Intrinsic vector-valued symmetric form for simple mechanical control systems in the nonzero velocity setting

We obtain an intrinsic vector-valued symmetric bilinear form that can be associate with an underactuated simple mechanical control system. We determine properties of the form which serve as necessary conditions for driving underactuated simple mechanical control systems to rest. We also determine properties of the form that serve as sufficient conditions for driving a simple mechanical systems underactuated by one control to an epsiv-neighborhood of rest from an arbitrary initial configuration and velocity. These conditions are computable and coordinate invariant. We focus on the case where the symmetric form is real-valued and indefinite on the entire configuration manifold. Our technical results give rise to a nonlinear control law that drives these systems to an epsiv-neighborhood of rest given an arbitrary initial configuration, velocity and epsiv > 0.