## 2 Modeling of Tethered UAV Dynamics
### Dynamics Modeling of UAV Platforms
#### 2.1.4 Dynamical Modeling  
For dynamic modeling of UAV platforms, they are often thought of as rigid bodies. The UAV platform dynamics equation consists of a position and attitude dynamics equation. The traditional methods of attitude description are Euler angles, rotation matrices, quaternions, etc. Most of the dynamic modeling methods adopt traditional methods such as the Newton-Euler method [16], the Lagrange method [17], and the Kane method [18]. The dynamic model established by these methods has different expressions, but the final results of the dynamic characteristics of the described objects are the same. In this paper, the kinematics and dynamics equations of the UAV platform based on the Euler-Poincare equation are as follows [19]:  
\[\begin{cases}\dot{\mathbf{q}}=V(\mathbf{q})\mathbf{p}\\ \mathbf{M}(\mathbf{q})\dot{\mathbf{p}}+\mathbf{C}(\mathbf{q},\mathbf{p})\mathbf{p}+\mathbf{F}(\mathbf{p},\mathbf{q},\bm {u})=\mathbf{0}\end{cases} \tag{1}\]  
where \(\mathbf{p}\) is the pseudo-velocity vector and \(\mathbf{q}\) is the generalized coordinate vector. Their expressions are as follows:  
\[\mathbf{p} =[u,v,w,p,q,r]^{T} \tag{2}\] \[\mathbf{q} =[x,y,z,\varphi,\theta,\psi]^{T} \tag{3}\]  
Figure 4: Force analysis of the UAV platform.  
\(\mathbf{V(q)}\) kinematic transition matrix in the form of the following:  
\[\mathbf{V(q)}=\begin{bmatrix}R^{\text{A}}_{\text{b\to g}}&\mathbf{0}\\ \mathbf{0}&R^{\text{P}}_{\text{b\to g}}\end{bmatrix} \tag{4}\]  
where \(R^{\text{P}}_{b\to g}\) represents the position transition matrix from the body coordinate system to the ground coordinate system and \(R^{\text{A}}_{\text{b\to g}}\) represents the attitude transition matrix from the body coordinate system to the ground coordinate system. The specific forms are:  
\[\mathbf{R}^{\text{P}}_{\text{b\to g}}=\begin{pmatrix}c\theta c\varphi&c\theta s \psi&-s\theta\\ s\varphi s\theta c\psi-c\varphi s\psi&s\varphi s\theta s\psi+c\varphi c\psi&s \varphi c\theta\\ c\varphi s\theta c\psi+s\varphi s\psi&c\varphi s\theta c\psi-s\varphi s\psi&c \varphi c\theta\end{pmatrix} \tag{5}\]  
\[R^{\text{A}}_{\text{b\to g}}=\begin{pmatrix}1&0&0\\ s\varphi s\theta/c\theta&c\varphi&s\varphi/c\theta\\ c\varphi s\theta/c\theta&-s\varphi&c\varphi/c\theta\end{pmatrix} \tag{6}\]  
where \(s\varphi\) is \(sin\varphi\); \(s\theta\) is \(sin\theta\); \(s\psi\) is \(sin\psi\). The same goes for \(cos\) and \(tan\).  
\(M(q)\) is an inertia matrix of the form:  
\[\mathbf{M(q)}=\begin{pmatrix}I_{x}&0&0&0&0&0\\ 0&I_{y}&0&0&0&0\\ 0&0&I_{z}&0&0&0\\ 0&0&0&m&0&0\\ 0&0&0&0&m&0\\ 0&0&0&0&0&m\end{pmatrix} \tag{7}\]  
where \(m\) is the mass of the UAV platform.  
\(\mathbf{C(q,p)}\) is a gyroscopic matrix of the form:  
\[\mathbf{C(q,p)}=\begin{pmatrix}0&-I_{y}r&I_{z}q&0&-mw&mv\\ I_{x}r&0&-I_{z}p&mw&0&-mu\\ I_{x}q&I_{y}p&0&-mv&mu&0\\ 0&0&0&0&0&mq\\ 0&0&0&mr&0&mp\\ 0&0&0&-mq&mp&0\end{pmatrix} \tag{8}\]  
\(\mathbf{F(p,q,\mu)}\) is a fusion term containing aerodynamic forces, gravity, control inputs, and tether cable tension. Its specific form is as follows:  
\[\mathbf{F(p,q,u)}=\begin{pmatrix}-\text{L}U_{2}\\ -\text{L}U_{3}\\ -\text{L}U_{4}\\ \text{mgs}\theta-U_{1}(c\theta c\
u s\theta+s\varphi s\psi)\\ -mgc\theta s\varphi-U_{1}(s\varphi c\psi+c\varphi s\theta s\psi)\\ -mgc\theta c\varphi-U_{1}c\varphi c\theta\end{pmatrix} \tag{9}\]  
In addition, the lift and torque of the motor are calculated as follows [20]:  
\[\left\{\begin{array}{ll}F_{\rm i}=\frac{1}{2}\rho_{\rm a}C_{\rm T}\Lambda\rm{R}^ {2}\Omega_{\rm i}^{2},&(i=1,2,3,4)\\ M_{\rm i}=\frac{1}{2}\rho_{\rm a}C_{\rm Q}\Lambda\rm{R}^{3}\Omega_{\rm i}^{2}, &(i=1,2,3,4)\end{array}\right. \tag{11}\]  
where \(\rho_{\rm a}\) is the air density; \(A=\pi R^{2}\) is the rotating area of the paddle; \(R\) is the radius of the paddle; \(\Omega\) is the rotational speed of the paddle.