## Abstract

The rotary axis is the basis for rotational motion. Its motion errors have critical effects on the accuracy of the related equipment, such as a five-axis computer numerical control machine tool. There are several difficult problems in the implementation of high-precision and fast measurement of the multi-degree-of-freedom motion errors of a rotary axis. In this paper, a novel method for the simultaneous measurement of five-degree-of-freedom motion errors of a rotary axis is proposed, which uses a single-mode fiber-coupled laser with a full-circle measuring range. It has the advantages of high efficiency, low cost, and it requires no decoupling calculation. An experimental system was built and a series of experiments were performed. The standard deviation of stability for 60 min of the five-degree-of-freedom measurement is 0.05 arcsec, 0.06 arcsec, 0.04 μm, 0.03 μm, and 0.19 arcsec, respectively. The repeatability deviation of measuring an indexing table is ± 3.4 arcsec, ± 4.6 arcsec, ± 2.6 μm, ± 2.4 μm, and ± 3.2 arcsec. The maximum deviation of comparison is 3.9 arcsec and 3.2 arcsec. These results demonstrate the effectiveness of the proposed method; thus, a new approach of simultaneous measurement of the multi-degree-of-freedom motion errors of a rotary axis is provided.

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

## 1. Introduction

The rotary axis is the basis of rotational motion, and it is the most important moving component of machining, measuring, or control equipment. A change in its attitude or motion error has critical effects on the accuracy of the related equipment, such as the machining accuracy of a multi-axis CNC machine tool and machining center [1–3], the measuring accuracy of a three-dimensional laser scanner and laser tracker [4-5], and the control accuracy of a multi-degree-of-freedom (DOF) manipulator and antenna pedestal of a radar or satellite [6]. With the rapid development of high-precision and high-efficiency equipment, the demand for accuracy of the rotary axis has continuously increased. By measuring the motion errors of the rotary axis accurately and determining the main sources of errors, the design and manufacturing processes of a rotary axis can be optimized and the errors of motion can be compensated during operation and maintenance. Ultimately, the aim of improving and ensuring the high-performance operation of the equipment can be achieved.

According to the international standard ISO230-7 [7], when the axis rotates around the Z-axis, there are six DOF of geometric motion errors (Fig. 1), including the angular positioning error around the Z-axis *ε _{z}*(

*θ*), the radial motion errors along the X-axis

*δ*(

_{x}*θ*) and Y-axis

*δ*(

_{y}*θ*), the axial motion error along the Z-axis

*δ*(

_{z}*θ*), the tilt motion errors around the X-axis

*ε*(

_{x}*θ*) and Y-axis

*ε*(

_{y}*θ*).

Research on the measurement of single-DOF motion error of a rotary axis, especially the angular position error, started a long time ago. A variety of practical and effective methods have been developed, such as the angular positioning error measurement method based on polygon prism and collimation system [8], the laser interferometry method based on double retro-reflectors [8-9] and the method based on birefringent crystal [10-11]. However, these methods still have some limitations in terms of measuring range, application environment, or installation and adjustment. For example, the number of angular positions, which can be measured by the first method, is limited by the number of facets of the polygon prism (generally the maximum is 72 facets). Furthermore, when using the second method, the laser beam must not be blocked at any moment during the measurement.

In recent years, significant progress has been made in step-by-step methods of measuring multi-DOF motion errors of a rotary axis. Park et al. used reflection grating and position sensitive detector (PSD) to measure six DOF motion errors through two steps [12]. He et al. proposed the dual optical path measurement method (DOPMM), which can measure six DOF motion errors through five steps [13]. Obviously, both single-DOF and step-by-step multi-DOF methods are highly time consuming and labor-intensive. The accuracy of the measurement cannot be ensured due to the changes of the environment and the actual state of the rotary axis. Therefore, there is an increasing demand for the simultaneous measurement of multi-DOF motion errors of a rotary axis with high precision and high efficiency. Jywe et al. realized the simultaneous measurement of four DOF motion errors by measuring several orders of diffraction light from the reflection grating [14]. Chen et al. proposed a method of simultaneously measuring six DOF motion errors based on a pyramid-polygon-mirror [15-16]. Hiroshi et al. developed a system for simultaneously measuring five DOF motion errors using a rod and a ball lens, which is suitable for a high-speed micro-spindle [17-18]. The above mentioned methods use multiple sets of light sources and detectors, assembled separately, which leads to large volume, high cost, and difficulties in the accurate assembling of all parts. Moreover, they are all indirect measurement methods, which have complicated error models and require large decoupling calculations [19].

