Distributed Control for Full-DOF Motion of Similarity-Based Rigid Formation

Compared with the distance-based or bearing-based rigid formation, the similarity-based rigid formation allows full-DoF (abbr. degree of freedom) trivial infinitesimal motions (i.e., translation, rotation and scaling), which plays an important role in maneuvers within multi-obstacle scenarios. To this end, this paper focuses on the distributed control development for using the configuration maneuverability of the similarity-based rigid formation with mixed angle and distance ratio constraints. First of all, a basic coordinate-free distributed control composed by the gradient descent item and the velocity consensus item is proposed for the combined flocking and similar configuration stabilization with local exponential convergence. Next, the distributed control is modified for the velocity tracking of the similarity-based rigid formation to a reference that is only available to the leader agents. Besides, another modified distributed control is built for the orientation and scaling adjustment of the similarity-based rigid formation by just tuning the orientation and scaling of the regulator agents according to references. Moreover, it is shown that the minimal number of the leaders for the velocity tracking is one, and the minimal number of the regulators for the orientation and scaling adjustment is two for the planar formation or three for the spatial formation. By combining two modified distributed controls, the full-DoF motion adjustment of the similarity-based rigid formation is finally achieved. Simulation and experiment are both given to verify the adjustment performance.