NISTIR 7725

The NIST 66 m Ball Range

A One Dimensional Artifact to Test Three 

Dimensional Imaging Systems

Bruce Borchardt

Craig Shakarji
The NIST 66 m Ball Range

A One Dimensional Artifact to test Three Dimensional 

Imaging Systems

Bruce Borchardt

Craig Shakarji

Precision Engineering Division

Manufacturing Engineering Laboratory

October 2010

U.S. Department of Commerce

Gary Locke, Secretary

National Institute of Standards and Technology

Patrick D. Gallagher, Director

The NIST 66 m Ball Range

A One Dimensional Artifact to Test Three Dimensional Imaging Systems

Bruce Borchardt, Craig Shakarji, Large Scale Metrology Group

Abstract

NIST (the National Institute of Standards and Technology) has instituted a 66 m long 

range for testing of distance-measuring systems using noncooperative targets. The range 

consists of titanium balls mounted along a line on stands at unequal intervals. This paper 

describes this range, including the uncertainty of the calibrated ball positions. The range 

has been shown to be sufficiently stable and repeatable that it, when calibrated from day 

to day, is capable of being used as a workable test range. The overall array of kinematic 

nests holding titanium balls can be used as a facility for performing range tests of threedimensional imaging systems with uncertainties in the test distances of (10 + 1.1 L) μm, 

where L is the distance in meters. Continued monitoring of the positions has shown some 

interesting systematic behaviors that do not interfere with using the range for its intended 

purpose.

Background and History

While still at the Van Ness Street site in Washington, DC, NBS (The National Bureau of 

Standards, since renamed NIST, the National Institute of Standards and Technology) 

built a tape tunnel in order to meet the demand for the calibration of metal tapes. The 

facility was reconstructed in Gaithersburg between 1964 and 1965. Tapes were the 

primary method of measuring long distances, and included surveyors’ “chains” used in 

geodesy and all types of surveying. Until 1960, the base length unit of the SI was the 

physical meter bar at BIPM (Bureau International des Poids et Mesures) in Paris, realized 

in this country by an array of platinum bars directly traceable to that bar. The facility at 

NBS was designed to transfer the meter to a master tape to be used on a bench for 

comparing tapes.

The facility consisted of 22 microscope stands on piers. (These piers form the mounts for 

the current range as discussed later in this document.) The microscopes were used in the 

process of calibrating master tapes. The stands were laid out in a particular pattern. For 

metric tapes only, the pattern would have been fairly straightforward: a stand every 5 m, 

with one 5 m interval divided into 1 m sections. The 1 m section is needed to transfer a 1 

m line standard to a 5 m standard. Complications arose from the need at the time to 

include 50 foot measured intervals for the calibration of 200 foot tapes in English (US 

customary) units. To cover 200 feet, four extra stands were required. Since 50 feet is 

exactly 15.24 m, the extra stands were put at multiples of 15.24 m from a reference point. 

To get the total number of stands: every 5 m from 0 m to 65 m requires 14 stands, plus 4 

stands for the 1 m sections, plus 4 stands for the 50 foot intervals, making a total of 22 

stands.
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Table 1: Stand positions

Distance from 

end stand, m

Comments

0 End Stand, farthest west

5 Reference for English 200 foot range

10

15

20

20.24 50 feet from 5 m stand

25

26 1 m section

27 1 m section

28 1 m section

29 1 m section

30

35

35.48 100 feet from 5 m stand

40

45

50

50.72 150 feet from 5 m stand

55

60

65

65.96 200 feet from 5 m stand

The system of transferring to tapes from line standards remained in use even after the 

redefinition of the meter in 1960, since interferometry over long distances was 

problematic and the Krypton lamp was difficult to use as a working standard. Practical 

laser interferometers became commercially available in the early 1970s, and the 

microscope stands and line standard carriages fell into general disrepair, as the calibration 

of tapes was done on the tape bench with laser interferometers as standards.

Adaptive Re-use

In 2005, the need became apparent for a facility to test three-dimensional (3D) imaging 

systems such as Lidar. An appropriate range would consist of suitable objects at known 

distances. To test an instrument, a typical procedure would be to use it to measure the 

objects; the errors of the instrument would be reported as the result of the measurements 

minus the calibrated values. The uncertainty of the calibrated values would then 

contribute to the uncertainty of the test result. 
3

The choice of a suitable object depends to some extent on the type of instrument. In 

general, the choice for traditional metrological artifacts would be well-finished hardened 

steel. However, the mirror-like surface of such an object is normally not suitable for any 

system relying on reflected light.

