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.

翻译：密度分布方法因其对群体规模变化的适应性，近期已成为一种前景广阔的研究范式。然而，现有研究在实现复杂形态表征与去中心化实施方面面临实际挑战。这促使我们开发一种完全去中心化的、基于分布的控制策略，该策略兼具形成复杂形态与适应群体规模变化的双重能力。具体而言，我们首先提出一种定义在一组采样点上的离散质量分布函数，用以建模群体编队。与连续密度分布方法相比，我们的模型无需定义连续密度函数——这对于复杂形态而言是一项困难的任务。其次，我们设计了一种去中心化的均值漂移控制律，通过反馈质量估计值，协调群体的全局分布以拟合采样点分布。所有采样点的质量估计由机器人通过所设计的质量估计器以去中心化方式实现。理论分析表明，采样点的质量估计值能够渐近收敛至真实的全局值。为验证所提策略，我们进行了全面的仿真与实物实验，以评估其在复杂形态形成效率及对群体规模变化的适应性方面的性能。