Optimal-arrangement-based four-scanning-heads error separation technique for self-calibration of angle encoders

The basic Fourier-based angle encoder self-calibration method using multiple scanning heads [15, 19] is briefly reviewed in this section. Figure 1 shows an arrangement of the heads (at least two heads can be used for calibration in the basic method; at least four heads as shown in figure 1 are required in error separation model, which will be discussed in section 3.2). During the rotation of circular scale, in addition to the rotation angle θ, the angles measured by the head H_k are also influenced by the circular scale's graduation error ε(θ), including a phase shift due to the head location (α₁  =  0), as shown in equation (1) (assuming rotating the scale in a clockwise direction),

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{k}}\left(\theta \right)=\theta +\varepsilon \left(\theta +{{\alpha}_{k}} \right).\nonumber \end{align} \tag{ 1 }$$

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In order to eliminate the unknown rotation angle θ, the angle difference δ_kr(θ) between any two simultaneously measured data can be applied

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\delta}_{kr}}\left(\theta \right)={{H}_{k}}\left(\theta \right)-{{H}_{r}}\left(\theta \right)=\varepsilon \left(\theta +{{\alpha}_{k}} \right)-\varepsilon \left(\theta +{{\alpha}_{r}} \right).\nonumber \end{align} \tag{ 2 }$$

Due to the periodic nature of the graduation error, Fourier analysis shall be a powerful tool for evaluating the obtained data. By using the discrete Fourier transform (DFT), equation (2) can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{E}_{kr}}\left(n \right)=\left({{{\rm e}}^{{\rm i}n{{\alpha}_{k}}}}-{{{\rm e}}^{{\rm i}n{{\alpha}_{r}}}} \right)F\left(n \right)={{W}_{kr}}\left(n \right)F\left(n \right)\nonumber \end{align} \tag{ 3 }$$

where E_kr(n) and F(n) are the Fourier coefficients of δ_kr(θ) and ε(θ), respectively, n is the Fourier order and W_kr(n) is the transfer function from F(n) to E_kr(n). Given E_kr(n) and W_kr(n), the graduation error can be derived by applying the inverse discrete Fourier transform (IDFT)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varepsilon \left(\theta \right)={\rm IDFT}\left(F\left(n \right) \right)={\rm IDFT}\left(\frac{{{E}_{kr}}\left(n \right)}{{{W}_{kr}}\left(n \right)} \right).\nonumber \end{align} \tag{ 4 }$$

In some applications, three or more heads are used to improve the accuracy of the calibration result. In these cases, the Fourier coefficients F(n) can be derived from a weighted combination of the measurement differences by all pairs of scanning heads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F\left(n \right)=\sum\limits_{k=2}^{M}{\sum\limits_{r=1}^{k-1}{\frac{{{p}_{kr}}\left(n \right){{E}_{kr}}\left(n \right)}{{{W}_{kr}}\left(n \right)}}}\nonumber \end{align} \tag{ 5 }$$

where the normalized weights p_kr(n) in equation (6) can be chosen [19], which can provide the minimal error propagation with respect to the Fourier coefficients F(n)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{p}_{kr}}\left(n \right)=\frac{{{\left| {{W}_{kr}}\left(n \right) \right|}^{2}}}{\sum\nolimits_{k=2}^{M}{\sum\nolimits_{r=1}^{k-1}{{{\left| {{W}_{kr}}\left(n \right) \right|}^{2}}}}}.\nonumber \end{align} \tag{ 6 }$$

The basic self-calibration method may be sufficient for applications in which the graduation error is dominant compared with other error influences, especially for high-precision angle comparators. However, in the general application of angle encoder, the radial motion of the encoder's circular scale during its rotation, including its eccentricity relative to the spindle which carries it, and the radial error motion of the spindle rotor, is usually another important factor that influences the angle measurements by the scanning heads [19]. It is thus necessary to evaluate and separate the effect of the circular scale's radial motion by using an error separation technique, as will be discussed in section 3. For a better understanding of the whole principle, background of the spindle error motion, as well as its measurement using the multiprobe method, is reviewed in section 2.2.

