# Error Averaging Effect in Parallel Mechanism Coordinate Measuring Machine
## 4 Property of Slider Motion Errors in PCMM
### Straightness Motion Error  
As shown in Figure 4, three sliders also generate motion straightness errors in \(Y\) and \(Z\) directions, namely \(\delta_{Y}(x)\) and \(\delta_{Z}(x)\), respectively [20,21]. Three sliders moving one after another not only cause the moving platform to produce positioning errors, but also result in straightness and orientation changes. Thus, the change of the probe position leads to measurement error. Considering the influence of \(\delta_{Y}(x)\) and \(\delta_{Z}(x)\), we can modify Equation (1) to:  
\[\left\{ \begin{array}{l}\left(x-x_{1}\right)^{2}+\left(y-\delta y_{1}\right)^ {2}+\left(z-\delta z_{1}\right)^{2}={l_{1}}^{2}\\ \left(x-x_{1}\right)^{2}+\left(y+g-c-\delta y_{1}\right)^{2}+\left(z-\delta z _{1}\right)^{2}={l_{2}}^{2}\\ \left(x+e-x_{2}\right)^{2}+\left(y+g+f-b-c-\delta y_{2}\right)^{2}+\left(z- \delta z_{2}\right)^{2}={l_{3}}^{2}\\ \left(x+d+e-a-x_{2}\right)^{2}+\left(y+g+f-b-c-\delta y_{2}\right)^{2}+\left(z -\delta z_{2}\right)^{2}={l_{4}}^{2}\\ \left(x+d+2e-x_{3}\right)^{2}+\left(y+g-c-\delta y_{3}\right)^{2}+\left(z- \delta z_{3}\right)^{2}={l_{5}}^{2}\\ \left(x+d+2e-x_{3}\right)^{2}+\left(y-\delta y_{3}\right)^{2}+\left(z-\delta z _{3}\right)^{2}={l_{6}}^{2}\end{array} \right. \tag{7}\]  
The analytical solution of the above set of equations is difficult to find. Only a numerical solution can be obtained by numerical iteration analysis. Other parallel mechanisms are also generally encountered. Any type of numerical algorithm in the calculation process consists of attempt, approximation, compromise, and convergence. In real parallel mechanism motion, the position of the moving platform is determined by the plurality of bars in common. In this process, mechanical tensile, compression, torsion, and elastic deformation of involved parts occur. When a certain position can achieve balance among these factors, this is the actual position that the moving platform reaches. Error averaging effect happens during this moving process. Reflecting on mathematics, it is iteration and convergence. Thus, we find that if any iterative algorithm has weak error averaging effect, it will be unsuitable for use in parallel mechanism calculation. As mentioned by [PERSON] in 1979 [12], the slider motion straightness errors are different when different points on the slider are measured. We are mainly interested in the error of the point where two universal joints center on the slider. Therefore, when a dual-frequency laser straightness interferometer was used to detect slider motion straightness error, the interference mirror (Wollaston prism) required adjustment to be as close as possible to the center point of two joints on the moving slider. The large reflection mirror was statically fixed to the guide support. The experimental setup is shown in Figure 8.  
Figure 7: Variation of error caused by Abbe error. Straightness errors in \(Y\) and \(Z\) directions were captured by the interferometer (Renihaw XL-80, Renishaw, New Mills, UK). The overall trend of three slider straightness motion errors is highly similar. Figure 9 shows the straightness motion errors of first slider in two directions. The horizontal axis is the coordinate \(x_{1}\) of the first slider. The interferometer picks up \(\delta_{Y}(x)\) once after the slider moves 50 mm in one direction, thus, the \"forward\" curve can be established. The measurement process was repeated in the reverse direction to established the \"reverse\" curve. It can be seen that the two directional measurements are quite consistent, and the average value of the data is then used as the straightness motion error of the slider in the \(Y\) direction, as shown in Figure 9 (above). For the straightness motion error of the slider in the \(Z\) direction, we repeated the detection process, but the interference mirror and the reflection mirror needed to be rotated \(90^{\circ}\) in advance. Figure 9 (below) shows the variation of \(\delta_{Z}(x)\). The value difference among six sets of data of straightness motion errors is extremely small.