A convex polynomial model for planar sliding mechanics

We propose a polynomial model for planar sliding mechanics. For the force-motion mapping, we treat the set of generalized friction loads as the 1-sublevel set of a polynomial whose gradient directions correspond to generalized velocities. The polynomial is confined to be convex even-degree homogeneous in order to obey the maximum work inequality, symmetry, shape invariance in scale, and fast invertibility. We present a simple and statistically efficient model identification procedure using a sum-of-squares convex relaxation. We then derive the kinematic contact model that resolves the contact modes and instantaneous object motion given a position controlled manipulator action. The inherently stochastic object-to-surface friction distributions are modeled by sampling polynomial parameters from distributions that preserve sum-of-squares convexity. Thanks to the model smoothness, the mechanics of patch contact is captured while being computationally efficient without mode selection at support points. Simulation and robotic experiments on pushing and grasping validate the accuracy and efficiency of our approach.