time control, aided with a high-precision six-step extrapolated-backward discretization rule, a novel discretized zeroing neural network model is proposed to resolve the proposed scheme, which has higher precision than the existing neural network models. Finally, numerical and physical experiments are conducted to substantiate the efficacy, superiority, and practicability of the proposed scheme and model.

This research first proposes the general expression of Zhanget al.discretization (ZeaD) formulas to provide an effective general framework for finding various ZeaD formulas by the idea of high-order derivative simultaneous elimination. Then, to solve the problem of future equality-constrained nonlinear optimization (ECNO) with various noises, a specific ZeaD formula originating from the general ZeaD formula is further studied for the discretization of a noise-perturbed continuous-time advanced zeroing neurodynamic model. Subsequently, the resulting noise-perturbed discrete-time advanced zeroing neurodynamic (NP-DTAZN) algorithm is proposed for the real-time solution to the future ECNO problem with various noises suppressed simultaneously. Moreover, theoretical and numerical results are presented to show the convergence and precision of the proposed NP-DTAZN algorithm in the perturbation of various noises. Finally, comparative numerical and physical experiments based on a Kinova JACO2robot manipulator are conducted to further substantiate the efficacy, superiority, and practicability of the proposed NP-DTAZN algorithm for solving the future ECNO problem with various noises.

In this article, the new jerk-level configuration adjustment (JLCA) schemes are proposed to achieve the configuration adjustment of constrained redundant robots. Specifically, by applying the zeroing neurodynamics design rule three times, the new JLCA performance index is first derived; then, together with the joint physical constraints incorporated, the new JLCA schemes are obtained for dual-arm and single-arm redundant robots, respectively. For comparison purposes, three other configuration adjustment schemes are also presented. Moreover, the comparative simulative experiments based on a planar dual-arm redundant robot (i.e., five-link dual-arm robot) and a spatial single-arm redundant robot (i.e., Kinova JACO$^2$robot) are performed to verify the efficacy and superiority of the proposed JLCA schemes, as compared with the three other configuration adjustment schemes. At last, the comparative physical experiments are conducted on the real Kinova JACO$^2$robot to substantiate the practicability and excellent performance of the proposed JLCA scheme for single-arm redundant robots.

In this work, the LQ decomposition problem of dynamic matrix is investigated. First, by applying Zhang neural network (ZNN) method as well as the Kronecker-product and vectorization techniques, a ZNN model is proposed to solve the LQ decomposition problem of dynamic matrix. Then, two simulative examples are provided to verify the validity of the proposed ZNN model. Finally, an example of the mobile object localization based on the angle-of-arrival (AoA) technique is provided to illustrate the applicability of the proposed ZNN model.

In this article, we derive the general explicit linear four-step (ELFS) rule with fifth-order precision systematically, together with a group of specific ELFS rules provided. Afterwards, we formulate and investigate a new and challenging discrete-time dynamic problem with relatively complex structure and future unknownness, which is simply termed future system of bounded inequalities and nonlinear equations (SBINE). With the aid of the general ELFS rule, the general ELFS-type discretized zeroing neural network (DZNN) model is proposed to solve the future SBINE. Moreover, theoretical and numerical results are presented to show the validity and high precision of the proposed general ELFS-type DZNN model. Finally, comparative numerical experiments based on a wheeled mobile robot containing several additional constraints are further performed to substantiate the applicability, validity, and superiority of the proposed general ELFS-type DZNN model.

In this work, time-dependent linear and nonlinear inequalities system (TDLNIS) is studied and solved. First, using zeroing neural network (ZNN) method twice, a continuous time-dependent ZNN (CTDZNN) model is proposed to solve the continuous TDLNIS. Subsequently, explicit linear dual-multistep methods, i.e., explicit linear dual-4-step, dual-3-step, and dual-2-step methods, are presented and studied. Afterwards, by applying the explicit linear dual-4-step method to the proposed CTDZNN model, a 4-step discrete time-dependent ZNN (4S-DTDZNN) model is proposed to solve the discrete TDLNIS. For comparison, 3-step discrete time-dependent ZNN (3S-DTDZNN) and 2-step discrete time-dependent ZNN (2S-DTDZNN) models are also developed for solving the discrete TDLNIS. In addition, theoretical analyses and results indicate the effectiveness and superiority of the proposed 4S-DTDZNN model. Finally, numerical experimental results further substantiate the effectiveness and superiority of the proposed 4S-DTDZNN model.

