II.               Body Diagonal Displacement Measurement

To measure the displacement accuracy of a linear axis, a laser interferometer can be used.  The laser beam is aligned to be parallel to the motion of the linear axis and the position errors are measured at each increment.  Since the measurement direction is parallel to the direction of the movement, the measured displacement errors are not sensitive to the straightness errors, which are perpendicular to the displacement direction.

It is noted that for a quick check of volumetric accuracy, linear displacement measurement along 4 body diagonals is recommended by the B5.57 standard [7].  This is because the body diagonal displacement measurements are sensitive to all the errors such as the displacement errors, straightness errors, squareness errors and angular errors.  Hence it is a good check of volumetric accuracy.  However, if the measured errors are large, there is not enough data to identify the error sources.   

III.           Vector Measurement

The basic concept of the vector method is that the laser beam direction (or the measurement direction) is not parallel to the motion of the linear axis.  Hence, the measured displacement errors are sensitive to the errors both parallel and perpendicular to the direction of the linear axis.  More precisely, the measured linear errors are the vector sum of errors, namely, the displacement errors (parallel to the linear axis), the vertical straightness errors (perpendicular to the linear axis), and horizontal straightness errors (perpendicular to the linear axis and the vertical straightness error direction), projected to the direction of the laser beam.  Furthermore, collect data with the laser beam pointing in 3 different diagonal directions; all 9-error components can be determined.  Since the errors of each axis of motion are the vector sum of the 3 perpendicular error components, we call this measurement a “vector” method.

For conventional body diagonal displacement measurement all 3 axes move simultaneously, 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 x-axis, then along the y-axis and then along the z-axis.  The trajectory of the target or the retroreflector is not parallel to the diagonal direction.  The 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 large lateral deviation, a Laser Doppler Displacement Meter [6] using a single aperture laser head and a flat-mirror as 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 3 movements, the flat-mirror target will move back to the center of the diagonal again, hence the size of the flat-mirror has only to be larger than the largest increment.

In summary, in a conventional body diagonal measurement all 3 axes move simultaneously along a body diagonal and collect data at each preset increment.  In the vector measurement all 3 axes move in sequence along a body diagonal and collect data after each axis is moved.  Hence, not only 3 times more data are collected, the error due to the movement of each axis can also be separated.

IV.            Basic Theory

1)         Trajectory Model

The general motion of a rigid body starting from point A and ending at point B, as shown in Fig. 1, can be described by 6 degrees of freedom.  These are 1 linear position error, 2 straightness errors, and 3 angular errors.

Fig. 1  A rigid body moved from A to B

To simplify the analysis, pick a representative point Pa on the rigid body (such as the tool tip or probe tip), and move the coordinate such that at A, Pa is at the origin of the coordinate. Assume the motion is along the x-direction with an increment of X and move the origin of the new coordinate to X.  If there is no error, Pb should be at the origin of the new coordinate.  However, in general Pb is not at the origin.  As shown in Fig. 2, in the new coordinate at B, Pb = X ux + E(x),  (a bold letter indicates a vector quantity) where ux is the unit vector in the x-direction and E(x) is the vector position error (or volumetric error) due to the motion in x-direction.  In general E(x) can be expressed as,

          E(x) = Ex(x) ux + Ey(x) uy + Ez(x) uz              Eq. 1

Where ux, uy, and uz are unit vectors in the directions of x-, y-, and z-axis, and Ex(x) is the error component in x-direction due to the motion in x, Ey(x) and Ez(x) are the error components in y- and z-direction respectively due to the motion in x.  Please note that the Ex(x), Ey(x) and Ez(x) are the position error components due to all the motion errors including the displacement error, 2 straightness errors, 3 angular errors and even non-rigid body motion errors.  Similarly, the errors due to y-axis motion and z-axis motion are E(y), and E(z) respectively, and can be expressed as,

          E(y) = Ex(y) ux + Ey(y) uy +Ez(y) uz

                                                                                  Eq. 2

          E(z) = Ex(z) ux + Ey(z) uy + Ez(z) uz