Estimating the fastest trajectory is one of the main challenges in autonomous vehicle research. It is fundamental that the vehicle determines its path not only to minimize travel time, but to arrive at the destination safely by avoiding any obstacles that may be in collision route. In this paper, we consider estimating the trajectory and acceleration functions of the trip simultaneously with an optimization objective function. By approximating the trajectory and acceleration function with B-splines, we transform an infinite-dimensional problem into a finite-dimensional one. Obstacle avoidance and kinematic constraints are carried out with the addition of a penalization function that penalizes trajectories and acceleration functions that do not satisfy the vehicles’ constraints or that are in a collision route with other obstacles. Our approach is designed to model observations of the obstacles that contain measurement errors, which incorporates the realistic stochasticity of radars and sensors. We show that, as the number of observations increases, the estimated optimization function converges to the optimal one where the obstacles’ positions are known. Moreover, we show that the estimated optimization function has a minimizer and that its minimizers converge to the minimizers of the optimization function involving the true threat zones.

估计最快的轨迹是自动驾驶汽车研究的主要挑战之一。至关重要的是，车辆不仅要确定行驶路径，以减少行驶时间，而且还要避免碰撞路线中的任何障碍物，从而安全地到达目的地。在本文中，我们考虑与优化目标函数同时估计行程的轨迹和加速度函数。通过用B样条曲线逼近轨迹和加速度函数，我们将无穷维问题转化为有限维问题。通过添加罚分功能来对障碍物进行避开和运动学约束，该罚分功能对不满足车辆约束或与其他障碍物发生碰撞的轨迹和加速度函数进行罚分。我们的方法旨在对包含测量误差的障碍物的观察进行建模，其中包含了雷达和传感器的现实随机性。我们表明，随着观察次数的增加，估计的优化函数收敛到已知障碍物位置的最优函数。此外，我们表明估计的优化函数具有最小化器，并且其最小化器收敛到涉及真实威胁区域的优化函数的最小化器。