A new noise-tolerant and predefined-time ZNN model for time-dependent matrix inversion

Inverse-free zeroing neural network for time-variant nonlinear optimization with manipulator applications

In this paper, the problem of time-variant optimization subject to nonlinear equation constraint is studied. To solve the challenging problem, methods based on the neural networks, such as zeroing neural network and gradient neural network, are commonly adopted due to their performance on handling nonlinear problems. However, the traditional zeroing neural network algorithm requires computing the matrix inverse during the solving process, which is a complicated and time-consuming operation. Although the gradient neural network algorithm does not require computing the matrix inverse, its accuracy is not high enough. Therefore, a novel inverse-free zeroing neural network algorithm without matrix inverse is proposed in this paper. The proposed algorithm not only avoids the matrix inverse, but also avoids matrix multiplication, greatly reducing the computational complexity. In addition, detailed theoretical analyses of the convergence performance of the proposed algorithm is provided to guarantee its excellent capability in solving time-variant optimization problems. Numerical simulations and comparative experiments with traditional zeroing neural network and gradient neural network algorithms substantiate the accuracy and superiority of the novel inverse-free zeroing neural network algorithm. To further validate the performance of the novel inverse-free zeroing neural network algorithm in practical applications, path tracking tasks of three manipulators (i.e., Universal Robot 5, Franka Emika Panda, and Kinova JACO2 manipulators) are conducted, and the results verify the applicability of the proposed algorithm.

A novel fixed-time error-monitoring neural network for solving dynamic quaternion-valued Sylvester equations

This paper addresses the dynamic quaternion-valued Sylvester equation (DQSE) using the quaternion real representation and the neural network method. To transform the Sylvester equation in the quaternion field into an equivalent equation in the real field, three different real representation modes for the quaternion are adopted by considering the non-commutativity of quaternion multiplication. Based on the equivalent Sylvester equation in the real field, a novel recurrent neural network model with an integral design formula is proposed to solve the DQSE. The proposed model, referred to as the fixed-time error-monitoring neural network (FTEMNN), achieves fixed-time convergence through the action of a state-of-the-art nonlinear activation function. The fixed-time convergence of the FTEMNN model is theoretically analyzed. Two examples are presented to verify the performance of the FTEMNN model with a specific focus on fixed-time convergence. Furthermore, the chattering phenomenon of the FTEMNN model is discussed, and a saturation function scheme is designed. Finally, the practical value of the FTEMNN model is demonstrated through its application to image fusion denoising.

Design and Analysis of a Self-Adaptive Zeroing Neural Network for Solving Time-Varying Quadratic Programming

In order to solve the time-varying quadratic programming (TVQP) problem more effectively, a new self-adaptive zeroing neural network (ZNN) is designed and analyzed in this article by using the Takagi-Sugeno fuzzy logic system (TSFLS) and thus called the Takagi-Sugeno (T-S) fuzzy ZNN (TSFZNN). Specifically, a multiple-input-single-output TSFLS is designed to generate a self-adaptive convergence factor to construct the TSFZNN model. In order to obtain finite- or predefined-time convergence, four novel activation functions (AFs) [namely, power-bi-sign AF (PBSAF), tanh-bi-sign AF (TBSAF), exp-bi-sign AF (EBSAF), and sinh-bi-sign AF (SBSAF)] are developed and applied in the TSFZNN model for solving the TVQP problem. Both theoretical proofs and experimental simulations show that the TSFZNN model using PBSAF or TBSAF has the property of converging in a finite time, and the TSFZNN model using EBSAF or SBSAF has the property of converging in a predefined time, which have superior convergence performance compared to the traditional ZNN model.

Novel adaptive zeroing neural dynamics schemes for temporally-varying linear equation handling applied to arm path following and target motion positioning

While the handling for temporally-varying linear equation (TVLE) has received extensive attention, most methods focused on trading off the conflict between computational precision and convergence rate. Different from previous studies, this paper proposes two complete adaptive zeroing neural dynamics (ZND) schemes, including a novel adaptive continuous ZND (ACZND) model, two general variable time discretization techniques, and two resultant adaptive discrete ZND (ADZND) algorithms, to essentially eliminate the conflict. Specifically, an error-related varying-parameter ACZND model with global and exponential convergence is first designed and proposed. To further adapt to the digital hardware, two novel variable time discretization techniques are proposed to discretize the ACZND model into two ADZND algorithms. The convergence properties with respect to the convergence rate and precision of ADZND algorithms are proved via rigorous mathematical analyses. By comparing with the traditional discrete ZND (TDZND) algorithms, the superiority of ADZND algorithms in convergence rate and computational precision is shown theoretically and experimentally. Finally, simulative experiments, including numerical experiments on a specific TVLE solving as well as four application experiments on arm path following and target motion positioning are successfully conducted to substantiate the efficacy, superiority, and practicability of ADZND algorithms.