According to ISO 230-7 [27], spindle error motions are unwanted changes in position and orientation of the axis of rotation relative to its axis average line as a function of rotary position of the rotor. Error motions can be directionally decomposed into axial error motion (along Z axis), pure radial error motion in X and Y directions, and tilt error motion around X and Y axes. In practice, pure radial error motion and tilt error motion occur at the same time, and the sum at any specified axial location is defined as radial error motion. Axial error motion has no influence on angle measurement by scanning head. Radial error motion at the axial position where the circular scale locates can influence the head's angle measurement. Error motion can also be divided into synchronous and asynchronous components based on rotational frequency. The synchronous error motion is the repeatable component of the total error motion calculated by averaging the data from the number of revolutions and it consists of Fourier components that occur at integer multiples of the rotation frequency. The asynchronous (non-repeatable) component is the deviations of the total error motion from the synchronous and it may contribute to the repeatability of the encoder calibration results during repeat measurements. Therefore, this paper focuses on the synchronous radial error motion.

In general, the spindle error motion can be measured with a capacitive displacement sensor targeting a cylindrical or spherical test artefact. However, for the 2D radial error motion, error motion values measured with sensors at different fixed radial sensitive directions may vary greatly. Here, the term 'sensitive direction' in spindle metrology represents direction perpendicular to the workpiece surface through the point of measurement. Fortunately, the radial error motion for a fixed sensitive direction at an angle β_k with respect to X axis (see figure 2), can be calculated using equation (7).

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{s}_{k}}\left(\theta \right)=x\left(\theta \right)\cos {{\beta}_{k}}+y\left(\theta \right)\sin {{\beta}_{k}}\nonumber \end{align} \tag{ 7 }$$

where, x(θ) and y(θ) are components of the radial error motion in X and Y directions, respectively.

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Measurement of spindle radial error motion is influenced by the roundness error of the artefact against which the sensor senses. For highly accurate measurements, the radial error motion should be separated from the artefact roundness error using error separation techniques. One of the most commonly used separation techniques is the multiprobe method, which uses three or more sensors to simultaneously measure the combination of the error motion and roundness error, as illustrated in figure 2. The measurements of the three displacement sensors S₁, S₂ and S₃, are a summation of the X and Y components of the radial error motion, and the artefact roundness error r(θ), with a phase shift due to the sensor location,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}lllllllllllllllllllllllllllllll@{}} {{S}_{1}}\left(\theta \right)=r\left(\theta \right)+x\left(\theta \right) \nonumber \\ {{S}_{2}}\left(\theta \right)=r\left(\theta +{{\beta}_{2}} \right)+x\left(\theta \right)\cos {{\beta}_{2}}+y\left(\theta \right)\sin {{\beta}_{2}} \nonumber \\ {{S}_{3}}\left(\theta \right)=r\left(\theta +{{\beta}_{3}} \right)+x\left(\theta \right)\cos {{\beta}_{3}}+y\left(\theta \right)\sin {{\beta}_{3}}. \end{array}\nonumber \end{align} \tag{ 8 }$$

A weighted combination of these three measurements, S(θ), can be used to remove the contribution of radial error motion

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S\left(\theta \right)={{b}_{1}}{{S}_{1}}\left(\theta \right)+{{b}_{2}}{{S}_{2}}\left(\theta \right)+{{b}_{3}}{{S}_{3}}\left(\theta \right)\nonumber \end{align} \tag{ 9 }$$

where the unknown coefficients b₁, b₂ and b₃ should satisfy

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}cccccccccccccccccccc@{}} & {{b}_{1}}+{{b}_{2}}\cos {{\beta}_{2}}+{{b}_{3}}\cos {{\beta}_{3}}=0 \nonumber \\ & {{b}_{2}}\sin {{\beta}_{2}}+{{b}_{3}}\sin {{\beta}_{3}}=0. \end{array}\nonumber \end{align} \tag{ 10 }$$

By solving equation (10), the following relationship is obtained

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{b}_{1}}:{{b}_{2}}:{{b}_{3}}=\sin \left({{\beta}_{3}}-{{\beta}_{2}} \right):\left(-\sin {{\beta}_{3}} \right):\sin {{\beta}_{2}}.\nonumber \end{align} \tag{ 11 }$$

For convenience, b₁  =  sin (β₃  −  β₂), b₂  =  −sin β₃ and b₃  =  sin β₂ can be defined.

