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.

翻译：本研究聚焦于利用数值最优控制技术的自动驾驶车辆轨迹规划问题。研究将约束优化问题重新表述为非线性规划问题，并引入显式碰撞规避约束。我们提出了三种精确的新型公式来描述碰撞约束。第一个公式源于关于点与凸集分离的命题，并通过德摩根定律证明该分离命题。随后，基于超平面分离定理，我们提出了两种高效的重构公式。与现有的对偶公式及第一个公式相比，这两种新公式显著减少了非线性规划问题中待优化的辅助变量数量以及不等式约束数量。最后，在典型自动泊车场景中，通过与现有最优方法对比，验证了所提公式的有效性。为体现普适性，我们设计了三种初始猜测方案，以评估采用不同碰撞公式时收敛至解所需的计算量。结果表明：基于德摩根定律的公式与基于对偶公式的方案性能相当，而基于超平面分离定理的另外两种方案在降低计算资源需求方面展现出额外优势。