Modeling traffic flow at intersections is essential for the design, control, and management of intersections. A challenging feature of microscopic modeling vehicular movement at intersections is that drivers can choose among an infinite number of alternative traveling paths and speeds. This makes it fundamentally different from structured straight road sections with lanes. This study proposes a novel method to model the trajectories of vehicles in two-dimensional space and speed. Based on optimal control theory, it assumes drivers schedule their driving behavior, including the steering and acceleration, to minimize the predicted costs. The costs contain the running costs, which consist of the travel time and driving smoothness (longitudinally and laterally), and the terminal cost, which penalizes the deviations from the desired final state. Different than conventional methods, the vehicle motion dynamics are formulated in distance rather than in time. The model is solved by an iterative numerical solution algorithm based on the Minimum Principle of Pontryagin. The descriptive power, plausibility, and accuracy of the proposed model are investigated by comparing the calculated results under several cases, which can be solved from symmetry or analytically. The proposed model is further calibrated and validated using empirical trajectory data, and the quality of the predicted trajectory is confirmed. Qualitatively, the optimal trajectory changes in the range of the shortest path and smoothest path under different weights of the running cost. The proposed model can be used as a starting point and extended with more considerations of intersection operation in the real world for future applications.

对交叉路口的交通流进行建模对于交叉路口的设计，控制和管理至关重要。在交叉路口对车辆运动进行微观建模的一个挑战性特征是，驾驶员可以在无数种可选的行驶路径和速度之间进行选择。这从根本上不同于带车道的结构化直道路段。这项研究提出了一种在二维空间和速度中对车辆轨迹建模的新颖方法。基于最佳控制理论，它假设驾驶员安排他们的驾驶行为，包括转向和加速，以最大程度地降低预测成本。这些成本包含运行成本，该运行成本包括行驶时间和行驶平稳性（纵向和横向），以及最终成本，该成本会惩罚与所需最终状态的偏差。与常规方法不同，车辆运动动力学以距离而不是时间来表示。通过基于庞特里亚金最小原理的迭代数值解算法对模型进行求解。通过比较几种情况下的计算结果，研究了所提模型的描述能力，合理性和准确性，这可以通过对称或解析来解决。利用经验轨迹数据对所提出的模型进行进一步的校准和验证，并确认了预测轨迹的质量。定性地，在不同的运行成本权重下，最优轨迹在最短路径和最平滑路径的范围内变化。