Sub--Riemannian boundary value problems for Optimal Geometric Locomotion

TL;DR

This work develops a geometric, dissipation-based framework for energy-efficient locomotion driven entirely by shape changes. By formulating the problem on a principal fiber bundle of positioned shapes and employing a total $G$-invariant metric that combines inner (bending/strain) and outer (environmental) dissipation, the authors show that optimal locomotion paths are sub-Riemannian geodesics, i.e., horizontal lifts minimizing the dissipation $\mathcal{E}$. They analyze three boundary-condition classes (fixed start/end, isoholonomic periodic gaits, and $\chi$-isoholonomic), demonstrate numerical discretization and convergence, and validate the approach against known results such as Purcell’s swimmer while revealing gains from higher-dimensional shape spaces. The framework qualitatively reproduces snake- and spermatozoa-like motions and provides a versatile tool for designing efficient, boundary-condition-specific gaits across length scales, with potential extensions to collision avoidance and more complex shape representations.

Abstract

We propose a geometric model for optimal shape-change-induced motions of slender locomotors, e.g., snakes slithering on sand. In these scenarios, the motion of a body in world coordinates is completely determined by the sequence of shapes it assumes. Specifically, we formulate Lagrangian least-dissipation principles as boundary value problems whose solutions are given by sub-Riemannian geodesics. Notably, our geometric model accounts not only for the energy dissipated by the body's displacement through the environment, but also for the energy dissipated by the animal's metabolism or a robot's actuators to induce shape changes such as bending and stretching, thus capturing overall locomotion efficiency. Our continuous model, together with a consistent time and space discretization, enables numerical computation of sub-Riemannian geodesics for three different types of boundary conditions, i.e., fixing initial and target body, restricting to cyclic motion, or solely prescribing body displacement and orientation. The resulting optimal deformation gaits qualitatively match observed motion trajectories of organisms such as snakes and spermatozoa, as well as known optimality results for low-dimensional systems such as Purcell's swimmers. Moreover, being geometrically less rigid than previous frameworks, our model enables new insights into locomotion mechanisms of, e.g., generalized Purcell's swimmers. The code is publicly available.

Figures (20)

• Figure 1: A shape (snake) can be variedly positioned (left). All of these configurations---being different only by a orientation preserving Euclidean motion---belong to the same fiber (right) and project down to a single shape in $\mathcal{S}$.
• Figure 2: A cylic motion of a changing shape (left) amounts to a closed curve in $\mathcal{S}$ under the projection $\pi$ (right). The fact that the cyclic shape deformation results in net motion per cycle is seen in $\boldsymbol{\gamma}$ beginning and ending in different locations on the same fiber (right).
• Figure 3: Top: Outer dissipation accounts for energy expenditure due to external forces, such as viscous friction. In general we expect normal motion (left) to be more expensive while tangential motion (right) is less so. This anisotropy, which may vary along the length of the creature is responsible for deformations resulting in net motion. Bottom: Inner dissipation accounts for energy spent on internal shape changes and results from metabolism or from sources such as batteries. First order effects correspond to stretching and compression while bending is a second order phenomenon.
• Figure 4: Solution of the boundary value problem with prescribed initial and target body in world coordinates (Eq. (\ref{['eq:geodesic']})) for different bending weights and fixed strain weight $1.0$ controlling the inner dissipation (top), for different anisotropy ratios controlling the outer dissipation (bottom).
• Figure 5: Top: optimal motion path between initial $\gamma^0$ and final $\gamma^1$ positioned shapes; Middle: fixing only the displacement from some initial configuration without requiring a particular end shape and thus with non prescribed destination fiber; Bottom: optimal periodic motion path resulting in a rigid body motion (here a translation $g$) of the initial shape. Each of the drawings in the left column visualizes the boundary constraints from a sub-Riemannian point of view.