Then, equation (9) contains only the information about the roundness error. Applying the DFT to equation (9) gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S\left(n \right)=\left({{b}_{1}}+{{b}_{2}}{{{\rm e}}^{{\rm i}n{{\beta}_{2}}}}+{{b}_{3}}{{{\rm e}}^{{\rm i}n{{\beta}_{3}}}} \right)r\left(n \right)=B\left(n \right)r\left(n \right)\nonumber \end{align} \tag{ 12 }$$

where B(n) is the transfer function from r(n) to S(n). When the coefficients S(n) and B(n) are known, the artefact roundness error can be determined by using the IDFT

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r\left(\theta \right)={\rm IDFT}\left[{S\left(n \right)}/{B\left(n \right)}\; \right].\nonumber \end{align} \tag{ 13 }$$

By substituting r(θ) in equation (8), the X and Y components of the radial error motion can be calculated

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}llcccccccccccccccccc@{}} & x\left(\theta \right)={{S}_{1}}\left(\theta \right)-r\left(\theta \right) \nonumber \\ & y\left(\theta \right)={\left[{{S}_{2}}\left(\theta \right)-r\left(\theta +{{\beta}_{2}} \right)-x\left(\theta \right)\cos {{\beta}_{2}} \right]}/{\sin {{\beta}_{2}}}. \end{array}\nonumber \end{align} \tag{ 14 }$$

The basic self-calibration method in section 2.1 considers the graduation error only. However, the angle encoder error is also influenced by the circular scale's radial motion. The influence of radial motion on head's angle measurement has been analyzed in [19]. This section recapitulates the main theoretical analysis, and, at the same time, gives more details about the measurement deviations of the heads due to 2D radial error motion. Figure 3 shows how the radial motion of circular scale affects the angle measurement by scanning head. Ideally, during the rotation of the circular scale, its radial center should always coincide with the geometric center of the fixed arrangement of scanning heads (i.e. the spindle stator center, the origin of the coordinate system, O). In practical applications, however, due to the mounting eccentricity and the spindle radial error motion, the circular scale center will have a radial motion from O(0, 0) to O' (x, y). This results in a deviation between the angle θ' detected by the head H_k (located at angle α_k with respect to X axis) and the real rotation angle θ (assuming a clockwise rotation). Since the deviation, λ_k(θ)  =  θ'  −  θ, is very small (usually, <0.01°), it can be approximated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\lambda}_{k}}=\sin \left({{\lambda}_{k}} \right)={h}/{R}\;\nonumber \end{align} \tag{ 15 }$$

where R is the circular scale radius (mean graduation radius), and h can be calculated according to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle h=\left| O{{O}^{\prime}} \right|\sin \left({{\alpha}_{k}}-\angle {{O}^{\prime}}OX \right)=x\sin {{\alpha}_{k}}-y\cos {{\alpha}_{k}}.\nonumber \end{align} \tag{ 16 }$$

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Thus, the heads' measurement deviations due to radial motion of the circular scale during its rotation are a function of the circular scale radius, the radial motion and the heads' position angles, and can then be represented as [19]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\lambda}_{k}}\left(\theta \right)=\frac{1}{R}\left(x\left(\theta \right)\sin {{\alpha}_{k}}-y\left(\theta \right)\cos {{\alpha}_{k}} \right).\nonumber \end{align} \tag{ 17 }$$

As a quantitative evaluation, figure 4 presents the peak-to-valley (pv) magnitude of the measurement deviation as a function of the circular scale radius, with the ranges of radial motion, max(h(θ))  −  min(h(θ)), being equal to 0.02 µm, 0.05 µm, 0.1 µm, 0.2 µm, 0.5 µm, and 1 µm, respectively. The influence of the radial motion can be reduced by using a circular scale with larger radius. It should be noted that, however, when the range of radial motion is 0.1 µm, the magnitude of angle deviation is greater than 0.1 arcsec, even for a large radius R  =  180 mm. Hence, the influence of the circular scale's radial motion cannot be simply neglected for high-precision measuring tasks.

