Sensor-related disturbances and measurement uncertainties often degrade the performance of dynamic neural network methods in solving time-varying problems, especially the time-varying generalized matrix inversion (TVGMI) problem, which serves as the computational foundation for real-time control and signal reconstruction tasks. Traditional zeroing neural network (ZNN) models for handling TVGMI problems usually require matrix inversion or vectorization operations, leading to high computational complexity and poor robustness under sensor disturbance or time-varying perturbations. To overcome these challenges, this paper proposes a novel low-complexity zeroing neural network (LCZNN) model that achieves efficient and unified computation of time-varying matrix inversion and pseudoinverse without involving inverse matrix computation. To further enhance robustness, the LCZNN model is extended to handle multilayered sensor-related disturbances, giving rise to three variants: the state-disturbed LCZNN (SDLCZNN), the velocity-disturbed LCZNN (VDLCZNN), and the hybrid-disturbed LCZNN (HDLCZNN) models. Each variant introduces structured compensation dynamics that enable accurate convergence under different disturbance scenarios. Rigorous theoretical analyses establish their convergence and stability properties. Comprehensive numerical experiments on representative TVGMI problems validate the low computational burden, fast convergence, and superior multilayered disturbance tolerance of the proposed LCZNN framework. Moreover, discrete algorithms derived via Euler discretization are applied to the real-time path-tracking inverse-kinematics control of robotic manipulators. Simulation and physical experimental results confirm that the proposed LCZNN-based algorithms achieve high tracking precision, strong robustness, and computational efficiency, making them well suited for real-time robotic and control applications.

Time-variant convex optimization involving nonlinear inequality constraints (TVCOINICs) is a challenging problem due to the nonlinearity and time-variant nature of its objective function and multitype constraints. Differing from traditional slack variables and projection operator methods, this article proposes a novel differentiable transform function that is continuous and differentiable everywhere, while eschewing the need for additional tunable hyperparameters. On the basis of the Lagrange multiplier technique and Karush–Kuhn–Tucker (KKT) conditions, the initial time-variant optimization problem is converted into a time-variant nonlinear system comprising both equalities and inequalities. By employing the proposed differentiable transform function, this system is then further refined into an equivalent time-variant nonlinear system of equalities. Subsequently, the comprehensive design process of the differentiable transform function-based Zhang neural network (DTFZNN) model is delineated, which, through the comprehensive utilization of the time derivatives and error feedback information, is capable of adeptly addressing the TVCOINICs problem. Moreover, sliding mode control (SMC) is integrated into the proposed model to endow it with verified noise-suppression capability. The relevant theorems prove its convergence and robustness, and corresponding numerical experiments verify its effectiveness. Ultimately, to evaluate the practical efficacy of the proposed model, the proposed model is implemented to address the path tracking and repetitive motion problem associated with a robotic arm, thereby illustrating its superiority in solving practical problems.

In order to efficiently solve the time-varying QP (quadratic programming) problem, some researchers (e.g., Zhang et al) proposed the OIpU-94LVI algorithm. Being the basic idea of the algorithm, let us assume that the time-varying problem may not change significantly in a short period. Therefore, the time-varying QP problem is divided into multiple relatively “static” QP sub-problems using a sampling gap. This paper also investigates a specific MMPaC (minimum motion planning and control) scheme at the joint angular velocity layer based on the kinematic knowledge and Zhang equivalency (ZE), and successfully applies the OIpU-94LVI algorithm to the scheme for the minimum joint motion at the joint angular velocity layer, further confirming the effectiveness of the OIpU-94LVI algorithm in real-world applications. In addition, we use the OIpU-94LVI algorithm to perform computer simulations of real-time MMPaC scheme for the minimal joint motion of a planar redundant robotic arm (i.e., a four-link robotic arm used in this paper for two situations). The experimental tasks involve tracking a bee-shaped path and an epicycloid path. The experimental results are consistent with expectations.

Robot manipulator path tracking, recognized as a crucial aspect in robot manipulator control, has garnered significant attention from researchers. In this paper, to address the path tracking problem of robot manipulators with unknown models, a novel Jacobian pseudoinverse estimator is first proposed based on Zhang neurodynamics method. The estimator directly provides an efficient and accurate estimation of the Jacobian matrix pseudoinverse, avoiding the complicated operation of matrix pseudoinverse and preventing potential singularity phenomenon of the Jacobian matrix. By utilizing the Euler difference formulas, a discrete model-free and Jacobian-pseudoinverse-free Zhang neurodynamics algorithm is proposed. The proposed algorithm focuses on leveraging the available current and previous known information to predict the future unknown information. Detailed theoretical analyses and proofs ensure the convergence and stability of the proposed algorithm. Finally, comparative experiments with various effective model-free algorithms, and experimental validations on different types of robot manipulators (UR5, Franka Emika Panda, and Kinova Gen3 robot manipulators) using various experimental platforms (MATLAB, CoppeliaSim, and physical platforms) illustrate the effectiveness of the proposed algorithm. Note to Practitioners—This paper is motivated by addressing the prevalent challenge of unknown models in real-time path tracking for robot manipulators. In this paper, a novel discrete model-free and Jacobian-pseudoinverse-free Zhang neurodynamics algorithm is proposed. Different from the existing model-free algorithms, the proposed algorithm avoids the complicated operation of computing the pseudoinverse of matrix without compromising precision, significantly reducing the computational complexity and preventing potential singularity phenomenon of the Jacobian matrix. The average computation time per updating for the proposed algorithm is approximately$0.1\,{\mathrm { ms}}$, which is significantly less than the sampling gap of the operation. This allows it to effectively meet the real-time requirements for robot manipulator path tracking. In addition, the error of the proposed algorithm is approximately$2\,\mu {\mathrm { m}}$, which can meet the requirements of most practical application scenarios. Moreover, the accuracy of the algorithm is limited by the differential formula and sampling gap. Improving the accuracy and robustness of the algorithm by using more accurate difference formulas and filtering technique will be our future research direction.

Path tracking control of robotic arms is regarded as a fundamental problem in the field of robotics. However, obtaining an accurate model of the robotic arm in practical engineering poses significant challenges. As a result, model-free schemes have become a focus of investigation. In contrast to traditional model-free schemes used for estimating the Jacobian matrix of the robotic arm, in this work, a novel estimator directly for the pseudoinverse (PI) of the Jacobian matrix based on Zhang neurodynamics (ZN) is proposed for the first time. In addition, a novel model-free and PI-free ZN (MFPIFZN) scheme for path tracking control of robotic arms is proposed. The MFPIFZN scheme not only significantly reduces the operation complexity by eliminating the requirement to compute the PI of the Jacobian matrix but also enhances the accuracy by eliminating the potential errors that may arise from the computation of the PI. Theoretical analyses provide guarantees for the convergence and stability of the MFPIFZN scheme. Finally, experimental results conducted on planar four-link and Kinova Jaco2 robotic arms vividly illustrate the excellent performance of the MFPIFZN scheme. Comparison experiments with four other model-free schemes further confirm the superiority of the MFPIFZN scheme.

It is noteworthy that the Sylvester equation plays a pivotal role in the field of industrial intelligence control. To meet the demands of real-time applications, temporal-dependent Sylvester equations (TDSEs) are employed to formulate motion planning problems for robot manipulators. Traditionally, the classical Zhang neurodynamics (ZN) method is utilized to address the TDSE problems, which encounters challenges associated with temporal-dependent inverse matrix computations. In this article, we introduce an inverse-free approach based on energy zeroing, termed the reciprocal-kind ZN (RKZN) model, specifically designed to tackle the TDSE problem. Additionally, we propose a discrete RKZN (DRKZN) algorithm to address future Sylvester equation (FSE) problems and the motion planning challenges of robot manipulators. Furthermore, we conduct a thorough analysis of the convergence property and robustness of the RKZN method for addressing the TDSE problem. This analysis is grounded in the Lyapunov stability theory of nonlinear systems and a comparative method for nonlinear systems with temporal-dependent error-feedback-related uncertainty disturbances. Numerical experiments, simulations, and physical experiments substantiate the effectiveness and superiority of the developed RKZN method in addressing both the TDSE problem and the motion planning challenges of robot manipulators.

To obtain smoother kinematic control of minimum motion, a novel snap-layer minimum motion scheme, otherwise known as the minimum motion planning and control (MMPC) scheme for redundant robot arms, is proposed for the first time in this study. With the primary task of tracking planned paths and the consideration of satisfying five-layer physical limits, the snap-layer MMPC problem is transformed into a quadratic programming (QP) problem. Five-layer physical limits include angle-layer, velocity-layer, acceleration-layer, jerk-layer, and snap-layer limits, which are all considered and then transformed into a unified-layer bounded constraint through Zhang neural dynamics (ZND) equivalency. Furthermore, the snap-layer performance index and equation constraint are derived by utilizing the ZND formula. Therefore, the proposed snap-layer MMPC scheme is formulated as a standard QP that can avoid the potential physical damage of redundant robot arms. The snap-layer projection neural dynamics (PND) solver is presented and used to acquire the neural solution of the QP. Simulation results on a 6-degrees-of-freedom (DOF) planar redundant robot arm are presented to substantiate the effectiveness and superiority of the proposed snap-layer MMPC scheme by comparing it with the jerk-layer MMPC scheme and the minimum snap norm (MSN) scheme.

