## Abstract

In this paper, a compact four-degree-of-freedom (4-DOF) measurement system is presented. With a special optical configuration, the pitch error, yaw error, and two straightness errors of the moving target are able to be detected by only a single laser beam from a collimated laser diode. A 2D hybrid mirror angle steering mount is designed to perform the large angle turning for the axis alignment and very fine angle tuning by PZT actuators for real-time beam drift compensation. A series of calibration and comparison experiments have been carried out to verify the performance of the proposed system. The developed active compensation system could effectively suppress the beam’s angular drift to within ± 0.01 arc-sec in both of yaw and pitch directions. The developed 4-DOF measuring system is compact, low cost, and suitable for long distance geometric error measurement of linear stages.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As the basic performance criterion of production machines, the measurement technology of machine’s geometric errors plays an important role in precision machine assembly and accuracy evaluation. Any axis of motion, either linear or rotation, has inherent six-degree-of-freedom (6-DOF) geometric errors, namely three translational errors and three rotational errors in three orthogonal axes. Conventionally, testing instruments used in the linear stage assembly process include dial indicator, mechanical artefacts, precision level, autocollimator or multi-function laser interferometer, etc [1, 2]. It normally consumes a tedious time to measure all errors one-by-one. Various multi-degree-of-freedom measurement (MDFM) systems, which provide fast and accurate means of measurement, have been proposed since early 1990 [3]. Huang and Ni [4, 5] proposed a 5-DOF motion error simultaneously measuring system using He-Ne as the laser source. It had been proved effectively in the geometric error compensation of a coordinate measurement machine by comparison experiments.

Fan [6, 7] proposed methods for simultaneous on-line measurement of 6-DOF motion errors of linear stage and X-Y stage by combining multi laser Doppler scales and quadrant photo detectors. Kuang [8] developed a 4-DOF measuring system using single mode fiber laser as laser source. Huang [9] proposed a 5-DOF measuring method using monolithic prism to generate parallel laser beams to avoid the influence of assembly errors. Feng et al [10] developed a system for simultaneously measuring 6-DOF motion errors, which used a single polarization maintaining fiber-coupled dual-frequency laser. Yu [11–13] developed a single beam 6-DOF measurement system using the differential wave front sensing and optical roll sensing principle. Some commercial MDFM systems also have been launched, such API’s XD5 for 5-DOF measurement system [14], SIOS’s SP 15000 C5 for 5-DOF simultaneous measurement with all interferometric signals and Renishaw’s XM-60 for 6-DOF measurement simultaneously measurement system [15].

Those measurement systems mentioned above still have some shortages. First, some of their sizes are bulky due to the use of He-Ne laser with complex configuration. Second, the cost is hard to reduce due to the use of expensive laser source. Third, active compensation for beam drift caused errors is rarely considered, which is necessary for a long distance measurement. Fourth, parallelism of emitted beams is difficult to be adjusted in case of taking the roll error measurement. It is known that any linear stage inevitably has inherent 6-DOF geometric errors due to imperfect assembly. Among these, the positioning error can be measured only by the laser length interferometer. The roll error can be measured by the rotation of polarization state with a single beam, but the resolution and accuracy cannot be better than one arc-sec. Generating two reference parallel beams to be detected by two distant four quadrant photodetectors (QPD) or two position sensitive photodetectors (PSD) can measure roll error with high accuracy, but the adjustment of parallel beams is a key technique. The other 4-DOF errors, including two straightness errors in vertical and horizontal directions and two angular errors in pitch and yaw, can be measured by various optical configurations. The technique of 4-DOF error measurement is a fundamental from which 5-DOF and 6-DOF can be added up. No matter what configuration is designed, however, the four shortages indicated above have to be considered. In this paper, a novel and compact 4-DOF simultaneously measuring method is proposed. By using a laser diode as reference beam the cost is low. In addition, by integrating an active beam drift compensation mechanism into a commercial mirror angle adjusting mount, a novel hybrid mirror angle steering mount has been developed that cannot only adjust the coarse beam direction by manual knob and but also tune fine direction by PZT actuator. In the following parts of this paper, the principle of 4-DOF measurement is described in section 2. In section 3, the principle of active beam drift compensation is proposed. In section 4, a series of experiments have been conducted to verify the performance of the proposed system.

