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，机器人被视为通用运动生成机器。其设计主要基于运动学考量，而期望的动力学特性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，可首先考虑机器人的内在动力学特性，并依据任务需求对其进行优化。因此，需要更深入地理解机器人系统的无控内在动力学。本文聚焦于周期轨道这一基础动力学特性及其广泛的应用价值。代数拓扑与微分几何为周期轨道的存在性提供了若干基本定理。我们以最简单的多体系统——重力作用下的双摆为例，展示了该简单系统已具备丰富的周期轨道类别，并将其归纳为三类：环面轨道、盘面轨道及非线性模态。其中部分轨道通过几何直觉发现，部分则借助数值模拟与采样获得。