Non-robustness of gradient control for 3-D undirected formations with distance mismatch

Gradient control laws can be used for effectively achieving undirected formation shape, by assuming that interagent distances between a certain set of joint agent pairs can be accurately specified and measured. This paper examines the formation behavior in a 3-D space context in the case that the neighboring agent pairs have slightly differing views or estimates about the desired interagent distances they are tasked to maintain. It is shown, by using a tetrahedron formation example, that the final formation shape will be slightly distorted as compared to the desired one. Further, in general each agent's motion will be a combination of rotation and translation. Specifically, a helical movement can be observed in the presence of distance mismatch.