## 2. Principle of measurement system

This 4-DOF measurement system is designed for measuring two straightness errors and two angular errors of a moving target. The principle of straightness error measurement is to detect two lateral motions of the target using a four-quadrant photodetector (QPD) relative to a collimated laser beam. The principle of angular error measurement is based on 2D autocollimator. Both principles shall be integrated into the 4-DOF measurement system under the conditions of: (1) just using a single laser beam, (2) as less optics as possible, (3) no ghost spot happened and (4) self-compensation of beam drift for long distance measurement. The schematic diagram of the system’s working principle is thus proposed, as shown in Fig. 1. It includes a stationary unit fixed to the base and a moving unit mounted onto the moving table as the target. In the stationary unit, a low cost collimated diode laser (made by Huanic Co. model DA635) is used as the referenced straightness line, which, by proper selection, can provide a collimated laser with a divergence angle less than 0.2 mrad. A mirror (M1) is mounted onto a specially designed 2D angle steering mechanism, which not only can manually adjust the reflected beam angle in line with the moving axis of the target but also compensate the drifted laser beam by embedded two-piezoelectric actuators (PZT). After M1, the laser beam is split by a polarized beam splitter (PBS) to two paths. The reflected beam is used to detect the beam drift by the first set of autocollimator (AC1), comprising a focusing lens (FL1) and a QPD1 on the focal point of FL1. If the beam has an angular drift, its drifted angle can be immediately detected by AC1 so that PZT will be actuated to compensate for the drifted angle before the beam is emitted to the moving unit. The transmitted beam, after bent by the mirror M2, is used as a straight reference for measuring straightness and angular errors of the moving target. In the moving unit, the incoming reference beam is split by the beam splitter (BS) to two paths. The reflected light is used to measure pitch and yaw angular errors by the second set of autocollimator (AC2, including FL2 and QPD2) and the transmitted light is received by QPD3 to detect the horizontal and vertical straightness errors of the moving target.

It is noted that the QPD surface itself and its protection glass will reflect a weak light back along the original beam path. This unwanted weak beam would produce a ghost spot noise to the laser or detectors if the optical configuration is not proper. In this 4-DOF measurement system, the role of using the PBS and two quarter waveplates (QWP) is to stop the reflected light from all QPDs back to the LD and the placement of measuring sensors on the moving part is to avoid the generation of the ghost spot. The whole 4-DOF measuring system so designed features only one laser beam, which angular drift can be actively compensated.

#### 2.1 Measurement of straightness errors

It can be clearly seen from Fig. 2 that straightness errors of the moving target are directly detected by QPD3. The position shift of the beam spot on QPD3 is proportional to the differential current in the same direction. Let the straightness errors in X and Y directions be denoted by *δ _{x}* and

*δ*, respectively.

_{y}*x*

_{QPD}_{3}and Δ

*y*

_{QPD}_{3}are the spot movements on QPD3 in X and Y directions, respectively,

*i*

_{1},

*i*

_{2},

*i*

_{3},

*i*

_{4}indicate the output currents acquired by the QPD’s 1~4 quadrants, respectively, and

*k*,

_{δx}*k*are constants, which can be obtained by the calibration experiments. The LD is not a power stabilized light source. The differential current could possibly be affected by the intensity change of the LD. Therefore, the normalization technique of dividing the differential current by the sum of the light intensity is often used in QPD output signals. The algorithm used in Eq. (1) also applies to other QPDs in this paper.

_{δy}#### 2.2 Measurement of pitch and yaw errors

The measurement principle of yaw and pitch is shown in Fig. 3(a). Half of the incoming beam is reflected by BS and incident into FL2. QPD2 is located on the focal point of FL2. When the moving target has a pitch or yaw rotation, the whole moving part rotates the same angle. The laser beam will rotate *ε _{y}* angle relative to the tilted central line of BS and QPD2. The spot movement on QPD2 follows the principle of autocollimator, as shown in Fig. 3(b). When the angular error is very small, the relationship between the spot motion and the angle can be linear.

