  Mechanical systems naturally evolve on principal bundles describing their
inherent symmetries. The ensuing factorization of the configuration manifold
into a symmetry group and an internal shape space has provided deep insights
into the locomotion of many robotic and biological systems. On the other hand,
the property of differential flatness has enabled efficient, effective planning
and control algorithms for various robotic systems. Yet, a practical means of
finding a flat output for an arbitrary robotic system remains an open question.
In this work, we demonstrate surprising new connections between these two
domains, for the first time employing symmetry directly to construct a flat
output. We provide sufficient conditions for the existence of a trivialization
of the bundle in which the group variables themselves are a flat output. We
call this a geometric flat output, since it is equivariant (i.e. maintains the
symmetry) and is often global or almost-global, properties not typically
enjoyed by other flat outputs. In such a trivialization, the motion planning
problem is easily solved, since a given trajectory for the group variables will
fully determine the trajectory for the shape variables that exactly achieves
this motion. We provide a partial catalog of robotic systems with geometric
flat outputs and worked examples for the planar rocket, planar aerial
manipulator, and quadrotor.
