Discrete and periodic contact switching is a key characteristic of steady-state legged locomotion. This paper introduces a framework for modeling and analyzing this contact-switching behavior through the framework of geometric mechanics on a toy robot model that can make continuous limb swings and discrete contact switches. The kinematics of this model form a hybrid shape-space and by extending the generalized Stokes' theorem to compute discrete curvature functions called \textit{stratified panels}, we determine average locomotion generated by gaits spanning multiple contact modes. Using this tool, we also demonstrate the ability to optimize gaits based on the system's locomotion constraints and perform gait reduction on a complex gait spanning multiple contact modes to highlight the method's scalability to multilegged systems.

翻译：离散和周期性接触切换是稳态腿部运动的关键特征。本文通过几何力学框架，在一个能够实现连续肢体摆动和离散接触切换的玩具机器人模型上，引入了一种建模与分析该接触切换行为的方法。该模型的运动学构成了一个混合形状空间。通过将广义斯托克斯定理扩展至计算称为“分层面板”的离散曲率函数，我们确定了跨越多种接触模式的步态所产生的平均运动。利用这一工具，我们还展示了基于系统运动约束优化步态的能力，并对跨越多种接触模式的复杂步态进行了降维处理，以突出该方法对多足系统的可扩展性。