Meanshift Shape Formation Control Using Discrete Mass Distribution

The density-distribution method has recently become a promising paradigm owing to its adaptability to variations in swarm size. However, existing studies face practical challenges in achieving complex shape representation and decentralized implementation. This motivates us to develop a fully decentralized, distribution-based control strategy with the dual capability of forming complex shapes and adapting to swarm-size variations. Specifically, we first propose a discrete mass-distribution function defined over a set of sample points to model swarm formation. In contrast to the continuous density-distribution method, our model eliminates the requirement for defining continuous density functions-a task that is difficult for complex shapes. Second, we design a decentralized meanshift control law to coordinate the swarm’s global distribution to fit the sample-point distribution by feeding back mass estimates. The mass estimates for all sample points are achieved by the robots in a decentralized manner via the designed mass estimator. It is shown that the mass estimates of the sample points can asymptotically converge to the true global values. To validate the proposed strategy, we conduct comprehensive simulations and real-world experiments to evaluate the efficiency of complex shape formation and adaptability to swarm-size variations.

💡 Research Summary

The paper addresses the challenge of shape formation for large robot swarms in a fully decentralized manner, especially when the desired shapes are complex and the swarm size may vary over time. Traditional density‑distribution approaches model the target shape as a continuous density function over a region, which requires a mathematically tractable description of the shape and global position information from all robots. Both requirements become prohibitive for intricate shapes (e.g., fish‑bone, star, letters) and for large‑scale, dynamic swarms.

To overcome these limitations, the authors propose an entirely discrete representation: the desired shape is sampled into a finite set of points ({q_k}_{k=1}^m). For each sample point a “mass” (P_k) is defined as a kernel‑density estimate of the robot positions, \