Roll angle autocollimator measurement method based on a cylindrical cube-corner reflector with a high resolution and large range

Renpu Li; Linyi Xie; Yan Zhen; Han Xiao; Wei Wang; Junqi Guo; Igor Konyakhin; Mikhail Nikitin; Xiaolong Yu

doi:10.1364/OE.453979

1. Introduction

The spatial attitude of the object needs to measure six-degree-of-freedom, namely the straightness of the three axes and three angles of freedom. Recently, optical-based measurement methods have important measurement characteristics such as non-contact, high resolution, and high precision, and have become the main measurement methods for precision machining, precision manufacturing and precision measurement. However, since the change of the roll angle of the object does not change the inclination angle of the optical path and the optical axis, measured by optical method is difficult. In response to this difficulty, some roll angle measurement methods based on different principles and structures have been proposed, which can be roughly divided into Monocular vision method [1,2], interferometric method [3,4], polarization variation method [5,6], parallel beams method [7–9] and autocollimation method [10,11]. Among them, Monocular vision method is generally believed to be able to achieve large measurement range and complete independent identification of three free angles, but with low accuracy. Interferometic method has the performance of high resolution and super precision, but the interferometic signal measurement system is composed of a variety of optical elements, and there are many optical path deflection times, so the measurement Angle range is limited. Polarization variation method has large measurement range but low resolution, it is not suitable for high-precision measurement occasions. Parallel beams method has simple structure, high resolution and measurement accuracy, but the roll angle information is measured by the two-dimensional displacement of the light spot, hence the range performance is limited by the size and working distance of the photodetector (PSD).

With the advent of Industry 4.0 era, spatial angle traceable optical measurement and large-scale multi-degree-of-freedom measurement technologies are highly regarded, and this requires the measurement system to have performance advantages such as high resolution, ultra-precision, large range, long working distance, etc. [12–16]. In this context, the autocollimation method has the theoretical advantages of compact structure and sensitivity to the beam Angle deviation because it adopts the principle of optical common path design. According to the latest reports from Germany Physikalisch-Technische Bundesanstalt and VTT Technical Research Centre of Finland Ltd, Centre for Metrology MIKES, the calibrated commercial AC can achieve a resolution of 0.0005”, the overall accuracy of yaw and pitch is superior to the ultra high measurement performance of 0.006” [17,18]. And since the measurement light path of AC is parallel, existing technology can achieve remote measurement of 20-50 m and measurement range of 6-10° [19–21]. Therefore, the angle measurement method based on AC has prospect to become the future angle measurement technology.

A traditional autocollimation method consists of a collimating objective lens and a charge-coupled device or a photodiode used as the light spot position-sensing detector, whereby the angle of the planar reflector is measured by detecting the deviation between the reflected beam with respect to the axis of the projected beam. Since the change of the roll angle of the planar reflector neither changes the optical path nor the vector direction of the beam. Therefore, traditional autocollimation method only pitch and yaw can be measured based on the collimation of a light beam, which reflected by a mirror and focused on a 2D-PSD. Moreover, it is insensitive to roll angle. To break through the theoretical limitation of AC roll angle measurement, we expound two modified cube corner reflector (MCCR) design methods for AC three-degree-of-freedom angle measurement [22,23]. An ideal cube corner reflector can make the reflected beam always parallel to the incident beam, regardless of its own spatial angle. However, when the angles of each working face of cube corner reflector are changed along the direction of the spatial coordinate system according to certain rules, MCCR will be sensitive to roll angles. This law can be obtained by establishing a spatial coordinate vector relationship model between AC and reflector, and analyzing the vector changes of the reflected beam according to the order of different reflectors. However, the RAC measurement method based on these MCCR structures has two main disadvantages: 1. The Angle sensitivity of the reflector is small, which leads to the decrease of the resolution of the measurement system and the increase of random error. 2. The single reflected beam displacement in the measurement formula contains MCCR multi-free Angle information at the same time, resulting in serious multi-free Angle crosstalk.

For obtain a free angle measurement method suitable for the new industrial era, this article proposes AC measurement method based on CCCR. The non-standard CCCR is obtained by using the analytic method of space coordinate vector and the Euler space rotation model. When the collimated beam hits the reflector, an image of two reflected beams, circular and cross, will be formed on the imaging surface of the RAC. The circular image is from the specular reflection of the aperture surface of the CCCR, and the cross image is formed by the reflected light beam passing through the three reflecting surfaces inside the CCCR. When the non-standard CCCR changes along the roll angle, the cross image will only change the tilt angle and will not move out of the field of view of the measurement system, which makes the measuring system to obtain the measuring range of the roll angle greater than ±20°. The circular image can be used as a criterion for judging pitch and yaw angles, thereby avoiding the crosstalk influence of the other two degrees of freedom angles on the roll angle measurement effectively. Benefiting from the above advantages, this method can not only realize the large-range measurement of the roll angle, but also ensure the high resolution, ultra-precision, and long working distance of the traditional AC measurement.

2. Principle and analysis

2.1 MCCR structure for roll angle measurement

As shown in Fig. 1, the MCCR structure proposed by us was obtained by rotating the three reflection mirrors 1, 2, and 3 of the standard cube-corner at an Angle of δ₁, δ₂, δ₃ along the OX₀, OY₀, and OZ₀ axes respectively [22,23]. The included angles between the three reflection mirrors 1’, 2’ and 3’ of the MCCR structure are respectively L_1’-2’=90°-δ₁, L_2’-3’=90°-δ₂ and L_1’-3’=90°-δ₃. When these included angles are less than 90°, the twirl angles δ₁, δ₂ and δ₃ are defined as positive, and vice versa.