For this range, the objects chosen were titanium balls. Titanium was chosen because it 

can be made round to a high degree of accuracy, typically a few micrometers, and can at 

the same time retain a matte optical finish while having the added advantage of being 

relatively light in weight. The size of 101.6 mm was chosen as a compromise. Small 

spheres on the order of 20 mm to 30 mm would be easier to make and measure, but are 

less representative of typical objects measured by a scanning system. Larger spheres 

would be very expensive. The titanium spheres were obtained with a “satin” finish, 

which is optically diffuse. All of the spheres were measured for form and diameter. The 

results are summarized in Figures 1 to 3. The (k = 2) expanded uncertainty of the CMM 

that performed these measurements is 0.5 m. The lack of repeatability of these numbers 

reflects the fact that for an imperfect object the choice of points influences the value 

obtained. The uncertainty budget includes a term to account for this effect. 

Building the Mounts

Because the balls need to be continually mounted, dismounted, and remounted for the 

calibration process, kinematic seats were necessary. It has been shown that kinematic 

seats with hard, smooth surfaces can repeat at the nanometer level. A simple three-ball 

nest was chosen, since mounting a sphere needs only a three degree of freedom mount, 

i.e., the angular motion of the sphere does not change its center location (at the level of 

the sphere roundness). Figure 4 is a simplified drawing of a mount, and Figure 5 is a 

picture of a titanium sphere in a mount. The mounting points are hardened steel tooling 

balls, 9.525 mm in diameter, expoxied in place. The bolt circle for the mounts (78.5 mm 

diameter) was calculated to have the contact points subtend a 90 angle (measured as the 

cone angle at the sphere center). This gives both reasonable stability and access to some 

of the lower hemisphere. 

The kinematic nests were mounted on two-axis stages previously used to mount the 

microscopes that were used in the line standard calibration process. The stages were in 

various degrees of disrepair, with about half missing the handwheels to make 

adjustments. Fortunately, replacement handwheels were readily available, and loosening 

the gibs and lubricating the stages made all of them usable. The connection to the 

microscope was through three threaded studs, arranged in an equilateral triangle. 

To mount the kinematic nests, we used a simple triangular plate with three holes to match 

the studs. (See Figure 6.) There were two different patterns of mounting studs on the 

two-axis stages, for unknown reasons lost in history; the pre-existing bolt circle diameter 

for 15 stages was 125 mm, and for the other 7 stages, it was 88 mm. There was room 

inside the 125 mm circle for a 63.5 mm diameter column. We had on hand a supply of 

63.5 mm extrusion suitable for these 17 columns. The space in the smaller bolt circle 

could only fit 50.8 mm columns, so these were made of solid aluminum stock with the 
4

idea to prevent, as far as possible, losing some stiffness otherwise inevitable with the 

smaller column diameter. Both types of columns were made 155 mm high. This value 

was chosen so that the centers of the 101.6 mm diameter balls were at the same height as 

the laser interferometer axis located above the tape bench on the other side of the room.

The plate, column, and nest are screwed together with M10 studs. The joints between 

elements have tapped holes on each side. A stud is screwed into the two holes on each 

side of the joint. Using this assembly method means that the angular position of the three 

ball kinematic mount is not controlled, but this does not affect its performance.

The assembled array with titanium balls mounted is shown in Figure 7.

The 101.6 mm diameter SMR (Spherically Mounted Retroreflector)

Generally, the plan for calibrating the distances between the titanium balls is to rest an 

SMR in the nests in place of the titanium balls and use a laser tracker to measure the 

distances from one location to the next. 

Thus, the plan for calibration depended on obtaining an SMR with the same diameter as 

the titanium balls. After some difficulty, we found one manufacturer who was able to 

supply two of these vital items with acceptable specifications for diameter, roundness, 

and concentricity of the optical elements. 

The SMR tolerance specifications were:

Diameter 2.5 m

Roundness 0.4 m

Concentricity 2.5 m

There was one unforeseen complication when we actually put an SMR into our kinematic 

nest: it wasn’t balanced. The weight of the SMR outside sphere is not symmetrically 

placed, so when it’s in a nest and free to pivot, it turns so the retroreflector points up. 

This problem was solved by making a small clip that goes between the lower edge of the 

kinematic nest and the lip of the SMR and holds it close to horizontal. Precise horizontal 

alignment isn’t needed, as long as the tracker beam can see the center of the 

retroreflector. (See Figure 8.)

Setup and Alignment of the Nests

At this point, the array of nests was ready for alignment and calibration. The actual 

positions were arbitrary, since they could not be set significantly better than 0.2 mm 

and needed to be calibrated with an order of magnitude better uncertainty. The two end 

nests were set to be near the middle of their travel in X, Y, and Z. X and Y travel were 

easy to set as this just required turning the handwheel until the carriage was in the right 

place. Setting the Z axis was harder, since it is necessary to loosen the three upper 
5

(locking) nuts, move the three bottom nuts to the right places, and re-tighten the locking 

nuts. During the Z axis setting procedure, a small bubble level was used to keep the top 

of the kinematic mount close to horizontal. 

The laser tracker was set up along the center line, a few meters beyond the “0 m” nest, at 

the height of the center of an SMR. At this point, it was necessary to make the tracker 

start tracking on the 101.6 mm SMR. The normal procedure calls for putting a 38.1 mm 

diameter SMR into the home position (“bird bath”) of the tracker and then moving it, 

without letting the beam be interrupted, to all measurement points. Since the 101.6 mm 

SMR could not fit in the home position, an alternate procedure was needed. 

The tracker has a method of searching near a previously defined location. We made a 

temporary nest that held the 38.1 mm SMR up in line with the 5 m nest so the center of 

the SMR was close to the center point of the 101.6 mm SMR when it was put into the 

nest. We measured the temporary point and stored it as a measured point. The small 

SMR was put back in the home position, and the large one was put on the nest. The 

tracker was then instructed to “go location” to the measured point; it found the large 

SMR in the nest and began tracking it. 

We measured the two end points of the range. Using the tracker’s software, we defined a 

coordinate system whose X axis passed through those end points, with the Z axis vertical. 

At this point, it was just a matter of moving the SMR sequentially to each nest and 

adjusting Y and Z so the tracker Y and Z components of the measured position read a 

value near zero. The X readings were set to be close to the nominal values. For 18 nests, 

these were even meter values, but the 4 nests on 15.24 m (50 foot) centers needed to have 

meter equivalents calculated. At this time, we did not understand that the 5 m nest was 

meant to be the origin for the English ranges, so we set the 4 nests to an arbitrary value, 

making sure only that they were 15.24 m (50 feet) from each other. The first one was set 

to 20.2312 m from the end nest, and the others were set at 15.240 m intervals. To follow 

the designed intention, the first nest should have been set at 20.240 m and the others at 

15.240 m intervals from there. This 8.8 mm maladjustment will be corrected the first 

time the nests are moved for any other reason; until then it’s not worth doing. It’s 

important to keep the positions fixed for as long a period as possible for the purpose of 

monitoring the history of their locations.

See Figures 9 and 10 for the approximate Y and Z values (as measured by a laser tracker) 

plotted for several calibration runs over a few week time period. Since the values of the 

Y and Z coordinates are not used in any report or analysis, uncertainties were not 

calculated. Judging solely by the repeatability values as shown, their uncertainties would 

be in the tenths of millimeters. On these plots, the end points are set to zero so that the 

other points will show the deviation from a straight line. Note that the Y values are 

closer to the nominal zero; this is because fine adjustment is much easier to do on the Y 

axis. Even so, only three nests are farther than 1 mm from a straight line in the Z 

direction. Nonetheless, for use as a ranging facility, the Y and Z deviations do not impact 

our uncertainty, because we only measure in the X direction. Furthermore, as a ranging 

test facility, we can calibrate the reference distances on the same day as a test, so effects 
6

that only occur over longer time periods do not affect the uncertainty of the reference 

values.

After we set the nest locations, the gibs of the stages were locked down tightly to 

minimize inadvertent movement. At this point, a casual tweak of any of the handwheels 

would no longer produce any measurable movement in the ball.

Using Trackers to Measure Nest Positions

A laser tracker when used as a linear interferometer has very small sources of error. In 

order to use a tracker in this mode, it must be set up in such a way that there are only 

small changes in the horizontal and vertical angles during the measurement. This 

normally means that the tracker is placed so it is in line with the line of measurement. 

(There are possible alternatives using one or more mirrors to direct the measurement line, 

but these are not relevant in this discussion.) 

If the tracker is in line, the only significant errors are those of doing interferometry. 

These would include the laser wavelength, index of refraction compensation, and 

possible software errors that count the fringes incorrectly. (“Fringes” is used loosely; 

modern heterodyne laser systems have more subtle operating principles than the classical 

Michelson interferometer.) 

Comparisons between two trackers

During the course of monitoring the 66 m range, two different trackers were used to 

measure the ball nest positions. The manufacturer’s specifications are:

Tracker Tracker 1 Tracker 2

Max Range 70 m 35 m

MPE at max range 70 m 24 m, from formula: 10 m + 0.4 L (L in m) 

(L is the measured length. MPE = Maximum Permissible Error- note these specifications 

apply to linear measurements and not to three-dimensional coordinates)

Because of the different maximum ranges, two different measurement plans were used 

for the two trackers. Tracker 1 was placed in line at the end of the row and the positions 

were measured directly. Since the maximum range for the tracker 2 is insufficient to 

measure the entire 66 m at once, this necessitated a modified plan. The tracker was 

placed in line between the 30 m and 35 m positions, and the nests on each side were 

measured from that point. Because of this technique, the distance from 30 m to 35 m was 

significantly less accurate. Measurements taken with a tracker in two opposite directions 

include twice the zero-point setting uncertainty. This is also called “bird bath error” 

because the physical point on the tracker where the interferometer starts counting is a 

shallow kinematic nest, vaguely resembling a bird bath.

The presence of zero-point error required a separate measurement of the 30 m to 35 m 

gap. The tracker was moved to approximately the 38 m point, and the gap was measured 
7

directly with the interferometer. This value was taken to be correct, and the positions of 

all of the nests from 35 m to 66 m were adjusted by the difference between the “true 

value” and the initial measurement (taken when the tracker was between 30 m and 35 m). 

For example, the first set of data (with the zero-point error included) might show the 

distance from 30 m to 35 m to be 5000.010 mm. The independent measurement of the 

gap would give 50000.030 mm. This is a better value, since it does not include any zeropoint error. All of the X positions 35 m and above would have 0.020 mm added to them.

This procedure adds some additional uncertainty that must be accounted for in the 

uncertainty budget.

Traceability of the Tracker Measurements

At intervals during the measurements, each tracker was compared with the reference 

interferometer used with the tape bench. The reference interferometer is calibrated using 

the iodine stabilized laser to realize the SI meter.

Although the comparisons with the reference interferometer are not completely stable, 

they are within acceptable limits for present needs. When Tracker 1 was sent back to the 

manufacturer for repairs, its error decreased significantly, so the current error budget can 

reflect a lower value for this term. 

It should be noted that the two trackers have different procedures for correcting the 

effective wavelength for temperature and pressure. Tracker 1 requires the user to 

manually input values for temperature and pressure, and then calculates effective 

wavelength using some equation, presumably Edlén’s, but the process is hidden from the 

user. Tracker 2 has its own weather station, so it is important to put its temperature 

sensor at a point that approximates the temperature in the path of the laser beam.

The values used to correct Tracker 1 were taken from the reference interferometer’s 

extensive instrumentation, which includes 7 air sensors evenly spaced along 60 m of the 

tape bench. The average of those sensors is a very good estimate for the effective 

temperature along the bench. There is a known gradient from the bench to the line of 

nests of about -0.15 C (i.e., the air above the nests is colder than that measured over the 

bench). This creates a bias in the measured lengths of about 1.4 x 10-7 m/m, or about 

9 m in the total length of 66 m.

Another factor which comes into play at these levels of accuracy is humidity. The 

Tracker-1 software assumes a constant relative humidity of 50 %. If the humidity is 

actually 20 %, as might often happen in the winter, the relative error is about 2.5 x 10-7, 

with a direction sense making the measured length appear longer than actual. This value 

has been applied to the calibration of January 2009 in the table below. The appropriate 

adjustment has also been made to data taken after September 2008. 
8

Table 2: Tracker range errors

Date (month/year) Tracker Relative error (max) Comments

11/2006 1 +7 x 10-7

4/2008 2 +7 x 10-7

4/2008 1 +5 x 10

-7

6/2008 1 +2.7 x 10-7before warmup

6/2008 1 +1.5 x 10-7after warmup

1/2009 1 +2.5 x 10-7 with humidity correction

Results of Range measurements (uncertainty over the short term)

When using the facility to test 3D imaging systems, the procedure requires measuring the 

ball positions before and after the test. Therefore, the relevant factor to include in the 

uncertainty analysis would be the change in position from measurements taken before 

and after the 3D imaging test. Therefore, it’s important to find out how much the 

positions change in the course of a day. 

Short term changes of selected data are shown in Figures 11 to 21. (The runs were 

selected to highlight some significant aspects of stability.) For these plots, the time 

between runs was typically a day to several weeks. Error bars are not shown on these 

plots in order to make trends more visible. If they had been included, the uncertainties of 

all these values on these plots would be as calculated in the uncertainty budget below. 

