## Abstract

A laser straightness interferometer system with rotational error compensation and simultaneous measurement of six degrees of freedom error parameters is proposed. The optical configuration of the proposed system is designed and the mathematic model for simultaneously measuring six degrees of freedom parameters of the measured object including three rotational parameters of the yaw, pitch and roll errors and three linear parameters of the horizontal straightness error, vertical straightness error and straightness error’s position is established. To address the influence of the rotational errors produced by the measuring reflector in laser straightness interferometer, the compensation method of the straightness error and its position is presented. An experimental setup was constructed and a series of experiments including separate comparison measurement of every parameter, compensation of straightness error and its position and simultaneous measurement of six degrees of freedom parameters of a precision linear stage were performed to demonstrate the feasibility of the proposed system. Experimental results show that the measurement data of the multiple degrees of freedom parameters obtained from the proposed system are in accordance with those obtained from the compared instruments and the presented compensation method can achieve good effect in eliminating the influence of rotational errors on the measurement of straightness error and its position.

© 2015 Optical Society of America

## 1. Introduction

In recent times, with the development of precision manufacturing and measuring equipments such as machine tool and coordinate measuring machine, the demands of high accuracy for straightness measurement instruments have increased [1–4]. Current straightness measurement methods and techniques can be mainly divided into laser collimation and laser interference. Among the laser collimation, position sensitive detector (PSD) or quadrant detector (QD) is usually used as sensor to realize the measurement of straightness error. These methods have the advantages of low cost, simple compact and fast optical adjustment. For example, Jeremy J. Kroll *et al* designed a six degree of freedom optical sensor for dynamic measurement of linear axes that uses a position sensitive photodiode to measure straightness error and heterodyne interferometry to measure linear displacement accuracy [8]. J. Ni *et al* developed a multi-degree-of-freedom measuring system for CMM geometric errors based on the principles of laser alignment and autocollimator [9]. Wen-Yuh Jywe presented a multi-degree of freedoms measuring system and an error compensation technique for machine tools integrating a laser interferometer for measuring displacement and QD for detecting straightness error [10]. Compared with the laser collimation [2, 5–10] and other straightness measurement methods such as diffracted beams interference method using grating [11, 12] and polarization angle detecting method [13], laser interferometric straightness measurement has advantages of high accuracy of nanometer level and long travel of meter range [14–20].

Among laser interferometric straightness measurement, one is using conventional laser interferometer to realize the determination of straightness such as Ping Yang *et al* or Kuang-Chao Fan *et al* used multiple laser interferometers to measure the straightness error of linear stage [3, 14, 15], and the other is called as laser straightness interferometer that specially designed for measuring straightness error. The typical representative of the straightness interferometer, which was invented by Richard R. Baldwin in 1974, is the interferometer system for measuring straightness that employs a Wollaston prism to produce two separate measuring beams and a pair of V-shaped plane mirrors to reflect the two beams [16]. After that, some related research works has been reported. For example, David R. McMurtry and Raymond J. Chaney invented a straightness interferometer system that used a roof-top reflector-prism combination as the retroreflector instead of the pair of plane mirrors in Baldwin’s system [17], Shyh-Tsong Lin proposed a straightness interferometer that utilizes the combination of a corner cube and two right-angle prisms to reflect the measuring beams so that the resolution is doubled by comparing with conventional straightness interferometer [18], Qianghua Chen *et al* presented a straightness/coaxiality measurement system that combines a transverse Zeeman dual-frequency laser with a pair of Wollaston prisms [19], and Benyong Chen *et al* proposed a straightness interferometer that uses a combination of a nonpolarizing beam-splitter and a polarizing beam-splitter to obtain two separate measurement signals so as to realize the simultaneous measurement of the magnitude and the position of straightness error [20]. Shyh-Tsong 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 [21].

Each of these works mentioned above has its own advantages. But, there are still the following problems to be studied in this straightness interferometer: (1) the influence of the rotational error of measuring reflector on straightness error measurement: when measuring reflector is rotated angular errors by the measured object, an erroneous optical path change will be induced and then result in wrong measurement result. (2) the simultaneous measurement of multiple degrees of freedom error parameters: most of these straightness interferometers only give out a straightness error in one direction at once measurement. If the straightness error in another perpendicular direction as well as the rotational errors (yaw, pitch and roll) can be simultaneously determined in one straightness interferometer system, this will have important significance in the field of performance testing and calibration of stages and guideways.

