  Algorithmic solutions for the motion planning problem have been investigated
for five decades. Since the development of A* in 1969 many approaches have been
investigated, traditionally classified as either grid decomposition, potential
fields or sampling-based. In this work, we focus on using numerical
optimization, which is understudied for solving motion planning problems. This
lack of interest in the favor of sampling-based methods is largely due to the
non-convexity introduced by narrow passages. We address this shortcoming by
grounding the solution in differential geometry. We demonstrate through a
series of experiments on 3 Dofs and 6 Dofs narrow passage problems, how
modeling explicitly the underlying Riemannian manifold leads to an efficient
interior-point non-linear programming solution.
