What Can Algebraic Topology and Differential Geometry Teach Us About Intrinsic Dynamics and Global Behavior of Robots?

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.