# Affine Formation Maneuver Control for Multi-Agent Based on Optimal Flight System
## 2 Preliminaries
### Affine Transformation Description  
Based on the undirected graph \(G\) to describe the communication topology of multi-agent formation, denote the \(R=[R_{I},R_{f}]^{T}=[r_{1},r_{2},\ldots,r_{N}]^{T}\) as the nominal configuration and let \(S\) be the affine span. Let the dimension be \(d=2\) and \(P=\{p_{i}\}_{i=1}^{n}\) in \(\mathbb{R}^{2}\) be the affine image set and the associated definitions and lemmas about affine transformation are presented as follow [22]. **Definitions 1** (Affine Span): _A linear space can be obtained by any affine span. Thus, provided with the nominal configuration \(\{r_{i}\}_{i=1}^{N}\) for formation \(G\), the affine span \(S\) is:_  
\[S=\left\{\sum_{i=1}^{N}\alpha r_{i}:\alpha_{i}\in\mathbb{R}^{2}\text{ and }\sum_{i=1}^{N}\alpha_{i}=1\right\} \tag{7}\]  
**Lemma 1** (Affine Span with Rank condition): _The nominal configuration \(\{r_{i}\}_{i=1}^{N}\) affinely span \(\mathbb{R}^{2}\) if and only if \(N\geq 3\) and \(rank(\overline{R}(r))=3\), where \(\overline{R}(r)=[R,1_{N}]\in\mathbb{R}^{N\times 3}\)._  
**Definitions 2** (Affine Image): _The affine image set consisting of all the affine transformations of the nominal configuration \(\{r_{i}\}_{i=1}^{N}\), which is defined as follows:_  
\[I(r)=\left\{P=[p_{1}^{T},p_{2}^{T},\ldots,p_{N}^{T}]^{T}\in\mathbb{R}^{2N} \Big{|}p_{i}=A^{\prime}r_{i}+b\right\} \tag{8}\]  
_where \(I(R)\) is the predefined affine image and can be applied to all time-varying target configurations, and a linear subspace because it is closed under addition and scalar multiplication, \(A^{\prime}\in\mathbb{R}^{2\times 2}\) and \(b\in\mathbb{R}^{2}\) are the linear affine transformation._  
**Lemma 2** (General Rigid): _The nominal configuration is universally rigid if and only if the stress matrix is positive semi-definite and satisfies \(\text{rank}(\Omega)=N-3\)._  
**Definitions 3** (Stress Matrix): _The stress matrix can be viewed as generalized graph Laplacian matrices with both positive and negative edge weights. Let graph stress denote as the scalar \(\left\{\omega_{ij}\middle|\omega_{ij}=\omega_{ji},\omega_{ij}\in\mathbb{R} \right\}_{ij\in\mathbb{E}^{\prime}}\), \(\omega_{ij}>0\) represents the attractive force and \(\omega_{ij}<0\) the opposite, and denote if:_  
\[\sum_{j\in N_{i}}\omega_{ij}(\boldsymbol{p}_{j}-\boldsymbol{p}_{i})=0 \tag{9}\]  
_the equilibrium set \(\Omega\in\mathbb{R}^{N\times N}\) satisfies:_  
\[\left[\Omega\right]_{ij}=\left\{ \begin{array}{ll}0,&i\
eq j,(i,j)\
otin E, \\ -\omega_{ij}&i\
eq j,(i,j)\in E,\\ \sum_{k\in N_{i}}\omega_{ik}&i=j.\end{array} \right. \tag{10}\]  
Equation (10) can be expressed in a matrix form as \((\Omega\otimes I_{2})R=0\), denote the \(\overline{\Omega}=\Omega\otimes I_{2}\):  
\[\overline{\Omega}=\Omega\otimes I_{2}=\left[ \begin{array}{ll}\overline{ \Omega}_{ll}&\overline{\Omega}_{lf}\\ \overline{\Omega}_{f1}&\overline{\Omega}_{ff}\end{array} \right] \tag{11}\]  
**Definitions 4** (Affine Localizability). : _Denote the nominal formation \((G,r)\) as affinely localizable if and only if \(\{r_{i}\}_{i=1}^{N}\) affinely span \(\mathbb{R}^{2}\) (Lemma 1) and any \(p=[p_{I}^{T},p_{f}^{T}]^{T}\in I(r)\), \(p_{f}^{T}\) can be uniquely determined by \(p_{i}^{T}\)._  
**Lemma 3** (Affine Localizability of Nominal Formation). : _If the nominal configuration \(\{r_{i}\}_{i=1}^{N}\) affinely span \(\mathbb{R}^{2}\) and \(\Omega\) is positive semi-definite, \((G,r)\) is affinely localizable if and only if \(\overline{\Omega_{ff}}\) is nonsingular._  
From Figure 2, it can be seen that Figure 2b-d are obtained by rotating, scaling, and shearing the original formation Figure 2a, respectively. Figure 2e is obtained from Figure 2d by reducing the scale in the vertical direction. Figure 2f, where the four points are collinear, is obtained from Figure 2e by reducing the scale in the vertical direction to zero. Due to the simplification of the communication complexity, only leaders need to be aware of the affine transformation \((A,b)\) and controlled only.