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A strictly predefined-time convergent and anti-noise fractional-order zeroing neural network for solving time-variant quadratic programming in kinematic robot control

Yi Yang, Xiao Li, Xuchen Wang, Mei Liu, Junwei Yin, Weibing Li, Richard M. Voyles, Xin Ma

TL;DR

The paper addresses TVQP in robotic kinematic control by introducing the SPTC-AN-FOZNN, a strictly predefined-time convergent and anti-noise fractional-order ZNN that employs a conformable fractional derivative and a novel activation to achieve order-independent, robust convergence. Theoretical results prove strictly predefined-time convergence under bounded noise, using Lyapunov analysis with time-varying gains $\gamma(t)=\gamma t^{\alpha-1}$ and a piecewise activation ensuring timely settling at $t_c$. Numerical comparisons against five recent RNNs show superior convergence precision and noise immunity, while simulations and real-robot experiments (Panda and Flexiv Rizon) validate accurate, energy-efficient kinematic tracking under disturbances. These findings highlight a practical, hardware-friendly approach to robust, time-constrained TVQP solving with potential for energy-efficient control architectures in robotics.

Abstract

This paper proposes a strictly predefined-time convergent and anti-noise fractional-order zeroing neural network (SPTC-AN-FOZNN) model, meticulously designed for addressing time-variant quadratic programming (TVQP) problems. This model marks the first variable-gain ZNN to collectively manifest strictly predefined-time convergence and noise resilience, specifically tailored for kinematic motion control of robots. The SPTC-AN-FOZNN advances traditional ZNNs by incorporating a conformable fractional derivative in accordance with the Leibniz rule, a compliance not commonly achieved by other fractional derivative definitions. It also features a novel activation function designed to ensure favorable convergence independent of the model's order. When compared to five recently published recurrent neural networks (RNNs), the SPTC-AN-FOZNN, configured with $0<α\leq 1$, exhibits superior positional accuracy and robustness against additive noises for TVQP applications. Extensive empirical evaluations, including simulations with two types of robotic manipulators and experiments with a Flexiv Rizon robot, have validated the SPTC-AN-FOZNN's effectiveness in precise tracking and computational efficiency, establishing its utility for robust kinematic control.

A strictly predefined-time convergent and anti-noise fractional-order zeroing neural network for solving time-variant quadratic programming in kinematic robot control

TL;DR

The paper addresses TVQP in robotic kinematic control by introducing the SPTC-AN-FOZNN, a strictly predefined-time convergent and anti-noise fractional-order ZNN that employs a conformable fractional derivative and a novel activation to achieve order-independent, robust convergence. Theoretical results prove strictly predefined-time convergence under bounded noise, using Lyapunov analysis with time-varying gains and a piecewise activation ensuring timely settling at . Numerical comparisons against five recent RNNs show superior convergence precision and noise immunity, while simulations and real-robot experiments (Panda and Flexiv Rizon) validate accurate, energy-efficient kinematic tracking under disturbances. These findings highlight a practical, hardware-friendly approach to robust, time-constrained TVQP solving with potential for energy-efficient control architectures in robotics.

Abstract

This paper proposes a strictly predefined-time convergent and anti-noise fractional-order zeroing neural network (SPTC-AN-FOZNN) model, meticulously designed for addressing time-variant quadratic programming (TVQP) problems. This model marks the first variable-gain ZNN to collectively manifest strictly predefined-time convergence and noise resilience, specifically tailored for kinematic motion control of robots. The SPTC-AN-FOZNN advances traditional ZNNs by incorporating a conformable fractional derivative in accordance with the Leibniz rule, a compliance not commonly achieved by other fractional derivative definitions. It also features a novel activation function designed to ensure favorable convergence independent of the model's order. When compared to five recently published recurrent neural networks (RNNs), the SPTC-AN-FOZNN, configured with , exhibits superior positional accuracy and robustness against additive noises for TVQP applications. Extensive empirical evaluations, including simulations with two types of robotic manipulators and experiments with a Flexiv Rizon robot, have validated the SPTC-AN-FOZNN's effectiveness in precise tracking and computational efficiency, establishing its utility for robust kinematic control.

Paper Structure

This paper contains 14 sections, 4 theorems, 31 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary theory
  3. Problem formulation and preliminaries
  4. Traditional GNN and ZNN models
  5. GNN model
  6. ZNN model
  7. Scheme design and convergence analysis
  8. Design of SPTC-AN-FOZNN model
  9. Main results in theoretical analysis
  10. Numerical validation
  11. Kinematic control of robotic manipulators
  12. Simulation validation
  13. Experimental validation
  14. Conclusion

Key Result

Lemma 1

(ref28) $x^\ast(t)\in\mathbb{R}^n$ is the KKT point for the TVQP problem (eq2) if for every $\tau\rightarrow0_+$ there exist the Lagrangian multipliers $\phi^\ast(t)\in\mathbb{R}^m$ and $\varphi^\ast(t)\in\mathbb{R}^p$ satisfying where $\psi_{FB}^{\tau}$ represents the perturbed Fischer-Burmeister (FB) function.

Figures (10)

  • Figure 1: Profiles of neural states $x_1$ and neural state $x_2$ across six models for solving the TVQP problem (\ref{['eq25']}) under the following noise conditions: (a) and (d) without additive noise; (b) and (e) a low-frequency noise $\delta(t)=0.2\cos(t)$; (c) and (f) a high-frequency random noise $\delta(t)=0.5\bar{n}(t)$, where $\bar{n}(t)$ is a white noise signal bounded by $1$. (Our SPTC-AN-FOZNN model takes three distinct $\alpha$ values)
  • Figure 2: Profiles of the residual error $\|\epsilon(t)\|$ across six models for solving the TVQP problem (\ref{['eq25']}) under the following noise conditions: (a) without additive noise; (b) a low-frequency noise $\delta(t)=0.2 \cos(t)$; (c) a high-frequency random noism $\delta(t)=0.5\bar{n}(t)$, where $\bar{n}(t)$ is a white noise signal bounded by $1$. (Our SPTC-AN-FOZNN odel takes three distinct $\alpha$ values.)
  • Figure 3: The snapshots of the (a) initial, (b) intermediate, and (c) final phases for a simulated Franka Emika Panda robot during tracking a heart-shaped path, with the rendered (d) joint angles, (e) joint velocities, (f) absolute tracking errors by the SPTC-AN-FOZNN model, and (g) residual errors rendered by six different neural models with a bounded random noise $\delta(t)=\cos(t)+\bar{n}(t)$.
  • Figure 4: The snapshots of the (a) initial, (b) intermediate, and (c) final phases for a simulated Franka Emika Panda robot during tracking a Lissajous curve, with the rendered (d) joint angles, (e) joint velocities, (f) absolute tracking errors by the SPTC-AN-FOZNN model, and (g) residual errors rendered by six different neural models with a bounded random noise $\delta(t)=\cos(t)+\bar{n}(t)$.
  • Figure 5: The snapshots of the (a) initial, (b) intermediate, and (c) final phases for a simulated Franka Emika Panda robot during tracking a five-petal-plum-shaped path, with the rendered (d) joint angles, (e) joint velocities, (f) absolute tracking errors by the SPTC-AN-FOZNN model, and (g) residual errors rendered by six different neural models with a bounded random noise $\delta(t)=\cos(t)+\bar{n}(t)$.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • remark 1
  • Lemma 1
  • remark 2
  • remark 3
  • remark 4
  • ...and 7 more