Energy Shaping for Systems with Two Degrees of Underactuation

In this thesis we are going to study the energy shaping problem on controlled Lagrangian systems with degree of underactuation less than or equal to two. Energy shaping is a method of stabilization by designing a suitable feedback control force on the given controlled Lagrangian system so that the total energy of the feedback equivalent system has a non-degenerate minimum at the equilibrium. The feedback equivalent system can then be stabilized by a further dissipative force. Finding a feedback equivalent system requires solving a system of PDEs. The existence of solutions for this system of PDEs is guaranteed, under some conditions, in the case of one degree of underactuation. Higher degrees of underactuation, however, requires a more careful study on the system of PDEs, and we apply the formal theory of PDEs to achieve this purpose in the case of two degrees of underactuation. The thesis is divided into four chapters. First, we review the basic notion of energy shaping and state the results for the case of one degree of underactuation. We then devise a general scheme to solve the energy shaping problem with degree of underactuation equal to one, together with some examples to illustrate the general procedure. After that we review the tools from the formal theory of PDEs, as a preparation for solving the problem with two degrees of underactuation. We derive an equivalent involutive system of PDEs from which we can deduce the existence of solutions which suit the energy shaping requirement.