A super-high-accuracy angular index table

An index table controls the static angle position of a rotary table, which is built into machine tools. It is also used for calibrating polygon mirrors and theodolites [1, 2]. Therefore, the static angle position of the index table must be measured with high accuracy. An optical method has been proposed for this purpose [3–5]. The angle accuracy of an index table is commonly determined by an in-built rotary encoder.

A rotary encoder can measure the full range of angles (0°–360°) and is used to control the rotational position of a shaft [6]. It consists of a disk with scale lines arranged like a protractor scale, which is read by sensor units. As the angle signals of the sensor units include the angle error, the output position differs from the ideal angle position. The angle accuracy of a rotary encoder can be improved by two methods: direct reduction of the angle error factors (such as runout of the main shaft and the eccentricity between the center of the main shaft and the scale disk) using the high performance parts of the rotary encoder, and self-calibration that detects and self-corrects the angle error. The angle accuracy of these methods is 0.1–0.4 arcsecs in each rotation [7, 8]. Further, the angle accuracy of the national standard in Japan using these methods is 0.01 arcsecs [9, 10].

The static angle position is measured by the rotary encoder as described above. However, the angle accuracy of these methods is limited because their calibrations are difficult to be performed on a specific scale line in the static angle position. Therefore, the National Institute of Advanced Industrial Science and Technology (NMIJ/AIST) in Japan has developed an angular indexing apparatus that measures the static angle positions without these methods [11]. The apparatus is applied in the principle of super-high-accuracy angular indexing. However, this principle has not exactly derived the indexing angle because its equations were expressed qualitatively. Therefore, the principle must be quantitatively derived by formulating the angle signal positions, which satisfy the circular closure property [12]. After formulating the angle signal position output by the rotary encoder, we simulate the principle and present its results.

This section formulates the angle signal positions output by the rotary encoder. Based on this formulation, it derives the principle of the super-high-accuracy angular indexing.

At first, the angle signal position output by a rotary encoder is expressed by a mathematical equation.

Figure 1 displays an optical-type rotary encoder. This encoder consists of two sensor units, which are arranged in an angular position with $180^\circ + \alpha$ around the scale disk; the installation error of the sensor units $\alpha$ denotes the angular misalignment error between the ideal arrangement position of the base sensor units and the other sensor unit. Figure 2 shows the relationships of the angle signal positions output by the sensor units. As shown in figure 2, ${\theta _n}$ is the ideal angle position and is given as

$$\begin{equation}\begin{array}{*{20}{c}} {{\theta _n} = \frac{{2\pi }}{N}n\,,} \end{array}\end{equation} \tag{ 1 }$$

where $n\left( { = 1,2,3, \ldots ,N} \right)$ are the numbers of the scale lines, and $N$ is the total number of scale lines. The counterclockwise direction is taken as the positive direction. Additionally, let ${B_n}$ and ${G_n}$ be the angle signal positions output by two sensor units, expressed by equations (2) and (3) respectively:

$$\begin{equation}\begin{array}{*{20}{c}} {{B_n} = {\theta _n} + E{B_n},} \end{array}\end{equation} \tag{ 2 }$$

$$\begin{equation}\begin{array}{*{20}{c}} {{G_n} = {\theta _n} + E{G_n},} \end{array}\end{equation} \tag{ 3 }$$

where $E{B_n}$ and $E{G_n}$ denote the corresponding angle errors.

