## Abstract

A laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors is proposed for precision linear stage metrology. In this interferometer, the vertical straightness error and its position are measured by interference fringe counting, the yaw and pitch errors are obtained by measuring the spacing changes of interference fringe and the horizontal straightness and roll errors are determined by laser collimation. The merit of this interferometer is that four degrees of freedom motion errors are obtained by using laser interferometry with high accuracy. The optical configuration of the proposed interferometer is designed. The principle of the simultaneous measurement of six degrees of freedom errors including yaw, pitch, roll, two straightness errors and straightness error’s position of measured linear stage is depicted in detail, and the compensation of crosstalk effects on straightness error and its position measurements is presented. At last, an experimental setup is constructed and several experiments are performed to demonstrate the feasibility of the proposed interferometer and the compensation method.

© 2017 Optical Society of America

## 1. Introduction

The precision linear stage is a key motion component for precision manufacturing and measuring equipment such as computer numerical control machine tools and coordinate measuring machines. As major technique specification of a precision linear stage, the straightness error combining with other degree of freedom motion errors directly affects the accuracy of the precision equipment. Thus, the high accuracy of straightness measurement instruments is increasingly demanded for the performance evaluation of a precision linear stage.

Current straightness measurement methods can be mainly divided into laser collimation, laser grating diffraction and laser interferometry. Among the laser collimation, the laser beam is usually used as reference line and the position sensitive detector (PSD) or quadrant detector (QD) is utilized as sensors to measure the straightness error [1–7]. For example, K. C. Fan and Y. Zhao proposed a laser straightness measurement system using a QD to receive the laser beam [1]. Q. Feng *et al* employed a fiber-coupled laser to measure six degrees of freedom motion errors simultaneously, in which straightness error, pitch, yaw and roll are measured by using laser collimation, and displacement is obtained by using laser interferometry [5, 6]. A six-degree-of-freedom simultaneous measurement system for precision linear stage with fiber coupling was presented by X. Yu *et al*, the straightness error is measured by laser collimation, and yaw, pitch and displacement are measured by heterodyne interferometry and differential wavefront sensing technique [7]. The methods using laser collimation have the advantages of low cost, simple compact and fast optical adjustment. However, the accuracy of straightness measurement is low due to the limitation of the resolution of PSD or QD.

In the laser grating diffraction, the straightness error is measured using the diffraction rays produced by the grating which serves as a sensor moving with the measured object [8, 9]. W. Gao *et al* developed a two-degree-of-freedom linear encoder which comprises a reflective-type one-axis scale grating and an optical sensor head to measure the straightness error and displacement simultaneously [10]. C. Lee *et al* employed a single unit of an optical encoder to measure a six-degree-of-freedom posture in a linear stage, in which the straightness error is detected by separate PSDs [11]. A laser encoder is proposed by C.-H. Liu *et al*, in which the ± 1 order diffracted light interference is used for the measurement of linear displacement, and the ± 2 order diffracted light spots are used to calculate the straightness and other motion errors [12]. H.-L. Hsieh *et al* presented a grating-based interferometer which is based on the quasi-common optical path design and combined the merits of the heterodyne, Michelson and grating-shearing interferometries in order to measure the straightness and other degrees of freedom errors [13]. Although the grating diffraction methods have the advantage of high precision, the large travel of straightness measurement is limited by the size of the grating.

