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.

翻译：运动几何力学模型揭示了机器人和动物如何利用环境交互将内部形状变化转化为世界中的位移，并将这一关系编码在“运动性地图”中。这类运动性地图的一个关键类别源于作用在系统各个身体部件上的（可能是各向异性的）线性阻力，其通过系统各部件的运动上的黎曼度量进行形式化描述。运动性地图可通过在系统整体运动上施加一个子黎曼约束来生成，在该约束下，由给定形状速度引起的位置速度是使摩擦耗散功率最小的那个。此类系统的运动是“几何的”，即系统最终到达的位置仅取决于系统经过的形状序列，而与形状变化的速率无关。本文中，我们考虑一类更一般的系统，其中阻力不仅可以是各向异性的（对前后和左右运动具有不同系数），还可以是非对称的（对前后运动具有不同系数）。形式上，在摩擦中引入非对称性将身体部件上的黎曼度量替换为芬斯勒度量。我们证明，构建系统运动性地图的子黎曼方法可自然扩展为子芬斯勒方法，并确定与子黎曼系统的约束曲率类似的系统属性，从而实现对系统运动能力的表征。