# Spindle Error Movements and Their Measurement
## 5 Publications Related to the Measurement of Spindle Error Movements
### Measurement Methods with Optical Sensors
#### 5.1.2 Multi-Sensor Measurement with 2D Slope Sensors (Biaxial)  
The system enables measurement of spindle movement errors and circularity error. The sensors are disposed in the XY plane around a cylindrical object mounted in a spindle as it is shown in the Figure 21. In traditional methods (SEA, Optodyne), the accuracy of the pattern (cylinder or sphere) and especially the circularity error interfere with the measurement of radial error movements. Therefore, the pattern circularity error should be much smaller than the spindle movement error. This is difficult to achieve especially in the case of testing ultra-precise machines. In addition, it would also be beneficial to measure the movement error while the machine is running, when the workpiece tool is used instead of a precise pattern. In this case, to determine the accuracy of the measurements, it is necessary to separate the spindle error from the non-circularity of the pattern.  
Figure 21: Principle of the three-point angular measurement method [27].  
Figure 20: Setting the measuring set on the lathe [26].  
Slope sensors that measure the local tilt of a surface are successfully used to measure the profile of a workpiece. The described method of determining the spindle error movement consists in the use of three sensors for measuring the tilt in the plane located around the surface of the cylindrical object.  
Each of the sensors detects a two-dimensional local slope at a point on the workpiece surface, which is defined as follows:  
\[r^{\prime}_{z}(\theta)=\frac{dr(\theta,z)}{dz} \tag{4}\]  
\[r^{\prime}_{r}(\theta)=\frac{dr(\theta,z)}{R_{r}d\theta} \tag{5}\]  
where \(r(\theta z)\) is the surface profile of the cylindrical workpiece. \(\theta\) is the spindle's rotation angle, and \(R_{r}\) is the mean radius of the cylindrical surface of the workpiece. \(R(\theta)\) is related to the error of the circularity of the workpiece at the point \(z\). Then, the method to calculate the error movement in the radial and angular directions is given.  
Figure 22 shows the specially designed biaxial slope sensors used in the tests.  
The tests were carried out on a laboratory stand consisting of a pneumatic spindle with an angular encoder connected, and a cylinder with a diameter of 80 mm, turned with a diamond plate, was used as the reference object.  
The spindle was mounted on a cross table as it is shown in the Figure 23.  
Figure 22: Biaxial slope sensors [27].  
It should be noted that, for proper operation, a system calibration procedure is needed, requiring the object (pattern in the spindle) to be moved against the sensors. As a result of the conducted research, the value of the radial error movement in the X axis was approx. 2 \(\upmu\)m and, in the Y direction, was 3.3 \(\upmu\)m. The angular error movement was 9.9 arcseconds in the X direction and 12 s in the Y direction. The test results confirmed that the slope sensors used can measure both circularity errors and the value of spindle error movement. Work on improving the system is still ongoing.