2 Problem Statement
Consider a team of \(N\) mobile agents, each having a different identifying index in the set \(\mathcal{N}=\{1, 2, \ldots, N\}\). Agent index values are always interpreted modulo \(N\) in this letter, with values outside \(\mathcal{N}\) being wrapped around into \(\mathcal{N}\). \(D\) denotes the number (2 or 3) of spatial dimensions. Positions in \(D\)-dimensional Euclidean space are expressed in a fixed global Cartesian coordinate frame. The position of agent \(i\) is denoted by \(p_i \in \mathbb{R}^D\). The stack vector of the agents' positions, defined as \(p=[p_1^\intercal, \ldots , p_N^\intercal{]}^\intercal\) \(\in \mathbb{R}^{D\cdot N}\), is called the configuration of the team. We assume the agents have single-integrator dynamics, i.e., \(\dot{p}_i(t)=u_i(t)\) \(\forall i \in \mathcal{N}\), where \(u_i(t)\) is the control input and \(t \in \mathbb{R}_{\geq 0}\) denotes time.
2.1 Affine Formations for a Nominal Configuration
Our work is based on the framework of affine multiagent formation control. We use the same concept of what an affine formation is as in the related literature (
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). Defining a constant nominal configuration \(c=[c_1^\intercal, \ldots, c_N^\intercal{]}^\intercal \in \mathbb{R}^{D\cdot N}\) with position \(c_i \in \mathbb{R}^D\) corresponding to agent \(i\), the multiagent team is in an affine formation with respect to the nominal configuration \(c\) if there exist \(A \in \mathbb{R}^{D\times D}\) and \(r \in \mathbb{R}^{D}\) such that
\[\tag{1}
p_i = A c_i + r \hspace{10pt} \forall i \in \mathcal{N}.
\]
\(A\) and \(r\) represent, respectively, a general linear transformation and a general translation that are applied to the positions in the nominal configuration for all the agents. This specification can allow for flexible configurations \(p\) which still preserve certain geometric characteristics of the nominal configuration \(c\), such as collinearity and parallelism in the agents' relative positions, making affine formations useful in diverse tasks. We can express
(1) compactly as
\[\tag{2}
p = (\widetilde{C} \otimes I_D) \widetilde{v},
\]
where we define
\[
\widetilde{C}\hspace{-2pt}= \hspace{-3pt}
\begin{bmatrix}
c^\intercal_1 & 1 \\
\vdots & \vdots \\
c^\intercal_N & 1
\end{bmatrix}\hspace{-3pt}\in \mathbb{R}^{N\times (D+1)}\hspace{0pt},\hspace{3pt} \widetilde{v}\hspace{-2pt}=\hspace{-3pt}\begin{bmatrix} col_1(A) \\ \vdots \\ col_D(A) \\ r
\end{bmatrix}\hspace{-3pt}\in \mathbb{R}^{D\cdot(D+1)}\hspace{-2pt}.
\]
Note that \(\widetilde{C}\) contains the information of the constant nominal configuration. On the other hand, \(\widetilde{v}\) is a parameter vector containing the elements of \(A\) and \(r\) that characterizes, for the particular team configuration \(p\), the specific affine formation with respect to the nominal configuration \(c\). The concept of affine image (
Z. Lin et al., 2016;
S. Zhao, 2018) can be used to define the set of the configurations \(p\) that satisfy
(1) for a given \(c\).
To further enhance flexibility, in this letter we propose using multiple distinct nominal configurations, each of them representing a certain geometric pattern of agent positions. Suppose we have \(L\) distinct nominal configurations, with \(L\geq1\), denoted \(c_{(l)}=[c_{l,1}^\intercal, \ldots, c_{l,N}^\intercal{]}^\intercal \in \mathbb{R}^{D\cdot N}\) \(\forall l \in \{1, 2, \ldots, L\}\). \(c_{l,i} \in \mathbb{R}^D\) denotes the position for agent \(i\) in the nominal configuration \(c_{(l)}\). Our control goal will be the achievement of a sum of affine formations, in the sense of (1), with respect to these nominal configurations. Therefore, the condition is now that there exist \(A_l \in \mathbb{R}^{D\times D}\) and \(r_l\in \mathbb{R}^{D}\) \(\forall l \in \{1, 2, \ldots, L\}\), such that
\[
p_i = \sum_{l=1}^L (A_l c_{l,i} + r_l) = \sum_{l=1}^L A_l c_{l,i} + \sum_{l=1}^L r_l\hspace{10pt}\forall i \in \mathcal{N}.
\]
We can write an expression analogous to (2), with a parameter vector \(v\) \(\in \mathbb{R}^{D\cdot(D\cdot L+1)}\), and \(C \in \mathbb{R}^{N\times (D\cdot L+1)}\), as
\[\tag{3}
p = (C \otimes I_D) v,
\]
\[\tag{4}
\hspace{-12pt} \text{with} \hspace{4pt} C \hspace{-1pt}=\hspace{-2pt}\begin{bmatrix}
c^\intercal_{1,1} & \cdots & c^\intercal_{L,1} & 1\\
\vdots & \ddots & \vdots & \vdots \\
c^\intercal_{1,N} & \cdots & c^\intercal_{L,N} & 1
\end{bmatrix}\hspace{-2pt}, \hspace{5pt} v \hspace{-2pt}=\hspace{-2pt}\begin{bmatrix} col_1(A_1) \\ \vdots \\ col_D(A_1) \\ \vdots \\ col_1(A_L) \\ \vdots \\ col_D(A_L) \\ \sum_{l=1}^L r_l
\end{bmatrix}\hspace{-1pt}.
\]
Note that with \(L=1\), one has the case in Section 2.1. Based on (3), we now present our definition for the formation task.
The multiagent team is in an affine formation with respect to the nominal configurations \(c_{(1)}, c_{(2)}, \ldots, c_{(L)}\) if there exists \(v\) \(\in \mathbb{R}^{D\cdot(D\cdot L+1)}\) such that \(p = ( C \otimes I_D) v\).
This letter tackles the problem of designing a control approach for achieving multiagent formations in the sense of Definition 1. Next, we describe our proposed solution.