the x axis to X, stop and take a
measurement, then move the y axis to Y, stop and take a measurement,
then move the zaxis to
Z, stop and take a measurement. Here R=v(X2+Y2+Z2)
is the increment in the diagonal direction, and X, Y, and Z are the increments in the x, y, and
zdirections, respectively.
Compared to
the conventional body diagonal
measurement where only
one data point is collected at each
increment R, the vector measurement collects three data
points, one at X, one
at Y, and one at Z. Hence three times
more data are collected.
Furthermore, the data collected after X are due to x-axis movement
only, and data collected
after Y and Z
are due to y-axis and z-axis movement,
respectively. Hence the error
sources due to x-axis motion, y-axis motion, and z-axis motion can be
separated. Second, point
the laser beam into
another body diagonal direction and repeat the same process until all four body
diagonals are measured. Since
each body diagonal measurement collected 3
sets of data, there are 12 sets of data.
Hence, there are enough data to solve the
three displacement errors and six straightness errors. For conventional body diagonal measurement, the displacement
is a
straight line along
the body diagonal;
hence a laser
interferometer can be used to do the
measurement. However, for the vector
measurement described here, the displacements are along the xaxis,
then along the y axis, and then along the zaxis. The trajectory of the target or the
retroreflector is not parallel to the
diagonal direction. Deviations from
the body diagonal are proportional to the size of the increment, X, Y, or Z. A conventional laser interferometer will be way out of alignment even with an increment of a few
mm. To tolerate
such a large lateral
deviation, a laser Doppler
displacement meter7 using a single aperture
laser head and a flat mirror as the target
can be used. This is
because any lateral movement or movement perpendicular to the normal direction of
the flat mirror will not displace the laser
beam. Hence the alignment is maintained.
After three movements, the flat-mirror target will move back to the center of the diagonal again, hence the size of the flat mirror need only be
larger than the
largest increment. A schematic
showing the vector measurement setup is shown in Fig. 1. Here the flatmirror
target is mounted on the machine spindle
and it is perpendicular to the laser beam
direction. Compared to a
conventional body diagonal measurement all
three axes move
simultaneously along a body diagonal and
collect data at each
preset increment. In the vector measurement
all three axes move in sequence
along a body diagonal and collect data after each axis is moved. Hence, not
only three times more data are
collected, the error due
to the movement of each axis can
also be separated.
III. BASIC
THEORY A. Motion
of a rigid body The general
motion of a rigid body along one
axis can be described
by six degrees of freedom. These are
one linear, two straightness, one pitch, one yaw,
and one roll. For a three axis
machine, there are 18
degrees of freedom plus three squareness,
a total of 21 degrees of freedom.
For each axis
of motion, there are
linear position errors, straightness errors, and pitch, yaw, and roll angles in the
x, y, and zdirections. Hence, for x-axis movement,
there are linear position error dx(x), straightness
errors dy(x) and
dz(x), and pitch, yaw, and roll
angles ay(x), az(x), and
ax(x).Similarly for y-axis and
z-axis movement,
there are linear position errors dy(y)and dz(z),
straightness errors dx(y)dz(y),
dx(z),
and dy(z), and pitch, yaw, and roll angles
ax(y),az(y),
ay(y),
ax(z),
ay(z), and
az(z). The squareness between axes is xy , yz, and zx.B.
Assumptions To simplify the
analysis, the following assumptions are made. (a)
The motion is repeatable
to within certain uncertainty. The
accuracy of the method
is limited to the repeatability of the
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