Abbé Error

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Abbé error (pronounced ab-¯a) can be a significant source of error in positioning
applications. Named after Ernst Abbé, a noted optical designer, it refers to a linear
error caused by the combination of an underlying angular error (typically in the
ways which define the motion) and a dimensional offset between the object being
measured and the accuracy determining element (typically a leadscrew or encoder).
In open loop systems (or closed loop systems employing rotary feedback), the accuracy
is nominally determined by the precision of the leadscrew. Similarly, in systems
with linear encoders or interferometers, it is that device which determines the
accuracy. It is important, however, to recall exactly what information these devices
provide: Leadscrews really tell us nothing but the relative position of the nut
and screw, and encoders tell us only the position of the read head relative to the
glass scale. Extrapolating this to include the position of an item of interest,
despite its firm mechanical connection to the nut or encoder read-head, is ill founded.

To illustrate this, consider Figure 6, which shows a single-axis stage with a linear
encoder. The stage carries an offset arm which positions a probe over a sample.
The apparent distortion in the stage is intentional; it is intended to illustrate,
in exaggerated fashion, a stage whose ways have a curvature (in this case, yaw).
Someone using this stage, and in possession of appropriate test instruments, would
measure an error between the stage position as determined by the encoder read-head,
and the actual linear position of the probe.

Figure 6 - Abbé Error Example

Suppose the curvature is sufficient to produce an angle a'b in Figure 6 of 40 arc-seconds
(a' is drawn parallel to a). If the stage moves forward 300 mm, the probe at the
end of the arm will be found to have moved +300.100 mm, resulting in an X-axis error
of +100 microns. If the ways were, in fact, curved in a circular arc as shown, there
would also be a Y-axis shift of +25 microns. This Y-axis error would be eliminated
(while the X-axis error would remain) if the angular error were a purely local property
of the ways at the +300 mm location. 100 microns is quite a large error, and Abbé
error is accordingly important among the error sources to be considered.

Abbé error is insidious, and can best be countered by assuming the presence of angular
error in a system and then working to minimize both the underlying error and its
effect, through design optimization and appropriate placement of leadscrews, encoders,
etc. The best tool to analyze angular error is the laser interferometer, which,
when used with special dual path optics, measures pitch or yaw with 0.025 arc-second
(0.125 micro-radian) resolution. We measure roll using a rectangular optical flat,
in conjunction with an autocollimator or two capacitance gauges in differential
mode.

Sources of angular error include the following:

1. Curvature of ways

2. Entry and exit of balls or rollers in recirculating ways

3. Variation in preload along a way

4. Insufficient preload or backlash in a way

5. Contaminants between rollers and the way surface

6. Finite torsional stiffness in a way, leading to angular deflection driven by:

a. external forces acting on the load

b. overhang torques due to the load's travel

c. overhang torques due to stage components

d. an offset leadscrew mounting position

e. friction due to wipers in a linear encoder

7. Mounting the stage to an imperfectly flat surface

In the example shown in Figure 6, Abbé error could be lessened by moving the encoder
to the left side of the stage. Reducing the arm's length, or mounting the encoder
at the edge of the sample (with the read head connected to the arm) would be more
effective. Virtual elimination of Abbé error could be achieved by using a laser
interferometer and mounting the moving retroreflector on the probe assembly. Note
that the component positions shown in Figure 6 effectively control Abbé error due
to pitch error of the stage, since the height of the probe and encoder are roughly
equal. While the stage might exhibit a pitch error (rotation around the Y-axis),
there is no corresponding vertical (Z-axis) offset needed to produce Abbé error.
The third degree of rotational freedom, roll, corresponds in the illustration to
the rotation around the axis of motion (X-axis). This would result in the gap between
the probe and the sample varying as the stage moved.

In general, try to estimate or measure the magnitude of all three possible angular
errors (roll, pitch, and yaw) in any given system under actual load bearing conditions.
Then, look for any offsets between driving or measuring devices and the point of
interest on the load. Calculate the Abbé error, and if it proves unacceptable, optimize
the design to reduce either the offset or the underlying angular error. In general,
systems built using precision lapped granite and air bearings, which do not extend
the load beyond the table base at any point in the travel, are best at minimizing
angular errors.

To determine the magnitude of Abbé error, simply multiply the offset by the tangent
of the angle. In the example, this was: 500 mm x tan (40 arc-seconds) = 500 x tan
(0.011 degrees) = 500 x 0.000194 = 0.100 mm. If the angle is known in radians instead
of degrees, the problem is that much easier:

**Abbé error = angle x offset.**

For example, an angular error of 194 micro-radians, or 0.000194 radians, in conjunction
with an offset of 500 mm will result in an Abbé error of 0.000194 x 500 mm = 0.100
mm (not coincidentally, the angle of 194 micro-radians was chosen to match the 40
arc-seconds of the previous example). Finally, a helpful rule of thumb is that the
Abbé error will equal about 5 microns per meter of offset and arc-second of angular
error; in Imperial units, this amounts to about 5 micro-inches per inch of offset
and arc-second of angular error. Once again, 40 arc-seconds x 20 inches x 5 = 4000
micro-inches, or 0.004". The accompanying chart and figure may prove helpful in
determining which offsets produce Abbé error for a given angular error.

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