Distributed Spatial-Temporal Trajectory Optimization for Unmanned-Aerial-Vehicle Swarm

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.

💡 Research Summary

The paper addresses the challenging problem of spatial‑temporal trajectory optimization for large UAV swarms, where the final flight time of each vehicle cannot be predetermined. The authors propose a two‑level distributed framework called Distributed Parameterized DDP (D‑PDDP). At the lower level, each UAV solves a local optimal control problem using Parameterized Differential Dynamic Programming (PDDP), which simultaneously optimizes the control sequence and an unknown time‑parameter θ (the final flight time). PDDP extends classic DDP by incorporating second‑order Taylor expansions with respect to θ, thereby enabling free‑time trajectory optimization while preserving the fast quadratic convergence of DDP.

At the upper level, the Alternating Direction Method of Multipliers (ADMM) is employed to enforce consensus among UAVs on spatial‑temporal constraints such as collision avoidance, communication‑range maintenance, and prescribed arrival‑time relationships. Unlike earlier ADMM‑based swarm planners that rely on a central coordinator or require fixed terminal times, the proposed scheme introduces auxiliary primal variables so that only one‑hop neighbors exchange primal and dual variables, achieving a fully decentralized C‑ADMM structure.

A major contribution is an adaptive penalty‑parameter update for ADMM based on the Barzilai‑Borwein spectral gradient method. By balancing the primal and dual residuals, the penalty ρ is scaled dynamically, eliminating the need for manual tuning and reducing the number of ADMM iterations by roughly 30‑45 % in the authors’ experiments.

The algorithm proceeds iteratively: (1) each UAV runs a PDDP pass to propose new controls and a new θ; (2) neighboring UAVs exchange their primal variables and Lagrange multipliers; (3) ADMM updates the shared variables and the penalty ρ; (4) the process repeats until both the local PDDP convergence criteria and the ADMM consensus residuals are satisfied. Convergence analysis shows that the quadratic convergence of PDDP combined with the standard ADMM convergence guarantees leads the overall method toward a globally near‑optimal solution.

Simulation studies cover three mission types: (a) free‑time optimal flight, (b) synchronized arrival, and (c) sequential arrival with prescribed time gaps. Swarms of 20, 50, and 100 UAVs are tested in environments containing obstacles and inter‑UAV collision constraints. Results demonstrate that D‑PDDP achieves comparable or lower total cost than centralized Sequential Quadratic Programming (SQP) and distributed sequential convex programming, while requiring about one‑fifth of the computational effort. Even with 100 agents, the algorithm converges in an average of 0.12 seconds, and collision‑avoidance constraints are satisfied in all trials. The adaptive ρ scheme further cuts iteration counts, leading to noticeable runtime savings.

Limitations acknowledged include potential divergence when constraints are highly non‑convex, the assumption of ideal, loss‑free communication, and sensitivity of PDDP to the initial trajectory guess. Future work is suggested on asynchronous ADMM variants, robust multiplier updates, and real‑world flight tests to validate the approach under realistic network delays and sensor noise.

In summary, the paper delivers a novel, fully distributed spatial‑temporal optimization method that removes the need for pre‑specified final times, scales to large swarms, and accelerates convergence through an adaptive ADMM penalty strategy, marking a significant step forward for practical UAV swarm autonomy.