In this paper, a laser straightness interferometer system with simultaneous measurement of multiple degrees of freedom error parameters is proposed and the straightness error compensation is introduced. The optical layout of the laser straightness interferometer system is described. The principle of the simultaneous measurement of six degrees of freedom parameters including two straightness errors, position, yaw, pitch, and roll errors of a linear stage is depicted and analyzed in detail. And the compensation method for straightness error and its position due to the influence of the rotational errors of the measured object is presented. Finally, an experimental setup is constructed to demonstrate the feasibility of the laser straightness interferometer system.

## 2. Configuration

The laser straightness interferometer system with simultaneous measurement of six degrees of freedom error parameters is shown in Fig. 1. It consists of a laser interferometric measurement unit (LIMU) and a motion errors detecting unit (MEDU). LIMU, which is composed of a stabilized dual-frequency He-Ne laser, a nonpolarizing beam-splitter (NPBS_{3}), a Wollaston prism (WP), a retroreflector (RR) made up of upper and down right-angle prisms, a polarizing beam-splitter (PBS_{1}), two polarizers (P_{1}, P_{2}) and two photodetectors (D_{1}, D_{2}) [20], carries out the measurement of vertical straightness error along the *x* axis and the straightness error’s position (moving displacement along the *z* axis) based on heterodyne interferometry with two measurement signals from D_{1} and D_{2} and a reference signal from the laser. MEDU includes two parts. One, which is composed of two nonpolarizing beam-splitters (NPBS_{1}, NPBS_{2}), a mirror (MR) mounted on RR, a convex lens (CL) and a two-dimensional PSD, carries out the determination of yaw error around the *x* axis and pitch error around the *y* axis. Another, which is composed of a nonpolarizing beam-splitter (NPBS_{4}), a polarizing beam-splitter (PBS_{2}) and two QDs (QD_{1}, QD_{2}), carries out the determination of horizontal straightness error along the *y* axis and roll error around the *z* axis.

## 3. Principle

As shown in Fig. 2, when the moving stage slides along a linear guideway, there are six degrees of freedom motion parameters including three linear parameters of the vertical straightness error Δ*h*, the horizontal straightness error *w* and the straightness error’s position *s* and three rotational parameters of the yaw error *α* around the *x* axis, the pitch error *β* around the *y* axis and the roll error *γ* around the *z* axis. Among them, the rotational errors will affect the measurement accuracy of the straightness error and its position obtained from laser interferometry. Thus, in order to improve the measurement accuracy of the straightness error and its position, the three rotational errors must be detected to compensate the measurement result of the straightness error and its position.

In the proposed straightness interferometer system, two QDs and a PSD are employed to achieve the separation and measurement of three rotational errors and horizontal straightness error. The compensation of vertical straightness error and its position measured by using laser interferometry will be given in section 4.

#### 3.1 Determination of yaw and pitch errors

As shown in Fig. 1, the beam reflected by MR is sensitive to the yaw and pitch of the moving stage but not sensitive to the roll of the moving stage. The beam passes through BS_{2} and is focused on PSD which is situated at the focal plane of CL. According to the light spot deviations on PSD, the yaw and pitch errors of the moving stage can be expressed by

*f*is the focal length of CL, $\Delta {x}_{PSD}$ and $\Delta {y}_{PSD}$ are the light spot deviations on PSD along the horizontal and vertical directions, respectively.

#### 3.2 Determination of roll and horizontal straightness errors

In order to establish the mapping relationship between the light spot position changes on two QDs and the rotational errors of RR, three-dimensional ray tracing and geometric analysis methods are used. The coordinate definition of the analytical model of the proposed straightness interferometer system is shown in Fig. 3. The coordinate frames {R} and {M} are established respectively at the initial and moving positions of the moving stage. The origins o and o′ of the two coordinates are both defined at the mounting point that RR’s bracket mounts on the moving stage. It is assumed that the initial position has no motion errors and the coordinate frame {R} is static as a reference while the coordinate frame {M} changes with the movement of the moving stage. *θ* is a half of the divergent angle of WP. *L*_{1} and *L*_{2} represent the traveling optical paths of the two emergent beams with frequencies of *f*_{1} and *f*_{2} from WP during measurement, respectively. B is 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. *H* is the distance between the join point and the moving stage. *s*_{0} is the distance between WP and the axis of RR’s bracket at the initial position. Δ*h* is the vertical straightness error and *s* is the straightness error’s position.

As shown in Fig. 4, the two emergent beams with a divergent angle from WP project onto the upper and down right-angle prisms, respectively. When the moving stage moves to a location of the linear guideway and does not produce any rotational error, it is assumed that the incident points on the upper and down right-angle prisms are ${P}_{ui}$ and ${P}_{di}$ and the emergent points are ${P}_{uo}$ and ${P}_{do}$ in the coordinate frame {M}, respectively. According to the geometric relationship, the spatial coordinates of the two incident points are given by

*D*is the distance between the incident and emergent beams in the right-angle prism. And

*w*is the horizontal straightness error at the moving position.

