Constrained multiple shooting parameterized differential dynamic programming: Algorithm development and convergence analysis

Differential dynamic programming (DDP) is widely used for trajectory optimization due to its fast local convergence. Its parameterized extension, parameterized differential dynamic programming (PDDP), further enables the simultaneous optimization of unknown parameters and trajectory costs. However, in time-critical spatiotemporal planning problems with complex physical constraints, standard PDDP does not natively enforce state and path constraints such as waypoint requirements and no-fly zones. This paper presents a control barrier function (CBF)–embedded multiple-shooting PDDP algorithm (CBF-MSPDDP) to address this limitation. By integrating a multiple-shooting transcription with CBF-based constraint enforcement, the proposed method extends PDDP to handle nonlinear constraints directly during the optimization. A rigorous convergence analysis is also provided to support the proposed algorithm. Numerical studies on a nonlinear quadrotor model—including obstacle avoidance and multi-waypoint trajectory optimization—demonstrate the effectiveness of CBF-MSPDDP. Comparative results against GPOPS-II and penalty-based approaches show that CBF-MSPDDP achieves substantially higher computational efficiency while maintaining constraint satisfaction. In addition, Monte Carlo simulations quantify robustness to variations in initial conditions, indicating strong potential for real-world deployment.