惯性主导运动系统最优步态的几何学
The Geometry of Optimal Gaits for Inertia-dominated Kinematic Systems
摘要：
隔离的机械系统--例如，那些漂浮在太空中、自由落体中或无摩擦表面上的物体，即使没有净角动量，也能够通过周期性地改变它们的形状来实现净旋转。类似地，沉浸在"完美流体"中的游泳者能够使用周期性形状变化来平移和旋转，即使游泳者-流体系统没有净线性或角动量。最后，系统完全受直接非完整约束（例如，被动轮）可以推动这些约束以在世界上移动。先前的工作已经表明，这些形状变化引起的净位移对应于步态包围的 * 约束曲率 * 的量。然而，为了正确评估或优化步态的效用，我们还必须考虑执行步态所需的时间或资源：每个周期产生小位移但可以在短时间内执行的步态可能比每个周期产生大位移但在相同的平均瞬时努力下完成一个周期需要更长时间的步态产生更快的平均速度。在本文中，我们考虑两个努力为基础的成本函数评估与执行这些周期的成本。对于这些成本函数中的每一个，我们证明，固定的平均瞬时成本为一个单位值，使我们能够转换的努力成本到执行时间成本为任何给定的步态周期。然后，我们说明如何约束曲率和这些成本之间的相互作用，导致最佳周期的特征几何形状，其中的步态轨迹类似于弹性箍从内部扩张的内部压力。
英文摘要：
Isolated mechanical systems -- e.g., those floating in space, in free-fall, or on a frictionless surface -- are able to achieve net rotation by cyclically changing their shape, even if they have no net angular momentum. Similarly, swimmers immersed in "perfect fluids" are able to use cyclic shape changes to both translate and rotate even if the swimmer-fluid system has no net linear or angular momentum. Finally, systems fully constrained by direct nonholonomic constraints (e.g., passive wheels) can push against these constraints to move through the world. Previous work has demonstrated that the net displacement induced by these shape changes corresponds to the amount of *constraint curvature* that the gaits enclose. To properly assess or optimize the utility of a gait, however, we must also consider the time or resources required to execute it: A gait that produces a small displacement per cycle, but that can be executed in a short time, may produce a faster average velocity than a gait that produces a large displacement per cycle, but takes much longer to complete a cycle at the same average instantaneous effort. In this paper, we consider two effort-based cost functions for assessing the costs associated with executing these cycles. For each of these cost functions, we demonstrate that fixing the average instantaneous cost to a unit value allows us to transform the effort costs into time-to-execute costs for any given gait cycle. We then illustrate how the interaction between the constraint curvature and these costs leads to characteristic geometries for optimal cycles, in which the gait trajectories resemble elastic hoops distended from within by internal pressures.
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