## 2 Model Identification of 3-SRR/SRU  
### Mathematical Model  
In this section, the mathematical model devised by [PERSON] et al. [14] for the 3-SRR/SRU PM is recapitulated, where \(\underline{\mathrm{R}}\) designates an actuated revolute joint. The 3-SRR/SRU PM, depicted in Figure 1, was investigated as a stabilization platform for helicopters landing aboard ships safely. It consists of three SRR limbs, \(B_{1}C_{1}A_{1}\), \(B_{2}C_{2}A_{2}\), and \(B_{4}C_{4}A_{4}\), in addition to one passive SRU limb, \(B_{3}C_{3}A_{3}\). The moving plate has 2R1T 3-DOF motion [14], namely two independent rotations in addition to one translation. Points \(A_{i}\) and \(B_{i}\) (\(i=1,\cdots,4\)) denote the intersection between the moving plate and the base, respectively, and the joints implemented on them. Points \(P\), \(O\), and \(O_{s}\) express the centers of the moving plate, the base, and the ship, respectively, the base being installed on the ship. The relative position of the base with respect to the ship is expressed by \(m\), \(n\), and \(r\), namely, the coordinates of \(O\) with respect to \(O_{s}\).  
Figure 1: CAD model of 3-SRR/SRU parallel mechanism.  
The motion of the ship is represented by heavy \(\Delta Z\), roll \(\alpha\), and pitch \(\beta\) in the reference frame \(\{O_{s},x,y,z\}\), as illustrated in Figure 1. The moving plate and the base both have a square cross-section, as seen from the top. Thus, let \(l_{O}\) denote the distance between the center of the base, \(O\), and each of points \(B_{i}\) (\(i=1,\cdots,4\)). Accordingly, the coordinates of \(B_{i}\) relative to the ship's motion are as follows (in Equation (1), operators \"\(s\)\" and \"\(c\)\" represent the trigonometric functions \"\(\sin\)\" and \"\(\cos\)\", respectively):  
\[\begin{split} O_{SB_{1}}&=\begin{bmatrix}-l_{O}s \alpha s\beta+mc\beta+ns\alpha\beta+(r+\Delta Z)c\alpha s\beta\\ -l_{O}c\alpha+nc\alpha-(r+\Delta Z)s\alpha\\ -l_{O}s\alpha c\beta-ms\beta+ns\alpha c\beta+(r+\Delta Z)c\alpha c\beta\end{bmatrix} \\ O_{SB_{2}}&=\begin{bmatrix}l_{O}c\beta+mc\beta+ns\alpha s\beta+(r+\Delta Z)c \alpha s\beta\\ nc\alpha-(r+\Delta Z)s\alpha\\ l_{O}s\beta-ms\beta+ns\alpha c\beta+(r+\Delta Z)c\alpha c\beta\end{bmatrix} \\ O_{SB_{3}}&=\begin{bmatrix}l_{O}s\alpha s\beta+m\ c\beta+ns\alpha s\beta+(r+ \Delta Z)c\alpha s\beta\\ l_{O}c\alpha+nc\alpha-(r+\Delta Z)s\alpha\\ l_{O}s\alpha c\beta-ms\beta+ns\alpha c\beta+(r+\Delta Z)c\alpha c\beta\end{bmatrix} \\ O_{SB_{4}}&=\begin{bmatrix}-l_{O}c\beta+mc\beta+ns\alpha s\beta+(r+\Delta Z)c \alpha s\beta\\ nc\alpha-(r+\Delta Z)s\alpha\\ l_{O}s\beta-ms\beta+ns\alpha c\beta+(r+\Delta Z)c\alpha c\beta\end{bmatrix} \end{split} \tag{1}\]  
Similarly, let \(l_{P}\) express the distance between center of the moving plate and each of points \(A_{i}\) (\(i=1,\cdots,4\)). In addition, let \(Z_{0}\) represent the initial distance between the moving plate and the base. The initial coordinates of points \(A_{i}\) (\(i=1,\cdots,4\)) with respect to \(O_{S}\) can be written as follows:  
\(Z_{0}\) is the initial distance between the two plates, and \(l_{a}\) and \(l_{b}\) present the distance between the center frame \(O\) and the selected \(A\) (\(i=1,\) 2, 3, and 4). Recalling that the initial coordinates of each \(A\) with respect to \(O_{b}\) can be written as follows [14]:  
\[\begin{split} O_{SA_{1}}&=\begin{bmatrix}m\\ n\\ r+Z_{0}\end{bmatrix}+\begin{bmatrix}0\\ -l_{P}\\ 0\end{bmatrix}=\begin{bmatrix}m\\ n-l_{P}\\ r+Z_{0}\end{bmatrix}\\ O_{SA_{2}}&=\begin{bmatrix}m\\ n\\ r+Z_{0}\end{bmatrix}+\begin{bmatrix}l_{P}\\ 0\end{bmatrix}=\begin{bmatrix}m+l_{P}\\ n\\ r+Z_{0}\end{bmatrix}\\ O_{SA_{3}}&=\begin{bmatrix}m\\ n\\ r+Z_{0}\end{bmatrix}+\begin{bmatrix}0\\ l_{P}\\ 0\end{bmatrix}=\begin{bmatrix}m\\ n+l_{P}\\ r+Z_{0}\end{bmatrix}\\ O_{SA_{4}}&=\begin{bmatrix}m\\ n\\ r+Z_{0}\end{bmatrix}+\begin{bmatrix}-l_{P}\\ 0\\ 0\end{bmatrix}=\begin{bmatrix}m-l_{P}\\ n\\ r+Z_{0}\end{bmatrix}\end{split} \tag{2}\]