Table of Contents
Introduction
Major Uncertainty Contributors in Laser Feedback Systems
Quantifying Positioning Uncertainty in an Example Machine
Definition of the Measurand
Resolution and Feedback Noise
Laser Wavelength Compensation and Dead Path Errors
Differential Thermal Expansion of Machine Components
Summary and Expanded Standard Uncertainty
Introduction
Laser interferometers are commonly used as a measurement reference for machine correction and accuracy validation during the production of several high precision motion systems. Laser interferometer measurement can offer low measurement uncertainty compared to the achievable accuracy of most widely used motion control devices under controlled environmental conditions.
When processes require the highest level of precision, laser interferometer measurement close to the work point of the machine is frequently used as the feedback mechanism for machine control. In such cases, using laser interferometry to characterize the motion of the machine is not justified as the measurement uncertainty of the metrology system is higher than or equivalent to the motion error.
The motion accuracy of these machines should be equated to an uncertainty in the feedback system's measurement of the defined motion of work point.
Major Uncertainty Contributors in Laser Feedback Systems
There are various sources of uncertainty in motion systems that employ laser interferometric feedback devices. Although the effects of Abbe offsets, noise in the feedback loop, differential thermal expansion, and the physical setup of the system can also have considerable impacts, most of the key contributors are related to the environment. The ability to understand and reduce these influences on a specific measurement is critical to achieve ultraprecise motion in these systems.
The laser’s environment is the main concern when using interferometric feedback. The wavelength of light that is used as the link to the meter  the base metric of motion  varies as a function of the refractive index of the medium through which it is passing. In air, the refraction index considerably varies with changes in pressure, humidity, temperature, and even the local composition of the air.
Therefore, accurate positioning measurement needs accurate and continuous knowledge of these environmental factors for use in active wavelength compensation. Additionally, the type and rate of air flow over the laser beam causes local fluctuations in refractive index and results in measurement noise.
In addition to environmentrelated contributors, other influences on the collective uncertainty of machine motion include a number of elements. The alignment and flatness of the plane mirror optics employed in the interferometry system causes uncertainty in the work point’s motion.
Similarly, the influences of angular error motions on any small Abbe offset, present between the defined work point and the interferometers, can be a key contributor. In situations of low uncertainty budgets, calibrated accuracy and laser wavelength stability can become a nonnegligible contributor; frequency bleed through can also induce considerable measurement uncertainty if heterodyne lasers are used.
Finally, the thermal expansion of any number of components over an uncompensated distance introduces considerable uncertainty in the control of the work point’s motion, as seen in almost all precision machines. Figure 1 is a diagram of the significant uncertainty contributors to motion control in machines where homodyne laser interferometer feedback is used close to the work point.
Figure 1. Diagram of chief uncertainty contributors to the measurement of work point motion by homodyne laser interferometer feedback in precision motion control devices. Some contributors are left undeclared, including laser head temperature sensitivity, and the associated uncertainty in many of the nominal constants used in calculation.
Quantifying Positioning Uncertainty in an Example Machine
As shown in Figure 1, the influence of the uncertainty contributors is a function of each machine’s design, defined work point, length of travel, and environment. It is possible to estimate quantification under a casebycase basis, where specifics of machine are known.
Each machine’s design should have its own uncertainty analysis, even if it is similar to the earlier analyzed machines. Therefore, in this article, the motion of a machine vision inspection station employed as a metrology tool at the headquarters of Aerotech is presented as a specific example in the estimation of measurement uncertainty. The machine currently installed at the Aerotech’s Metrology Lab is shown in Figure 2.
Figure 2. An image of an example precision inspection machine installed in a temperaturecontrolled environment. Plane mirrors are integrated into the payload plate, providing laser interferometer feedback at the work point plane for motion control.
The inspection station shown in Figure 2 has been developed with a large payload plate that holds a vacuum chuck for substrate retention as well as two plane mirror reflectors for use in laser interferometer feedback at the work point. This machine shares all of the significant contributors outlined in Figure 1 when estimating the measurement uncertainty of work point motion by the lasers.
By working through these sources onebyone, a reasonable estimation of measurement uncertainty can be achieved for this specific machine’s application. Like all engineering tasks, certain assumptions have to be made in the quantification of uncertainty, and the uncertainty or error budget trying to be attained for a particular application dictate the validity of the assumptions.
This applicationspecific depth of analysis is another reason why each machine task and machine design must have its own uncertainty analysis.
Definition of the Measurand
Uncertainty analysis is undoubtedly a specific measurement. However, this is usually misunderstood. A device or machine itself cannot have an uncertainty  only the particular measurement of a defined quantity can have an uncertainty.
Therefore, the explicit definition of the quantity being measured, the measurand, will have a major effect on the quantification of uncertainty. In the case discussed here, the measurand is defined as 2D planar measurement of work point position through two independent plane mirror interferometers. The machine’s work point should be specifically defined for this measurand to be meaningful.
As shown in Figure 2, the origin or the work point of the machine vision inspection station is defined as the intersection of the optical axis of the camera with a plane parallel to the least squares planar fit of the mounting surface of the vacuum chuck, only 3 mm offset above the vacuum chuck.
The vertical position of this “floating” plane is dictated by the typical thickness of the parts the machine is designed to measure. Therefore, the vacuum chuck surface with a 3 mm offset defines the XY coordinate plane to nominally coincident with the upper surface of a standard part being tested.
The motion’s Xaxis is defined as the unit normal vector of the best fit plane to the surface of the X mirror, projected onto the XY coordinate plane as defined by the offset vacuum chuck surface.
Finally, the Yaxis of motion is defined as the cross product of the unit normal vector of the mounting plane of the vacuum chuck and the unit vector of the Xaxis, all attached using the intersection of the optical axis of the camera as the origin. Figure 3 displays the work point of the machine.
Figure 3. Illustration of the optical CMM’s work point. The XY plane is dictated by the vacuum chuck’s mounting surface. The origin is determined by the intersection of the camera’s optical axis with this XY plane, and the Xaxis direction is dictated by the unit normal vector of the X mirror’s surface. Finally, the Zaxis is the unit normal of the XY plane, and the Yaxis direction is the cross product of the Z and X unit vectors. This defines the work point and the measurement coordinate frame.
Resolution and Feedback Noise
The resolution and feedback noise floor of the measurement system are the first two significant uncertainty contributors that need to be measured. It is very simple to determine the interferometer resolution because it is governed by the manufacturer’s electronic hardware, and must be stated in the related operations manual.
