  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.
