Efficient control law simulation for multiple mobile robots

In this paper we consider the problem of simulating simple control laws involving large numbers of mobile robots. Such simulation can be computationally prohibitive if the number of robots is large enough, say 1 million, due to the O(N²) cost of each time step. This work therefore uses hierarchical tree-based methods for calculating the control law. These tree-based approaches have O(NlogN) cost per time step, thus allowing for efficient simulation involving a large number of robots. For concreteness, a decentralized control law which involves only the distance and bearing to the closest neighbor robot will be considered. The time to calculate the control law for each robot at each time step is demonstrated to be O(logN)