In this paper, a geometrical approach is developed to generate simultaneously optimal (or near-optimal) smooth paths for a set of non-holonomic robots, moving only forward in a 2D environment cluttered with static and moving obstacles. The robots environment is represented by a 3D geometric entity called Bump-Surface, which is embedded in a 4D Euclidean space. The multi-motion planning problem (MMPP) is resolved by simultaneously finding the paths for the set of robots represented by monoparametric smooth C2 curves onto the Bump-Surface, such that their inverse images onto the initial 2D workspace satisfy the optimization motion-planning criteria and constraints. The MMPP is expressed as an optimization problem, which is solved on the Bump-Surface using a genetic algorithm. The performance of the proposed approach is tested through a considerable number of simulated 2D dynamic environments with car-like robots.
