## 4 Kinetostatic Model of _n_-4R Compliant Parallel Pointing Mechanism
### Kinetostatic Model of \(n\)-4R Compliant Parallel Pointing Mechanism  
In this section, the kinetostatic model is extended from a 4-4R to a class of \(n\)-4R compliant parallel pointing mechanism. It can be seen from Equation (9) that only the parameters \(\mathbf{K}_{B2}^{O}\), \(\mathbf{K}_{B3}^{O}\), and \(\mathbf{K}_{B4}^{O}\) are associated with the number of mechanism's branches, and Equation (9) can thus be rewritten as follows:  
\[\left\{ \begin{array}{l}\left(\mathbf{K}_{\mathrm{CO}}\right)_{F_{1}}=\mathbf{K}_{B1B}^ {O}+\mathbf{K}_{4\text{-}4\text{R}}-\mathbf{K}_{B1}^{O}\\ \mathbf{K}_{F_{1}F_{1}}=\mathbf{K}_{B1B}^{F_{1}}+\mathbf{K}_{B1A}^{F_{1}}\\ \mathbf{K}_{OF_{1}}=-\left(\mathbf{A},\mathbf{d}_{\mathrm{O}}^{O}\right)^{-\mathsf{T}}\left[ \mathbf{K}_{B1B}^{O}\right]\left(\mathbf{A}\mathbf{d}_{\mathrm{O}}^{F_{1}}\right)^{-1}\\ \mathbf{K}_{F_{1}O}=-\left(\mathbf{A}\mathbf{d}_{\mathrm{O}}^{F_{1}}\right)^{-\mathsf{T}} \left[\mathbf{K}_{B1B}^{O}\right]\left(\mathbf{A}\mathbf{d}_{\mathrm{O}}^{O}\right)^{-1} \end{array} \right. \tag{18}\]  
where \(\mathbf{K}_{4\text{-}4\text{R}}\) denotes the overall stiffness of the 4-4R compliant parallel pointing mechanism, and \(\mathbf{K}_{B1}^{O}\) denotes the stiffness of branch 1. According to Equation (18), it is easy to derive the general equation of the mapping matrix \(\mathbf{C}_{TOF_{1}}\) between the force \(\mathbf{F}_{1}\) applied to the branch 1 and the displacement \(\mathbf{U}_{1}\) of the mobile platform center in the kinetostatic model of the \(n\)-4R compliant parallel pointing mechanism. \[\mathbf{C}_{TOF_{1}}=-\left(\left(\mathbf{K}_{\mathrm{OO}}\right)_{F_{1}}-\mathbf{K}_{OF_{ 1}}\mathbf{K}_{F_{1}F_{1}}^{-1}\mathbf{K}_{F_{1}O}\right)^{-1}\left(\mathbf{K}_{OF_{1}}\bm {K}_{F_{1}F_{1}}^{-1}\right) \tag{19}\]  
where:  
\[\left\{ \begin{array}{l}\mathbf{K}_{B1A}^{O}=\left(\mathbf{C}_{\mathrm{R}1}^{O} \right)^{-1},\mathbf{K}_{B1B}^{O}=\left(\mathbf{C}_{R2}^{O}+\mathbf{C}_{\mathrm{R}3}^{O}+ \mathbf{C}_{\mathrm{R}4}^{O}\right)^{-1}\\ \mathbf{K}_{\mathrm{B1}}^{O}=\left(\mathbf{C}_{\mathrm{B1}}^{O}\right)^{-1},\mathbf{K}_{n -4\text{R}}=\left(\mathbf{C}_{n-4\text{R}}\right)^{-1}\\ \mathbf{K}_{B1A}^{F_{1}}=\left(\mathbf{A}\mathbf{d}_{\mathrm{O}}^{F_{1}}\left(\mathbf{K}_{B1A} ^{O}\right)^{-1}\left(\mathbf{A}\mathbf{d}_{\mathrm{O}}^{F_{1}}\right)^{\mathsf{T}} \right)^{-1}\\ \mathbf{K}_{B1B}^{F_{1}}=\left(\mathbf{A}\mathbf{d}_{\mathrm{O}}^{F_{1}}\left(\mathbf{K}_{B1B} ^{O}\right)^{-1}\left(\mathbf{A}\mathbf{d}_{\mathrm{O}}^{F_{1}}\right)^{\mathsf{T}} \right)^{-1}\end{array} \right. \tag{20}\]  
where \(\mathbf{K}_{B1}^{O}\) denotes the stiffness of the branch 1 in the coordinate frame \(Oxyz\) and \(\mathbf{K}_{n\text{-}4\text{R}}\) denotes the overall compliance matrix of the \(n\)-4R mechanism. Similar to Equation (17), the mapping relationship \(\mathbf{C}_{TOF_{2}}\) between the force \(\mathbf{F}_{2}\) applied to branch 2 and the displacement \(\mathbf{U}_{2}\) of the mobile platform center can be simplified to a rotational transformation of \(\mathbf{C}_{TOF_{1}}\), and the angle of rotation is determined by the number of the mechanism's branches. \[\mathbf{C}_{TOF_{2}}=\left[\mathbf{A}\mathbf{d}_{R}\right]^{-\mathsf{T}}\left[\mathbf{C}_{TOF_ {1}}\right],\text{ where }\mathbf{A}\mathbf{d}_{R}\!\!=\!\!\left[ \begin{array}{cc}\mathbf{R}_{y,2\pi/n}&0\\ 0&\mathbf{R}_{y,2\pi/n}\end{array} \right] \tag{22}\]  
where \(\mathbf{R}_{y,2\pi/n}\) is the rotation matrix, indicating a rotation of \(2\pi/n\) angle around the \(y\)-axis of the global coordinate frame \(Oxyz\). According to the principle of superposition, the relationship between the displacement of the mobile platform center and the forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) can be obtained as follows:  
\[\mathbf{U}_{n-4\text{R}}=\left[\mathbf{C}_{TOF_{1}}\ \ \mathbf{C}_{TOF_{2}}\right] \begin{bmatrix} \mathbf{F}_{1}\\ \mathbf{F}_{2}\end{bmatrix}