Neurodynamic optimization approaches with finite/fixed-time convergence for absolute value equations

This paper proposes three novel accelerated inverse-free neurodynamic approaches to solve absolute value equations (AVEs). The first two are finite-time converging approaches and the third one is a fixed-time converging approach. It is shown that the proposed first two neurodynamic approaches converge to the solution of the concerned AVEs in a finite-time while, under some mild conditions, the third one converges to the solution in a fixed-time. It is also shown that the settling time for the proposed fixed-time converging approach has an uniform upper bound for all initial conditions, while the settling times for the proposed finite-time converging approaches are dependent on initial conditions. The proposed neurodynamic approaches have the advantage that they are all robust against bounded vanishing perturbations. The theoretical results are validated by means of a numerical example and an application in boundary value problems.

Model-free kinematic control of redundant manipulators with simultaneous joint-physical-limit and joint-angular-drift handling

Model-free tracking control methods have been developed for redundant robotic manipulators with unknown kinematic or dynamic models. However, most existing works have not considered the problems of joint physical limits and joint angular drift. Joint angular drift refers to the phenomenon that the joint angles of a redundant manipulator are different from their initial state when the end-effector of the manipulator returns to its initial position. Therefore, this article proposes a model-free scheme for the tracking control of redundant manipulators as well as the avoidance of joint physical limits and joint angular drift. The proposed method aims to overcome these challenges by formulating the inverse kinematics problem as a constrained optimization problem, which incorporates the avoidance of joint angular drift into the optimization objective and takes the joint physical limits as a constraint. At the same time, an online Jacobian estimator is designed to observe the state of the manipulator. Specifically, it can estimate the Jacobian matrix of a redundant manipulator in real-time during the operation by fully exploiting the sensory data, without knowing the analytic robot model. Then, the optimization problem is integrated with the Jacobian estimator and solved by a discrete-time algorithm. Theoretical analysis is conducted to prove the stability and convergence of the proposed method. Moreover, the efficacy, practicability and superiority of the proposed method are supported by simulations and experiments based on different redundant manipulators.

A direct discretization recurrent neurodynamics method for time-variant nonlinear optimization with redundant robot manipulators

Discrete time-variant nonlinear optimization (DTVNO) problems are commonly encountered in various scientific researches and engineering application fields. Nowadays, many discrete-time recurrent neurodynamics (DTRN) methods have been proposed for solving the DTVNO problems. However, these traditional DTRN methods currently employ an indirect technical route in which the discrete-time derivation process requires to interconvert with continuous-time derivation process. In order to break through this traditional research method, we develop a novel DTRN method based on the inspiring direct discrete technique for solving the DTVNO problem more concisely and efficiently. To be specific, firstly, considering that the DTVNO problem emerging in the discrete-time tracing control of robot manipulator, we further abstract and summarize the mathematical definition of DTVNO problem, and then we define the corresponding error function. Secondly, based on the second-order Taylor expansion, we can directly obtain the DTRN method for solving the DTVNO problem, which no longer requires the derivation process in the continuous-time environment. Whereafter, such a DTRN method is theoretically analyzed and its convergence is demonstrated. Furthermore, numerical experiments confirm the effectiveness and superiority of the DTRN method. In addition, the application experiments of the robot manipulators are presented to further demonstrate the superior performance of the DTRN method.

Explicit Linear Left-and-Right 5-Step Formulas With Zeroing Neural Network for Time-Varying Applications

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.

