Abstract
The purpose of this study is to develop a methodology for processing vehicle trajectory data which are presented as a series of discrete positions of vehicles recorded over consecutive time intervals. The framework combines vehicle trajectory smoothing and imputation, ensuring that speeds and higher-order derivatives of positions are consistently defined as symplectic differences in positions, while adhering to physically meaningful bounds determined by traffic laws, drivers’ behaviors, and vehicle characteristics.
To remove the outliers and high-frequency noises in speeds and higher-order derivatives, we incorporate some basic principles, including internal consistency, bounded speeds and higher-order derivatives, and minimum MAE between the raw and smoothed positions, based on physical properties and empirical observations. We propose an iterative method. One iteration comprises four types of calculations: differentiation, correction, smoothing, and integration. We adopt the adaptive average method for correction, the Gaussian filter for smoothing, and minimizing the MAEs as the objective in integration. The efficacy of the method is numerically shown with the NGSIM data. However, it is mathematically challenging to demonstrate when the iterations converge or even that the iterations can converge, leading us to develop more mathematically tractable techniques that can either be proved to converge or get rid of iterations.
We then propose a simplified iterative moving average method that makes the ranges of the smoothed speeds, acceleration rates, and jerks align with physical meaning, while preserving the average speeds or total travel distance for a specified time duration segment of a vehicle’s trajectory. Theoretically, we prove that without termination, the speed converges to a constant value after an infinite number of iterations, ensuring the termination of our method and physically meaningful ranges in speeds and their derivatives. Numerically, we demonstrate the advantages of the method in achieving physically and behaviorally meaningful ranges by applying it to the NGSIM dataset and comparing the results with manually re-extracted data and traditional filtering methods.
As another extension of the first smoothing method, We propose a two-step quadratic programming method that incorporates insights into human behavior, particularly the tendency to minimize jerks during motion, and integrates prior position errors derived from pixel length in video images. This method operates without the need for iterative processes, facilitating a single-round solution. Mathematically, we establish the existence and uniqueness of solutions to the quadratic programming problems, thus ensuring the well-defined nature of the method. Numerically, using NGSIM data, we compare the method with an existing approach with respect to the manually re-extracted ones and show the robustness of the method upon the highD data.
In addition, we investigate the scenarios involving missing portions of trajectories. In the last part of this dissertation, we consider segment scenarios where leading and trailing vehicles’ trajectories are obtainable through mobile sensors, while those of intermediate vehicles require imputation based on detected entering and exiting times from loop detectors, and propose a three-step quadratic programming method for longitudinal trajectory imputation of fully sampled vehicles. The method ensures maintaining safe inter-vehicle spacing and adheres to physically meaningful speed, acceleration, and jerk ranges. Using NGSIM and highD data, we demonstrate the great performance of the method in imputing trajectories for three-, four-, five-, and six-vehicle platoons and illustrate its successful application in capturing the true conditions of a mixed-traffic system including 10% connected vehicles (CVs) and 10% CAVs.
