1 Introduction
Formation control is a fundamental competence for teams of mobile agents in diverse types of missions. Numerous existing formation control schemes consider a geometric reference (e.g., a nominal configuration of the multiagent team) and design a motion strategy aimed at achieving and maintaining that reference, allowing for specific degrees of freedom to enhance flexibility. Translations, rigid transformations, and shape-preserving transformations are common examples of allowed degrees of freedom. Many schemes along these lines have been proposed, with formulations based on inter-agent relative positions (W. Ren, 2007; D. V. Dimarogonas and K. J. Kyriakopoulos, 2008; K. Fathian et al., 2021), distances (H. Garcia de Marina et al., 2015; F. Mehdifar et al., 2020), or bearings (S. Zhao and D. Zelazo, 2016). Affine formation control, which allows for more general types of transformations, has been studied in a significant number of recent works, such as (Z. Lin et al., 2016; S. Zhao, 2018; Y. Xu et al., 2019; Q. Yang et al., 2022; Y. Lin et al., 2022; M. Aranda and A. Perez-Yus, 2024). In these studies, the affine formations are defined with respect to a single nominal configuration. This letter presents a novel approach that defines affine formations with respect to multiple nominal configurations, considered simultaneously. By using multiple geometric references, this approach makes a team capable of adapting its shape more flexibly to mission requirements (e.g., avoiding obstacles, reacting to threats, or enclosing mobile targets) while still exhibiting the behavior of a formation. The proposed approach is based on grouping the agents in chained sets. We adopt this grouping because it has practical interest and it allows us to obtain formal guarantees. The agents in every set use motion vectors based on orthogonal projections to move toward partial affine formations, defined with respect to multiple nominal configurations. Every pair of consecutive sets along the considered chain share a number of agents that ensures that the parameters of the achieved partial formations are the same for both sets. As a result, the team as a whole achieves a consistent formation. This approach has interesting features: it is distributed, since it is based on local agent interactions, and it accommodates the use of measurements expressed in local reference frames. The type of chained formations we define is particularly useful when specific physical vicinities between the agents need to be kept; e.g., to form a contour enclosing a region, or to transport a deformable object.
Recent works (B. Zhang et al., 2023; M. Aranda, 2023; M. Aranda et al., 2024) used multiagent formations designed to fit environmental boundaries or to form low-frequency closed curves. In comparison with these approaches, our formulation is more general, as it encompasses different types of formations, and it does not require global information or communications-based estimators. Other related studies proposed placing robots at discrete samples of a circle (L. Briñón-Arranz et al., 2019; G. C. Maffettone et al., 2023) or of other virtual parametric curves (B. Zhang et al., 2024) or surfaces (M. Guinaldo et al., 2024). The method we propose provides higher flexibility in terms of the shapes that can be achieved. The approach in (X. Zhang et al., 2025) extends affine formation control to so-called linear formation control, enhancing flexibility by defining a nominal configuration in a space of higher dimension than the one where the agents move. Our strategy is different, as it is based on multiple nominal configurations, and it uses neither estimators nor pre-defined design matrices. We support and illustrate the benefits of our approach with formal analysis and numerical simulation.