It is very challenging to precisely control a redundant manipulator with an unknown model during the end-effector tracking task. Adata-driven approach offers a promising solution for manipulator control under such conditions. In this article, two data-driven discrete neurodynamics (DDDN) schemes are proposed for redundant manipulator tracking control. First, utilizing discrete neurodynamics (DN) principles, the DDDN-1 scheme with an adaptive Jacobian matrix is developed. Subsequently, the DDDN-2 scheme is further presented, which eliminates the need for the Jacobian matrix inversion operation. Detailed theoretical analyses verify the effectiveness of DDDN-1 and DDDN-2 schemes. Additionally, detailed comparisons with existing schemes have been provided. Finally, simulative and physical experiments conducted using the UR5 manipulator validate the theoretical analyses, demonstrating the effectiveness and superiority of DDDN-1 and DDDN-2 schemes.

Repetitive motion control of redundant manipulators typically requires precise kinematic models to construct Jacobian matrices. However, model-based approaches are inherently limited when manipulator parameters are unavailable or only partially known. This paper introduces a novel data-driven discrete zeroing neurodynamics (DDZN) model for repetitive motion control. Specifically, a Jacobian matrix estimation method based on data-driven technology is proposed, which eliminates the need for prior models by leveraging historical input-output information. By integrating the Jacobian matrix estimation with a discrete zeroing neurodynamics (DZN) model, the approach enables simultaneous trajectory tracking and repeatable configuration recovery without relying on structural parameters. Theoretical analysis verifies the performance of DDZN model under noise environment. Furthermore, abundant experiment results validate its reliability and superior performance compared with various models.

Precise motion tracking control with unknown structural knowledge and noise disturbance for redundant robots remains a critical and unresolved challenge. This article proposes a novel data-driven fuzzy discrete recurrent neural network (D2-FDRNN) model to address two fundamental limitations of existing models: dependency on known kinematic knowledge and fixed sampling schemes. First, a Jacobian pseudo-inverse estimator is developed to reconstruct the manipulator’s necessary kinematic knowledge using input and output data, eliminating the need for explicit Jacobian inversion. Second, a fuzzy logic-based adaptive sampling strategy dynamically adjusts the step size to balance computational efficiency and tracking precision. In addition, a Kalman filter algorithm is applied to reduce the impact of noise. Rigorous proofs confirm the model’s exponential convergence and noise immunity. To validate the proposed D2-FDRNN model, simulations and physical experiments are carried out. The source code is available at https://github.com/YingluckZ/DD-FDRNN.git.

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.

Redundant manipulators have shown great potential in the application of robots. These manipulators possess additional degrees of freedom beyond what is essential for completing specific tasks, presenting an opportunity to optimize energy usage. However, the existence of additional degrees of freedom also brings control challenges. Due to the ability to address problems of time-varying tracking, zeroing neural network (ZNN) is gradually widely used in the control of redundant manipulators. Discrete models are often used in engineering, and the sampling gap selected during discretization is an important factor that affects the tracking precision. Large sampling gaps require less computational consumption but yield lower tracking precision, whereas small sampling gaps result in higher precision but at a greater computational cost. In this article, a minimum energy consumption scheme (MECS) for the time-varying tracking control task of redundant manipulators is presented first. By applying the ZNN design formula, the continuous ZNN (CZNN) model is established to solve the MECS. Subsequently, Euler discretization formula is utilized to transform the CZNN model into its discrete form, known as the discrete ZNN (DZNN) model. Then a dual-input–single-output fuzzy control system is designed to obtain suitable sampling gaps. The fuzzy-control-aided ZNN (FCAZNN) model enables redundant manipulators to track desired paths with the expected precision. Finally, a series of experiments are carried out in this article to demonstrate the advantages of FCAZNN model, including both computer simulations and physical experiments.

The robot industry is developing rapidly, and how to control the redundant manipulators precisely and effectively has become a new hot topic in industry’s development. In recent years, many scholars in the industry have also proposed various control methods. However, most of these methods are proposed assuming that the Jacobian matrix is known. Actually, in practical applications, the detailed information of Jacobian matrix is often not precisely known. Therefore, this article develops a novel data-driven recurrent neural network (RNN) model that can update the Jacobian matrix and joint angles. By defining two dynamic error functions, two RNN designed formulas are used to obtain a continuous RNN (CRNN) model. Subsequently, the CRNN model is discretized by using Euler forward formula, and a discrete RNN (DRNN) model is generated. Then, a classic sliding mode control (SMC) algorithm is introduced, and DRNN-SMC model is further proposed. Moreover, the corresponding rigorous mathematical derivation and proof are carried out. In addition, simulation tests are carried out by using the Kinova Gen2 manipulator, comparing the DRNN model and PD controller, as well as the DRNN-SMC model and DRNN model, validating the precision of the DRNN-SMC model. Additionally, practical experiments using the Kinova Gen3 manipulator are performed to showcase the applicability and versatility of the DRNN-SMC model.

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.

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.

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.

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.

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 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.

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.

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 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.

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.