Flocking of Multi-Agent Systems on Lie Groups With Invariant Metrics

This research investigates flocking problem of multi-agent systems on Lie groups, addressing key challenges such as velocity synchronization, collision avoidance, and aggregation. Traditional models defined in Euclidean spaces struggle with the nonlinear structures and convergence complexities of Lie groups. To overcome these challenges, we propose an intrinsic Cucker-Smale (C-S) type flocking model that utilizes velocity errors between agents and the gradient of a potential energy function. This approach ensures the achievement of flocking on compact Lie groups or Lie groups with a constant sectional curvature, equipped with a left-invariant Riemannian metric. Additionally, we demonstrate that velocity synchronization on direct product Lie groups can be achieved if and only if it is successfully realized in each individual factor space. Leveraging the structure of connected Lie groups that admit a bi-invariant Riemannian metric, which are isomorphic to the Cartesian product of a compact group and a vector group, we show that the modified model realizes flocking in such settings. Our study focuses on three concrete Lie groups: the unit circle, the infinite cylinder, and the special orthogonal group, and we also consider an example of a Riemannian manifold with constant sectional curvature that is not a Lie group, namely the 2-dimensional unit sphere. Numerical simulations are conducted to verify the theoretical findings across all these manifolds.