Numerical optimization-based methods are among the prevalent trajectory planners for autonomous driving. In a numerical optimization-based planner, the nominal continuous-time trajectory planning problem is discretized into a nonlinear program (NLP) problem with finite constraints imposed on finite collocation points. However, constraint violations between adjacent collocation points may still occur. This study proposes a safety-guaranteed collision-avoidance modeling method to eliminate the collision risks between adjacent collocation points in using numerical optimization-based trajectory planners. A new concept called embodied box is proposed, which is formed by enlarging the rectangular footprint of the ego vehicle. If one can ensure that the embodied boxes at finite collocation points are collide-free, then the ego vehicle's footprint is collide-free at any a moment between adjacent collocation points. We find that the geometric size of an embodied box is a simple function of vehicle velocity and curvature. The proposed theory lays a foundation for numerical optimization-based trajectory planners in autonomous driving.

翻译：数值优化方法在自动驾驶轨迹规划领域应用广泛。在数值优化规划器中，名义上的连续时间轨迹规划问题被离散化为非线性规划问题，仅对有限配置点施加约束。然而，相邻配置点之间仍可能出现约束违规现象。本研究提出一种安全保障碰撞规避建模方法，以消除采用数值优化轨迹规划器时相邻配置点间的碰撞风险。本文提出具身箱体这一新概念，该概念通过扩展自车矩形足迹形成。若能确保有限配置点处的具身箱体无碰撞，则自车足迹在相邻配置点间的任意时刻均无碰撞。研究发现，具身箱体的几何尺寸可简化为车辆速度与曲率的函数。本理论为自动驾驶中基于数值优化的轨迹规划器奠定了理论基础。