Local equilibrium controllability of multibody systems controlled via shape change

We study local equilibrium controllability of shape controlled multibody systems. The multibody systems are defined on a trivial principal fiber bundle by a Lagrangian that characterizes the base body motion and shape dynamics. A potential dependent on an advected parameter, e.g., uniform gravitational potential, is considered. This potential breaks base body symmetries, but a symmetry subgroup is assumed to exist. Symmetric product formulas are derived and important properties are obtained for symmetric products of horizontal shape control vector fields and a potential vector field that is dependent on an advected parameter. Based on these properties, sufficient conditions for local equilibrium controllability and local fiber equilibrium controllability are developed. These results are applied to two classes of shape controlled multibody systems in a uniform gravitational field: multibody attitude systems and neutrally buoyant multibody underwater vehicles.