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Figure 3 shows the relationship between the angle errors in two sensor units for a case $\alpha = 0$ and the Fourier components of the phases and their difference. The top panel shows the angle errors indicated by $E{B_n}$ and $E{G_n}$. Moreover, two angle errors coincide at 0° and 360° and sum to zero because they possess the circle closure property [1, 12]. Therefore, any arbitrary function of the angle error can be expanded as a Fourier series because the angle error is the periodic function in one rotation. The bottom panels show the phase of the angle error in each Fourier component and its difference. As indicated in the difference, the phases in the odd-term Fourier components are 180° and the phases in the even-term Fourier components are 0°. Therefore, the phase of the angle error depends on the installation position of the two sensor units. Accordingly, the angle errors $E{B_n}$ and $E{G_n}$ can be expressed by equations (4) and (5), respectively:

$$\begin{equation}\begin{array}{*{20}{c}} {E{B_n} = \frac{1}{N}\mathop \sum \limits_{k = 0}^{N - 1} \left\{ {{c_k}\,{e^{ik{\theta _n}}}} \right\},} \end{array}\end{equation} \tag{ 4 }$$

$$\begin{equation}\begin{array}{*{20}{c}} {E{B_n} = \frac{1}{N}\mathop \sum \limits_{k = 0}^{N - 1} \left\{ {{c_k}\,{e^{ik\left( {{\theta _n} + \pi + \alpha } \right)}}} \right\},} \end{array}\end{equation} \tag{ 5 }$$

where ${c_k}$ is a complex number and $i$ is the imaginary unit.

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The respective angle signal positions are then expressed as

$$\begin{equation}\begin{array}{*{20}{c}} {{B_n} = {\theta _n} + \frac{1}{N}\mathop \sum \limits_{k = 0}^{N - 1} \left\{ {{c_k}\,{e^{ik{\theta _n}}}} \right\},} \end{array}\end{equation} \tag{ 6 }$$

$$\begin{equation}\begin{array}{*{20}{c}} {{G_n} = {\theta _n} + \frac{1}{N}\mathop \sum \limits_{k = 0}^{N - 1} \left\{ {{c_k}\,{e^{ik\left( {{\theta _n} + \pi + \alpha } \right)}}} \right\}.} \end{array}\end{equation} \tag{ 7 }$$

Taking the average of equations (6) and (7), the angle signal position output by the rotary encoder is computed as

$$\begin{equation}\begin{array}{*{20}{c}} \begin{aligned} {V_n} = & \frac{1}{2}\left( {{B_n} + {G_n}} \right) \\ = & {\theta _n} + \frac{1}{N}\mathop \sum \limits_{k = 0}^{N - 1} \left\{ {{c_k}\,\frac{{1 + {e^{ik\left( {\pi + \alpha } \right)}}}}{2}\,{e^{ik{\theta _n}}}} \right\}. \\ \end{aligned} \end{array}\end{equation} \tag{ 8 }$$

Secondly, the angle signal positions output by the apparatus shown in figure 4 are expressed by a mathematical equation.

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The apparatus has two scale disks (Scale A and Scale B) with the same total scale line $N$, two coaxial independent shafts (index shaft and motor shaft) and two sets of sensor units that are arranged in 180° angular position. Encoder B consists of Scale B and the sensor unit attached to the housing of the apparatus. Encoder A also consists of Scale A and the sensor unit attached to the index shaft that extends below the index table. As indicated in the top face of Encoder A, the sensor units of Encoder A are indexed and rotated in sync with the indexing angle $\varphi$ of the index table. Moreover, the installation error of the sensor units $\alpha '$ denotes the angular misalignment error between the ideal arrangement position of the base sensor units and the other sensor unit. Figure 5 shows the relationships between the angle signal positions output from the apparatus in figure 4.

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The sensor units of Encoder B are fixed in the housing of the apparatus. Encoder B outputs the angle signals triggered by its original signal (see figure 2). Therefore, the angle signal positions output by Encoder B are given by ${V_n}$ in equation (8).

As shown in figure 5, Encoder A outputs the angle signals triggered by Encoder B's original signal. We first explain the original signal position $O{'_\varphi }$ output by Encoder A. $O{'_\varphi }$ is the angle between the original signal positions of Encoder A and Encoder B, and changes with the indexing angle $\varphi$. Therefore, $O{'_\varphi }$ is given as

$$\begin{equation}\begin{array}{*{20}{c}} {{{O'}_\varphi } = \varphi + {\varphi _0},} \end{array}\end{equation} \tag{ 9 }$$

where ${\varphi _0}$ is the initial original signal position output by Encoder A at $\varphi \, = \,0^\circ$. Additionally, ${\varphi _0}$ can be eliminated by calculating more than one value of $\varphi$ because ${\varphi _0}$ is a constant value.

