Distributed Framework Construction for Affine Formation Control

In affine formation control problems, the construction of the framework with universal rigidity and affine localizability is a critical prerequisite, but it has not yet been well addressed, especially when additional agents join the formation or link/agent failures emerge. Motivated by this observation, we investigate the problem of constructing affine formation frameworks in three scenarios, including vertex addition, edge deletion and vertex deletion. Our approach starts from the original affine formation and uses geometric methods to locally adjust the structure of the weighted graph to describe the topology, so that the modified framework maintains the universal rigidity and affine localizability. Notably, the developed strategies only utilize local measurements and exhibit distributed characteristics, laying the foundation for applications in multi-agent systems. To demonstrate the compatibility with affine formation control proposals, we present a case study on affine formation tracking in a multi-UAV formation, demonstrating the effectiveness of our algorithms in constructing eligible frameworks in aforementioned scenarios. Moreover, a comparative simulations is also conducted to highlight the low time complexity of our distributed algorithm relative to the centralized optimization-based method.