For the straightness measurement using laser interferometry, the typical representative is the straightness interferometer which measures the straightness error by detecting the optical path difference between two separated measuring beams which are produced by a Wollaston prism or other optical splitter [14–16]. C. Yin *et al* adopted a pair of Wollaston prisms as sensors which meets the principle of self-adaptation to overcome the atmospheric disturbance, and the straightness error in a long measurement range of 16 m was measured [17]. C.-M. Wu proposed three construction principles of a straightness interferometer to reduce periodic nonlinearity, and developed a generalized periodic nonlinearity-reduced interferometer [18]. S.-T. Lin *et al* designed a calibrator utilizing a low-coherent light source straightness interferometer that adopts an approach of driving the Wollaston prism back a lateral displacement to compensate the influence of rotational error until the zero-order fringe appears at the original location [19]. Compared with the laser collimation, laser grating diffraction and other straightness measurement methods, the laser interferometric straightness measurement has advantages of high measurement accuracy, sensitivity, stability and large measurement range. However, a common disadvantage in these instruments is that they cannot provide the position where the straightness error was measured. To solve this problem, we proposed a laser heterodyne straightness interferometer that realizes the simultaneous measurement of the straightness error and its position in our previous work [20]. In this framework, a laser heterodyne straightness interferometer system with simultaneous measurement of six degree of freedom errors was developed, and the compensation method for straightness error and its position due to the influence of the rotational errors of the measured object is presented [21]. However, the three rotational errors including yaw, pitch, and roll errors and the horizontal straightness error of a linear stage were still measured with laser collimation, meaning that it has the same shortcoming as laser collimation method described above. Moreover, the periodic nonlinearity of heterodyne interferometer, being higher than that of homodyne interferometer [22, 23], limits the improvement of measurement accuracy of the straightness error and its position.

To make full use of laser interferometry with advantage of high accuracy to realize the measurement of multiple degrees of freedom parameters, a laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors is proposed in this paper. In this interferometer, the vertical straightness error and its position are measured based on interference fringe counting, the yaw and pitch errors are obtained by interferometric fringe image processing and the horizontal straightness and roll errors are determined by laser collimation. The optical configuration of the interferometer is designed in section 2. The principle of the simultaneous measurement of six degrees of freedom errors including yaw, pitch, roll, two straightness errors and straightness error’s position of a linear stage is presented, and the compensation of crosstalk effects on straightness error and its position measurements is depicted in section 3. In section 4, an experimental setup is constructed and several experiments are performed to demonstrate the feasibility of the interferometer and the compensation method. Finally, the measurement range and resolution of the interferometer are discussed in section 5.

## 2. Configuration

The laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors is shown in Fig. 1. A linearly polarized beam emitted from a He-Ne laser is adjusted to 45° polarization direction with respect to the *x*-axis by a half wave plate (HWP). The main sensing unit based on homodyne interferometry includes a Wollaston-prism-type straightness interferometer and a Michelson interferometer. There are three parts in the detecting unit of the proposed interferometer. Part I is composed of a CCD camera, a plane mirror (MR_{1}) and a nonpolarizing beam-splitter (NPBS_{2}), which is used to determine the yaw error around the *x* axis and the pitch error around the *y* axis based on homodyne interference fringe pattern analysis. Part II is composed of a polarizer (P), two nonpolarizing beam-splitters (NPBS_{4,} NPBS_{5}), two quarter-wave plates (QWP_{1}, QWP_{2}), three polarizing beam-splitters (PBS_{1}, PBS_{2}, PBS_{3}) and five photodetectors (D_{1}-D_{5}), which serves for the determination of vertical straightness error along the *x* axis and its position along the *z* axis with homodyne interferometry. And Part III is composed of a polarizing beam-splitter (PBS_{4}) and two quadrant photodetectors (QD_{1}, QD_{2}), which is used for the determination of horizontal straightness error along the *y* axis and the roll error around the *z* axis.

## 3. Principle

#### 3.1 Determination of the yaw and pitch errors

The schematic of the yaw and pitch errors measurement employing two-dimensional small angle measurement method based on homodyne interference fringe pattern analysis is shown in Fig. 2. The linearly polarized beam reflected by NPBS_{1} is split into a reference beam and a measuring beam by NPBS_{2}. The measuring beam is reflected by a measuring mirror (MR_{2}) which is mounted on retro-reflector (RR), back into NPBS_{2} and recombined with the reference beam which is reflected by MR_{1}. The recombined beams interfere with each other and the homodyne interference fringe patterns are captured by CCD. The spacing of interference fringe in pattern varies with the measuring beam propagating direction [24–26] which is determined by the yaw and pitch errors of MR_{2}.

