  We introduce a mathematical technique based on modifying a given Riemannian
metric and we investigate its applicability to controlling and stabilizing
constrained mechanical systems. In essence our result is based on the
construction of a complete Riemannian metric in the modified space where the
constraint is included. In particular this can be applied to the controlled
Lagrangians technique Bloch et al. [2000b, 2001] modifying its metric to
additionally cover mechanical systems with configuration constraints via
control. The technique used consists of approximating incomplete Riemannian
metrics by complete ones, modifying the evolution near a boundary and finding a
controller satisfying a given design criterion.
