## 2 Dynamic Model of the Feed System
### Calculation of Restoring Force for Ball Screw
#### 2.1.1 Restoring Force for the Screw-Nut Left Section  
The structure of a single nut ball screw with preload applied by variable lead is shown in Figure 3. Considering that the ball contact force direction for the screw-nut left and right sections is different, this leads to the different restoring force function for the screw-nut left and right sections. Therefore, it is necessary to describe the position of the raceway centers  
Figure 1: Structure of ball screw feed system. Figure 2: The relationship between different coordinate systems. in CS\({}_{0}\), respectively. The position vectors (\(\mathbf{R}_{L\text{s}}\), \(\mathbf{R}_{L\text{n}}\)) of the raceway centers corresponding to the \(i\)th ball for the screw-nut left section in CS\({}_{2}\) can be given by:  
\[\mathbf{R}_{L\text{s}}=\left[ \begin{array}{cc}0&-(R_{\text{s}}-R_{\text{b}}) \cos\alpha_{\text{sn0}}&(R_{\text{s}}-R_{\text{b}})\sin\alpha_{\text{sn0}}&1 \end{array} \right]^{\text{T}} \tag{3}\]  
\[\mathbf{R}_{L\text{n}}=\left[ \begin{array}{cc}0&(R_{\text{n}}-R_{\text{b}}) \cos\alpha_{\text{sn0}}&-(R_{\text{n}}-R_{\text{b}})\sin\alpha_{\text{sn0}}&1 \end{array} \right]^{\text{T}} \tag{4}\]  
where \(R_{\text{s}}\) and \(R_{\text{n}}\) are the raceway radius of the screw and nut; \(R_{\text{b}}\) is the ball radius; \(\alpha_{\text{sn0}}\) is the initial contact angle. Combining Equations (1), (3) and (4), the coordinates of the raceway centers corresponding to the \(i\)th ball for the screw-nut left section in CS\({}_{S}\) and CS\({}_{W}\) can be expressed as:  
\[\left[ \begin{array}{cccc}X_{\text{L}si}^{S}&Y_{\text{L}si}^{S}&Z_{\text{L}si }^{S}&1\end{array} \right]^{\text{T}}=\text{Trans}(0,0,L_{\text{L}bi})\cdot \mathbf{T}_{1-2}\cdot\mathbf{R}_{L\text{s}} \tag{5}\]  
\[\left[ \begin{array}{cccc}X_{\text{L}ni}^{W}&Y_{\text{L}ni}^{W}&Z_{\text{L}ni }^{W}&1\end{array} \right]^{\text{T}}=\text{Trans}(-H_{W},0,L_{\text{L}bi})\cdot \mathbf{T}_{1-2}\cdot\mathbf{R}_{L\text{n}} \tag{6}\]  
where \(H_{W}\) is the distance along the \(x\)-axis direction between the CS\({}_{W}\)'s origin and the nut centerline; \(L_{\text{L}bi}\) is the displacement of the \(i\)th ball center for the screw-nut left section along the \(z\)-axis direction in CS\({}_{S}\) (CS\({}_{W}\)), given by:  
\[L_{\text{L}bi}=-\frac{L_{D}}{2}+(i-1)\Delta L \tag{7}\]  
where \(L_{D}\) is axial distance between the first ball and last ball; \(\Delta L\) is the distance along the \(z\)-axis direction between adjacent balls. Considering that the center position of the nut raceway changes with the worktable posture variation, this affects the functions of the screw-nut restoring force. We assumed that at the geometric center of the worktable exists three translational displacements \((\delta_{W\text{x}},\delta_{W\text{y}},\delta_{W\text{z}})\) along \(x_{W}\), \(y_{W}\) and \(z_{W}\) axes under the action of excitation force. According to the homogeneous coordinate transformation, the coordinates of the nut raceway center in CS\({}_{0}\) are:  
\[\left[ \begin{array}{cccc}x_{\text{L}wi}^{W}\\ y_{\text{L}wi}^{W}\\ z_{\text{L}ni}^{W}\\ 1\end{array} \right]=\text{Trans}(H_{W},0,L_{OW})\cdot\text{Trans}\left(\delta_{ W\text{x}},\delta_{W\text{y}},\delta_{W\text{z}}\right)\left[ \begin{array}{cccc}X_{ \text{L}wi}^{W}\\ Y_{\text{L}wi}^{W}\\ Z_{\text{L}ni}^{W}\\ 1\end{array} \right] \tag{8}\]  
where \(L_{OW}\) is the axial distance between CS\({}_{0}\) and CS\({}_{S}\) (CS\({}_{W}\)). Because the screw shaft at the screw-nut interface is short, we assumed that this section of screw shaft is rigid. Similarly, the geometric center of screw shaft at the screw-nut interface exits three translational displacements \((\delta_{SN\text{x}},\delta_{SN\text{y}},\delta_{SN\text{z}})\) along \(x_{S}\), \(y_{S}\) and  
Figure 3: Structure of a single nut ball screw under preload. directions caused by the axial and bending deformations of screw shaft. The coordinates of the screw raceway center in CS\({}_{0}\) are:  
\[ \begin{bmatrix}x_{Issi}^{S}\\ y_{Issi}^{S}\\ z_{Issi}^{S}\\ 1\end{bmatrix}