Next, we explain the relationship between ${O'_\varphi }$ and angle signal output by Encoder B. As indicated in figure 5, ${\theta _z}$ is the ideal angle signals of Encoder B and can be expressed as

$$\begin{equation}\begin{array}{*{20}{c}} {{\theta _z} = {{O'}_\varphi } + {d_\varphi },} \end{array}\end{equation} \tag{ 10 }$$

where $z\,\left( { = 1,2,3, \ldots ,N} \right)$ is the positive integer indicating the number of the scale line output by Encoder B after Encoder A detects its own original signal. Then, replacing ${O'_\varphi }$ with the scale number expanding to a decimal range, we get

$$\begin{equation}\begin{array}{*{20}{c}} {z - 1 < \frac{N}{{2\pi }}{{O\prime}_\varphi }\leqq z.} \end{array}\end{equation} \tag{ 11 }$$

Therefore, the number of scale line $z$ can be expressed as

$$\begin{equation}\begin{array}{*{20}{c}} {z = \left\lfloor {\frac{N}{{2\pi }}{{O'}_\varphi }} \right\rfloor ,} \end{array}\end{equation} \tag{ 12 }$$

where $\left\lfloor {} \right\rfloor$ is the floor function.

Accordingly, ${d_\varphi }$ is a function of the indexing angle $\varphi$ and is specifically given by

$$\begin{equation}\begin{array}{*{20}{c}} {{d_\varphi } = \frac{{2\pi }}{N}\left\{ {\left\lfloor {\frac{N}{{2\pi }}{{O'}_\varphi }} \right\rfloor - \frac{N}{{2\pi }}{{O'}_\varphi }} \right\}.} \end{array}\end{equation} \tag{ 13 }$$

We now explain the angle signal positions ${V'_n}$ output by Encoder A. ${V'_n}$ can be derived from the following three facts and is expressed by equation (14).

• ${V'_n}$ takes the form of equation (8) because the sensor units of Encoder A are installed in diametrically opposite positions as in Encoder B.
• The phase of the angle error in ${V'_n}$ changes with the indexing angle $\varphi$ of the index table because the installation positions of the sensor units in Encoder A are indexed synchronously with $\varphi$.
• Like the original signal position of Encoder A, $V{'_n}$ includes a term ${d_\varphi }$ because Encoder A outputs the angle signals triggered by Encoder B's original signal:

$$\begin{equation}\begin{array}{*{20}{c}} {{{V'}_n} = {\theta _n} + {d_\varphi } + \frac{1}{N}\mathop \sum \limits_{k = 0}^{N - 1} \left\{ {{{c'}_k}\,\frac{{1 + {e^{ik\left( {\pi + \alpha '} \right)}}}}{2}\,{e^{ik\left( {{\theta _n} + \varphi } \right)}}} \right\},} \end{array}\end{equation} \tag{ 14 }$$

where ${c'_k}$ is a complex number.

Finally, deriving the indexing angle $\varphi$ of the index table. The angle signal positions of Encoder A and Encoder B are given by equations (14) and (8), respectively. The indexing angle $\varphi$ can be directly calculated from ${\theta _z}$ and ${d_\varphi }$ by equation (10) because ${O'_\varphi }$ of equation (10) is the function of $\varphi$. ${\theta _z}$ can be calculated by counting the z angle signals detected by Encoder A between the original signals of Encoder A and Encoder B. Next, ${d_\varphi }$ is obtained directly as follows:

• (i)  
  Let ${\delta _n}$ be the difference between equations (8) and (14), given by

$$\begin{equation}\begin{aligned} {\delta _n} = & {{V'}_n} - {V_n} \\ = & {d_\varphi } + \frac{1}{N}\mathop \sum \limits_{k = 0}^{N - 1} \left\{ {{{c'}_k}\,\frac{{1 + {e^{ik\left( {\pi + \alpha '} \right)}}}}{2}{e^{ik\varphi }}} \right. \\ & \quad - \left. {{c_k}\,\frac{{1 + {e^{ik\left( {\pi + \alpha } \right)}}}}{2}} \right\}{e^{ik{\theta _n}}}. \\ \end{aligned} \end{equation} \tag{ 15 }$$

• (ii)  
  Taking the average difference over the $N$ scale lines, equation (15) becomes

$$\begin{equation}\begin{array}{*{20}{c}} {\bar \delta = \frac{1}{N}\mathop \sum \limits_{n = 1}^N {\delta _n} = {d_\varphi } + {{c'}_0} - {c_0},} \end{array}\end{equation} \tag{ 16 }$$

where ${c'_0}$ and ${c_0}$ are the offsets of the angle errors.

These offsets are constants and independent of the indexing angle $\varphi$. Equation (16) provides a direct derivation of ${d_\varphi }$ because the offsets are eliminated for any $\bar \delta$.

To confirm that the offsets in equation (16) indeed cancel and that equation (16) matches equation (13), simulations were carried out under the conditions shown in table 1. As indicated in table 1, the simulation ranges and its steps are determined at the values that are confirmed to calculate less than the resolution of the conditions. The total number of scale lines are the same values in the two simulation, and then the steps between the scale lines are 20 μm. The initial indexing angles are zero for ease during simulations. The installation errors $\alpha$ and $\alpha '$ are determined by values that are calculated using the case of the misalignment with 200 μm and 400 μm in the x direction, respectively, as shown in figure 6. Figure 7 plots ${d_\varphi }$ calculated from equation (13) using the parameters of the simulation conditions.

Table 1. Simulation conditions.

	Simulation 1	Simulation 2
Simulation range of $\varphi$ (arcsecs)	$0 \leqslant \varphi \leqslant 72$ (1 step)	$36 \leqslant \varphi \leqslant 36.5$
Step width (arcsec)	2	0.01
Total number of scale lines N	18 000	18 000
Initial indexing angle ${\varphi _0}$	0	0
Installation error $\alpha \,\;(^\circ )$	0.1	0.1
Installation error $\alpha '\;(^\circ )$	0.2	0.2

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As indicated in figure 7, the horizontal axis is angle position $\varphi$ as shown in table 1 and the vertical axis is the ${d_\varphi }$ calculated from equation (13). Moreover, equation (13) is confirmed to have the proportional relationship regarding the angle position $\varphi$ as same as its theory.

The coefficients ${c_k}$ and ${c'_k}$ in equations (8) and (14) were computed by equations (17) and (18) respectively, and their values are listed in table 2. Figure 8 plots the simulated results of equations (17) and (18) using the parameters of table 2:

$$\begin{equation}\begin{array}{*{20}{c}} {{c_k} = \frac{{{A_k}}}{2}\,\left( {\cos {\beta _k} + i\,\sin {\beta _k}} \right),} \end{array}\end{equation} \tag{ 17 }$$

$$\begin{equation}\begin{array}{*{20}{c}} {{{c'}_k} = \frac{{{{A'}_k}}}{2}\,\left( {\cos {{\beta '}_k} + i\,\sin {{\beta '}_k}} \right).\,} \end{array}\end{equation} \tag{ 18 }$$

Table 2. Angle error parameters calculated by equations (17) and (18).