Temporal-variant linear equations (TVLEs) are widely acknowledged for their pivotal role in various engineering fields, offering a potent means to model dynamic processes and evolving relationships over time. A conventional approach involves leveraging the zeroing neural network (ZNN) model for tackling TVLE problems. In response to challenges associated with inverse matrix computations and infinite-time convergence constraints, we introduce an innovative single inverse-free finite-time reciprocal ZNN (FRZNN) model constructed to effectively address TVLE problems without using the activation functions. The convergence property and robustness of the FRZNN model are thoroughly examined adopting Lyapunov stability method of the nonlinear system and a comparative approach for nonlinear perturbed systems. Through two numerical experiments and an Angle-of-Arrival (AOA) simulation, the performance of the FRZNN model is thoroughly evaluated, revealing its validity and superior effectiveness when compared to state-of-the-art approaches. In detail, the performance improvement ratio (PIR) of the FRZNN model in addressing the AOA problem is 60.52%, and under a noise environment, the PIR of the FRZNN model is 99.99%.

Time-dependent Lyapunov matrix equation (TDLME) plays a central role in the control of linear and nonlinear systems. Existing models, including the classical zeroing neural dynamics (ZNDs) model and its variants, have been used to address the TDLME problem. However, those models require time-dependent matrix inversion, which is computationally demanding, and they primarily focus on measurement-related noise, overlooking other sources of system uncertainty. To overcome these challenges, we propose an inverse-free reciprocal-type ZND (RTZND) model. This model integrates an energy-based error function with the ZND framework, eliminating the need for matrix inversion and incorporating error-feedback-related noise through its closed-loop control structure. We establish the convergence and robustness of the RTZND model using Lyapunov stability theory and assess its performance under external disturbances. Numerical simulations confirm its effectiveness and improved computational efficiency in solving the TDLME problem. We further confirm its applicability through two case studies, a time-dependent linear system and a nonlinear system modeled by the single machine infinite bus (SMIB) system, highlighting the RTZND model’s practical value in addressing TDLME problems.

Solving the time-varying optimization with nonlinear equation constraint (TVONEC) remains a significant challenge due to the complexities introduced by temporal variability and nonlinearity, which is less explored in existing research. To address this challenge, a novel nonlinear activation function (AF), the sign-exponential AF (SEAF), is first designed in this article. Building on the SEAF, we propose two advanced zeroing neurodynamics (ZN) models: the finite and fixed-time convergent ZN (FFTCZN) and the arbitrary-time convergent ZN (ATCZN). Unlike traditional models, the FFTCZN model reaches a convergence state within a fixed and finite time frame, while the ATCZN model achieves arbitrary-time convergence, enabling it to reach the convergence state within an arbitrary time frame. Their respective convergence properties, including superior convergence speed and robustness, are verified through rigorous mathematical analyses and experimental validations. Furthermore, two novel controllers are developed to achieve the drift-free path tracking for the manipulator, exhibiting excellent tracking precision and strong resilience to persistent noise.

The GRC (global reserve currency) plays a crucial role on the development of the global (i.e., world) economy and the evolution of the international situation. Over the past 500 years, it has undergone five transitions. Scientifically predicting the time of the next (i.e., coming) transition help us make corresponding preparations in advance. In this paper, we split the transition year into two parts: the thousand and hundred (TH) digits as well as the ten and unit (TU) digits. For the former, we employ linear regression for prediction, while, for the latter, we utilize the moment estimation and maximum likelihood estimation based on the uniform distribution’s characteristics to predict. We also verify that maximum likelihood estimation is more effective than moment estimation under unbiased situation. After theoretical derivation and experimental analysis, we finally predict the time interval for the occurrence of new GRC narrowly as [2024, 2040] and generally [2k00, 2k50], where k represents a specific century.

Vision and Language Navigation (VLN) is a challenging task that requires agents to understand instructions and navigate to the destination in a visual environment. One of the key challenges in outdoor VLN is keeping track of which part of the instruction was completed. To alleviate this problem, previous works mainly focus on grounding the natural language to the visual input, but neglecting the crucial role of the agent's spatial position information in the grounding process. In this work, we first explore the substantial effect of spatial position locating on the grounding of outdoor VLN, drawing inspiration from human navigation. In real-world navigation scenarios, before planning a path to the destination, humans typically need to figure out their current location. This observation underscores the pivotal role of spatial localization in the navigation process. In this work, we introduce a novel framework, Locating before Planning (Loc4Plan), designed to incorporate spatial perception for action planning in outdoor VLN tasks. The main idea behind Loc4Plan is to perform the spatial localization before planning a decision action based on corresponding guidance, which comprises a block-aware spatial locating (BAL) module and a spatial-aware action planning (SAP) module. Specifically, to help the agent perceive its spatial location in the environment, we propose to learn a position predictor that measures how far the agent is from the next intersection for reflecting its position, which is achieved by the BAL module. After this locating process, we propose the PSA module to associate visual observations After the locating process, we propose the SAP module to incorporate spatial information to ground the corresponding guidance and enhance the precision of action planning. Extensive experiments on the Touchdown and map2seq datasets show that the proposed Loc4Plan outperforms the SOTA methods.

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.

As the zeroing neural network (ZNN) approach needs human intervention to handle temporally variant quadratic program (TVQP) problem, which results in less flexibility and convenience, especially considering the situation of tolerated precision given. Aiming at these weaknesses, in this article, we propose a type of adaptive ZNN (AZNN) approach on solving TVQP problem. Specifically, we design and propose a novel continuous-time AZNN model and two adaptive time-discretization techniques with different precision to obtain two discrete-time AZNN (DAZNN) solvers, making them more readily implemented on digital computers. The convergence analyses for these model and solvers are theoretically proved. Comparative experiments among the proposed DAZNN solvers and four existing solvers are conducted to verify the efficacy and superiority of the DAZNN solvers. Furthermore, simulative and physical experiments for the position tracking of JACO2 mechanical arm are successfully performed. Finally, based on the AZNN approaches, a novel adaptive kinematics planner and two adaptive solvers are proposed and applied for the pose tracking of the Franka Emika Panda mechanical arm, further substantiating their efficacy and effectiveness.

To solve the time-variant Sylvester equation, in 2013, Li et al. proposed the zeroing neural network with sign-bi-power function (ZNN-SBPF) model via constructing a nonlinear activation function. In this article, to further improve the convergence rate, the zeroing neural network with coefficient functions and adjustable parameters (ZNN-CFAP) model as a variation in zeroing neural network (ZNN) model is proposed. On the basis of the introduced coefficient functions, an appropriate ZNN-CFAP model can be chosen according to the error function. The high convergence rate of the ZNN-CFAP model can be achieved by choosing appropriate adjustable parameters. Moreover, the finite-time convergence property and convergence time upper bound of the ZNN-CFAP model are proved in theory. Computer simulations and numerical experiments are performed to illustrate the efficacy and validity of the ZNN-CFAP model in time-variant Sylvester equation solving. Comparative experiments among the ZNN-CFAP, ZNN-SBPF, and ZNN with linear function (ZNN-LF) models further substantiate the superiority of the ZNN-CFAP model in view of the convergence rate. Finally, the proposed ZNN-CFAP model is successfully applied to the tracking control of robot manipulator to verify its practicability.

Time-dependent linear system (TDLS) is usually encountered in scientific research, which is the mathematical formulation of many practical applications. Different from conventional inverse-need models, by utilizing zeroing neural network (ZNN) method twice, an inverse-free continuous ZNN (CZNN) model is developed for solving TDLS. For conveniently practical use, a discrete model is naturally desired. Superior to conventional discretization methods, a general linear six-step (LSS) method with the seventh-order precision and five variable parameters is proposed for the first time. Constraints about five variable parameters are theoretically analyzed to guarantee the efficacy of the general LSS method. Within constraints, 12 specific LSS methods are further developed. Aided with the general LSS method, an inverse-free discrete ZNN (DZNN) is proposed and termed DZNN-LSS model, and its precision is greatly improved compared with conventional discrete models. For comparison, three conventional discretization methods are also utilized to generate DZNN models. Detailed theoretical analyses are provided to prove the efficacy of relevant models. In addition, a specific TDLS example is considered to show the effectiveness and superiority of the DZNN-LSS model. More than that, applications to manipulator control and sound source localization are conducted to illustrate the applicability of the DZNN-LSS model.