*ε*is the pitch error,

_{y}*ε*is the yaw error, Δ

_{x}*x*

_{QPD}_{2}and Δ

*x*

_{QPD}_{2}are the corresponding beam spot shifts on QPD2,

*f*

_{2}is the focal length of FL2.

## 3. Laser beam drift feedback compensation method

Laser beam drift has been recognized as one of critical error sources in geometric error measurement, especially the straightness and angular errors in a long-distance motion. Several methods have been proposed to cope with this issue in the past. Ni adopted a beam expander to reduce the drifted angle [11, 12]. Tan’s group proposed a passive method using common-path principle in dealing with laser beam drifts and compensated by off-line calculation [16, 17]. Zhao firstly applied an active drifted angle compensation method by using a PZT-actuated turning mirror [18, 19]. Considering that the laser beam has to be in line with the moving axis of the target by a manual type 2-D angle adjusting mirror and also maintained the angle by an active compensating mechanism during the measurement process, a novel active compensation mechanism is thus proposed in this study that integrates the functions of manual type angle turning and PID controlled fine angle motion. Details are expressed as follows.

#### 3.1 Measurement and compensation of beam drifted angle

Assuming the beam drift is entirely caused by the laser source, the drifted angle can be detected by the autocollimator principle. In the stationary part of the measuring system, an active beam drift compensation system is proposed. Figure 4(a) shows the concept that a part of emitted laser beam is reflected by the PBS and incident into a small angle measurement unit (AC1) consisting of FL1 and QPD1.

In order to keep the beam angle unchanged, a turning mirror M1 mounted on a 2-D angle adjustment mechanism is used to control the reflected laser beam so that the focused spot on QPD1 remains at the same position. Two PZT actuators are used to tune the reflected beam angles corresponding to the pitch and yaw directions. A classical PID control strategy has been developed including the measurement unit, the signal conditioner unit and the controller unit, which control loop is schematically shown in Fig. 4(b).

#### 3.2 Analysis of beam vector for drift compensation

In order to calculate the vector of each beam in the optical path, the ray tracing calculation needs to be performed according to the law of reflection. Let the setting angle of mirror M1 be 45° in the reference coordinate system as given in Fig. 4 (a). The unit normal vector of the reflecting surface in the reference coordinate system can be expressed by a vector form of

the transformation matrix of the reflecting surface M1 can be expressed by

The vector of the incident light from LD is${\overrightarrow{n}}_{\text{LD}}\text{=}{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{T}$. The vector of the reflected light of M1 is

the direction vector of the PBS and M2 can be expressed as

then, the reflection matrices of PBS and M2 can be obtained similar to Eq. (4) as

The direction vector of incident light of AC1 (${\overrightarrow{n}}_{AC1}$) and the emitted light of M2 (${\overrightarrow{n}}_{rm2}$) in the ideal condition can be obtained as

*ε*and

_{dy}*ε*are the beam drift angles of laser beam in X (yaw) and Y (pitch) directions, respectively. The direction vectors of incident light of AC1 and the emitted light of M2 under drifted angles can be calculated as

_{dp}These drifted angles can be detected by AC1. Similar to Eq. (2), the relationship between output of QPD1 and drifted angle can be expressed by

*x*

_{QPD}_{1}and Δ

*y*

_{QPD}_{1}are the corresponding beam spot shifts on QPD1 in X and Y directions, respectively, and

*f*

_{1}is the focal length of FL1.

Therefore, by using the output of QPD1 as feedback signals, the drifted angle can be compensated by adjusting the normal vector of M1 through PID control. Under this condition, the actual normal vector of M1 can be expressed as

*T*

_{m}_{1}is the homogeneous transformation matrix (HTM) of M1 due to the rotational motion,

*α*

_{m}_{1}and

*β*

_{m}_{1}are the corresponding rotated angles of M1 in yaw and pitch directions, respectively. The compensated reflection matrix of M1 can be gained similar to Eq. (4) as