Fig. 1. (a) Standard cube-corner mirror; (b) modified MCCR structure.

Download Full Size | PDF

If δ₁, δ₂, δ₃ satisfy the following relationship:

(1)$${\delta _1}\textrm{ = }0;{\delta _2}\textrm{ = }{\delta _3}\textrm{ = }\delta$$

According to the order of different reflection mirrors in MCCR (as shown in Fig. 2), the reflected light path vector in the order of 1-2-3 reflection mirrors can be expressed as Eq. (2).

(2)$${B_{123}} = \left( {\begin{array}{*{20}{l}} \begin{array}{l} - 2\sqrt 2 \delta \cos ({\Theta _1})\sin ({\Theta _3}) - 2{\delta^2}\cos {({\Theta _3})^2}\sin (2{\Theta _2})\\ + 2{\delta^2}\sin ({\Theta _1})\sin (2{\Theta _3})\cos (2{\Theta _2}) + 2{\delta^2}\sin (2{\Theta _2})\sin {({\Theta _1})^2}\sin {({\Theta _3})^2} \end{array}\\ \begin{array}{l} 2\sqrt 2 \delta \cos ({\Theta _2})\cos ({\Theta _3}) + 2\sqrt 2 \delta \sin ({\Theta _1})\sin ({\Theta _2})\sin ({\Theta _3})\\ + 2{\delta^2}(\cos ({\Theta _2})\sin (2{\Theta _1})\sin {({\Theta _3})^2} - \sin ({\Theta _2})\sin (2{\Theta _3})) \end{array}\\ \begin{array}{l} - 4{\delta^2}\cos {({\Theta _1})^2}\cos {({\Theta _2})^2}\sin {({\Theta _3})^2} - 4{\delta^2}\cos {({\Theta _2})^2}\cos (2{\Theta _3})\\ - 2{\delta^2}\sin ({\Theta _1})\sin (2{\Theta _2})\sin (2{\Theta _3}) - 4{\delta^2}\sin {({\Theta _3})^2} + 1 \end{array} \end{array}} \right)$$

Fig. 2. Sequential reflected light path via MCCR 1-2-3 reflection mirrors.

Download Full Size | PDF

The first column of Eq. (2) represents the change vector of the reflected beam along the X-axis, when the δ angle is small, the displacement of the reflected beam along the X-axis can be expressed as:

(3)$${X_{123}} = f \cdot ( - 2\sqrt 2 \delta \cdot {\Theta _3} - 4{\delta ^2} \cdot {\Theta _2})$$

where f is the focus of the AC.

It can be found from the Eq. (3) that the displacement of the reflected light path along the horizontal-axis in the order of 1-2-3 mainly corresponds to the roll angle information Θ₃ of the MCCR, so the MCCR structure can be used for measuring the roll angle of the object. But the element in Eq. (3) that represents the angle sensitivity of the MCCR is ${K_{MCCR}}\textrm{ = }2\sqrt 2 \delta$, The condition for the formulation is that δ<1°, so MCCR is usually K_MCCR<0.048, which means that the resolution of the AC measurement system will decrease by about 42 times compared with the K_mirror=2 which is sensitive to the reflection Angle of the plane mirror.

2.2 CCCR structure for roll angle measurement

According to the conclusion in section 2.1, to make the designed reflector suitable for roll angle measurement, the CCCR should follow the angle relationship of the reflection mirrors in Eq. (1), i. e. L_1’-2’=90°, L_2’-3’=L_1’-3’=90°-δ. According to this relationship, the third reflection mirrors in Fig. 2 can be oriented and transformed into cylindrical mirror to realize the design of a CCCR, and the design of the cylindrical mirror should follow these principles:

1. The cylindrical axis O'O'‘ is perpendicular to the intersecting line of the 1 and 2 sides of the cube-corner mirror, which is perpendicular to the OZ₀-axis of the Angle cone mirror.
2. The cylindrical axis O'O'‘ forms an equal angle with the reflection mirrors 1 and 2 respectively.

The structure of the CCCR formed according to the above principles is shown in Fig. 3. The angle between the cylindrical section and the OX-axis of the corner cube is α = 45°.

Fig. 3. CCCR: (a) Theoretical structure diagram; (b) Design view.

Download Full Size | PDF

Obviously, the normal vectors N₁ and N₂ of mirror 1 and mirror 2 are the unit vectors of the OX₀ and OY₀ axes of the CCCR, and are determined by the following expressions:

(4)$${N_1} = {\left[ {\begin{array}{*{20}{c}} 1&0&0 \end{array}} \right]^T};{N_2} = {\left[ {\begin{array}{*{20}{c}} 0&1&0 \end{array}} \right]^T}$$

The normal vector N₃ of the third cylinder can be expressed as:

(5)$${N_3} = \left( {\begin{array}{*{20}{c}} {\frac{{\sqrt 2 \sin (\beta )}}{2}}\\ {\frac{{\sqrt 2 \sin (\beta )}}{2}}\\ {\cos (\beta )} \end{array}} \right)$$