For dealing with the temporally variant optimization with nonlinear equation constraints (TVONECs), a novel Zhang neural net (ZNN) model is proposed in this work. Two continuous-time computer numerical simulations are constructed to testify the feasibility and correctness of the continuous-time ZNN (CZNN) model. To facilitate the implementation of numerical algorithms on computer, a novel 11-instant time discretization (TD) formula is proposed in this article, and a discrete-time ZNN (DZNN) algorithm (i.e., 11-instant DZNN algorithm) is thus obtained. Besides, theoretical analyses prove the superiority as well as the feasibility of the DZNN11I algorithm. For comparison, other three TD formulas and corresponding discrete-time algorithms (i.e., 2-instant DZNN, 3-instant DZNN, and 7-instant DZNN algorithms) are presented. Finally, numerical experiments and an application to Kinova Jaco2 manipulator control are conducted to illustrate the superiority of the proposed model and algorithm.

It is common knowledge that the future quadratic programming (FQP) problem is a challenging and widely applicable topic in mathematical science and many engineering fields. In this article, we systematically derive a new specific explicit linear left and right six-step (ELSS) rule with order-7 precision. With the help of the ELSS rule, the ELSS-type discretized Zhang neurodynamics (ELSS-DZNs) algorithm that develops from continuous Zhang neurodynamics (CZNs) model is proposed for tackling the FQP problem subject to linear equation (LE), while the inverse-free continuous hybrid gradient Zhang neurodynamics (IFCHGZNs) model, which combines the Getz–Marsden dynamic system (GMDS) model 2 for time-variant inverse computing (TVIC) and CZN model for tackling the time-variant QP (TVQP) problem subject to LE, develops the ELSS-type inverse-free discretized hybrid gradient Zhang neurodynamics (ELSS-IFDHGZN) algorithm for dealing with the FQP problem subject to LE. Theoretically, three theorems are presented to show the convergent properties of the proposed general CZN and IFCHGZN models by Lyapunov stability theory, while three theorems and five corollaries are presented to show the truncation error pattern of the ELSS-DZN and ELSS-IFDHGZN algorithms by using the stability and convergence theory of linear multistep formulas. Moreover, two numerical experiments are adopted to demonstrate the availability and convergent properties of the proposed general ELSS-DZN and ELSS-IFDHGZN algorithms. In the end, numerical and physical experiments based on two kinds of robot manipulators are additionally performed to verify the applicability and validity of the proposed general ELSS-DZN and ELSS-IFDHGZN algorithms.

In this article, being different from conventional time-discretization (simply called discretization) formulas, explicit linear left-and-right 5-step (ELLR5S) formulas with sixth-order precision are proposed. The general sixth-order ELLR5S formula with four variable parameters is developed first, and constraints of these four parameters are displayed to guarantee the zero stability, consistence, and convergence of the formula. Then, by choosing specific parameter values within constraints, eight specific sixth-order ELLR5S formulas are developed. The general sixth-order ELLR5S formula is further utilized to generate discrete zeroing neural network (DZNN) models for solving time-varying linear and nonlinear systems. For comparison, three conventional discretization formulas are also utilized. Theoretical analyses are presented to show the performance of ELLR5S formulas and DZNN models. Furthermore, abundant experiments, including three practical applications, that is, angle-of-arrival (AoA) localization and two redundant manipulators (PUMA560 manipulator and Kinova manipulator) control, are conducted. The synthesized results substantiate the efficacy and superiority of sixth-order ELLR5S formulas as well as the corresponding DZNN models.

Optimization problems are frequently encountered in various fields. In this article, the unconstrained time-variant convex optimization (UTVCO) problem is investigated. Generally, gradient neural network (GNN) is a traditional and effective method for solving time-invariant problems by making use of the gradient information. However, GNN is less effective on time-variant problems. On the other hand, Zhang neural network (ZNN) performs well on time-variant problems by exploiting the time-derivative information. In order to solve the UTVCO problem effectively and quickly with the help of the gradient information, inspired by the two methods, gradient-feedback ZNN (GZNN) is presented by taking both advantages of GNN and ZNN to solve the UTVCO problem. The main contributions are presented as follows. 1) The GZNN model for solving the UTVCO problem is proposed and analyzed theoretically. 2) Comparisons among GZNN, ZNN, GNN, and other models are presented with detailed discussions. Suggestions are provided on how to choose a model for solving the UTVCO problem better. 3) Tracking control of the robot manipulator is formulated as a UTVCO problem and studied with the GZNN model. According to the simulative and physical experiments, the task of tracking control is accomplished excellently by using the GZNN model.

The effectiveness of continuous-time zeroing neural network (ZNN) (CZNN) in continuum robot control has been preliminarily verified. However, CZNN is not friendly to digital devices and hardware implementation. Although discrete ZNN (DZNN) has been investigated in other tasks, all the existing DZNNs are designed with fixed time steps, which cannot meet the needs of different tasks. This motivates us to develop a generic DZNN model with variable number of time steps and apply it to continuum robot control. In this article, we present a data-driven scheme based on CZNNs to learn the unknown kinematics of continuum robots and solve the kinematic control problem. Furthermore, a unified$m$-step forward discretization formula is derived to discretize the CZNN-based scheme into DZNN-based control algorithms. Finally, we take the 1-step and the 3-step algorithms as examples to show the discretization process, and verify their efficacy by simulations and experiments.

Equivalency is a powerful approach that can transform an original problem into another problem that is relatively more ready to be resolved. In recent years, Zhang neurodynamics equivalency (ZNE), in the form of neurodynamics or recurrent neural networks (RNNs), has been investigated, abstracted, and proposed as a process that can equivalently solve equations at different levels. After long-term research, we have noticed that the ZNE can not only work with equations, but also inequations. Thus, the ZNE of inequation type is proposed, proved, and applied in this study. The ZNE of inequation type can transform different-level bound constraints into unified-level bound constraints. Applications of the jerk-level ZNE of bound constraints, equation constraints, and objective indices ultimately build up effective time-varying quadratic-programming schemes for cyclic motion planning and control (CMPC) of single and dual robot-arm systems. In addition, as an effective time-varying quadratic-programming solver, a projection neural network (PNN) is introduced. Experimental results with single and dual robot-arm systems substantiate the correctness and efficacy of ZNE and especially the ZNE of inequation type. Comparisons with conventional methods also exhibit the superiorities of ZNE.

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.

Differing from the common linear matrix equation, the future different-level linear matrix system is considered, which is much more interesting and challenging. Because of its complicated structure and future-computation characteristic, traditional methods for static and same-level systems may not be effective on this occasion. For solving this difficult future different-level linear matrix system, the continuous different-level linear matrix system is first considered. On the basis of the zeroing neural network (ZNN), the physical mathematical equivalency is thus proposed, which is called ZNN equivalency (ZE), and it is compared with the traditional concept of mathematical equivalence. Then, on the basis of ZE, the continuous-time synthesis (CTS) model is further developed. To satisfy the future-computation requirement of the future different-level linear matrix system, the 7-instant discrete-time synthesis (DTS) model is further attained by utilizing the high-precision 7-instant Zhang et al. discretization (ZeaD) formula. For a comparison, three different DTS models using three conventional ZeaD formulas are also presented. Meanwhile, the efficacy of the 7-instant DTS model is testified by the theoretical analyses. Finally, experimental results verify the brilliant performance of the 7-instant DTS model in solving the future different-level linear matrix system.

Obstacle avoidance is usually an additional task for a redundant manipulator when the end-effector tracking task is performed, which guarantees the safety of the redundant manipulator. By formulating and combining the end-effector tracking task and the obstacle avoidance task using zeroing neural network (ZNN), in this article, a concise continuous ZNN (CZNN) controller is first proposed. To develop discrete controllers for practical control, a second-order discrete formula and a third-order discrete formula are introduced. Therefore, by utilizing two discrete formulas to discretize the CZNN controller, two concise discrete ZNN (DZNN) controllers are proposed. Detailed theoretical analyses guarantee the effectiveness of task formulations, CZNN controller, and DZNN controllers. In addition, some comparisons with existing studies are presented in details. Furthermore, three groups of simulative experiments on the basis of UR5 manipulator and two groups of physical experiments on the basis of Kinova manipulator are conducted to illustrate the effectiveness, superiority, and practicability of two DZNN controllers.

The problem of solving linear equations is considered as one of the fundamental problems commonly encountered in science and engineering. In this article, the complex-valued time-varying linear matrix equation (CVTV-LME) problem is investigated. Then, by employing a complex-valued, time-varying QR (CVTVQR) decomposition, the zeroing neural network (ZNN) method, equivalent transformations, Kronecker product, and vectorization techniques, we propose and study a CVTVQR decomposition-based linear matrix equation (CVTVQR-LME) model. In addition to the usage of the QR decomposition, the further advantage of the CVTVQR-LME model is reflected in the fact that it can handle a linear system with square or rectangular coefficient matrix in both the matrix and vector cases. Its efficacy in solving the CVTV-LME problems have been tested in a variety of numerical simulations as well as in two applications, one in robotic motion tracking and the other in angle-of-arrival localization.

