Controlled Invariants and Trajectory Planning for Underactuated Mechanical Systems

We study the problem of motion planning for underactuated mechanical systems. The idea is to reduce complexity by imposing via feedback a sufficient number of invariants and then to compute a projection of the dynamics onto an induced invariant sub-manifold of the closed-loop system. The inspiration comes from two quite distant methods, namely the method of virtual holonomic constraints, originally invented for planning and orbital stabilization of gaits of walking machines, and the method of controlled Lagrangians, primarily invented as a nonlinear technique for stabilization of (relative) equilibria of controlled mechanical systems. Both of these techniques enforce the presence of particular invariants that can be described as level sets of conserved quantities induced in the closed-loop system. We link this structural feature of both methods to a procedure to transform a Lagrangian system with control inputs via a feedback action into a control-free Lagrangian system with a sufficient number of first integrals for the full state space or an invariant sub-manifold. In both cases, this transformation allows efficient (analytical) description of a new class of trajectories of forced mechanical systems appropriate for further orbital stabilization. For illustration purposes, we approach the challenging problem for a controlled mechanical system with two passive degrees of freedom: planning periodic (or bounded) forced upper-hemisphere trajectories of the spherical pendulum on a puck. Another example of the technique is separately reported in [21].