## Abstract

A novel method for simultaneously measuring six degree-of-freedom (6DOF) geometric motion errors is proposed in this paper, and the corresponding measurement instrument is developed. Simultaneous measurement of 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser is accomplished for the first time to the best of the authors’ knowledge. Dual-frequency laser beams that are orthogonally linear polarized were adopted as the measuring datum. Positioning error measurement was achieved by heterodyne interferometry, and other 5DOF geometric motion errors were obtained by fiber collimation measurement. A series of experiments was performed to verify the effectiveness of the developed instrument. The experimental results showed that the stability and accuracy of the positioning error measurement are 31.1 nm and 0.5 μm, respectively. For the straightness error measurements, the stability and resolution are 60 and 40 nm, respectively, and the maximum deviation of repeatability is ± 0.15 μm in the x direction and ± 0.1 μm in the y direction. For pitch and yaw measurements, the stabilities are 0.03″ and 0.04″, the maximum deviations of repeatability are ± 0.18″ and ± 0.24″, and the accuracies are 0.4″ and 0.35″, respectively. The stability and resolution of roll measurement are 0.29″ and 0.2″, respectively, and the accuracy is 0.6″.

© 2016 Optical Society of America

## 1. Introduction

A linear guide is the key component in ultra-precision machining and measuring equipment such as computerized numerical control machines and coordinate measuring machines (CMMs). However, owing to deviations in manufacturing and assembly, the linear guide inherently possesses six degree-of-freedom (6DOF) geometric motion errors, including three translational errors (positioning, horizontal straightness, and vertical straightness errors) and three angular errors (pitch, yaw, and roll).

High-precision measurement of 6DOF geometric motion errors is a prerequisite for performance evaluation and error compensation of the ultraprecision equipment mentioned above. The conventional method for measuring the 6DOF geometric motion errors of a linear guide employs a laser interferometer, which is a single-parameter measurement device. In general, it takes a few days or even a week to measure the 21 geometric errors of a typical three-axis machining center in this manner [1, 2]. The measurement accuracy cannot be ensured because of changes in the ambient environment during the time-consuming measurement process. Therefore, the primary need and development trend in this field is to develop a system that can simultaneously measure 6DOF geometric motion errors to significantly improve the measurement efficiency and accuracy.

An early work by Shimizu et al. provided a method for simultaneously measuring the 6DOF geometric motion errors of machine tools [3]. The developed set-up is large and has cable connections in the moving unit, which is not convenient for *in situ* measurements. Ni et al. proposed a measurement method for simultaneously measuring 6DOF geometric motion errors of CMMs [4,5]. The method provided an accuracy of 1 μm for straightness error measurement and 0.5″ for angular error measurement, but the optical configuration was very complicated. Liu et al. achieved 6DOF geometric error measurement by splitting the beam of a fiber-coupled laser interferometer into three parallel beams, which were reflected back by the corresponding error sensing elements [6]. The method provided an accuracy of 0.6 μm for straightness error measurement and 0.3″ for angular error measurement in the range of 120 mm. Feng et al. developed a simple system for simultaneously measuring 6DOF geometric motion errors [7]. The common-path compensation method is integrated into the system, and the beam angular drift can be compensated in real time to improve the measurement accuracy. All the above-mentioned methods focused on a combination of laser interferometry and laser collimation. Fan et al. integrated three laser Doppler displacement meters and two quadrant detectors to measure 6DOF geometric motion errors [8,9]. The system configuration is complex and the total cost is high. The measurement method based on the laser grating interferometer is another important approach [10–13]. It is generally used for measuring medium-sized and small machines, as it is difficult to manufacture large gratings for practical applications. Thus far, the measurement system from Automated Precision Inc. is the only commercially available instrument for 6DOF geometric error measurement [14]. In this system, however, the roll is measured by an electronic level, which cannot be used for vertical axis measurement.

