Fibered structures for discontinuous geometrical control of mechanical systems

Considers a mechanical system with symmetry, applying a discontinuous control law theory to the reduced equations. We begin with the general theory, without reduction: we consider a mechanical system (M,g), where M is the configuration manifold and g is the kinetic energy Riemannian metric. We introduce the control fibration π:M→N, where N⊂M describes some of the system´s degrees of freedom, so that the π-fibers represent the possible configurations of the system when some of the degrees of freedom are locked. The kinematic control law is a path u(t)∈N and a kinematic evolution of the controlled system compatible with the assigned control law is a path q(t)∈M that projects on u(t). By supposing that the control law can be physically implemented by suitable active constraints, we turn (M,g,π) into an under-actuated control system. Often, the solution of an optimal control problem for (M,g,π) is an optimal path u(t) which is a discontinuous function. We introduce the dynamic equations for (M,g,π) and we discuss the procedure to implement a discontinuous control law through a control completion. The discontinuous control is applicable to (M,g,π) if g satisfies a certain condition with respect to π. We then consider the case of a simple mechanical system with symmetry (Q,G,K), where G is a Lie group acting by isometries with respect to the kinetic energy metric K. We want to control the evolution of part of the shape variables in the shape space S=Q/G. Using Lagrangian reduction theory, we introduce equations describing the reduced dynamics on the reduced configuration space Q/G_m; these equations are written in terms of the Routhian function. Discontinuous control can be implemented if the same condition is satisfied by the reduced kinetic energy with respect to the fibration π_c:S→N