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Modeling and Control of Intrinsically Elasticity Coupled Soft-Rigid Robots

Zach J. Patterson, Cosimo Della Santina, Daniela Rus

TL;DR

The paper addresses control of intrinsically elastically coupled, underactuated soft-rigid robots by introducing four simple coupling models and a provably stable PD controller with gravity compensation. It presents a Lyapunov-based zero-dynamics analysis and a PD regulator that ensures convergence without requiring elastic dominance, supported by simulations and hardware experiments. A sensorless force-control approach leverages elastic coupling as an actuator-like force estimator, enabling force regulation without direct sensing. Hardware validation demonstrates model identification (Neo-Hookean performing best) and effective force control with modest errors, indicating practical benefits of elastically coupled designs for embodied intelligence.

Abstract

While much work has been done recently in the realm of model-based control of soft robots and soft-rigid hybrids, most works examine robots that have an inherently serial structure. While these systems have been prevalent in the literature, there is an increasing trend toward designing soft-rigid hybrids with intrinsically coupled elasticity between various degrees of freedom. In this work, we seek to address the issues of modeling and controlling such structures, particularly when underactuated. We introduce several simple models for elastic coupling, typical of those seen in these systems. We then propose a controller that compensates for the elasticity, and we prove its stability with Lyapunov methods without relying on the elastic dominance assumption. This controller is applicable to the general class of underactuated soft robots. After evaluating the controller in simulated cases, we then develop a simple hardware platform to evaluate both the models and the controller. Finally, using the hardware, we demonstrate a novel use case for underactuated, elastically coupled systems in "sensorless" force control.

Modeling and Control of Intrinsically Elasticity Coupled Soft-Rigid Robots

TL;DR

The paper addresses control of intrinsically elastically coupled, underactuated soft-rigid robots by introducing four simple coupling models and a provably stable PD controller with gravity compensation. It presents a Lyapunov-based zero-dynamics analysis and a PD regulator that ensures convergence without requiring elastic dominance, supported by simulations and hardware experiments. A sensorless force-control approach leverages elastic coupling as an actuator-like force estimator, enabling force regulation without direct sensing. Hardware validation demonstrates model identification (Neo-Hookean performing best) and effective force control with modest errors, indicating practical benefits of elastically coupled designs for embodied intelligence.

Abstract

While much work has been done recently in the realm of model-based control of soft robots and soft-rigid hybrids, most works examine robots that have an inherently serial structure. While these systems have been prevalent in the literature, there is an increasing trend toward designing soft-rigid hybrids with intrinsically coupled elasticity between various degrees of freedom. In this work, we seek to address the issues of modeling and controlling such structures, particularly when underactuated. We introduce several simple models for elastic coupling, typical of those seen in these systems. We then propose a controller that compensates for the elasticity, and we prove its stability with Lyapunov methods without relying on the elastic dominance assumption. This controller is applicable to the general class of underactuated soft robots. After evaluating the controller in simulated cases, we then develop a simple hardware platform to evaluate both the models and the controller. Finally, using the hardware, we demonstrate a novel use case for underactuated, elastically coupled systems in "sensorless" force control.
Paper Structure (17 sections, 2 theorems, 24 equations, 8 figures)

This paper contains 17 sections, 2 theorems, 24 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Modeling Parallel Elasticity
  3. Linear Coupling
  4. Distance-based coupling
  5. Rejection-based coupling
  6. Neo-Hookean Model
  7. Underactuated Dynamics
  8. System Properties
  9. Controllers
  10. Zero Dynamics
  11. PD Regulator
  12. Simulations
  13. Experiments
  14. Model Validation
  15. Coupled Force Control
  16. ...and 2 more sections

Key Result

Theorem 1

For any initial state and any $q_{\mathrm{a}} = \overline{q}_{\mathrm{a}}$, the trajectories of (eq:zero_dynamics) are bounded and converge to $(q_{\mathrm{u}},\dot{q}_{\mathrm{u}}) = (q_{\mathrm{u,eq}},0)$ where $q_{\mathrm{u,eq}}$ is found by solving

Figures (8)

  • Figure 1: Three examples of systems with coupled stiffness. a) Simple hardware implementation inspired by flippers. Two rigid 3D printed links are embedded in a silicone matrix. b) A "finger" with torsional springs on all joints and a coupling between the second and third joint. $q_{\mathrm{a}}$ denotes actuated degrees of freedom (DOF) and $q_{\mathrm{u}}$ denotes unactuated. c) Flipper inspired structure with a rigid "skeleton" embedded in a soft flipper
  • Figure 2: Depictions of the qualitative aspects of the models. Left: Linear model. Middle: Distance and Rejection models. Right: Neo-Hookean shear model.
  • Figure 3: A simulated demo on a simplified finger model under gravity. Solid lines designate our regulator whereas dash-dot lines designate an identical PD regulator without the coupling compensation. a) Shows the trajectories with an inset graphic of the model. b) Shows the control torques.
  • Figure 4: A simulated demo of our controllers on a simplified "flipper" model. a) Shows the trajectories with an inset image of the real-life version of the idealized system. b) Shows snapshots taken from the simulated model along the trajectory.
  • Figure 5: Models comparison on a subset of the data.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof