Equilibrium Unit Based Localized Affine Formation Maneuver for Multi-agent Systems

TL;DR

The paper tackles the challenge of achieving affine localizability in multi-agent systems without relying on a generic nominal configuration or centralized construction. It introduces the equilibrium unit concept and layerable directed graphs, providing a distributed equilibrium-unit construction (EUC) method to build affine-localizable nominal frameworks from the ground up. It further develops a localized communication criterion (LCC) and a localized sensing based affine formation maneuver control (LSAFMC) protocol to enable self-reconstruction when nodes are added or removed, with stability guarantees via Riccati-based design and Lyapunov analysis. The proposed framework is validated through simulations showing robust formation maneuvering with node addition/removal and obstacle avoidance, demonstrating scalability and real-world applicability for dynamic robot swarms.

Abstract

Current affine formation maneuver of multi-agent systems (MASs) relys on the affine localizability determined by generic assumption for nominal configuration and global construction manner. This does not live up to practical constraints of robot swarms. In this paper, an equilibrium unit based structure is proposed to achieve affine localizability. In an equilibrium unit, existence of non-zero weights between nodes is guaranteed and their summation is proved to be non-zero. To remove the generic assumption, a notion of layerable directed graph is introduced, based on which a sufficient condition associated equilibrium unit is presented to establish affine localizability condition. Within this framework, distributed local construction manner is performed by a designed equilibrium unit construction (EUC) method. With the help of localized communication criterion (LCC) and localized sensing based affine formation maneuver control (LSAFMC) protocol, self-reconstruction capability is possessed by MASs when nodes are added to or removed from the swarms.

Figures (9)

• Figure 1: Example to illustrate the nominal framework with affine localizability in $\mathbb{R}^{2}$. Specifically, $\mathcal{V}_{1}$, $\mathcal{V}_{2}$ and $\mathcal{V}_{3}$ are set as leaders and $\mathcal{V}_{4}$, $\mathcal{V}_{5}$ and $\mathcal{V}_{6}$ are set as followers. Positions of nodes are presented in (a), thus, we can conclude that leaders affinely span $\mathbb{R}^{2}$. Then, condition 1) in Definition 3 is satisfied. According to the associated Laplacian matrix presented in (b), it is easy to obtain that $\Omega_{ff}$ is invertible. Then, condition 2) in Definition 3 is satisfied. Note that the nominal configuration of the nominal framework is not generic because there exist collinear nodes, i.e., $\mathcal{V}_{4}$, $\mathcal{V}_{5}$ and $\mathcal{V}_{6}$ collineate in $\mathbb{R}^{2}$.
• Figure 2: (a) Representation element in $\mathcal{U}^{3}(\mathbb{R}^{2})$. (b) Representation element in $\mathcal{U}^{2}(\mathbb{R}^{2})$.
• Figure 3: (a) Representation element in $\mathcal{U}^{4}(\mathbb{R}^{3})$. (b) Representation element in $\mathcal{U}^{3}(\mathbb{R}^{3})$. (c) Representation element in $\mathcal{U}^{2}(\mathbb{R}^{3})$.
• Figure 4: Illustration of EUC procedure proposed in Theorem 1 ($\mathbb{R}^{2}$ case).
• Figure 5: (a) FIA of a new node $\mathcal{V}_{9}$ (green). (b) FOA of an old node $\mathcal{V}_{5}$ (yellow). Specifically, node $\mathcal{V}_{7}$ is shifted forward to replace node $\mathcal{V}_{5}$ and node $\mathcal{V}_{8}$ is shifted forward to replace $\mathcal{V}_{7}$.