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Uncertainty investigation of grating interferometry in six degree-of-freedom motion error measurements

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Abstract

An uncertainty analysis of grating interferometry in six degree-of-freedom (DOF) motion error measurements in a linear stage is presented. Six-DOF refers to the translational and rotational motion of a rigid body in a three-dimensional space. The optical measurement method and sensor were constructed and applied to the ballscrew-driven stage for validation. With a single traversal of the stage along the X direction, the optical sensor measured the motion in high resolution along each axis: less than 0.03 arcsec, 20 nm, and 0.4 nm for the rotational, Y and Z directions, and X direction, respectively. A laser interferometer and autocollimator were used for comparison. In conjunction with a performance evaluation for the proposed optical sensor, the measurement uncertainty (k=1) was estimated from the error sources, which included installation error, optics error, electronics error, environmental effects, and motion-induced error (with the exception of the driving axis). Under a given experimental condition, the environmental error is a major error source for the X direction, the optics error strongly affects the Y and Z directions, and environmental effects affect the three rotational directions. Furthermore, the motion error in the high precision stage is less sensitive to the measurement error, but it has to be taken into account in case of a general-purpose stage.

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Abbreviations

DOF:

degree-of-freedom

QPD i :

i-th quadrant photodetector

PD i :

i-th photodetector

BS i :

i-th beam splitter

PBS i :

i-th polarized beam splitter

QWP i :

i-th quarter waveplate

λ :

wavelength of light

CPIT:

circularly-polarizing interferometric technique

HTM:

homogeneous transformation matrix

{R}:

reference coordinate

{D}:

scale coordinate

{M i,j }:

QPD coordinate (i: diffraction order, j: x, y, z)

{V i,j }:

coordinate of beam position at QPD (i: diffraction order, j: direction)

i R j :

orientation matrix from {i} to {j} coordinate.

i L j :

position matrix from {i} to{j} coordinate.

\(^i \mathop {Q_j }\limits_{i \to } \) :

position vector from {i} to {j} coordinate

u j,k :

propagation vector of an incident laser beam (i: coordinate, j: coordinate, k: x, y, z)

ɛ X :

roll error

ɛ Y :

pitch error

ɛ Z :

yaw error

u i :

horizontal displacement at i-th QPD coordinate

v i :

vertical displacement at i-th QPD coordinate

Λ :

pitch of the grating

m :

diffraction order

θ :

diffraction angle

Δφ :

phase

f i :

i-th Doppler frequency

U :

uncertainty

α i :

sensitivity of uncertainty source i

U i :

standard deviation for uncertainty source i

RMS :

root mean square

ɛ :

uncertainty source

δ :

distance between PD and beam spot

F :

frequency of stabilized He-Ne laser

L S :

length of scale

L C :

distance from laser to PSD

L O :

dimension of optical components

N :

number of optical components

D :

diameter of the beam at the grating scale

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Authors and Affiliations

  1. School of Mechatronics, Gwangju Institute of Science and Technology, 1 Oryong-dong, Buk-gu, Gwangju, South Korea, 500-712

    Cha Bum Lee, Gyu-Ha Kim & Sun-Kyu Lee

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  1. Cha Bum Lee
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  2. Gyu-Ha Kim
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  3. Sun-Kyu Lee
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Correspondence to Sun-Kyu Lee.

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Lee, C.B., Kim, GH. & Lee, SK. Uncertainty investigation of grating interferometry in six degree-of-freedom motion error measurements. Int. J. Precis. Eng. Manuf. 13, 1509–1515 (2012). https://doi.org/10.1007/s12541-012-0199-8

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  • DOI: https://doi.org/10.1007/s12541-012-0199-8

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