  Hessian information speeds convergence substantially in motion optimization.
The better the Hessian approximation the better the convergence. But how good
is a given approximation theoretically? How much are we losing? This paper
addresses that question and proves that for a particularly popular and
empirically strong approximation known as the Gauss-Newton approximation, we
actually lose very little--for a large class of highly expressive objective
terms, the true Hessian actually limits to the Gauss-Newton Hessian quickly as
the trajectory's time discretization becomes small. This result both motivates
it's use and offers insight into computationally efficient design. For
instance, traditional representations of kinetic energy exploit the generalized
inertia matrix whose derivatives are usually difficult to compute. We introduce
here a novel reformulation of rigid body kinetic energy designed explicitly for
fast and accurate curvature calculation. Our theorem proves that the
Gauss-Newton Hessian under this formulation efficiently captures the kinetic
energy curvature, but requires only as much computation as a single evaluation
of the traditional representation. Additionally, we introduce a technique that
exploits these ideas implicitly using Cholesky decompositions for some cases
when similar objective terms reformulations exist but may be difficult to find.
Our experiments validate these findings and demonstrate their use on a
real-world motion optimization system for high-dof motion generation.
