# Affine Formation Maneuver Control for Multi-Agent Based on Optimal Flight System
## 2 Preliminaries
### Kinematic and Dynamic Models of UAV  
The multi-agent formation is denoted as \((G,p)\), where the vertex \(v_{i}\) within graph \(G\) is mapped to point \(p_{i}\). In addition, it is assumed that the first agents \(n_{l}\) are leaders and the rest \(n_{f}\) are followers. Let \(V_{l}=\left\{v_{1},v_{2},\ldots,v_{l}\right\}\) and \(V_{f}=\left\{v_{l+1},v_{l+2},\ldots,v_{n}\right\}\) be the sets of leaders and followers, respectively, where \(l,f\in n\). A mass-point [18; 36] can be used to describe the UAV agent and the motion of the formation. The fixed-wing agent model is described as Figure 1 shows:  
It is shown that all these variables are presented to depict the agent motion from the inertial coordinate frame \((x,y,z)\); thus, the kinematic motion equations for UAV can be expressed as  
\[ \begin{array}{l}\dot{x}_{i}=V_{i}\cos\gamma_{i}\cos\chi_{i}\\ \dot{y}_{i}=V_{i}\cos\gamma_{i}\sin\chi_{i}\\ \dot{h}_{i}=V_{i}\sin\gamma_{i}\end{array}  \tag{1}\]  
Figure 1: Fixed-wing agent model. Meanwhile, the dynamic of UAV is given by  
\[ \begin{array}{l}\dot{V}_{i}=\frac{T_{i}-D_{i}}{\pi_{i}}-g\sin\gamma_{i}\\ \dot{\gamma}_{i}=\frac{\dot{\psi}_{i}}{\dot{\psi}_{i}}(n_{gi}\cos\phi_{i}-\cos \gamma_{i})\\ \dot{\chi}=\frac{L_{i}\sin\phi_{i}}{m_{i}V_{i}\cos\gamma_{i}}\end{array} , \tag{2}\]  
where \(x_{i}\) and \(y_{i}\) are the range displacement projected on the \(X\) and \(Y\) axes separately, \(h_{i}\) is the flight altitude, and \(V_{i}\) is the speed which is assumed as ground speed in the paper, \(\chi_{i}\) and \(\gamma_{i}\) are the heading angle and the flight one. \(T_{i}\) is the engine thrust vector directed with the velocity vector, \(D_{i}\) is a general term for drag. Additionally, \(m_{i}g\) is used to describe the gravity of the agent, the \(g\) load \(n_{gi}=L_{i}/m_{i}g\) is contained by an elevator, \(\phi_{i}\) the banking angle and the engine thrust \(T_{i}\) are considered as the control inputs. Therefore, by the use of the feedback linearization technology, the non-linear UAV model Formulation (1) and (2) can be simplified to a linear constant system  
\[\left\{ \begin{array}{l}x_{i}(t)=u_{xi}(t)\\ y_{i}(t)=u_{yi}(t)\\ h_{i}(t)=u_{hi}(t)\end{array} \right. \tag{3}\]  
However, \(u_{xi}(t)\), \(u_{yi}(t)\), and \(u_{hi}(t)\) are virtual control inputs to describe the accelerations in each way. The actual control input is given as  
\[ \begin{array}{l}\phi_{i}=\tan^{-1}[\frac{u_{gi}\cos\chi_{i}-u_{xi}\sin\chi_{ i}}{\cos\gamma_{i}(u_{gi}+g)\sin\gamma_{i}(u_{gi}\cos\chi_{i}-u_{gi}\sin\chi_{i})}]\\ n_{gi}=\frac{\cos\gamma_{i}(u_{hi}+g)-\sin\gamma_{i}(u_{gi}\cos\chi_{i}-u_{gi }\sin\chi_{i})}{g\cos\phi_{i}}\\ T_{i}=[\sin\gamma_{i}(u_{hi}+g)+\cos\gamma_{i}(u_{xi}\cos\chi_{i}+u_{yi}\sin \chi_{i})]m_{i}+D_{i}\end{array} , \tag{4}\]  
where the heading angle \(\chi_{i}\) can be expressed as \(\tan\chi_{i}=\dot{y}_{i}/\dot{x}_{i}\), the fight one \(\gamma_{i}\) is \(\sin\gamma_{i}=\dot{h}_{i}/V_{i}\). Thus, in terms of the state-space representation, denoting \(\mathbf{p}_{i}=[x_{i}\ y_{i}\ h_{i}]\), \(v_{i}=[v_{xi}\ y_{xi}\ u_{hi}]\), and \(\mathbf{u}_{i}=[u_{xi}\ u_{yi}\ u_{hi}]\), which are the position, velocity, and acceleration control vector, the double-integrator system is reduced as  
\[\left\{ \begin{array}{l}\dot{\mathbf{p}}_{i}(t)=\mathbf{v}_{i}(t)\\ \dot{\mathbf{v}}_{i}(t)=\mathbf{u}_{i}(t)\end{array} \right. \tag{5}\]  
where \(\mathbf{p}_{i}\in R^{n}\), \(\mathbf{v}_{i}\in R^{n}\), \(\mathbf{u}_{i}\in R^{n}\). Consider a flock of \(n\) agents in \(R^{d}\) and \(\mathbf{p}_{i}\in R^{d}\) as the position of agent \(i\) and \(\mathbf{p}=[p_{1}^{T},p_{2}^{T},\ldots,p_{n}^{T}]\in R^{dn}\) is the configuration of all the agents, where \(d\geq 2\) and \(n\geq d+1\). Let \(m\) be the number of edges in an undirected graph and the orientation of which is the direction assignment to each edge.