Sub--Riemannian boundary value problems for Optimal Geometric Locomotion

2026-02-12

AI Summary

This study investigates the optimal locomotion trajectories of slender, shape-changing swimmers—such as snakes and sperm—in resistive environments, jointly accounting for external fluid dissipation and internal energy costs associated with shape deformation. By formulating the problem as a boundary-value problem in sub-Riemannian geometry and applying the principle of least dissipation in a Lagrangian framework, the authors derive efficient swimming strategies through sub-Riemannian geodesics. The key innovation lies in the first unified integration of external displacement dissipation and internal metabolic or actuation energy, yielding a geometric locomotion model that better reflects biological and soft robotic systems and accommodates diverse boundary conditions. Combining variational principles, spatiotemporally consistent discretization, and numerical optimization, the method successfully reproduces classical results such as the Purcell swimmer and generates optimal gaits that closely match real biological motion, revealing new mechanisms for generalized low-Reynolds-number swimming.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• sub-Riemannian geometry
• optimal locomotion
• shape change
• energy dissipation
• boundary value problems

Innovation

• sub-Riemannian geometry
• optimal locomotion
• shape-change-induced motion
• least-dissipation principle
• geometric mechanics