Stabilization of Mechanical Systems on Semidirect Product Lie Groups with Broken Symmetry via Controlled Lagrangians

Abstract

Motivated by the problem of stabilizing bottom-heavy underwater vehicles, we find the matching condition for controlled Lagrangians via the kinetic shaping for mechanical systems on a semidirect product Lie group with broken symmetry. Of particular interest is the problem of stabilizing a steady translational motion of an underwater vehicle. We show that the control by the kinetic shaping stabilizes the equilibrium, and also that an additional dissipative control renders it asymptotically stable.