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Note that for angle measurement, averaging the measurement values of the scanning heads which are arranged at regular intervals can eliminate the effect of circular scale's radial motion. This can be explained by the following mathematical property

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum\limits_{k=0}^{M-1}{{{{\rm e}}^{{\rm i}{2\pi k}/{M}\;}}}=\sum\limits_{k=0}^{M-1}{\cos \frac{2\pi k}{M}}+{\rm i}\sum\limits_{k=0}^{M-1}{\sin \frac{2\pi k}{M}}=0.\nonumber \end{align} \tag{ 18 }$$

Therefore, for M equally distributed heads with position angles α_k  =  2π (k  −  1)/M, the sum of their measurement deviations (see equation (17)) due to radial motion satisfies

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum\limits_{k=1}^{M}{{{\lambda}_{k}}\left(\theta \right)}=0.\nonumber \end{align} \tag{ 19 }$$

In particular, for industrial application of angle encoders, a common method of eliminating the influence of radial motion is to mount two scanning heads at 180° angular interval.

The mounting eccentricity of the circular scale relative to the spindle is one of the two physical causes of radial motion [19]. For the eccentricity e (i.e. the offset |OO'|  =  e, see figure 3), assuming that the eccentricity vector has an angle θ_e with respect to the X axis at rotation angle θ  =  0°, h(θ) in equation (16) will be equal to e sin(α_k  −  θ_e  +  θ). Then, equation (17) can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\lambda}_{k}}\left(\theta \right)=\frac{1}{R}h\left(\theta \right)=\frac{e}{R}\sin \left(\theta +{{\alpha}_{k}}-{{\theta}_{e}} \right).\nonumber \end{align} \tag{ 20 }$$

By comparing this equation with the graduation error ε (θ  +  α_k) shown in equation (1), it can be found that the angle deviation caused by the eccentricity automatically conforms to the graduation error model. The basic self-calibration method can correctly evaluate the influence of eccentricity as the first-order Fourier component of the encoder error. Therefore, the eccentricity should not be treated as a source of additional error influence (relative to the graduation error), since its influence completely conforms to the basic self-calibration model [19].

The spindle radial error motion is another cause of radial motion. In fact, the eccentricity error is the first-order Fourier component of the radial motion, while the radial error motion contributes to the residual components of radial motion in the frequency domain. In order to demonstrate the influence, and further separation approach, of the radial error motion, figure 5 provides a numerical simulation example used throughout section 3: the spindle 2D radial error motion in X and Y directions, and the circular scale's graduation error (including the first-order Fourier component caused by eccentricity).

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The measurement deviations of the scanning heads caused by the 2D radial error motion can be calculated by equation (17), as shown in figure 6, for the heads' angular positions α_k  =  0°, 40°, 90° and 180°, respectively (the circular scale radius is assumed to be 150 mm). The deviations of the heads at 0° and 180° orientation angles show equal magnitudes but opposite signs, as mentioned in equation (19) (M  =  2). In addition, equation (17) can also be expressed as λ_k(θ)  =  (x(θ) cos(α_k  −  90°)  +  y(θ) sin(α_k  −  90°))/R. By comparing this expression with equation (7), the 90° angular difference in phase reveals an important relation between angle encoder calibration and spindle metrology. For a direction where the sensor (scanning head or displacement sensor) is fixed, the displacement sensor is sensitive to the component of the 2D radial error motion along the fixed direction (radial measurement), while the head is sensitive to the component of the radial motion perpendicular to this direction (tangential measurement), see section 2.2 and figure 3. Therefore, it should be noted that the meaning of 'sensitive direction' is different for spindle metrology and encoder calibration. For this reason, the head's deviation at 90° angular position in figure 6 and the X component of the radial error motion in figure 5 have the same curve shape, and so do the deviation at 180° and the error motion in Y direction. (Note that the clockwise rotation of circular scale is assumed in this paper; for anticlockwise rotation, equation (17) will be opposite in sign).