Then the direction vector of the compensated incident beam of AC1 and the emitted beam to the moving unit can be expressed as

through the feedback control, the output of QPD1 should be remained to 0. Hence, the compensated direction vector of the incident beam of AC1 should be the same as its ideal condition, i.e. Eq. (15) is equal to Eq. (8), yielding to

It can be seen that the adjusted mirror angle for the compensation of drifted angle in the yaw direction is half of the drifted angle and that for compensation in pitch direction is equal to the drifted angle. The difference in the sign relates to the direction of the coordinate system. Substituting *β _{m}*

_{1}and

*α*

_{m}_{1}into Eq. (16), the vector of the compensated output laser beam to the moving unit will be the same as its ideal condition, i.e.

through the above analysis, it is proved that the use of feedback control can compensate the drifted angle of the emitted laser beam.

#### 3.3 Design of a 2-D steering mirror mechanism for coarse and fine motions

The reflection mirror M1 in Fig. 4(a) has two missions. The primary mission is to provide manual adjustment of output beam angle to be exactly in line with the moving target at the alignment stage, and the additional mission is to actuate fine PZT tuning to compensate for the instant drifted angle of the laser beam. Normally, to meet these two requirements, separate manual and PZT driven stages are needed to stack up, resulting in a bulky mechanism. A novel active compensation mechanism is thus designed that integrates the functions of manual angle turning and PID controlled extremely fine angle tuning into a single mechanism. The design of this hybrid mirror angle steering mount is shown in Fig. 5(a). A commercial small-sized angle mirror mount (made by DHT Co. in China, Model GCM080205M) was adopted for use as the base frame. It original threaded shaft was purposely drilled a large hole in the front so as to insert in a mini-sized PZT actuator (TOKIN Ltd., model AL1, size: 1.65x1.65x5 mm), which is preloaded by the contact pin. There is also a small hole drilled through the back of the threaded shaft so as to allow the wires of PZT to connect to the connector. The threaded shaft is screwed into an inner threaded bronze ring that is force fitted into the hole of the base frame. The shaft and the connector are fixed by an outer nylon sleeve, which plays the role of manual turning the thread so as to push or pull the contact pin, resulting in turning the mirror angle to achieve the beam alignment with the moving target. The PZT tunes the mirror angle in very fine steps with PID control for the drifted beam angle compensation. The commanded displacement of PZT is determined by the drifted angle of the laser beam. According to Eq. (17) and Fig. 5(a), the commanded PZT strokes for drifted angle compensation are

*d*and

_{x}*d*are the commanded motions of PZT1 and PZT2, respectively.

_{y}*L*and

_{x}*L*are the offset lengths between the pin point and the center of the pivotal ball. It can be seen that when the adjusted angle is very small, the relationship between the controlled amount of the PZT and the angle of the exit light is linear. The control algorithm is a classical incremental PID control, and the PID parameters are obtained by relay feedback self-tuning method [20]. The experimental results will be presented in section 4.

_{y}Figure 5(b) shows the photo of the prototype of developed 2D steering mirror mechanism. The area of the tilting plate is about 20 × 20 (mm) and *L _{x}*,

*L*are about 16 mm. The maximum stroke of PZT is about 3.7 μm. Therefore, the moving range of this steering mirror is about ± 40 arc-sec in pitch direction and ± 20 arc-sec in yaw direction. These adjustable ranges are much larger than the actual beam drifted angles. The minimum step of PZT is about 1 nm and the corresponding stepped angle is about 0.012 arc-sec in pitch and 0.006 arc-sec in yaw.

_{y}## 4. Experimental results

All experiments were carried out in a non-environmental controlled open laboratory in which disturbance caused by temperature change and air flow during the experiment cannot be avoidable. The experimental setup of the developed 4-DOF measuring system is shown in Fig. 6. The stationary part is fixed to the base table and the moving part is mounted onto the carrier that is sliding along a pair of parallel linear guideways. The maximum travel is about one meter. The initial laser beam alignment with respect to the moving axis of QPD3 was done by turning the nylon sleeve until the readings of QPD3 at two end positions were nearly zero. Then, the carrier was moved to the first target position and reset all QPDs to zero. During the measurement, if the QPD1 detected the motion of the focused spot, the drifted beam angles in pitch and yaw directions were calculated by Eq. (12). Then the commanded displacement of the corresponding PZT was obtained by Eq. (19).

