1. Introduction
Polygons and rotary encoders are considered to be full-circle angle standards. A polygon can be treated as a circumference divided by an equal angle, such as 15°, 30°, and 90°. These are named 24-faced, 12-faced, or 4-faced polygons, respectively. A rotary encoder is made up of a glass disc with a suitable reader. The reader can determine the angular positions of the glass disc. Calibration of polygons and rotary encoders is essential to determine their accuracy at most national standards laboratories worldwide. The calibration values are standard angle values to transfer other instruments. Polygons and rotary encoders are implemented for various tasks such as accuracy testing and calibration of angle-measuring instruments, rotary tables, and so on. The calibration is based on the circle closure principle: the sum of all angle errors of a full circle must be zero. Therefore, the circle closure principle is widely used to calibrate polygons and rotary encoders.
For the calibration of rotary encoders, many previous papers proposed using self-calibration. For the self-calibration method, it is unnecessary to use an external standard. Sensor heads and a rotary encoder are installed inside a rotary table. The rotary encoder is rotated, and the sensor heads measure angle values at different angle positions corresponding to the rotary encoder. According to the circle closure principle, the rotary encoder can self-calibrate. Many self-calibration methods have been proposed [
1,
2,
3]. Probst et al. [
4,
5] developed an angle comparator that was constructed using a rotary encoder and 16 read heads to calibrate all angle errors of the rotary encoder. Watanabe et al. [
6,
7,
8,
9,
10,
11] developed the SelfA rotary encoder for self-calibrating the rotary encoder. The SelfA rotary encoder consists of a rotary encoder and several groups of sensor heads. The SelfA rotary encoder can calibrate all angle errors of the rotary encoder. In contrast to self-calibration, using an external standard is another method to calibrate a rotary encoder. Many studies have demonstrated the use of a polygon as an external standard [
12,
13]. Kim et al. [
14] and Huang et al. [
15] used a 36-faced polygon, Jia et al. [
16] used a 24-faced polygon, and Pisani et al. [
17] used a 12-sided polygon and a 6-sided polygon. This method has to use an autocollimator to measure the polygon. The problem with this method is that the polygon must be calibrated before calibrating a rotary encoder.
For the calibration of polygons, many papers proposed using a rotary encoder as an external standard [
18,
19]. The rotary encoder has to calibrate by using self-calibration to transfer standard angle values. These standard angle values can calibrate a polygon. This method also has the disadvantage of using two experiments to calibrate the rotary encoder and the polygon. Using two autocollimators is another method to only calibrate a polygon [
20,
21,
22]. The polygon is mounted on a rotary table. The two autocollimators measure the polygon. The angle errors of the rotary table do not influence the calibration of the polygon. Cross-calibration is used simultaneously, calibrating a rotary encoder and a polygon [
23]. With this method, a rotary encoder is the external angle standard, and a polygon is the calibrated standard. All angle errors of the rotary encoder and all pitch angle deviations of the polygon are unknown. Based on the circle closure principle, the rotary encoder and the polygon can be calibrated simultaneously. However, the measurement cycle depends on how many faces the polygon has. Hsieh et al. [
24] proposed a 24-sided polygon as the calibrated angle standard. The SelfA rotary encoder is the external angle standard. The polygon and SelfA rotary encoder can be calibrated simultaneously. This method is time-consuming depending on how many faces the polygon has. The 24-sided polygon is used so the measurement cycle is 24. Therefore, we propose the two-autocollimator method. The experiment setup is the same as in the previous studies [
20,
21,
22]. These previous studies only proposed the calibration of the polygon. However, the proposed method can simultaneously calibrate the polygon and the rotary encoder. The measurement cycle is one.
This paper presents the new two-autocollimator method for calibrating the rotary encoder and the polygon. First, the introduction section presents the related research papers and techniques. Next, the configuration of the two-autocollimator method shows two autocollimators, a 24-faced polygon, and the SelfA rotary encoder. The theory of the two-autocollimator method is described in detail using the circle closure principle. In the experiment, the two-autocollimator method, using the two autocollimators and the 24-faced polygon, is conducted to calibrate the SelfA rotary encoder. A comparison of calibration results, the uncertainty evaluation, and the En-value calculation are presented. Finally, we present our discussion and conclusions.