In this paper, a novel method for simultaneously measuring five DOF motion errors of a rotary axis is proposed. Such method is based on our previous research for a linear axis [20]. However, the motion error measurements of a rotary axis can be quite difficult by comparing to that of the linear axis. In order to solve the problem, our method includes one reference rotary axis and two retro-reflectors. For one of the retro-reflectors, the half of its bottom surface was coated with a beam-splitting film. The two retro-reflectors were used as the error-sensitive element, and a single-mode fiber-coupled laser diode was used to ensure stability and accuracy [21]. Our method is indeed a direct measurement method with a full-circle measuring range (i.e., 360°). It does not require the decoupling calculation and can accurately determine the origin of the error. The main components and the relevant principles for simultaneously measuring five DOF motion errors of a rotary axis will be described in section 2. In section 3, the experimental device will be provided. Our experimental results are scientifically valid in order to confirm the proposed method in this work.

## 2. Measurement principles

The proposed method for simultaneously measuring five DOF motion errors of a rotary axis is based on the principle of laser-beam fiber-collimation. A schematic diagram is shown in Fig. 2, which consists of a laser source and a fiber coupling unit, a measurement unit, an error-sensitive unit, and a reference rotary axis.

The laser source and fiber coupling unit is composed of a semiconductor laser, a fiber-coupled lens, and a single-mode fiber. The application of the single-mode fiber not only improves the spatial stability and the energy distribution of the measurement laser beam significantly, but also improves the thermal stability as the heat output from the laser source is isolated. The error-sensitive unit is mainly composed of two corner-cube retro-reflectors. As it has small size, light weight, and no electronic connections, the error-sensitive unit can be fixed on a high-precision reference rotary axis with low load. The reference rotary axis and the target rotary axis are coaxially mounted together.

The laser beam, emitted from the semiconductor laser, is coupled to a single-mode fiber by the coupling lens, C-lens, and transmitted to the collimator in the measurement unit. The collimated laser beam is split into two parallel beams by the prism group M1, they are the reflected beam and the transmitted beam respectively. In the following principles, the beam path is described relying on the corresponding error.

#### 2.1 Angular positioning error and tilt motion error around the Y-axis

As shown in Fig. 2, for the angular positioning error, beam-splitting film BS1 coated on half of the bottom surface of retro-reflector RR1 is the sensitive element. The reflected beam from M1 passes through the prism group M2 and arrives at RR1. The beam that reflected by BS1 is then twice reflected by M2 and focused on PSD1 by the lens L1. The whole process is known as the laser auto-collimation measurement system, and it can precisely measure small angles [22]. In order to expand the measuring range to 360°, a precision reference rotary axis is mounted between the error-sensitive unit and the target rotary axis. When the target rotary axis rotates a nominal angle *θ* clockwise, the real rotation angle is *θ + ε _{z}*(

*θ*), because of the presence of the angular positioning error

*ε*(

_{z}*θ*). The beam reflected by BS1 may travels out of the measurement unit. At this time, let the reference rotary axis rotate with the same nominal angle anticlockwise; thus, the reflected beam will be focused again on PSD1. The angular positioning error can be obtained by measuring the change in position of the laser spot on PSD1, which can be expressed as

*Y*

_{PSD1}is the change in position of the spot on PSD1 in the Y-direction and

*f*

_{1}is the focal length of L1.

Similarly, the tilt motion error around the Y-axis can be expressed as

where Δ*Z*

_{PSD1}is the change in position of the spot on PSD1 in the Z-direction.

In fact, the main purpose of the reference rotary axis is to keep the reflected beam imaged on PSD1. Therefore, the reverse rotation angle is not limited to nominal angle *θ*, especially when the angular positioning error of the target rotary axis is large and out of the linear response range of PSD1. Thus, the reference rotary axis can rotate with an angle larger or smaller than *θ*, and the angular positioning error can be expressed as

*θ*

_{ref}is the rotation angle of the reference rotary axis.

#### 2.2 Radial and axial motion errors

As shown in Fig. 2, both RR1 and RR2 can be used as the sensitive element for measuring the radial and axial motion errors [23]. For example, the beam that passes through BS1 is backward reflected by RR1 and enters the quadrant detector QD1. According to the characteristics of the RRs, when the radial motion error *δ _{y}*(

*θ*) is present, the change in position of the spot on QD1 in the Y-direction is doubled, which is

Similarly, the axial motion error *δ _{z}*(

*θ*) can be expressed as

*Y*

_{QD1}, Δ

*Y*

_{QD2}and Δ

*Z*

_{QD1}, Δ

*Z*

_{QD2}denote the changes in position of the spots on QD1, QD2 and in the Y- and Z-direction, respectively.

