Controlled Lagrangians and Stabilization of Euler-Poincaré Mechanical Systems with Broken Symmetry

We extend the method of Controlled Lagrangians to Euler-Poincaré mechanical systems with broken symmetry, and find asymptotic stabilizing controls of unstable equilibria of such mechanical systems. Our motivating example is a top spinning on a movable base: The gravity breaks the symmetry with respect to the three-dimensional rotations and translations of the system, and also renders the upright spinning equilibrium unstable. We formulate the system as Euler-Poincaré equations with advected parameters using semidirect Lie group $\mathrm{SE}(3)⋉(R4)\ast$, and find a control that is applied to the base to asymptotically stabilize the equilibrium.