This study proposes a unified optimization-based planning framework that addresses the precise and efficient navigation of a controlled object within a constrained region, while contending with obstacles. We focus on handling two collision avoidance problems, i.e., the object not colliding with obstacles and not colliding with boundaries of the constrained region. The object or obstacle is denoted as a union of convex polytopes and ellipsoids, and the constrained region is denoted as an intersection of such convex sets. Using these representations, collision avoidance can be approached by formulating explicit constraints that separate two convex sets, or ensure that a convex set is contained in another convex set, referred to as separating constraints and containing constraints, respectively. We propose to use the hyperplane separation theorem to formulate differentiable separating constraints, and utilize the S-procedure and geometrical methods to formulate smooth containing constraints. We state that compared to the state of the art, the proposed formulations allow a considerable reduction in nonlinear program size and geometry-based initialization in auxiliary variables used to formulate collision avoidance constraints. Finally, the efficacy of the proposed unified planning framework is evaluated in two contexts, autonomous parking in tractor-trailer vehicles and overtaking on curved lanes. The results in both cases exhibit an improved computational performance compared to existing methods.

翻译：本研究提出一种统一的基于优化的规划框架，用于解决受控对象在受限区域内避开障碍物时的精确高效导航问题。我们重点处理两类碰撞规避问题：即对象不与障碍物发生碰撞，且不超出受限区域的边界。对象或障碍物被表示为凸多面体与椭球体的并集，受限区域则表示为此类凸集的交集。基于这些表示，碰撞规避可通过构建两类显式约束实现：分离两个凸集的约束（称为分离约束），以及确保一个凸集包含于另一凸集的约束（称为包含约束）。我们提出利用超平面分离定理构建可微分的分离约束，并采用S-过程和几何方法构建光滑的包含约束。相较于现有技术，所提出的约束表达形式可显著降低非线性规划问题的规模，并减少用于构建碰撞规避约束的辅助变量几何初始化需求。最后，该统一规划框架的有效性在牵引-挂车车辆的自主泊车与弯道超车两个场景中进行了评估。结果表明，两种场景下的计算性能均优于现有方法。