Smooth approximation of polyhedral potential field in NMPC for obstacle avoidance

The sum function notion allows one to define a continuous, piecewise affine over a polyhedral support, surface which accurately penalizes the closeness to polyhedral obstacles. This, in turn, leads to a continuous and piecewise description of the potential field surface further used into an NMPC (Nonlinear Model Predictive Control) motion planning problem. We introduce and analyze their smooth versions to show significant computational speedup.We analyze the links between the piecewise and smooth surfaces (magnitude and location of critical points). The results are validated in simulation and shown to compare favourably with previous mixed-integer based formulations.