## 4 Kinetostatic Model of _n_-4R Compliant Parallel Pointing Mechanism
### Relationship between Input Force \(F_{1}\) and Output Displacement \(U_{1}\)  
For the convenience of analysis, the mechanism is simplified as an equivalent spring system, and the concept of equivalent stiffness is introduced [24; 27]. As shown in Figure 10, \(\mathbf{K}_{\text{B1A}}\) is defined as the equivalent stiffness matrix of the part between the loading position on branch 1 and the fixed platform. \(\mathbf{K}_{\text{B1B}}\) is defined as the equivalent stiffness matrix of the part between the loading position on branch 1 and the mobile platform. \(\mathbf{K}_{\text{B2}}\), \(\mathbf{K}_{\text{B3}}\), and \(\mathbf{K}_{\text{B4}}\) are the equivalent stiffness matrices of branch 2, 3, and 4, respectively. Therefore, according to [PERSON]'s law, the governing equations of the elastic deformation of the system is expressed as:  
\[ \begin{bmatrix}\left(\mathbf{K}_{\text{OO}}\right)_{F_{1}}&\mathbf{K}_{\text{OF}_{1} }\\ \mathbf{K}_{F_{1}O}&\mathbf{K}_{F_{1}F_{2}}\end{bmatrix}\begin{bmatrix}\mathbf{U}_{1}\\ \mathbf{U}_{F_{1}}\end{bmatrix} = \begin{bmatrix}\mathbf{F}_{\text{O}}\\ \mathbf{F}_{1}\end{bmatrix}  \tag{8}\]  
Figure 9: (a) Complete overview of the 4-4R compliant parallel pointing mechanism to analyze; (b) coordinate frame setting and loading position of force \(\mathbf{F}_{1}\). where \(\mathbf{U}_{1}\) represents the displacement of the mobile platform center relative to the global coordinate frame \(Oxyz\), \(\mathbf{U}_{F1}\) represents the displacement of the action point of force \(F_{1}\) relative to coordinate frame \(F_{1}x_{F1}y_{F1}z_{F1}\), and \(\mathbf{F}_{O}\) represents the force acting on the mobile platform center. The stiffness matrices in the governing equations of spring system are computed by Equation (9). \[\left\{ \begin{array}{l}\left(\mathbf{K}_{\mathrm{OO}}\right)_{F_{1}}=\mathbf{K}_{B1B} ^{O}+\mathbf{K}_{B2}^{O}+\mathbf{K}_{B3}^{O}+\mathbf{K}_{B4}^{O}\\ \mathbf{K}_{F_{1}F_{1}}=\mathbf{K}_{B1A}^{F_{1}}+\mathbf{K}_{B1B}^{F_{1}}\\ \mathbf{K}_{\mathrm{OO}F_{1}}=-\left(\mathbf{Ad}_{\mathrm{O}}^{O}\right)^{-1}\left[\bm {K}_{B1B}^{O}\right]\left(\mathbf{Ad}_{\mathrm{O}}^{F_{1}}\right)^{-1}\\ \mathbf{K}_{F_{1}O}=-\left(\mathbf{Ad}_{\mathrm{O}}^{F_{1}}\right)^{-1}\left[\mathbf{K}_{ B1B}^{O}\right]\left(\mathbf{Ad}_{\mathrm{O}}^{O}\right)^{-1}\end{array} \right. \tag{9}\]  
where the superscripts \(O\) and \(F_{1}\) of each stiffness matrix indicate that the stiffness matrix is relative to coordinate frame \(Oxyz\) and \(F_{1}x_{F1}y_{F1}z_{F1}\), respectively. \[\mathbf{Ad}_{\mathrm{O}}^{O}=\mathbf{I}_{\delta\times 6},\mathbf{Ad}_{\mathrm{O}}^{F_{1}}= \begin{bmatrix}\mathbf{R}_{\mathrm{O}}^{F_{1}}&0\\ \mathbf{T}_{O}^{F_{1}}\mathbf{R}_{\mathrm{O}}^{F_{1}}&\mathbf{R}_{\mathrm{O}}^{F_{1}}\end{bmatrix} , \text{ where }\mathbf{T}_{O}^{F_{1}}= \begin{bmatrix}0&d_{3}&d_{2}\\ -d_{3}&0&-d_{1}\\ -d_{2}&d_{1}&0\end{bmatrix} ,\,\mathbf{R}_{\mathrm{O}}^{F_{1}}=\mathbf{I}_{3\times 3}\]  
where \(\mathbf{I}\) is the unit matrix. \(D_{1}\), \(d_{2}\), and \(d_{3}\) represent loading position of force \(\mathbf{F}_{1}\) relative to global coordinate frame \(Oxyz\). The stiffness matrices in Equation (9) can be calculated from Equations (2), (3), and (7):  
\[\left\{ \begin{array}{l}\mathbf{K}_{B1A}^{O}=\left(\mathbf{C}_{\mathrm{R}}^{O}\right) ^{-1},\mathbf{K}_{B1B}^{O}=\left(\mathbf{C}_{R2}^{O}+\mathbf{C}_{R3}^{O}+\mathbf{C}_{R4}^{O} \right)^{-1}\\ \mathbf{K}_{B2}^{O}=\left(\mathbf{C}_{B2}^{O}\right)^{-1},\mathbf{K}_{B3}^{O}=\left(\mathbf{C} _{B3}^{O}\right)^{-1},\mathbf{K}_{B4}^{O}=\left(\mathbf{C}_{B4}^{O}\right)^{-1}\\ \mathbf{K}_{B1A}^{F_{1}}=\left(\mathbf{Ad}_{\mathrm{O}}^{F_{1}}\left(\mathbf{K}_{B1A}^{O} \right)^{-1}\left(\mathbf{Ad}_{\mathrm{O}}^{F_{1}}\right)^{-1}\right)\\ \mathbf{K}_{B1B}^{F_{1}}=\left(\mathbf{Ad}_{\mathrm{O}}^{F_{1}}\left(\mathbf{K}_{B1B}^{O} \right)^{-1}\left(\mathbf{Ad}_{\mathrm{O}}^{F_{1}}\right)^{T}\right)^{-1}\end{array} \right.