Period $k$	${c_k}$		$c{'_k}$
	${A_k}$ ('')	${\beta _k}$ (°)	$A{'_k}$ ('')	$\beta {'_k}$ (°)
0	10	0	20	0
1	50	30	70	10
2	40	60	60	40
3	30	90	50	70
4	25	120	45	100
5	20	150	30	130
6	15	180	25	160
7	10	210	20	190
8	5	240	15	220
9	2	270	10	250
10	1	300	5	280

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Figure 8 plots the real-part and the imaginary-part in each equation (17) and equation (18) because equations (17) and (18) are defined as the complex number. To plot the complex number to figure 8, we can obtain the relationship between the amounts of the amplitude in each period of equations (17) and (18). ${A_k}$ and ${A'_k}$ in table 2 are used the values that decrease toward the higher periods in the lower periods range as same as the real angle error detected by the rotary encoder. Specifically, the eccentricity as the angle error factors gives the large influence to one period component in an angle error. Moreover, the runout gives the influence to the lower periods range. The scale disk also gives same influence as the runout.

Figure 9 plots the ${d_\varphi }$ and the angle error calculated from equations (8) and (14) for various $\varphi$ in simulation 1. Figure 10 also plots the ${d_\varphi }$ and the angle error calculated from equations (8) and (14) for various $\varphi$ in simulation 2. As indicated in the two figures, the horizontal axis is the ideal angle position and the vertical axis denotes the angle error and equation (13) from the ideal angle position in equations (8) and (14). Moreover, equation (14) has the different offsets about various $\varphi$ and includes not only even-order periodic components of the angle error but also odd-order periodic components because two simulations have considered the installation error of the sensors.

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Figure 11 plots the ${\delta _n}$ in equation (15) calculated by approach (i) in section 2 under simulation 1, and figure 12 also plots the ${\delta _n}$ in equation (15) calculated by approach (i) in section 2 under simulation 2. Then, the horizontal axis is the ideal angle position and the vertical axis is the difference values between equations (8) and (14). Moreover, equation (15) also has the different offsets about various $\varphi$. As the actual analysis scene, we can first analyze equation (15) indicated in figures 11 and 12, and at second calculate the indexing angle $\varphi$.

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Figure 13 shows the $\bar \delta$ in equation (16) calculated by approach (ii). The vertical axis shows the average values for equation (15) in various $\varphi$.

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Comparing the simulation results of figures 7 and 13, we confirm that equations (13) and (16) are similar in sign. Moreover, $\bar \delta$ in equation (16) indicated in figure 13 decreases in proportion to the indexing angle $\varphi$ between adjacent scale lines as same as figure 7. Additionally, equation (16) clearly describes the offset and is consistent with the theory. This offset can be eliminated in the analysis because the all values calculated using equation (16) are shifted. Therefore, we can obtain equation (13) directly through approaches (i) and (ii) in section 2. Additionally, we can match ${O'_\varphi }$ in equation (10) to the indexing angle $\varphi$ shown in table 1 through equations (1) and (16). Accordingly, the principle of the super-high-accuracy angular indexing is validated and quantitatively derived. Moreover, the angular indexing accuracy is expected to improve to 0.01 arcsecs because it is not limited by the resolution and accuracy of the rotary encoder; moreover, it eliminates the angle error accompanying the angle signals.

Highly accurate measurements of the indexing angle of an index table are necessary for improving the processing accuracy of machining tools. In previous research, the indexing angle was measured by optical methods using an interferometer. Here we developed a special optical method that measures this angle using two rotary encoders. We call this method the principle of super-high-accuracy angular indexing. This paper presented a formal derivation of the principle and validated it in a simulation study. The following conclusions were drawn from the results:

• The principle was derived by formulating the angle signal positions' output by the rotary encoders.
• This formulation was expressed as a Fourier series expansion of the angular errors of the installation positions of the sensor units built into the rotary encoder. The errors have circular closure.
• The simulation results were exactly matched. Therefore, the simulation confirmed that the mathematical equation quantifies the principle of super-high-accuracy angular indexing.
• The characteristics of the principle is that it can remove the angle error having the angle signals' output by a rotary encoder.
• The advantage of the principle is that it can measure all indexing angles in theory. Its disadvantage is that it cannot measure in real-time because the measurement needs one rotation.

In future work, we will compare the theoretical values with actual values.