Control of triangular formations with a time-varying scale function

This paper considers a novel problem of how to choose the scale of the final geometry for three agents in a triangular formation. Instead of assigning a set of desired side lengths, here the only requirement for the desired geometry is a triangle without any location, rotation and, most importantly, scale constraints. We set up a cost function that corresponds to the geometries degree of similarity with respect to the desired shape during convergence, and the cost value is compared between a system with a time varying scale function and the one with a constant scale. A fixed structure nonlinear control law on the positions of agents and the scale function is developed to drive the three agents exponentially converge to a triangle that matches the desired one in a cooperative manner. The control algorithms are validated on three AirRobots. It is shown that system with the proposed time-varying scale function outperforms the one with a constant scale.