Swarm trajectory optimization problems are a well-recognized class of multi-agent optimal control problems with strong nonlinearity. However, the heuristic nature of needing to set the final time for agents beforehand and the time-consuming limitation of the significant number of iterations prohibit the application of existing methods to large-scale swarm of Unmanned Aerial Vehicles (UAVs) in practice. In this paper, we propose a spatial-temporal trajectory optimization framework that accomplishes multi-UAV consensus based on the Alternating Direction Multiplier Method (ADMM) and uses Differential Dynamic Programming (DDP) for fast local planning of individual UAVs. The introduced framework is a two-level architecture that employs Parameterized DDP (PDDP) as the trajectory optimizer for each UAV, and ADMM to satisfy the local constraints and accomplish the spatial-temporal parameter consensus among all UAVs. This results in a fully distributed algorithm called Distributed Parameterized DDP (D-PDDP). In addition, an adaptive tuning criterion based on the spectral gradient method for the penalty parameter is proposed to reduce the number of algorithmic iterations. Several simulation examples are presented to verify the effectiveness of the proposed algorithm.

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