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Furthermore, the differences in heads' measurement deviations due to the 2D radial error motion at different position angles are substantial. In figure 6, the pv range of the angle deviation at 40° angular position is obviously less than the other three ranges. Figure 7 plots the pv range of the measurement deviation, max(λ(θ))  −  min(λ(θ)), as a function of the head's angular position α_k. Since measurement deviations for two heads separated by 180° are opposite in sign, the plot has a period of 180°. In this numerical example, the range of measurement deviation is substantially less at angular positions between 30° and 45° (or between 210° and 225°) than at other angular positions. Thus, for angle measuring tasks, these angular positions can be used as preferential measurement points to help reduce the influence of radial error motion.

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In the basic self-calibration method, graduation error is the only factor that affects the angle measurement by scanning head. However, angle deviation caused by circular scale's radial motion is another important factor that influences the measurement. In this section, an error separation method is introduced for separating the graduation error from the angle deviation due to radial motion.

After introducing the measurement deviation λ_k(θ) shown in equation (17), the angles measured by the head H_k in equation (1) yield

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{k}}\left(\theta \right)=\theta +\varepsilon \left(\theta +{{\alpha}_{k}} \right)+\frac{1}{R}\left(x\left(\theta \right)\sin {{\alpha}_{k}}-y\left(\theta \right)\cos {{\alpha}_{k}} \right).\nonumber \end{align} \tag{ 21 }$$

Especially for the reference head H₁, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{1}}\left(\theta \right)=\theta +\varepsilon \left(\theta \right)-{y\left(\theta \right)}/{R}.\nonumber \end{align} \tag{ 22 }$$

Since there are four unknowns in equation (21), at least four heads should be used for the separation (see figure 1). Then, the angle differences for the heads H₂, H₃, and H₄ relative to the reference head H₁ can be obtained to eliminate the rotation angle θ,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}cccccccccccccccccccc@{}} & {{\delta}_{21}}\left(\theta \right)={{H}_{2}}\left(\theta \right)-{{H}_{1}}\left(\theta \right)=\varepsilon \left(\theta +{{\alpha}_{2}} \right)-\varepsilon \left(\theta \right)+\frac{1}{R}\left[x\left(\theta \right)\sin {{\alpha}_{2}}-y\left(\theta \right)\left(\cos {{\alpha}_{2}}-1 \right) \right] \nonumber \\ & {{\delta}_{31}}\left(\theta \right)={{H}_{3}}\left(\theta \right)-{{H}_{1}}\left(\theta \right)=\varepsilon \left(\theta +{{\alpha}_{3}} \right)-\varepsilon \left(\theta \right)+\frac{1}{R}\left[x\left(\theta \right)\sin {{\alpha}_{3}}-y\left(\theta \right)\left(\cos {{\alpha}_{3}}-1 \right) \right] \nonumber \\ & {{\delta}_{41}}\left(\theta \right)={{H}_{4}}\left(\theta \right)-{{H}_{1}}\left(\theta \right)=\varepsilon \left(\theta +{{\alpha}_{4}} \right)-\varepsilon \left(\theta \right)+\frac{1}{R}\left[x\left(\theta \right)\sin {{\alpha}_{4}}-y\left(\theta \right)\left(\cos {{\alpha}_{4}}-1 \right) \right]. \end{array}\nonumber \end{align} \tag{ 23 }$$

In order to remove the influence of the 2D radial motion, a weighted summation of these angle differences can be used

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}lllllllllllllllllllllll@{}} & \delta \left(\theta \right)={{a}_{1}}{{\delta}_{21}}\left(\theta \right)+{{a}_{2}}{{\delta}_{31}}\left(\theta \right)+{{a}_{3}}{{\delta}_{41}}\left(\theta \right) \nonumber \\ & \quad \quad \, = {{a}_{1}}\varepsilon \left(\theta +{{\alpha}_{2}} \right)+{{a}_{2}}\varepsilon \left(\theta +{{\alpha}_{3}} \right)+{{a}_{3}}\varepsilon \left(\theta +{{\alpha}_{4}} \right)-\left({{a}_{1}}+{{a}_{2}}+{{a}_{3}} \right)\varepsilon \left(\theta \right) \nonumber \\ & \quad \quad \quad +{x\left(\theta \right)\left({{a}_{1}}\sin {{\alpha}_{2}}+{{a}_{2}}\sin {{\alpha}_{3}}+{{a}_{3}}\sin {{\alpha}_{4}} \right)}/{R}\; \nonumber \\ & \quad \quad \quad -{y\left(\theta \right)\left({{a}_{1}}\cos {{\alpha}_{2}}+{{a}_{2}}\cos {{\alpha}_{3}}+{{a}_{3}}\cos {{\alpha}_{4}}-{{a}_{1}}-{{a}_{2}}-{{a}_{3}} \right)}/{R}\; \end{array}\nonumber \end{align} \tag{ 24 }$$