2. Configuration and Theory of the Two-Autocollimator Method
Figure 1a shows a schematic of the calibration setup. A 24-faced polygon is mounted at the center of a rotary encoder. The 24-faced polygon and the rotary encoder are rotated at the same time. In this paper, the 24-faced polygon is used to illustrate the related theory and conduct experiments. A polygon with any number of faces can be used. This polygon has 24 measuring faces. Two autocollimators are placed and fixed in front of the rotary encoder. The no. 1 autocollimator is collimated to the first face of the polygon. The no. 2 autocollimator is collimated to the second face of the polygon.
As shown in
Figure 1b, the polygon has the measuring faces (R) and the nominal faces (N). In the ideal polygon, the measuring faces are the same as the nominal faces. Between the measuring face and the nominal face is a slight deviation, called the pitch angle deviation,
βi. Each measuring face has a different pitch angle deviation. For the 24-faced polygon, one full circle is assumed to be divided into 24 nominal faces and 24 measuring faces. The angle interval between two consecutive nominal faces is 15°, called the nominal pitch angle. According to the circle closure principle, the summation of all pitch angle deviations must be zero:
The rotary encoder is calibrated to determine the angle error α (θ), which is an angle deviation between the measured and true angles. The angle error α (θ) is the angle error of the rotary encoder at θ. The 24-faced polygon can be measured at rotary encoder angles θ of ((i − 1)*15°), where i is from 1 to 24.
Figure 2 shows the calibration value determination. The measured values contain the angle error of the rotary encoder, the pitch angle deviation of the 24-faced polygon, and the setup error. The setup error is the misalignment between the rotary encoder and the polygon, which remains constant at different angles. Different autocollimators have different setup errors. The setup error
U1 is for the no. 1 autocollimator, and the setup error
U2 is for the no. 2 autocollimator. As shown in
Figure 2a, the rotary encoder is rotated at the starting angle of 0°. The first face of the 24-faced polygon is adjusted to a starting angle of 0°. The no. 1 autocollimator measures the angle error
α (0°), the pitch angle deviation
β1, and the setup error
U1. The rotary encoder is rotated in one step to an angle of 15°. The no. 1 autocollimator collimates the second face of the 24-faced polygon. The no. 1 autocollimator then measures the angle error
α (15°), the summation of the pitch angle deviations (
β1 +
β2), and the setup error
U1. The rotary encoder is rotated in two steps to an angle of 30°. The no. 1 autocollimator collimates the third face of the 24-faced polygon. The no. 1 autocollimator then measures the angle error
α (30°), the summation of the pitch angle deviations (
β1 +
β2 +
β3), and the setup error
U1. Overall, the rotary encoder is rotated in 23 steps to an angle of 345°. The no. 1 autocollimator collimates the 24th face of the 24-faced polygon. The no. 1 autocollimator measures the angle error
α (345°), the summation of the pitch angle deviations of the first to the 24th faces, and the setup error
U1. As shown in
Figure 2b, the rotary encoder is at the starting angle of 0°. The no. 2 autocollimator collimates the second face of the 24-faced polygon. The no. 2 autocollimator measures the angle error
α (0°), the pitch angle deviation
β2, and the setup error
U2. The rotary encoder is rotated in one step at 15°. The no. 2 autocollimator collimates the third face of the 24-faced polygon. The no. 2 autocollimator then measures the angle error
α (15°), the summation of the pitch angle deviations (
β2 +
β3), and the setup error
U2. The rotary encoder is rotated in two steps at 30°. The no. 2 autocollimator collimates the fourth face of the 24-faced polygon. The no. 2 autocollimator then measures the angle error
α (30°), the summation of the pitch angle deviations (
β2 +
β3 +
β4), and the setup error
U2. At the final angle, the rotary encoder is rotated in 23 steps to an angle of 345°. The no. 2 autocollimator collimates the first face of the 24-faced polygon. The no. 2 autocollimator measures the angle error
α (345°), the summation of the pitch angle deviations of the first to the 24th faces, and the setup error
U2. The measured value contains the angle error of the rotary encoder, the summation of the pitch angle deviations, and the setup error, as follows:
where
εi,j is the value measured by the autocollimator at an angle of i × 15°, and i is the number of steps of rotation of the rotary encoder. The value of i equals 1 at a starting angle of 0°; j denotes the no. j autocollimator;
βk is the pitch angle deviation at
kth face of the 24-faced polygon; and
Uj denotes the setup error for the no. j autocollimator.