At the initial position, the spacing of interference fringe as shown in Fig. 2(b) can be expressed as

*X*,

_{M}*X*,

_{R}*Y*and

_{M}*Y*are all small values which denote

_{R}*x*and

*y*components of the direction vectors corresponding to the measuring and reference beams, respectively.

When MR_{2} rotates with a moving stage, the spacing of interference fringe varies correspondingly as shown in Fig. 2(c) and can be expressed as

*α*and

*β*denote the yaw error and pitch error, respectively.

According to Eqs. (1) and (2), the yaw error and pitch error can be expressed as

#### 3.2 Determination of the vertical straightness error and its position

The schematic of vertical straightness error and its position measurement with homodyne interferometry is shown in Fig. 3. The linearly polarized beam transmitted by NPBS_{1} is divided into two beams by NPBS_{3}. The transmitted beam of NPBS_{3} is split into two orthogonal linearly polarized beams (*p*-polarized light and *s*-polarized light) by a Wollaston prism (WP). The two divergent measuring beams are reflected by RR which is made up of two right-angle prisms and placed on the moving stage, back into WP and recombined at another point. The two recombined beams can be expressed with Jones matrix as

*E*

_{0}denotes the amplitude and

*f*denotes the frequency of the two beams,

*t*denotes the time,

*φ*

_{1}and

*φ*

_{2}denote the initial phases of the two beams, respectively.

The measuring beam containing *Ĵ*_{1} and *Ĵ*_{2} is split into two beams by NPBS_{4}, then the reflected beam is transmitted through QWP_{2} whose fast axis is at 45°with respect to the polarization directions of *Ĵ*_{1} and *Ĵ*_{2}, and transformed into a linearly polarized beam whose polarization direction is decided by the phases of *Ĵ*_{1} and *Ĵ*_{2.} This linearly polarized beam can be expressed as

*φ*/2 denotes the polarization direction of

*Ĵ*

_{3}with respect to the

*x*axis.

Then *Ĵ*_{3} is split by NPBS_{5}, the reflected beam passes through P which is at 45° with respect to the *x* axis, and reaches D_{1} which generates an interference signal that can be expressed as

*A*denotes the amplitude of the interference signal. The transmitted beam is split into two orthogonal linearly polarized beams by PBS

_{3}, which reach D

_{2}and D

_{3}, respectively. Two interference signals are generated and can be expressed as

According to *I _{D}*

_{1},

*I*

_{D}_{2}and

*I*

_{D}_{3}, the sinusoidal and cosine signals are given by

*φ*is the phase difference of

*Ĵ*

_{1}and

*Ĵ*

_{2}which can be calculated by combining the sinusoidal and cosine signal. The difference between the optical path differences (OPDs) corresponding to the two measuring beams

*Ĵ*

_{1}and

*Ĵ*

_{2}can be expressed aswhere

*λ*is the wavelength of the laser.

The transmitted beam of NPBS_{4} reaches PBS_{2} and meets the reference beam reflected by NPBS_{3}. The *p*-polarized light of the measuring beam and *s*-polarized light of the reference beam are transmitted through PBS_{2} and transformed into a right circularly polarized and a left circularly polarized beams, respectively, after passing through the QWP_{2}. The two circularly polarized beams are transformed to two transmitted *p*-polarized beams and two reflected *s*-polarized beams by PBS_{1}. Then two interference signals are generated by D_{4}, D_{5} and can be expressed as

*ϕ*is the phase of the interference signal,

*A*

_{1}and

*A*

_{2}are the DC components,

*B*

_{1}and

*B*

_{2}are the signal amplitudes.

