This paper presents an optoelectronic measurement system for measuring 6 degree-of-freedom (DOF) motion error of rotary parts. It comprises a pyramid-polygon-mirror, three laser diodes and three 2-axis position sensing detectors (PSD). The laser/PSD pairs are arranged evenly around the pyramid-polygon-mirror, which is mounted rigidly on and aligned axially with the rotary part to be measured. Laser rays from the laser diodes are reflected off the respective mirrors to the respective PSDs. The incidence point of the laser ray on the PSD’s surface varies with the pose of the pyramid-polygon-mirror, allowing the PSD to register variation in the mirror and, thereby, the rotary part. With appropriate orientation of the lasers and PSDs, this system can measure variation (error) during rotation of a rotary part. By use of skew-ray tracing and first order Taylor series expansion, the system achieves measurement of translational and rotational motion errors for each Cartesian axis. To validate the proposed methodology, a laboratory prototype system is built. System verification and stability tests are conducted to evaluate its performance. Stability test results show that measurement errors and maximum crosstalk are within ±1 μm in translation and ±1.5 arc sec in rotation.
© 2007 Optical Society of America
The mechanical elements of machines are connected most commonly by either rotary or translational parts, so there is increasing demand for accurate measurement and monitoring of the motion of these two mechanical parts. Many devices have been proposed for this task. Previous research includes multi-DOF error measurement system , 5-DOF motion error measurement system , 6-DOF geometric error measurement system  and 6-DOF motion measurement system [4, 5]. This present study develops and verifies a high-accuracy optoelectronic measurement system for 6-DOF motion error in rotary parts.
Many methods have been developed to provide motion and positioning data for the numerous revolving mechanical elements that exist in modern technology. High-accuracy angular positioning commonly is obtained by autocollimators, although laser interferometers sometimes perform this function at greater cost and difficulty. Autocollimators project a collimated light beam (typically a laser) onto a prismatic polygonal mirror that is rigidly attached to and axially aligned with the rotary part to be measured. The light beam reflects from the mirror and onto a position sensing detector (PSD). Motion errors of that mirror cause the light beam to change its position on the PSD, which then outputs the angular deviation of the mirror (rotary part). Autocollimators use a pair of lenses between the mirror and the PSD to increase sensitivity to beam deviation, but also increasing system cost, size and making system accuracy dependant on lens accuracy. Autocollimator resolution is high, but the angular range measurable per polygon mirror face is small, typically in the range of 400 arc seconds (0.1°). Importantly, a single autocollimator can measure only 2-DOF motion error since the reflection process is a function of the unit normal vector of the mirror. Nevertheless, a rigid body in 3D space has 6 DOF. Consequently, a rotary part may have three angular motion errors and three translational motion errors. To measure all of these six motion errors simultaneously and accurately is a challenging task, especially in a small practical system. Presently available PSDs are planar devices which output 2 1-axis readings capable of measuring at most 2 DOF. Consequently, some arrangement of multiple sensors, lasers and mirrors is required to detect 6-DOF motion error in a rotary part. Literature is meager for literature related to 6-DOF rotary system measurement but includes 5-DOF motion error measurement of translational parts , 4-DOF motion measurement of an indexing (rotary) table [6, 7] and 6-DOF measurement of vibration .
The optoelectronic hardware of the system presented in this paper is evolved from the autocollimator described above, but employs 3 laser diodes and 3 2-axis PSDs. Also, the prismatic polygon mirror in the autocollimator is replaced by a pyramid-polygon-mirror of mn sides, where m is the number of laser/PSD pairs in the system and n is an integer ≥1. The use of n allows for multiplying of measurement faces for greater sampling of a rotational period, while m maintains a minimum of one face for each laser/PSD pair since simultaneous measurement of all the laser/PSD pairs is desired. A design example of the proposed system is seen in Fig. 2, in which m=3 and n=2 (i.e. the number of faces is 6). For 6-DOF measurement, the minimum number of laser/PSD pairs is 3, though more could be used. Notably, in an autocollimator, incident and reflected light beams are coincident and require use of a beam splitter, but in our proposed system the incident and reflected light beams are separated naturally by different physical locations of laser and PSD. The beam-to-beam angle in the example shown below is set at 90° for easy computation. Furthermore, the 3 laser diodes and 3 2-axis PSDs are arranged evenly around the circumference of the pyramid-polygon-mirror which is mounted rigidly and centered axially on the rotary part to be measured. The light rays from the laser diodes are projected onto the respective mirrors, then reflected to and detected by the respective PSDs. Each PSD outputs the position of the incident light beam on the surface of the PSD along the axis in question. Consequently, the 6-DOF motion error of a rotary part can be measured.
2. Skew-ray tracing at flat boundary surfaces
This section reviews as much of our algebraic skew-ray tracing technique as is necessary to model the 6-DOF motion error measurement system. The proposed optoelectronic system contains only flat reflecting boundary surfaces, so only skew-ray tracing at flat boundary surfaces  (Fig. 1) and the reflection process will be considered. The ith flat boundary surface iri of an optical system can be obtained by rotating its generating curve [βi, 0 0 1]T (βi ≥ 0) about its yi axis, i.e.
where Rot is a rotation that is defined in the Appendix, and the symbols S and C denote sine and cosine respectively. The unit normal vector ini of the flat boundary surface can be obtained from
where si is set to +1 or -1, the choice being made such that the cosine of the incident angle is greater than zero, i.e. Cθi > 0 .
Equations (1) and (2) are parametric equations of the flat boundary surface iri and its unit normal vector ini, respectively. However, most derivations in the paper are built with respect to the world coordinate frame (xyz)0. Therefore, the following pose matrix of the world coordinate frame (xyz)0 with respect to (xyz)i is needed:
Then the unit normal vector ni referred to (xyz)0 is given by
In Fig. 1, a light ray originating at point P i-1 =[P i-1x P i-1y P i-1z 1]T and traveling along the unit directional vector ℓ i-1 =[ℓ i-1x ℓ i-1y ℓ i-1z 0]T is reflected at the flat boundary surface ri. When the ray hits the boundary surface, the incidence point Pi is
Incidence point parameters αi and βi are not of interest, since the reflection process is independent of the location of the incidence point. According to Snell’s Law , the reflected unit directional vector ℓi can be expressed as
After reflection at ri, the light ray proceeds onward with Pi as its new source and ℓi as its new unit directional vector.
3. Modeling of 6-DOF motion error measurement system
Shown in Fig. 2 is the proposed system. Laser rays emitted from the laser diodes strike the respective mirrors and then are reflected toward the respective PSDs. These relations can be expressed by Snell’s Law such that the unit directional vectors of a reflected laser ray depends on the unit normal vector of the mirrors. Figure 3 shows a sub-system of Fig. 2, i.e. 1 laser diode, 1 mirror and 1 PSD. Applying the skew-ray tracing of Section 2 to this sub-system, the mirror and PSD are respectively labeled i = 1 and i = 2 . A coordinate frame (xyz)i (i=1,2) is assigned to each boundary surface ri with yi parallel to the normal vector of ri. The laser ray source 1 P 0=[f/√2 -f/√2 0 1]T is positioned in the x 1 y 1 plane and projects a laser ray along 1 ℓ 0 =[-1/√2 1/√2 0 0]T onto the mirror with an incident angle of 45°. Note that the origin of (xyz)1 (i.e., 1 P 1 = [0 0 0 1]T) is the incident point at the mirror in this arrangement. If the z 2 axis of (xyz)2 (the coordinate frame of PSD) is parallel to z 1, then the pose matrix of (xyz)2 with respect to (xyz)1 is given by 1A2 = Rot (z, 135°) Trans (0,d,0).
To model the 6-DOF measurement system, three pairs (labeled respectively as a, b and c) of the sub-system in Fig. 3 are placed evenly around the circumference of the pyramid-polygon-mirror. One should note that there are two coordinate frames, (xyz)0 and (xyz)0', in Fig. 2, where (xyz)0' is the coordinate frame imbedded in the pyramid-polygon-mirror and (xyz)0 is the world coordinate frame. Any motion in the pyramid-polygon-mirror causes these two coordinate frames to deviate. After integrating the sub-systems with the pyramid-polygon-mirror, the pose matrices of the mirrors with respect to (xyz)0' (also with respect to (xyz)0 if there is no error motion in the pyramid-polygon-mirror) are respectively given by
where (tax,tay,taz) , (tbx,tby,tbz) , (tcx,tcy,tcz) , ωay , ωby and ωcy are parameters to be determined. ϕ is the semi-conic angle of a pyramid-polygon-mirror. Now the positions of the laser ray sources with respect to the world coordinate frame (xyz)0 can be obtained by transforming 1 P 0 to (xyz)0, i.e.,
Similarly, the incidence points at the mirrors, expressed with respect to (xyz)0 when there is no error motion in the pyramid-polygon-mirror, are given by
The unit directional vectors of the laser ray sources with respect to (xyz)0 are given by
The three sub-systems are placed evenly around the circumference of the pyramid-polygon-mirror. The unit directional vectors ℓ 0b and ℓ 0c of the laser sources can be obtained by rotating ℓ 0a around axis z 0 through 120° and 240°, i.e.
Similarly, the incidence points P 1b and P 1c can be obtained by rotating P 1a around axis z 0 through 120° and 240°, i.e.
where P 1a = [tax 0 taz 1]T , the centroid of the trapezium mirror, is the incidence point on the mirror of the first sub-system. One has ωay = ωby = ωcy after equating Eqs. (11) and (12), and tbx = tcx = -tax/2, tby = -tcy = √3ax/2, taz = tbz = tcz after equating Eqs. (10) and (13).
4. Modeling of system equations
The prior section discussed the positions and orientations of laser ray sources and incidence points when the pyramid-polygon-mirror is stationary. When the pyramid-polygon-mirror is mounted on a rotary part having 6-DOF motion error described by 0 A 0' = Trans(δx,δy,δz)Rot(z,ηz)Rot(y,ηy)Rot(x,ηx) with respect to the world coordinate frame (xyz)0 , then the coordinate frame (xyz)0' (built in the pyramid-polygon-mirror) deviates from (xyz)0. Their relative pose is given by the pose matrix
where Trans(δx,δy,δz) translates the pyramid-polygon-mirror by the vector δxi⃗ + δyj⃗ + δz k⃗ and Rot(z,ηz), Rot(y,ηy), Rot(x,ηx) rotate the pyramid-polygon-mirror about the z, y and x axes, respectively. Note that hereafter position motions and angular motions will be abbreviated as δ̱ and η̱ respectively.
To perform the skew-ray tracing of Section 1, we need the pose matrices 1aA0 0 = 1aA0' 0'A0, 2a A0 = 2a A1a 1aA0, 1b A0 = 1b A0' 0' A0, 2b A0 = 2b A1b 1bA0, 1c A0 = 1cA0' 0'A0 and 2c A0 = 2c A1c 1cA0. The matrices 1aA0', 1bA0' and 1b A0' can be obtained respectively from the inverse matrices of Eqs. (8), i.e.
Note that 0' A 0 = Rot(x,-ηx)Rot(y,-ηy)Rot(z,-ηz)Trans(-δx,-δy,-δz) is the inverse matrix of 0 A 0' and 2a A 1a = 2b A 1b = 2c A 1c = Trans(0,-d,0) Rot(z,-135°) is the inverse matrix of 1A2. The required pose matrices (1a A 0, 1b A 0, 1c A 0, 2a A 0, 2b A 0,2c A 0) in skew-ray tracing described in Section 2 can therefore be obtained from the matrix products.
Now having the pose matrices (1a A 0, 1b A 0, 1c A 0, 2a A 0, 2b A 0,2c A 0) of the boundary surfaces, the laser ray sources (P 0a,P 0b ,P 0c) and their unit directional vectors (ℓ 0a ,ℓ 0b,ℓ 0c), one can perform skew-ray tracing according to Section 1. By setting i=1, one obtains the incidence points and the unit directional vectors of the laser rays undergoing reflection at the mirrors from Eqs. (5) and (7):
where B 1a, B 1b, B 1c, λ 1a, λ 1b and λ 1c are defined by Eq. (6).
One can also obtain the incident points on the PSD surfaces from Eq. (5) by setting i=2, expressed as
where λ 2a, λ 2b, and λ 2c are defined in Eq. (6).
Note that Eq. (18) expresses the incidence points with respect to (xyz)0. The PSD readings can be obtained from 2a P 2a = 2a A 1a A 0 P 2a, 2b P 2b = 2b A 1b 1b A 0 P 2b and 2c P 2c = 2c A 1c 1c A 0 P 2c, given by
Equations (19a)–(19c) indicate the readings as functions of the 6-DOF motion errors.
In the real world, it is impossible for exact placement of any component at a specified position and orientation. Consequently, setting errors exist in every system. In order to remove reading errors due to setting errors, the proposed system first measures an initially stationary surface by placing the pyramid-polygon-mirror on that surface. By substituting δ̱ = η̱ = 0̱ into Eq. (19), one has the following recorded readings of PSDs when this stationary surface is measured:
If the readings [Xa Za]T , [Xb Zb]T and [Xc Zc]T are known and Eqs. (21a–c) are independent, then the 6-DOF motions, δ̱ and η̱, can be determined numerically.
5. Linearization of system equations
Numerical iteration is needed to solve for the 6-DOF motions δ̱ and η̱ for the PSD readings of Eq. (21), since Eq. (21) is in implicit nonlinear form. Note that the δ̱ and η̱ are usually very small. Therefore we can use a first order Taylor series expansion to expand Eq. (21) at δ̱ = η̱ = 0̱ to obtain the linear form of the system equations. One obtains the following linear equations from Eq. (21) when Sϕ = 1/√3, Cϕ = √2/3 , tax = 2h/(3√6) and tax = h/(3√3) are used to perform the first order-Taylor series expansion of Eq. (21):
Note that when ωay = 0° or ωay =180° , then Eq. (22) only provides five independent equations, since Za + Zb + Zc = 0 , i.e. in this case the system can only measure 5-DOF error motions. Consequently, ωay = 0° or ωay = 180° must be avoided when constructing the proposed system.
The following parameters are used in the construction of our laboratory prototype of the system: ωay = -90°, d = 70mm. Then the 6-DOF motions are respectively given by:
Table 1 shows the differences of δ̱ and η̱ as calculated from Eqs. (23) and (21) for several sets of simulated sensor readings. It is shown from this table that when the angular motion is less than ±0.5°, the linear equation Eq. (23) can provide accurate δ̱ and η̱ from the sensor readings.
6. Experimental verification and discussion
Figure 4 shows the lab-built prototype system. The outer mirrored surfaces of a commercial corner cube are used as a pyramid-polygon-mirror of m=3, n=1. Because no system can be fabricated without manufacturing errors, each of the lasers and PSDs and the holder of the pyramid-polygon-mirror have pose adjustment mechanisms so that, during initial set up, the laser rays can be centered in their respective PSDs at a table rotation of 0°. Components not shown in Fig. 4 include a conventional desktop PC connected to an A/D card connected in turn to commercial signal conditioning circuit (On-trak OT-301, USA) optimized for use with PSDs, that connects finally with the PSDs themselves. Table 2 lists the component specifications. Mathematically, the resolution of position motions δ̱ and angle motions η̱ are 0.106 μm and 0.274 arc sec, respectively, when an Advantech PCI-1716 AD card is used. Each of the 6 DOFs is tested independently to verify starting system function. Linear motion verification uses a 5-DOF manual-stage that translates along the x-, y- and z-axes and rotates around the y- and z-axes by manual adjustment, with an accuracy of 10 μm. Angular verification uses a manual rotary table and manual goniometer. The rotary table rotates around the z-axis with an accuracy of 1°. The goniometer has an accuracy of 5 arc min. Input-output curves for each DOF are obtained by manually making a series of stepwise changes to the test table so as to adjust the position and angle motions δ̱ and η̱ over the ranges of 3 mm and ±0.5°, respectively. These changes result in the PSD output changing over about half the possible working area for linear motion and about 1/5 for angular motion. Figure 5(a–f) show the verification results for δ̱ and η̱. It can be seen that each individual DOF shows good linearity for motion within the tested range. Figure 5(a–f) also give standard deviations for system uncertainty, indicating accuracies of 0.5 μm and 0.4 arc sec for position motion and angular motion, respectively.
General system stability was evaluated under normal laboratory conditions (i.e. no special temperature or vibration isolation) by setting table rotation to 0°, warming up the system for about 15 minutes and then continuously recording the output signal for 5 minutes. The results of this test can be seen in Fig. 6, which shows that δ̲ and η̲ remain within ±1 μm and ±1.5 arc sec, respectively, over 300 sec. Finally, this prototype system (Fig. 7) was employed to measure 6-DOF position/angle motion of a rotary table (NewPort PM500-360). Figures 8(a–f) show the measured motion errors, δx, δy, δz, ηx, ηy, and ηz, of a rotary table from the lab-built system. Figure 8(e) and Fig. 9(a) show the measured angular motion ηy for our prototype system and our laboratory autocollimator (NewPort LDS Vector, measurement range: 400 arc sec), respectively. The flattened curve in Fig. 9(a) is an artifact resulting from the curve exceeding the scale of the system. It should be remembered that our laboratory prototype data is for a 3-sided mirror while the autocollimator data is for a 24-sided mirror; hence the higher number of data points in the autocollimator’s curve, and demonstrating the significance of increasing the number of polygon faces. Comparison of the two plots (Fig. 8(e) and Fig. 9(a)) shows significantly different amplitude and shape. We assume this difference is the result of the translation-stage used during verification of our prototype system but not used for the autocollimator. Measured angular motion ηz can be seen in Figs. 8(f) and 9(b) for our prototype and the commercial autocollimator, respectively, again validating the presented modeling.
The foregoing has demonstrated and verified an optoelectronic system capable of high-precision measurement of motion errors for all 6-DOF of a rotating object. If we were to employ enhanced data collection and correlation, our presented system could easily yield rotational direction (slope of the PSD curve), angular velocity (rate of change of the slope), the momentary position of the rotating part as well as basic error motion. An increased number of polygon faces could allow constant monitoring of all these parameters, regardless of whether the rotary part was moving or static. If the mirror were integrated on the surface of the rotary part and a miniaturized laser/PSD array were included in a housing built around the rotary part, a highly useful basic component would result. Various factors would need to be considered and optimized such as the width of the PSD, distance from the PSD to mirror, needed resolution, etc. These various issues go beyond the scope of this present study, but may be pursued in our future work.
This paper has presented and verified a 6-DOF optoelectronic motion error measurement system constructed from 3 laser-diode/PSD pairs and a pyramid-polygon-mirror. For high accuracy, analytic skew-ray tracing was used to model the system and determine the system equations for expressing the 6-DOF motion. To improve computational speed for this proof-of-concept paper, we employed a first-order Taylor series expansion to obtain a linear form of the system equations. The proposed system was validated using a laboratory-built prototype to perform calibration and stability experiments. Calculations showed position and angular motion measurement ranges of ±3.5 mm and ±2.5° , respectively. Stability testing for 5 minutes of continuous operation showed position and angular measurement variance in the range of±1 μm and ±1.5 arc-sec, respectively.
The authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under grant number (NSC 96-2221-E-006-208-).
In this paper the ith position vector Pix i + Piy j + Piz k + is written as a column matrix jPi = [Pix Piy Piz 1]T, where the fourth component is a scale factor. The pre-superscript “j” of jPi indicates this ith vector is referenced to coordinate frame (xyz)j. Given a point jPi, its transformation kPi is represented by the matrix product kPi = kAj jPi. Here, kAj is a 4×4 pose (positional and rotational data) matrix which defines the pose of the coordinate frame (xyz)j with respect to the coordinate frame (xyz)k. These notation rules are also applicable to the ith unit directional vector jℓi= [ℓix ℓiy ℓiz 0]T. If a vector is given with respect to the world coordinate frame (xyz)0, then its superscript “0” is omitted for simplicity.
The pose matrix of a coordinate frame with respect to another coordinate frame can be defined by a sequence of rotations and translations about the x, y, or z axes. The transformation matrices corresponding to rotations about x, y, or z axes by an angle θ are respectively given by
The transformation corresponding to a translation by a vector tx i⃗ + ty j⃗ + tz k⃗ is
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