The measured values of the no. 1 autocollimator at different rotated angles are as follows:
The measured values of the no. 2 autocollimator at different rotated angles are as follows:
The difference measured values between the measured values of the no. 1 autocollimator and the measured values of the no. 2 autocollimator at the same rotated angles are as follows:
For Equation (6), all difference measured values reduce the difference measured value,
ε24,2 −
ε24,1, at the final rotated angle. Equation (6) can be rewritten as follows:
The summation of Equation (7) is as follows:
According to the circle closure principle, the summation of all the pitch angle deviations must be zero, as in Equation (1). The first term on the right-hand side of Equation (8) must be zero. The first pitch angle deviation
β1 is calculated. According to Equation (7), the pitch angle deviations from
β2 to
β24 are calculated. Until this calculation, all pitch angle deviations from
β1 to
β24 are calculated. Then, the summation of Equation (4) is as follows:
According to Equation (2), the summation of all angle errors is zero. As in the previous discussion, all pitch angle deviations are known in the second term on the right-hand side of Equation (9). In the third term on the right-hand side of Equation (9), the setup error of the no. 1 autocollimator is calculated.
The summation of Equation (5) is as follows:
According to Equation (2), the summation of all angle errors is zero. As in the previous discussion, all pitch angle deviations are known in the second and third terms on the right-hand side of Equation (10). In the fourth term on the right-hand side of Equation (10), the setup error of the no. 2 autocollimator is calculated. Until this calculation, all pitch angle deviations from β1 to β24, the setup error of the no. 1 autocollimator , and the setup error of the no. 2 autocollimator are calculated. According to Equations (4) and (5), all angle errors of the rotary encoder from α (0°) to α (345°) are calculated. As in the previous calculation procedure, all angle errors of the rotary encoder and all pitch angle deviations are calculated simultaneously by using measured values of the two autocollimators.
5. Conclusions
The proposed two-autocollimator method uses two autocollimators to calibrate the 24-sided polygon and the SelfA rotary encoder. The two autocollimators are set up outside the SelfA rotary encoder. The 24-faced polygon is on the SelfA rotary table. As discussed in the Theory of The Two-Autocollimator Method section, all pitch angle deviations of the 24-faced polygon and angle errors of SelfA rotary encoder can be calibrated simultaneously. To verify the proposed two-autocollimator method, the shift-angle method, based on cross-calibration, is used to measure the same 24-faced polygon, and self-calibration is used to measure the same SelfA rotary encoder. According to the shift-angle method, based on cross-calibration, an autocollimator calibrates all pitch angle deviations of the 24-faced polygon. The difference in pitch angle deviations is smaller than ±0.28”. The SelfA rotary encoder comprises 12 read heads and calibrates using self-calibration. Angle errors of SelfA rotary encoder can be calibrated using the proposed two-autocollimator method and self-calibration. The difference in angle errors is smaller than ±0.27”.
Following the ISO Guide to the Expression of Uncertainty in Measurement [
25], the evaluated expanded uncertainty of the proposed two-autocollimator method is 0.46”. For a 95% confidence level, the coverage factor is 2.00. The difference in pitch angle deviations and expanded uncertainty are used in the
En-value for the proposed two-autocollimator and shift-angle methods. The maximum
En-value is 0.58. The difference in angle errors and expanded uncertainty are used in the
En-value to compare the proposed two-autocollimator method and self-calibration. The maximum
En-value is 0.59. Both the
En-values are less than 1, meaning that the proposed two-autocollimator method is practical.