Generalized Serret-Andoyer transformation and applications for the controlled rigid body

The Serret-Andoyer transformation is a classical method for reducing the fr ee rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimension al Hamiltonian flow. First, we show that this transformation is the computa tion, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein) reduction associated with the lifted left-action of SO(3) on T*SO(3)-a gene ralization and extension of Noether's theorem far Hamiltonian systems with symmetry. In fact, we go on to generalize the Serret-Andoyer transformation to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperrg ular Hamiltonian functions. Interpretations of the Serret-Andoyer variables , both as Eulerian coordinates and as canonical coordinates of the co-adjoi nt orbit, are given. Next, we apply the result obtained to the controlled r igid body with momentum wheels. For the class of Hamiltonian controls that preserve the symmetry on T*SO(3), the closed-loop motion of the main body c an again be reduced to canonical form. This simplifies the stability proof for relative equilibria, which then amounts to verifying the classical Lagr ange-Dirichlet criterion. Additionally, issues regarding numerical integrat ion of closed-loop dynamics are also discussed. Part of this work has been presented in [16].