## 3 Calibration Method for the Angular Positioning Deviation of the Rotary Table
### Calibration of The Angular Positioning Deviation of the Rotary Table
#### 3.3.2 Calibration of the Angular Positioning Deviation of the Rotary Table  
As shown in Figure 3, the angle \(\theta_{j}^{\prime}\) between adjacent stations of the laser tracer is  
\[\theta^{\prime}_{j}=\arccos\frac{|P_{j-1}{}^{\prime}O|^{2}+|P_{j}{}^{\prime}O |^{2}-|P_{j-1}{}^{\prime}P_{j}{}^{\prime}|^{2}}{2|P_{j-1}{}^{\prime}O|\cdot|P_{ j}{}^{\prime}O|} \tag{32}\]  
where \(P_{j-1}{}^{\prime}\) is the coordinate of the j-th station optimized by the laser tracer; \(P_{j}{}^{\prime}\) is the coordinate of the j-th station optimized by the laser tracer, and \(\theta_{j}^{\prime}\) is the actual rotation angle of the rotary table.  
Figure 3: Schematic diagram of the relationship between the position of the laser tracer and the rotating axis.  
Set the theoretical rotation angle of the rotary table as \(\theta_{j}\). When the number of the coordinates of the laser tracer stations is \(m\), the rotary table rotates \((m-1)\) times. According to Equation (32), the angular positioning deviation \(\Delta\theta_{j}\) is obtained  
\[\Delta\theta_{j}=\theta^{\prime}{}_{j}-\theta_{j} \tag{33}\]  
where \(\theta_{j}\). is the angle of theoretical rotation of the rotary table, \(j=1\),\(2\),\(3\)... \(m-1\).  
According to the multimeasurement averaging method, the angular positioning deviation \(\overline{\sigma}_{\Delta\theta}\) of the rotary table based on the laser tracer multi-station measurement system is calibrated as  
\[\overline{\sigma}_{\Delta\theta}=\pm\frac{1}{2}\sqrt{\frac{\sum\limits_{j=1}^ {m-1}\Delta\theta_{j}^{2}}{m(m-1)}} \tag{34}\]  
Algorithm Error of the Calibration Method of the Angular Positioning Deviation of the Rotary Table Based on the Laser Tracer Multi-Station Measurement System  
In the calibration process of the angular positioning deviation of the rotary table, it is assumed that the rotating axis of the rotary table is perpendicular to the plane of the table of the CMM. However, due to the squareness error of the X-axis or the Y-axis of the CMM, the rotating axis of the rotary table is not perpendicular to the plane of the CMM working plane. By comparing the working plane of the rotary table obtained by the coordinates fitting of the laser tracer multi-station with the CMM working plane, the algorithm error of the calibration method of the angular positioning deviation of the rotary table based on the laser tracer multi-station measuring system can be calculated.  
The CMM working plane is \(\beta\), as shown in Figure 4. The normal vector of the theoretical plane equation of the \(\beta\) plane is \(\overrightarrow{n_{1}}=(0,0,1)\). The normal vector of the \(\gamma\) plane is \(\overrightarrow{n_{2}}=(a,b,c)\). The angle between the \(\beta\) plane and the \(\gamma\) plane is \(\alpha\), Then, we can get  
\[\cos\alpha=\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}} \tag{35}\]  
where \(a\) is the coefficient of the \(X\)-axis direction of the fitting plane equation \(\gamma\); \(b\) is the coefficient of the \(Y\)-axis direction of the fitting plane equation \(\gamma\); \(c\) is the coefficient of the \(Z\)-axis direction of the fitting plane equation \(\gamma\).  
According to the geometric relationship, the algorithm error \(\Delta\) of the calibration method of the angular positioning deviation of the rotary table based on the laser tracer multi-station measurement system can be expressed as  
Figure 4: Schematic diagram of the angle between the rotating axis and the normal line of the working plane of the rotary table.  
\[\Delta=\pm(\theta^{\prime}-\theta)=\pm(ac\tan(\tan\theta\cos\alpha)-\theta) \tag{36}\]