Using the hardware method, the analog signals of *I _{D}*

_{4}and

*I*

_{D}_{5}can be converted into digital signals by an analog-to-digital converter (ADC) and then processed by a field-programmable gate array (FPGA) to obtain the phase of the interference signal

*ϕ*. With the maximum, minimum and average values of

*I*

_{D}_{4}and

*I*

_{D}_{5}which are detected and updated periodically,

*I*

_{D}_{4}and

*I*

_{D}_{5}are blocked of DC to eliminate

*A*

_{1}and

*A*

_{2}, respectively, then are normalized to eliminate

*B*

_{1}and

*B*

_{2}, respectively. The signal processing of sin

*ϕ*and cos

*ϕ*includes integral fringe counting and fraction fringe counting. For integral fringe counting, sin

*ϕ*and cos

*ϕ*are converted into square waves by means of a trigger circuit and the number of sine square waves is counted by a bi-directional counter. For fractional fringe counting, the arctangent calculation is used to obtain the phase change corresponding to less than one integral fringe. The phase of the interference signal

*ϕ*can be calculated with the combination of integral and fractional fringe counting. Then the OPD of measuring light

*Ĵ*

_{1}can be expressed as

*N*and

*ε*are the integral and fractional number of the interference fringe, respectively.

As shown in Fig. 3(b), assuming that the distance of RR to WP is *s*_{0} at the initial position. Then RR moves a distance of *s* to the current position which has a straightness error of Δ*h*. According to the geometric relationship, the straightness error Δ*h* and its relative position *s* with respect to the initial position can be expressed as

*θ*is a half of the divergent angle of WP. Δ

*L*and

*L*

_{1}can be obtained from Eqs. (11) and (13), respectively.

#### 3.3 Determination of the horizontal straightness and roll errors

The schematic of horizontal straightness and roll errors measurement with laser collimation is shown in Fig. 4. As the measuring beams for the horizontal straightness and roll errors, the two orthogonal linearly polarized beams from WP are reflected by RR and recombined at WP. The recombined beam is reflected by NPBS_{6} and split into two orthogonal linearly polarized beams by PBS_{4} which are received by QD_{1} and QD_{2}, respectively.

As the moving stage moves along the *z* direction, the horizontal straightness error and the roll error of the stage will result in a shift of RR in the *y* direction and a rotation of RR around the *z* direction, respectively. The distance of the shift is the horizontal straightness error (*w*) as shown in Fig. 4(a), and the angle of rotation is the roll error (*γ*) as shown in Fig. 4(b). In Fig. 4(c), with the horizontal straightness and roll errors, the emergent points of upper and down right-angle prisms of RR change from ${P}_{uo}$ and ${P}_{do}$ to ${{P}^{\prime}}_{uo}$ and ${{P}^{\prime}}_{do}$, respectively. As a result, the positions of the laser spot at QDs will change, and the changes in the *x* direction of the QDs can be expressed as

*y*coordinate values of ${P}_{uo}$ and ${P}_{do}$, and ${{y}^{\prime}}_{uo}$ and ${{y}^{\prime}}_{do}$ are

*y*coordinate values of ${{P}^{\prime}}_{uo}$ and ${{P}^{\prime}}_{do}$. According to the geometric relationship, the roll error

*γ*can be expressed as

*x*coordinate values of ${P}_{uo}$ and ${P}_{do}$,

*B*denotes the distance between the axis of RR’s bracket and the join point of the right-angle sides of the upper and down right-angle prisms as shown in Fig. 3(b), and

*W*denotes the width of the hypotenuse of rectangular prism, which is shown in Fig. 4(c).

And combining with *γ* in Eq. (18), the horizontal straightness error *w* can be expressed as

*H*denotes the distance between the join point and the surface of the moving stage.

#### 3.4 Crosstalk effects compensation

All measurements of six degrees of freedom motion errors are influenced by crosstalk effects with respect to each other. The vertical straightness error and its position are coupled to pitch error because the OPDs corresponding to the two measuring beams change as the pitching of the moving stage. And the horizontal straightness and roll errors are coupled to the yaw error because the positions of the laser spot in the *x* direction of QDs change as the yawing of the moving stage.

