## 2 Dynamic Modeling of Trans-Medium Process
### Dynamic Modeling
#### 2.4 The HAUV Dynamic Model  
Based on the above analysis, a dynamic model can be obtained according to the Newton-Euler formula. Owing to the underactuated characteristics of the coaxial HAUV, the coaxial HAUV model is divided into a double-ring structure [30], namely a translation subsystem and a rotation subsystem, as shown in the following formulae:\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{4}x_{6}\left(I_{yw}-I_{zw}\right)/I_{xw}\,+\,u_{2}/\,I_{xw}\,-\, \left(k_{s}k_{qw}+k_{sa}k_{qa}\right)|x_{2}|x_{2}/\,I_{xw}\,+\,d_{\phi}\\ \dot{x}_{3}=x_{4}\\ \dot{x}_{4}=x_{2}x_{6}\left(I_{zw}-I_{xw}\right)/I_{yw}\,+\,u_{3}/\,I_{yw}\,- \,\left(k_{s}k_{\theta w}+k_{sa}k_{\theta a}\right)|x_{4}|x_{4}/\,I_{yw}\,+\,d_ {\theta}\\ \dot{x}_{5}=x_{6}\\ \dot{x}_{6}=x_{2}x_{4}\left(I_{xw}-I_{yw}\right)/I_{zw}\,+\,u_{4}/\,I_{zw}\,- \,\left(k_{s}k_{\psi w}+k_{sa}k_{\psi a}\right)|x_{6}|x_{6}/\,I_{zw}\,+\,d_{ \phi}\end{array}\right. \tag{14a}\] \[\left\{\begin{array}{l}\dot{x}_{7}=x_{8}\\ \dot{x}_{8}=u_{1}/m_{w}\left(cx_{1}\cdot sx_{3}\cdot cx_{3}+sx_{1}\cdot sx_{5} \right)-\,\left(k_{s}k_{xw}+k_{sa}k_{xu}\right)|x_{8}|x_{8}|x_{8}/m_{w}\,+\,d_ {x}\\ \dot{x}_{9}=x_{10}\\ \dot{x}_{10}=u_{1}/m_{w}\left(cx_{1}\cdot sx_{3}\cdot sx_{5}-sx_{1}\cdot cx_{ 5}\right)-\,\left(k_{s}k_{yw}+k_{sa}k_{yu}\right)|x_{10}|x_{10}/\,m_{w}\,+\,d_ {y}\\ \dot{x}_{11}=x_{12}\\ \dot{x}_{12}=u_{1}/m_{w}cx_{1}\cdot cx_{3}-g+\,k_{s}F_{bw}/m_{w}\,-\,\left(k_ {s}k_{zw}+k_{sa}k_{za}\right)|x_{12}|x_{12}/m_{w}\,+\,d_{z}\end{array}\right. \tag{14b}\]  
where \(x=\left[\phi,\dot{\phi},\theta,\dot{\theta},\psi,\dot{p},x,\dot{x},y,\dot{y},z, \dot{z}\right]^{\mathrm{T}}\in R^{12}\) is defined as a state vector. The rotational dynamic model and the translational dynamic model can be considered second-order systems with perturbations. \(d=\left[d_{x}\quad d_{y}\quad d_{z}\quad d_{\phi}\quad d_{\phi}\right]^{\mathrm{ T}}\) is the lumped uncertainty, including model uncertainty and environmental disturbance.  
**Remark 5**.: _Since the vehicle body can always maintain a small attitude-angle change operation during the vertical takeoff and landing process, the small angle assumption can be further made. Based on the small angle hypothesis, we have reasonably simplified the mathematical model [19]._  
**Assumption 3**.: _Obviously, lumped uncertainties are not divergent, and there exists a positive number satisfying the inequation, \(\|d\|\leq D\), where \(D\) is the upper bound of lumped uncertainty. \(D\) can be expressed as \(D\leq a_{0i}+a_{1i}|e_{i}|+a_{2i}|e_{i+1}|,i=1,3,5,\cdots,11\)._  
Since the whole system is divided into inner and outer loops, the position of virtual control is defined as \(V=\left[v_{x}\quad v_{y}\quad v_{z}\right]^{\mathrm{T}}\).  
The virtual position controller is designed according to Equation (15) [31]:  
\[\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\end{bmatrix}=\left[\begin{array}{l}u_{1}/m_{w}\left(c\phi\cdot s\theta \cdot c\psi+s\phi\cdot s\psi\right)\\ u_{1}/m_{w}\left(c\phi\cdot s\theta\cdot s\psi-s\phi\cdot c\psi\right)\\ u_{1}/m_{w}\left(c\phi\cdot c\theta-g+k_{s}F_{bw}/m_{w}\right)\end{array}\right] \tag{15}\]  
Hence, the expected roll and pitch can be calculated through the virtual position controller. Meanwhile, the total lift force can be calculated as follows:  
\[\left\{\begin{array}{l}u_{1}=m\sqrt{v_{x}^{2}+v_{y}^{2}+\left(v_{z}+g-k_{s} F_{bw}/m_{w}\right)^{2}}\\ \phi_{d}=\tan^{-1}\left(\cos\theta_{d}\left(\frac{v_{x}\sin\phi_{d}-v_{y}\cos \phi_{d}}{v_{z}+g-k_{s}F_{bw}/m_{w}}\right)\right)\\ \theta_{d}=\tan^{-1}\left(\frac{v_{x}\cos\psi_{d}+v_{y}\sin\phi_{d}}{v_{z}+g-k _{s}F_{bw}/m_{w}}\right)\end{array}\right. \tag{16}\]