The spatial coordinates of the two emergent points ${P}_{uo}$ and ${P}_{do}$ on the upper and down right-angle prisms are given by

When the moving stage moves to a location of the linear guideway and produces rotational errors *α*, *β* and *γ*, the incident points on the upper and down right-angle prism interfaces and the directions inside the prisms of the incident beams will change. In order to study the relationship of the variations of the incident points and beams with the rotational errors, two sub-coordinate frames {U} and {D} are established with respect to the coordinate frame {M} as shown in Fig. 5. M_{i} (i = 0-5) denotes the corresponding plane of the upper and down right-angle prisms.

The sub-coordinate frames {U} and {D} are the analysis coordinate system of the upper and down right-angle prisms, respectively. They share the same origin point which is the join point of the right-angle sides of the upper and down right-angle prisms. The *z* axes of {U} and {D} are the right-angle sides of the upper and down right-angle prisms. Consider the upper right-angle prism, the transformation matrix ${}^{U}T{}_{M}$ representing the transformation of {U} to {M} can be expressed as

Assuming the incident point after RR rotation is ${{P}^{\prime}}_{ui}$(${x}_{ui}^{U}$,${y}_{ui}^{U}$,${z}_{ui}^{U}$) in {U}, the spatial coordinates ${{P}^{\u2033}}_{ui}$(${x}_{ui}^{M}$,${y}_{ui}^{M}$,${z}_{ui}^{M}$) of this point in {M} can be expressed as

In {M}, the incident beam *I _{0}* does not change whether there is rotation error or not. So the incident point ${P}_{ui}$(${x}_{ui}$,${y}_{ui}$,${z}_{ui}$) before RR rotation and the incident point ${{P}^{\u2033}}_{ui}$(${x}_{ui}^{M}$,${y}_{ui}^{M}$,${z}_{ui}^{M}$) after RR rotation are both on the incident beam ${I}_{0}$. And the point ${{P}^{\prime}}_{ui}$(${x}_{ui}^{U}$,${y}_{ui}^{U}$,${z}_{ui}^{U}$) is also on the plane ${M}_{0}$. According to the equations of the incident beam ${I}_{0}$ and the plane ${M}_{0}$, the coordinate value of the incident point ${{P}^{\prime}}_{ui}$ in {U} can be derived as

*L*is the hypotenuse length of the right-angle prism.

In {U}, the direction vector of the incident beam ${I}_{ui}$ after RR rotation is

According to the refraction law, the direction vector of the beam ${I}_{1}$, which the incident beam ${I}_{ui}$ enters into the upper right-angle prism, can be gotten by

*n*is the refractive index of RR’s material and the refractive index of air is taken as 1.

The normal vectors to the right-angle reflection planes ${M}_{1}$ and ${M}_{2}$ of the upper right-angle prism are ${N}_{M1}={[\begin{array}{ccc}0& 1& 0\end{array}]}^{T}$ and ${N}_{M2}={[\begin{array}{ccc}1& 0& 0\end{array}]}^{T}$, respectively. The functionary matrix of the reflection plane can be given by [22]

Assuming that the beam ${I}_{1}$ incidents onto ${M}_{1}$ at the point ${P}_{1}$. After being reflected by ${M}_{1}$, the beam ${I}_{1}$ becomes the beam ${I}_{2}$. The beam ${I}_{2}$ incidents onto ${M}_{2}$ at the point ${P}_{2}$. And then after being reflected by ${M}_{2}$, the beam ${I}_{2}$ becomes the beam ${I}_{3}$. The beam ${I}_{3}$ incidents onto ${M}_{0}$ at the point ${{P}^{\prime}}_{uo}$(${x}_{uo}^{U}$,${y}_{uo}^{U}$,${z}_{uo}^{U}$). According to the refraction law, the direction vector of the emergent beam ${I}_{uo}$ can be gotten by

Using the transformation matrix ${}^{U}T{}_{M}$, the coordinate value of the emergent point ${{P}^{\u2033}}_{uo}$(${x}_{uo}^{M}$, ${y}_{uo}^{M}$, ${z}_{uo}^{M}$) in {M} can be gotten by

In {M}, the direction vector of the emergent beam ${I}_{3}^{R}$ is expressed as

Using similar analysis method for the down right-angle prism, the coordinate value of the emergent point ${{P}^{\u2033}}_{do}$ and the direction vector of the emergent beam ${I}_{7}^{R}$ with respect to the incident beam ${I}_{4}$ in {M} can be gotten by