The time periods are shown in the figure captions. There are several interesting features 

to notice in these plots. First, the changes are almost always linear; that is, the ball 

positions change proportionally with distance, subject to day-to-day random repeatability 

noise of about 5 m. There are a few exceptions to this rule which will be pointed out 

later. 

Figures 12 and 13 are plotted in a somewhat unusual way; the plotted values are all 

referenced to the 5 m point. Note that the 0 m point has a wide range of values, from 

-14 m to 26 m. During these runs, the weight of the tracker rested on the wood floor 

adjacent to the steel rails that the line scale carriage runs on. When the operator picked 

up the SMR from the 0 m pillar, it was necessary to walk on the wood floor. 

Consequently, the tracker would move slightly. Once the 0 m measurement was finished, 

no one walked near the tracker, and so after that the numbers became more stable and 

self-consistent. Once this phenomenon was recognized, a tracker mount that put the 

weight onto the steel rails was built, and the problem was corrected. The runs taken 

during the time the mount was less stable were plotted with the 5 m mount as the 

reference point.

Figure 12 also shows the biggest jump observed over the entire 4 years of measurements. 

Between runs 78 and 79, all of the positions moved significantly, even beyond the 0 m 

instability noted above. Every ball nest changed position by 10 m to 35 m. The two 

runs were on 26 May and 30 May of 2006, before and after the Memorial Day holiday 

weekend. We have no definite explanation for this unique behavior. Intriguingly enough, 

there was a major earthquake in Java, Indonesia, at about 7 pm on May 26. The 
9

longitude of the epicenter was at 110 East, about 173 of longitude from Maryland. This 

is unlikely to be a full explanation, especially since there were several significantly larger 

earthquakes elsewhere in the world over the time span of the data, and they didn’t cause 

any noticeable changes. The unlikelihood of this explanation has been underscored by 

the occurrence of a magnitude 3.6 earthquake in Gaithersburg on the morning of July 16, 

2010, which didn’t cause any noticeable changes.

Figure 14 shows a small jump in the 20 m position; this change is only about 12 m and 

may well be due to someone bumping into the pillar.

In Figure 19, a change with a known cause is shown. Just before run 470, it was found 

that the 20 m nest had come slightly unscrewed. The runs from 464 to 469 show the 

position was around +0.1 mm. For run 470, the nest was tightened down and the position 

shifted to -0.09 mm. Run 470 became the new reference, and subsequent runs are shown 

as the difference from that run.

One conclusion to be drawn from this information is that the individual ball positions are 

stable to approximately 10 m over a short time interval (a day or two), with rare 

anomalies where greater changes are possible. Another conclusion is that the overall 

length of the range changes slowly over a time span of weeks to months, and that this 

drift is uniformly distributed along the 66 m length of the room. 

Results of Range measurements (long term stability)

Even though we anticipate doing frequent recalibrations of the range so that the short 

term stability is the only important term, it is interesting to investigate the long term 

stability. Figure 22 shows all of the history of the values for the total length of the range 

(from the 0 m position to the 65.951 m position). Because we have shown that the 

positions move uniformly proportional to length, this plot is indicative of the scale 

changes of the whole range. To assure ourselves that the last ball is representative of the 

range, we have also checked the position of the 65 m pillar, and it tracks the changes in 

the 65.951 m position very closely. 

Inspection of the plot shows some interesting features. The most obvious is the seasonal 

variation. Every summer the range gets longer, and every winter it shrinks again. The 

seasonal changes exceed 10 m/m. This would require a change of 10 C in air, and 2 C 

in most types of rock. Since we continually monitor the tunnel, ensuring it to be 

20.0 ± 0.2 C during a measurement, this can be eliminated as an explanation, especially 

considering that the bases of the columns are five meters underground.

However, we note that temperature variations, when they occur, can produce the effect 

seen. This is underlined by the results from late 2009, after the February to September 

timeframe when the air conditioning system was undergoing extensive renovation, with 

the result that the temperature was entirely uncontrolled. During this time, the average air 

temperature frequently exceeded 25 C and approached 30 C. (No measurements were 

made during this time period.) When the air temperature was brought under control 
10

around September 11, the first check of the range was longer than the historical 

maximum, 0.7 mm longer than nominal. However, it began to shrink rapidly, and by 

December 10 it was down to 0.077 mm, in line with the previous year’s values. The 

values continued to decline all winter, reaching a minimum on March 1, 2010. The 

minimum value, -0.428 mm from nominal, was the lowest value ever. By June 29, the 

range length had rebounded to -0.007 mm, and by October was up to +0.010 mm, entirely 

in accord with past history.

