Inertia-dominated mechanical systems can achieve net displacement by 1) periodically changing their shape (known as kinematic gait) and 2) adjusting their inertia distribution to utilize the existing nonzero net momentum (known as momentum gait). Therefore, finding the gait that most effectively utilizes the two types of locomotion in terms of the magnitude of the net momentum is a significant topic in the study of locomotion. For kinematic locomotion with zero net momentum, the geometry of optimal gaits is expressed as the equilibria of system constraint curvature flux through the surface bounded by the gait, and the cost associated with executing the gait in the metric space. In this paper, we identify the geometry of optimal gaits with nonzero net momentum effects by lifting the gait description to a time-parameterized curve in shape-time space. We also propose the variational gait optimization algorithm corresponding to the lifted geometric structure, and identify two distinct patterns in the optimal motion, determined by whether or not the kinematic and momentum gaits are concentric. The examples of systems with and without fluid-added mass demonstrate that the proposed algorithm can efficiently solve forward and turning locomotion gaits in the presence of nonzero net momentum. At any given momentum and effort limit, the proposed optimal gait that takes into account both momentum and kinematic effects outperforms the reference gaits that each only considers one of these effects.

翻译：惯性主导的机械系统可通过以下方式实现净位移：1）周期性改变自身形状（运动学步态），2）调整惯性分布以利用存在的非零净动量（动量步态）。因此，从净动量幅值角度寻找最有效利用这两种运动模式的步态，是运动学研究中的重要课题。针对零净动量的运动学运动，最优步态的几何特性表现为步态所围曲面上的系统约束曲率通量平衡态，以及度量空间中执行步态的关联代价。本文通过将步态描述提升至形状-时间空间中的时间参数化曲线，揭示了具有非零净动量效应的最优步态几何特性。我们提出了对应提升几何结构的变分步态优化算法，并识别出由运动学与动量步态是否同心决定的两种最优运动模式。包含与不含附加流体质量的系统示例表明，所提算法能有效求解存在非零净动量时的前进与转向步态。在给定动量和能耗约束下，同时考虑动量与运动学效应的最优步态性能优于仅考虑单一效应的基准步态。