In summary, measurement methods based on a combination of laser interferometry and laser collimation are widely used in related fields because they have the advantages of a large measurement range, high accuracy, and high integration. However, they still have several shortcomings, which can be summarized as follows. First, the measurement unit is large because a He-Ne laser source is usually placed directly inside it. Second, the measurement accuracy is heavily influenced by the beam drift and thermal-mechanical interaction, which is caused by the heat generated by the laser head. Third, the homodyne-interferometry-based positioning error measurement is sensitive to changes in the ambient environment and light intensity.

To overcome these limitations, a novel method for simultaneously measuring 6DOF geometric motion errors is proposed, and the corresponding measurement instrument is developed in this study. The measurement method is based on a combination of heterodyne interferometry and laser fiber collimation. Simultaneous measurement of 6DOF geometric motion errors using a polarization maintaining fiber (PMF)-coupled dual-frequency laser is accomplished for the first time to the best of the authors’ knowledge. It has the following advantages. First, the heat generated by the He-Ne laser source is removed from the measurement unit by a fiber connection, improving the thermal stability of the measurement instrument. Second, a high-quality laser beam can be obtained through fiber collimation, and can be used as the measuring datum for collimation measurements. At the same time, the influence of changes in the ambient environment on the dual-frequency orthogonally polarized measuring beams is inhibited in common mode, and the measurement accuracy can be enhanced as well. Finally, the heterodyne-interferometry-based positioning error measurement of the proposed system is insensitive to changes in light intensity.

## 2. Measurement system and developed instrument

Figure 1 shows the schematic diagram of the proposed system, which consists of a fiber coupling and transmitting unit, a measurement unit, and a moving unit.

The fiber coupling and transmitting unit consists of dual-frequency He-Ne laser, quarter-wave plate (QWP1), beam splitter (BS1), polarizer (P1), photodetector (D1), coupling lens (C-lens), and PMF. The output beams from the dual-frequency laser are right and left circularly polarized beams with a constant frequency difference and are transformed into two orthogonal linear polarized beams by QWP1. The two orthogonal linear polarized beams, which are the measuring datum for the 6DOF geometric motion errors, are transmitted by the PMF to maintain the stability of the frequency difference and polarization state.

The two orthogonal linear polarized beams are directed at the measurement unit by the PMF and are split by a polarization beam splitter (PBS1). The transmitted p-polarized light is used as the measuring beam for measurement of the straightness error, positioning error, and beam drift. The s-polarized light, which is totally reflected, reaches a beam splitter (BS2). The light that is transmitted through BS2 is the reference beam for positioning error measurement, and the beam reflected by BS2 serves as the measuring beam for measurement of the straightness error, pitch, and yaw. The photodetectors D2, QD1, QD2, QD3, and QD4 in the measurement unit receive the measurement beams and perform photoelectric conversions. The measurement principle for each geometric error is described in detail in the following section.

The moving unit, which moves along a linear guide, consists of two cube corner reflectors (RR2 and RR3) and a beam splitter (BS5). The moving unit has a small volume and non-cable connections, which are convenient for in situ measurements.

The corresponding measurement instrument was developed, and the fiber coupling and transmitting unit can be placed at any convenient position in practical applications. As the measurement unit and the machine tool are fixed on the identical benchmark during the measurement process, the influence of the ground vibration on measurement accuracy can be ignored. The flexible connection ensured by PMF transmission removes the laser source from the measurement unit. In this manner, the volume of the measurement unit can be reduced. In addition, the influence of the heat generated by the laser source on the measurement accuracy can be eliminated.

## 3. Measurement principle for single geometric error

#### 3.1 Positioning error measurement

The schematic diagram of positioning error measurement based on heterodyne interferometry is shown in Fig. 2. The dual-frequency laser generated by the He-Ne laser source is transmitted by a single PMF, and heterodyne interference can be realized; this method differs from that of the conventional heterodyne interferometer [15].

