Fig. 4: Comparison between the proposed method, the decentralized assignment-based method in ref. ¹² and the decentralized assignment-free method in ref. ²¹.

a Snapshots and trajectories of the shape assembly processes by the three methods. There are 300 robots assembling the shape of “N''. b Statistics of convergence rate and convergence time. Each average value is calculated based on 10 trials. For the proposed method and the assignment-based one in ref. ¹², the convergence rate is defined as the ratio between the number of robots that are inside the shape and the total number of robots. The convergence time is defined as the time when the convergence rate is equal to 100%. For the assignment-free method in ref. ²¹, the convergence rate is defined as the ratio between the number of robots that have reached the edge and the total number of robots. The width of the border, a parameter in the method in ref. ²¹, is set to zero so that the robots can aggregate evenly around the shape as much as possible. Since it is difficult for all the robots evenly distribute along the edge, the convergence time is defined as the time when the convergence rate reaches 70%. The parameters of the three algorithms are given in Supplementary Table 2.