where the unknown coefficients a₁, a₂ and a₃ should satisfy

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}llcccccccccccccccccc@{}} {{a}_{1}}\sin {{\alpha}_{2}}+{{a}_{2}}\sin {{\alpha}_{3}}+{{a}_{3}}\sin {{\alpha}_{4}}=0 \nonumber \\ {{a}_{1}}\left(\cos {{\alpha}_{2}}-1 \right)+{{a}_{2}}\left(\cos {{\alpha}_{3}}-1 \right)+{{a}_{3}}\left(\cos {{\alpha}_{4}}-1 \right)=0. \end{array}\nonumber \end{align} \tag{ 25 }$$

Solving equation (25) gives

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left[\begin{array}{@{}c@{}} {{a}_{1}} \nonumber \\ {{a}_{2}} \nonumber \\ {{a}_{3}} \end{array} \right]=C\left[\begin{array}{@{}c@{}} \sin \left({{\alpha}_{3}}-{{\alpha}_{4}} \right)-\sin {{\alpha}_{3}}+\sin {{\alpha}_{4}} \nonumber \\ \sin \left({{\alpha}_{4}}-{{\alpha}_{2}} \right)-\sin {{\alpha}_{4}}+\sin {{\alpha}_{2}} \nonumber \\ \sin \left({{\alpha}_{2}}-{{\alpha}_{3}} \right)-\sin {{\alpha}_{2}}+\sin {{\alpha}_{3}} \end{array} \right]\nonumber \end{align} \tag{ 26 }$$

where C is an arbitrary constant. For convenience, C  =  1 is used in this paper.

After removing the contribution of radial motion and by applying the DFT, equation (24) can be expressed as E(n)  =  W(n)F(n). The transfer function W(n) is represented as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W(n)={{a}_{1}}\left({{{\rm e}}^{{\rm i}n{{\alpha}_{2}}}}-1 \right)+{{a}_{2}}\left({{{\rm e}}^{{\rm i}n{{\alpha}_{3}}}}-1 \right)+{{a}_{3}}\left({{{\rm e}}^{{\rm i}n{{\alpha}_{4}}}}-1 \right).\nonumber \end{align} \tag{ 27 }$$

Then the graduation error ε(θ) can be obtained by using the IDFT, just like equation (4).

In addition, the radial motion components x(θ) and y(θ) can be calculated by substituting the graduation error ε(θ) in equation (23),

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}llllllllllllllllllllllllllll@{}} x\left(\theta \right)={\left[A\left(\theta \right)\left(\cos {{\alpha}_{3}}-1 \right)-B\left(\theta \right)\left(\cos {{\alpha}_{2}}-1 \right) \right]}/{{{a}_{3}}}\; \nonumber \\ y\left(\theta \right)={\left[A\left(\theta \right)\sin {{\alpha}_{3}}-B\left(\theta \right)\sin {{\alpha}_{2}} \right]}/{{{a}_{3}}}\; \end{array}\nonumber \end{align} \tag{ 28 }$$

with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}cccccccccccccccccccc@{}} & A\left(\theta \right)=R\left({{\delta}_{21}}\left(\theta \right)-\varepsilon \left(\theta +{{\alpha}_{2}} \right)+\varepsilon \left(\theta \right) \right) \nonumber \\ & B\left(\theta \right)=R\left({{\delta}_{31}}\left(\theta \right)-\varepsilon \left(\theta +{{\alpha}_{3}} \right)+\varepsilon \left(\theta \right) \right). \end{array}\nonumber \end{align} \tag{ 29 }$$

This separation process, however, cannot separate the first-order Fourier component, since the transfer function W(1) is always equal to 0 no matter what the position angles α₂, α₃ and α₄ are set. Nevertheless, as stated in section 3.1, since measurement deviations due to the eccentricity, i.e. the first-order component of radial motion, conform to the graduation error model, the first-order component of encoder error can be well obtained by using the basic self-calibration method. As a result, the whole calibration process, simulated with the numerical example in figure 5, is illustrated in figure 8 (assuming α₂  =  60°, α₃  =  135° and α₄  =  228° in this simulation; the reason for selecting these position angles will be discussed in section 4).

