  This research focuses on trajectory planning problems for autonomous vehicles
utilizing numerical optimal control techniques. The study reformulates the
constrained optimization problem into a nonlinear programming problem,
incorporating explicit collision avoidance constraints. We present three novel,
exact formulations to describe collision constraints. The first formulation is
derived from a proposition concerning the separation of a point and a convex
set. We prove the separating proposition through De Morgan's laws. Then,
leveraging the hyperplane separation theorem we propose two efficient
reformulations. Compared with the existing dual formulations and the first
formulation, they significantly reduce the number of auxiliary variables to be
optimized and inequality constraints within the nonlinear programming problem.
Finally, the efficacy of the proposed formulations is demonstrated in the
context of typical autonomous parking scenarios compared with state of the art.
For generality, we design three initial guesses to assess the computational
effort required for convergence to solutions when using the different collision
formulations. The results illustrate that the scheme employing De Morgan's laws
performs equally well with those utilizing dual formulations, while the other
two schemes based on hyperplane separation theorem exhibit the added benefit of
requiring lower computational resources.
