Sparse-Graph-Enabled Formation Planning for Large-Scale Aerial Swarms

TL;DR

The paper tackles the computational bottleneck of formation planning for large-scale 3D UAV swarms by moving from dense complete-graph constraints to a carefully designed sparse-graph framework. It introduces a sparsification mechanism that preserves global rigidity and a good sparse-graph construction method based on submatrix selection of the Laplacian, enabling scalable planning while maintaining formation fidelity. The approach leverages a Laplacian-based formulation $L = D - A$ and achieves significant computational savings by reducing per-drone constraint connectivity to $O((\varrho_c N)^2)$ with a chosen connection rate $\varrho_c$, exemplified by $\varrho_c \approx 0.30$. Extensive simulations with up to 72 drones in cluttered environments show comparable formation error to complete graphs while delivering roughly an order of magnitude improvement in planning efficiency, supported by ablation studies on graph sparsification and a systematic benchmark of sparse-graph variants.

Abstract

The formation trajectory planning using complete graphs to model collaborative constraints becomes computationally intractable as the number of drones increases due to the curse of dimensionality. To tackle this issue, this paper presents a sparse graph construction method for formation planning to realize better efficiency-performance trade-off. Firstly, a sparsification mechanism for complete graphs is designed to ensure the global rigidity of sparsified graphs, which is a necessary condition for uniquely corresponding to a geometric shape. Secondly, a good sparse graph is constructed to preserve the main structural feature of complete graphs sufficiently. Since the graph-based formation constraint is described by Laplacian matrix, the sparse graph construction problem is equivalent to submatrix selection, which has combinatorial time complexity and needs a scoring metric. Via comparative simulations, the Max-Trace matrix-revealing metric shows the promising performance. The sparse graph is integrated into the formation planning. Simulation results with 72 drones in complex environments demonstrate that when preserving 30\% connection edges, our method has comparative formation error and recovery performance w.r.t. complete graphs. Meanwhile, the planning efficiency is improved by approximate an order of magnitude. Benchmark comparisons and ablation studies are conducted to fully validate the merits of our method.

Figures (10)

• Figure 1: Simulation results of the sparse-graph-enabled formation planning. (a) The visualization of spelling “ZJU” with 72 drones while avoiding obstacles. (b)-(d) The illustration of keeping different geometric shapes with 48 drones in cluttered environments.
• Figure 2: Illustration of graph rigidity. (a) A flexible graph deforms under disturbances. (b) A rigid graph corresponds to multiple geometric shapes. (c) A globally rigid graph ensures the stability and uniqueness.
• Figure 3: A diagram of our spare graph construct architecture.
• Figure 4: Illustration of submatric selection. (a) Laplacian matrix of complete graphs with six vertices. (b) Laplacian matrix of sparse graphs by submatrix selection, and the element in dashed box corresponds to the connected edge in the sparse graph with the same color. Four columns of the matrix are selected to form the submatrix, resulting in that vertices 1, 2, 3, & 5 are selected as the base set $\boldsymbol v^{bas}$. Vertices 1, 2, 3, & 5 connect with each other using undirected edges to establish a complete graph, and vertices 4 & 6 connect with $\boldsymbol v^{bas}$ via directed edges.
• Figure 5: Comparative simulation on four candidate matrix-revealing metrics. (a) We simulate drones flying in formation from left side in a cluttered map to right side with a velocity limit of 2m/s. The map is generated randomly. The global formation trajectory is planned from the start point to end point. (b) Three formation configurations are tested. (c) Comparison results of formation error $\bar{e}_{dist}$ along the global trajectory are provided.