# Design and Accuracy Analysis of a Micromachine Tool with a Co-Planar Driving Mechanism
## 3 Error Model and Analysis  
### Error Model  
With the use of the co-planar multi-DOF machine kinematic model, the relative position of the working table with respect to the tool-end point can be determined (Figure 7). The model describes the relationship of motion between the tilt-servo mechanism and the platform. After obtaining the relative coordinate position of the working table with respect to the tool-end point, according to the parameters of the motion model, the error sources of the co-planar multi-DOF machine tool were defined. Subsequently, the defined error sources were combined with the kinematic model to obtain the total error model for the co-planar multi-DOF machine tool. The error sources in the coordinate transformation include translation errors (\(\delta\)x, \(\delta\)y, \(\delta\)z) in the three axes, rotation angle errors (rx, ry, rz) in the three axes, and the motion errors generated by the ballscrew nut movement (\(\Delta\)x, \(\Delta\)y, \(\Delta\)z). Firstly, the error parameters of each key component were defined. The translation error and rotation angle error are shown in Table 1. \(\delta\)x\({}_{1}\), \(\delta\)y\({}_{1}\), and \(\delta\)z\({}_{1}\) in the table respectively represent the translational error in the X, Y, and Z directions when the coordinate frame O\({}_{0}\) moves to the coordinate frame O\({}_{1}\), and rx\({}_{1}\), ry\({}_{1}\), and rz\({}_{1}\), respectively, represent the rotation angle errors of coordinate frame O\({}_{0}\) to frame O\({}_{1}\) about the X-, Y-, and Z-axes. The motion error parameters of the platform due to the movement errors of the ballscrew nut are shown in Table 2.  
Figure 5: Platform linear travel vs. tilt-drive angle. Figure 6: Rotation resolution of platform vs. rotation angle. After defining the error parameters, the error model was derived. The main structure of the co-planar multi-DOF horizontal machine tool includes a bottom co-planar multi-DOF platform for X-axis and Z-axis movement and a tilt-drive mechanism for Y-axis movement. The actual total coordinate transformation matrix of the horizontal co-planar multi-DOF machine tool is the result of multiplying the associated transformation matrices. Meanwhile, the translation errors (\(\delta\)x, \(\delta\)y, \(\delta\)z) and rotation angle errors (rx, ry, rz) need to be included in the transformation matrix. The rotation errors at the three axes are equivalent to rotating the three axis coordinates at the same time, as shown in Equation (7). \begin{table}
\begin{tabular}{c c} \hline \hline
**Motion Errors** & **Description** \\ \hline \(\Delta\)x\({}_{2}\), \(\Delta\)y\({}_{3}\) & X, Y direction motion error when the D ballscrew moves \\ \(\Delta\)x\({}_{5}\), \(\Delta\)z\({}_{5}\) & X, Z direction motion error when the C ballscrew moves \\ \(\Delta\)x\({}_{7}\), \(\Delta\)z\({}_{10}\) & X, Z direction motion error when the B ballscrew moves \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Motion error parameters of platform due to ballscrew nut movement. Figure 7: Coordinate frames defined for the machine. \[ \begin{array}{c}\begin{array}{c}1\\ \mathrm{Srx}_{5}\mathrm{Sry}_{5}+\mathrm{Srz}_{5}\\ -\mathrm{Srx}_{5}\mathrm{Sry}_{5}\mathrm{Srz}_{5}+1\\ -\mathrm{Sry}_{5}+\mathrm{Srx}_{5}\mathrm{Srz}_{5}\end{array}\begin{array}{ c}-\mathrm{Srx}_{5}\mathrm{Sry}_{5}\mathrm{Srz}_{5}+1\\ -\mathrm{Srx}_{5}\mathrm{Sry}_{5}\mathrm{Srz}_{5}+\mathrm{Srx}_{5}\end{array} \begin{array}{c}1\\ -\mathrm{Srz}_{5}+\mathrm{DZ}_{5}+\delta z_{5}+\Delta z_{5}\\ 0\end{array}\end{array} \end{array}\]  
where \(\Delta\)E represents volumetric error; \(\frac{\partial\Delta\text{E}_{\text{P}}}{\partial\Delta\text{W}_{\text{k}}}\) represents the sensitivity parameter; and \(\text{W}_{\text{k}}\) represents error sources.