In this paper, a six-point discretization (6PD) formula is presented to discretize continuous-time models. Then, by using the 6PD formula and introducing the adaptivity/variability of parameter, a new-type discrete-time gradient (DTG) model is proposed to solve a discrete dynamic nonlinear equation system (DDNES) with occasionally-singular Jacobian matrix. For comparative purposes, based on the 6PD formula, a 6PD-type discrete-time zeroing (DTZ) model and an old-type (i.e., conventional) DTG model are also presented to handle the same problem. Finally, comparative numerical experiments, including an application to the discrete-time motion control of a robot manipulator, are conducted to substantiate the validity and superiority of the new-type DTG model for solving the DDNES with occasionally-singular Jacobian matrix. That is, when the Jacobian matrix of DDNES occasionally becomes singular as time evolves, the new-type DTG model can provide a feasible and effective solution to the singular Jacobian problem, whereas the other presented models fail to achieve such a solution.

In this article, a future different-layer nonlinear and linear equation system (DLNLES) is investigated. First, based on a zeroing neural network (ZNN) method, a zeroing equivalency theorem is proposed. Then, a continuous ZNN (CZNN) model is developed for continuous DLNLES solving. Next, a new eight-node Zhang et al. discretization formula is proposed to discretize the CZNN model, and thus, an eight-node discrete ZNN (DZNN) model is proposed for the future DLNLES solving. Five-node and four-node DZNN models are also developed for the same problem solving. Besides, numerical experiments are executed to substantiate the validity and superiority of the proposed eight-node DZNN model. Finally, the path-tracking control problem of a four-link redundant robot arm is formulated as a specific future DLNLES problem and can, thus, be solved by the three DZNN models. Comparative numerical results further indicate that the proposed eight-node DZNN model is much superior to the other two DZNN models.

In this article, a novel and challenging problem called future different-level system of nonlinear inequality and linear equation (FDLSNILE) is proposed and investigated. To solve FDLSNILE, the corresponding continuous different-level system of nonlinear inequality and linear equation (CDLSNILE) is first analyzed, and then, a continuous combined zeroing neural network (CCZNN) model for solving CDLSNILE is proposed. To obtain a discrete combined zeroing neural network (DCZNN) model for solving FDLSNILE, a high-precision general 7-instant Zhang et al. discretization (ZeaD) formula for the first-order time derivative approximation is proposed. Furthermore, by applying the general 7-instant ZeaD formula to discretize the CCZNN model, a general 7-instant DCZNN (7IDCZNN) model is thus proposed for solving FDLSNILE. For comparison, by using three conventional ZeaD formulas, three conventional DCZNN models are also developed. Meanwhile, theoretical analyses and results guarantee the efficacy and superiority of the general 7IDCZNN model compared with the other three conventional DCZNN models for solving FDLSNILE. Finally, several comparative numerical experiments, including the motion control of a 5-link redundant manipulator, are provided to substantiate the efficacy and superiority of the general 7-instant ZeaD formula and the corresponding 7IDCZNN model.

In this paper, a high-precision general discretization formula using six time instants is first proposed to approximate the first-order derivative. Then, such a formula is studied to discretize two continuous-time neurodynamic models, both of which are derived by applying the neurodynamic approaches based on neural networks (i.e., zeroing neurodynamics and gradient neurodynamics). Originating from the general six-instant discretization (6ID) formula, a specific 6ID formula is further presented. Subsequently, two new discrete-time neurodynamic algorithms, i.e., 6ID-type discrete-time zeroing neurodynamic (DTZN) algorithm and 6ID-type discrete-time gradient neurodynamic (DTGN) algorithm, are proposed and investigated for online future matrix inversion (OFMI). In addition to analyzing the usual nonsingular situation of the coefficient, this paper investigates the sometimes-singular situation of the coefficient for OFMI. Finally, two illustrative numerical examples, including an application to the inverse-kinematic control of a PUMA560 robot manipulator, are provided to show respective characteristics and advantages of the proposed 6ID-type DTZN and DTGN algorithms for OFMI in different situations, where the coefficient matrix to be inverted is always-nonsingular or sometimes-singular during time evolution.