The He-Ne laser generates right and left circularly polarized beams with the same amplitude and a constant frequency difference by the Zeeman effect. After passing through QWP1, the two circularly polarized beams are transformed into two orthogonal linear polarized beams, which can be expressed as

where *E _{0}* denotes the amplitude,

*f*and

_{1}*f*denote the frequencies of the two beams,

_{2}*t*denotes the time, and

*ϕ*and

_{1}*ϕ*denote the initial phases of the two beams.

_{2}To avoid polarization mixing during fiber transmission, the polarization directions of ** E_{1}** and

**are aligned with the optical axes of the PMF by rotating QWP1. The light transmitted through QWP1 is split by BS1 and the reflected light passes through P1, which is at an angle of 45° with respect to the polarization directions of**

*E*_{2}**and**

*E*_{1}**. Subsequently,**

*E*_{2}**and**

*E*_{1}**reach D1 and generate an interference signal that can be expressed as**

*E*_{2}The AC signal with a frequency difference of *f _{1}−f_{2}* is extracted by a band pass filter to serve as the reference signal for positioning error measurement. The reference signal can be expressed as

The light transmitted through BS1 is coupled into the PMF by C-lens, and the light emerging from the PMF is collimated by a collimator. To avoid polarization mixing in PBS1, the polarization directions of ** E_{1}** and

**are aligned with PBS1 by rotating a half-wave plate (HWP1).**

*E*_{2}**, which is totally reflected by PBS1, is used as the reference beam, and**

*E*_{1}**, which is totally transmitted, serves as the measuring beam. The reference beam gains a constant phase shift**

*E*_{2}*ϕ*after traveling the reference arm and is denoted by

_{ref}**. The measuring beam is denoted by**

*E*_{1}’**after traveling the measurement arm and gains a phase shift**

*E*_{2}’*ϕ*, which is proportional to the displacement of RR2 in the moving unit. Subsequently,

_{meas}**and**

*E*_{1}’**are recombined in PBS1 and pass through polarizer P2, which is at 45° with respect to the polarization direction of**

*E*_{2}’**and**

*E*_{1}’**. An interference signal is generated on D2 and can be expressed as**

*E*_{2}’Similarly, the AC signal with a frequency difference of *f _{1}−f_{2}* is extracted by the band pass filter to serve as the measuring signal. The measuring signal can be expressed as

where *Δϕ* denotes the phase difference between the reference beam and the measuring beam, and can be calculated by combining *I _{r}* in Eq. (3) and

*I*in Eq. (5). The relationship between

_{m}*Δϕ*and

*L*, which is the displacement of RR2, can be expressed as

Here, *n* is the refractive index of the medium and *λ* is the laser wavelength in vacuum. *ΔZ*, the positioning error of the linear guide, can be obtained as

Here, *L′* is the nominal displacement of the driven linear guide.

The proposed positioning error measurement method retains the advantages of heterodyne interferometry, such as high accuracy, high signal-to-noise ratio, and strong immunity to changes in the ambient environment. Simultaneously, the laser head moves out of the interferometry path without losing optical connections because of the flexible connections. In this manner, the influence of the heat generated by the laser head on the measurement accuracy can be eliminated.

#### 3.2 Straightness error measurement

Straightness error measurement is based on laser collimation, as shown in the schematic diagram in Fig. 3 [16]. RR2 and RR3 in the moving unit are sensitive to the straightness error. The transmitted light from PBS1, which is the measuring beam for the straightness error, is reflected by RR2 and BS3 and is received by QD1. The light reflected by PBS1 is then reflected by BS2 and is used as the measuring beam for the straightness error. The light reflected by BS2 is further reflected by RR3 and received by QD2.

When the moving unit moves along the linear guide in the *z* direction, the straightness error of the linear guide causes translation of RR2 and RR3 in the *x* and *y* directions, respectively. The position of the laser spot on QD1 and QD2 changes correspondingly, and this change can be expressed as

where *ΔX* and *ΔY* are the straightness errors of the linear guide in the *x* and *y* directions, respectively; *ΔX _{QD1}* and

*ΔX*denote the position change of the spot on QD1 and QD2 in the

_{QD2}*x-*direction, respectively; and

*ΔY*and

_{QD1}*ΔY*denote the position change of the spot on QD1 and QD2 in the

_{QD2}*y-*direction, respectively.

