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Differential flatness enables efficient planning and control for
underactuated robotic systems, but we lack a systematic and practical means of
identifying a flat output (or determining whether one exists) for an arbitrary
robotic system. In this work, we leverage recent results elucidating the role
of symmetry in constructing flat outputs for free-flying robotic systems. Using
the tools of Riemannian geometry, Lie group theory, and differential forms, we
cast the search for a globally valid, equivariant flat output as an
optimization problem. An approximate transcription of this continuum
formulation to a quadratic program is performed, and its solutions for two
example systems achieve precise agreement with the known closed-form flat
outputs. Our results point towards a systematic, automated approach to
numerically identify geometric flat outputs directly from the system model,
particularly useful when complexity renders pen and paper analysis intractable.
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