By comparing Eq. (4) with Eq. (13) and Eq. (5) with Eq. (15) before and after RR rotation, respectively, the relationship between the coordinate changes of the emergent points on the upper and down right-angle prisms (URP and DRP) and the rotational errors can be expressed as

From Eq. (14) and Eq. (16), by comparing to the direction vectors ${[\theta ,0,1]}^{T}$ and ${[-\theta ,0,1]}^{T}$of the emergent beams with respect to the incident beams ${I}_{0}$ and ${I}_{4}$ before RR rotation, the emergent beams ${I}_{3}^{R}$ and ${I}_{7}^{R}$ only rotate the angle of $2\beta $ around the *y* axis after RR rotation. Thus, due to the angle changes of the emergent beams, when higher-order small quantities are neglected, the spot deviations on the interface of WP can be gotten by

When RR produces translational displacements without rotational errors at the measured position, the vertical straightness error and the straightness error’s position do not affect the spot deviation on two QDs and the horizontal straightness error will generate 2*w* spot deviation on two QDs. When RR produces translational displacements and rotational errors at the measured position, the spot deviations on two QDs relative to the initial position can be expressed as

_{1}and WP, ${s}_{QW2}$is the distance between QD

_{2}and WP, and 2

*w*is the spot deviation on the horizontal direction of two QDs caused by the horizontal straightness error.

According to Eqs. (21) and (22), the roll error *γ* and the horizontal straightness error *w* can be gotten by

In summary, combining the measurement of the vertical straightness error Δ*h* and the straightness error’s position *s* by using LIMU, the expressions of six degrees of freedom error parameters of the moving stage can be gotten by

## 4. Error compensation

From the last two formulas in Eq. (25), it is known that the accurate measurement of the vertical straightness error and its position depends on the accuracy of the measurement of *L*_{1} and *L*_{2} because the other four motion errors, especially the rotational errors, of the moving stage will deteriorate the measurement value of *L*_{1} and *L*_{2}. So it is important to find out the functional relationship between the motion errors and the value of *L*_{1} and *L*_{2} in order to compensate the measuring data and to improve the accuracy of the vertical straightness error and its position.

The traveling optical paths *L*_{1} and *L*_{2} of the two emergent beams from WP during measurement can be divided into two parts outside and inside RR. For influence of the horizontal straightness error *w*, as shown in Fig. 6, the optical paths outside and inside RR do not change when this error occurs. According to the geometrical relationship, the optical paths inside RR can be derived as

For the three rotational errors, the changes of *L*_{1} and *L*_{2} inside RR are analyzed firstly. As shown in the Fig. 5, for example, consider the upper right-angle prism, the angle formed between each of the three direction vectors ${I}_{1}$, ${I}_{2}$ and ${I}_{3}$ and the plane ${{o}^{\u2033}}_{u}{{x}^{\u2033}}_{u}{{y}^{\u2033}}_{u}$ can be derived as

The three vectors ${I}_{1}$, ${I}_{2}$ and ${I}_{3}$ are projected onto the plane ${{o}^{\u2033}}_{u}{{x}^{\u2033}}_{u}{{y}^{\u2033}}_{u}$ shown in Fig. 7. Assuming that the yaw of *Lin1*′ is ${\alpha}^{\prime}$. According to the geometrical relationship, the optical path *L*_{1} inside URP can be derived as

From the direction vector of ${I}_{1}$, $\mathrm{cos}{\alpha}^{\prime}$ is expressed by

So $L{{1}^{\prime}}_{in}$ becomes

Similarly, the optical path *L*_{2} inside DRP can be gotten by

Comparing Eqs. (30) and (31) with Eq. (26), the changes of the optical paths *L*_{1} and *L*_{2} inside RR can be neglected when the three rotational errors are small.

For the optical paths *L*_{1} and *L*_{2} outside RR, the emergent beams ${I}_{3}^{R}$ and ${I}_{7}^{R}$ only rotate the angle of $2\beta $ around the *y* axis after RR rotation relative to before RR rotation and have no change in the *x* axis and the *z* axis. In other words, only the pitch error will deteriorate the optical paths *L*_{1} and *L*_{2} outside RR. Figure 8 shows the influence of the pitch error on the optical paths *L*_{1} and *L*_{2} outside RR.