A Barrier Varying-Parameter Dynamic Learning Network for Solving Time-Varying Quadratic Programming Problems With Multiple Constraints

Many scientific research and engineering problems can be converted to time-varying quadratic programming (TVQP) problems with constraints. Thus, TVQP problem solving plays an important role in practical applications. Many existing neural networks, such as the gradient neural network (GNN) or zeroing neural network (ZNN), were designed to solve TVQP problems, but the convergent rate is limited. The recent varying-parameter convergent-differential neural network (VP-CDNN) can accelerate the convergent rate, but it can only solve the equality-constrained problem. To remedy this deficiency, a novel barrier varying-parameter dynamic learning network (BVDLN) is proposed and designed, which can solve the equality-, inequality-, and bound-constrained problem. Specifically, the constrained TVQP problem is first converted into a matrix equation. Second, based on the modified Karush-Kuhn-Tucker (KKT) conditions and varying-parameter neural dynamic design method, the BVDLN model is conducted. The superiorities of the proposed BVDLN model can solve multiple-constrained TVQP problems, and the convergent rate can achieve superexponentially convergence. Comparative simulative experiments verify that the proposed BVDLN is more effective and more accurate. Finally, the proposed BVDLN is applied to solve a robot motion planning problems, which verifies the applicability of the proposed model.

Solving Complex-Valued Time-Varying Linear Matrix Equations via QR Decomposition With Applications to Robotic Motion Tracking and on Angle-of-Arrival Localization

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.

A Variable-Parameter Noise-Tolerant Zeroing Neural Network for Time-Variant Matrix Inversion With Guaranteed Robustness

Matrix inversion frequently occurs in the fields of science, engineering, and related fields. Numerous matrix inversion schemes are often based on the premise that the solution procedure is ideal and noise-free. However, external interference is generally ubiquitous and unavoidable in practice. Therefore, an integrated-enhanced zeroing neural network (IEZNN) model has been proposed to handle the time-variant matrix inversion issue interfered with by noise. However, the IEZNN model can only deal with small time-variant noise interference. With slightly larger noise interference, the IEZNN model may not converge to the theoretical solution exactly. Therefore, a variable-parameter noise-tolerant zeroing neural network (VPNTZNN) model is proposed to overcome shortcomings and improve the inadequacy. Moreover, the excellent convergence and robustness of the VPNTZNN model are rigorously analyzed and proven. Finally, compared with the original zeroing neural network (OZNN) model and the IEZNN model for matrix inversion, numerical simulations and a practical application reveal that the proposed VPNTZNN model has the best robust property under the same external noise interference.

Design and Analysis of Anti-Noise Parameter-Variable Zeroing Neural Network for Dynamic Complex Matrix Inversion and Manipulator Trajectory Tracking

Dynamic complex matrix inversion (DCMI) problems frequently arise in the territories of mathematics and engineering, and various recurrent neural network (RNN) models have been reported to effectively find the solutions of the DCMI problems. However, most of the reported works concentrated on solving DCMI problems in ideal no noise environment, and the inevitable noises in reality are not considered. To enhance the robustness of the existing models, an anti-noise parameter-variable zeroing neural network (ANPVZNN) is proposed by introducing a novel activation function (NAF). Both of mathematical analysis and numerical simulation results demonstrate that the proposed ANPVZNN model possesses fixed-time convergence and robustness for solving DCMI problems. Besides, a successful ANPVZNN-based manipulator trajectory tracking example further verifies its robustness and effectiveness in practical applications.

Recurrent Neural Dynamics Models for Perturbed Nonstationary Quadratic Programs: A Control-Theoretical Perspective

Recent decades have witnessed a trend that control-theoretical techniques are widely leveraged in various areas, e.g., design and analysis of computational models. Computational methods can be modeled as a controller and searching the equilibrium point of a dynamical system is identical to solving an algebraic equation. Thus, absorbing mature technologies in control theory and integrating it with neural dynamics models can lead to new achievements. This work makes progress along this direction by applying control-theoretical techniques to construct new recurrent neural dynamics for manipulating a perturbed nonstationary quadratic program (QP) with time-varying parameters considered. Specifically, to break the limitations of existing continuous-time models in handling nonstationary problems, a discrete recurrent neural dynamics model is proposed to robustly deal with noise. This work shows how iterative computational methods for solving nonstationary QP can be revisited, designed, and analyzed in a control framework. A modified Newton iteration model and an improved gradient-based neural dynamics are established by referring to the superior structural technology of the presented recurrent neural dynamics, where the chief breakthrough is their excellent convergence and robustness over the traditional models. Numerical experiments are conducted to show the eminence of the proposed models in solving perturbed nonstationary QP.

