Rigid Motions of 3-D Undirected Formations With Mismatch Between Desired Distances

Use of a gradient descent control law has been a popular method to effectively stabilize undirected rigid formations, by assuming that interagent distances between a certain set of neighboring agent pairs can be accurately specified and measured. This paper examines the collective motion behavior for an infinitesimally rigid formation in a three-dimensional ambient space, in the case that neighboring agent pairs have slightly differing views or measurements of the desired interagent distances they are tasked to maintain. It is shown that the formation shape will converge exponentially fast to a rigid one, while additional rigid helical motions of the final formation will occur. We further discuss the convergence to the equilibrium motions, and derive certain motion parameter formulas to describe the rigid formation movements by employing the angular momentum concept from classical mechanics. Finally, we explain how the idea can be used for steering a rigid formation to move as a rigid body.