Convergence analysis for rigid formation control with unrealizable shapes: The 3 agent case

We study the outcome of using a gradient descent control law for a minimally rigid formation consisting of N agents, in which each agent is modeled by a single integrator and the desired interagent distances are specified though they are not realizable. We first formulate the problem for formations of N ≥ 3 agents and derive a condition in terms of the rigidity matrix which the final formation must satisfy. Special attention will be given to the triangular formation for which the desired distances fail to satisfy the triangle inequality. In this case, we show the formation converges to a straight line. Detailed analysis is provided to describe the stability properties in the unrealizable triangle shape control problem.