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。通过将含显式避碰约束的优化问题重新表述为非线性规划问题，我们提出了三种精确描述碰撞约束的新颖公式。第一种公式基于点与凸集分离的命题推导，并通过德摩根定律对该分离命题进行了证明。进而，借助超平面分离定理提出两种高效重表述。相较于现有对偶公式及第一种公式，这两种新表述显著减少了非线性规划问题中需优化的辅助变量数量与不等式约束数量。最后，在典型自动泊车场景下与现有最优方法对比，验证了所提公式的有效性。为体现通用性，我们设计了三种初始猜测来评估不同碰撞约束公式收敛至解所需的计算成本。结果表明：基于德摩根定律的方案与采用对偶公式的方案性能相当，而另外两种基于超平面分离定理的方案则额外具有更低计算资源消耗的优势。