3. Dynamics of Direct Locomotion: The Kinematic Case

Based on the geometric concepts introduced earlier, the time evolution of the shape of an SMM defines a motion on the manifold S, while that of the reference body defines a motion in the Lie group G. Consequently, to solve the direct dynamics of shape-driven locomotion, we must establish a relationship between these two types of motion on the principal fiber bundle G × S. In general, such relations are given by a dynamic model linking forces and accelerations. In this case, the dynamics of the contacts between the system and its environment must be solved, as we will discuss in the next section. However, there is a particularly elegant case where locomotion is entirely governed by kinematics, that is, by relations between shape velocities and ensemble velocities. In this case, the contact model is encoded in what we will call a connection on the principal fiber bundle of configurations...