  We extend the method of controlled Lagrangians with kinetic shaping to those
mechanical systems on semidirect product Lie groups with broken symmetry, more
specifically to the Euler--Poincar\'e equations with advected parameters. We
find a matching condition for the controlled Lagrangian for such systems whose
configuration manifold is a general semidirect product Lie group $\mathsf{G}
\ltimes V$. Our motivating examples are a bottom-heavy underwater vehicle and a
top spinning on a movable base. Their configuration space is the special
Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \mathbb{R}^{3}$, where
the $\mathsf{SE}(3)$-symmetry is broken by the gravity. The controls resulting
from the matching condition stabilize unstable equilibria of these examples.
Furthermore, the matching helps us find additional dissipative controls that
asymptotically stabilize those unstable equilibria.
