For the development of the aviation industry, machine tools are becoming large and travel long distances, making optical alignment setup difficult. An auto-tracking laser interferometer (ATLI) is proposed and researched in this paper for the squareness error measurement of machine tools or coordinate-measuring machines (CMMs). The procedure involves measurement of only one line of an axis, and the measurement results provide us information about not only the positioning errors but also the squareness errors. This specially designed interferometer instrument can be useful in checking industrial machine tools in a short time.
Machine tools and coordinate measuring machines (CMMs) with 3 - 5 axes have played fundamental roles in industrial development. As seen in
Geometric errors are caused by straightness errors of the guideways [
A laser interferometer and an auto-tracking laser interferometer (ATLI) can be used to perform the seven lines tests. However, the time consumed when using a laser interferometer is approximately three times that when an ATLI is used. The laser tracker [
The first step in using ATLI, following ISO 230-2 and 230-6, is to fix the ATLI onto the machine tool carriage [
The second step in the ISO 230-2 and 230-6 test procedures, as mentioned in Section 2.1, is determining the ATLI’s coordinates on the machine tool, which can be calculated using a six-point measurement and the following formula:
[ x t y t z t L 0 ξ ] = [ 2 x 1 2 y 1 2 z 1 2 Δ L 1 − 1 2 x 2 2 y 2 2 z 2 2 Δ L 2 − 1 2 x 3 2 y 3 2 z 3 2 Δ L 3 − 1 2 x 4 2 y 4 2 z 4 2 Δ L 4 − 1 2 x 5 2 y 5 2 z 5 2 Δ L 5 − 1 2 x 6 2 y 6 2 z 6 2 Δ L 6 − 1 ] + [ x 1 2 + y 1 2 + z 1 2 − Δ L 1 2 x 2 2 + y 2 2 + z 2 2 − Δ L 2 2 x 3 2 + y 3 2 + z 3 2 − Δ L 3 2 x 4 2 + y 4 2 + z 4 2 − Δ L 4 2 x 5 2 + y 5 2 + z 5 2 − Δ L 5 2 x 6 2 + y 6 2 + z 6 2 − Δ L 6 2 ] (1)
where the residual measurement errors are ζ = x t 2 + y t 2 + z t 2 − L 0 2 and the “+” symbol represents the pseudo-inverse operator. The six stop points ( p = 1 , 2 , ⋯ , 6 ) of the cat’s eye reflector are (xp, yp, zp). These points are independent and should be known. (xt, yt, zt) is the LT coordinate to be determined. L0 is the initial distance from the LT to the cat’s eye reflector, which is unknown. ΔLi is the measured distance deviation from the LT.
The measurement of ATLI requires only the length information. The motion of a three-axis machine tool comprising 21 error terms results in the ideal and actual distance differences in the ATLI’s coordinate determination.
In
Δ L i + 1 , i = ( x i + 1 2 + y i + 1 2 + z i + 1 2 ) − ( x i 2 + y i 2 + z i 2 ) (2)
Δ L ′ i + 1 , i = ( x ′ i + 1 2 + y ′ i + 1 2 + z ′ i + 1 2 ) − ( x ′ i 2 + y ′ i 2 + z ′ i 2 ) (3)
For the effects of positioning and angular error, we consider
O0 is the home position of the machine, O1 is the position of target and Ot is the location of the auto-tracking ranging system, which represents the center of
the reference ball inside the instrument. L0 + ΔL is the measured distance between the ranging instrument and the moving target because, here, incremental interferometer is considered.
Let [sx sy sz]T be the ideal position and LV be the measuring line along the distance on the V-axis; subsequently, the simulated position [ŝx ŝy ŝz]T with errors can be expressed as
[ s ^ x s ^ y s ^ z ] = [ s x + δ x x s y + δ y x s z + δ z x ] (4)
Considering a coordinate position shift [εx εy εz]T and scale errors gx of the position errors, the movement can be described as follows:
[ s ^ x s ^ y s ^ z ] = [ g x s x + ε x + g x δ x x s y + ε y + δ y x s z + ε z + δ z x ] (5)
Finally, the effects of positioning and angular error can be described as follows:
[ δ x x δ y x δ z x ] = [ cos β x 0 sin β x 0 1 0 − sin β x 0 cos β x ] [ cos γ x − sin γ x 0 sin γ x cos γ x 0 0 0 1 ] [ s x 0 0 ] − [ s x 0 0 ] (6)
here, γx is the squareness error of the x-y axis and βx is the squareness angle of the x-z axis.
The experimental setup was based on a three-axis CMM (Leitz PMM-C) in the National Measurement Laboratory in Taiwan; it is a gantry structure with an x-axis of 1400 mm, a y-axis of 700 mm, and a z-axis of 600 mm. The simulation software was developed in VB.NET. Considering Equations (5) and (6), we set the squareness error of the x-y axis to 0.1˚, which is a common scale for squareness in machine tools or CMM assembly lines. The simulated and experimental results are plotted in
This research focused on simulating the measurement results of an auto-tracking ranging system and analyzing trends in the deviation results. The experimental results indicated a significant difference between positioning and squareness error when only one axis was measured using an ATLI. From the plotted trends in the simulated and experimental results, we can identify and quantify linear and squareness errors, and thus ensuring that ATLIs can be used to perform machine tool inspection more easily and conveniently. Compared with traditional interferometers, auto-tracking facilitates easier optic alignment setup, especially for diagonal line measurements of machine tools.
The work was supported by the standard maintenance and services project from Bureau of Standards, Metrology and Inspection (BSMI), Ministry of Economic Affairs (MOEA).
The authors declare no conflicts of interest regarding the publication of this paper.
Chen, J.-R., Ho, B.-L., Lee, H.-W., Pan, S.-P. and Hsieh, T.-H. (2018) Research on Geometric Errors Measurement of Machine Tools Using Auto-Tracking Laser Interferometer. World Journal of Engineering and Technology, 6, 631-636. https://doi.org/10.4236/wjet.2018.63039