β is the Angle formed between the normal line of the third cylinder and the OZ₀-axis of the CCCR. The angle of β ranges from 0° to β_max, and the value of β_max will be determined by the size of the third cylinder and the radius r of the cylinder. And the Angle L_1-3 between mirror 1 and mirror 3 can be expressed as:

(6)$${L_{1 - 3}} = {N_1} \cdot {N_3} = \left( {\sqrt 2 /2} \right) \cdot \beta$$

Since L_1-3 =(90°+δ), then according to Eqs. (1) and (6):

(7)$$\sin (\delta )={-} \left( {\sqrt 2 /2} \right) \cdot \beta ;{\delta _3} = \delta ={-} \arcsin \left( {\left( {\sqrt 2 /2} \right) \cdot \beta } \right)$$

In such a CCCR, the normal vector N₃ of the third cylinder has an equal projection vector on the coordinate axes OX₀ and OY₀, so the angle between the mirror 2 and the mirror 3 is L_2-3=L_1-3, also satisfies Eq. (7). Therefore, the CCCR meets the structure of the MCCR described in section 2.1, and can realize the measurement of the roll angle Θ₃.

2.3 Reflection matrix of CCCR

According to the different order of the three reflection mirrors in the Cube-corner, there are six reflected beams in different order (1-2-3 and 3-2-1, 2-1-3 and 3-1-2, 2-3-1 and 1-3-2). The reflection matrices M₁₂₃ and M₃₂₁, M₂₁₃ and M₃₁₂, M₂₃₁ and M₁₃₂ of the CCCR can be calculated according to the Eqs. (17)–(19) from [22], where the reflection matrix M₁₂₃ can be expressed as:

(8)$${M_{123}} = \left( {\begin{array}{*{20}{c}} { - 1}&0&{2\delta }\\ 0&{ - 1}&{2\delta }\\ { - 2\delta }&{ - 2\delta }&{ - 1} \end{array}} \right)$$

The reflection matrix M₃₂₁ can be expressed as:

(9)$${M_{321}} = \left( {\begin{array}{*{20}{c}} { - 1}&0&{ - 2\delta }\\ 0&{ - 1}&{ - 2\delta }\\ {2\delta }&{2\delta }&{ - 1} \end{array}} \right)$$

And since the angle between mirrors 1 and 2 is 90°, 1-2 and 2-1 are a pair of mutually reversible reflection sequence combinations, and their corresponding reflection matrices have the same expression form:

(10)$${M_{123}} = {M_{213}};{M_{321}} = {M_{312}}$$

The reflection matrices M₂₃₁ and M₁₃₂ for the reflection order of 2-3-1 and 1-3-2 are:

(11)$${M_{231}} = \left( {\begin{array}{*{20}{c}} { - 1}&0&{2\delta }\\ 0&{ - 1}&{ - 2\delta }\\ { - 2\delta }&{2\delta }&{ - 1} \end{array}} \right);{M_{132}} = \left( {\begin{array}{*{20}{c}} { - 1}&0&{ - 2\delta }\\ 0&{ - 1}&{2\delta }\\ {2\delta }&{ - 2\delta }&{ - 1} \end{array}} \right)$$

The Eqs. (8)–(11) are the reflection matrix form calculated for the coordinate system X₀Y₀Z₀ of the CCCR, but for the AC measurement end part as shown in Fig. 4(a), there is another measurement coordinate system X₁Y₁Z₁. When the CCCR does not rotate at the roll angle and the initial measurement state is maintained, the coordinate systems XYZ of the AC and X₁Y₁Z₁ are parallel to each other.

Fig. 4. The dual coordinate system of the AC measurement system: (a) the CCCR and the AC measurement system; (b) the two coordinate systems in the case of the rotation of the CCCR.

Download Full Size | PDF

When the CCCR produces three free angle changes (pitch or yaw or roll), the two coordinate systems will be relatively tilted as shown in Fig. 4(b). the OZ₀-axis of the CCCR and the OY₁-axis of the AC measurement end part will form an Angle φ:

(12)$$\varphi = \arcsin (\sqrt 2 /\sqrt 3 )$$

The reflection matrix of the cube-corner mirror for the coordinate system X₁Y₁Z₁ of the AC measurement system should be:

(13)$${M_{{x_1}{y_1}{z_1}}} = {R_1} \cdot {M_{xyz}}{R_1}^T$$

M_X0Y0Z0 is the reflection matrix of a cylindrical angular mirror in its own coordinate system, R₁ is the change matrix from coordinate system X₀Y₀Z₀ to coordinate system X₁Y₁Z₁.

According to Eq. (12) and take ϑ=0, $\Psi ={-} \frac{{3\pi }}{4}$, the change matrix R₁ can be expressed as:

(14)$${{\mathbf R}_\textrm{1}}\textrm{ = }\left( {\begin{array}{*{20}{c}} { - \frac{{\sqrt 2 }}{2}}&{\frac{{\sqrt 2 }}{2}}&0\\ {\frac{{\sqrt 2 \cdot \sqrt 3 }}{6}}&{\frac{{\sqrt 2 \cdot \sqrt 3 }}{6}}&{\frac{{\sqrt 2 \cdot \sqrt 3 }}{3}}\\ {\frac{{\sqrt 3 }}{3}}&{\frac{{\sqrt 3 }}{3}}&{\frac{{\sqrt 3 }}{3}} \end{array}} \right)$$

