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We apply the method of controlled Lagrangians by potential shaping toEuler--Poincar\'e mechanical systems with broken symmetry. We assume that theconfiguration space is a general semidirect product Lie group $\mathsf{G}\ltimes V$ with a particular interest in those systems whose configurationspace is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes\mathbb{R}^{3}$. The key idea behind the work is the use of representations of$\mathsf{G} \ltimes V$ and their associated advected parameters. Specifically,we derive matching conditions for the modified potential exploiting therepresentations and advected parameters. Our motivating examples are a heavytop spinning on a movable base and an underwater vehicle with non-coincidentcenters of gravity and buoyancy. We consider a few different control problemsfor these systems, and show that our results give a general framework thatreproduces our previous work on the former example and also those of Leonard onthe latter. Also, in one of the latter cases, we demonstrate the advantage ofour representation-based approach by giving a simpler and more succinctformulation of the problem.

Controlled Lagrangians and Stabilization of Euler--Poincar\'e Mechanical Systems with Broken Symmetry II: Potential Shaping