In this article, the repetitive motion control of redundant manipulators is investigated. First, a repetitive motion control scheme is presented, and a continuous zeroing neural network (CZNN) model is obtained for solving the scheme. Meanwhile, the development of a discrete zeroing neural network (DZNN) model is desired for convenient computational processing. Based on this, this article proposes a 6-step discretization formula, which has high precision. By using the 6-step discretization formula and the 4-step backward difference formula, a 6-step DZNN (6SDZNN) model is further proposed to handle the repetitive motion control scheme. Theoretical analyses verify the efficacy of the 6SDZNN model. Additionally, some discrete forms of conventional models are developed for comparison. Computer simulations on the basis of the 4-link redundant manipulator are carried out, verifying the theoretical analyses and showing the efficacy of the 6SDZNN model. Finally, physical experiments on the basis of the Kinova Jaco2manipulator substantiate the practicability of the 6SDZNN model.

In this article, time-varying matrix equation problems, including the Lyapunov equation, matrix inversion, and generalized matrix inversion are investigated in a future (or say, discrete time-varying) perspective. Then, in order to develop a unified solution model for the above three future problems, a future matrix equation (FME) is investigated. The discrete-time unified solution (DTUS) model, which is based on the zeroing neural dynamics (ZND) method and a new nine-instant Zhanget al.discretization (ZeaD) formula, is thus proposed and termed the nine-instant DTUS (9IDTUS) model. Meanwhile, theoretical analyses on the stability and precision of the 9IDTUS model are provided. In addition, conventional DTUS models obtained from the Euler forward formula, Taylor–Zhang discretization formula, and a seven-instant discretization formula are also presented for comparisons. Furthermore, numerical experiments including the robot motion generation, are conducted and analyzed to substantiate the efficacy and superiority of the proposed 9IDTUS model.

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 recent years, the robotics industry has been a hotspot and provided much convenience in people's daily life. For further research on robots, we introduce the problem of velocity-level minimum joint motion planning and control in this paper. Firstly, by using the Zhang neural network (ZNN) method and the Lagrange multiplier method, a continuous-time ZNN model is presented to solve the problem. Besides, an advanced ten-instant time-discretization formula with higher precision is presented, and five discrete-time ZNN (DTZNN) models are listed because of the digital hardware's requirements. At last, computer and robot experiments verify the effectiveness and feasibility of the presented DTZNN models.

In this study, a continuous-time gradient-Zhang neural dynamics model is proposed for computing pseudoinverses (also termed as Moore-Penrose inverses) of time-varying matrices on the basis of gradient and Zhang neural dynamics. For hardware realization, the discrete-time gradient-Zhang neural dynamics (DTGZND) models, including DTGZND-1 and DTGZND-2 models, are proposed by using Zhang et al. discretization (ZeaD) and extrapolation formulas. Compared with five conventional Zhang neural dynamics models in numerical experiments, the proposed DTGZND-1 and DTGZND-2 models own high precision, avoid computing inverse explicitly, and also work out for challenging Z-type matrices. Besides, the DTGZND-1 model is applied to robot manipulator motion planning and control via simulative and physical experiments. The experimental results substantiate the correctness and effectiveness of the proposed models.

Because of its extensive appearance and application in scientific research and industrial production, the matrix square root problem has received massive attention and study. In this paper, based on our previous work, by using zeroing neural dynamics (ZND) method, a continuous-time dynamic matrix square root (CTDMSR) model is given at first. Besides, a general ten-instant Zhang et al. discretization (ZeaD) formula is derived, constructed and investigated, and the corresponding theoretical analysis is provided. Next, by applying this general formula to discretize the CTDMSR model, a general ten-instant discrete-time dynamic matrix square root (DTDMSR) model with sixth-order precision is further obtained. For comparison purposes, four DTDMSR models, with the second-, third-, fourth-, and fifth-order precision, are also acquired and presented, respectively, by using other ZeaD formulas. At last, the effectiveness and correctness of the proposed DTDMSR models for dynamic matrix square root finding are further substantiated by numerical experimental results.

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.

Time-varying matrix pseudoinverse (TVMP) problem has been investigated by many researchers in recent years, but a new class of matrix termed Zhang matrix has been found and not been handled by some conventional models, e.g., Getz-Marsden dynamic model. On the other way, future matrix pseudoinverse (FMP), as a more challenging and intractable discrete-time problem, deserves more attention due to its significant role-playing on some engineering applications, such as redundant manipulator. Based on the zeroing neural network (ZNN), this article concentrates on designing new discrete ZNN models appropriately for computing the FMPs of all matrices of full rank, including the Zhang matrix. First, an inverse-free continuous ZNN model for computing TVMP is derived. Subsequently, Zhang et al. discretization (ZeaD) formulas and equidistant extrapolation formulas are used to discretize the continuous ZNN model to two discrete ZNN models for computing FMPs with different truncation errors. The numerical experiments are conducted for the five conventional discrete models and two new discrete ZNN models. Distinct numerical results substantiate the effectiveness and choiceness of newly proposed models. Finally, one of the newly proposed models is implemented on simulating and physical instances of robot manipulators, respectively, to show its practicability.

Future quadratic programming (FQP) is an interesting and challenging topic due to its unknown future information and time-dependent feature. In this paper, a continuous-time zeroing neurodynamics (CTZN) model for quadratic programming is first obtained via zeroing neurodynamics (ZN) method. Then, a general four-step Zhang et al. discretization formula is presented and adopted to discretize the above CTZN model, and thus the general four-step discrete-time ZN (DTZN) algorithm for the FQP is developed. For comparison, a three-step DTZN algorithm and a one-step DTZN algorithm for the FQP are also presented. It is worth noting that there is an important parameter termed stepsize in the DTZN algorithms, which is closely related to their stability. If the value of stepsize is outside its effective interval, the DTZN algorithms are impossible to achieve convergence in terms of residual errors, which leads to failure of the FQP problem solving. By utilizing bilinear transformation and Routh stability criterion, the effective stepsize interval of the general four-step DTZN algorithm is confirmed via theoretical proof. Besides, numerical results substantiate the effectiveness and superiority of the general four-step DTZN algorithm as well as the accuracy of the effective stepsize interval. Finally, the general four-step DTZN algorithm is applied to fulfill the path-tracking control of different robot manipulators, with the effectiveness and superiority further validated.

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 recent years, many models with high precision for redundant manipulator tracking control have been proposed based on precise kinematics equations. Nevertheless, without precise kinematic equations, developing a model with high precision for tracking control is meaningful. With the help of zeroing neural dynamics (ZND), a continuous ZND model with adaptive Jacobian matrix is obtained. For better computer operation and easier understanding, developing corresponding discrete ZND (DZND) model is also significant. Therefore, two DZND models (termed DZND-I model and DZND-II model) are proposed in this article on the basis of two discretization formulas, respectively. Meanwhile, theoretical analyses are conducted to ensure the efficacy of DZND-I model and DZND-II model. Finally, the efficacy of the two DZND models with adaptive Jacobian matrix is substantiated by experimental results on the basis of the four-link manipulator, UR5 manipulator, and Jaco2 manipulator, respectively.

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.

Different-level dynamic linear system (DLDLS) is an interesting and challenging topic due to its complicated structure and time-variant characteristic. To solve this difficult problem, the equivalency of solutions at different levels is analyzed and obtained via zeroing neural network (ZNN) method. Based on the equivalency, a continuous ZNN model is proposed to solve the continuous DLDLS. For easier hardware realization, a new Zhang et al. discretization formula with high precision is proposed for the continuous ZNN model discretization, and the corresponding new discrete ZNN (NDZNN) model is proposed to solve discrete DLDLS. Note that the NDZNN model satisfies the requirement of real-time computation because it has the online ability to predict the solution for the future instant. Furthermore, the problems of robot manipulator control with additional restrictions (e.g., joint damage) are formulated as specific discrete DLDLS, and the proposed NDZNN model is employed to solve such problems. Simulation results substantiate the effectiveness of NDZNN model.

In this paper, a new one-step-ahead numerical differentiation rule termed 5-instant discretization formula is proposed for the first-order derivative approximation with higher computational precision. Then, by exploiting the proposed formula to discretize the continuous-time zeroing neural network [or termed, continuous-time Zhang neural network (ZNN)] models, two new discrete-time zeroing neural network [or termed, discrete-time ZNN (DTZNN)] models are proposed, analyzed and investigated for solving systems of discrete time-variant inequalities, including the system of discrete time-variant linear inequalities and the system of discrete time-variant nonlinear inequalities. For comparative purposes, the recently developed Taylor-type DTZNN models and the widely used Euler-type DTZNN models are also presented. Theoretical analyses show that the proposed DTZNN models are convergent, and their steady-state residual errors have an O(g4) pattern with g denoting the sampling gap. Comparative numerical experimental results further substantiate the efficacy and superiority of the proposed DTZNN models for solving the systems of discrete time-variant inequalities.

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.

Different from general linear inequality or equality, the problem of future different-level linear inequality and equality (FDLLIE) is investigated, which is much more interesting and challenging. In order to solve this difficult FDLLIE, continuous different-level linear inequality and equality (CDLLIE) is first considered. A zeroing equivalency theorem is proposed based on the zeroing neural network method, and then a continuous solution model is, thus, obtained for CDLLIE solving. Furthermore, a new discrete-solution (NDS) model is developed for FDLLIE solving by using a proposed new 7-instant Zhang et al. discretization (ZeaD) formula to discretize the continuous solution model. Meanwhile, theoretical analyses and results are presented to show the excellent properties of the NDS model. Numerical results illustrate the effectiveness and superiority of the NDS model for solving FDLLIE. Furthermore, application experiments for motion planning of robot manipulator are conducted to substantiate the efficacy of the NDS model for FDLLIE solving.

