Highlights • Formulate a two-stage optimization model to optimize platoon split trajectories. • Extend the time geography theory to analytically solve the first-stage problem. • Propose a heuristic approach to solve the second-stage problem instantaneously. • Verify the superiority of the heuristic by comparing it with benchmarks.

Abstract Autonomous modular vehicle (AMV) technology allows for the flexible adjustment of vehicle length en-route per application needs, e.g., docking multiple short vehicles into one long vehicle or, conversely, splitting a long vehicle into multiple shorter ones. AMV docking is an extreme case of autonomous vehicle (AV) platooning in that AMVs are physically connected with zero gaps. This paper studies the trajectory planning for platoon split operations. A two-stage optimization problem is proposed to design AMV or platooned AV split operations trajectories. The first-stage objective minimizes the split operation time duration for operation efficiency. The second-stage objective minimizes the sum of squared acceleration to identify the smoothest trajectories for riding comfort and fuel efficiency. A feasible cone method is proposed by extending the time geography to reveal theoretical properties on the solution feasibility and analytically solve the first-stage problem. An exact solution approach based on quadratic programming and a heuristic solution approach based on Pontryagin's maximum principle are proposed to solve the second-stage problem. The original feasible region of the second-stage problem is greatly reduced through the feasible region analyses. Numerical experiments show that the heuristic solution approach can always solve the second-stage problem instantaneously without any or with a slight loss of the operation optimality to satisfy real-time applications needs, whereas the exact solution approach with a state-of-the-art solver may take a much longer solution time that may impose challenges to certain real-time applications. It is also noted that the exact solution time is significantly reduced after accommodating the reduced feasible region. This is critical for applications requiring absolute optimality when the heuristic solution approach fails to reach the exact optimum. The comparison between the heuristic solution approach and a customized benchmark approach reveals the superiority of the heuristic solution approach in optimizing vehicle split trajectories. Results from sensitivity analyses on key parameters provide managerial insights into engineering implementations. The generalizability of the heuristic solution approach in solving another trajectory smoothing objective function considering both acceleration and jerk is proven by extensive numerical experiments. The generalizability of the heuristic solution approach in large-scale applications and when considering surrounding traffic is also discussed.