Lie theory, Riemannian geometry, and the dynamics of coupled rigid bodies

In this article we formulate, in a Lie group setting, the equations of motion for a system of n coupled rigid bodies subject to holonomic constraints. A mapping f : M --> N is constructed, where M is the m-dimensional configuration manifold of the system, and N = Se(3) x...x SE(3) (n copies) is endowed with the left-invariant Riemannian metric h corresponding to the total kinetic energy of the system, where SE(3) is the special Euclidean group. The generalized inertia tensor of the system is given by the pullback metric f*h; the equations of motion are then the geodesic equations on M with respect to this metric. We show how this coordinate-free formulation leads directly to a factorization of the generalized inertia tensor of the form (SLHLS)-L-T-H-T, where S is a constant block-diagonal matrix consisting only of kinematic parameters, H is a constant block-diagonal matrix consisting only of inertial parameters, and L is a block lower-triangular matrix composed of Adjoint operators on se(3). Such a factorization is useful for various multibody system dynamics applications, e.g., inertial parameter identification, adaptive control, and design optimization. We also show how in many practical situations N can be reduced to a submanifold, thereby considerably simplifying the derivation of the equations of motion. Our geometric formulation not only suggests ways to choose the best coordinates for analysis and computation, but also provides high-level insight into the structure of the equations of motion.