Asymmetric Friction in Geometric Locomotion

TL;DR

This paper extends geometric locomotion from symmetric, linear drag (Riemannian) models to asymmetric, direction-dependent friction captured by Finsler metrics. By formulating sub-Finsler motility maps and introducing a linear-conical decomposition into $\mathbf{\bar{B}}$ and $\mathbf{\tilde{B}}$, it reveals how nonreciprocal effects dominate net motion at small amplitudes while nonconservative contributions govern larger gaits. The authors define a nonreciprocal constraint curvature $\boldsymbol{\delta}\mathbf{B}$ to quantify asymmetry-driven locomotion and provide practical approximations for predicting gait outcomes via curvature integrals and center-point evaluations. This framework enables gait planning and analysis of asymmetry-driven locomotion, with applications to robot feet design and directional adhesion in complex environments.

Abstract

Geometric mechanics models of locomotion have provided insight into how robots and animals use environmental interactions to convert internal shape changes into displacement through the world, encoding this relationship in a ``motility map''. A key class of such motility maps arises from (possibly anisotropic) linear drag acting on the system's individual body parts, formally described via Riemannian metrics on the motions of the system's individual body parts. The motility map can then be generated by invoking a sub-Riemannian constraint on the aggregate system motion under which the position velocity induced by a given shape velocity is that which minimizes the power dissipated via friction. The locomotion of such systems is ``geometric'' in the sense that the final position reached by the system depends only on the sequence of shapes that the system passes through, but not on the rate with which the shape changes are made. In this paper, we consider a far more general class of systems in which the drag may be not only anisotropic (with different coefficients for forward/backward and left/right motions), but also asymmetric (with different coefficients for forward and backward motions). Formally, including asymmetry in the friction replaces the Riemannian metrics on the body parts with Finsler metrics. We demonstrate that the sub-Riemannian approach to constructing the system motility map extends naturally to a sub-Finslerian approach and identify system properties analogous to the constraint curvature of sub-Riemannian systems that allow for the characterization of the system motion capabilities.

Figures (15)

• Figure 1: Cross-country skier as a model system for sub-Finsler locomotion. A simple model for cross-country skiing is that the skis have asymmetric viscous friction with the ground (making it harder to slide them backwards than to slide them forwards), and that the skier can modulate this friction by shifting weight onto one or other of the skis. A similar skier with symmetric friction on the skis can be modeled using extant sub-Riemannian geometric methods. A key result from this prior work is that the net motion is proportional to the product of the weight and ski oscillations---growing quadratically if the oscillation amplitudes are increased together---and that its sign depends on the relative phase of the oscillations. Conversely, geometrically modeling a skier with only asymmetric friction requires the new sub-Finslerian analysis introduced in this paper. A key result of this analysis is that the net motion grows linearly with the sum of oscillation amplitudes, and its direction is independent of the relative phases of the oscillations. Combining these effects produces a locomotion model in which the full skier can use weight-shifting to boost or counteract the positive motion induced by the asymmetric friction.
• Figure 2: Planarized cross-country skier model. The two blocks in contact with the ground are each able to slide along the $x$ axis and are connected by a linear actuator. A second block (which contains all of the system mass) is connected to a rail between these blocks. The two blocks are subject to viscous friction, whose coefficient depends both on a base value $c$ and the proportion of the sliding mass whose weight the friction-block is supporting.
• Figure 3: Locomotion of the planarized skier. (a) The skier's motility map $\mathbf{A}$ evaluated over its shape space, together with a pair of box-shaped gaits at different scales and an elliptical gait with the same amplitude as the bigger box. The small cartoons of the skier along the axes illustrate the geometric meaning of the $\alpha_{1}$ and $\alpha_{2}$ variables. (b) The net displacement induced by the gaits grows approximately quadratically with the gait amplitude, with an approximately $\pi/4$ ratio between the box and elliptical gaits. (c) The constraint curvature $D\mathbf{A}$ (plotted with density of the stippling points indicating its magnitude). The slight subquadratic growth in the net displacement corresponds to bigger gaits enclosing regions where $D\mathbf{A}$ is less rich, and the $>\pi/4$ ratio between displacement from large ellipses and large boxes corresponds to the areas "given up" by the ellipses being biased towards the low-richness regions of the shape space.
• Figure 4: Example of spines generating a asymmetric friction force. When applied to low ply industrial carpet, these spines have many times the friction sliding in one direction vs. the opposite direction.
• Figure 5: Geometry of vector norms under Riemannian and Finsler metrics. (a) Riemannian metric norms are positive-homogeneous (scaling the vector scales the norm proportionally), and their level sets are centered around the origin. (b) In more than one dimension, the level sets of the metric are ellipsoidal, but remain centered around the origin. (c) Finsler metric norms are positive-homogeneous, but their level sets do not need to be symmetric around the origin. (d) In more than one dimension, Finsler metric norms are functions whose level sets are convex, concentric, evenly spaced, and enclose the origin. The illustrated metric is the union of two half-ellipses, and could plausibly represent the friction generated by the spines in [S]fig:spines. See [S]exp:singleblockriemanfriction, exp:singleblockriemanfriction2d for mathematical representations of the Riemannian metrics, and [S]exp:finsler1dsingleblock,exp:finsler2dsingleblock for representations of the Finsler metrics.

Theorems & Definitions (24)

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