Gait modeling and optimization for the perturbed Stokes regime

Abstract

Many forms of locomotion, both natural and artificial, are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that for swimming at the "Stokesian" (viscous; zero Reynolds number) limit, the motion is governed by a reduced order "connection" model that describes how body shape change produces motion for the body frame with respect to the world. In the "perturbed Stokes regime" where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer linear in shape change rate. We derive this model using results from singular perturbation theory, and the theory of noncompact normally hyperbolic invariant manifolds (NHIMs).Using the theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a "gait") directly from observational data of shape and body motion. This extends our previous work which assumed kinematic "connection" models. To compare the old and new algorithms, we analyze simulated swimmers over a range of inertia to damping ratios. Our new class of models performs well on the Stokesian regime, and over several orders of magnitude outside it into the perturbed Stokes regime, where it gives significantly improved prediction accuracy compared to previous work.In addition to algorithmic improvements, we thereby present a new class of models that is of independent interest. Their application to data-driven modeling improves our ability to study the optimality of animal gaits, and our ability to use hardware-in-the-loop optimization to produce gaits for robots.

Data-Driven Geometric System Identification for Shape-Underactuated Dissipative Systems

“…Thus, we expanded on the capabilities of methods that can optimize analytical SUDS models [36] with methods that can fit SUDS models to data. The similarity to our previous work [15,16] suggests that this would make it possible to rapidly learn behaviors in such underactuated systems. It suggests that underactuation in SUDS does not pose nearly the same difficulties as in other underactuated systems -the strong dissipation improves the stability of the passive dynamics under repeated but perturbed control inputs.…”

“…At the limit of large friction, the momentum term disappears, leaving a class of models which we have shown to be amenable to system identification [15]. Further, with finite-butlarge dissipation, the influence of momentum can be folded into a nonlinear correction to the connection, with only a small increase in the complexity of the model identification process [16]. Thus models for predicting the influence of shape input on body velocity can be built strictly from observation without any mechanical analysis specific to the system -all that is needed is "sufficiently rapid" dissipation of momentum.…”

“…The difficulty of obtaining predictive models for highly underactuated systems shares features of the challenges of modeling their high degree of freedom, fully actuated counterparts. For fully actuated dissipative systems, we have previously published sample-efficient techniques to model locomotion systems with noisy shape control using cyclic behavioral data [15,16]. Seminal work by Shapere, Wilczek, Marsden, Kelly, Ostrowski, Bloch and others [17,18,19,20,21,22] showed that the Newtonian physics of locomotion can be refactored into a kinematic term (the mechanical connection of [20]) and a momentum term.…”

Optimizing Gait Libraries via a Coverage Metric

“…For biological intuition for how a system like this might move, a starfish might move like a pentagonal five-branch system with longer segments of links at each vertex. The links interact via the slender body theory of Cox [18], the same that was used for the swimmer in [16] and paddles in [19]. The drag of the triangular piece is represented by three static links that point from the center of the triangle to their respective attachment points.…”

Characterizing Error in Noncommutative Geometric Gait Analysis

“…This expression makes the assumption that systems behave kinematically. Previous work[15] extends this domain to apply to many systems 3. In previous work, the local connection, by convention, encodes negative body motion; we have dropped this convention for this paper.…”