Asymmetric Friction in Geometric Locomotion

2025-12-27

AI Summary

Classical geometric kinematics models friction as isotropic or merely anisotropic (e.g., left/right or forward/backward asymmetry), failing to capture realistic asymmetric dissipation where forward and backward resistance coefficients differ. Method: We introduce asymmetric friction into the theoretical framework by modeling it as a Finsler metric—generalizing classical sub-Riemannian locomotion theory to the sub-Finsler setting—and develop a Finsler-geometric locomotion mapping theory. We propose a novel “quasi-constraint curvature” metric to quantitatively characterize system mobility. Results: Our theory rigorously characterizes the reachable configuration set and mobility boundary under asymmetric friction, revealing how asymmetry fundamentally governs net displacement direction, energetic efficiency, and optimal gait topology. This establishes a universal geometric mechanics foundation for soft robotics, biological locomotion modeling, and other underactuated systems exhibiting directional dissipation.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

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.

Problem

• Model locomotion with asymmetric friction using Finsler metrics
• Extend sub-Riemannian approach to sub-Finslerian for motility maps
• Characterize motion capabilities under anisotropic and asymmetric drag

Innovation

• Using Finsler metrics for asymmetric friction modeling
• Extending sub-Riemannian to sub-Finslerian motility map construction
• Characterizing motion via constraint curvature analog properties