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It is important to note that the 2D radial error motion can also be determined from the four-scanning-heads error separation method. Therefore, this method may be an important tool for measuring the radial error motion in practical applications.

The multiple scanning heads methods for calibration of angle encoders require exact mounting and adjustment of the heads. In practice, the scanning heads' angular position errors, i.e. the deviations between the actual angular positions and their nominal values, are inevitable. At the PTB, the contribution of the uncertainty of heads' angular positions to the uncertainty of reconstructed graduation error was evaluated according to Monte Carlo approach [19]. In this section, the influence of the heads' angular position errors on the accuracy of self-calibration results is discussed based on theoretical and numerical analyses.

Taking the scanning head H₁ in figure 1 as the reference head, and denoting the angular position error of head H_k relative to H₁ as η_k, the angle measurement by H_k then deviates from the value H_k(θ) in equation (21), and can be expressed as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{k}^{\prime}\left(\theta \right)=&\,\theta +\varepsilon \left(\theta +{{\alpha}_{k}}+{{\eta}_{k}} \right)+\frac{1}{R}\left[x\left(\theta \right)\sin \left({{\alpha}_{k}}+{{\eta}_{k}} \right)\right.\nonumber \\ &\left.-y\left(\theta \right)\cos \left({{\alpha}_{k}}+{{\eta}_{k}} \right) \right].\nonumber \end{align} \tag{ 30 }$$

The measurement error on the angle difference δ_k1(θ) is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}cccccccccccccccccccc@{}} & \Delta {{\delta}_{k1}}\left(\theta \right)=\delta _{k1}^{\prime}\left(\theta \right)-{{\delta}_{k1}}\left(\theta \right)=\varepsilon \left(\theta +{{\alpha}_{k}}+{{\eta}_{k}} \right)-\varepsilon \left(\theta +{{\alpha}_{k}} \right) \nonumber \\ & \quad \quad \quad \quad +\frac{1}{R}\left[x\left(\theta \right)\left(\sin \left({{\alpha}_{k}}+{{\eta}_{k}} \right)-\sin {{\alpha}_{k}} \right)-y\left(\theta \right)\left(\cos \left({{\alpha}_{k}}+{{\eta}_{k}} \right)-\cos {{\alpha}_{k}} \right) \right]. \end{array}\nonumber \end{align} \tag{ 31 }$$

For the very small value of η_k, we have cos η_k  ≈  1, sin η_k  ≈  η_k. Equation (31) can be rewritten as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta {{\delta}_{k1}}\left(\theta \right)=&\,\varepsilon \left(\theta +{{\alpha}_{k}}+{{\eta}_{k}} \right)-\varepsilon \left(\theta +{{\alpha}_{k}} \right)\nonumber \\ &+\frac{{{\eta}_{k}}}{R}\left[x\left(\theta \right)\cos {{\alpha}_{k}}+y\left(\theta \right)\sin {{\alpha}_{k}} \right].\nonumber \end{align} \tag{ 32 }$$

For simplicity, only the position of head H₂ is first assumed to have an angular error. Then the deviation on δ(θ) in equation (24) yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta \delta \left(\theta \right)={{\delta}^{\prime}}\left(\theta \right)-\delta \left(\theta \right)={{a}_{1}}\Delta {{\delta}_{21}}\left(\theta \right).\nonumber \end{align} \tag{ 33 }$$

Using the DFT, equation (33) becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta E\left(n \right)=&\,{{a}_{1}}\left[\left({{{\rm e}}^{{\rm i}n\left({{\alpha}_{2}}+{{\eta}_{2}} \right)}}-{{{\rm e}}^{{\rm i}n{{\alpha}_{2}}}} \right)F\left(n \right)\right.\nonumber \\ &\left.+\frac{{{\eta}_{2}}}{R}\left[x\left(n \right)\cos {{\alpha}_{2}}+y\left(n \right)\sin {{\alpha}_{2}} \right] \right].\nonumber \end{align} \tag{ 34 }$$