#### 4.1 Calibration of 4-DOF measuring system

The accuracy calibrations of two straightness errors and two angular errors were conducted separately. The reference for straightness error calibration was a Laser Doppler Displacement Meter (LDDM, made by Optodyne Inc. USA, model LICS-100) and the experimental setup for horizontal straightness is shown in Fig. 7(a). The distance between the stationary part and moving part in all the calibration process was 450 mm. A comparison test was carried out between the QPD3 and corner cube retroreflector (CCR) of LDDM by moving the horizontal stage in prescribed steps. For the calibration of vertical straightness error, the laser beam of LDDM was bent down 90 degrees from the top side by a pentaprism and the vertical stage was moved in steps, as shown by the subgraph in the lower right side of Fig. 7(a). The reference for angular error calibration was an autocollimator (made by AutoMAT Co. model 5000U). The setup for pitch and yaw error calibration is shown in Fig. 7(b). Two angular stages, including pitch stage and yaw stage, provided commanded steps which were comparatively read by the 4-DOF measuring system and autocollimator. Figure 8 shows the calibrated results that the residuals of two straightness errors for the range of ± 100 μm were all within ± 0.5 μm, and the residuals of two angular errors for the range of ± 100 arc-sec were all within ± 0.6 arc-sec. The performance of this developed 4-DOF measuring system was acceptable for use.

#### 4.2 Tested results of a linear slide

A tested linear slide was assembled on an optical bench with the carrier’s moving range to 900 mm and the stationary part as well as the moving part of the developed 4-DOF measuring system was installed in, as shown in Fig. 6. The carrier was manually moved approximately 50 mm in steps to 900 mm long. Testing of the straightness errors of the linear slide was made by comparison between the developed 4-DOF measuring system and a commercial Laser Straightness Measuring System (LSMS made by 3DFamily, Taiwan, accuracy ± 0.5 arc-sec) with both moving parts mounted on the carrier. The LSMS is also a QPD detected system. The experimental set-up is shown in Fig. 9(a). Two QPDS, respectively detected by LSMS and 4-DOF system, are separated by a distance, being a non-common path condition. According to the Bryan principle [21], the straightness error measured at point A is different from that measured at point B, as shown in Fig. 9(b). Therefore, the angular induced error has to be corrected in association with the offset, expressed by

where,*δ*(

_{Ax}*z*) and

*δ*(

_{Ay}*z*) denote the converted horizontal and vertical straightness error at point A from measured errors at point B (

*δ*(

_{Bx}*z*),

*δ*(

_{By}*z*)),

*L*(

_{BAx}*z*) and

*L*(

_{BAy}*z*) are Bryan offsets from point B to point A in X and Y directions, respectively, and

*ε*is the roll angular error of the carrier (in Z axis) measured by a commercial electronic level (made by Qianshao Co. China, model WL-2, accuracy ± 0.5 arc-sec). It is noted that Eq. (20) is a simplified expression that ignores the induced errors by pitch and yaw rotations, because the offset in Z direction was almost zero.

_{z}(z)Each experiment was repeated 5 times to show the repeatability. Figure 10(a) plots the horizontal straightness error measured by the 4-DOF system, Fig. 10(b) shows the horizontal straightness error measured by the LSMS and converted to the measured point of 4-DOF system. Two error curves are very close to each other and the repeatability of each curve is quite good. A little separation among five curves nearby the position of 200 mm in each Fig was apparent. It was due to the fact that manual movement of the carrier could not guarantee exact positioning at each target. This explanation can be justified by the phenomenon of curve separation that appears similarly in both of the 4-DOF system and the LSMS. Figure 10(c) shows that the averaged error curve of 4-DOF system almost coincides with the averaged error curve of LSMS, and the residuals are within ± 2 μm to the travel of 900 mm long. The reason why the residuals were still a bit large could be explained that the original horizontal straightness error of the tested linear slide was too large to the magnitude of 60 μm. In practice, most machine tools have much better straightness errors than this tested linear slide, normally less than ± 10 μm. It could be estimated that, if tested on a machine tool, the comparison results between the 4-DOF system and LSMS would be better than ± 1 μm. Figures 10(d), 11(e) and 11(f) show the comparative measured results of vertical straightness errors. Similar satisfied results were obtained.

