# Design and Modelling of an Amphibious Spherical Robot Attached with Assistant Fins
## 3 Dynamic Model of the Robot
### Kinematics
\tag{8}\]  
The angular velocity \(\mathbf{\omega}_{p}\) then can be deduced from \(S_{p}\) reversely:  
\[\mathbf{\omega}_{p}=\left(\dot{\phi}c\psi c\theta-\hat{\theta}s\psi,\dot{\theta}c \psi+\dot{\phi}s\psi c\theta,\dot{\psi}-\dot{\phi}s\theta\right). \tag{9}\]  
Note that \(\mathbf{\omega}_{p}\) is not a generalized velocity but a quasi-velocity. The angular velocity of the IDU platform with respect to the \(O_{D}\)-\(X_{D}Y_{D}Z_{D}\) frame can be obtained by coordinate transformation  
\[{}^{D}\mathbf{\omega}_{p}=\overset{U}{D}\mathbf{R}^{\mathrm{T}}\mathbf{\omega}_{p}=\left( \dot{\phi}-\dot{\psi}s\theta,\dot{\theta}c\phi+\dot{\psi}s\phi c\phi,-\dot{ \theta}s\phi+\dot{\psi}c\phi c\theta\right). \tag{10}\]  
To deduce the angular velocity of the spherical shell, let \(S_{s}:=\overset{U}{S}\mathbf{R}\overset{U}{S}\mathbf{R}^{\mathrm{T}}\), where \(\overset{U}{S}\mathbf{R}\) is the rotation matrix between the \(U\)-\(XYZ\) frame and the \(O_{S}\)-\(X_{S}Y_{S}Z_{S}\) frame. \(\overset{U}{S}\mathbf{R}\) can be obtained by matrix multiplication, \(\overset{U}{S}\mathbf{R}=\overset{U}{D}\mathbf{R}\overset{D}{S}\mathbf{R}\), where \(\overset{D}{S}\mathbf{R}\) is the rotation matrix between the \(O_{D}\)-\(X_{D}Y_{D}Z_{D}\) frame and the \(O_{S}\)-\(X_{S}Y_{S}Z_{S}\) frame. \(\overset{D}{S}\mathbf{R}\) can be expressed as  
\[\overset{D}{S}\mathbf{R}=\left[ \begin{array}{ccc}\alpha s_{s}&0&\mathrm{s}s_{s} \\ 0&1&0\\ -\mathrm{s}\alpha_{s}&0&\mathrm{c}\alpha_{s}\end{array} \right]. \tag{11}\]  
Substituting \(\overset{D}{S}\mathbf{R}\) into the equation of \(\overset{U}{S}\mathbf{R}\), \(\overset{U}{S}\mathbf{R}\) then is  
\[\overset{U}{S}\mathbf{R}=\left[ \begin{array}{ccc}c\psi c\theta c\alpha_{s}-(s \psi s\phi+c\psi c\phi s\theta)\mathrm{s}\alpha_{s}&c\psi s\theta s\phi-s\psi c \phi&c\psi c\phi c\mathrm{s}\alpha_{s}+(s\psi s\phi+c\psi c\psi s\theta) \mathrm{c}\alpha_{s}\\ s\psi c\phi c\alpha_{s}+(c\psi s\phi-s\phi s\psi c\phi)\mathrm{s}\alpha_{s}&c \psi c\phi+s\psi s\theta s\phi&\psi c\theta c\mathrm{s}\alpha_{s}+(s\psi s\psi c \phi-c\psi s\phi)\mathrm{c}\alpha_{s}\\ -s\theta c\alpha_{s}-c\phi c\phi s\alpha_{s}&c\theta s\phi&c\phi c\phi c\alpha _{s}-s\theta s\alpha_{s}\end{array} \right]. \tag{12}\]  
The angular velocity of the spherical shell with respect to the \(U\)-\(XYZ\) frame, \(\mathbf{\omega}_{s}=\left(\omega_{sx},\omega_{sy},\omega_{sz}\right)\), can be deduced from the matrix \(S_{s}\), whose three components are  
\[\begin{split}\omega_{sx}&=\dot{\phi}c\psi c\phi-\hat{\theta}s \psi+\dot{\mathrm{\dot{\mathrm{\dot{\mathrm{\dot{\mathrm{\dot{\mathrm{\dot{ \mathrm{\dot{\mathrm{\dot{\mathrm{\dot  
Substituting \( \begin{smallmatrix}D\\ A\end{smallmatrix} \) into the equation of \( \begin{smallmatrix}U\\ A\end{smallmatrix} \)\(R\), we get  
\[ \begin{smallmatrix}U\\ A\end{smallmatrix} R=\left[ \begin{array}{cccc}\text{c}\psi\text{c}\delta \alpha_{s}-(\text{s}\psi\text{s}\phi+\text{c}\psi\text{c}\phi\text{s}\theta) \text{s}\alpha_{s}&\text{c}\psi\text{s}\theta\text{s}\phi-\text{s}\psi\text{c }\phi&\text{c}\psi\text{c}\delta\text{s}\alpha_{s}+(\text{s}\psi\text{s}\phi+ \text{c}\psi\text{c}\phi\text{s}\theta)\alpha_{s}\\ \text{s}\psi\text{c}\delta\alpha_{s}+(\text{c}\psi\text{s}\phi-\text{s}\psi \text{s}\psi\text{c}\phi)\text{s}\alpha_{s}&\text{c}\psi\text{c}\psi+\text{s} \psi\text{s}\theta\text{s}\phi&\text{s}\psi\text{c}\delta\alpha_{s}+(\text{s} \theta\text{s}\psi\text{c}\phi-\text{c}\psi\text{s}\phi)\alpha_{s}\\ -\text{s}\theta\text{c}\alpha_{s}-\text{c}\theta\text{c}\phi\text{s}\alpha_{s} &c\theta\text{s}\phi&\text{c}\theta\text{c}\phi\text{c}\alpha_{s}-\text{s} \theta\text{s}\alpha_{s}\end{array} \right]. \tag{16}\]  
The angular velocity of the rotator with respect to the \(U\)-\(XYZ\) frame, \(\omega_{r}=\left(\omega_{rx},\omega_{ry},\omega_{rz}\right)\), can be deduced from the matrix \(S_{a}\), its three components are  
\[