Consensus-based formation of a swarm of quadrotors interacting over ring digraphs

This work proposes a cooperative strategy for a group of quadrotors interacting over ring digraphs with macro-vertices of size two. Consensus for a group of general double integrators has been initially investigated, and it has been proved that through a suitable choice of a single controller parameter, consensus and stability of the resulting networked dynamical system can be ensured. This further opens up the possibility of achieving a desired formation and to move a swarm of quadrotors, interacting over ring digraphs, at a desired flight velocity, using a single controller gain. An analysis of achievable velocities is performed. Examples have been provided to offer deeper insights into the obtained analytical results. Simulation studies clearly demonstrate that a desired formation is achieved, starting from arbitrary initial positions, while also ensuring convergence to a final desired flight velocity.

💡 Research Summary

The paper addresses the problem of coordinating a swarm of quadrotors that interact over a directed ring graph whose macro‑vertices consist of two agents each. By modeling the translational dynamics of each quadrotor as a double integrator (justified by the large time‑scale separation between position and attitude dynamics), the authors develop a distributed consensus‑based control law that uses only relative position and velocity information from neighboring agents.

The underlying communication topology is a “ring digraph” as defined in prior work: each macro‑vertex contains two nodes connected by bidirectional edges, and successive macro‑vertices are linked by a single directed edge, forming a directed cycle. The graph Laplacian L of this topology has a special block‑circulant structure. Using Fourier diagonalisation of block‑circulant matrices, the authors obtain an explicit expression for the eigenvalues of L. They prove (Theorem 3) that all non‑zero eigenvalues of L have positive real parts if and only if the intra‑macro‑vertex gain k satisfies

 k > −2 + √2 cos(π/m)

where m is the number of macro‑vertices (i.e., N = 2m agents). This condition guarantees that −L has all non‑zero eigenvalues in the left‑half complex plane, a prerequisite for stability of the second‑order consensus dynamics.

The control input for each agent i is

 u_i = −α ∑{j∈N_i} a{ij}(p_i−p_j) − β ∑{j∈N_i} a{ij}(v_i−v_j)

with positive scalar gains α (position coupling) and β (velocity coupling). Stacking all positions and velocities yields the state‑space model

 ẋ =