In this work, we present a nonlinear dynamics perspective on generating and connecting gaits for energetically conservative models of legged systems. In particular, we show that the set of conservative gaits constitutes a connected space of locally defined 1D submanifolds in the gait space. These manifolds are coordinate-free parameterized by energy level. We present algorithms for identifying such families of gaits through the use of numerical continuation methods, generating sets and bifurcation points. To this end, we also introduce several details for the numerical implementation. Most importantly, we establish the necessary condition for the Delassus' matrix to preserve energy across impacts. An important application of our work is with simple models of legged locomotion that are often able to capture the complexity of legged locomotion with just a few degrees of freedom and a small number of physical parameters. We demonstrate the efficacy of our framework on a one-legged hopper with four degrees of freedom.

翻译：在本文中，我们提出了一种非线性动力学视角，用于生成和连接能量守恒腿式系统模型中的步态。特别地，我们证明了守恒步态集合构成了步态空间中局部定义的一维子流形的连通空间。这些流形以能量水平为无坐标参数化。我们提出了通过使用数值延拓方法、生成集和分岔点来识别此类步态族的算法。为此，我们还介绍了数值实现中的若干细节。最重要的是，我们建立了德拉克萨斯矩阵在碰撞中保持能量守恒的必要条件。我们工作的一项重要应用是腿式运动的简单模型，这些模型通常能够仅通过少数自由度和少量物理参数捕捉腿式运动的复杂性。我们在一个具有四个自由度的单腿跳跃器上展示了我们框架的有效性。