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解决自动驾驶车辆的轨迹规划问题。研究将含显式碰撞规避约束的优化问题重构为非线性规划问题，提出了三种描述碰撞约束的新型精确表达范式。第一种表述源自点与凸集分离命题，通过德摩根定律给出该分离命题的证明；进而基于超平面分离定理提出两种高效重构方法。相较于现有对偶表述及第一种方法，新方法显著减少了非线性规划问题中待优化辅助变量与不等式约束的数量。最后，在典型自动泊车场景中验证了所提方法的有效性。为评估不同碰撞约束表达方式收敛解的计算成本，我们设计了三种通用初始猜测条件。结果表明：采用德摩根定律的方案与基于对偶表述的方案性能相当，而基于超平面分离定理的两种方案在降低计算资源消耗方面具有显著优势。