It appears that permanent mountings are not as permanent as one might believe. They 

need recalibration regularly; at the highest level, they must be calibrated before and after 

each use.

We have a very good site; it is underground, has had 40 years to settle, and the room has 

excellent temperature stability. Despite all of these factors, there are influences beyond 

our control. Given this data, a range with its pillars set in surface earth or sited out of 

doors could well be much worse than this. 

A long-term study such as this is valuable to point out the necessity of monitoring things 

that one might not think need monitoring.

Uncertainty Budget

The standard uncertainty in the calibrated distances between the centers of mounted 

titanium spheres (when using Tracker 1) is given by the general equation: 

2

humidity

2

pressure

2

temp gradient

2

tracker comparison

2

reference

2

form

2

ucal urep u u u u u u ,

where the components have values as follows (length-dependent values use L, the 

measured length, in meters): 

urep

= (3.0 + 0.3L) μm Repeatability studies over very short intra-day time periods 

were performed as well as reproducibility studies over oneday and many-day time intervals in order to capture several 

effects including stand instability, SMR imperfections, drift, 

etc. For the short time periods, several distances (on every 

stand) were measured twice (with the SMR removed and 

then placed on the nest again) to capture the short term 

effect. For the longer time periods, reproducibility studies 

were performed over several one-day and multi-day periods

(displayed in Figures 11 to 22). The formula for 

urep

capturing these effects is given as (3.0 + 0.3L) μm for 

time durations of one full day (including overnight). (The 

primary interest lies in time periods of one day or less, due to 

the time between measurement and performing a test under

normal testing). Under normal testing, the distances between 
11

nests would be measured both before and after a test, 

virtually eliminating the chance that an undetected anomaly 

occurs (as is rarely seen in Figures 11 to 22).While it is 

possible this term is not completely independent of other 

uncertainty contributors, the inclusion of its full value leads 

to only a mild overvaluation of the uncertainty.

form u

= 3.0 μm The form of the titanium spheres can affect the center 

location when seated in the kinematic nest. Simulations were 

run using the form magnitudes shown in Figure 1 to arrive at 

the value used here. 

reference u

= (2.5 + 0.2L) μm The reference interferometer has been compared to an 

iodine-stabilized laser many times and is used as part of a 

ranging test facility having a well characterized uncertainty. 

See [1, 2]. The uncertainty of the reference is dominated by 

atmospheric compensation on the tape bench line, 

temperature gradients along the bench line, and the 

mechanical effects of the carriage (e.g., Abbe errors). 

utracker comparison

= 0.40L μm The value for the back-to-back comparison between the 

tracker including atmospheric temperature compensation and 

the reference interferometer is based on the results shown in 

Table 2 converted to one standard deviation, taken to be 

large enough to cover all variations.

utemp-gradient

= 0.2L μm Computed from the Edlén Equation using the effect of a 0.15 

°C mean difference between the temperature sensors on the 

tape bench and the actual ball range. The mean temperature 

difference will likely be reduced in future calculations due to 

upgrades to the thermal environment.

upressure

(negligible) The effects of pressure gradients (up to 0.02 hPa)—as 

calculated by the Edlén Equation—are dominated by other 

terms in this uncertainty calculation [2].

hum idity u

(negligible) The effect of a non-50 % humidity is corrected for in the 

calibrated value. The effect of the uncertainty of the 

humidity—as calculated by the Edlén Equation—is 

dominated by other terms in this uncertainty calculation [2].

Combining these terms gives:

2

ucal 24.3 2.8L 0.33L

μm.
12

Or, written in a more understood way (mildly overvalued):

cal u

= (5.0 + 0.57L) μm.

The expanded (k = 2) uncertainty is found as:

Ucal

= (10 + 1.1L) μm.

Using this budget, the k = 2 uncertainty for a 66 m range (our maximum) for a one-day 

duration is 83 m.

Conclusions

The existing pillars and kinematic mounts have been shown to be sufficiently stable and 

repeatable and, if calibrated from day-to-day, are capable of being used as an accurate

test range.

The overall array of kinematic nests holding titanium balls can be used as a facility for 

performing range tests of scanning instruments with an expanded uncertainty (k = 2) of 

(10 + 1.1 L) μm. To achieve this level requires checking the coordinates of the ball nests 

before and after the range test, preferably on the same day. If the time between 

calibrations is a few days to a week, data shows the uncertainty increases about 50 %. 