When the moving stage produce pitch error, the variations of the optical paths *L*_{1} and *L*_{2} outside RR can be derived as

In the actual measurement, the displacements acquired by processing the interference signals on the photodetectors (D_{1} and D_{2}) are the displacement deviations of URP and DRP at the moving position relative to the initial position, namely ${{L}^{\prime}}_{1}$ and ${{L}^{\prime}}_{2}$. Because of the rotational errors, the measuring data of ${{L}^{\prime}}_{1}$ and ${{L}^{\prime}}_{2}$ are deteriorated. According to the analysis above, they should be compensated as follows

Substituting Eqs. (34) and (35) into the last two formulas in Eq. (25), the compensated expressions of the vertical straightness error and its position are given by

## 5. Experiments and results

In order to verify the feasibility of the proposed straightness interferometer system for simultaneous measurement of six degrees of freedom parameters and the proposed compensation of the vertical straightness error and its position, an experimental setup was constructed as shown in Fig. 9. The laser source is a dual-frequency stabilized He-Ne laser (5517A, Agilent Co., USA) which emits a pair of beams with the frequency difference of 1.7 MHz and the wavelength of λ = 632.99137 nm. RR and WP with a divergent angle of 1.5° are a short range straightness measurement kit (A-8003-0443, Renishaw Co., UK). The two-dimensional PSD is a position sensitive detector (PDP90A, Thorlabs Co., USA) with the resolution of 0.675 μm. And the two QDs are two quadrant detectors (Spoton u-type, Duma Optronics Ltd., Israel) with the resolution of 0.75 μm and the accuracy of ± 1 μm. The measured stage is a precision linear stage (M-531.DD, Physik Instrumente Co., Germany) with the travel range of 300 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 laser interferometer system (XL80, Renishaw Co., UK) was used to test the same stage for comparison. The straightness resolution is 0.01μm while the accuracy is ± 0.005A ± 0.5 ± 0.15M^{2}μm, the angular measurement resolution is 0.1μm/m while the accuracy is ± 0.002A ± 0.5 ± 0.1M μm/m and the linear measurement resolution is 0.001μm while the accuracy is ± 0.5 ppm of the Renishaw interferometer where A is the straightness measuring data and M is the measuring distance.

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

In this experiment, a CL with the focal length of 190 mm was used, the laser beam with diameter of 3 mm was adjusted to project on the center of PSD at one end of the M-531.DD stage as the initial position, RR and the measuring angle reflector of Renishaw interferometer were mounted on the moving element of the stage. During the experiments, the element moved to the other end of the stage with a step displacement of 5mm and a velocity of 1 mm/sec, the yaw of the stage and the pitch of the stage were measured simultaneously with the proposed system and the Renishaw interferometer, respectively. The experimental results are shown in Fig. 10. It shows that the results with the proposed system are consistent with those obtained from the Renishaw interferometer. The deviations are the differences between the measurement results with the proposed system and the Renishaw interferometer. For the yaw errors, the maximum deviation is 0.842 arcsec while the standard deviation is 0.352 arcsec. For the pitch errors, the maximum deviation is 1.586 arcsec and the standard deviation is 0.568 arcsec.

#### 5.2 Measurement experiment of roll error

In this experiment, an electronic level (WL11, AVIC Qianshao Precision Machinery Co., China) with the resolution of 0.2 arcsec was used for comparison. At the beginning of the experiment, the laser beam was adjusted to project on the centers of two QDs at one end of the M-531.DD stage, RR and the WL11 level were mounted on the moving element of the stage. During the measurement, the element moved to the other end of the stage with a step displacement of 5 mm, the roll of the stage were determined simultaneously by the proposed system and the WL11 level. The experimental results are shown in Fig. 11. It shows that the results with the proposed system and the WL11 level are in basic agreement. The deviations are the differences between the measurement results with the proposed system and the WL11 level. The maximum deviation of roll error between the proposed system and the WL11 level is 2.692 arcsec while the standard deviation is −0.846 arcsec.

#### 5.3 Measurement experiment of horizontal straightness error

In this experiment, the laser beam was adjusted to project on the centers of PSD and QDs at one end of the M-531.DD stage, RR and the measuring straightness reflector of Renishaw interferometer were mounted on the moving element of the stage. The element moved to the other end of the stage with a step displacement of 5 mm, the yaw, pitch and roll of the stage were measured simultaneously with the proposed system, and then substituting the measurement results of the rotational errors into the fourth formula in Eq. (25) can obtain the horizontal straightness errors. At same time, the Renishaw straightness interferometer also measured the horizontal straightness errors of the stage. Using the least square method, the experimental results are shown in Fig. 12. It shows that the results with the proposed system are consistent with those obtained from the Renishaw interferometer. The horizontal straightness obtained with the proposed system is 8.55 µm while that obtained with the Renishaw interferometer is 8.26 µm. The deviations are the differences between the measurement results with the proposed system and the Renishaw interferometer. The maximum deviation of the horizontal straightness error is 1.869 µm while the standard deviation is 0.735 µm.

