## 3 Dynamic Modeling of CDPDR
### Dynamic Model  
In severe sea conditions, the disturbance of the waves to the upper working space of the hull is quite large, which interferes with the normal work of the CDPDR. Therefore, the dynamic model must consider the motion of the hull relative to the inertial frame. Reference frames and dynamic parameters of CDPDR are shown in Figure 3.  
Figure 3: Reference frames and dynamic parameters of CDPDR. The position vector of the mass center of the motion platform in the inertial frame can be expressed as  
\[x^{O}_{PO}=x^{O}_{SO}+R^{O}_{S}x^{S}_{PS} \tag{13}\]  
where \(x^{S}_{PS}=x^{C}_{PC}+x^{S}_{CS}\). Taking the derivation of Equation (13), the velocity vector of the mass center of the motion platform relative to the inertial frame can be obtained as  
\[\dot{x}^{O}_{PO}=\dot{x}^{O}_{SO}+\left[\omega^{O}_{SO}\times\right]R^{O}_{S}x^{ S}_{PS}+R^{O}_{S}\dot{x}^{S}_{PS} \tag{14}\]  
where \(\left[\omega^{O}_{SO}\times\right]\) is an obliquely symmetric matrix of \(\omega^{O}_{SO}\). Taking the derivation of Equation (14), the acceleration of the mass center of the motion platform in the inertial frame is expressed as  
\[\ddot{x}^{O}_{PO}=\ddot{x}^{O}_{SO}+\left[\dot{\omega}^{O}_{SO}\times\right]R^ {O}_{S}x^{S}_{PS}+\left[\omega^{O}_{SO}\times\right]\left[\omega^{O}_{SO}\times \right]R^{O}_{S}x^{S}_{PS}+2\left[\omega^{O}_{SO}\times\right]R^{O}_{S}\dot{x} ^{S}_{PS}+R^{O}_{S}\dot{x}^{S}_{PS} \tag{15}\]  
The angular velocity of the motion platform relative to the inertial frame is  
\[\omega^{O}_{PO}=R^{O}_{S}\omega^{S}_{SO}+R^{O}_{S}\omega^{S}_{PS} \tag{16}\]  
Taking the derivation of Equation (16), the angular acceleration of the motion platform relative to the inertial frame is  
\[\dot{\omega}^{O}_{PO}=R^{O}_{S}\dot{\omega}^{S}_{SO}+\left[\omega^{O}_{SO} \times\right]R^{O}_{S}\omega^{S}_{PS}+R^{O}_{S}\dot{\omega}^{S}_{PS} \tag{17}\]  
According to the rigid body dynamic theory  
\[\omega^{P}_{PS}=I_{PS}\dot{\psi}_{PS} \tag{18}\]  
where \(I_{PS}= \begin{bmatrix}1&0&-\sin\theta_{PS}\\ 0&\cos\phi_{PS}&\sin\phi_{PS}\cos\theta_{PS}\\ 0&-\sin\phi_{PS}&\cos\phi_{PS}\cos\theta_{PS}\end{bmatrix} \). The angular acceleration of the frame \(P\) relative to the frame \(S\) is obtained by further derivation as  
\[\dot{\omega}^{P}_{PS}=I_{PS}\ddot{\psi}_{PS}+\dot{I}_{PS}\dot{\psi}_{PS} \tag{19}\]  
Before establishing the dynamic model of CDPDR, make the following assumptions:  
1. The cable mass is much smaller than the load weight and is not considered when modeling. 2. Cables are generally elastic and will deform elastically when subjected to force. However, the elastic deformation is very small compared to the cable length, so the cable elasticity is ignored. The system dynamic model is established based on the Newton-Euler method. The resultant force and moment acting on the motion platform are  
\[\left\{ \begin{aligned} m\ddot{x}^{O}_{PO}=\sum\limits_{i=1}^{m} \mathbf{u}_{i}t_{i}+m\mathbf{g}+\mathbf{f}_{r}\\ I\dot{\omega}^{O}_{PO}+\omega^{O}_{PO}\times\left(I\omega^{O}_{PO}\right)= \sum\limits_{i=1}^{m}t_{i}(\mathbf{p}_{i}\times\mathbf{u}_{i})+\mathbf{m}_{r}\end{aligned} \right. \tag{20}\]  
where m is the mass of the motion platform. \(I\) is the inertia matrix of the motion platform relative to the center of mass in the inertial frame, \(\mathbf{I}=R^{O}_{S}R^{C}_{P}I_{PP}\left(R^{C}_{P}\right)^{T}\left(R^{O}_{S} \right)^{T}\). \(I_{PP}= \begin{bmatrix}I_{xx}&-I_{xy}&-I_{yz}\\ -I_{yx}&I_{yy}&-I_{yz}\\ -I_{zx}&-I_{zy}&I_{zz}\end{bmatrix} \) is the inertia matrix in the motion platform frame. Since the motion platform frame \(P\) coincides with the inertial main axis of the motion platform, the inertial product \(I_{xy}=I_{xz}=I_{yz}=0\). t\({}_{i}\) is the \(i\)-th cable tension. \(f_{r}\) is the external disturbance force, and \(m_{r}\) is the external disturbance torque.