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.

翻译：我们提出了一种用于细长运动体（例如在沙地上滑行的蛇）最优形状变化诱导运动的几何模型。在这些场景中，物体在世界坐标系中的运动完全由其采用的形状序列决定。具体而言，我们将拉格朗日最小耗散原理表述为边值问题，其解由亚黎曼测地线给出。值得注意的是，我们的几何模型不仅考虑了物体在环境中位移所耗散的能量，还包含了动物新陈代谢或机器人执行器为诱发弯曲和拉伸等形状变化所消耗的能量，从而全面捕捉了整体运动效率。我们的连续模型结合一致的时间和空间离散化，能够数值计算三种不同类型边界条件下的亚黎曼测地线，即固定初始与目标形体、限制为周期性运动，或仅规定物体位移与朝向。所得最优变形步态在定性上匹配了蛇类和精子等生物体的观测运动轨迹，以及如珀塞尔游泳者等低维系统的已知最优性结果。此外，由于在几何上比先前框架更具灵活性，我们的模型能够为广义珀塞尔游泳者等系统的运动机制提供新的见解。相关代码已公开。