Previous works provide a few effective discretization formulas for zeroing neural network (ZNN), of which the precision is a square pattern. However, those formulas are separately developed via many relatively blind attempts. In this paper, general square-pattern discretization (SPD) formulas are proposed for ZNN via the idea of the second-order derivative elimination. All existing SPD formulas in previous works are included in the framework of the general SPD formulas. The connections and differences of various general formulas are also discussed. Furthermore, the general SPD formulas are used to solve future optimization under linear equality constraints, and the corresponding general discrete ZNN models are proposed. General discrete ZNN models have at least one parameter to adjust, thereby determining their zero stability. Thus, the parameter domains are obtained by restricting zero stability. Finally, numerous comparative numerical experiments, including the motion control of a PUMA560 robot manipulator, are provided to substantiate theoretical results and their superiority to conventional Euler formula.

In this brief, future equality-constrained quadratic programming (FECQP) is studied. Via a zeroing neurodynamics method, a continuous-time zeroing neurodynamics (CTZN) model is presented. By using Taylor-Zhang discretization formula to discretize the CTZN model, a Taylor-Zhang discrete-time zeroing neurodynamics (TZ-DTZN) model is presented to perform FECQP. Furthermore, we focus on the critical parameter of the TZ-DTZN model, i.e., stepsize. By theoretical analyses, we obtain an effective range of the stepsize, which guarantees the stability of the TZ-DTZN model. In addition, we further discuss the optimal value of the stepsize, which makes the TZ-DTZN model possess the optimal stability (i.e., the best stability with the fastest convergence). Finally, numerical experiments and application experiments for motion generation of a robot manipulator are conducted to verify the high precision of the TZ-DTZN model and the effective range and optimal value of the stepsize for FECQP.

Tracking control of robot manipulators is a fundamental and significant problem in robotic industry. As a conventional solution, the Jacobian-matrix-pseudo-inverse (JMPI) method suffers from two major limitations: one is the requirement on known information of the robot model such as parameter and structure; the other is the position error accumulation phenomenon caused by the open-loop nature. To overcome such two limitations, this paper proposes a novel Jacobian-matrix-adaption (JMA) method for the tracking control of robot manipulators via the zeroing dynamics. Unlike existing works requiring the information of the known robot model, the proposed JMA method uses only the input-output information to control the robot with unknown model. The solution based on the JMA method transforms the internal, implicit, and unmeasurable model information to the external, explicit, and measurable input-output information. Moreover, simulation studies including comparisons and tests substantiate the efficacy and superiority of the proposed JMA method for the tracking control of robot manipulators subject to unknown models.

Improvement of the real-time performance of tracking control is increasingly desirable. It is a routine for most conventional algorithms that the control input at current time instant is to track the current desired output. However, lagging errors resulting from computational time and the fluctuation of the desired output exist for the tracking control. Different from conventional algorithms, a look-ahead scheme of zeroing dynamics (ZD) is established in this paper to achieve the real-time tracking control of both serial and parallel manipulators. With the exploitation of data at current time and that in history, the control inputs generated by the proposed ZD algorithms never lead to lagging errors with the source from the inevitable computational time. To tackle prediction errors for ZD algorithms, a new high-precision discretization formula, as an essential part of ZD algorithms, is presented to confine the prediction error in an ignorable range in comparison with lagging errors.

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.

This paper proposes a novel robust zeroing neural-dynamics (RZND) approach as well as its associated model for solving the inverse kinematics problem of mobile robot manipulators. Unlike existing works based on the assumption that neural network models are free of external disturbances, four common forms of time-varying disturbances suppressed by the proposed RZND model are investigated in this paper. In addition, theoretical analyses on the antidisturbance performance are presented in detail to prove the effectiveness and robustness of the proposed RZND model with time-varying disturbances suppressed for solving the inverse kinematics problem of mobile robot manipulators. That is, the RZND model converges toward the exact solution of the inverse kinematics problem of mobile robot manipulators with bounded or zero-oriented steady-state position error. Moreover, simulation studies and comprehensive comparisons with existing neural network models, e.g., the conventional Zhang neural network model and the gradient-based recurrent neural network model, together with extensive tests with four common forms of time-varying disturbances substantiate the efficacy, robustness, and superiority of the proposed RZND approach as well as its time-varying disturbances suppression model for solving the inverse kinematics problem of mobile robot manipulators.

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.

In this paper, to achieve the fault-tolerant capability for redundant manipulators, a dimension-reduction method is presented and investigated at the joint-acceleration level. By incorporating such a dimension-reduction method and the limits of joint angle, joint velocity as well as joint acceleration (i.e., the physical constraints on joints), an acceleration fault-tolerant scheme for redundant manipulator motion planning and control (or say, motion-planning-and-control, MPaC) is thus proposed and investigated. The scheme is then reformulated as a quadratic program (QP) subject to equality and bound constraints. For the online solution of the proposed scheme, the PLPE (piecewise-linear projection equation) oriented numerical algorithm is adopted to obtain the final QP solver. The derived QP resulting from the proposed scheme combines the abilities of fault tolerance, joints limits avoidance and repetitive motion, which can achieve the repetitive motion before and after the fault tolerance.

In this paper, a hybrid multi-objective scheme is proposed to complete simultaneously four objectives, i.e., the specified primary task for the end-effector, obstacle avoidance, joint-physical limits avoidance, and repetitive motion of redundant robot manipulators. In addition, corresponding theoretical analysis is given, which guarantees the validity of the proposed scheme. Then, the proposed hybrid multi-objective scheme is reformulated as a dynamical quadratic program (DQP) problem. The optimal solution of the DQP problem is found by the PLPE (piecewise-linear projection equation) neural network, i.e., PLPENN, and also by the corresponding numerical algorithm implemented on the computer. Furthermore, simulation and comparison based on a six-link planar redundant robot manipulator substantiate the effectiveness and accuracy of the proposed scheme. At last, a hardware experiment is conducted on a six-link physical robot manipulator system, which substantiates the physical realizability, operational stability, and safety of the proposed hybrid multi-objective scheme.

Redundancy resolution is a critical problem in the control of robotic manipulators. Recurrent neural networks (RNNs), as inherently parallel processing models for time-sequence processing, are potentially applicable for the motion control of manipulators. However, the development of neural models for high-accuracy and real-time control is a challenging problem. This paper identifies two limitations of the existing RNN solutions for manipulator control, i.e., position error accumulation and the convex restriction on the projection set, and overcomes them by proposing two modified neural network models. Our method allows nonconvex sets for projection operations, and control error does not accumulate over time in the presence of noise. Unlike most works in which RNNs are used to process time sequences, the proposed approach is model-based and training-free, which makes it possible to achieve fast tracking of reference signals with superior robustness and accuracy. Theoretical analysis reveals the global stability of a system under the control of the proposed neural networks. Simulation results confirm the effectiveness of the proposed control method in both the position regulation and tracking control of redundant PUMA 560 manipulators.

The solving of nonlinear equation systems (e.g., complex transcendental dispersion equation systems in waveguide systems) is a fundamental topic in science and engineering. Davidenko method has been used by electromagnetism researchers to solve time-invariant nonlinear equation systems (e.g., the aforementioned transcendental dispersion equation systems). Meanwhile, Zhang dynamics (ZD), which is a special class of neural dynamics, has been substantiated as an effective and accurate method for solving nonlinear equation systems, particularly time-varying nonlinear equation systems. In this paper, Davidenko method is compared with ZD in terms of efficiency and accuracy in solving time-invariant and time-varying nonlinear equation systems. Results reveal that ZD is a more competent approach than Davidenko method. Moreover, discrete-time ZD models, corresponding block diagrams, and circuit schematics are presented to facilitate the convenient implementation of ZD by researchers and engineers for solving time-invariant and time-varying nonlinear equation systems online. The theoretical analysis and results on Davidenko method, ZD, and discrete-time ZD models are also discussed in relation to solving time-varying nonlinear equation systems.

Matrix inversion often arises in the fields of science and engineering. Many models for matrix inversion usually assume that the solving process is free of noises or that the denoising has been conducted before the computation. However, time is precious for the real-time-varying matrix inversion in practice, and any preprocessing for noise reduction may consume extra time, possibly violating the requirement of real-time computation. Therefore, a new model for time-varying matrix inversion that is able to handle simultaneously the noises is urgently needed. In this paper, an integration-enhanced Zhang neural network (IEZNN) model is first proposed and investigated for real-time-varying matrix inversion. Then, the conventional ZNN model and the gradient neural network model are presented and employed for comparison. In addition, theoretical analyses show that the proposed IEZNN model has the global exponential convergence property. Moreover, in the presence of various kinds of noises, the proposed IEZNN model is proven to have an improved performance. That is, the proposed IEZNN model converges to the theoretical solution of the time-varying matrix inversion problem no matter how large the matrix-form constant noise is, and the residual errors of the proposed IEZNN model can be arbitrarily small for time-varying noises and random noises. Finally, three illustrative simulation examples, including an application to the inverse kinematic motion planning of a robot manipulator, are provided and analyzed to substantiate the efficacy and superiority of the proposed IEZNN model for real-time-varying matrix inversion.