Data shown indicates that for time periods of six months or more, the uncertainty can rise 

several-fold.

We are continuing to monitor the ball nest positions, when circumstances permit. 

Acknowledgements

The staff of the Large Scale Coordinate Metrology Group have given extensive help in 

this project. In particular Dan Sawyer and Tyler Estler were integral to setting up the 

range. They, in addition to Chris Blackburn, frequently measured the ball range to keep 

the data string as intact as possible. Steve Phillips oversaw the project and was 

responsible for funding the work.

References

[1] W.T. Estler, D.S. Sawyer, B. Borchardt, S. D. Phillips, Large-Scale Metrology 

Instrument Performance Evaluations at NIST, The Journal of the CMSC, Vol. 1, 

No. 2, Autumn 2006, p. 31.

[2] W.T. Estler, Uncertainty Analysis for NIST Tape Calibrations, NIST Large Scale 

Coordinate Metrology internal white paper, September 2008.
13

List of Figures:

1. Form Error of Titanium Spheres. The second data points are repeated measurements. 

Note that not all spheres have repeated data. See text for a discussion of CMM 

uncertainty.

2. Diameter error of Titanium Spheres. The second data points are repeated 

measurements. Note that not all spheres have repeated data. See text for a discussion 

of CMM uncertainty.

3. Titanium Spheres, all measured points. The point numbers increase from pole to 

equator, with 36 points per line of latitude. Note that only four spheres have data 

points outside the 5 m lines: numbers 21, 23, 24, and 28. See text for a discussion 

of CMM uncertainty.

4. Drawing of kinematic mount (dimensions in mm).

5. Photograph of kinematic mount with titanium sphere.

6. Drawing of mounting plate (dimensions in mm). Note that the two versions differ 

only in the size of the bolt circle of the three through holes. Plate thickness is 9.5 

mm.

7. The whole array with all 22 titanium balls mounted. The apparent varying tilt of the 

columns is an optical illusion exacerbated by the straight fluorescent lights.

8. Photograph of SMR holding clip.

9. Y positions of monuments. The values shown are raw numbers from the tracker 

software. Since the values of the Y and Z coordinates (Figures 9 and 10) are not used 

in any report or analysis, uncertainties were not calculated. Judging by the 

repeatabilities shown, they would exceed 0.5 mm. 

10. Z positions of monuments.

11. Runs 11 to 31 over 35 days: 23 January 2006 to 27 February 2006. Note that error 

bars are omitted from Figures 11 to 21 for clarity. If shown, they would follow the 

formula (10 + 1.0 L) m (see uncertainty analysis).

12. Runs 32 to 79 over 21 days: 9 May 2006 to 30 May 2006.

13. Runs 80 to 102 over 34 days: 30 May 2006 to 3 July 2006.

14. Runs 108 to 122 over 6 days: 26 July 2006 to 1 August 2006.

15. Runs 123 to 171 over 98 days: 1 August 2006 to 7 November 2006. There is a 64 day 

gap (24 August to 27 October) between the top group and the lower group of runs.

16. Runs 209 to 259 over 50 days: 13 December 2007 to 1 February 2008.

17. Runs 314 to 356 over 33 days: 7 May 2008 to 9 June 2008.

18. Runs 357 to 410 over 60 days: 9 June 2008 to 8 August 2008.

19. Runs 411 to 471 over 89 days: 8 August 2008 to 5 November 2008.

20. Runs 472 to 493 over 25 days: 11 November 2008 to 1 December 2008.

21. Change in the measured value of the 66 m length from 24 January 2006 to 6 October 

2010. Error bars are k = 2 expanded uncertainty.
14

Form Error of Titanium Spheres

6

5

3

2

7

8

10

11 13

26

25

24

23

22

21

27

28

30

31

32

33

34

6

5

3

2

7

8

10 11

13

26

25

24

23

22

21

0

5

10

15

20

25

Sphere number Form, μm

Figure 1. Form Error of Titanium Spheres as measured on the M48 CMM using 225 

points. The second data points shown are repeated measurements. Note that not all 

spheres have repeated data. See text for a discussion of CMM uncertainty.
15

Diameter Error of Titanium Spheres

6

5

3

2

7 8 10 11

13

26

25

24

23

22

21 27

28

30

31

32

33

34

6

5

3

2

7 8 10 11

13

26

25

24

23 22

21

-120

-100

-80

-60

-40

-20

0

20

Sphere number Deviation from 101.6 mm, μm

Figure 2. Diameter error of Titanium Spheres as measured on the M48 CMM using 225 

