This paper presents a Lie group setting for the problem of control of formations, as a natural outcome of the analysis of a planar two-vehicle formation control law.

The vehicle trajectories are described using planar Frenet-Serret equations of motion, which capture the evolution of both the vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The set of all possible (relative) equilibria for arbitrary G-invariant curvature controls is described (where G = SE(2) is a symmetry group for the control law). A generalization of the control law for n vehicles is presented, and the corresponding (relative) equilibria are characterized. Work is on-going to discover stability and convergence results for the n-vehicle problem.

The practical motivation for this work is the problem of formation control for meter-scale UAVs; therefore, an implementation approach consistent with UAV payload constraints is also discussed.