In this paper, a four-point one-step-ahead finite-difference formula is presented, which obtains higher computational precision in approximating the first-order derivative. Then, the formula is used for the discretization of the continuous-time Zhang neural network (CTZNN), and it can greatly overcome the limitation of the conventional formulas in CTZNN discretization. Based on this formula, a new-type discrete-time Zhang neural network (DTZNN) model is proposed and investigated for time-variant matrix pseudoinversion. Numerical experiments further validate the feasibility, effectiveness, and superiority of the proposed new-type DTZNN model for solving the time-variant matrix pseudoinversion. Moreover, the proposed new-type DTZNN model is applied to the control of a robot manipulator. Physical experiment performed on a four-link planar robot manipulator is presented to demonstrate physical realizability and effectiveness of the proposed new-type DTZNN model.

In this brief, a new one-step-ahead numerical differentiation rule called six-instant -cube finite difference (6I CFD) formula is proposed for the first-order derivative approximation with higher precision than existing finite difference formulas (i.e., Euler and Taylor types). Subsequently, by exploiting the proposed 6I CFD formula to discretize the continuous-time Zhang neural network model, two new-type discrete-time ZNN (DTZNN) models, namely, new-type DTZNNK and DTZNNU models, are designed and generalized to compute the least-squares solution of dynamic linear equation system with time-varying rank-deficient coefficient in real time, which is quite different from the existing ZNN-related studies on solving continuous-time and discrete-time (dynamic or static) linear equation systems in the context of full-rank coefficients. Specifically, the corresponding dynamic normal equation system, of which the solution exactly corresponds to the least-squares solution of dynamic linear equation system, is elegantly introduced to solve such a rank-deficient least-squares problem efficiently and accurately. Theoretical analyses show that the maximal steady-state residual errors of the two new-type DTZNN models have an pattern, where denotes the sampling gap. Comparative numerical experimental results further substantiate the superior computational performance of the new-type DTZNN models to solve the rank-deficient least-squares problem of dynamic linear equation systems.

Discrete time-varying problems are frequently encountered in mathematics and engineering fields, such as numerical analysis, signal processing and computer computing. However, conventional algorithms mainly solve time-invariant problems. Employed for discrete time-varying problems solving, conventional algorithms may generate quite large and unacceptable lagging errors. In this paper, discrete time-varying four fundamental operations (DTVFFOs) are studied. In order to eliminate the lagging errors, based on the zeroing dynamics (ZD) and Taylor-Zhang discretization formula, discrete computing models, which are termed T-Z-K and T-Z-U models, are proposed and investigated. Note that the aforementioned models have an error pattern of O(g3), where g denotes the sampling gap. For comparison, Euler-type discrete models and Newton iteration (NI) models are also presented. Eventually, illustrative numerical experiments are displayed to testify the great performances of the proposed Taylor-Zhang discrete ZD models.

The following topics are dealt with: power grids; invertors; voltage control; electric current control; control system synthesis; distributed power generation; power convertors; machine control; power generation control; switching convertors.

This paper primarily demonstrates the effectiveness of the Z-type methodology for solving the problem of time-variant matrix inverse (termed Zhang matrix inverse, ZMI). As a case study of ZMI with examples, the online solution of ZMI is investigated in this paper. Specifically, different Zhang functions (ZFs), which lead to different effective Z-type models (i.e., Zhang neural nets), are proposed and implemented as the error basis functions for ZMI. Meanwhile, a specific relationship between the Z-type model and others' model/method [i.e., the Getz and Marsden (G-M) dynamic system] is presented. Eventually, the MATLAB Simulink modeling and simulative verifications with examples using such different Z-type models are further researched. Both theoretical analysis and modeling results demonstrate the efficacy of the proposed Z-type models which originate from different ZFs for ZMI.