  We apply the method of controlled Lagrangians by potential shaping to
Euler--Poincar\'e mechanical systems with broken symmetry. We assume that the
configuration space is a semidirect product $\mathsf{G} \ltimes V$ of a Lie
group $\mathsf{G}$ and a vector space $V$ with a particular interest in those
systems whose configuration space 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 the representations and advected parameters.
Our motivating examples are a heavy top spinning on a movable base and an
underwater vehicle with non-coincident centers of gravity and buoyancy. We
consider a few different control problems for these systems, and show that our
results give a general framework that reproduces our previous work on the
former example and also those of Leonard on the latter. Also, in one of the
latter cases, we demonstrate the advantage of our representation-based approach
by giving a simpler and more succinct formulation of the problem.