#### 5.4 Measurement and compensation experiments of vertical straightness error

In order to verify the effectiveness of the compensation of the vertical straightness error proposed in section 4, the comparison and compensation experiments were performed.

Firstly, the comparison experiment of measuring vertical straightness error was conducted as follows: RR and the straightness reflector of Renishaw interferometer were mounted on the moving element of the stage; the vertical straightness error was measured simultaneously with the proposed system and the Renishaw interferometer when the stage was moving with a step displacement of 5 mm. The experimental results are shown in Fig. 13. The deviations are the differences between the measurement results with the proposed system and the Renishaw interferometer. The maximum deviation of the vertical straightness error is 5.56 µm while the standard deviation is 2.51 µm. The vertical straightness obtained with the proposed system is 41.85 µm while that obtained with the Renishaw interferometer is 44.57 µm. This indicates that the results with the proposed system and the Renishaw interferometer are in basic agreement. However, these results seriously deviate from the parameter of straightness given in the datasheet of the stage because RR and the straightness reflector of Renishaw interferometer were both affected by the rotational errors of the stage. Therefore, in order to eliminate the influence of the rotational errors when the measuring reflector is mounted on the moving element, it is necessary to compensate the measurement result to reveal the true value of straightness of the measured stage (guideway).

Secondly, the compensation experiment of measuring vertical straightness error was conducted as follows: RR and the Wollaston prism of Renishaw interferometer were mounted on the moving element of the stage, this arrangement of Renishaw straightness interferometer is that considering Wollaston prism as measuring mirror can minimize the influence of the rotational errors; the vertical straightness error was measured simultaneously with the proposed system and the Renishaw interferometer. The experimental results are shown in Fig. 14(a). The vertical straightness obtained with the proposed system is 43.62 µm before compensation while that obtained with the Renishaw interferometer is 7.26 µm. To eliminate the influence of the rotational errors, substituting the measurement data of the pitch of the stage determined simultaneously by the proposed system into Eq. (36), the compensated result is shown in Fig. 14(b). The vertical straightness obtained with the proposed system is 9.85 µm after compensation. The maximum deviation of the vertical straightness error is 2.075 µm while the standard deviation is 0.934 µm. This demonstrated the effectiveness of the proposed compensation method of straightness error.

#### 5.5 Measurement and compensation experiments of straightness error’s position

In order to verify the effectiveness of the compensation of straightness error’s position proposed in section 4, the comparison and compensation experiments were performed.

In this experiment, RR and the measuring displacement reflector of Renishaw interferometer were mounted on the moving element of the stage; the displacement (straightness error’s position) of the stage was measured simultaneously with the proposed system and the Renishaw interferometer when the stage was moving. The experimental results are shown in Fig. 15(a). It shows that the results with the proposed system and the Renishaw interferometer are in agreement. The maximum error between the displacements obtained by the proposed system and those of the stage is 10.23 µm and the standard deviation is 3.67 µm. Meanwhile, the maximum error between the displacements obtained by the Renishaw interferometer and those of the stage is 10.09 µm and the standard deviation is 3.97 µm. However, these results seriously deviate from the unidirectional repeatability of ± 0.1µm given in the datasheet of the stage. This is because of the influence of the rotational errors of the stage. Using the compensation method mentioned in Eq. (37), the compensated result is shown in Fig. 15(b). After compensation, the maximum error is 0.71 µm while the standard deviation is 0.33 µm obtained with the proposed system. This demonstrated the effectiveness of the proposed compensation method of straightness error’s position.

#### 5.6 Simultaneous measurement and repeatability experiments

In order to verify the effectiveness of the proposed system for simultaneous measurement of six degrees of freedom error parameters, three measurements were performed by testing the stage. The experimental results are shown in Fig. 16 and Table 1. Figures 16(a)–16(f) and Table 1 show a good repeatability in simultaneously measuring yaw, pitch, roll, horizontal straightness, vertical straightness and position with the proposed system.

## 6. Discussion

#### 6.1 Analysis of the measurement resolution and the stability of the proposed system

According to the measurement principle and optical setup of the proposed system, the resolution and stability are mainly affected by the factors such as the stability of laser source, the quality of spots on PSD and QDs, the interference signals on photodetectors, the nonlinear error of polarization state, the fluctuation of refractive index of air and so on.

