Data-Driven Soft Robot Control via Adiabatic Spectral Submanifolds

TL;DR

This work tackles controlling soft robots by exploiting adiabatic spectral submanifolds (aSSMs) to derive low-dimensional, data-driven reduced-order models that remain accurate along large, nonlinear trajectories. By leveraging slow-fast time-scale separation, the authors develop an MPC scheme that combines slow control inputs with bounded fast deviations, yielding a reduced model that can be learned purely from data. They demonstrate the approach on high-fidelity soft-trunk FEM models and a Cosserat-rod soft arm, showing 4–6 dimensional aSSM reductions provide up to an order of magnitude improvement over linear baselines across diverse tracks and constraints. The results highlight the practical potential for real-time, nonlinear model-based control of soft robots, with strong generalization and favourable computation times, suggesting viability for hardware implementation.

Abstract

The mechanical complexity of soft robots creates significant challenges for their model-based control. Specifically, linear data-driven models have struggled to control soft robots on complex, spatially extended paths that explore regions with significant nonlinear behavior. To account for these nonlinearities, we develop here a model-predictive control strategy based on the recent theory of adiabatic spectral submanifolds (aSSMs). This theory is applicable because the internal vibrations of heavily overdamped robots decay at a speed that is much faster than the desired speed of the robot along its intended path. In that case, low-dimensional attracting invariant manifolds (aSSMs) emanate from the path and carry the dominant dynamics of the robot. Aided by this recent theory, we devise an aSSM-based model-predictive control scheme purely from data. We demonstrate our data-driven model's effectiveness in tracking dynamic trajectories across diverse tasks, validated on a high-fidelity, high-dimensional finite-element model of a soft trunk robot and a Cosserat rod-based elastic soft arm. Notably, we find that five- or six-dimensional aSSM-reduced models outperform the tracking performance of other data-driven modeling methods by a factor up to $10$ across all closed-loop control tasks.

Figures (24)

• Figure 1: (a) For $\epsilon =0$, critical limit of the adiabatic SSM geometry in the phase and actuation space. (b) For $\epsilon >0$ and slow input $\mathbf{u}(\epsilon t)$, the leading order adiabatic SSM geometry of $\mathcal{A}_\epsilon$ anchored to the target $\mathbf{S}(\mathbf{u}(\epsilon t))$. (c) For $\mathbf{u}_d(t) \not \equiv 0$, the perturbed aSSM geometry of $\tilde{\mathcal{A}}_{\epsilon}$.
• Figure 2: Three step procedure for modeling and control of soft robots using aSSMs. Step 1, involves data collection of decaying and controlled data about random static configurations of the soft robot. Step 2, learns the aSSM geometry and dynamics using existing SSMLearn algorithm (see cenedese22b), for specific details see Section \ref{['sec:learn_aSSMs']}. Step 3, fuses the learned aSSM-reduced model in a model predictive control scheme, for specific details see Section \ref{['sec:fh_horizon']}.
• Figure 3: (a) For $\delta =0$, comparison of different aSSM approximations with the full system response. (b) For $\delta =1.2$, comparison plots in the time window $[50,150]$. (c) The same for $\delta =3.2$. (d) aSSM snapshot in the phase space at $t=20.2 \text{ [s]}$ for $\delta =1.2$. (e) aSSM snapshot at a later time $t=40.2 \text{ [s]}$.
• Figure 4: (a) The trunk at rest. (b) The trunk in a curved static configuration. (c) The trunk in a S-shaped static configuration
• Figure 5: (a) The workspace of the trunk robot. Static steady state are shown in gray and a target trajectory in green. (b) $x_{ee}$ and $z_{ee}$ test predictions for a $5$D static SSM model anchored at the origin

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7