In this paper, a new Taylor-type numerical differentiation formula is first presented to discretize the continuous-time Zhang neural network (ZNN), and obtain higher computational accuracy. Based on the Taylor-type formula, two Taylor-type discrete-time ZNN models (termed Taylor-type discrete-time ZNNK and Taylor-type discrete-time ZNNU models) are then proposed and discussed to perform online dynamic equality-constrained quadratic programming. For comparison, Euler-type discrete-time ZNN models (called Euler-type discrete-time ZNNK and Euler-type discrete-time ZNNU models) and Newton iteration, with interesting links being found, are also presented. It is proved herein that the steady-state residual errors of the proposed Taylor-type discrete-time ZNN models, Euler-type discrete-time ZNN models, and Newton iteration have the patterns of O(h(3)), O(h(2)), and O(h), respectively, with h denoting the sampling gap. Numerical experiments, including the application examples, are carried out, of which the results further substantiate the theoretical findings and the efficacy of Taylor-type discrete-time ZNN models. Finally, the comparisons with Taylor-type discrete-time derivative model and other Lagrange-type discrete-time ZNN models for dynamic equality-constrained quadratic programming substantiate the superiority of the proposed Taylor-type discrete-time ZNN models once again.

With the superiority of owning more degrees of freedom than ordinary robot manipulators, redundant robot manipulators have gotten much attention in recent years. In order to control the trajectory of the robot end-effector with a desired velocity, it is very popular to apply the inverse kinematics approaches, such as pseudo-inverse scheme. However, calculating the inverse of Jacobian matrix requires a lot of time. Thus base on gradient neural dynamics (GND), an inverse-free scheme is proposed at the joint-velocity level. The scheme is named G1 type as it uses GND once. In addition, two path tracking simulations based on five-link and six-link redundant robot manipulators illustrate the efficiency and the accuracy of the proposed scheme. What is more, the physical realizability of G1 type scheme is also verified by a physical experiment based on the six-link planar redundant robot manipulator hardware system.

In this paper, to remedy the joint-angle drift phenomenon for manipulation of two redundant robot arms, a novel scheme for simultaneous repetitive motion planning and control (SRMPC) at the joint-acceleration level is proposed, which consists of two subschemes. To do so, the performance index of each SRMPC subscheme is derived and designed by employing the gradient dynamics twice, of which a convergence theorem and its proof are presented. In addition, for improving the accuracy of the motion planning and control, position error, and velocity, error feedbacks are incorporated into the forward kinematics equation and analyzed via Zhang neural-dynamics method. Then the two subschemes are simultaneously reformulated as two quadratic programs (QPs), which are finally unified into one QP problem. Furthermore, a piecewise-linear projection equation-based neural network (PLPENN) is used to solve the unified QP problem, which can handle the strictly convex QP problem in an inverse-free manner. More importantly, via such a unified QP formulation and the corresponding PLPENN solver, the synchronism of two redundant robot arms is guaranteed. Finally, two given tasks are fulfilled by 2 three-link and 2 five-link planar robot arms, respectively. Computer-simulation results validate the efficacy and accuracy of the SRMPC scheme and the corresponding PLPENN solver for synchronous manipulation of two redundant robot arms.

In this brief, a discrete-time Zhang neural network (DTZNN) model is first proposed, developed, and investigated for online time-varying nonlinear optimization (OTVNO). Then, Newton iteration is shown to be derived from the proposed DTZNN model. In addition, to eliminate the explicit matrix-inversion operation, the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is introduced, which can effectively approximate the inverse of Hessian matrix. A DTZNN-BFGS model is thus proposed and investigated for OTVNO, which is the combination of the DTZNN model and the quasi-Newton BFGS method. In addition, theoretical analyses show that, with step-size h=1 and/or with zero initial error, the maximal residual error of the DTZNN model has an O(τ(2)) pattern, whereas the maximal residual error of the Newton iteration has an O(τ) pattern, with τ denoting the sampling gap. Besides, when h ≠ 1 and h ∈ (0,2) , the maximal steady-state residual error of the DTZNN model has an O(τ(2)) pattern. Finally, an illustrative numerical experiment and an application example to manipulator motion generation are provided and analyzed to substantiate the efficacy of the proposed DTZNN and DTZNN-BFGS models for OTVNO.

We propose a dual-arm cyclic-motion-generation (DACMG) scheme by a neural-dynamic method, which can remedy the joint-angle-drift phenomenon of a humanoid robot. In particular, according to a neural-dynamic design method, first, a cyclic-motion performance index is exploited and applied. This cyclic-motion performance index is then integrated into a quadratic programming (QP)-type scheme with time-varying constraints, called the time-varying-constrained DACMG (TVC-DACMG) scheme. The scheme includes the kinematic motion equations of two arms and the time-varying joint limits. The scheme can not only generate the cyclic motion of two arms for a humanoid robot but also control the arms to move to the desired position. In addition, the scheme considers the physical limit avoidance. To solve the QP problem, a recurrent neural network is presented and used to obtain the optimal solutions. Computer simulations and physical experiments demonstrate the effectiveness and the accuracy of such a TVC-DACMG scheme and the neural network solver.

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.

In this paper, a novel weighted-combination-of-components (WCC) method Is proposed for modeling and forecasting trend and seasonal time series, and such a method is based on decomposition model which regards the time series as the weighted combination of trend, seasonality and other components. Specifically, the Holt's two-parameter exponential smoothing (HTPES) method is improved (for short, the IHTPES method) to evaluate the trend with linearly declining increments; and the multiple sine functions decomposition (MSFD) method is developed to evaluate the seasonality. Then the weighted combination of the evaluations is obtained to estimate the global time series. Numerical experiment results substantiate the effectiveness and superiority of the proposed WCC method in terms of modeling and forecasting time series from the NN3 competition.

Zhang neural network (ZNN) is a novel class of recurrent neural network with superior solution ability and convergence performance. For real-time solution of Moore-Penrose pseudoinverses of time-varying matrices based on continuous-time recurrent neural network, this paper proposes three different ZNN models, each of which is derived from a specifically-chosen Zhang function (ZF). Theoretical analyses guarantee the global convergence of the three different ZNN models and their fast convergence rate. Besides, the proposed ZNN models show additional great advantages when used to deal with matrices with contrasting numbers of rows and columns. Computer simulations and experiments further verify the theoretical results, vividly demonstrating the effectiveness and efficiency of the proposed ZNN models.

This paper illustrates theoretical analysis and simulative verification on the performance of the linear-variatlonal inequality based primal-dual neural network (LVI-PDNN), which was designed originally for static quadratic programming (QP) problem solving but Is now applied to time-varying QP problem solving. It Is theoretically proved that the LVI-PDNN for solving the time-varying QP problem subject to equality, Inequality and bound constraints simultaneously could only approximately approach the time-varying theoretical solution, Instead of converging exactly. In other words, the steady-state error of the realtime solution can not decrease to zero. In order to better evaluate the time-varying situation, we Investigate the upper bound of such an error and the global exponential convergence rate for the LVI-PDNN approaching Its loose error bound. Computer simulations further substantiate the performance analysis of the LVI-PDNN exploited for real-time solution of the time-varying QP problem.

This paper proposes and investigates a weighted velocity and acceleration minimization scheme to prevent the occurrence of high joint velocity and joint acceleration caused by the minimum acceleration norm (MAN) scheme in redundant robot manipulators. The proposed scheme considers minimum kinetic energy (MKE) and MAN criterions via two weighting factors, thus guaranteeing the final joint velocity of motion to be near zero, which is acceptable for engineering applications. Joint physical constraints (i.e., joint angle limits, joint velocity limits, and joint acceleration limits) are incorporated in the formulation of the proposed scheme. The proposed scheme is reformulated as a quadratic program and then calculated by using a numerical algorithm based on linear variational inequality. Computer simulation results of a PUMA560 robot manipulator verify the efficacy and flexibility of the proposed scheme for redundancy resolution in robot manipulators. Experimental verifications conducted on a six-link planar robot manipulator demonstrate the effectiveness and physical realizability of the proposed scheme.

A mobile manipulator is a robotic device composed of a mobile platform and a stationary manipulator fixed to the platform. To achieve the repetitive motion control of mobile manipulators, the mobile platform and the manipulator have to realize the repetitive motion simultaneously. To do so, a novel quadratic performance index is, for the first time, designed and presented in this paper, of which the effectiveness is analyzed by following a neural dynamics method. Then, a repetitive motion scheme is proposed by combining the criterion, physical constraints, and integrated kinematical equations of mobile manipulators, which is further reformulated as a quadratic programming (QP) subject to equality and bound constraints. In addition, two important Bridge theorems are established to prove that such a QP can be converted equivalently into a linear variational inequality, and then equivalently into a piecewise-linear projection equation (PLPE). A real-time numerical algorithm based on PLPE is thus developed and applied for the online solution of the resultant QP. Two tracking-path tasks demonstrate the effectiveness and accuracy of the repetitive motion scheme. In addition, comparisons between the nonrepetitive and repetitive motion further validate the superiority and novelty of the proposed scheme.

In this paper, for online solution of time-varying linear matrix inequality (LMI), such an LMI is first converted to a time-varying matrix equation by introducing a time-varying matrix, of which each element is greater than or equal to zero. Then, by employing Zhang et al.'s neural dynamic method, a special recurrent neural network termed Zhang neural network (ZNN) is proposed and investigated for solving online the converted time-varying matrix equation as well as the time-varying LMI. Such a ZNN model showed in an explicit dynamics exploits the time-derivative information of time-varying coefficients. In addition, theoretical analysis and results of the proposed ZNN model are discussed and presented to show its excellent performance on solving the time-varying LMI. Computer simulation results further demonstrate the efficacy of the proposed ZNN model for online solution of the time-varying LMI and the converted time-varying matrix equation.

As a special class of recurrent neural network, Zhang neural network (ZNN) has been recently proposed since 2001 for solving various time-varying problems, and has shown high efficiency and excellent performance for solving the problems in the real domain. In this paper, to solve online the time-varying complex generalized inverse (in most cases, the pseudoinverse) problem in the complex domain, a new type of complex-valued ZNN is further proposed and investigated. The design of such a complex ZNN is based on a complex Zhang function (ZF) which is indefinite and quite different from the usual error function (specially, the scalar-valued energy function) in the studies of conventional algorithms. By introducing five different complex ZFs, five different complex ZNN models (termed complex ZNN-I, ZNN-II, ZNN-III, ZNN-IV, and ZNN-V models) are proposed, developed, and investigated for the online solution of the time-varying complex generalized inverse matrices. Theoretical results of convergence analysis are presented to show the desirable properties of complex ZNN models. In addition, we discover the link between the proposed complex ZNN models and the Getz-Marsden dynamic system in the complex domain. Computer-simulation results further demonstrate the effectiveness of complex ZNN models based on different complex ZFs for the time-varying complex generalized inverse matrices.

Zhang dynamics (ZD) and gradient dynamics (GD) are both effective methods for online problems solving. By combining ZD and GD methods, an innovative ZG (Zhang-gradient) control method is thus proposed and investigated in this paper, which is applied to ship course tracking for the first time. Firstly, for a constant parameter setting, we design a ZD-based controller to solve the tracking-control problem of a ship course system with no singularity appearing. Then, for a time-varying parameter setting, the ZG method is applied generally to solve the singularity-containing tracking-control problem of such a system. Simulation results further demonstrate and verify the feasibility and superiority of the unified ZG method in fulfilling the tracking-control task while conquering the singularity problem for the ship course system.

This paper proposes a novel multi-input Bernoulli-polynomial neuronet (MIBPN) on the basis of function approximation theory. The MIBPN is trained by a weights-and-structure-determination (WASD) algorithm with twice pruning (TP). The WASD algorithm can obtain the optimal weights and structure for the MIBPN, and overcome the weaknesses of conventional BP (back-propagation) neuronets such as slow training speed and local minima. With the TP technique, the neurons of less importance in the MIBPN are pruned for less computational complexity. Furthermore, this MIBPN can be extended to a multiple input multiple output Bernoulli-polynomial neuronet (MIMOBPN), which can be applied as an important tool for classification. Numerical experiment results show that the MIBPN has outstanding performance in data approximation and generalization. Besides, experiment results based on the real-world classification data-sets substantiate the high accuracy and strong robustness of the MIMOBPN equipped with the proposed WASD algorithm for classification. Finally, the twice-pruning aided WASD neuronet of Bernoulli-polynomial type in the forms of MIBPN and MIMOBPN is established, together with the effective extension to robust classification.

Since 2001, a novel type of recurrent neural network called Zhang neural network (ZNN) has been proposed, investigated, and exploited for solving online time-varying problems in a variety of scientific and engineering fields. In this paper, three discrete-time ZNN models are first proposed to solve the problem of time-varying quadratic minimization (TVQM). Such discrete-time ZNN models exploit methodologically the time derivatives of time-varying coefficients and the inverse of the time-varying coefficient matrix. To eliminate explicit matrix-inversion operation, the quasi-Newton BFGS method is introduced, which approximates effectively the inverse of the Hessian matrix; thus, three discrete-time ZNN models combined with the quasi-Newton BFGS method (named ZNN-BFGS) are proposed and investigated for TVQM. In addition, according to the criterion of whether the time-derivative information of time-varying coefficients is explicitly known/used or not, these proposed discrete-time models are classified into three categories: 1) models with time-derivative information known (i.e., ZNN-K and ZNN-BFGS-K models), 2) models with time-derivative information unknown (i.e., ZNN-U and ZNN-BFGS-U models), and 3) simplified models without using time-derivative information (i.e., ZNN-S and ZNN-BFGS-S models). The well-known gradient-based neural network is also developed to handle TVQM for comparison with the proposed ZNN and ZNN-BFGS models. Illustrative examples are provided and analyzed to substantiate the efficacy of these proposed models for TVQM.

In this paper, two simple-structure neural networks based on the error back-propagation (BP) algorithm (i.e., BP-type neural networks, BPNNs) are proposed, developed, and investigated for online generalized matrix inversion. Specifically, the BPNN-L and BPNN-R models are proposed and investigated for the left and right generalized matrix inversion, respectively. In addition, for the same problem-solving task, two discrete-time Hopfield-type neural networks (HNNs) are developed and investigated in this paper. Similar to the classification of the presented BPNN-L and BPNN-R models, the presented HNN-L and HNN-R models correspond to the left and right generalized matrix inversion, respectively. Comparing the BPNN weight-updating formula with the HNN state-transition equation for the specific (i.e., left or right) generalized matrix inversion, we show that such two derived learning-expressions turn out to be the same (in mathematics), although the BP and Hopfield-type neural networks are evidently different from each other a great deal, in terms of network architecture, physical meaning, and training patterns. Numerical results with different illustrative examples further demonstrate the efficacy of the presented BPNNs and HNNs for online generalized matrix inversion and, more importantly, their common natures of learning.

This paper proposes a new inequality-based criterion/constraint with its algorithmic and computational details for obstacle avoidance of redundant robot manipulators. By incorporating such a dynamically updated inequality constraint and the joint physical constraints (such as joint-angle limits and joint-velocity limits), a novel minimum-velocity-norm (MVN) scheme is presented and investigated for robotic redundancy resolution. The resultant obstacle-avoidance MVN scheme resolved at the joint-velocity level is further reformulated as a general quadratic program (QP). Two QP solvers, i.e., a simplified primal-dual neural network based on linear variational inequalities (LVI) and an LVI-based numerical algorithm, are developed and applied for online solution of the QP problem as well as the inequality-based obstacle-avoidance MVN scheme. Simulative results that are based on PA10 robot manipulator and a six-link planar robot manipulator in the presence of window-shaped and point obstacles demonstrate the efficacy and superiority of the proposed obstacle-avoidance MVN scheme. Moreover, experimental results of the proposed MVN scheme implemented on the practical six-link planar robot manipulator substantiate the physical realizability and effectiveness of such a scheme for obstacle avoidance of redundant robot manipulator.

In this paper, a cyclic-motion generation (CMG) scheme at the acceleration level is proposed to remedy the joint-angle drift phenomenon of redundant robot manipulators which are controlled at the joint-acceleration level or torque level. To achieve this, a cyclic-motion criterion at the joint-acceleration level is exploited. This criterion, together with the joint-angle limits, joint-velocity limits, and joint-acceleration limits, is considered into the scheme formulation. In addition, the neural-dynamic method of Zhang is employed to explain and analyze the effectiveness of the proposed criterion. Then, the scheme is reformulated as a quadratic program, which is solved by a primal-dual neural network. Furthermore, four tracking path simulations verify the effectiveness and accuracy of the proposed acceleration-level CMG scheme. Moreover, the comparisons between the proposed acceleration-level CMG scheme and the velocity-level scheme demonstrate that the former is safer and more applicable. The experiment on a physical robot system further verifies the physical realizability of the proposed acceleration-level CMG scheme.

By following Zhang design method, a new type of recurrent neural network [i.e., Zhang neural network (ZNN)] is presented, investigated, and analyzed for online solution of time-varying linear inequalities. Theoretical analysis is given on convergence properties of the proposed ZNN model. For comparative purposes, the conventional gradient neural network is developed and exploited for solving online time-varying linear inequalities as well. Computer simulation results further verify and demonstrate the efficacy, novelty, and superiority of such a ZNN model and its method for solving time-varying linear inequalities.