Typically, the straightness error measurement method employs only one cube corner reflector as the error-sensing element, and an error compensation model is required to compensate for the systematic error caused by error crosstalk [17]. However, in the developed measurement system, this type of systematic error can be corrected automatically by adding the measurement results of QD1 and QD2 because the yaw and the roll have opposite effects on the positions of RR2 and RR3. Therefore, Eq. (8) can be rewritten as

#### 3.3 Pitch and yaw measurement

The schematic of pitch and yaw measurement, which is based on auto collimation, is shown in Fig. 4 [18]. BS5 in the moving unit is used as the error-sensing element for pitch and yaw measurement. The reflected light from BS2, which is the measuring beam for the pitch and yaw, is totally transmitted through the polarization beam splitter PBS2 by rotating the half wave plate HWP2. The light transmitted by PBS2 is p-polarized and is transformed into circularly polarized light after passing through the quarter wave plate QWP2, whose fast axis is set at 45° with respect to the vertical plane. The light reflected by BS5 is transformed into s-polarized light after passing through QWP2 and is totally reflected by PBS2. Subsequently, the light reflected from mirror M is focused by lens L1 onto the photodetector QD3, which is positioned at the focal plane of L1.

When BS5 moves along the linear guide, the angular change in the light reflected by BS5 is twice the angular error of the linear guide. The relationship between the angular error and the position change of the spot on QD3 can be expressed as

where *α* and *β* denote the pitch and yaw of the linear guide, respectively; *ΔX _{QD3}* and

*ΔY*denote the position change of the spot on QD3 in the

_{QD3}*x*and

*y*directions,respectively; and

*f*denotes the focal length of L1. Theoretically, other 4DOF geometric motion errors have no crosstalk in the measurement of the pitch and yaw because the normal direction of BS5 is not sensitive to the positioning error, straightness error, and roll.

_{1}#### 3.4 Roll measurement

A roll measurement method using a rhombic prism was proposed in an early work [7]. In the proposed system, the light transmitted from PBS1 and the light reflected by BS2 are parallel to each other and serve as the measuring datum for roll measurement. The combination of RR2 and RR3 is adopted as the error-sensing component instead of the rhombic prism to reduce the volume of the moving unit.

O2 and O3 in Fig. 5 are the centers of the incident planes of RR2 and RR3, respectively; P1 and P2 are the incident points of the measuring beam on RR2 and RR3, respectively; and Q1 and Q2 are the emergent points of the reflected light. Q1 and Q2 are in the same position in the *y* direction in the absence of roll. If any roll error occurs, the positions of Q1 and Q2 change, and the roll can be obtained as

Here, *h* is the distance between the measuring beams, *γ* is the roll, and *ΔY _{1}* and

*ΔY*are the position changes of O2 and O3 in the

_{2}*y*direction, respectively. The proposed roll measurement method has a simple configuration and can be easily integrated into the 6DOF error simultaneous measurement system. The systematic error induced by the parallelism error of the two measuring beams can be eliminated by the error compensation model to further improve the measurement accuracy [19].

#### 3.5 Common-path compensation for laser beam drift

A common-path compensation method is integrated into the proposed system to eliminate the influence of laser beam drift on the measurement accuracy [7].

The schematic diagram of straightness error measurement with common-path compensation is given in Fig. 6. The beam angular drift can be obtained as

where *Δα* and *Δβ* denote the beam angular drift in the *y* and *x* directions, respectively; *ΔX _{QD4}* and

*ΔY*denote the position change of the spot on QD4 in the

_{QD4}*x*and

*y*directions; and

*f*denotes the focal length of L2.

_{2}The modified equation for straightness error measurement can be expressed as

Here, *l* is the moving distance of the kinetic pair, which can be measured by the heterodyne interferometer integrated into the system.

