Many motion planning algorithms for automated driving require estimating the probability of collision (POC) to account for uncertainties in the measurement and estimation of the motion of road users. Common POC estimation techniques often utilize sampling-based methods that suffer from computational inefficiency and a non-deterministic estimation, i.e., each estimation result for the same inputs is slightly different. In contrast, optimization-based motion planning algorithms require computationally efficient POC estimation, ideally using deterministic estimation, such that typical optimization algorithms for motion planning retain feasibility. Estimating the POC analytically, however, is challenging because it depends on understanding the collision conditions (e.g., vehicle's shape) and characterizing the uncertainty in motion prediction. In this paper, we propose an approach in which we estimate the POC between two vehicles by over-approximating their shapes by a multi-circular shape approximation. The position and heading of the predicted vehicle are modelled as random variables, contrasting with the literature, where the heading angle is often neglected. We guarantee that the provided POC is an over-approximation, which is essential in providing safety guarantees. For the particular case of Gaussian uncertainty in the position and heading, we present a computationally efficient algorithm for computing the POC estimate. This algorithm is then used in a path-following stochastic model predictive controller (SMPC) for motion planning. With the proposed algorithm, the SMPC generates reproducible trajectories while the controller retains its feasibility in the presented test cases and demonstrates the ability to handle varying levels of uncertainty.

翻译：自动驾驶的许多运动规划算法需要估计碰撞概率（POC），以考虑道路使用者运动测量与估计中的不确定性。常见的POC估计技术通常采用基于采样的方法，这些方法存在计算效率低和估计结果非确定性的问题，即相同输入对应的每次估计结果略有差异。相比之下，基于优化的运动规划算法需要计算高效的POC估计，理想情况下应采用确定性估计，以使典型的运动规划优化算法保持可行性。然而，解析地估计POC具有挑战性，因为它依赖于对碰撞条件（例如车辆形状）的理解以及对运动预测不确定性的表征。本文提出一种方法，通过使用多圆形形状近似对两车形状进行过近似来估计它们之间的POC。与现有文献中常忽略航向角不同，我们将预测车辆的位置和航向建模为随机变量。我们保证所提供的POC是过近似估计，这对于提供安全保障至关重要。针对位置和航向存在高斯不确定性的特定情况，我们提出一种计算高效的POC估计算法。该算法随后被应用于运动规划中的路径跟踪随机模型预测控制器（SMPC）。通过所提出的算法，SMPC能够生成可复现的轨迹，同时控制器在所示测试案例中保持可行性，并展现出处理不同不确定性水平的能力。