## 4 Kinematic Model
### Forward Kinematics of 3(Rc)PU PM  
The position parameters {\(X\), \(Y\), \(Z\), \(\alpha\), \(\beta\), \(\gamma\)} of the moving platform are obtained from the input parameters {\(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), \(l_{1}\), \(l_{2}\), \(l_{3}\)} of the actuated joint, which is the forward kinematics analysis of the 3(Rc)PU PM. The geometric parameter constraint equation based on the moving platform in the fixed coordinate system \(O_{n}\)-\(XYZ\) is established. According to the cosine theorem, the geometric constraint equation on the moving platform is  
\[\left\{\begin{array}{l}\left(X_{O_{n}B_{1}}-X_{O_{n}B_{2}}\right)^{2}+\left(Y _{O_{n}B_{1}}-Y_{O_{n}B_{2}}\right)^{2}+\left(Y_{O_{n}B_{1}}-Y_{O_{n}B_{2}} \right)^{2}=\left(\sqrt{s}r\right)^{2}\\ \left(X_{O_{n}B_{2}}-X_{O_{n}B_{3}}\right)^{2}+\left(Y_{O_{n}B_{2}}-Y_{O_{n}B_ {3}}\right)^{2}+\left(Y_{O_{n}B_{2}}-Y_{O_{n}B_{3}}\right)^{2}=\left(\sqrt{s}r \right)^{2}\\ \left(X_{O_{n}B_{3}}-X_{O_{n}B_{1}}\right)^{2}+\left(Y_{O_{n}B_{3}}-Y_{O_{n}B_ {1}}\right)^{2}+\left(Y_{O_{n}B_{3}}-Y_{O_{n}B_{1}}\right)^{2}=\left(\sqrt{s}r \right)^{2}\end{array}\right. \tag{23}\]  
By substituting the vector Equation (20) in the fixed coordinate system \(O_{n}\)-\(XYZ\) into Equation (23) and substituting the position parameters in the equation with cos\(\theta_{i}\) = (1 - \(t_{i}^{\ 2}\))/ (1+ \(t_{i}^{\ 2}\)), sin\(\theta_{i}\) = 2\(t_{i}\)/(1 + \(t_{i}^{\ 2}\)), a system of equations can be obtained:  
\[\left\{\begin{array}{l}f_{1}(1,\ t_{1}^{\ 2},\ t_{2}^{\ 2},\ t_{1}t_{2},\ t_{ 1}^{\ 2}t_{2}^{\ 2})=0\\ f_{2}(1,\ t_{2}^{\ 2},\ t_{3}^{\ 3},\ t_{2}t_{3},\ t_{2}^{\ 2}t_{3}^{\ 2})=0\\ f_{3}(1,\ t_{1}^{\ 2},\ t_{3}^{\ 2},\ t_{1}t_{3},\ t_{1}^{\ 2}t_{3}^{\ 2})=0 \end{array}\right. \tag{24}\]  
where \(f_{i}\)(\(\cdot\)) is a linear equation for \(t_{i}\), whose coefficients depend only on known structure parameters. The Sylvester junction elimination method was used to eliminate some parameters, and the solutions of all equation parameters (\(t_{1}\), \(t_{2}\), \(t_{3}\)) were obtained. The results were substituted into the geometric constraint equation \(\theta_{i}\) = 2 arctan(\(t_{i}\)), and the angles between each limb and the static platform plane were obtained; these angles can also be regarded as the output variables of the PM. The forward kinematic solution of the 3(Rc)PU PM is as follows:  
\[\left\{\begin{array}{l}\mathbf{w}=\left(\mathbf{O}_{n}B_{2}-\mathbf{O}_{n}B_{1}\right) \times\left(\mathbf{O}_{n}B_{3}-\mathbf{O}_{n}\mathbf{B}_{1}\right)/\left\|\left(\mathbf{O}_{n }\mathbf{B}_{2}-\mathbf{O}_{n}\mathbf{B}_{1}\right)\times\left(\mathbf{O}_{n}B_{3}-\mathbf{O}_{n} \mathbf{B}_{1}\right)\right\|\\ \mathbf{u}=\left(\mathbf{O}_{n}B_{2}-\mathbf{O}_{n}B_{3}\right)/\left\|\left(\mathbf{O}_{n}B_{ 2}-\mathbf{O}_{n}B_{3}\right)\right\|\\ \mathbf{v}=\left(\mathbf{w}\times\mathbf{u}\right)\\ \mathbf{R}=\left(\mathbf{u},\mathbf{v},\mathbf{w}\right)\\ \mathbf{P}=\mathbf{O}_{n}\mathbf{B}_{1}+r\mathbf{v}\end{array}\right. \tag{25}\]  
where \(\mathbf{R}\) is the attitude matrix of the moving platform (when the mechanism is in 3T motion mode, the rotation matrix is zero matrix), and \(\mathbf{P}\) is the position vector of the center of the moving platform shape in the fixed coordinate system \(O_{n}\)-\(XYZ\).