# Error Averaging Effect in Parallel Mechanism Coordinate Measuring Machine
## 4 Property of Slider Motion Errors in PCMM
### Abbe Error and Angular Motion  
The Abbe principle, which was proposed in 1890, is often referred to as the first principle for instrument design. This principle could avoid or eliminate the first-order measuring error caused by angular motion error. The Abbe error also plays a key role in 3-PUU PCMM. As shown in Figure 4, point P is the center of one joint on the slider, while point A is the center of grating reading head. Two points are located at the different corners of the cuboids diagonal. Distance \(H\) and \(S\) are Abbe offsets, which are perpendicular to each other in \(Y\) and \(Z\) directions, respectively. Pitch \(\beta(x)\) and yaw \(\gamma(x)\) of the slider produce Abbe error of point P with respect to point A. When the slider moves to a point \(x_{i}\), its yaw angle is \(\gamma(x_{i})\). This means that the slider will rotate a small angle around the \(Z\) axis, which changes the position of reading head to linear grating; this is one kind of Abbe error which is equal to Abbe offset \(H\) times \(\gamma(x_{i})\). The pitch \(\beta(x_{i})\) also produces similar results with another Abbe offset \(S\). Thus, total Abbe error can be expressed as Equation (5). This issue can be analyzed as an input error. The method is similar to the analysis of the sensor reading error in part 3 [18; 19; 20]. We substitute the grating reading error in Equation (4) with Abbe error. Here, \(\mathrm{d}x_{i}\) can be expressed by:  
\[\mathrm{d}x_{i}=-H_{i}\mathrm{tan}\gamma(x_{i})-S_{i}\mathrm{tan}\beta(x_{i}) \tag{5}\]  
As the pitch and yaw angle is very small, Equation (5) can be simplified as:  
\[\mathrm{d}x_{i}=-H_{i}\gamma(x_{i})-S_{i}\beta(x_{i}) \tag{6}\]  
In an experiment, the pitch and yaw angular motion errors of three sliders (HIWIN Technologies Corp., Taichung, China) were measured by an autocollimator (Jingda Measurement Technology Co. Ltd., Jiujiang, China), as shown in Figure 5. The overall trends of the three sliders' angle errors were very similar. Figure 6 shows the angular motion data of the left slider, which contains 37 points. Angular data are collected once after the slider moves 50 mm. Thus, the serial number of the horizontal axis in Figure 6 is corresponds to the different positions of the left slider. Several models of spline curve interpolation for the angular motion error of three sliders were established after obtaining experimental data, which was important for completing the subsequent error calculation. As shown in Table 2, 3 cross-sections (\(x\) = 750, 1000, 1250) in the measurement space are selected initially, and 9 points are selected as feature points in each section. Therefore, 27 points are distributed in the measurement space. Corresponding to these points are each slider's position coordinate, which can be determined through the inverse kinematic solution of PCMM using Equation (2). Thus, the pitch and yaw angle error value of every slider can be achieved through the angle error model. Putting measured angle errors of Figure 6 into in Equations (3) and (6), we can determine the contribution of Abbe errors of the three sliders to the probe's positional errors in the measurement space as listed in Table 2.  
Figure 4: Slider motion in PCMM. Based on the calculation results, for 27 selected feature points, it can be seen that the contribution of the angular motion errors to measuring accuracy of the probe in the \(X\) direction is the smallest. Error variation fluctuates from \(-7.6\) to \(+5.9\)\(\upmu\)m. The error contribution to the \(Y\) direction is the largest,