Mixed integer/LMI programs for low-level path planning

Trajectory generation methods use a finite dimensional parameterization of solutions and impose constraints on a finite grid in time. They result in large feasibility problems and are at best only sufficient. We show that there is no need to grid if the admissible trajectories are restricted to polynomials of degree no larger than a specified bound. A finite dimensional necessary and sufficient condition, whose size depends only the assumed degree bound, is presented. We also give a systematic procedure to approximate non-convex constraints. The procedure leads to mixed integer/LMI programs. The techniques are used to improve some existing trajectory generation methods and develop new approximation methods. A numerical example is provided for illustration.