# Spindle Error Movements and Their Measurement
### Theoretical Basis for Determining the Spindle Error Movements  
As previously described, each rotating shaft (spindle) has one desired degree of freedom--the ECC, rotation around the stationary \(Z\) reference axis--and possible error  
Figure 1: An exemplary schematic drawing of a CNC lathe spindle headstock assembly with a marked structural axis (blue) and two exemplary positions of the instantaneous axis of rotation (red). movements in the remaining five degrees of freedom. Standards [2,3] define the types and directions of shifts of the real axis of rotation in relation to the Z reference axis. Figure 2. presents the method of spindle error measurement [4]. * EXC--radial displacement in the direction of the X reference axis;
* EYC--radial displacement in the direction of the Y reference axis;
* EZC--axial displacement in the Z axis;
* EAC--rotational movement in regard to the X reference axis;
* EBC--rotational movement in regard to the Y reference axis. These error movements can be measured using five contactless sensors (capacitive or inductive) close to the mandrel mounted on the spindle: two in the Z-X and Z-Y planes and one in the spindle axis (ZR) (Figure 2). Measurements made during fast spindle revolutions (several dozen or more measurements during one revolution) inform about spindle error movements in a stationary reference frame. Meanwhile, from the point of view of machining accuracy, displacements in the so-called sensitive directions are important. For turning, this direction is stationary (like the tool in Figure 3a), while for milling, it rotates with the tool (Figure 3b). It follows that, for a turning spindle, you can limit yourself to three sensors and measure the displacements (error movements) of the spindle in one plane. The procedure for face milling is shown in Figure 4.  
Figure 3: Sensitive directions for: (**a**) turning and (**b**) face milling. Figure 2: The method of error movements measurement [4]. Determination of the error radial motions is based on the radii of the circles determining the ideal and real motions of the rotating spindle. At outset, for example, by the least squares method, a mean circle (LSC (Least Squares Circle) is defined that represents the stationary axis of rotation of the spindle, and its centre, along with the graph, moves to the centre of the coordinate system. Its radius results from the gauge plunger eccentricity with respect to the axis of rotation and is not related to error movements. This eccentricity can be introduced to enable determination of the instantaneous rotation angle of the spindle \(\theta\) (Figures 2 and 4). The radius of the LSC can also be increased or even created virtually (mathematically) to make the graph more readable. The results of measurements in XY coordinates must be projected onto the sensitive direction and presented as a function of the angle of rotation according to the following formula (Figure 4):  
\[r(\theta)=\Delta X(\theta)\cos(\theta)+\Delta Y(\theta)\sin(\theta) \tag{1}\]  
Error movement data are plotted graphically in polar diagrams (Figure 5) as originally suggested in 1959 by spindle metrology pioneer [PERSON] [5]. In the polar diagram of error movements, radial deviations from the multi-revolution equilibrium position relative to the rotation angle \(\theta\) of the spindle axis position are indicated by radial lines. This graph may be the subject of further analysis, i.e., determination of total, synchronous (correlated with rotation) and asynchronous errors. Figure 4: Method of calculating radial error motion. Figure 5: A polar diagram of error movements in the radial direction. The current description of spindle measurement issues is dealt with in standards [2; 3; 6; 7; 8]. These standards are mainly intended for OEMs (Original Equipment Manufacturer) and end-users; they update the methods testing and describing the spindle movement and reflect the progress in measuring and describing the movement of spindles. A key role in new measurement technologies is played by contactless capacitive sensors.