The measurements of pitch and yaw angular errors of the tested linear slide could be done simultaneously in one travel. The setup is similar to Fig. 7(b) but no need of the angular stages. The operational procedure was the same as the straightness error measurement. Results are plotted in series in Fig. 11. The performance of 4-DOF system is as good as the commercial autocollimator of AutoMAT 5000U. Both obtained similar results, as shown in Figs. 11(a) and 11(b), with the difference within ± 2 arc-sec up to 900 mm long. It is believed that if the original angular errors were smaller, the difference would be smaller.

Regarding the specifications of the 4-DOF system, the resolution was evaluated by the steady-state fluctuation of the signal in a short time (12 secs in this work). the uncertainty is difficult to assess from Figs. 10 and 11. It is because the linear slide is moved by hands so that each target position cannot be located precisely among 5 cycles. The original straight error of the slide is large. A slight position deviation might cause large measurement error. However, the comparison error between our 4-DOF system and the reference instrument (LSMS or AUTOMAT) at each position is small. The uncertainty can be evaluated from the standard deviation of the residual. The performance of the developed 4-DOOF system is summarized in Table 1. The measurement range is obtained from the calibration in Sec. 4.1.

#### 4.3 Stability and beam drift compensation experiments

In order to verify the effectiveness of the active compensation system, the pointing stability experiments with and without the beam drift error compensation were carried out under the same ambient condition without any temperature control. The distance between the moving part and the stationary part was 450 mm and both parts were fixed to the same optical bench to avoid the influence of external disturbances. The readings of the 4-DOF system were automatically recorded at every 1 second and the total experiment time lasted 1.5 hours. The comparison results are shown in Fig. 12.

The beam drift in yaw angle was more serious than the pitch angle, as seen in Figs. 12(a) and 12(b). However, both drifted angles could be compensated to within ± 0.01 arc-sec. The corresponding measured angles by AC2 on the moving part can be seen from Figs. 12(e) and 12(f). Both trends and magnitudes are quite consistent with the drifted angle in Fig. 12(b). It seems that the noise of Fig. 12(f) is a bit large, it is due to the smaller scale factor. The measured horizontal and vertical straightness errors by QPD3 were propositional to the corresponding angular errors by a factor of the distance, i.e. 450 mm, as shown in Figs. 12(c) and 12(d). A significant fluctuation in the uncompensated pitch and vertical straightness data could be due to large ground vibration by other disturbance, because the yaw and horizontal were not affected. This example demonstrates that the active compensation is very effective to any disturbance that shifts the beam spot. Because the environment was not controlled, the residuals after beam drift compensation in Figs. 12(d) and 12(f) could be attributed to the temperature change and air turbulence. From our point of view, after beam drift compensation, the remaining error contains two components. The first one is the temperature disturbance caused form error that is gradually increased up to about 1 μm in the vertical straightness direction and up to about 0.4 arc-sec in the vertical pitch direction. The second component is the air disturbance caused sudden variations along the form curves of both errors that happened almost at the same times with minor effects. This assumption can be seen from Fig. 12(d) and Fig. 12(f). It can verify that the proposed feedback compensation system can effectively suppress the measurement error caused by laser drift in the common environment.

## 5. Conclusions

In this paper, a 4-DOF measurement system with active feedback beam drift compensation is proposed. The feasibility of the measurement system was verified by calibration and comparison experiments. The comparison results of pointing stability measurement showed that the proposed active feedback compensation system can effectively suppress the beam drift error. The developed 4-DOF measuring system needs only one single laser beam so that the cost is low and the size is compact. Future works will expand to 5-DOF and 6-DOF system for on machine tool measurements. Major concerns will be the parallelism adjustment of parallel beams.

## Funding

The National Key Research and Development Program of China (2017YFF0204800) and The Fundamental Research Funds for the Central Universities (DUT16TD20).

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