4. Dynamics of Direct Locomotion: The General Case

As mentioned earlier, a dynamic model is generally required to solve the direct model of shape-driven locomotion. Due to the principal fiber bundle structure, the derivation of this model warrants special attention. The Lie group structure allows us to replace the standard variational calculus (from which the Lagrange equations are derived), applied to the charts of any manifold, with an intrinsic calculus applied directly to the group. Such a calculus has the advantage of providing a formulation of the dynamics with a minimum of geometric nonlinearities. Indeed, in this approach, all nonlinearities introduced by rigid motions are due to the curvature of the group (which can be intuitively regarded as the geometric manifestation of noncommutativity on the algebraic side) and not to the parameterization. Euler exploited these virtues in his study of the spinning top before Lie groups were...