Renishaw RLD double pass plane mirror interferometers are utilized by this machine. They provide a digital measurement resolution of 10 nm, which is assumed to be a square distribution width.
The feedback noise floor not only dictates the minimum effective resolution of the machine, but also sets a floor for the uncertainty in measuring the planar motion of the work point. In this case, the feedback noise (Figure 1) is dominated by electronic noise, control loop stability, mechanical vibration, and most significantly, air turbulence across the open interferometer beams.
The influence of mechanical vibration on the interferometers is reduced in this machine through the use of a highcompliance, passive air isolation system. Similarly, the coordinate measuring machine (CMM) itself is surrounded by an environmental enclosure to reduce air currents across the laser beams. This machine’s overall implementation produces longterm (>3 hours) laser stability of about 25 nm rootmeansquare.
Laser Wavelength Compensation and Dead Path Errors
As stated, the connection to the meter’s base metric is the wavelength of the light employed in the system. The wavelength is changed by deviations in the refraction index within the medium through which it travels, and these changes need to be compensated.
The requirement for compensation of wavelength change as a function of the local environment of a laser has been long documented^{1}, and it is an automated feature in a wide range of commercially available interferometer systems.
In this case, humidity and air composition changes are small enough to overlook. However, the accuracy/uncertainty of wavelength compensation, and as a result the interferometric measurement itself, is dictated by the ability to accurately measure changes in the remaining environmental parameters of pressure and temperature.
When applied to distance measurement, as in this machine, interferometric measurement is carried out relative to some reference location. Therefore, wavelength compensation is actively applied to the number of waves, or fringes, counted by the detector in a relative move away from the reference location.
In practice, the laser heads cannot be positioned in contact with the mirrors when at the reference location, where some relief has to be provided. A small part of laser beam is always left unaccounted for in the wavelength compensation. When there is a change in local environment, this beam length is uncompensated and causes small shifts in the zero location of the system.
This is called dead path error. As the errors contributed from the omitted wavelength compensation are dependent on length, reducing the amount of space between each laser head and the location of the mirrors at the reference point will reduce the uncertainty caused by dead path errors.
Laser wavelength compensation and dead path effects on uncertainty in the work point’s measurement are dependent on length. Their influence over the greatest operating distance will be considered for this example of quantification.
In this machine, the weather station used to adjust environmentallyrelated wavelength changes has an absolute pressure accuracy of 1 mbar and an absolute temperature accuracy of 0.2 °C. The relative change of pressure and temperature over a 3hour test, which influences dead path error, has been observed to be below 0.5 mbar and 0.1 °C, respectively.
To make it simple, the influences of these values on measurement uncertainty can be quantified via the latest laser scale correction sensitivity coefficients per ASME standards^{2,3}. This prevents the need for performing propagation of uncertainty analysis according to the Guide to the Expression of Uncertainty in Measurement (GUM) ^{4}, and the corrected Edlen equations^{4}.
Assuming the operation of the machine around standard pressure, temperature, and humidity, the uncertainty contribution of the weather station’s pressure and temperature measurement accuracy is 5 nm and 29 nm, respectively, over the entire 300 mm travel of the machine.
Similarly, the relative change in pressure and temperature during a given test contributes 1 nm and 5 nm of measurement uncertainty, respectively, over the 96 mm of dead path per axis. All of these contributions are considered square distribution widths. A more detailed explanation of dead path errors and laser wavelength compensation can be found in Renishaw technical paper TE3296.
Differential Thermal Expansion of Machine Components
Finally, the differential thermal expansion of different machine components can drastically influence measurement uncertainty. For instance, according to the workpoint’s definition in this instrument, there is a large length of granite between the work point and each laser head that is left to freely expand and contract with the changing temperature.
During a test, the granite bridge that holds the camera would expand or contract and move the optical axis relative to the laser heads if the temperature of the granite machine base changes. However, the interferometer system’s feedback control would retain the same distance between each laser head and each mirror.
This would cause the work point to move without seeing a change in position reading, and therefore, it would lead to the measurement uncertainty over the course of a measurement. However, in this machine design there is a smaller length of aluminum between the work point and the mirror faces, in the form of payload plate, left to contract and expand with the change in temperature.
By design, the lengths of aluminum and granite contribute opposite directions of workpoint motion relative to a position that is being measured on a grid plate when they expand and contract.
So, as long as the ratio of the respective lengths of granite and aluminum are equal to the ratio of their respective thermal expansion coefficients, the growth of machine components contributes an insignificant amount of measurement uncertainty to the measurement of a part under test.
It must be noted that in this analysis, thermal gradients are neglected. Assuming simultaneous bulk temperature change is inaccurate, as the granite’s thermal mass is much larger compared to the aluminum payload plate.
As a result, the components of the machine would not expand and contract at the same time, voiding their cancelation of induced work point motion. However, these dynamic thermal expansion effects give negligible uncertainty in the controlled environment of this machine.
Summary and Expanded Standard Uncertainty
When combining the uncertainty contributions of uncorrelated inputs, the estimated uncertainty of their distributions can be calculated in quadrature. This is a simplified application of the propagation of uncertainty explained in the GUM^{4}, and is applicable for this example.
Table 1 shows a summary of the significant contributors considered in this analysis, their uncertainty contributions, and the expanded and combined uncertainty existing in the positioning measurement of the system. Summaries of uncertainty as shown in Table 1 can provide insight into the efforts needed to be focused on further improving measurement.
It should be noted that throughout this uncertainty analysis, numerous assumptions have been made. Similarly, a mass of underlying principles and background knowledge have been used to guide this analysis.
As mentioned before, each machine design must be accompanied by its own uncertainty budget and analysis. The key contributors, shown in Figure 1, may not always be adequately exhaustive for other cases, nor the simplified assumptions made throughout suitable for lower levels of desired measurement uncertainty.
The Guide to the Expression of Uncertainty in Measurement (GUM) and its supplementary materials^{4}, freely available at the International Bureau of Weights and Measures’ website^{7}, are a good reference to achieve a better understanding of measurement uncertainty and its estimation.
Table 1. Summary of measurement uncertainty contributors and an estimation of expanded standard uncertainty for work point position measurement
Uncertainty Contributor 

