# Uncertainty of Machine Positioning with Laser Interferometer Feedback

## 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

Mirror Flatness, Orthogonality, and Cosine Errors

Abbe Offset and Angular Error

Differential Thermal Expansion of Machine Components

Summary and Expanded Standard Uncertainty

## Introduction

Laser interferometers are used as a measurement reference for machine correction and accuracy validation in the manufacture of a number of high precision motion systems. Under controlled environmental conditions, laser interferometer measurement can provide low measurement uncertainty relative to the achievable accuracy of most frequently used motion control devices. As such, when processes require the highest precision, laser interferometer measurement near the machine’s work point is often used as the feedback mechanism for machine control. In these instances, the application of laser interferometry to characterize the machine’s motion is unfounded because the measurement uncertainty of the metrology system is higher than or equivalent to the motion error. The accuracy of these machines' motion must be equated to an uncertainty in the feedback system's measurement of the defined work point's motion.

The purpose of this article is to explain the key contributors to machine positioning uncertainty in systems with laser interferometer feedback near the work point. An example to quantify these uncertainties in a real implementation of a laser-feedback-driven machine will be used. This is not meant to be an introduction to measurement uncertainty, error/uncertainty budgeting, laser interferometer feedback, or rigid body error motions. It is meant to be an overview of the level of work point position measurement uncertainty that can be obtained in a particular machine design.

## Major Uncertainty Contributors in Laser Feedback Systems

There are a number of sources of uncertainty in motion systems that use laser interferometric feedback devices. Most of the main contributors are environment related, although the effects of differential thermal expansion, quality of optics, Abbe offsets, noise in the feedback loop, and the physical setup of the system can have considerable impacts as well. The ability to understand and curtail these influences on a specific measurement is crucial to attaining ultra-precise motion in such systems.

The key concern when using interferometric feedback is the laser's environment. The wavelength of light, which is used as the link to the base metric of motion, the meter, differs as a function of the refractive index of the medium through which it is passing. In air, the index of refraction differs considerably with changes in temperature, humidity, pressure and even the air’s local composition. Consequently, accurate positioning measurement requires continuous and accurate knowledge of these environmental factors for use in active wavelength compensation. Moreover, the rate and type of air flow over the laser beam will cause local variations in refractive index, resulting in what is considered as measurement noise.

Besides environment related contributors, other impacts on the united uncertainty of machine motion include several elements. The flatness and alignment of the plane mirror optics used in the interferometry system add to the uncertainty in the motion of the work point. Similarly, the effects of angular error motions over any small Abbe offset present between the interferometers and the defined work point can be a chief contributor. In cases of low uncertainty budgets, laser wavelength stability and calibrated accuracy can be a non-negligible contributor, and when heterodyne lasers are employed, frequency bleed through can induce major measurement uncertainty as well.

Finally, as in nearly all precision machines, the thermal expansion of any number of components over an uncompensated distance will introduce considerable uncertainty in the regulation of the work point’s motion. Figure 1 shows a diagram of the key uncertainty contributors to motion control in machines where homodyne laser interferometer feedback is used near 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

The impact of the uncertainty contributors defined in Figure 1 is a function of each machine’s design, environment, length of travel, and its distinct work point. Quantification can only be estimated on a case-by-case basis where machine specifics are identified. Each machine design must have its own uncertainty analysis, even if similar to formerly analyzed machines. Thus, this article presents the motion of a machine vision inspection station used as a metrology tool at Aerotech’s headquarters as a particular example in measurement uncertainty estimation. Figure 2 shows the machine presently installed in Aerotech’s Metrology Lab.

**Figure 2. **An image of an example precision inspection machine installed in a temperature-controlled 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 built 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. When assessing the uncertainty in the measurement of work point motion by the lasers, this machine shares all of the key contributors defined in Figure 1. By working through these sources one-by-one, a rational estimate of measurement uncertainty can be attained for this particular machine’s application. As in all engineering tasks, some assumptions have to be made in the quantification of uncertainty, and their validity is defined by the uncertainty or error budget trying to be realized for a particular application. This application specific depth of analysis is another crucial reason why each machine design and, more specifically, machine task must have its own uncertainty analysis.

## Definition of the Measurand

Uncertainty analyses are always of a specific measurement. This is a commonly misunderstood point, and is worth repeating. A machine or device itself cannot have an uncertainty – only the particular measurement of a defined quantity can have an uncertainty. Thus, the explicit definition of the quantity under measurement, the measurand, will have a great impact on uncertainty quantification. In the example mentioned here, the measurand is defined as the two dimensional (2D) planar measurement of work point position via two independent plane mirror interferometers. The machine’s work point must be specially defined for this measurand to be meaningful.

The work point, or origin, of the machine vision inspection station illustrated in Figure 2 above, is defined as the intersection of the camera’s optical axis with a plane parallel to the least squares planar fit of the vacuum chuck’s mounting surface, only offset 3 mm above the vacuum chuck. This “floating” plane’s vertical position is established by the typical thickness of the parts the machine is designed to measure. The vacuum chuck surface plus a 3 mm offset, thus, defines the XY coordinate plane to be nominally coincident with the upper surface of a standard part under test.