In our previous work [21], the functional relationship between the rotational errors and the OPDs corresponding to two measuring beams is presented along with the mapping relationship between the light spot positions change on two QDs and the rotational errors of RR. However, the changes of optical path of the beams inside WP resulting from the rotational errors are neglected in the functional relationship. In fact, the variations inside WP should be considered and introduced into the analytical model. The schematic of the influence of the pitch error on the OPDs corresponding to the two measuring beams is shown in Fig. 5. *M*_{1} and *M*_{2} are the contour planes which are determined by the manufacturing parameters of WP. The optical paths between the contour plane and the emergent points on WP are equivalent to the optical path inside WP corresponding to measuring beams. Then $\overline{AC}$ and $\overline{BE}$ serve as the optical paths including the part inside WP corresponding to two measuring beams, respectively. When the moving stage produces pitch error, the round-trip optical path of *p*-polarized beam changes to $\overline{A{C}^{\prime}}$ and $\overline{{A}^{\prime}D}$, and the round-trip optical path of *s*-polarized beam changes to $\overline{B{E}^{\prime}}$ and $\overline{{B}^{\prime}F}$. Thus the compensated OPD *L*_{1} and the difference Δ*L* between the OPDs corresponding to the two measuring beams can be expressed as

*H*

_{1}and

*H*

_{2}denote the distances of two incident light spots to the intersected line of the oblique planes of the upper and down right-angle prisms, respectively. ${{L}^{\prime}}_{1}$ and Δ

*L'*denote the measured values with rotational errors, which are acquired by processing the interference signals.

Combining Eqs. (20) and (21), the compensated vertical straightness error and its position can be expressed as

*s'*and Δ

*h'*denote the measured values before compensation. Equations (22) and (23) are also the amendment of Eqs. (36) and (37) in our previous work [21].

On the other hand, according to the mapping relationship between the light spot position changes on two QDs and the rotational errors of RR, the compensated laser spot position changes $\Delta {x}_{QD1}$ and $\Delta {x}_{QD2}$ on the *x* direction of QDs can be expressed as

From Eqs. (24) and (25), it is found that the yaw error has opposite influences on $\Delta {x}_{QD1}$ and $\Delta {x}_{QD2}$. Substituting Eqs. (24) and (25) into Eq. (18), thus the roll error *γ* is unchanged. Substituting Eqs. (24) and (25) into Eq. (19), the horizontal error is compensated and can be recalculated by

*n*denotes the refractive index of the air.

## 4. Experiments and results

In order to verify the feasibility of the proposed laser homodyne straightness interferometer for simultaneously measuring six degrees of freedom motion errors of precision linear stage, an experimental setup was constructed as shown in Fig. 6. The laser source is a single-frequency stabilized He-Ne laser (XL80, Renishaw) which emits a laser beam with the wavelength of λ = 632.990577 nm. RR and WP with a divergent angle of 1.5° are a short range straightness measurement kit (A-8003-0443, Renishaw). The CCD camera (VLG-20M, Baumer) employs a Sony ICX274 sensor with the resolution of 1624 × 1228 px and the pixel size of 4.4 × 4.4 µm. Two QDs are two quadrant photodetectors (Spoton u-type, Duma Optronics) with the position accuracy of ± 1.5 μm. The measured stage is a precision linear stage (M-521.DD, Physik Instrumente) with the travel range of 200 mm, the straightness per 100 mm of 1 μm, the displacement resolution of 0.1 μm and the pitch/yaw of ± 50 μrad ( ± 10.31arcsec). And a comparison interferometer is a calibration laser system (XL80, Renishaw) with the straightness resolution of 0.01μm while the accuracy is ± 0.005A ± 0.5 ± 0.15M2μm, the angular measurement resolution of 0.1μm/m while the accuracy is ± 0.002A ± 0.5 ± 0.1M μm/m and the linear measurement resolution of 0.001μm while the accuracy is ± 0.5 ppm, where A is the straightness measuring data and M is the measuring distance. The comparison of the measurement results of the proposed interferometer with the Renishaw interferometer for each degree of freedom motion error was presented.