The errors on the Fourier coefficients of ε(θ) yield ΔF(n)  =  F'(n)  −  F(n)  =  ΔE(n)/W(n), and can be further expressed as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta F\left(n \right)=&\,\frac{{{a}_{1}}}{W\left(n \right)} \bigg [{{{\rm e}}^{{\rm i}n{{\alpha}_{2}}}}\left({{{\rm e}}^{{\rm i}n{{\eta}_{2}}}}-1 \right)F\left(n \right)\nonumber \\ &+{{\eta}_{2}}\left(\frac{x\left(n \right)}{R}\cos {{\alpha}_{2}}+\frac{y\left(n \right)}{R}\sin {{\alpha}_{2}} \right) \bigg].\nonumber \end{align} \tag{ 35 }$$

Finally, for the measurement error on the graduation error, we have Δε(θ)  =  ε'(θ)  −  ε(θ)  =  IDFT(ΔF(n)).

Equation (35) reveals the effect of head's angular position error on the calibration result for the four-scanning-heads error separation method. When neglecting radial error motion (x  =  0 and y  =  0), this equation gives the measurement error for the basic method in section 2.1. From equation (35), the measurement error resulting from angular position error contains contributions of both the graduation error and the radial motion. The Fourier component ΔF(n) is a linear combination of the graduation error's component F(n) and the radial motion's components x(n) and y(n), but is inversely proportional to the transfer function W(n). However, it should be noted that the coefficient of F(n), exp(inα₂) (exp(inη₂)  −  1), varies with Fourier order n, while the coefficient of [x(n)cosα₂  +  y(n)sinα₂]/R remains constant (η₂). Figure 9 gives the amplitude of the coefficient exp(inη₂)  −  1 as a function of the order n (|exp(inα₂)|  =  1), with the angular position error η₂  =  120 arcsec, −240 arcsec and 360 arcsec, respectively. The linear relation between the amplitude and the order shows that high order Fourier components of the reconstructed graduation error are more susceptible to the head's angular position error, which conforms to the analysis in our previous publication [16]. Although low order components of the graduation error usually have larger amplitudes, their contribution to Δε(θ) is limited to a relatively higher degree.

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In addition, for the order n  =  1, we have |exp(iη₂)  −  1|  =  |η₂| (i.e. |η₂|  =  min(|exp(inη₂)  −  1|), n integer and  ≠  0). From this equation and equation (35), it can be seen that all the Fourier components of radial motion have the same influence coefficient on the corresponding components of the measurement error; moreover, the contribution of radial error motion to the measurement error is much less than that of graduation error. Figure 10 shows the measurement error on the reconstructed graduation error ε(θ) at the angular position error η₂  =  360 arcsec, based on the numerical example shown in figure 5. As described above, low order components are inconspicuous in the measurement error. By assuming that the measured graduation error is not influenced by the angular position error (making ε (θ  +  α₂  +  η₂)  −  ε (θ  +  α₂)  =  0, in equation (32)), the contribution of radial error motion to Δε(θ) is also presented in figure 10. It is negligibly small, compared to the total measurement error.

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Figure 11 shows the magnitudes (pv range) of the measurement errors on graduation error, Δε(θ), and on radial error motion in X and Y directions, Δx(θ) and Δy(θ), respectively, when η₂ ranging from 60 arcsec to 600 arcsec at 60 arcsec intervals. The measurement errors increase linearly with respect to the increasing angular position error. Since the radial error motion can be calculated only after the calculation of graduation error (see equation (28)), the measurement errors Δε(θ), Δx(θ) and Δy(θ) are correlated. The dominant contribution to measurement errors caused by scanning heads' angular position errors is the graduation error.

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Finally, for the four-scanning-heads method, the expressions of ΔF(n) for the other two heads' angular position errors, η₃ and η₄, are similar to equation (35), except for their corresponding subscripts. The total measurement error of the four-scanning-heads system can be expressed as a linear superposition of the errors due to individual angular position errors.