Formation movements in minimally rigid formation control with mismatched mutual distances

When a gradient descent control law is employed for stabilizing undirected minimally rigid formations, mismatched desired distances between neighboring agent pairs will cause additional motions of the whole formation. By reviewing and extending the results in [1], [2], we show that in general rotational motions and helical motions will occur for 2-D formations and 3-D formations, respectively. We then consider the problem of how to compute the formulas for the motions caused by constant mismatches. A novel idea based on the angular-momentum concept in rigid body dynamics is proposed for deriving the formation formulas, e.g., angular velocity, rotational radius, etc. in terms of the distance mismatch terms. This has implications on steering and controlling rigid formation motions.