Substituting Eqs. (8) and (14) into Eq. (13), the reflection matrices M'₁₂₃ and M'₂₁₃ of the CCCR in the X₁Y₁Z₁ coordinate system are obtained:

(15)$$M{^{\prime}_{123or213}} = \left( {\begin{array}{*{20}{c}} { - 1}&0&0\\ 0&{ - 1}&\Delta \\ 0&{ - \Delta }&{ - 1} \end{array}} \right)$$

Substituting Eqs. (9) and (14) into Eq. (13), the reflection matrices M'₃₂₁ and M'₃₁₂ of the CCCR in the X₁Y₁Z₁ coordinate system are obtained:

(16)$$M{^{\prime}_{321or312}} = \left( {\begin{array}{*{20}{c}} { - 1}&0&0\\ 0&{ - 1}&{ - \Delta }\\ 0&\Delta &{ - 1} \end{array}} \right)$$

In Eqs. (15) and (16), $\varDelta \textrm{ = }2\sqrt 2 \cdot \delta \cdot n$, Where n is the refractive index of cube-corner, and according to Eqs. (7), $\varDelta \textrm{ = }2 \cdot \beta \cdot n$.

Similarly, substituting Eqs. (11) and (14) into Eq. (13), the reflection matrices M'₁₃₂ and M'₂₃₁ of the CCCR in the order of 1-3-2 and 2-3-1 can be obtained:

(17)$$M{^{\prime}_{132}} = \left( {\begin{array}{*{20}{c}} { - 1}&{ - \frac{{2 \cdot \Delta }}{{n\sqrt 6 }}}&{ - \frac{\Delta }{{\sqrt 3 }}}\\ {\frac{{2 \cdot \Delta }}{{n\sqrt 6 }}}&{ - 1}&0\\ {\frac{\Delta }{{\sqrt 3 }}}&0&{ - 1} \end{array}} \right);M{^{\prime}_{231}} = \left( {\begin{array}{*{20}{c}} { - 1}&{\frac{{2 \cdot \Delta }}{{n\sqrt 6 }}}&{\frac{\Delta }{{\sqrt 3 }}}\\ { - \frac{{2 \cdot \Delta }}{{n\sqrt 6 }}}&{ - 1}&0\\ { - \frac{\Delta }{{\sqrt 3 }}}&0&{ - 1} \end{array}} \right)$$

2.4 Reflected beam imaging

The reflected beam vector B in the AC measurement system is determined by the following expression:

(18)$$B = M \cdot A$$

M is the reflector matrix of the CCCR in the coordinate system XYZ of the AC measurement system, and A is the collimated incident beam vector in Fig. 4.

Since the incident beam is parallel to the optical axis of the AC measurement system, that is, the incident beam is parallel to OZ₁-axis, then the beam vector A is defined as:

(19)$$A = \left( {\begin{array}{*{20}{c}} 0\\ 0\\ { - 1} \end{array}} \right)$$

By expressing the reflection matrix in Eq. (18) with Eqs. (15) and (19), the reflected beam vector B₁₂₃ or B₂₁₃ can be obtained:

(20)$${B_{123or213}} = \left( {\begin{array}{*{20}{c}} {{B_{123or213x}}}\\ {{B_{123or213y}}}\\ {{B_{123or213z}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0\\ { - \Delta }\\ 1 \end{array}} \right)$$

If Eqs. (16) and (19) are substituted into Eq. (18), the reflected beam vector B₃₂₁ or B₃₁₂ can be obtained:

(21)$${B_{321or312}} = \left( {\begin{array}{*{20}{c}} {{B_{321or312x}}}\\ {{B_{321or312y}}}\\ {{B_{321or312z}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0\\ \Delta \\ 1 \end{array}} \right)$$

As above, if Eqs. (17) and (19) are substituted into Eq. (18), the reflected beam vector B₁₃₂ or B₂₃₁ can be obtained:

(22)$${B_{\textrm{132}}} = \left( {\begin{array}{*{20}{c}} {{B_{132x}}}\\ {{B_{132y}}}\\ {{B_{132z}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{\Delta }{{\sqrt 3 }}}\\ 0\\ 1 \end{array}} \right);\,{B_{231}} = \left( {\begin{array}{*{20}{c}} {{B_{231x}}}\\ {{B_{231y}}}\\ {{B_{231z}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - \frac{\Delta }{{\sqrt 3 }}}\\ 0\\ 1 \end{array}} \right)$$

From the expressions (20), (21) and (22), the reflected beam imaging of the CCCR is shown in Fig. 5. The reflected beam imaging in different orders will form several groups of bands according to the angle Δ = 2·β, if β has a small value, the band-shaped light spot is approximately linear and has a length:

(23)$${X_{132or231}} = {B_{132or231x}} \cdot f;{Y_{123or321}} = {B_{123or321y}} \cdot f$$

where f is the focal length of the collimating lens.

Fig. 5. Imaging of the reflected beam through the internal reflection surface of the CCCR: 1-The beam imaging with reflection sequence 1-2-3, 2-1-3; 2-The reflection beam imaging with reflection sequence 3-2-1, 3-1-2; 3-Reflection sequence 1-3-2 the reflected beam and 4-reflection order 2-3-1 for its imaging.

Download Full Size | PDF

CCCR cut with glass material has an aperture mirror that can independently complete the reflection of the beam in addition to the three internal reflection mirrors, as shown in Fig. 6(a). Therefore, there will be two reflections on the image plane of the AC, one from the reflection plane and the cylinder inside the conical mirror, the other from the direct reflection of the aperture reflection surface. However, it must be emphasized that the above theories are derived based on the light source vector of Eq. (19), that is point light source imaging, which is not applicable to AC with cross aperture. Thus, the image of the complete reflected beam of the CCCR is shown in Fig. 6(b).

Fig. 6. (a) All reflection mirrors of CCCR; (b) complete reflected beam imaging.

Download Full Size | PDF

2.5 Measuring formula of RAC

To measure the rotation of the object, the non-standard CCCR mounted on it rotates Θ₃ roll relative to the OZ₁-axis as shown in Fig. 4. In actual measurement, the object may also rotate Θ₁ pitch along the OX₁-axis and Θ₂ yaw angles on the OY₁-axis. And the free angle change of the object to be measured, relative to the coordinate system of the AC measurement system XYZ, the vector B of the reflected beam is expressed as:

(24)$$B^{\prime} = R \cdot M^{\prime} \cdot {R^T} \cdot A$$

where M’ is the reflection matrix of the CCCR relative to the coordinate system XYZ of the AC measurement system, A is the incident beam vector, and R is the rotation matrix around the three axes of the AC measurement system, and the rotation matrix factor φ = Θ₁; ϑ = Θ₂; ψ = Θ₃.

According to Eqs. (19) and (24), and by substituting Eqs. (15), (16), (17) into Eq. (24) respectively, the vector change of the reflected beam caused by the rotation of roll angle of CCCR can be determined. Where the reflected beam vectors 1-2-3 and 2-1-3 in reverse order are:

(25)$$B{^{\prime}_{123or213}} = \left( {\begin{array}{*{20}{c}} {B{^{\prime}_{123or213x}}}\\ {B{^{\prime}_{123or213y}}}\\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\Delta \cos ({\Theta _1})\sin ({\Theta _3}) + \Delta \cos ({\Theta _3})\sin ({\Theta _1})\sin ({\Theta _2})}\\ { - \Delta \cos ({\Theta _2})\cos ({\Theta _3})}\\ 1 \end{array}} \right)$$

The reflected beam vector in the reflection order 1-3-2 and 2-3-1 is:

(26)$$B{^{\prime}_{132or231}} = \left( {\begin{array}{*{20}{c}} {B{^{\prime}_{132or231x}}}\\ {B{^{\prime}_{132or231y}}}\\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{{\Delta (\cos ({\Theta _1})\cos ({\Theta _3}) + \sqrt 2 \cos ({\Theta _2})\sin ({\Theta _1}) - \sin ({\Theta _1})\sin ({\Theta _2})\sin ({\Theta _3}))}}{{\sqrt 3 }}}\\ {\frac{{\Delta (\sqrt 2 \sin ({\Theta _2}) + \cos ({\Theta _2})\sin ({\Theta _3}))}}{{\sqrt 3 }}}\\ 1 \end{array}} \right)$$

The reflected beam vectors with reflection order 3-2-1 and 3-1-2 have a matrix element representation with a negative sign to Eq. (25). The reflected beam vector matrix of reflection order 2-3-1 and 1-3-2 is the negative matrix of Eq. (26).

When the object to be measured rotates along the three free angles of pitch Yaw and roll, according to Eqs. (25) and (26), Eqs. (27) and (28) of the imaging change law of the reflected beam through the three internal reflectors of the non-standard CCCR can be derived. However, it is worth emphasizing that according to section 5.3 of the literature [23], Eq. (28) also needs to take into account the influence of the refractive index n of the CCCR material:

(27)$$\Xi = a\tan \left( {\frac{{B{^{\prime}_{123or213x}} \cdot f}}{{B{^{\prime}_{123or213y}} \cdot f}}} \right) = a\tan \left( { - \frac{{\cos ({\Theta _1})\sin ({\Theta _3}) + \cos ({\Theta _3})\sin ({\Theta _1})\sin ({\Theta _2})}}{{\cos ({\Theta _2})\cos ({\Theta _3})}}} \right)$$

(28)$$\displaystyle\Psi = a\tan \left( {\frac{{B{^{\prime}_{132y}} \cdot f}}{{B{^{\prime}_{132x}} \cdot f}}} \right) = a\tan \left( {\frac{{\sqrt 2 \sin ({\Theta _2}) + n\cos ({\Theta _2})\sin ({\Theta _3})}}{{n\cos ({\Theta _1})\cos ({\Theta _3}) + \sqrt 2 \cos ({\Theta _2})\sin ({\Theta _1}) - n\sin ({\Theta _1})\sin ({\Theta _2})\sin ({\Theta _3})}}} \right)$$

According to the description of Eqs. (27) and (28), the three-degree-of-freedom change of the non-standard CCCR will cause the reflected beam imaging change as shown in Fig. 7:

1. The Angle deviation of the Ξ is obtained by imaging the reflected beams in the sequence of 1-2-3 and 2-1-3 reflections.
2. The reflection beam imaging according to the reflection order of 1-3-2 and 2-3-1 will produce the ψ Angle skew.
3. At the same time, the image of the beam reflected from the aperture plane will be displaced. $(29)$$dx = f \cdot (2 \cdot {\Theta _1})$$$ $(30)$$dy = f \cdot (2 \cdot {\Theta _2})$$$

Fig. 7. Imaging changes of the reflected beam during rotation of a non-standard CCCR.

Download Full Size | PDF

According to the inclination angle of the internal reflected beam and reflected beam imaging of aperture surfaces in Fig. 7, and from Eqs. (29), (30), (27) and (28), the roll Angle Θ₃ of CCCR can be measured.

3. Experiment and results

In this section, the performance of the RAC is tested for resolution, range, and accuracy under multi-axis crosstalk.

3.1 Experimental platform

The experimental device of this paper as shown in Fig. 8(a)and (b). The RAC of the platform is based on Nikon 6D autocollimator of Japan prototype. The RAC is equipped with a SONY MI-20 CMOS sensor, which has a resolution of 5496×3672 and a pixel size of 2.4×2.4 μm². The cross mercerized stop in front of the light source is changed to a circular stop with a radius of 0.5 mm, with a focal length of 700 mm, an aperture of 70 mm, measurement range is 30’, and the resolution is 0.003”.

Fig. 8. experimental device: (a) Schematic diagram; (b) photo-realistic

Download Full Size | PDF

CCCR is made of glass material with refractive index n = 1.5163 (when the wavelength is 0.89 μm), and the clear aperture is 68 mm. The angle between mirror 1 and mirror 2 is L_1-2=90° ±0.5”, and the cylindrical radius of mirror 3 r=6065 mm ±0.1 mm.

We mainly use two orthogonal autocollimators AC1 and AC2 as the measurement reference of the RAC. AC1 is CA860, produced by Anshan Guang zhun Technology Co., LTD. Its measuring range is ±350”, resolution is 0.01” and accuracy is 0.3”. AC2 is based on the Japanese Nikon 6D with a MI-20CMOS sensor. AC1 and AC2 are respectively aligned with the two orthogonal planes of the cube mirror to ensure that their optical axes coincide with the OY and OZ axes of the turntable, respectively. This design enables AC1 to measure the roll angle of the turntable, and AC2 to detect the pitch Angle Θ₁ and yaw Θ₂ changes of the turntable, thus effectively avoiding the multi-axis crosstalk between Θ₁ and Θ₂ on the rotation direction of the turntable Θ₃.

In addition, in the experiment, the PT5 three-degree-of-freedom precision turntable made in China was mainly used to drive the CCCR to rotate, and the rotation range of three-degrees-of-freedom is greater than 30°, and the minimum angular displacement is 1”. A three-degree-of-freedom small turntable with an angle range of ±20° is also used. The small turntable is installed between the cube mirror and the precision turntable. When the rotation angle of the precision turntable exceeds the measurement range of AC1, rotate the small turntable to adjust the deflection angle between the cube mirror and the reference AC until the angle of AC1 is reset, so that the reference AC has a sufficient measurement range.

3.2 Measurement data acquisition

In the process of measuring the roll angle of the reflector by the AC, we convert the rotation angle of the imaging into the linear displacement (i. e. height change) of the image and the original coordinate system for measurement. The measurement method is shown in Fig. 9.

Fig. 9. Schematic diagram of data collection

Download Full Size | PDF

According to Eqs. (27) and (28), the measurement equation can be transformed into (31) and (32).

(31)$$\tan \Xi ={-} \frac{{{h_2}}}{{y^{\prime}}}\textrm{ = } - \frac{{\cos ({\Theta _1})\sin ({\Theta _3}) + \cos ({\Theta _3})\sin ({\Theta _1})\sin ({\Theta _2})}}{{\cos ({\Theta _2})\cos ({\Theta _3})}}$$

(32)$$\tan \Psi = \frac{{{h_1}}}{{x^{\prime}}}\textrm{ = }\frac{{\sqrt 2 \sin ({\Theta _2}) + n\cos ({\Theta _2})\sin ({\Theta _3})}}{{n\cos ({\Theta _1})\cos ({\Theta _3}) + \sqrt 2 \cos ({\Theta _2})\sin ({\Theta _1}) - n\sin ({\Theta _1})\sin ({\Theta _2})\sin ({\Theta _3})}}$$

where x’ and y’ are the projections of the imaging beams 4 and 1 on the X-axis and the Y-axis, respectively, h₁ and h₂ are the vertical heights of the imaging beams 4 and 1 to the X-axis and the Y-axis, respectively.

3.3 Resolution test of RAC

In the process of rotating the cross imaging relative to the CMOS coordinate system OXY, since the size of the image is fixed, the projection x’, and y’ of the image on the OXY coordinate system will become shorter with the increase of the rotation angle of the reflector. According to Eqs. (29) and (30), the minimum resolution Angle Θ_3min of RAC will also decrease accordingly. Therefore, in order to verify the minimum resolution Θ_3min of the RAC, the range selected in the experiment at this stage is Θ₃=20° to 20° -20”, where 20° is the maximum range of RAC, and ±20” is the optimal measurement accuracy range of AC1. The specific experimental steps are as follows:

(1) The turntable rotates 20° along the Z₁-axis, while the small turntable is used to modulate the measuring range of AC1 to the Z-axis to 20° ±350”, which is recorded as the initial position of measurement.
(2) The turntable rotates around the Z₁-axis in the range of 20° to 20° -20”, and the rotation angle step is 1”.
(3) In the rotation of the turntable, AC2 is used to ensure that the turntable rotates only around Z₁-axis.
(4) The angle Θ₃ of the turntable rotation about Z₁-axis is recorded as actual value by AC1.
(5) In each step of rotation of the turntable, the projection lengths X and Y of imaging beams 4 and 1 at the initial position (i.e., 20° of the turntable around Z₁-axis.) are taken as measurement datum. CMOS of RAC is used to measure image beams 4 and 1 with vertical heights h₁ and h₂ of X and Y axes respectively.
(6) According to the average result of Eqs. (29) and (30), calculate the roll angle change value Θ₃ measured by the RAC.
(7) The RAC made 20 measurements during the residence time of 10s per step. The above process was repeated 6 times.

The test results are shown in Fig. 10. The results show that the output of roll Angle Θ₃ can be clearly displayed every 1” interval, indicating that the system resolution of RAC is less than 1”. The measuring principle of the traditional photoelectric AC is to convert the angle information of the reflector into the displacement information of the image, and the angular resolution Θ_min is determined by the relationship between the minimum displacement resolution d_min of the photoelectric sensor and the focal length f of the collimating lens, Θ_min= d_min/2f. The RAC measures the height variation Δh₁ and Δh₂ of the image beam relative to the initial image coordinate system OXY, and its minimum resolution angle is Θ_3min= Δh_1min or Δh_2min/y’ or x’. Since the minimum height Δh_1min, Δh_2min and d_min are the minimum displacement resolution of photoelectric sensor, the minimum resolution ratio between RAC and traditional AC is Θ_{3 /}Θ_3min= 2f/x’ or y’, in theory. After adding a CMOS camera, the focal length of the Nikon 6D is 700 mm, and the resolution reaches 0.003”, and when the CCCR is rotated by 20°, the projections of the imaging beams 4 and 1 on the OX and OY axes x’ and y’, is 4.13 mm. Therefore, the theoretical angle measurement resolution of the RAC is 1.0169”, which is close to the experimental value.

Fig. 10. Results of the resolution test.

Download Full Size | PDF

3.4 Measuring range and multi-axis crosstalk characteristics of RAC

To discuss the measurement accuracy of the full range of RAC, the crosstalk effect of pitch Angle Θ₁ and yaw Angle Θ₂ on roll Angle measurement should be considered. Therefore, in this stage of the experiment, the turntable was used to modulate the CCCR into the following three states: Θ₁=0”, Θ₂=0” (no crosstalk state), Θ₁=0”, Θ₂=350” and Θ₁=350”, Θ₂=0"(two-axis crosstalk state), Θ₁=350”, Θ₂=350” (three-axis crosstalk state). The CCCR rotates around the OZ₁-axis at the roll angle Θ₃ within the range of ±20° with each step of 300” under the four groups of combined angles of the three states. The operation details of this part of the experiment are basically similar to those of the previous part. The difference is that AC1 needs to be reset to zero by using a small turntable after each step of rotation, so the residence time after each step of rotation is not strictly defined. The RAC is measured as the average of 20 measurements at each step. In addition, the entire measurement process was repeated six times.

As shown in Fig. 11, when the measurement range is within ±20°, the maximum average measurement errors of the RAC under no crosstalk, two-axis crosstalk, and three-axis crosstalk are ±5.8”, ±6.1”, ±5.9” and ±5.3”, respectively, and the range of difference is only 0.8”, which shows that the technology has better anti-multi-axis crosstalk performance. In addition, with the increase of the measurement Angle, the repeatability of the RAC becomes worse, reaching the maximum uncertainty range of ±39.7”, which may be caused by the superposition of operating errors in the range expansion of the measurement datum AC1 through a small turntable.

Fig. 11. Range and multi-axis crosstalk test results

Download Full Size | PDF

4. Conclusion

In this paper, RAC based on a CCCR is proposed. The CCCR is obtained by deforming a reflection surface of the standard Cube-corner into a cylindrical surface. This design does not need to change the other measurement structure of the AC, and only uses the deduced measurement for equation to calculate the rotation Angle of the reflected imaging beam. High-precision, high-resolution, wide-range roll angle measurement can be achieved. In the experiment, we built an experimental verification benchmark platform with sufficient measurement performance, accuracy and sufficient range, and obtained the minimum resolution performance of RAC of 1”. In addition, by pre-setting the turntable to four sets of combined angles in three different states, the excellent anti-multi-axis crosstalk characteristics of the RAC are verified, and the accuracy is better than 6.1” in the measurement range of ±20°.

Funding

National Key Research and Development Program of China (2020YFC2003301); National Natural Science Foundation of China (62005033); Basic Research Project of Chongqing Science and Technology Commission (CSTC-2020jcyj-msxm0603); Grant of Russian Federation (Priority 2030); Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202000609).

Disclosures

The authors declare no conflict of interest.

Data availability

The datasets are available from the corresponding author on reasonable request.