From Eqs. (25), (36) and (37), the uncertainty of the pitch, yaw, roll, horizontal straightness, vertical straightness and straightness error’s position can be derived as follows:

*s*. $\delta {L}_{1}$ and $\delta {L}_{2}$ are the uncertainties of ${L}_{1}$ and ${L}_{2}$, respectively.

When $\delta \Delta {x}_{PSD}=0.\text{675}$ μm, $\delta \Delta {y}_{PSD}=0.\text{675}$ μm, $f=\text{19}0$mm, $\delta \Delta {x}_{QD1}=0.\text{75}$μm, $\delta \Delta {x}_{QD2}$ = 0.75μm, ${s}_{0}$ = 1000mm, $B$ = 50mm, $\theta $ = 0.75°, $H$ = 100mm, $L$ = 30mm, *n* = 1.516, *s* = 150mm, $\delta {L}_{1}$ = 30nm, $\delta {L}_{2}$ = 30nm, the results of the uncertainty of the proposed system is shown in Table 2. Compared with Table 1, it is consistent with the measurement results obtained in the above experiments.

#### 6.2 Discussion of influence of the polarization state change caused by right-angle prism

The measurement of the vertical straightness error and its position in the proposed system is based on heterodyne interferometry which involves nonlinear periodic error problems. Because the nonlinear periodic errors about heterodyne interferometer have been discussed in many other research works [23–26], the same problems about these nonlinear errors are not discussed here. The nonlinear error induced by the polarization state change caused by right-angle prism is analyzed as follows.

When a linearly polarized beam incidents on to a right-angle prism, the emergent beam changes to a elliptically polarized beam. In the proposed system, the polarization state changes of the emergent beams ${{f}^{\prime}}_{1}$ and ${{f}^{\prime}}_{2}$ from RR are shown in Fig. 17. The *y* components of the emergent beams will transmit PBS_{1} to generate the first measurement signal with the beam ${f}_{2}$ on D_{1} and the *x* components of the emergent beams will be reflected by PBS_{1} to generate the second measurement signal with the beam ${f}_{1}$ on D_{2} shown in Fig. 1.

As an example, the nonlinear error of the first measurement signal is analyzed as follows. The electric fields of the beam ${f}_{2}$ and the *y* components of the emergent beams ${{f}^{\prime}}_{1}$ and ${{f}^{\prime}}_{2}$ from RR can be expressed as

*o*-${y}^{\prime}$, ${\epsilon}_{1}$ and ${\epsilon}_{2}$ are the angles between the

*y*axis and the major axis of each ellipse, ${\delta}_{1}$ and ${\delta}_{2}$ are the phase delays between the major and minor axes of each ellipse.

Then the measurement signal is expressed as

Firstly, using the Zemax to simulate the influence of the transitional errors of RR on the change of the polarization state, the simulation result shows that the main error source results from the roll error of RR. Then, according to Eq. (41), a simulation to determine the influence of the polarization state change on the measurement signal is shown in Fig. 18. The curve with $\gamma $ = 0° represents the case without roll error. When RR has roll errors of 0.5° and 1°, the phase errors compared with the measurement signal without roll error are 0.0148° and 0.0302°, respectively. That is, the corresponding displacement errors are 0.03nm and 0.05nm, respectively. Thus, the influence of the polarization state change caused by right-angle prism can be neglected because the rotational error of precision stage or guideway is below several hundreds arcsec.

## 7. Conclusion

In this paper, a laser straightness interferometer system with rotational error compensation and simultaneous measurement of six degrees of freedom error parameters is proposed. The optical layout of this system is described. The principle of the simultaneous measurement of six degrees of freedom parameters of a linear stage including three rotational parameters of the yaw, pitch and roll errors and three linear parameters of the horizontal straightness error, the vertical straightness error and the straightness error’s position is depicted and analyzed in detail. Meanwhile, to solve the influence of the rotational errors of measuring reflector on the measurement of straightness error and its position in laser straightness interferometer, the compensation methods for straightness error and its position are presented. In order to verify the feasibility of the proposed system, a serious of measurement and compensation experiments were done. Firstly, the separate measurement experiments of the yaw, the pitch, the roll, the horizontal straightness error, the vertical straightness error and the straightness error’s position of a precision linear stage were performed simultaneously with the proposed system and a commercial interferometer for comparison (the roll comparison with an electric level). These experiments show that the results with the proposed system are consistent with those obtained from the comparison interferometer (or the electric level). Secondly, the compensation experiments of the influence of rotational errors of the measured object on vertical straightness error and its position show the effectiveness of the presented compensation method. Finally, the simultaneous measurement and repeatability experiments indicate a good repeatability in simultaneously measuring six degrees of freedom parameters of the yaw, pitch, roll, horizontal straightness, vertical straightness and position with the proposed system. All these demonstrate that the proposed system could be applied in the field of performance testing and calibration of precision stages and guideways.

