Parametrization of Catmull-Clark subdivision surfaces for posture generation

In this paper we propose a method to build a smooth and non-singular map from the unit sphere to a Catmull-Clark subdivision surface. We use a tailored ray-casting algorithm to associate a point of the surface to each point of the sphere. This allows us to have smooth approximate representations for a large variety of (possibly non-convex) meshes that we can use to write numerically well-behaved constraints in gradient-based optimization. In particular, we address the writing of contact constraints between a robot and objects of its environment. We first detail our algorithm, taking care of speed and robustness, then show several use-cases for posture generations with a humanoid robot or a hand.