## 2 Dynamic Model of a Tilt-Rotor Quadcopter (TRQ) with Various Actuation Modes
### Rotational and Translational Dynamics  
The rotation matrix \({}^{B}R_{P_{i}}\) from the ith rotor frame to the body frame is  
\[{}^{B}R_{P_{i}}=R_{Z}(\beta_{i})R_{X}(\alpha_{i}) \tag{1}\]  
\[R_{X}(\alpha_{i})= \begin{bmatrix}1&0&0\\ 0&\cos\alpha_{i}&-\sin\alpha_{i}\\ 0&\sin\alpha_{i}&\cos\alpha_{i}\end{bmatrix} ,\]  
and  
\[R_{Z}(\beta_{i})= \begin{bmatrix}\cos\beta_{i}&-\sin\beta_{i}&0\\ \sin\beta_{i}&\cos\beta_{i}&0\\ 0&0&1\end{bmatrix} ,\ \beta_{i}=(i-1)\frac{\pi}{2},\ i=1,\ 2,\ 3,\ 4.\]  
The angular velocity is  
\[w_{Pi}={}^{P_{i}}R_{B}w_{B}+ \begin{bmatrix}\dot{a}_{i}&0&w_{i}\end{bmatrix} ^{T} \tag{2}\]  
We define  
\[sgn(x)= \begin{cases}1&if\ x\ >0\\ -1&if\ x\ <0\end{cases} \]  
where  
\[w_{1}<0,\ w_{3}<0\,\ w_{2}>0,\ w_{4}>0\]  
The rotational dynamics of the TRQ can be formulated as:  
\[\tau_{B}=I_{B}\dot{w}_{B}+w_{B}\times I_{B}w_{B}+\sum_{i=1}^{4}{}^{B}R_{P_{i}} \tau_{Pi} \tag{3}\]  
where  
\[\tau_{Pi}=I_{Pi}\dot{w}_{Pi}+w_{Pi}\times I_{Pi}w_{Pi}-\tau_{di} \tag{4}\]  
\[\tau_{di}= \begin{bmatrix}0&0&-k_{m}w_{i}^{2}sgn(w_{i})\end{bmatrix} ^{T} \tag{5}\]  
The force in the rotor frame is  
\[T_{pi}= \begin{bmatrix}0&0&k_{f}w_{i}^{2}\end{bmatrix} ^{T} \tag{6}\]  
and the toque in body frame is  
\[\tau_{B}=\sum_{i=1}^{4}({}^{B}O_{P_{i}}\times{}^{B}R_{P_{i}}T_{Pi}) \tag{7}\]  
The transform between body angular rates to the Euler rates is  
\[\dot{r}=R_{T}w_{B} \tag{8}\]  
where  
\[R_{T}= \begin{bmatrix}1&s_{\phi}\tan\theta&c_{\phi}\tan\theta\\ 0&c_{\phi}&-s_{\phi}\\ 0&s_{\phi}\sec\theta&c_{\phi}\sec\theta\end{bmatrix}  \tag{9}\]  
and \(r= \begin{bmatrix}\phi&\theta&\psi\end{bmatrix} ^{T}\in R^{3}\) is the attitude vector of the roll, the pitch, and the yaw angle. We denote \(s_{q_{i}}=\sin q_{i}\) and \(c_{q_{i}}=\cos\ q_{i}\). Taking the derivative of (8) and ignoring \(I_{Pi}\) in (4), we have  
\[\bar{r}=\dot{R}_{T}R_{T}^{-1}\dot{r}+R_{T}I_{B}^{-1}(\tau_{B}-w_{B}\times I_{B }w_{B}+\sum_{i=1}^{4}{}^{B}R_{P_{i}}\tau_{di}) \tag{10}\]We denote  
\[\tau=\sum_{i=1}^{4}(^{B}O_{P_{i}}\times{}^{B}R_{P_{i}}T_{Pi})+\sum_{i=1}^{4}{}^{B }R_{P_{i}}\tau_{di} \tag{11}\]  
Define the transform matrix  
\[\Psi=R_{T}^{-1} \tag{12}\]  
From (8), we have  
\[w_{B}=\Psi\dot{r} \tag{13}\]  
It follows from (10) and (12) that  
\[ \begin{array}{c}\Psi^{T}I_{B}\Psi\ddot{r}=-\Psi^{T}I_{B}\dot{\Psi}\dot{r}- \Psi^{T}w_{B}\times I_{B}w_{B}+\Psi^{T}\;\tau\\ =-\Big{[}\Psi^{T}I_{B}\dot{\Psi}+\Psi^{T}\big{(}\Psi\dot{r}\times I_{B}\Psi\big{)} \Big{]}\dot{r}+\Psi^{T}\tau\end{array}  \tag{14}\]  
Considering the perturbation torque \(\tau_{d}\), we obtain  
\[H(r)\ddot{r}+C\big{(}r,\dot{r}\big{)}\dot{r}=\Psi^{T}(\tau+\tau_{d}) \tag{15}\]  
where  
\[H(r)=\Psi^{T}I_{B}\Psi \tag{16}\]  
\[C\big{(}r,\dot{r}\big{)}=\Big{[}\Psi^{T}I_{B}\dot{\Psi}+\Psi^{T}\big{(}\Psi \dot{r}\times I_{B}\Psi\big{)}\Big{]} \tag{17}\]  
\[\Psi(r)= \begin{bmatrix}1&0&-s_{\theta}\\ 0&c_{\phi}&c_{\theta}s_{\phi}\\ 0&-s_{\phi}&c_{\theta}c_{\phi}\end{bmatrix}  \tag{18}\]  
and \(H(r)\;\in\;R^{3\times 3}\) is the inertia matrix, \(C\big{(}r,\dot{r}\big{)}\dot{r}\in R^{3}\) represents the centrifugal and Coriolis forces. \(\tau= \begin{bmatrix}\tau_{\phi}&\tau_{\theta}&\tau_{\phi}\end{bmatrix} ^{T}\in R ^{3}\) is the vector of torques and \(\tau_{d}\in R^{3}\) is the perturbation torque. The velocity in \(F_{w}\) is  
\[\dot{p}={}^{W}R_{B}V_{B} \tag{19}\]  
The derivative of velocity in \(F_{B}\) can be expressed as  
\[\dot{V}_{B}=-w_{B}\times V_{B}+{}^{W}R_{B}^{T}\big{[}0\quad 0\quad-g\big{]}^{T}+ \frac{f}{m} \tag{20}\]  
where  
\[{}^{W}R_{B}= \begin{bmatrix}c_{\theta}c_{\phi}&s_{\phi}s_{\phi}c_{\phi}-c_{ \phi}s_{\psi}&c_{\phi}s_{\phi}c_{\psi}+s_{\phi}s_{\psi}\\ c_{\theta}s_{\psi}&s_{\phi}s_{\theta}s_{\psi}+c_{\phi}c_{\psi}&c_{\phi}s_{ \theta}s_{\psi}-s_{\phi}c_{\psi}\\ -s_{\theta}&s_{\phi}c_{\theta}&c_{\phi}c_{\theta}\end{bmatrix} \]  
and  
\[f=\sum_{i=1}^{4}{}^{B}R_{P_{i}}T_{Pi} \tag{21}\]  
where \(p= \begin{bmatrix}x&y&z\end{bmatrix} ^{T}\in R^{3}\). \(f= \begin{bmatrix}f_{x}&f_{y}&f_{z}\end{bmatrix} ^{T}\in R^{3}\) is the vector of forces.