Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Predefined-Time Convergent and Noise-Tolerant Zeroing Neural Network Model for Time Variant Quadratic Programming With Application to Robot Motion Planning

Yi Yang, Xuchen Wang, Richard M. Voyles, Xin Ma

TL;DR

This work tackles time-variant quadratic programming (TVQP) in robotic kinematic control under measurement noise. It introduces a predefined-time convergent, noise-tolerant fractional-order zeroing neural network (PTC-NT-FOZNN) that employs a conformable fractional derivative and a novel predefined-time activation to ensure rapid convergence within a user-specified time \\(t_c\\), while mitigating additive noise. The authors provide rigorous convergence proofs, derive residual bounds, and show that smaller orders \\(\\alpha\\) improve settling time and accuracy. The method is validated through numerical comparisons against six ZNNs, cyclic motion planning simulations, and real-world experiments on a Flexiv Rizon robot, demonstrating high tracking precision and energy-efficient operation with guaranteed settling times.

Abstract

This paper develops a predefined-time convergent and noise-tolerant fractional-order zeroing neural network (PTC-NT-FOZNN) model, innovatively engineered to tackle time-variant quadratic programming (TVQP) challenges. The PTC-NT-FOZNN, stemming from a novel iteration within the variable-gain ZNN spectrum, known as FOZNNs, features diminishing gains over time and marries noise resistance with predefined-time convergence, making it ideal for energy-efficient robotic motion planning tasks. The PTC-NT-FOZNN enhances traditional ZNN models by incorporating a newly developed activation function that promotes optimal convergence irrespective of the model's order. When evaluated against six established ZNNs, the PTC-NT-FOZNN, with parameters $0 < α\leq 1$, demonstrates enhanced positional precision and resilience to additive noises, making it exceptionally suitable for TVQP tasks. Thorough practical assessments, including simulations and experiments using a Flexiv Rizon robotic arm, confirm the PTC-NT-FOZNN's capabilities in achieving precise tracking and high computational efficiency, thereby proving its effectiveness for robust kinematic control applications.

A Predefined-Time Convergent and Noise-Tolerant Zeroing Neural Network Model for Time Variant Quadratic Programming With Application to Robot Motion Planning

TL;DR

This work tackles time-variant quadratic programming (TVQP) in robotic kinematic control under measurement noise. It introduces a predefined-time convergent, noise-tolerant fractional-order zeroing neural network (PTC-NT-FOZNN) that employs a conformable fractional derivative and a novel predefined-time activation to ensure rapid convergence within a user-specified time \, while mitigating additive noise. The authors provide rigorous convergence proofs, derive residual bounds, and show that smaller orders \ improve settling time and accuracy. The method is validated through numerical comparisons against six ZNNs, cyclic motion planning simulations, and real-world experiments on a Flexiv Rizon robot, demonstrating high tracking precision and energy-efficient operation with guaranteed settling times.

Abstract

This paper develops a predefined-time convergent and noise-tolerant fractional-order zeroing neural network (PTC-NT-FOZNN) model, innovatively engineered to tackle time-variant quadratic programming (TVQP) challenges. The PTC-NT-FOZNN, stemming from a novel iteration within the variable-gain ZNN spectrum, known as FOZNNs, features diminishing gains over time and marries noise resistance with predefined-time convergence, making it ideal for energy-efficient robotic motion planning tasks. The PTC-NT-FOZNN enhances traditional ZNN models by incorporating a newly developed activation function that promotes optimal convergence irrespective of the model's order. When evaluated against six established ZNNs, the PTC-NT-FOZNN, with parameters , demonstrates enhanced positional precision and resilience to additive noises, making it exceptionally suitable for TVQP tasks. Thorough practical assessments, including simulations and experiments using a Flexiv Rizon robotic arm, confirm the PTC-NT-FOZNN's capabilities in achieving precise tracking and high computational efficiency, thereby proving its effectiveness for robust kinematic control applications.
Paper Structure (12 sections, 4 theorems, 25 equations, 4 figures, 2 tables)

This paper contains 12 sections, 4 theorems, 25 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary Theory
  3. Problem Formulation
  4. Traditional ZNN model
  5. Scheme Design and Main Results
  6. Design of PTC-NT-FOZNN Model
  7. Main Results in Convergence Analysis
  8. Experiments and Comparison
  9. An Illustrative Example
  10. Simulative Verification
  11. Practical Verification
  12. Conclusion

Key Result

Lemma 1

31 For the TVQP problem (eq2), $\boldsymbol{y}(t) \in$$\mathbb{R}^{n}$ is the KKT point if, for any $\varepsilon \rightarrow 0_{+}$there exist Lagrangian multipliers $\boldsymbol{\lambda}_{\mathbf{1}}(t) \in \mathbb{R}^{m}$ and $\boldsymbol{\lambda}_{\mathbf{2}}(t) \in \mathbb{R}^{l}$ to make the fo where $\phi_{F B}^{\varepsilon}$ denotes the perturbed Fischer-Burmeister (FB) function 31, which i

Figures (4)

  • Figure 1: Neural states (a) $y_{1}$, (b) $y_{2}$, and their respective errors (c) $\left|y_{1}^{*}-y_{1}\right|$, (d) $\left|y_{2}^{*}-y_{2}\right|$ for six established models, compared to our PTC-NT-FOZNN at four $\alpha$ values when the additive noise $\boldsymbol{\delta}(t)=$$0.1 \sin (t)$ is considered, as tackling the case from Section 4.1.
  • Figure 2: Residual error $\|\boldsymbol{\epsilon}(t)\|_{2}$ for six established models compared to our PTC-NT-FOZNN at four $\alpha$ values under various noise conditions: (a) no additive noise, (b) $\boldsymbol{\delta}(t)=0.1 \sin (t)$, (c) constant noise $\boldsymbol{\delta}(t)=$ 0.5, and (d) $\boldsymbol{\delta}(t)=\sin (t)$, as tackling the case from Section 4.1.
  • Figure 3: Illustration of the captured phases: (a) initialization, (b) transitional state, and (c) cycling-back state for a simulated Flexiv robot tracking a clover-shaped path, and the simulated (d) joint angles, (e) joint velocities, (f) absolute tracking errors calculated using the PTC-NT-FOZNN model, and (g) residual errors evaluated by seven different neural models under an uncertain noise $\boldsymbol{\delta}(t)=\sin (t)+\cos (t)+\bar{n}(t)$.
  • Figure 4: Illustration of (a) an image of the experimental setup, (b) a comparison between the actual trajectory and the intended clovershaped path, along with (c) joint angles, (d) joint velocities, (e) absolute tracking errors, and (f) real-time joint torques as measured during experiments using the PTC-NT-FOZNN model on the physical test platform.

Theorems & Definitions (7)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Remark 1