This paper presents a discrete competitive Hopfield neural network (HNN) (DCHNN) based on the estimation of distribution algorithm (EDA) for the maximum diversity problem. In order to overcome the local minimum problem of DCHNN, the idea of EDA is combined with DCHNN. Once the network is trapped in local minima, the perturbation based on EDA can generate a new starting point for DCHNN for further search. It is expected that the further search is guided to a promising area by the probability model. Thus, the proposed algorithm can escape from local minima and further search better results. The proposed algorithm is tested on 120 benchmark problems with the size ranging from 100 to 5000. Simulation results show that the proposed algorithm is better than the other improved DCHNN such as multistart DCHNN and DCHNN with random flips and is better than or competitive with metaheuristic algorithms such as tabu-search-based algorithms and greedy randomized adaptive search procedure algorithms.

Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm. A quantum-behaved particle swarm optimization (QPSO) is also proposed by combining the classical PSO philosophy and quantum mechanics. These algorithms have been very successful in solving the global continuous optimization, but their applications to combinatorial optimization have been rather limited. Estimation of distribution algorithm (EDA) samples new solutions from a probability model which characterizes the distribution of promising solutions. This paper proposes a novel discrete QPSO based on EDA for the combinatorial optimization problem. The proposed algorithm combines global statistical information extracted by EDA with local information obtained by discrete QPSO to create promising solutions. To demonstrate the performance of the proposed algorithm, experiments are carried out on the unconstrained binary quadratic programming problem which numerous hard combinatorial optimization problems can be formulated as. The results show that the discrete QPSO based on EDA have superior performance to other algorithms.

Based on polynomial interpolation and approximation theory, a special feed-forward neural network using power activation functions is constructed in this paper. The neural model employs a three-layer structure with the hidden-layer neurons activated by a group of order-increasing power functions (while other layers’ neurons use linear activation functions). In addition, the weights-updating formula for such a neural network could be derived from the standard BP training method. A pseudoinverse-based method (or termed, weights-direct-determination / one-step-weights-determination method) is then established to determine immediately the neural-network weights without lengthy iterative BP-training. It is shown that such a power-activated feed-forward neural network could perform effectively and efficiently for nonlinear system identification. Computer-simulation results further substantiate the benefits of its weights-direct-determination method.

Recently, a special kind of recurrent neural networks has been proposed by Zhang et al for online solution of Sylvester equation with time-varying coefficients. Their neural dynamics are elegantly introduced by defining a matrix-valued error function rather than the usual scalar-valued norm-based error function, so that the computational error can vanish to zero globally and exponentially. The resultant Zhang neural networks (ZNN), perform much better on solving time-varying problems in comparison with gradient-based neural networks. MATLAB Simulink is a software package for model-based design and multi-domain simulation of dynamic systems. By using click-and-drag mouse operations, it is much easier to model and simulate complex neural systems as compared to MATLAB coding. This paper investigates the MATLAB Simulink modeling and simulative verification of ZNN models for timevarying Sylvester equation solving. Computer-simulation results substantiate the ZNN efficacy on solving online the time-varying problems (specifically, the time-varying Sylvester equation).

To obtain the inverses of time-varying matrices in real time, a special kind of recurrent neural networks has recently been proposed by Zhang et al. It is proved that such a Zhang neural network (ZNN) could globally exponentially converge to the exact inverse of a given time-varying matrix. To find out the effect of time-derivative term on global convergence as well as for easier hardware-implementation purposes, the ZNN model without exploiting time-derivative information is investigated in this paper for inverting online matrices. Theoretical results of both constant matrix inversion case and time-varying matrix inversion case are presented for comparative and illustrative purposes. In order to substantiate the presented theoretical results, computer-simulation results are shown, which demonstrate the importance of time derivative term of given matrices on the exact convergence of ZNN model to time-varying matrix inverses.

This paper presents a simplified primal-dual neural network based on linear variational inequalities (LVI) for online repetitive motion planning of PA10 robot manipulator. To do this, a drift-free criterion is exploited in the form of a quadratic function. In addition, the repetitive-motion-planning scheme could incorporate the joint limits and joint velocity limits simultaneously. Such a scheme is finally reformulated as a time-varying quadratic program (QP). As a QP real-time solver, the simplified LVI-based primal-dual neural network (LVI-PDNN) is designed based on the QP-LVI conversion and Karush-Kuhn-Tucker (KKT) conditions. It has a simple piecewise-linear dynamics and could globally exponentially converge to the optimal solution of strictly-convex quadratic-programs. The simplified LVI-PDNN model is simulated based on PA10 robot arm, and simulation results show the effective remedy of the joint angle drift problem of PA10 robot.

Following the idea of using first-order time derivatives, this paper presents a general recurrent neural network (RNN) model for online inversion of time-varying matrices. Different kinds of activation functions are investigated to guarantee the global exponential convergence of the neural model to the exact inverse of a given time-varying matrix. The robustness of the proposed neural model is also studied with respect to different activation functions and various implementation errors. Simulation results, including the application to kinematic control of redundant manipulators, substantiate the theoretical analysis and demonstrate the efficacy of the neural model on time-varying matrix inversion, especially when using a power-sigmoid activation function.

In this paper, for joint torque optimization of redundant manipulators subject to physical constraints, we show that velocity-level and acceleration-level redundancy-resolution schemes both can be formulated as a quadratic programming (QP) problem subject to equality and inequality/bound constraints. To solve this QP problem online, a primal-dual dynamical system solver is further presented based on linear variational inequalities. Compared to previous researches, the presented QP-solver has simple piecewise-linear dynamics, does not entail real-time matrix inversion, and could also provide joint-acceleration information for manipulator torque control in the velocity-level redundancy-resolution schemes. The proposed QP-based dynamical system approach is simulated based on the PUMA560 robot arm with efficiency and effectiveness demonstrated.

One important issue in the motion planning and control of kinematically redundant manipulators is the obstacle avoidance. In this paper, a recurrent neural network is developed and applied for kinematic control of redundant manipulators with obstacle avoidance capability. An improved problem formulation is proposed in the sense that the collision-avoidance requirement is represented by dynamically-updated inequality constraints. In addition, physical constraints such as joint physical limits are also incorporated directly into the formulation. Based on the improved problem formulation, a dual neural network is developed for the online solution to collision-free inverse kinematics problem. The neural network is simulated for motion control of the PA10 robot arm in the presence of point and window-shaped obstacle.

One important issue in motion planning and kinematic control of redundant manipulators is the real-time obstacle avoidance. Following the previous researches, a new problem formulation has been proposed in the sense that the collision avoidance scheme is described by dynamically-updated inequality constraints, and that physical constraints such as joint limits are also incorporated in the formulation. For real-time computation, the dual neural network is applied for the online solution of obstacle-avoidance inverse-kinematic control problem, and then simulated based on the PA10 robot manipulator in the presence of obstacles.

In this paper, a recurrent neural network called the dual neural network is proposed for online redundancy resolution of kinematically redundant manipulators. Physical constraints such as joint limits and joint velocity limits, together with the drift-free criterion as a secondary task, are incorporated into the problem formulation of redundancy resolution. Compared to other recurrent neural networks, the dual neural network is piecewise linear and has much simpler architecture with only one layer of neurons. The dual neural network is shown to be globally (exponentially) convergent to optimal solutions. The dual neural network is simulated to control the PA10 robot manipulator with effectiveness demonstrated.

Presents a recurrent neural network for solving the Sylvester equation with time-varying coefficient matrices. The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation. Theoretical results of convergence and sensitivity analysis are presented to show the desirable properties of the recurrent neural network. Simulation results of time-varying matrix inversion and online nonlinear output regulation via pole assignment for the ball and beam system and the inverted pendulum on a cart system are also included to demonstrate the effectiveness and performance of the proposed neural network.

Global exponential stability is the most desirable stability property of recurrent neural networks. The paper presents new results for recurrent neural networks applied to online computation of feedback gains of linear time-invariant multivariable systems via pole assignment. The theoretical analysis focuses on the global exponential stability, convergence rates, and selection of design parameters. The theoretical results are further substantiated by simulation results conducted for synthesizing linear feedback control systems with different specifications and design requirements.

A dual neural network is presented for the bi-criteria kinematic control of redundant manipulators. To diminish the discontinuity of minimum infinity-norm solutions, the kinematic-control problem is formulated in the bi-criteria of the infinity and Euclidean norms. Physical constraints such as joint limits and joint velocity limits are also incorporated simultaneously into the proposed kinematic control scheme. The single-layer dual neural network model with a simple structure is developed for bi-criteria redundant resolution of redundant manipulators subject to robot physical constraints. The dual neural network is shown to be globally convergent to optimal solutions in the bi-criteria sense, and is demonstrated to be effective in controlling the PA10 robot manipulator.

A dual neural network is presented for the real-time joint torque optimization of kinematically redundant manipulators, which corresponds to global kinetic energy minimization of robot mechanisms. Compared to other computational strategies on inverse kinematics, the dual network is developed at the acceleration level to resolve redundancy of limited-joint-range manipulators. The dual network has a simple architecture with only one layer of neurons and is proved to be globally exponentially convergent to optimal solutions. The dual neural network is simulated with the PUMA 560 robot arm to demonstrate effectiveness.