Inverse kinematics of redundant manipulators with guaranteed performance

In this paper, the inverse kinematics (IK) of redundant manipulators is presented and studied, where the performance of end-effector path planning is guaranteed. A new Jacobian pseudoinverse (JP)-based IK method is proposed and studied using a typical numerical difference rule to discretize the existing IK method based on JP. The proposed method is depicted in a discrete-time form and is theoretically proven to exhibit great performance in the IK of redundant manipulators. A discrete-time repetitive path planning (DTRPP) scheme and a discrete-time obstacle avoidance (DTOA) scheme are developed for redundant manipulators using the proposed method. Comparative simulations are conducted on a universal robot manipulator and a PA10 robot manipulator to validate the effectiveness and superior performance of the DTRPP scheme, the DTOA scheme, and the proposed JP-based IK method.

Recurrent neural network with noise rejection for cyclic motion generation of robotic manipulators

Recurrent neural network (RNN), as a kind of neural network with outstanding computing capability, improvability, and hardware realizability, has been widely used in various fields, especially in robotics. In this paper, an RNN with noise rejection is deliberately constructed to remedy the issue of joint-angle drift frequently occurring during the cyclic motion generation (CMG) of a manipulator in a noisy environment. Different from general RNNs, the proposed RNN possesses inherent noise immunity, especially for time-varying polynomial noises. Besides, proofs on the convergence of the proposed RNN in the absence and presence of noises are given. Furthermore, we carry out simulations on manipulators PUMA 560 and UR5 to demonstrate the reliability of the proposed RNN in remedying joint-angle drift, and comparison simulations under different noisy conditions further verify its superiority. In addition, experiments are conducted on manipulator FRANKA Panda to elucidate the realizability of the proposed RNN.

Design and Comprehensive Analysis of a Noise-Tolerant ZNN Model With Limited-Time Convergence for Time-Dependent Nonlinear Minimization

Zeroing neural network (ZNN) is a powerful tool to address the mathematical and optimization problems broadly arisen in the science and engineering areas. The convergence and robustness are always co-pursued in ZNN. However, there exists no related work on the ZNN for time-dependent nonlinear minimization that achieves simultaneously limited-time convergence and inherently noise suppression. In this article, for the purpose of satisfying such two requirements, a limited-time robust neural network (LTRNN) is devised and presented to solve time-dependent nonlinear minimization under various external disturbances. Different from the previous ZNN model for this problem either with limited-time convergence or with noise suppression, the proposed LTRNN model simultaneously possesses such two characteristics. Besides, rigorous theoretical analyses are given to prove the superior performance of the LTRNN model when adopted to solve time-dependent nonlinear minimization under external disturbances. Comparative results also substantiate the effectiveness and advantages of LTRNN via solving a time-dependent nonlinear minimization problem.

Model-free motion control of continuum robots based on a zeroing neurodynamic approach

As a result of inherent flexibility and structural compliance, continuum robots have great potential in practical applications and are attracting more and more attentions. However, these characteristics make it difficult to acquire the accurate kinematics of continuum robots due to uncertainties, deformation and external loads. This paper introduces a method based on a zeroing neurodynamic approach to solve the trajectory tracking problem of continuum robots. The proposed method can achieve the control of a bellows-driven continuum robot just relying on the actuator input and sensory output information, without knowing any information of the kinematic model. This approach reduces the computational load and can guarantee the real time control. The convergence, stability, and robustness of the proposed approach are proved by theoretical analyses. The effectiveness of the proposed method is verified by simulation studies including tracking performance, comparisons with other three methods, and robustness tests.

A robust zeroing neural network for solving dynamic nonlinear equations and its application to kinematic control of mobile manipulator

Nonlinear phenomena are often encountered in various practical systems, and most of the nonlinear problems in science and engineering can be simply described by nonlinear equation, effectively solving nonlinear equation (NE) has aroused great interests of the academic and industrial communities. In this paper, a robust zeroing neural network (RZNN) activated by a new power versatile activation function (PVAF) is proposed and analyzed for finding the solutions of dynamic nonlinear equations (DNE) within fixed time in noise polluted environment. As compared with the previous ZNN model activated by other commonly used activation functions (AF), the main improvement of the presented RZNN model is the fixed-time convergence even in the presence of noises. In addition, the convergence time of the proposed RZNN model is irrelevant to its initial states, and it can be computed directly. Both the rigorous mathematical analysis and numerical simulation results are provided for the verification of the effectiveness and robustness of the proposed RZNN model. Moreover, a successful robotic manipulator path tracking example in noise polluted environment further demonstrates the practical application prospects of the proposed RZNN models.