This research addresses the increasing demand for advanced navigation systems capable of operating within confined surroundings. A significant challenge in this field is developing an efficient planning framework that can generalize across various types of collision avoidance missions. Utilizing numerical optimal control techniques, this study proposes a unified optimization-based planning framework to meet these demands. 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：本研究旨在应对日益增长的、能够在受限环境中运行的高级导航系统需求。该领域的一个重要挑战是开发一种能够泛化于各类避障任务的高效规划框架。利用数值最优控制技术，本研究提出了一种统一的基于优化的规划框架以满足这些需求。我们重点处理两类避障问题：物体不与障碍物碰撞以及不与约束区域的边界碰撞。物体或障碍物被表示为凸多面体与椭球体的并集，约束区域则表示为这类凸集的交集。基于这些表示，避障问题可通过构建显式约束来处理：分离两个凸集，或确保一个凸集包含于另一凸集中，分别称为分离约束与包含约束。我们提出使用超平面分离定理构建可微分离约束，并利用S-过程与几何方法构建光滑包含约束。我们指出，相较于现有技术，所提公式能显著减少非线性规划问题规模，并降低用于构建避障约束的辅助变量中基于几何的初始化难度。最后，所提统一规划框架的有效性在两种场景中得到验证：铰接式车辆的自主泊车与弯道超车。两种案例的结果均显示，相较于现有方法，计算性能得到显著提升。