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# Accuracy

## Positioning system accuracy can be conveniently divided into two categories: the accuracy of the way itself, and the linear positioning accuracy along the way

The former describes the degree to which the ways (ball and rod, crossed roller, air bearing, etc.) provide an ideal single-axis translation, while the latter is concerned with the precision of incremental motion along the axis (typically related to the leadscrew, linear encoder, or other feedback device).

##### Way Accuracy

Any moving object has six available degrees of freedom (Figure 1). These consist of translation, or linear movement, along any of three perpendicular axes (X, Y, and Z), as well as rotation around any of those axes (qx, qy, and qz). The function of a linear positioning way is to precisely constrain the movement of an object to a single translational axis only (typically described as the X axis). Any deviations from ideal straight line motion along the X axis are the result of inaccuracy in the way assembly.

There are five possible types of way inaccuracy, corresponding to the five remaining degrees of freedom (Figure 2): translation in the Y axis; translation in the Z axis; rotation around the X axis (roll); rotation around the Y axis (pitch); and rotation around the Z axis (yaw). Since there are interrelations between these errors (angular rotation, for example, produces a translational error at any point other than the center of rotation), it is worthwhile to carefully examine the effects of each type of error and its method of measurement.

##### Way Translation Errors

Pure translational errors from straight line motion (that is, without any underlying angular error) are usually minor, since all useful methods of producing linear motion average over a number of points (due to multiple balls or rollers, or the area of an air bearing). An exaggerated sine wave error in rolling element ways could achieve a pure translational error without rotation, as would the case of each roller in a way running over a contaminant particle at the same time; both of these cases are never encountered in practice. If a rolling element stage has been subjected to a large impact, the ways may be brinelled (dented) at each ball or roller location; this can result in a pure translational error that occurs periodically along the travel.Positioning tables do nonetheless, exhibit some vertical and horizontal runout (typically referred to as errors of flatness and straightness, respectively), which can be divided into two categories:

1. A low frequency component
This is rarely a “pure” translational error, but is rather a consequence of the underlying angular errors (pitch, roll, and yaw) in the ways. Since the moving portion of the stage follows (at some level) a curved trajectory, there is a corresponding linear deviation from a straight line. The angular and linear errors correlate quite well, and one can be obtained from the other by the process of integration or differentiation.
2. Higher frequency components, which can arise from a variety of sources, not necessarily errors of the ways.
A once-per-revolution rise and fall of the table top can occur near each end of travel if a ball screw is used. The use of flexurally coupled nuts and/or friction nuts can reduce this effect. Additional sources of higher frequency flatness errors can include microstructure in the ways or rolling elements, drive and/or motor induced vibration, and structural resonances in the stage top.Since a number of optical positioning applications have limited depths of field, it is important to understand the magnitude of each of the above effects, and to modify the stage design so as to minimize the effects. The use of air bearings and linear motors can reduce total errors of flatness and straightness.
##### Way Angular Errors

The angular errors of roll, pitch, and yaw (qx, qy, and qz, respectively) are always present at some level in positioning tables, and degrade performance in several ways. Their direct effect is to vary the angular orientation of a user payload; due to the relative ease with which these errors can be maintained at low levels (1 – 50 arc-seconds, depending on stage technology), the effects of changing payload angle are of little consequence in many applications. Certain optical positioning tasks, however, may be directly impacted by angular errors.

Of somewhat greater concern are the translational errors resulting from underlying angular errors. The simple pitch error of �16.5 arc-seconds shown in Figure 3, corresponding to a radius of curvature of 1 kilometer, will produce a Z axis translation of 20 microns in a half meter travel stage at either end of travel, relative to its centered position. Such simple pitch errors are typically found in non-recirculating table designs, due to the overhanging nature of the load at both extremes of travel. More complex curvatures, involving roll, pitch, and yaw, as well as multiple centers of curvature can also be encountered.

The worst impact of angular errors is the resulting Abbé (offset) error, which affects linear positioning accuracy. Unlike the simple translational error described in the above example, Abbé error increases as the distance between the precision determining element and the measurement point increases. This effect is described in detail in the Abbé Error section.

Way angular errors are easily affected by the method of mounting the positioning stage (see Mounting Issues). In general, air bearings provide the ultimate in angular accuracy, as they have an inherently averaging effect, and their reference surfaces can be made very flat. The best stages can hold angular errors to as low as 1 arc-second per 250 mm.

Angular errors of a way assembly can best be measured using a laser interferometer. We employ a dual path optical assembly to eliminate sensitivity to linear translation, while providing 6.5 milli-arc-second (32 nano-radian) resolution for either pitch or yaw. The measurement of roll requires the use of a rectangular optical flat and either an autocollimator or a pair of capacitance gauges operated differentially.