Similarly, the equation for calculating the pitch and yaw can be modified as

According to Eqs. (13) and (14), the measurement errors caused by beam angular drift which results mainly from variation of ambient condition and other factors can be compensated.

## 4. Experimental results and analysis

A series of experiments were performed to evaluate the reliability and effectiveness of the proposed system.

#### 4.1 Stability experiments

Stability experiments were performed under laboratory conditions and the variation range of the air temperature is about ± 1°C. The moving unit was placed 400 mm away from the measurement unit and the experimental results were automatically recorded. The beam angular drift was compensated in real time by common-path compensation. The experimental results are shown in Fig. 7.

In Fig. 7, the standard deviation for the positioning error and straightness error measurements are 31.1 and 60 nm, respectively, and the standard deviation for the pitch, yaw, and roll measurements are 0.03″, 0.04″, and 0.29″, respectively.

Compared with the experimental results of our previous work [7, 16–19], the stability error of the proposed system is reduced by nearly one order of magnitude, indicating advatanges of combination of heterodyne interferometry and laser fiber collimation, which include the high thermal stability, the high-quality laser beam used as the measuring datum for collimation measurements, the common-mode rejection of the ambient condition changes, and the insensitivity to light intensity variations.

#### 4.2 Resolution of straightness and angular error measurement

A piezo nanopositioner (PI, P-611.1, resolution 2 nm, repeatability <10 nm) was used to check the resolution of straightness error measurement of the proposed system. The moving unit of the proposed system was fixed on the translational stage of the piezo nanopositioner. The translational stage was driven to move from 0 to 200 nm and back to the starting point at intervals of 40 nm; the measurement results of the straightness error were recorded simultaneously. As shown in Fig. 8, the measurement results responded quite well to the 40 nm step. Therefore, the resolution of straightness error measurement of the proposed system is estimated to be approximately 40 nm. Theoretically, it is possible to improve the measurement resolution by reducing the beam radius [17] or by using multi-reflections and lens combination [7]. However, better environmental conditions are required in order to obtain higher resolution in practice.

The focal length of L1 in the proposed system is 200 mm. Therefore, the resolution of the pitch and yaw measurements is 0.02″, according to Eq. (10). For roll measurement, the distance between the measuring beams is 40 mm, and the roll measurement resolution is 0.2″, according to Eq. (11).

#### 4.3 Repeatability and comparison experiments

Repeatability and comparison experiments were performed using the developed instrument and certain standard measurement instruments to simultaneously measure the same air-bearing linear guide thrice. The practical application of the developed instrument is shown in Fig. 9.

In the proposed system, the quadrant detector possesses good linearity in the range of ± 100μm. Therefore, the measurement range of the straightness errors in both the x and y directions is ± 100μm. The focal length of L1 in the proposed system is 200mm, thus the measurement range of the pitch and yaw are both ± 100˝, according to Eq. (10). For roll measurement, the distance between the measuring beams is 40mm, thus the measurement range is ± 500˝ according to Eq. (11). The measurement range of positioning error measurement depends on the polarization and collimation characteristics of the laser source, and it is not less than 5m for the proposed system.

Positioning error and straightness error measurement of the proposed system were compared with a laser interferometer (SIOS, resolution 1nm) and a 5D measurement system (Automated Precision Inc, resolution 0.1 μm, accuracy 0.5 μm), respectively. The standard measurement instrument for pitch and yaw comparison was the photoelectric autocollimator (Acrobeam, resolution 0.02″, accuracy 0.2″). The total moving distance of the kinetic pair was 400 mm, and the moving interval was 50 mm. The standard measurement instrument for roll comparison was also the photoelectric autocollimator, and the roll error can be measured by scanning the surface of a plane optical flat mounted on the kinetic pair [20]. Because of the limited length of the plane optical flat, the measurement range of the roll error was 250 mm and the step was 50 mm. The experimental results are shown in Fig. 10.