The X-axis of motion is defined as the unit normal vector of the best fit plane to the X mirror’s surface, projected onto the XY coordinate plane as defined by the offset vacuum chuck surface. Lastly, the Y-axis of motion is established as the cross product of the unit normal vector of the vacuum chuck’s mounting plane and the X-axis’ unit vector, all fastened using the intersection of the camera’s optical axis as the origin. An illustration of the machine’s work point is illustrated in Figure 3.

**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 X-axis direction is dictated by the unit normal vector of the X mirror’s surface. Finally, the Z-axis is the unit normal of the XY plane, and the Y-axis 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 first two chief uncertainty contributors to be quantified are the measurement system’s resolution and feedback noise floor. The interferometer resolution is easy to establish as it is dictated by the manufacturer’s electronic hardware, and should be stated in the related operations manual. This machine utilizes Renishaw RLD double pass plane mirror interferometers. They have a digital measurement resolution of 10 nm, which is assumed to be a square distribution width.

The feedback noise floor will dictate the machine’s minimum effective resolution, and also sets a floor for the uncertainty in measuring the work point’s planar motion. As illustrated in Figure 1, the feedback noise in this case is dominated by mechanical vibration, electronic noise, control loop stability, and most significantly, air turbulence across the open interferometer beams. In this machine, the impact of mechanical vibration on the interferometers is reduced through the use of a high compliance passive air isolation system. Similarly, the CMM itself is surrounded by an environmental enclosure to reduce air currents across the laser beams. The total implementation of this machine creates long term (>3 hours) laser stability of about 25 nm root-mean-square.

## Laser Wavelength Compensation and Dead Path Errors

As stated, the connection to the base metric of the meter is the wavelength of the light used in the system. The wavelength is altered by deviations in index of refraction within the medium through which it travels, and those variations must be compensated. The need for compensation of wavelength change as a function of a laser’s local environment has been well documented^{1}, and is an automated feature in most commercially available interferometer systems. Here, humidity and air composition variations are small enough to overlook. However, the accuracy/uncertainty of wavelength compensation, and thus the interferometric measurement itself, is dictated by the ability to accurately measure variations in the remaining environmental parameters of pressure and temperature.

When applied to distance measurement, as in this machine, interferometric measurement is done relative to some reference location. Consequently, wavelength compensation is only actively applied to the number of fringes, or waves, counted by the detector in a relative move away from the reference location. In practice, it is not possible to place the laser heads in contact with the mirrors when at the reference location, where some relief should be provided. There is always a small portion of laser beam that is not accounted for in the wavelength compensation. When the local environment varies, this length of beam is uncompensated, and causes small shifts in the system’s zero location. This is known as dead path error. Since the errors contributed from omitted wavelength compensation are dependent on length, decreasing the amount of space between each laser head and the location of the mirrors at the reference point will decrease induced uncertainty from dead path errors.

Both laser wavelength compensation and dead path effects on uncertainty in the measurement of the work point are dependent on length. As such, their impact over the maximum operating distance will be measured for this quantification example. The weather station used to correct for environmentally related wavelength variations in this machine has an absolute pressure accuracy of 1 mbar and an absolute temperature accuracy of 0.2 °C. The relative change of temperature and pressure over a three hour test, which impacts dead path error, has been observed to be less than 0.1 °C and 0.5 mbar, respectively. For the sake of simplicity, 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 avoids the need to perform propagation of uncertainty analysis per the Guide to the Expression of Uncertainty in Measurement (GUM)^{4}, and the corrected Edlen equations^{5}. Assuming machine operation around standard temperature, humidity and pressure, the uncertainty contribution of the weather station’s temperature and pressure measurement accuracy is 29 nm and 5 nm, respectively, across the entire 300 mm travel of the machine. Similarly, the relative change in temperature and pressure during a given test contributes 5 nm and 1 nm of measurement uncertainty, respectively, across the 96 mm of dead path per axis. These contributions are all said to be square distribution widths. A more detailed explanation of both laser wavelength compensation and dead path errors is given in Renishaw technical paper TE3296

## Mirror Flatness, Orthogonality, and Cosine Errors

Compared to the consequences of laser wavelength variations, the effects of mirror flatness and mechanical alignment on measurement uncertainty are more direct. The flatness of each mirror directly creates positioning uncertainty of the work point in the form of straightness errors. Furthermore, the alignment of the mirrors’ surface normals to the defined XY coordinate plane will trigger cosine errors in the definition of the X and Y axis scales. Similarly, but autonomous of the mirrors’ alignment, is the induced cosine error from the alignment of the laser beams to each mirror normal. Lastly, orthogonality of the mirror surfaces to each other will impart a squareness error on the position measurement of the work point.