Distribution & Full Width 
Standard Uncertainty 
Measurement Resolution 

10 nm, Square 
3 nm 
Feedback Noise Floor 

100 nm, Square 
25 nm 
Laser Wavelength Compensation 
Temp. Accuracy = 0.2 °C 
58 nm, Square 
17 nm 
Press. Accuracy = 1 mbar 
11 nm, Square 
3 nm 
Air Dead Path Errors 
Temp. Change = 0.1 °C 
9 nm, Square 
3 nm 
Press. Change = 0.5 mbar 
2 nm, Square 
0.6 nm 
Mirror Flatness 

64 nm, Guassian 
16 nm 
Cosine Errors 
Mirror Alignment = 50 µrad 
0.5 nm, Biased 
0.1 nm 
Laser Alignment = 50 µrad 
0.5 nm, Biased 
0.1 nm 
Mirror Orthogonality = 0.05 µrad 
15 nm, Triangle 
3 nm 
Abbe Offset & Angular Errors 

0 nm 
0 nm 
Differential Thermal Expansion 

0 nm 
0 nm 
Combined Standard Uncertainty 
35 nm 
Expanded Uncertainty, K = 2 
70 nm 
References
 Edlen, B. The refractivity of air. Metrologia, 1966.
 ASME B5.57. Methods for performance evaluation of computer numerically controlled lathes and turning centers. American Society of Mechanical Engineers, New York, 1998.
 ASME B89.1.8. Performance evaluation of displacement measuring laser interferometers. American Society of Mechanical Engineers, New York, 1998.
 Bureau International des Poids et Mesures, Commission électrotechnique internationale, and Organisation internationale de normalisation. Guide to the Expression of Uncertainty in Measurement. International Organization for Standardization, 1995.
 Birch, K. P., and M. J. Downs. "Correction to the updated Edlén equation for the refractive index of air." Metrologia 31.4 (1994): 315.
 Chapman, M.V. TE329: Environmental compensation of linear laser interferometer readings. Renishaw Technical White Papers.
 Bureau International des Poids et Mesures. <<http://www.bipm.org/en/publications/guides/> >
This information has been sourced, reviewed and adapted from materials provided by Aerotech, Inc.
For more information on this source, please visit Aerotech, Inc.