#### 2.3 Tilt motion error around the X-axis

As shown in Fig. 2, the reflected beam and the transmitted beam from M1 are backward reflected by RR1 and RR2, and enters QD1 and QD2, respectively. When the tilt motion error around the X-axis is present, both the incident point and exit point on the bottom surfaces of RR1 and RR2 change. There are corresponding changes in positions of the spots on QD1 and QD2 [24]. Therefore, the tilt motion error around the X-axis *ε _{x}*(

*θ*) can be obtained as

*h*is the distance between the two centers of the bottom surfaces of RR1 and RR2.

#### 2.4 Measurement and compensation of the laser beam drift

The key component of the proposed method is the collimation laser beam. Both the parallel drift and the angular drift of the laser beam influence the measurement accuracy. Owing to the use of a single-mode fiber, the measurement unit and the laser are isolated, which not only excludes thermal-mechanical interactions, but also reduces the parallel and angular drifts significantly. The main factors causing drift of the collimation beam, emerging from the collimator during the subsequent measurement process, are slow mechanical deformation and air disturbance, which cause the angular drift, random tremble, or bending of the beam. Therefore, in order to further increase the measurement accuracy, a common-path compensation module is integrated into the method.

The laser beam that originally comes from the collimator is indicated by the red solid line in Fig. 3. The two beams entering QD2 and PSD2 have a common optical path. The laser beam is indicated by the blue dotted line when only the radial motion error is present, and is indicated by the green dashed line when both the angular drift and the radial motion error are present. Thus, the angular drift of the laser beam can be obtained as

*β*and ∆

*γ*denote the angular drift in the Y- and Z- directions; ∆

*Y*

_{PSD2}and ∆

*Z*

_{PSD2}denote the changes in position of the spot on PSD2 in the Y- and Z- directions; and

*f*

_{2}is the focal length of L2.

Thus, Eqs. (1) and (2) of the angular positioning error and tilt motion error around the Y-axis can be modified as

Similarly, Eqs. (4) and (5) of the radial and axial motion errors can be modified as

*l*denotes the transmission distance of the laser beam.

Because the tilt motion error around the X-axis can be obtained from the axial motion errors measured by QD1 and QD2, the influence due to the beam drift can also be eliminated after the compensation of the axial motion errors according to Eq. (9).

## 3. Experiments and analysis

Considering the measurement principles mentioned above, an experimental system of simultaneously measuring five DOF motion errors of a rotary axis was built. The main components are two QDs (Pacific Silicon Sensor, QP50-6SD2-DIAG, resolution of 0.08 μm), two PSDs (First Sensor, DL16-7-PCBA3, resolution of 0.5 μm), other optical elements including two RRs (Edmund, N-BK7, 3 arcsec beam deviation), and a reference rotary axis, which consisted of a step-rotary table (BOCI, MRS201, resolution of 0.33 arcsec) and a circular-grating angle-encoder (Heidenhai, RON886, accuracy of ± 1 arcsec, resolution of ± 0.2 arcsec). As shown in Fig. 4, an indexing table (HSD-200RT, accuracy of <60 arcsec, indexing angle of 30°) was chosen as the target rotary axis. The distance between the measurement unit and the sensitive unit was 250 mm. A photoelectric auto-collimator was used for comparison.

**Measurement resolution:** According to the parameters of the PSDs, Eqs. (1) and (2), the longer the focal length, the higher the resolution of the angular positioning error and the tilt motion error around the Y-axis, but the smaller the measuring range. In fact, the voltage outputs of the PSD is determined by the centroid location of the light spot power density. Hence, theoretically, the light spot size does not influence the measurement resolution, but it influences the repetitive positioning error of the PSD [25]. The smaller the spot size, the higher the positioning accuracy. In order to obtain a small spot size, a short focal length is preferred. To balance the resolution, the spot size and the compactness of our experimental setup, the two compound lenses with 200 mm in the focal length were chosen as L1 and L2, and the resolution can reach 0.26 arcsec. The PSDs can be replaced with QDs for higher resolution without considering the measuring range [22]. According to Eq. (6), the longer the distance *h* between the centers of the incident planes of RR1 and RR2, the higher the resolution of the tilt motion error around the X-axis, but the smaller the measuring range. Taking the compactness and the common-path of the two parallel beams into considerations, the distance *h* is chosen to be 30 mm, and the resolution can be as high as 0.27 arcsec. Referring to the parameters of the QDs, Eqs. (4) and (5), the resolution of the radial and axial motion errors is 0.04 μm. The influence of the angular drift can be remarkably increased due to the increase of the transmission distance *l* of the laser beam, as it is indicated in Eq. (9). Thus, it is important to determine a proper working distance between the measurement unit and the sensitive unit based on the size of the target rotary axis.