### LINEAR POSITIONING ACCURACY

A variety of techniques are available to incrementally position a user payload along a linear axis. Leadscrews and ball screws are by far the most common, although linear motors are also used. Linear positioning accuracy is simply the degree to which commanded moves match internationally defined units of length.

##### Leadscrew-Based Systems

Low to moderate accuracy systems typically depend on a leadscrew or ball screw to provide accurate incremental motion. Such systems are often operated open loop via stepping motors; if closed loop operation is employed, it is frequently with a rotary encoder. In either case, the screw is a principal accuracy determining element. screws exhibit a cumulative lead error, which is usually monotonic in nature, together with a periodic component, which is cyclic and varies over each revolution of the screw. In addition, there can be backlash in a leadscrew nut, which will reveal itself upon direction reversal. Precision positioning stages generally employ either a preloaded ball screw, or a leadscrew with an anti-backlash friction nut. Ball screws are preferred for high speed applications, and offer a high natural frequency due to their inherent stiffness. Leadscrews with anti-backlash nuts provide very high repeatability at modest cost, and are appropriate for most applications. DOVER leadscrews are available in both commercial and precision grades, with cumulative lead errors of 0.0001″/inch (1 micrometer/cm) for the precision grade, and 0.0004″/inch (4 micrometers/cm) for the commercial grade. Periodic error values are 0.0004″ (10 micrometers) and 0.001″ (25 micrometers) respectively. The above cumulative lead errors correspond to 100 and 400 ppm for precision and commercial grades, respectively.

It is important to realize that use of a leadscrew with a specified cumulative lead error, periodic error, and repeatability does not ensure that the positioning table will provide that level of accuracy. Among the factors which conjoin to degrade overall performance are thermal expansion, due both to ambient temperature changes and nut-friction induced heating, and Abbé error. Both of the latter effects produce different error values, depending on the location on the user payload. In the case of leadscrew thermal expansion, the position of the nut relative to the stage duplex bearing is important, while for Abbé error, it is the distance from the leadscrew centerline to the customer payload.

##### Geometry and Multi-Axis Errors

As mentioned above, angular errors in the stage ways degrade linear positioning accuracy through Abbé error. X-Y Tables have an additional parameter that impacts accuracy to a substantial degree: orthogonality, or the degree of squareness between the two axes. This parameter is held to less than 50 arc-seconds on our commercial grade tables, and less than 20 arc-seconds for precision models. For the latter case, a 300 mm travel corresponds to 30 microns of error due to orthogonality alone. We can, upon request, prepare tables which are square to within 10 arc-seconds; note, however, that trying to get the level of orthogonality lower than the value for yaw has limited meaning. Custom systems (typically air bearing designs) can hold orthogonality errors to below 2 arc-seconds. Another error source in systems with two or more axes is opposite axis error, which results when one axis has a straightness error. It is the job of the leadscrew or encoder on the other axis to provide accuracy in this direction, but since they are on two separate axes, this error is not corrected. Cosine error, or inclination of the leadscrew or encoder to the ways, is usually slight, but grows in importance with short travel, interferometer based stages. All of the above geometry errors are amenable to cancellation through mapping.

##### Linear Encoder-Based Systems

Use of a linear encoder eliminates concern over the leadscrew cumulative and periodic error, as well as friction induced thermal expansion. In many systems, the leadscrew can be dispensed with altogether and replaced with a non-contacting linear motor. With intrinsic accuracies on the order of 5 microns per meter, linearly encoded stages offer a significant increase in accuracy over leadscrew based systems, as well as much higher resolution (typically 0.1 to 1 micron). A number of error sources remain, however, and are often overlooked when specifying an encoder. The single largest error is often Abbé error, which can easily degrade accuracy by tens of microns. With a thermal expansion coefficient of ~10 ppm/degree C, linear encoders must be carefully controlled thermally to utilize their potential accuracy. An ambient temperature change of 1 degree C produces a 10 micron per meter error, double the encoders intrinsic 5 micron per meter accuracy. Contacting encoders are convenient, but read-head wind-up can be about half a micron, and higher if rubber sealing wipers are left in place. Non-contact encoders eliminate read-head wind-up, but can have tighter alignment requirements during installation. The encoder resolution itself defines an error source; a 1 micron resolution encoder moving from zero to +5 microns may display +2 microns when the read-head is actually at +2.7 microns, resulting in a 0.7 micron worst case error. Increasing the resolution below 2-5 microns generally requires electronic interpolation, which can also contribute low-level errors. In X-Y tables, each encoder fails to detect horizontal run-out in the other axis (opposite axis error), thereby ignoring translation along its measurement axis of potentially large magnitude (1 to 10 microns, depending on stage design, precision, and travel). Linear encoders are also incapable of correcting for orthogonality errors, which can range from 1 to 20 microns, again dependent on stage design, precision, and travel. Properly specified, linear encoders can significantly improve system accuracy, particularly if mapping is employed, but their limitations are frequently understated.

