Multicluster Design and Control of Large-Scale Affine Formations

Abstract

Conventional affine formation control (AFC) empowers a network of agents with flexible but collective motions - a potential which has not yet been exploited for large-scale swarms. One of the key bottlenecks lies in the design of an interaction graph, characterized by the Laplacian-like stress matrix. Efficient and scalable design solutions often yield suboptimal solutions on various performance metrics, e.g., convergence speed and communication cost, to name a few. The current state-of-the-art algorithms for finding optimal solutions are computationally expensive and therefore not scalable. In this work, we propose a more efficient optimal design for any generic configuration, with the potential to further reduce complexity for a large class of nongeneric rotationally symmetric configurations. Furthermore, we introduce a multicluster control framework that offers an additional scalability improvement, enabling not only collective affine motions as in conventional AFC but also partially independent motions naturally desired for large-scale swarms. The overall design is compatible with a swarm size of several hundred agents with fast formation convergence, as compared to up to only a few dozen agents by existing methods. Experimentally, we benchmark the performance of our algorithm compared with several state-of-the-art solutions and demonstrate the capabilities of our proposed control strategies.

Figures (9)

• Figure 1: Examples of frameworks in $\mathbb{R}^2$ with increasing rigidity lin2015necessary
• Figure 2: A few examples of rotationally symmetric configurations. Note that the optimal stress for these configurations also applies to their affine transformations.
• Figure 3: Stress matrices compared with EDMs using an octagon in Fig. \ref{['fig: rot-symm configs']}(a) (upper) and a cuboctahedron in Fig. \ref{['fig: rot-symm configs']}(c) (lower), which illustrate the equivalence of the proposed optimization (Generic $\mathcal{P}_2$) and the unique stress identifier (USI) method $\mathcal{P}_3$ on rotationally symmetric configurations. The stress matrices for the octagon are symmetric, Toeplitz, and circulant. The diagonals of matrices are made empty as they do not represent distinct edge classes but are determined by the others. Matching colors represent the same stress values for the first two columns, and the same stress classes to compare with the EDM.
• Figure 4: An illustration example of the mean dynamics model (\ref{['equ: global mean dynamics']}) with $6$ nodes over 2 clusters. The dimension is set to $D=1$ for simplicity.
• Figure 5: An illustration of local flexibility introduced by rank-deficient bridging nodes. The shaded area outlines the desired configuration and the dashed geometry shows the potential ambiguity.

Theorems & Definitions (14)

• Theorem 1
• Definition 1: Rotationally symmetric configurations
• Definition 2: Permutation invariance of feasible sets of $\mathcal{P}_1$
• Definition 3: Permutation invariance of objectives of $\mathcal{P}_1$
• Lemma 1: Permutation invariance of $\mathcal{P}_1$
• Definition 4: Permutation-invariant stress matrices $\bm{\Omega}_\mathrm{pi}$
• Theorem 2: Optimality of permutation-invariant stress
• Corollary 1: Edge equivalence in the stress matrix
• Remark 1: Stress structure for circular configurations
• Theorem 3: Convergence of mean dynamics