The mirrors used in this machine are stated to have 63 nm of peak-to-valley flatness over their total area with a 95% confidence interval in this specification’s validation report. Through mechanical fixturing and machining tolerancing, the alignment of the mirror normals to the XY coordinate plane is less than 25 µrad, and the alignment of the laser beams to the mirrors is also less than 25 µrad per the level of signal difference detected through travel. Lastly, the orthogonality of the two mirrors to each other is software amended through a square reference artifact and reversal method. Uncertainty in the reversal matches to a less than 0.05 µrad squareness error between the mirrors. Thus, the mirror flatness specification contributes 16 nm of measurement uncertainty. The cosine error of both mirror and laser alignment contributes 0.1 nm each over the machine’s total travel. The orthogonality of the two mirrors will also add 15 nm of a triangular uncertainty distribution over the machine’s range.

## Abbe Offset and Angular Error

The effects of angular error motions on the measurement uncertainty of a work point position are usually some of the most significant. This is because of the frequency of including Abbe offsets between the measurement point and the defined work point of interest. One of the main reasons for using interferometric feedback at or near the work point is to eliminate Abbe offset, and thus eliminate the effects of angular error motions on work point measurement. In this specific example, there is nominally zero Abbe offset between the positioning of the laser beams, the feedback mechanism, and the defined work point. Thus, there is a minor amount of measurement uncertainty contributed by the angular error motions of the bearing ways. However, as a demonstration of the significance of Abbe offset minimization, a mere 10 mm of Abbe offset can trigger 250 nm of measurement uncertainty for just 25 µrad of angular error motion.

## Differential Thermal Expansion of Machine Components

Lastly, the differential thermal expansion of many machine components can have a major impact on measurement uncertainty. For instance, due to the work point’s definition in this machine, there is a large length of granite between each laser head and the work point that is left to expand and contract freely with varying temperature. If the temperature of the granite machine base would alter during a test, the granite bridge holding the camera would contract or expand, moving the optical axis relative to the laser heads.

However, the feedback control of the interferometer system would maintain the same distance between each mirror and each laser head. This would cause the work point to shift without seeing a variation in position reading, and would thus add to the measurement uncertainty over the course of a measurement. However, in this specific machine design, there is also a lesser length of aluminum between the mirror faces and the work point, in the form of the payload plate, left to expand and contract with temperature variation. By design, the lengths of granite and aluminum contribute opposite directions of work point motion relative to a position being measured on a grid plate when both expand and contract. Consequently, as long as the ratio of the respective lengths of aluminum and granite are equivalent to the ratio of their corresponding coefficients of thermal expansion, the growth of each machine component adds a tiny amount of measurement uncertainty to the measurement of a part under test.

It is crucial to note that thermal gradients are ignored in this analysis. Assuming simultaneous bulk temperature change is incorrect, as the thermal mass of the granite is a lot larger than that of the aluminum payload plate. Thus, the machine components would not expand and contract in unison, voiding their termination of induced work point motion. Yet, in the controlled environment of this machine, these dynamic thermal expansion effects contribute minor uncertainty.

## Summary and Expanded Standard Uncertainty

When combining the uncertainty contributions of uncorrelated inputs, it is possible to sum the estimated uncertainty of their distributions in quadrature. This is a basic application of the propagation of uncertainty described in the GUM^{4}, and is valid for this example. Table 1 is a summary of the key contributors considered in this analysis, their uncertainty contributions, and the combined and expanded uncertainty enclosed in the system’s positioning measurement. Uncertainty summaries such as Table 1 can provide an understanding on where efforts should be focused to further enhance measurement.

It is essential to note that there have been many assumptions made during this uncertainty analysis. Similarly, there is a mass of underlying principles and background knowledge that has been used to guide this analysis. As stated before, each machine design should be supplemented by its own uncertainty budget and analysis. The key contributors defined in Figure 1 may not always be adequately exhaustive for other cases, nor the streamlining assumptions made throughout suitable for lower levels of desired measurement uncertainty.

To attain a better understanding of measurement uncertainty and its estimation, a good reference is the Guide to the Expression of Uncertainty in Measurement (GUM) and its supplementary materials^{4}, available for free at the International Bureau of Weights and Measures’ website^{7}.

**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, Gaussian | 25 nm | |

Laser Wavelength Compensation | Temp. Accuracy = 0.2 °C Press. Accuracy = 1 mbar |
58 nm, Square 11 nm, Square |
17 nm 3 nm |

Air Dead Path Errors | Temp. Change = 0.1 °C Press. Change = 0.5 mbar |
9 nm, Square 2 nm, Square |
3 nm 0.6 nm |

Mirror Flatness | 64 nm, Guassian | 16 nm | |

Cosine Errors | Mirror Alignment = 50 µrad Laser Alignment = 50 µrad |
0.5 nm, Biased 0.5 nm, Biased |
0.1 nm 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**

This information has been sourced, reviewed and adapted from materials provided by Aerotech, Inc.

For more information on this source, please visit Aerotech, Inc.