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Fast Whole-Body Strain Regulation in Continuum Robots

Lekan Molu

TL;DR

The paper tackles real-time, real-world strain control for multiphysics, multiscale continuum soft robots by formulating a two-time-scale singularly perturbed model that separates fast and slow dynamics with a perturbation parameter $\epsilon$. It designs nonlinear backstepping controllers for each subdynamic and proves stability of the fast, slow, and interconnected subsystems using Lyapunov analysis, culminating in a composite Lyapunov function for the full system. The authors validate the approach through fast numerical simulations on an Octopus-like soft arm, achieving substantially faster whole-body strain regulation than prior single-layer PD methods, while operating in a decentralized, GPU/CPU parallel setup. This work enhances practical applicability of soft robotics by enabling real-time, scalable, and embodied-control-driven regulation of continuum robots, which is critical for broad deployment in automation and manipulation tasks.

Abstract

We propose reaching steps towards the real-time strain control of multiphysics, multiscale continuum soft robots. To study this problem fundamentally, we ground ourselves in a model-based control setting enabled by mathematically precise dynamics of a soft robot prototype. Poised to integrate, rather than reject, inherent mechanical nonlinearities for embodied compliance, we first separate the original robot dynamics into separate subdynamics -- aided by a perturbing time-scale separation parameter. Second, we prescribe a set of stabilizing nonlinear backstepping controllers for regulating the resulting subsystems' strain dynamics. Third, we study the interconnected singularly perturbed system by analyzing and establishing its stability. Fourth, our theories are backed up by fast numerical results on a single arm of the Octopus robot arm. We demonstrate strain regulation to equilibrium, in a significantly reduced time, of the whole-body reduced-order dynamics of an infinite degrees-of-freedom soft robot. This paper communicates our thinking within the backdrop of embodied intelligence: it informs our conceptualization, formulation, computational setup, and yields improved control performance for infinite degrees-of-freedom soft robots.

Fast Whole-Body Strain Regulation in Continuum Robots

TL;DR

The paper tackles real-time, real-world strain control for multiphysics, multiscale continuum soft robots by formulating a two-time-scale singularly perturbed model that separates fast and slow dynamics with a perturbation parameter . It designs nonlinear backstepping controllers for each subdynamic and proves stability of the fast, slow, and interconnected subsystems using Lyapunov analysis, culminating in a composite Lyapunov function for the full system. The authors validate the approach through fast numerical simulations on an Octopus-like soft arm, achieving substantially faster whole-body strain regulation than prior single-layer PD methods, while operating in a decentralized, GPU/CPU parallel setup. This work enhances practical applicability of soft robotics by enabling real-time, scalable, and embodied-control-driven regulation of continuum robots, which is critical for broad deployment in automation and manipulation tasks.

Abstract

We propose reaching steps towards the real-time strain control of multiphysics, multiscale continuum soft robots. To study this problem fundamentally, we ground ourselves in a model-based control setting enabled by mathematically precise dynamics of a soft robot prototype. Poised to integrate, rather than reject, inherent mechanical nonlinearities for embodied compliance, we first separate the original robot dynamics into separate subdynamics -- aided by a perturbing time-scale separation parameter. Second, we prescribe a set of stabilizing nonlinear backstepping controllers for regulating the resulting subsystems' strain dynamics. Third, we study the interconnected singularly perturbed system by analyzing and establishing its stability. Fourth, our theories are backed up by fast numerical results on a single arm of the Octopus robot arm. We demonstrate strain regulation to equilibrium, in a significantly reduced time, of the whole-body reduced-order dynamics of an infinite degrees-of-freedom soft robot. This paper communicates our thinking within the backdrop of embodied intelligence: it informs our conceptualization, formulation, computational setup, and yields improved control performance for infinite degrees-of-freedom soft robots.
Paper Structure (18 sections, 3 theorems, 41 equations, 3 figures, 1 table)

This paper contains 18 sections, 3 theorems, 41 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. SoRo Configuration
  4. Continuous Strain Vector and Twist Velocity Fields
  5. Discrete Cosserat-Constitutive PDEs
  6. Singularly Perturbed Dynamics
  7. Soft Robots' Dynamics Separation
  8. Fast subsystem dynamics extraction
  9. Slow sub-dynamics
  10. Hierarchical Controller Synthesis
  11. Stability analysis of the fast strain subdynamics
  12. Stability analysis of the strain twist subdynamics
  13. Stability analysis of the slow strain subdynamics
  14. Stability of the singularly perturbed interconnected system
  15. Numerical Results
  16. ...and 3 more sections

Key Result

Theorem 1

The control law is sufficient to guarantee an exponential stability of the origin of $\bm{\theta}^\prime = \bm{\nu}$ such that for all $t_f \ge 0$, $\bm{q}_\text{fast}(t_f) \in S$ for a compact set $S \subset \mathbb{R}^{6N}$. That is, $\bm{q}_\text{fast}(t_f)$ remains bounded as $t_f \rightarrow \infty$.

Figures (3)

  • Figure 1: Simplified configuration of an Octopus arm, reprinted from LekanSoRoPD.
  • Figure 2: Backstepping control on the singularly perturbed soft robot system with 10 discretized pieces, divided into 6 fast and 4 slow pieces. For a tip load of $\bm{\mathcal{F}}_p^y=10\,N$, the backstepping gains were set as $\bm{K}_p = 10$, $\bm{K}_d = 2.0$ for a desired joint configuration $\xi^d = [0, 0, 0, 1, 0.5, 0]^\top$ and ${\eta}^d = \bm{0}_{6\times 1}$ that is uniform throughout the robot sections.
  • Figure 3: Backstepping control on the singularly perturbed soft robot system with $10$ pieces $4$ slow and $6$ fast sections.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof