1. Introduction
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 [
11. J. Ni and S. M. Wu, “An on-line measurement technique for machine volumetric error compensation,” ASME J. Eng. Indus. 115, 85–92 (1993).
], 5-DOF motion error measurement system [
22. P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” ASME J. Dyn. Syst. 118, 425–433 (1996). [CrossRef]
], 6-DOF geometric error measurement system [
33. S. W. Lee, R. Mayor, and J. Ni, “Development of a six-degree-of-freedom geometric error measurement System for a Meso-Scale Machine Tool,” ASME J. Manu. Sci. Eng. 127, 857–865 (2005). [CrossRef]
] and 6-DOF motion measurement system [
44. E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement system for milli-structures,” Mea. Sci. Technol. 12, 1495–1502 (2001). [CrossRef]
,
55. J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom-displacement systems using cooperative targets,” Precis. Eng. 26, 99–104 (2002). [CrossRef]
]. 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 [
22. P. D. Lin and K. F. Ehmann, “Sensing of motion related errors in multi-axis machines,” ASME J. Dyn. Syst. 118, 425–433 (1996). [CrossRef]
], 4-DOF motion measurement of an indexing (rotary) table [
66. C. H. Liu, W. Y. Jywe, L. H. Shyu, and C. J. Chen, “Application of a diffraction grating and position sensitive detectors to the measurement of error motion and angular indexing of an indexing table,” Precis. Eng. 29, 440–448 (2005). [CrossRef]
,
77. W. Y. Jywe, C. J. Chen, W. H. Hsieh, P. D. Lin, H. H. Jwo, and T. Y. Yang, “A novel simple and low cost 4 degree of freedom angular indexing calibrating techniques for a precision rotary table,” Int. J. Mach. Tool Manu. 47, 1978–1987 (2007). [CrossRef]
] and 6-DOF measurement of vibration [
88. E. H. Bokelberg, H. J. Sommer III, and M. W. Trethewey, “A six-degree-of-freedom laser vibormeter, part I and II,” J. Sound Vib. 178, 643–667 (1994). [CrossRef]
].
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 [
99. P. D. Lin and T. T. Liao, “Analysis of Optical Elements with Flat Boundary Surfaces,” J. Appl. Opt. 42, 1191–1202 (2003). [CrossRef]
] (
Fig. 1) and the reflection process will be considered. The
ith flat boundary surface
^{i}r_{i} of an optical system can be obtained by rotating its generating curve [
β_{i}, 0 0 1]
^{T} (
β_{i} ≥ 0) about its
y_{i} 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 ^{i}n_{i} of the flat boundary surface can be obtained from
where s_{i} is set to +1 or -1, the choice being made such that the cosine of the incident angle is greater than zero, i.e. Cθ_{i} > 0 .
Equations (
1) and (
2) are parametric equations of the flat boundary surface
^{i}r_{i} and its unit normal vector
^{i}n_{i}, 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 n_{i} 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
r_{i}. When the ray hits the boundary surface, the incidence point
P_{i} is
where
Incidence point parameters
α_{i} and
β_{i} are not of interest, since the reflection process is independent of the location of the incidence point. According to Snell’s Law [
99. P. D. Lin and T. T. Liao, “Analysis of Optical Elements with Flat Boundary Surfaces,” J. Appl. Opt. 42, 1191–1202 (2003). [CrossRef]
], the reflected unit directional vector
ℓ_{i} can be expressed as
After reflection at r_{i}, the light ray proceeds onward with P_{i} as its new source and ℓ_{i} as its new unit directional vector.
Fig. 1. Light ray at a flat reflective boundary surface.
3. Modeling of 6-DOF motion error measurement system
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 (t_{ax},t_{ay},t_{az}) , (t_{bx},t_{by},t_{bz}) , (t_{cx},t_{cy},t_{cz}) , ω_{ay} , ω_{by} and ω_{cy} 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.,
Fig. 2. Schematic diagram of a 6-DOF motion measurement system with a 6-sided pyramid-polygon-mirror.
Fig. 3. A laser/PSD sub-system of the
Fig. 2 measurement system.
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} = [
t_{ax} 0
t_{az} 1]
^{T} , the centroid of the trapezium mirror, is the incidence point on the mirror of the first sub-system. One has
ω_{ay} =
ω_{by} =
ω_{cy} after equating Eqs. (
11) and (
12), and
t_{bx} =
t_{cx} = -
t_{ax}/2,
t_{by} = -
t_{cy} = √3
_{ax}/2,
t_{az} =
t_{bz} =
t_{cz} 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(δ_{x},δ_{y},δ_{z})Rot(z,η_{z})Rot(y,η_{y})Rot(x,η_{x}) 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(δ_{x},δ_{y},δ_{z}) translates the pyramid-polygon-mirror by the vector δ_{x}i⃗ + δ_{y}j⃗ + δ_{z}
k⃗ and Rot(z,η_{z}), Rot(y,η_{y}), Rot(x,η_{x}) 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.
Note that ^{0'}
A
_{0} = Rot(x,-η_{x})Rot(y,-η_{y})Rot(z,-η_{z})Trans(-δ_{x},-δ_{y},-δ_{z}) 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
^{1}A_{2}. The required pose matrices (^{1a}
A
_{0}, ^{1b}
A
_{0}, 1_{c}
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.
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:
The readings of the three PSDs are obtained by subtracting Eq. (
20) from Eq. (
19), i.e.
If the readings [
X_{a}
Z_{a}]
^{T} , [
X_{b}
Z_{b}]
^{T} and [
X_{c}
Z_{c}]
^{T} are known and Eqs.
(21a–c) are independent, then the 6-DOF motions,
δ̱ and
η̱, can be determined numerically.
Table 1:. The differences of δ̱ and η̱ calculated from Eqs. (23) and (21) (units: deg. or mm) |
| |
5. Linearization of system equations
Note that when
ω_{ay} = 0° or
ω_{ay} =180° , then Eq. (
22) only provides five independent equations, since
Z_{a} +
Z_{b} +
Z_{c} = 0 , i.e. in this case the system can only measure 5-DOF error motions. Consequently,
ω_{ay} = 0° or
ω_{ay} = 180° must be avoided when constructing the proposed system.
The following parameters are used in the construction of our laboratory prototype of the system: ω_{ay} = -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.
Table 2:. Component details |
| |
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.
Fig. 4. Photograph of lab-built 6-DOF motion error measurement system using the mirrored exterior of the commercial corner cube as a 3-sided pyramid-polygon-mirror: (a) set up for verification of angular motion; (b) set up for verification of linear motion.
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,
δ_{x},
δ_{y},
δ_{z},
η_{x},
η_{y}, and
η_{z}, of a rotary table from the lab-built system.
Figure 8(e) and
Fig. 9(a) show the measured angular motion
η_{y} 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
η_{z} can be seen in
Figs. 8(f) and
9(b) for our prototype and the commercial autocollimator, respectively, again validating the presented modeling.
Fig. 5.(a). Verification results of δ_{x}.
Fig. 5.(b). Verification results of δ_{y}.
Fig. 5.(c). Verification results of δ_{z}.
Fig. 5.(d). Verification results of η_{x}
Fig. 5. (e). Verification results of η_{y}
Fig. 5. (f). Verification results of η_{z}.
Fig. 6. Results of system stability test (sample rate 1000 Hz): (a) translational parameters δ_{x}, δ_{y} and δ_{z}; (b) rotational parameters η_{x}, η_{y} and η_{z}.
Fig. 7. Photograph of lab-built 6-DOF rotary table measurement system.
Fig. 8. The measured results of the 6-DOF motion error of a rotary table from the lab-built system (square, circle and triangle each equal one rotation of the rotary table).
Fig. 9. Autocollimator results (with 24-sided mirror) for rotary table measurement (flattened curve is artifact caused by curve exceeding measurement range).
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.
7. Conclusion
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.
8. Acknowledgments
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-).
9. Appendix
In this paper the ith position vector P_{ix}
i + P_{iy}
j + P_{iz}
k + is written as a column matrix ^{j}P_{i} = [P_{ix}
P_{iy}
P_{iz} 1]^{T}, where the fourth component is a scale factor. The pre-superscript “j” of ^{j}P_{i} indicates this ith vector is referenced to coordinate frame (xyz)_{j}. Given a point ^{j}P_{i}, its transformation ^{k}P_{i} is represented by the matrix product ^{k}P_{i} = ^{k}A_{j}
^{j}P_{i}. Here, ^{k}A_{j} 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 ^{j}ℓ_{i}= [ℓ_{ix}
ℓ_{iy}
ℓ_{iz} 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 t_{x}
i⃗ + t_{y}
j⃗ + t_{z}
k⃗ is