## 3 Dynamic Modeling of CDPDR  
### Kinematic Model  
Generally, the pose conversion relationship between frames can be expressed as \(\mathbf{X}_{ij}^{j}=\left[\mathbf{x}_{ij}^{j}\quad\gamma_{ij}^{j}\right]^{T}\). In this article, the vector superscript \(j\) represents the reference frame. If the vector has no superscript, the default reference frame is the inertial frame \(O\). For any vector \(\mathbf{v}\) in space, the frame transformation can be completed through the rotation matrix, which satisfies the following relationship:  
\[\mathbf{v}^{j}=\mathbf{R}_{i}^{j}\mathbf{v}^{j} \tag{1}\]  
where \(\mathbf{R}_{i}^{j}\) is the rotation matrix from the \(i\) frame to the \(j\) frame. \[\mathbf{R}_{i}^{j}= \begin{bmatrix}\cos\psi\cos\theta&\cos\psi\sin\theta\sin\phi- \sin\psi\cos\phi&\cos\psi\sin\theta\cos\phi+\sin\psi sin\phi\\ sin\psi\cos\theta&sin\psi\sin\theta\sin\phi+\cos\psi\cos\phi&sin\psi\sin\theta \cos\phi-\cos\psi\sin\phi\\ -\sin\theta&\cos\theta\sin\phi&\cos\theta\cos\phi\end{bmatrix}  \tag{2}\]  
Reference frames and kinematic parameters of CDPDR are shown in Figure 2. \(O\) is the inertial frame. The origin \(O\) is a point in the inertial space, and the coordinate axes \(\mathbf{x}_{o}\), \(y_{o}\) and \(z_{o}\) are consistent with the direction of the hull on the initial calm sea surface. \(S\) is the hull frame. The origin \(S\) is located at the bottom of the hull, the coordinate axis \(\mathbf{x}_{s}\) is the baseline from the stern to the bow, the coordinate axis \(y_{s}\) points from starboard to port, and the direction of the axis \(z_{s}\) is determined by the right-hand rule. \(C\) is the CDPDR pulley frame, and its origin position is related to the location of derusting operation surface and ship structure parameters. The directions of \(\mathbf{x}_{c}\), \(y_{c}\) and \(z_{c}\) are the same as the hull frame, and the coordinate origin coincides with \(C_{4}\). \(P\) is the motion platform frame. The coordinate origin \(P\) is the mass center of the motion platform. Figure 1: (**a**) Schematic diagram of working principle; (**b**) 3D model of the prototype. The excitation exerted by the waves on the hull can be summed up as a 6-DOF motion, defined as \(\mathbf{X}_{wave}= \begin{bmatrix}R_{x}&R_{y}&R_{z}&T_{x}&T_{y}&T_{z}\end{bmatrix} ^{T}\). Among them \(R_{x}\), \(R_{y}\) and \(R_{z}\) are respectively the roll angle, pitch angle and yaw angle of the ship. \(T_{x}\), \(T_{y}\) and \(T_{z}\) correspond to the ship's surge, sway and heave motions respectively. \(\mathbf{p}_{i}\) and \(\mathbf{c}_{i}\) (\(i=1,\cdots,m\)) represent the connection point of the cable with the motion platform and the CDPDR pulley respectively. \(l_{i}\) represents the cable length vector of the \(i\)-th cable, specifically expressed as  
\[\mathbf{l}_{i}=\mathbf{c}_{i}-(\mathbf{x}_{PO}+\mathbf{p}_{i}) \tag{3}\]  
where \(\mathbf{c}_{i}=\overset{\rightarrow}{OC}_{i}\) and \(\mathbf{p}_{i}=\overset{\rightarrow}{P\mskip-10.0 mu \rightarrow}_{P \mskip-10.0 mu \rightarrow}\). Further, Equation (3) is expressed in the frame C:  
\[\mathbf{l}_{i}^{C}=\mathbf{c}_{i}^{C}-(\mathbf{x}_{PC}^{C}+\mathbf{R}_{P}^{C}\mathbf{p}_{i}^{P}) \tag{4}\]  
where \(\mathbf{p}_{i}^{P}\) is the connecting point of the motion platform in the frame \(P\), which is related to the volume of the derusting tools and is also a known constant. \(\mathbf{x}_{PC}^{C}\) is the input trajectory. In order to reduce the kinematic error of CDPDR, the influence of the pulley model on the cable length should be considered. Define \(\mathbf{c}_{i}^{C}\) as the outgoing cable point on the pulley, \(k_{i}^{C}= \begin{bmatrix}k_{ix}^{C}&k_{iy}^{C}&k_{iz}^{C}\end{bmatrix} \) as the center point of the pulley and \(r_{pully}\) as the radius of the pulley. The position of point \(\mathbf{c}_{i}^{C}\) can be obtained:  
\[\left\{