where, D is the linear error, subscript is the error direction
and the position coordinate is inside the
parenthesis, A is the angular error, subscript is the axis of rotation and
the position coordinate is inside the
parenthesis. III.
Body diagonal
displacement errors The performance
or the accuracy of a CNC machine tool is determined by the 3 D
volumetric positioning errors,
which includes the linear displacement
error, the straightness
error, the angular error and the
thermal induced error. A complete
measurement of those errors is very complex
and time consuming, for those reasons the
body diagonal displacement error defined in the ASME B5.54 [2] or ISO 230-6
[3] standard is a good quick check of the
volumetric error. This is because all the errors, including 3 displacement errors, 6 straightness errors,
3 squareness errors and some angular
errors, will contribute to the 4 body diagonal displacement errors [4].
Hence it
is a good and efficient measurement of
the volumetric error. The B5.54
body diagonal displacement tests
have been used by Boeing Aircraft Company and many others for many years with very good results and success.
Briefly, similar to a laser linear displacement measurement,
instead of pointing the laser beam
in the axis direction, pointing the
laser beam in the body diagonal
direction. Mount a
retroreflector on the spindle and move the spindle in the body
diagonal direction from the lower corner
(X=0 Y=0 Z=0) to the opposite upper corner (Xmax, Ymax, Zmax). Starting from the zero position and
at each increment of the three axes, which
are moved together to reach the new position along the diagonal,
the displacement error is measured.
The key question is, what is the correlation between the body
diagonal displacement errors and the true
3D volumetric errors. IV.
Definitions of volumetric errors
There are various definitions on 3D
volumetric errors. One conventional definition is the root mean square of the 3 axes displacement error.
Here, the linear displacement error
of each axis is Dx(x), Dy(y), and Dz(z). The maximum error of each
axis is X-axis error, ELx = Max[Dx(x)] – min[Dx(x)],
Y-axis error, ELy = Max[Dy(y)] – min[Dy(y)],
(1)
Z-axis error, ELz = Max[Dz(z)] – min[Dz(z)].
Hence
the volumetric error ELv can be defined as the root mean square sum of all the
3 linear displacement errors in Eq. 1.
That is, ELv = SQRT[ ELx * ELx + Ely * Ely + ELz * ELz].
(2)
This
definition is reasonable as long as the dominate errors are the 3 displacement
errors (or lead
screw pitch errors). Now, most of
the lead screw errors have been
reduced by
linear encoder or error compensation.
Hence, the largest machine tool
3