#### 4.1 Measurement experiment of yaw and pitch errors

In this experiment, the laser beam was adjusted to project on the center of CCD at one end of the M-521.DD stage which is taken as the initial position. RR and the measuring angle reflector of the Renishaw interferometer were mounted on the moving component of the stage. During the experiment, the component moved to the other end of the stage with the step increment of 2 mm and the velocity of 1 mm/s, the yaw and the pitch errors of the stage were measured simultaneously by the proposed interferometer and the Renishaw interferometer, respectively. The experimental results are shown in Fig. 7. It can be seen that the results with the proposed interferometer are consistent with those obtained from the Renishaw interferometer. The deviations are the differences between the measurement results with the proposed interferometer and the Renishaw interferometer. For the yaw errors, the maximum deviation is 0.937 arcsec with a standard deviation of 0.412 arcsec. For the pitch errors, the maximum deviation is 0.976 arcsec and the standard deviation is 0.353 arcsec.

#### 4.2 Measurement experiment of roll error

In this experiment, an electronic level (WL11, AVIC Qianshao Precision Machinery) with the resolution of 0.2 arcsec was used for comparison. The laser beam was adjusted to project on the centers of two QDs at the initial position. RR and the WL11 level were mounted on the moving component of the stage. The roll error of the stage was determined simultaneously by the proposed interferometer and the WL11 level. The experimental results are shown in Fig. 8. It shows that the results with the proposed interferometer and the WL11 level are in basic agreement. The maximum deviation of roll errors between the proposed interferometer and the WL11 level is 2.868 arcsec while the standard deviation is 0.839 arcsec.

#### 4.3 Measurement experiment of horizontal straightness error

In this experiment, the laser beam was adjusted to project on the center of CCD and QDs at the initial position, RR and the Wollaston prism of the Renishaw interferometer were mounted on the moving component of the stage. The horizontal straightness error can be obtained by using Eq. (23) with the measurement results of the yaw and roll errors which were measured simultaneously by the proposed interferometer. And the Renishaw straightness interferometer also measured the horizontal straightness errors of the stage simultaneously. The experimental results after the least square fitting are shown in Fig. 9. It shows that the results with the proposed interferometer are basically consistent with those obtained from the Renishaw interferometer. The horizontal straightness obtained with the proposed interferometer is 3.940 μm while that obtained with the Renishaw interferometer is 4.116 μm. The maximum deviation of the horizontal straightness errors is 2.542 μm with a standard deviation of 0.886 μm.

#### 4.4 Measurement experiment of vertical straightness error and its position

In this experiment, RR of the proposed interferometer and the Wollaston prism of the Renishaw interferometer were mounted on the moving component of the stage. The vertical straightness error, pitch error and the corresponding position of these errors were measured simultaneously by the proposed interferometer, while the Renishaw straightness interferometer measured the vertical straightness error as comparison. The experimental results of the vertical straightness error after the least square fitting are shown in Fig. 10. The vertical straightness obtained with the proposed interferometer before compensation is 10.663 μm while that obtained with the Renishaw interferometer is 2.187 μm, and the maximum deviation of vertical straightness error is 5.689 μm with a standard deviation of 1.773 μm. According to Eq. (23), the vertical straightness is 2.017 μm and the maximum deviation is 1.271 μm while the standard deviation is 0.392 μm after compensation. Compared with those before compensation, the experimental results of the proposed interferometer after compensation are more consistent with the results obtained from the Renishaw interferometer. This demonstrates the effectiveness of the proposed interferometer and compensation method for vertical straightness error measurement.

In the meantime, the displacement comparison experimental results are shown in Fig. 11. It shows that the results with the proposed interferometer are in good agreement with those provided by the stage itself. The maximum deviation of displacements obtained by the proposed interferometer before compensation and those provided by stage is 1.897 μm while the standard deviation is 0.564 μm. According to Eq. (22), the maximum deviation of displacements is 0.795 μm while the standard deviation is 0.328 μm after compensation. This demonstrates the effectiveness of the proposed interferometer and compensation method for displacement measurement.