#### 3.1 Calibration experiments

The errors measured by our experimental system can be classified into two categories: displacement errors measured by the QDs and angular errors measured by the PSDs. Thus, a grating ruler (LG-50, accuracy of ± 0.1 μm, resolution of ± 0.05 μm) and a photoelectric auto-collimator (Collapex EXP, accuracy of ± 0.2 arcsec, resolution of ± 0.01 arcsec) were used to calibrate the two types of errors. The calibration results are shown in Fig. 5. In the range of ± 80 μm, the linear-fit determination coefficient between the grating ruler and the QDs is up to 0.9996, the standard deviation is 0.76 μm, and the range of fluctuation is from −1.2 μm to 1.0 μm. In the range of ± 120 arcsec, the linear-fit determination coefficient between the auto-collimator and the PSDs is up to 0.9999, the standard deviation is 0.58 arcsec, and the range of fluctuation is from −1.2 arcsec to 0.7 arcsec.

#### 3.2 Stability experiments

Stability experiments were performed under laboratory conditions and the variation range of the environmental temperature was about 20 ± 1 °C. The experimental system was fixed on the optical platform and the distance between the measurement unit and the sensitive unit was 250 mm. The experimental results were auto-recorded in a computer by using a data acquisition board, and plotted in Fig. 6 after being extracted once every 10 s for 60 min. The standard deviation of the angular positioning error and tilt motion error around the Y-axis is 0.05 arcsec and 0.06 arcsec, respectively. The standard deviation of the radial and axial motion errors are 0.04 μm and 0.03 μm, respectively. The standard deviation of the tilt motion error around the X-axis is 0.19 arcsec.

#### 3.3 Repeatability and comparison experiments

**Repeatability experiments:** The results of measuring the five DOF motion errors of the indexing table for three times are shown in Fig. 7. The deviation of repeatability of the angular positioning error is ± 3.4 arcsec [Fig. 7(a)]. The deviation of the tilt motion error around the Y-axis is ± 4.6 arcsec [Fig. 7(c)]. The deviations of the radial and axial motion errors are ± 2.6 μm and ± 2.3 μm, respectively [Figs. 7(e) and 7(f)]. The deviation of the tilt motion error around the X-axis is ± 3.2 arcsec [Fig. 7(g)]. Thus, our experimental system has a good repeatability for measuring the five DOF motion errors of a rotary axis.

**Comparison experiments:** A photoelectric auto-collimator was used for the comparisons, with its sensitive unit (a plane mirror) fixed on the back of the sensitive unit of the experimental system. The two systems simultaneously measured the angular positioning error and the tilt motion error around the Y-axis of the indexing table. As shown in Figs. 7(b) and 7(d), the deviations of repeatability of the auto-collimator are ± 0.6 arcsec for the angular positioning error and ± 5.6 arcsec for the tilt motion error around the Y-axis, and the maximum deviations of comparison between the two systems are 3.9 arcsec and 3.2 arcsec, respectively.

**Analysis of the results:** The experimental system is capable of simultaneously measuring five DOF motion errors of a rotary axis and exhibits good repeatability. However, there are certain deviations compared to the auto-collimator. The main reason is the error crosstalk, arising when simultaneously measuring five DOF motion errors. Owing to the limitation in length, the details of error modelling and compensation are discussed later.

The experimental system and the auto-collimator exhibits similar deviations of repeatability for measuring the tilt motion error around the Y-axis. This is because the indexing table itself exhibits obvious deviation of repeatability. Furthermore, with elapsing time, environmental parameters such as the temperature are changed, and the fixed or structural components of the systems will be slowly shifted or deformed such that deviations are introduced. In the future, the automatic control of the measurement process can be realized, on the one hand, to further improve the measurement efficiency, and, on the other hand, to eliminate the interference of human factors and to further improve the accuracy.

## 4. Conclusion

In this paper, a novel and simple method for the simultaneous measurement of the five DOF motion errors of a rotary axis is proposed. It belongs to the direct measurement methods and it requires no decoupling calculation, which is greatly useful to determine the origin of the error. The application of a single-mode fiber-coupled semiconductor laser not only ensured the stability and accuracy of the measurement, but also significantly reduced the costs. An experimental system was built and a series of experiments were performed including calibration experiments, stability experiments, repeatability experiments, and comparison experiments. The results demonstrate that the five DOF motion errors of a rotary axis in the range of 0–360° can be simultaneously measured by the proposed method, which exhibits good stability and repeatability, and relatively high accuracy. In addition, the method also has the advantages of low cost and high efficiency. It is expected to achieve real-time dynamic measurements of multi-DOF motion errors of a rotary axis by using a direct-drive servo reference rotary axis. After analyzing the error crosstalk, the installation errors of the components and the manufacturing errors of the optical elements, the measurement accuracy can be further improved by error modeling and compensation, which is the main focus of our following work.

## Funding

National Natural Science Foundation of China-Major Program (51527806); Fundamental Research Funds for the Central Universities (S16RC00030).

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