Optimization-based methods are commonly applied in autonomous driving trajectory planners, which transform the continuous-time trajectory planning problem into a finite nonlinear program with constraints imposed at finite collocation points. However, potential violations between adjacent collocation points can occur. To address this issue thoroughly, we propose a safety-guaranteed collision-avoidance model to mitigate collision risks within optimization-based trajectory planners. This model introduces an embodied footprint, an enlarged representation of the vehicle's nominal footprint. If the embodied footprints do not collide with obstacles at finite collocation points, then the ego vehicle's nominal footprint is guaranteed to be collision-free at any of the infinite moments between adjacent collocation points. According to our theoretical analysis, we define the geometric size of an embodied footprint as a simple function of vehicle velocity and curvature. Particularly, we propose a trajectory optimizer with the embodied footprints that can theoretically set an appropriate number of collocation points prior to the optimization process. We conduct this research to enhance the foundation of optimization-based planners in robotics. Comparative simulations and field tests validate the completeness, solution speed, and solution quality of our proposal.

翻译：基于优化的方法通常应用于自动驾驶轨迹规划器，其将连续时间轨迹规划问题转化为一个有限非线性规划问题，并在有限配置点处施加约束。然而，相邻配置点之间可能存在潜在的违规现象。为彻底解决这一问题，我们提出了一种安全保证的碰撞避免模型，以减轻基于优化轨迹规划器中的碰撞风险。该模型引入了一种具身足迹，即车辆名义足迹的放大表示。若具身足迹在有限配置点处不与障碍物碰撞，则可保证自车名义足迹在相邻配置点之间的任意无限时刻均无碰撞。根据我们的理论分析，我们将具身足迹的几何尺寸定义为车辆速度和曲率的简单函数。特别地，我们提出了一种采用具身足迹的轨迹优化器，其可在优化过程之前从理论上设定适当的配置点数量。本研究旨在增强机器人学中基于优化规划器的基础。对比仿真和现场测试验证了我们提出的完整性、求解速度和解质量。