## 2 Measurement, Evaluation and Compensation Model for Spatial Symmetry Errors  
### Measurement and Evaluation Model for Spatial Symmetry Errors  
Spatial symmetry errors in herringbone gears come from deviations in the starting machining position between the upper and lower teeth. If the involute curve of a herringbone gear is enlarged indefinitely, the shape of the herringbone gear will be similar to an oblique gear rack, and the actual spatial symmetry error shown in Figure 1 will be the offset error of the center position for the upper and lower teeth. Thus, the spatial symmetry error of the herringbone gear shown in Figure 2 can be assumed to be a circular angle error. Therefore, in order to identify a herringbone gear's spatial symmetry error, it is important to calculate and acquire the circular angle error of the upper and lower teeth through measurement and calculation using a mathematic model. Figure 1: Symmetry error of oblique gear rack. The phase error at the symmetrical position of the upper and lower helical teeth of the herringbone gear is detected by the probe. After many tests, the measurement of the spatial symmetry error value of the herringbone gear is completed. To realize the detection and compensation of a herringbone gear's spatial symmetry error, the symmetry plane of the left and right revolving gear is defined as the symmetry center plane O. As shown in Figure 3, the position measurements of planes A, B, C and D are completed with the touch probe, and the \(n\) random points can be written as \(A_{1}\sim A_{n}\), \(B_{1}\sim B_{n}\), \(C_{1}\sim C_{n}\) and \(D_{1}\sim D_{n}\). Further, the position coordinates of midpoints connected with corresponding points of the same height can be expressed as \(A_{i}(X_{Ai}\), \(Y_{Ai}\), \(Z_{Ai})\), \(B_{i}(X_{Bi}\), \(Y_{Bi}\), \(Z_{Bi})\), \(C_{i}(X_{Ci}\), \(Y_{Ci}\), \(Z_{Ci})\) and \(D_{i}(X_{Di}\), \(Y_{Di}\), \(Z_{Di})\), \(i\in[1,n]\). Using a plane fitting method, the fitting symmetry plates E and F based on the position coordinates of the midpoints are calculated respectively. Using a spatial projection method, the projection line equations of plane E and F projecting on the plane O are calculated, respectively. As shown in Figure 4, we ensure the two projection lines are parallel through approximate processing of the projection line equations, and then calculate the linear distance \(e\) of the two projection lines. In fact, the spatial symmetry error is usually relatively small in the machining process. Thus, the deflection chord length, that is, spatial symmetry error, is infinitely equivalent to the linear distance \(e\). Figure 3: Schematic diagram of detected spatial symmetry error. Figure 2: Spatial symmetry error of herringbone gear. The sets of position coordinate midpoints connected with plane \(A\) and \(B\) corresponding to points of the same height can be written as follows:  
\[(x_{i},y_{i},z_{i})=\left\{\left(\frac{X_{Ai}+X_{Bi}}{2},\frac{Y_{Ai}+Y_{Bi}}{2}, \frac{Z_{Ai}+Z_{Bi}}{2}\right)\right\}\,i\in[1,n] \tag{1}\]  
In the same way, the sets of position coordinate midpoints connected with plane \(C\) and \(D\) corresponding to points of the same height can be written as follows:  
\[\left(x^{\prime}_{i},y^{\prime}_{i},z^{\prime}_{i}\right)=\left\{\left(\frac{ X_{Ci}+X_{Di}}{2},\frac{Y_{Ci}+Y_{Di}}{2},\frac{Z_{Ci}+Z_{Di}}{2}\right) \right\}\,i\in[1,n] \tag{2}\]  
The fitting symmetry plate E and F can be calculated based on spatial analytic geometry and the calculated midpoint sets. The equations of plane E and F can be formulated as follows:  
\[z=a_{0}x+a_{1}y+a_{2} \tag{3}\]  
\[z^{\prime}=a^{\prime}_{0}x+a^{\prime}_{1}y+a^{\prime}_{2} \tag{4}\]  
Utilizing LSM to calculate the fitting symmetry planes, the results can be calculated as follows:  
\[ \begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\end{bmatrix} = \begin{pmatrix}\sum x_{i}^{2}&\sum x_{i}y_{i}&\sum x_{i}\\ \sum x_{i}y_{i}&\sum y_{i}&\sum y_{i}\\ \sum x_{i}&\sum y_{i}&n\end{pmatrix} ^{-1} \begin{pmatrix}\sum x_{i}z_{i}\\ \sum y_{i}z_{i}\\ \sum z_{i}\end{pmatrix}