## 3 Parametric Modeling and Model Validation
### Parametric Model Validation  
Although the structural parameters of the model can be changed quickly through parametric modeling, unverified models can lead to unreliable designs. Therefore, it is essential to compare the simulation test results with the theoretical calculation results to verify the validity of the parametric model before the optimization design is carried out. Selecting the drive branch chain as the research object, when the distribution circle radius \(r\) of the geometric center of the spherical hinge of the mobile platform and the distribution circle radius \(R\) of the geometric center of the universal joint of the fixed platform are known, the position vector \(\mathbf{A_{i}}\) at the connection between the mobile platform and the spherical hinge and the position vector \(\mathbf{B_{i}}\) at the connection between the fixed platform and the universal joint can be represented on the respective coordinate systems to determine the length vector \(\mathbf{L_{i}}\) of the drive branch chain. \[\mathbf{L_{i}}=\mathbf{R}\mathbf{A_{i}}+\mathbf{E_{i}}-\mathbf{B_{i}}\;(i=1,2,\cdots,6) \tag{1}\]  
where \(\mathbf{L_{i}}\) is the length vector of the drive branch chain; \(\mathbf{R}\) is the rotation transformation matrix obtained after three rotation transformations; \(\mathbf{A_{i}}\) is the position vector at the connection between the mobile platform and the spherical hinge; \(\mathbf{E_{i}}\) is the unit matrix; \(\mathbf{B_{i}}\) is the position vector at the connection between the fixed platform and the universal joint. The coordinate transformation diagram of the mechanism is shown in Figure 4. Among them, the body-fixed coordinate system (BCS) \(O_{a}\)-\(X_{a}Y_{a}Z_{a}\) on the mobile platform can be regarded as obtained by the inertial coordinate system (ICS) \(O_{b}\)-\(X_{b}Y_{b}Z_{b}\) on the fixed platform after three translations and three rotations. The ICS is translated in the order along the \(X\) axis, \(Y\) axis, and \(Z\) axis, so that the origin \(O_{b}\) of the ICS coincides with the origin \(O_{a}\) of the BCS to obtain the translation transformation matrix \(\mathbf{T}\):  
\[\mathbf{T}= \begin{bmatrix}x&y&z\end{bmatrix} ^{\mathrm{T}} \tag{2}\]  
where \(x\), \(y\), and \(z\) are the displacements of the BCS relative to the \(X\), \(Y\), and \(Z\) axes of the ICS, respectively. The ICS is rotated in the order around the \(X\) axis, \(Y\) axis, and \(Z\) axis, so that the coordinate axes of the ICS and the BCS are completely coincident to obtain the rotation transformation matrix \(\mathbf{R}\):  
\[\mathbf{R}= \begin{bmatrix}\varsigma\gamma\varsigma\psi&\varsigma\gamma\psi s \varphi-s\gamma\varsigma\varphi&s\gamma s\varphi+c\varphi c\gamma\varsigma\psi \\ s\gamma\varsigma\psi&\varsigma\gamma\varsigma\varphi-s\sigma s\gamma\varsigma \psi&s\gamma\varsigma\psi c\varphi-c\gamma s\varphi\\ -s\psi&c\psi s\varphi&c\psi c\varphi\end{bmatrix}  \tag{3}\]  
Figure 3: Parametric model of the 6-UPS parallel mechanism. where s represents the sine function \(\sin x\); c represents the cosine function \(\cos x\); \(\varphi\), \(\psi\), and \(\gamma\) are the rotation angles of the BCS relative to the \(X\), \(Y\), and \(Z\) axes of the ICS, respectively. Let \(A_{i}\) be the absolute coordinates of the spherical hinge on the mobile platform and \(B_{i}\) be the absolute coordinates of the universal joint on the fixed platform, then  
\[\left\{\begin{array}{ccc}A_{i}=\left[ \begin{array}{cc}x_{ia}&y_{ia}&z_{ia} \\ B_{i}=\left[\begin{array}{cc}x_{ib}&y_{ib}&z_{ib}\end{array}\right]^{\rm T} \end{array} \right.\left(i=1,2,\cdots,6\right) \tag{4}\]  
Substitute into Formula (1) to obtain  
\[\mathbf{L_{i}}= \begin{bmatrix}l_{ix}\\ l_{iy}\\ l_{iz}\end{bmatrix} =