Optimal gait design for nonlinear soft robotic crawlers

TL;DR

This work addresses gait design for a minimal soft crawler composed of two segments connected by a viscoelastic link and actuated by symmetric intersegment forces under nonlinear friction. Through describing-function analysis, it proves a resonance principle: for sinusoidal actuation the average forward speed is maximized when the forcing frequency matches the body’s undamped natural frequency $\omega_n$. Building on this, the authors formulate an Optimal Periodic Control (OPC) problem to synthesize arbitrary periodic actuation profiles, trading center-of-mass speed against actuator effort and body strain, and solve it with a hill-climbing algorithm based on first-order optimality conditions and direct-collocation, initialized from a resonant harmonic. A case study demonstrates that, when penalties on effort and strain are included, the optimizer favors gaits whose period is an integer multiple of the natural frequency, illustrating how resonance and waveform design interact in soft crawling. These results provide a framework to design efficient, safe gaits for more complex multi-segment soft crawlers and guide future extensions to decentralized feedback control.

Abstract

Soft robots offer a frontier in robotics with enormous potential for safe human-robot interaction and agility in uncertain environments. A stepping stone towards unlocking their potential is a control theory tailored to soft robotics, including a principled framework for gait design. We analyze the problem of optimal gait design for a soft crawling body - the crawler. The crawler is an elastic body with the control signal defined as actuation forces between segments of the body. We consider the simplest such crawler: a two-segmented body with a passive mechanical connection modeling the viscoelastic body dynamics and a symmetric control force modeling actuation between the two body segments. The model accounts for the nonlinear asymmetric friction with the ground, which together with the symmetric actuation forces enable the crawler's locomotion. Using a describing-function analysis, we show that when the body is forced sinusoidally, the optimal actuator contraction frequency corresponds to the body's natural frequency when operating with only passive dynamics. We then use the framework of Optimal Periodic Control (OPC) to design optimal force cycles of arbitrary waveform and the corresponding crawling gaits. We provide a hill-climbing algorithm to solve the OPC problem numerically. Our proposed methods and results inform the design of optimal forcing and gaits for more complex and multi-segmented crawling soft bodies.

Figures (2)

• Figure 1: Nonlinear anisotropic friction models in the crawler dynamics. (A) Smooth ($\sigma$) and piece-wise constant ($\sigma_{\text{DF}}$) anisotropic friction input-output maps, as defined in \ref{['eq: friction']} and \ref{['eq: piecewise constant friction (negative)']}, respectively. The piece-wise constant model $\sigma_{\text{DF}}$ is used in the describing function analysis of Section §\ref{['sec:describing_function_analysis']} for the sake of analytical tractability. (B) Sinusoidal speed input. Horizontal axis in panels A and B are the same. (C) Respective outputs of the friction models in panel A when the input is a sinusoidal local speed --- as depicted in B. Input and output are in phase. Color-code is the same in panels A and C. (D) Block diagram of the system \ref{['eq:ND_dynamical equations']}. Blocks shaded in blue capture the nonlinearity in the dynamics of the soft crawler. The low-pass filtering structure of the linear component of the crawler dynamics makes describing function analysis a suitable technique. $s$ denotes the Laplace transform variable and the $s$-block corresponds to temporal differentiation. (E) Schematic of the soft crawler and its different components.
• Figure 2: Optimal solution to the OPC problem \ref{['optimizationProblem']}, numerically obtained using Algorithm \ref{['alg:hillClimbing_OPC']}. Parameter values are as described in § \ref{['subsec:case_study']}. (A) Initial ($\mathsf{f}_0$) and optimized ($\mathsf{f}_*$) actuation force profiles in time. $\mathsf{f}_0$ is chosen according to Proposition \ref{['prop:forcing_frequency']}. Algorithm \ref{['alg:hillClimbing_OPC']} converges to a periodic force profile, its period being an integer multiple of the crawler's natural frequency. (B) Positions of the soft crawler's head and tail in time, illustrating the optimal crawling gait. (C) Periodic trajectories of the co-states $\boldsymbol{\lambda}$, solution to \ref{['eq:costate_dynamics_all']}.

Theorems & Definitions (2)

• Proposition III.1: Optimal frequency for sinusoidal forcing
• proof