Assignment Algorithms for Variable Robot Formations

This paper describes algorithms to perform optimal assignment of teams of robots translating in the plane from an initial formation to a variable goal formation. We consider the case when each robot is to be assigned a goal position, the individual robots are interchangeable, and the goal formation can be scaled or translated. We compute the costs for all candidate pairs of initial, goal robot assignments as functions of the parameters of the goal formation, and partition the parameter space into equivalence classes invariant to the cost order using computational geometry techniques. We compute a minimum completion time assignment for an equivalence class by formulating it as a linear bottleneck assignment problem (LBAP). To improve efficiency, we solve the LBAP problem for each equivalence class by incrementally updating the solution as the formation parameters are varied. This work is motivated by applications that include the motion of droplet formations in digital microfluidic lab-on-a-chip devices, and of robot and drone formations in the plane.