  This paper proposes a new set of conditions for exactly representing
collision avoidance constraints within optimization-based motion planning
algorithms. The conditions are continuously differentiable and therefore
suitable for use with standard nonlinear optimization solvers. The method
represents convex shapes using a support function representation and is
therefore quite general. For collision avoidance involving polyhedral or
ellipsoidal shapes, the proposed method introduces fewer variables and
constraints than existing approaches. Additionally the proposed method can be
used to rigorously ensure continuous collision avoidance as the vehicle
transitions between the discrete poses determined by the motion planning
algorithm. Numerical examples demonstrate how this can be used to prevent
problems of corner cutting and passing through obstacles which can occur when
collision avoidance is only enforced at discrete time steps.
