## 2 Measurement, Evaluation and Compensation Model for Spatial Symmetry Errors
### Compensation Model of Spatial Symmetry Error  
On the basis of the proposed measurement and evaluation model for a herringbone gear' spatial symmetry error, the spatial symmetry error is translated into displacement deviation \(e\). In fact, the herringbone gear's spatial symmetry error cannot be compensated through rotating the gear or controlling the worktable's rotating axis. To achieve compensa  
Figure 4: Schematic diagram of evaluated spatial symmetry error.  
tion of the spatial symmetry error, the controlled target is translated into the gear shaping cutter according to the generating machining method. Thus, the movements involved in the gear shaping cutter need to be analyzed.  
As shown in Figure 5, the movements of the gear shaping cutter mainly include the principal cutting movement (P), circular cutting motion of numerical control (NC) axis C2 (C2), cutter back-off motion (B) and movement of oblique knife (S). Apparently, the angle error compensation of the tangential direction cannot be realized through controlling movement P; moreover, the simplex control of movement C2 can make tangential displacement in the gear machining process, and the motion B only involves the axial direction because the spatial symmetry error is translated into displacement deviation \(e\) at the X-axis coordinate. Therefore, the method of controlling movement S is proposed to compensate for the displacement deviation \(e\), and this method is viable to compensate for the tangential displacement through approximation in theory. The compensation model of the herringbone gear's spatial symmetry error is shown in Figure 6. Finally, the whole closed loop system of measurement and compensation machining on the herringbone gear's spatial symmetry error is realized.  
On the basis of the proposed compensation model for the herringbone gear's spatial symmetry error, the spatial symmetry error is translated into displacement deviation \(e\) and can be written as follows. Further, the movement of the oblique knife is adjusted, and then  
Figure 5: Schematic diagram of movements involved in gear shaping cutter.  
Figure 6: Compensation model of spatial symmetry error.  
the gear shaping cutter should be adjusted to confirm the mesh after compensating for the spatial symmetry error. The angle \(\alpha_{1}\) can be written as follows:  
\[e\approx mz\frac{\sin\gamma}{\cos\beta}=\frac{d\sin\gamma}{2\cos\beta} \tag{7}\]  
\[\alpha_{1}\approx arcsin\frac{e\cos\beta}{MZ}=arcsin\frac{2c\cos\beta}{D} \tag{8}\]  
where \(m\), \(z\), \(d\), and \(\gamma\) are the herringbone gear's module, tooth number, pitch diameter and angle error, respectively, where \(M\), \(Z\), \(D\) and \(\alpha_{1}\) are the gear shaping cutter's module, tooth number, pitch diameter and angle error, respectively.  
Therefore, by setting up the measurement and evaluation and compensation model of the herringbone gear's spatial symmetry error, the machining process for on-line detection and compensation of the herringbone gear's spatial symmetry error can be intuitively displayed, providing theoretical support for instance machining.