## Acknowledgment

This work was supported in part by the National Natural Science Foundation of China (NSFC) under grants No.51375461, No.51475435 and No.51205365, the Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) under grant No. IRT13097 and the 521 talents training program of Zhejiang Sci-Tech University.

## References and links

**1. **O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, “Performance evaluation of a new taut wire system or straightness measurement of machine tools,” Precis. Eng. **38**(3), 492–498 (2014). [CrossRef]

**2. **F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng, and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express **21**(22), 25805–25819 (2013). [CrossRef] [PubMed]

**3. **P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage,” Precis. Eng. **35**(3), 424–430 (2011). [CrossRef]

**4. **J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, “Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer,” Precis. Eng. **38**(3), 578–588 (2014). [CrossRef]

**5. **J. J. Kroll, “Six degree of freedom optical sensor for dynamic measurement of linear axes,” PhD thesis, University of North Carolina, Charlotte (2003).

**6. **J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measuring system for CMM geometric errors,” J. Eng. Ind. **114**(3), 362–369 (1992).

**7. **W. Y. Jywe, C. H. Liu, W. H. Shien, L. H. Shyu, T. H. Fang, Y. H. Sheu, T. H. Hsu, and C. C. Hsieh, “Developed of a multi-degree of freedoms measuring system and an error compensation technique for machine tools,” J. Phys. Conf. Ser. **48**(1), 761–765 (2006). [CrossRef]

**8. **I. Rahneberg, H.-J. Büchner, and G. Jäger, “Optical system for the simultaneous measurement of two-dimensional straightness errors and the roll angle,” Proc. SPIE **7356**, 73560S (2009). [CrossRef]

**9. **K. C. Fan and Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. **40**(14), 2073–2081 (2000). [CrossRef]

**10. **C. H. Liu, J. H. Chen, and Y. F. Teng, “Development of a straightness measurement and compensation system with multiple right-angle reflectors and a lead zirconate titanate-based compensation stage,” Rev. Sci. Instrum. **80**(11), 115105 (2009). [CrossRef] [PubMed]

**11. **A. Kimura, W. Gao, and L. J. Zeng, “Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder,” Meas. Sci. Technol. **21**(5), 054005 (2010). [CrossRef]

**12. **C. H. Liu and C. H. Cheng, “Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light,” Sen. Actuators A. **181**(7), 87–93 (2012). [CrossRef]

**13. **L. J. Zhu, L. Li, J. H. Liu, and Z. H. Zhang, “A method for measuring the guideway straightness error based on polarized interference principle,” Int. J. Mach. Tools Manuf. **49**(3–4), 285–290 (2009). [CrossRef]

**14. **P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precis. Eng. **35**(4), 686–692 (2011). [CrossRef]

**15. **K. C. Fan and M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. **24**(1), 15–23 (2000). [CrossRef]

**16. **R. R. Baldwin, “Interferometer system for measuring straightness and roll,” U.S. Patent, 3790284 (1974).

**17. **D. R. McMurtry and R. J. Chaney, “Straightness interferometer system,” U.S. Patent, 5026163 (1991).

**18. **S. T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. **33**(3), 195–199 (2001). [CrossRef]

**19. **Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. **16**(10), 2030–2037 (2005). [CrossRef]

**20. **B. Chen, E. Zhang, L. Yan, C. Li, W. Tang, and Q. Feng, “A laser interferometer for measuring straightness and its position based on heterodyne interferometry,” Rev. Sci. Instrum. **80**(11), 115113 (2009). [CrossRef] [PubMed]

**21. **S. T. Lin, S. L. Yeh, C. S. Chiu, and M. S. Huang, “A calibrator based on the use of low-coherent light source straightness interferometer and compensation method,” Opt. Express **19**(22), 21929–21937 (2011). [CrossRef] [PubMed]

**22. **Y. S. Zhai, Q. B. Feng, and B. Zhang, “A simple roll measurement method based on a rectangular-prism,” Opt. Laser Technol. **44**(4), 839–843 (2012). [CrossRef]

**23. **R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Pachard J. **34**, 10 (1983).

**24. **D. J. Loner, B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Annals-Manufac, Technol. **52**(1), 439–442 (2003).

**25. **C. M. Wu and C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. **7**(1), 62–68 (1996). [CrossRef]

**26. **W. M. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. **30**(3), 337–346 (2006). [CrossRef]