It can be seen from Fig. 10 that the deviation of the positioning error measurement for the developed instrument and the interferometer is within ± 0.5 μm. For straightness error measurement, the maximum deviation of the repeatability experiments is ± 0.15 μm in the x direction and ± 0.11 μm in the y direction. In particular, the proposed system possesses much better repeatability than the API 5D measurement system. For pitch and yaw measurement, the maximum deviations in the repeatability experiments are ± 0.18″ and ± 0.24″, respectively, and the deviations for the developed instrument and the photoelectric autocollimator are within ± 0.4″ and ± 0.35″, respectively. For roll measurement, the maximum deviation in the repeatability experiments is ± 0.35″, and the comparison deviation between the developed instrument and the photoelectric autocollimator is within ± 0.6″.

Comparing the experimental results in Fig. 10 in this paper with those in Fig. 13 in Ref [7], it can be seen that the repeatability error for straightness error measurement is reduced from ± 1.2 μm to ± 0.15 μm, and the repeatability error for pitch, yaw, and roll measurement are reduced from ± 1.0″, ± 0.8″, and ± 1.8″ to ± 0.18″, ± 0.24″, and ± 0.35″, respectively. Therefore, the measurement accuracy and thermal stability are obviously improved. In addition, the volume of the measurement unit is reduced as well, thus adding convenience in practical application.

According to ISO Standard 230 for machine tool tests, 6DOF geometric motion errors must be measured step by step. Compared with the laser interferometer, which can measure only one error component at a time, the proposed system can simultaneously measure six error components during one measurement process. Therefore, the measurement efficiency can be enhanced by six times. In addition, readjustment of different measurement apparatuses and attachments is no longer needed and the measurement efficiency can be further improved. According to the above-mentioned analysis and our practical experiments, the measurement efficiency can be enhanced by about ten times.

#### 4.4 Error analysis

The experimental results in Fig. 10 exhibit the consistent repeatability of the proposed system. The primary reasons for the repeatability error are beam bending caused by air turbulence or an irregular air index distribution and the random errors of the proposed system. In addition, it is hard to locate the moving unit at the same position in different measurement processes, which also contributes to the repeatability error.

For positioning error measurement, the main constraint for the comparison deviation between the proposed system and the laser interferometer is the nonlinear error that arises because of polarization mixing, which is caused mainly by the imperfection of the PMF and misalignment of the optical components. Polarization mixing can be effectively reduced by using a PMF with a higher extinction ratio and by carefully aligning the laser with the optical axes of the optical components. Beam drift and bending are also important factors in the comparison deviation.

For measurement of the other 5DOF geometric motion errors, the primary constraint on the comparison deviation is that the proposed system and the standard instruments can rarely measure the same point of the linear guide because their targets are fixed at different positions on the kinetic pair. This problem can be solved by using smaller targets and by careful assembly to avoid the Abbe offset. Fabrication and installation deviation of the optical elements and the error crosstalk also contribute to the comparison deviation; these deviations can be compensated by mathematic analysis and an error compensation model.

As just discussed above, the errors of the 6DOF measurement mainly include the random error, the fabrication and installation deviation and the error crosstalk, and they are very complex and cannot be given in detail in this paper due to the limitation of length. We will analyze these errors and establish a compensation model in future works.

## 5. Conclusion

A novel method for simultaneously measuring 6DOF geometric motion errors was proposed in this paper, and the corresponding measurement instrument was developed. The flexible connection ensured by a single PMF realized separation of the laser head and the measurement unit, and it satisfied the demand for miniaturization, high integration, and high thermal stability. Orthogonal linear polarized beams were adopted as the measuring datum. Simultaneous measurement of 6DOF geometric motion errors using a PMF-coupled dual-frequency laser was accomplished for the first time to the best of the authors’ knowledge. The effectiveness of the proposed system was verified through a series of experiments. Future works will focus on system optimization and accuracy improvement.

## Acknowledgments

The authors acknowledge support from the Chinese National Natural Science Foundation Major Pr ogram under Grant No.51527806 and the Key Program under Grant No. 51027006.

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