# Error Averaging Effect in Parallel Mechanism Coordinate Measuring Machine
## 3 Analysis of Grating Reading Error  
It can be seen from Equation (3) that the measuring result depends on the positions of three reading heads. The reading error of the grating scale, which is often called \"input motion error\" in the parallel mechanism, affects the measuring error. The sensitivity of reading error to measuring error, called uncertainty analysis, can be derived from Equation (3) with differential coefficient theory.  
\[\left[\begin{array}{c}\delta_{1}x\\ \delta_{1}y\\ \delta_{1}z\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial f_{1}} {\partial x_{1}}&\frac{\partial f_{1}}{\partial x_{2}}&\frac{\partial f_{1}}{ \partial x_{3}}\\ \frac{\partial f_{2}}{\partial x_{3}}&\frac{\partial f_{2}}{\partial x_{2}}& \frac{\partial f_{2}}{\partial x_{3}}\\ \frac{\partial f_{3}}{\partial x_{1}}&\frac{\partial f_{3}}{\partial x_{2}}& \frac{\partial f_{3}}{\partial x_{3}}\end{array}\right]\left[\begin{array}{ c}dx_{1}\\ dx_{2}\\ dx_{3}\end{array}\right] \tag{4}\]  
where, \([\delta x_{p}\), \(\delta y_{p}\), \(\delta z_{p}]^{\mathrm{T}}\) are measuring errors while \([dx_{1}\), \(dx_{2}\), \(dx_{3}]^{\mathrm{T}}\) are reading errors. The middle \(3\times 3\) matrix is called the error transfer coefficient matrix.  
Putting practical values of the PCMM structure parameters into Equations (2) and (3), we can derive the error transfer coefficient matrix of every point in the measuring space. By conducting substantial simulation calculation, we found that the PCMM averaged the original reading head errors through link rods. Table 1 lists the error transfer coefficients of two randomly selected points in the measuring space. Other points generally have the same characteristics.  
As given in Equation (2), the \(x\) coordinate of point \(e_{1}\) is determined only by the first and the third reading head data. The value of the second reading head is not related. The error transfer coefficient of the first and the third reading head are the same at 0.5. The error transfer coefficient of the second reading head is zero. In other words, if the length of the first and the third link rod are equal (i.e., \(l_{1}=l_{2}=l_{5}=l_{6}\)), then \(x\) coordinate of point \(e_{1}\) will be equal to \((x_{1}+x_{3}-2e-d)/2\). This phenomenon is very close to what we conduct in angle measurement, with two reading heads installed \(180^{\circ}\) apart on the circular grating. In the \(Y\) and \(Z\) directions, positive and negative error transfer coefficients  
\begin{table}
\begin{tabular}{c c c c} \hline Serial & \((\partial f_{i}/\partial x_{1})/\mathrm{mm}\) & \((\partial f_{i}/\partial x_{2})/\mathrm{mm}\) & \((\partial f_{i}/\partial x_{3})/\mathrm{mm}\) \\ Number & \((x,y,z)=(600,420,240)\) & \\ \hline \(i=1\) & 0.5 & 0 & 0.5 \\ \(i=2\) & 0.13 & 0.16 & \(-0.43\) \\ \(i=3\) & \(-0.11\) & \(-1.24\) & 1.21 \\ \hline \multicolumn{4}{c}{\((x,y,z)=(600,360,290)\)} \\ \hline \(i=1\) & 0.5 & 0 & 0.5 \\ \(i=2\) & \(-1.13\) & 1.26 & 0.13 \\ \(i=3\) & 1.71 & \(-0.86\) & \(-0.23\) \\ \hline \end{tabular}
\end{table}
Table 1: Error transfer coefficient of two points.  
Figure 2: Geometrical model of PCMM.  
alternately appear, and the sum tends to zero. Therefore, we can conclude that three grating reading head errors have been averaged in the PCMM. In other words, the error average effect of the parallel mechanism is embodied in the motion input errors.  
Using the same total differential method, we have studied the length manufacturing error of six link rods and its influence on the measuring error. A similar conclusion was determined. For the \(X\) direction, the error transfer coefficient could realize compensation by itself. In \(Y\) and \(Z\) direction, the error averaging effect still exists, but is weaker than that in the \(X\) direction [11].