  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 general semidirect product Lie group $\mathsf{G}
\ltimes V$ 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.