#### 4.5 Simultaneous measurement and repeatability experiments

Using the proposed interferometer, the simultaneous measurement of six degrees of freedom error parameters of the M-521.DD stage was performed, and to verify the repeatability of the proposed interferometer, this experiment was done repeatedly for three times. The experimental results are shown in Fig. 12 and summarized in Table 1. Figures 12(a)-12(f) and Table 1 show a good repeatability in simultaneously measuring the yaw, pitch, roll, two straightness errors and straightness error’s position of the stage with the proposed interferometer.

## 5. Discussion of the measurement range and resolution

The measurement range and resolution of the proposed interferometer are determined by the measurement method and optical setup. For the measurement of the yaw and pitch errors, the image processing technique is employed to detect the spacing of interference fringe. The image of homodyne interference fringe patterns captured by CCD is shown in Fig. 13(a). A greyscale threshold is applied to the image for eliminating ambient light effect, and the feature pixels of fringes are retained as shown in Fig. 13(b). A group of optimal fringes of the feature pixels is picked up and fitted into lines by the least square method as shown in Fig. 13(c). With the fitted lines, the spacing of interference fringe can be obtained with the average of all the distances between adjacent lines. According to the CCD resolution of 1624 × 1228 px, the CCD pixel size of 4.4 × 4.4 µm and the laser beam diameter of 6 mm, the maximum spacing of two adjacent fringes is 2701.6 µm (614 px) with the minimum spacing of 4.4 µm (1 px). However, too large or too small spacing will lead to poor fitting results, and affect the accuracy of the spacing detection. After the experimental test, the appropriate parameters are set for the measurement of the yaw and pitch errors. At the initial position, the spacing of interference fringe in the horizontal and vertical directions are adjusted to about 120 px, respectively. The number of the fitted lines is set to six. Based on the mentioned above, the measurement ranges of the yaw error and the pitch error are 200 arcsec with the measurement resolutions of 0.069 arcsec.

The measurement ranges of the vertical straightness error and its position are determined by the short range straightness measurement kit (A-8003-0443, Renishaw) including RR and WP with a divergent angle of 1.5°, and the measurement resolutions are determined by the electronic subdivision of the signal processing with the FPGA. According to the geometric relationship between the position of laser spot on RR and the divergent angle of WP, the measurement range of the straightness error is 2.5 mm and the measurement range of straightness error’s position is 0.3-4.0 m. Based on the performance of five photodetectors and the Cyclone IV FPGA with 8-channel 16-bit ADC, the measurement resolutions of the straightness error and its position are 0.01 µm.

The measurement ranges of the horizontal straightness and roll errors are limited by a combination of laser beam diameter, QD sensor size and RR size. Based on the beam diameter of 6 mm, QD dimensions of 10 × 10 mm and RR hypotenuse width of 40 mm, the measurement range of the horizontal straightness error is 4 mm with the measurement resolution of 0.75 µm, and the measurement range of the roll error is 100 arcsec with the measurement resolution of 2.94 arcsec. However, the measurement accuracy will deteriorate as the laser spot leaves the center of QDs. In future studies, PSDs with higher resolution will be used instead of QDs to improve measurement accuracy.

## 6. Conclusion

A laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors for precision linear stage metrology is presented. The optical configuration and the principle of the simultaneous measurement of six degrees of freedom motion errors are depicted and analyzed in detail. The yaw and pitch errors are obtained by the two-dimensional small angle measurement method based on homodyne interference fringe pattern analysis. The vertical straightness error and its position are obtained by homodyne interferometry. And the horizontal straightness error and roll error are determined by using laser collimation. The compensation method of crosstalk effects on the straightness error and its position measurements is presented. Finally, a series of comparison experiments and repeatability experiments demonstrated the effectiveness of the proposed interferometer and the compensation method, and the measurement range and resolution of the proposed interferometer are discussed. The experimental results and the discussion results show that the proposed interferometer can be used for precision linear stage metrology.

## Funding

National Natural Science Foundation of China (NSFC) (51375461, 51527807 and 51475435); Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (IRT13097).

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