Traditionally, robots are regarded as universal motion generation machines. They are designed mainly by kinematics considerations while the desired dynamics is imposed by strong actuators and high-rate control loops. As an alternative, one can first consider the robot's intrinsic dynamics and optimize it in accordance with the desired tasks. Therefore, one needs to better understand intrinsic, uncontrolled dynamics of robotic systems. In this paper we focus on periodic orbits, as fundamental dynamic properties with many practical applications. Algebraic topology and differential geometry provide some fundamental statements about existence of periodic orbits. As an example, we present periodic orbits of the simplest multi-body system: the double-pendulum in gravity. This simple system already displays a rich variety of periodic orbits. We classify these into three classes: toroidal orbits, disk orbits and nonlinear normal modes. Some of these we found by geometrical insights and some by numerical simulation and sampling.

翻译：传统上，机器人被视为通用的运动生成机器。其设计主要基于运动学考虑，而期望的动力学行为则通过强力执行器与高频率控制回路来强制实现。作为一种替代方案，我们可以优先考虑机器人的内在动力学特性，并使其与目标任务相适应。因此，我们需要更深入地理解机器人系统的固有且不受控的动力学特性。本文聚焦于周期性轨道这一具有广泛应用前景的基础动力学特性。代数拓扑与微分几何为周期性轨道的存在性提供了若干基础性论断。我们以最简单的多体系统——重力场中的双摆为例，展示了其丰富的周期性轨道类型。这些轨道被分为三类：环形轨道、盘状轨道及非线性模态。其中部分轨道通过几何学洞察发现，其余则通过数值仿真与采样获得。