References

1. H. X. Dong, Q. Fu, X. Zhao, Q. Quan, and R. F. Zhang, “Practical rotation angle measurement method by monocular vision,” Appl. Opt. 54(3), 425–435 (2015). [CrossRef]

2. W. Li, J. Jin, X. Li, and B. Li, “Method of rotation angle measurement in machine vision based on calibration pattern with spot array,” Appl. Opt. 49(6), 1001–1006 (2010). [CrossRef]

3. K. Shi, J. Su, and W. Hou, “Roll angle measurement system based on differential plane mirror interferometer,” Opt. Express 26(16), 19826–19834 (2018). [CrossRef]

4. E. Zhang, Q. Hao, B. Chen, L. Yan, and Y. Liu, “Laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect,” Opt. Express 22(21), 25587–25598 (2014). [CrossRef]

5. J. Qi, Z. Wang, J. Huang, Q. Wang, and J. Gao, “Heterodyne interferometer with two parallel-polarized input beams for high-resolution roll angle measurement,” Opt. Express 27(10), 13820–13830 (2019). [CrossRef]

6. S. R. Gillmer, X. Yu, C. Wang, and J. D. Ellis, “Robust high-dynamic-range optical roll sensing,” Opt. Lett. 40(11), 2497–2500 (2015). [CrossRef]

7. Y. D. Cai, B. H. Yang, and K. C. Fan, “Robust roll angular error measurement system for precision machines,” Opt. Express 27(6), 8027–8036 (2019). [CrossRef]

8. W. Ren, J. Cui, and J. Tan, “Parallel beam generation method for a high-precision roll angle measurement with a long working distance,” Opt. Express 28(23), 34489–34500 (2020). [CrossRef]

9. Y. Fan, Z. Lou, and Y. Huang, “Self-compensation method for dual-beam roll angle measurement of linear stages,” Opt. Express 29(17), 26340–26352 (2021). [CrossRef]

10. Y. L. Chen, Y. Shimizu, J. Tamada, Y. Kudo, S. Madokoro, K. Nakamura, and W. Gao, “Optical frequency domain angle measurement in a femtosecond laser autocollimator,” Opt. Express 25(14), 16725–16738 (2017). [CrossRef]

11. F. Zhu, J. Tan, and J. Cui, “Common-path design criteria for laser datum based measurement of small angle deviations and laser autocollimation method in compliance with the criteria with high accuracy and stability,” Opt. Express 21(9), 11391–11403 (2013). [CrossRef]

12. F. Zhang, H. Zhang, and Y. Qi, “Long dynamic range spread spectrum optical domain reflectometer,” Optoelectron Lett. 17(11), 651–655 (2021). [CrossRef]

13. J. Mou, J. Su, and L. Miao, “Research on Field Application Technology of Dynamic Angle Measurement Based on Fiber Optic Gyroscope and Autocollimator,” IEEE Sensors J. 21(13), 15308–15317 (2021). [CrossRef]

14. G. Wang, Y. Gao, and S. Zhang, “Physical simulation of trajectory tracking for tracking performance evaluation of photoelectric turntable,” Optoelectron Lett. 16(4), 272–278 (2020). [CrossRef]

15. J. Luo, Z. Wang, Z. Wen, M. Li, S. Liu, and C. Shen, “Reflector automatic acquisition and pointing based on auto-collimation theodolite,” Rev. Sci. Instrum. 89(1), 015101 (2018). [CrossRef]

16. J. G. Bergues, C. Schürrer, and N. Brambilla, “Uncertainty determination of the set nikon 6b autocollimator + visual interface,” IEEE T. Instrum. Meas. 67(5), 1058–1064 (2018). [CrossRef]

17. V. Heikkinen, V. Byman, I. Palosuo, B. Hemming, and A. Lassila, “Interferometric 2D small angle generator for autocollimator calibration,” Metrologia 54(3), 253–261 (2017). [CrossRef]

18. M. Schumann, R. DGeckeler, and M. Krause, “The spatial angle autocollimator calibrator: Optimised model, uncertainty budget and experimental validation,” Metrologia 56(1), 015011 (2019). [CrossRef]

19. R. P. Li, I. Konyakhin, and Q. Zhang, “Error compensation for long-distance measurements with a photoelectric autocollimator,” Opt. Eng. 58(10), 1 (2019). [CrossRef]

20. Y. L. Chen, Y. Shimizu, Y. Kudo, S. Ito, and W. Gao, “Mode-locked laser autocollimator with an expanded measurement range,” Opt. Express 24(14), 15554–15569 (2016). [CrossRef]

21. I. Konyakhin, R. P. Li, and M. Zhou, “A 2D quadrangular pyramid photoelectric autocollimator with extended angle measurement range,” Optoelectron Lett. 17(8), 468–474 (2021). [CrossRef]

22. R. P. Li, M. Zhou, I. Konyakhin, K. Di, Y. Lu, J. Guo, and Y. Liu, “Cube-corner autocollimator with expanded measurement range,” Opt. Express 27(5), 6389–6403 (2019). [CrossRef]

23. R. P. Li, Y. Zhen, and K. Di, “Three-degree-of-freedom autocollimator with large angle-measurement range,” Meas. Sci. Technol. 32(11), 115005 (2021). [CrossRef]

Roll angle autocollimator measurement method based on a cylindrical cube-corner reflector with a high resolution and large range

Abstract

1. Introduction

2. Principle and analysis

2.1 MCCR structure for roll angle measurement

2.2 CCCR structure for roll angle measurement

2.3 Reflection matrix of CCCR

2.4 Reflected beam imaging

2.5 Measuring formula of RAC

3. Experiment and results

3.1 Experimental platform

3.2 Measurement data acquisition

3.3 Resolution test of RAC

3.4 Measuring range and multi-axis crosstalk characteristics of RAC

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Equations (32)

Optics Express