##### Laser Interferometer-Based Systems

Laser interferometers are the ultimate position feedback device. They offer very high resolution, typically 10 nanometers in single pass and 5 nanometers in double pass. Intrinsic accuracy is better than 1 ppm for unstabilized sources, and as high as 0.01 ppm for stabilized designs. Abbé error can be virtually eliminated by appropriate location of the retroreflector or plane mirrors. Opposite axis error and table orthogonality error, intrinsic to encoders, can be eliminated in X-Y tables by the use of two plane mirrors (see Interferometer Feedback Systems). Among the barriers to achieving the very high intrinsic accuracy possible with laser interferometers is the variability of the speed of light in air. This value, constant only in a vacuum, is a function of atmospheric pressure, temperature, and humidity, as well as the concentration of other trace gases. The impact amounts to about 1 ppm per degree Centigrade, 0.4 ppm per mm-Hg pressure, and 0.1 ppm per 10% change in relative humidity. In actuality, the relationship (the Edelin equation) is non-linear, but the above linear approximations are valid for small changes near S.T.P. (760 mm-Hg, and 20 degrees C). Compensation for varying atmospheric conditions can be performed by manual entry, or by automatic sensing and correction term calculation, using precision environmental sensors and the system computer. Since atmospheric effects influence the entire air path between the polarizing beamsplitter and retroreflector (or plane mirror), it is important to minimize the “dead path” between the positioning table and the stationary beamsplitter.

Assuming that the beam path has been chosen so as to eliminate Abbé error, the remaining error sources (other than atmospheric effects) are thermal expansion of the user’s part, the positioning table parts, and the base which mounts the table relative to the optics; differential flexing of the table top as it travels; cosine error; and imperfect squareness and flatness of the plane mirrors in X-Y assemblies. The use of “L” mirrors can replace two adjustable plane mirrors with a single glass L mirror; while this avoids concern about misadjustment, neither case can readily assure squareness below the ±1 arc-second level. This limits X-Y systems to a minimum of 5 ppm inaccuracy due to this effect alone; over a 300 mm travel, this accumulates to 1.5 microns. Single-axis systems, which do not have squareness to contend with, can achieve overall accuracies approaching several ppm (1-3 microns/meter), assuming exacting thermal management and atmospheric compensation, as well as beam angle trimming to minimize cosine error. At this level, positioning system design becomes a fairly elaborate exercise in HVAC (heating, ventilation, and air conditioning).

To illustrate the degree to which thermal issues complicate system design, consider a 300 mm travel single-axis table which seeks to achieve “tenth micron accuracy”. One tenth of a micron over 300 mm is equal to 0.3 ppm. Recall that atmospheric compensation for laser interferometers is 1 ppm per degree C and 0.4 ppm per mm-Hg pressure. There will also be ~350 mm of base material (we will presume granite) between the table center and the stationary beamsplitter. The thermal expansion coefficient of granite is 6.3 ppm per degree C. If we choose to allocate our “error budget” of 0.3 ppm, assigning 0.1 ppm to atmospheric temperature, 0.1 ppm to atmospheric pressure, and 0.1 ppm to granite thermal expansion, then we have the following result: air temperature must be measured with 0.1 degree C absolute accuracy; pressure must be measured to within 0.25 mm-Hg accuracy; and the granite must be maintained at a constant temperature within 0.02 degrees C.

This analysis neglects thermal expansion of the user’s part or the positioning table top, as well as cosine error, humidity changes, table top differential flexure, etc. Temperature changes in the interferometer optics alter the path length of the reference beam, introducing another error source, although specialized optics are available which reduce this effect. If the user’s part is not maintained at exactly 20 degrees C, back correcting to that temperature requires precise knowledge of its thermal expansion coefficient, which is rarely available. Proper estimation and inclusion of all these error sources further exacerbates the thermal control requirements, often raising them to largely unachievable levels. Given that a fairly expensive laser interferometer fails to approach the needed accuracy levels in this application, the application of appropriate skepticism to advertising claims for stage accuracy is warranted.