points. The second data points are repeated measurements. Note that not all spheres have 

repeated data. See text for a discussion of CMM uncertainty.
16

Titanium Ball Measurements

-0.010

-0.005

0.000

0.005

0.010

0.015

1 37 73 109 145 181 217

Point number, pole to equator

36 points per line of latitude Deviation, mm

Ball #6 Ball #5 Ball #3 Ball #2 Ball #7 Ball #8

Ball #10 Ball #11 Ball #13 Ball #26 Ball #25 Ball #24

Ball #23 Ball #22 Ball #21 Ball #27 Ball #28 Ball #30

Ball #31 Ball #32 Ball #33 Ball #34 19.05 master 25.4 master

Figure 3. Titanium Spheres, all measured points. The point numbers increase from pole 

to equator, with 36 points per line of latitude. Note that only four spheres have data 

points outside the 5 m lines: numbers 21, 23, 24, and 28. See text for a discussion of 

CMM uncertainty.
17

Figure 4. Drawing of kinematic mount (dimensions in mm)

101.6

3 holes, 8.2 diam,

on 78.5 bolt circle 

M10 tapped hole

68

38.1
18

Figure 5. Photograph of kinematic mount with titanium sphere.
19

Figure 6. Drawing of mounting plate (dimensions in mm). Note that the two versions 

differ only in the size of the bolt circle of the three through holes. Plate thickness is 9.5 

mm.

140.3

M10 tap

3 holes, 11.9 diam,

on 109 (or 88) bolt circle 
20

Figure 7. The array with all titanium balls mounted. The apparent varying tilt of the 

columns is an optical illusion exacerbated by the straight fluorescent lights.
21

Figure 8. Photograph of SMR holding clip.
22

Y Positions of Monuments- after setting

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 10 20 30 40 50 60 70

X position, m Y deviation, mm

Figure 9. Y positions of monuments measured over a 14 week time period. Since the 

values of the Y and Z coordinates (Figures 9 and 10) are not used in any report or 

analysis, uncertainties were not calculated. 
23

Z Positions of Monuments- after setting

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70

X position, m Z deviation, mm

Figure 10. Z positions of monuments measured over a 14 week time period. Since the 

values of the Y and Z coordinates (Figures 9 and 10) are not used in any report or 

analysis, uncertainties were not calculated. 
24

X Position Change- runs 11 to 31

-0.15

-0.10

-0.05

0.00

0.05

0.10

0 10 20 30 40 50 60 70

X position, m Difference from run 10, mm

Figure 11. Runs 11 to 31 over 35 days: 23 January 2006 to 27 February 2006. Note that 

error bars are omitted from Figures 11 to 21 for clarity. If shown, they would follow the 

formula (10 + 1.1 L) m (k = 2 expanded uncertainty; see uncertainty analysis).
25

X Position Change- runs 32 to 79

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0 10 20 30 40 50 60 70

X position, m Difference from run 31, mm

Run 79

Run 78

Figure 12. Runs 32 to 79 over 21 days: 9 May 2006 to 30 May 2006
26

X Position Change- runs 80 to 102

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50 60 70

X position, m Difference from run 79, mm

Figure 13. Runs 80 to 102 over 34 days: 30 May 2006 to 3 July 2006.
27

X Position Change- runs 108 to 122

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40 50 60 70

X position, m Difference from run 107, mm

Figure 14. Runs 108 to 122 over 6 days: 26 July 2006 to 1 August 2006.
28

X Position Change- Runs 123 to 171

-0.15

-0.10

-0.05

0.00

0.05

0.10

0 10 20 30 40 50 60 70

Position, m Difference from run 122, mm

Figure 15. Runs 123 to 171 over 98 days: 1 August 2006 to 7 November 2006. There is 

a 64 day gap (24 August to 27 October) between the top group and the lower group of 

runs.
29

X Position Change- runs 209 to 259

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0 10 20 30 40 50 60 70

Position, m Difference from run 208, mm

Figure 16. Runs 209 to 259 over 50 days: 13 December 2007 to 1 February 2008.
30

X Position Change- runs 314 to 356

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50 60 70

Position, m Difference from run 313, mm

Runs 352 & 353 (6/5/08)

Figure 17. Runs 314 to 356 over 33 days: 7 May 2008 to 9 June 2008.
31

X Position Change- runs 357 to 410

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 10 20 30 40 50 60 70

Position, m Difference from run 356, mm

Figure 18. Runs 357 to 410 over 60 days: 9 June 2008 to 8 August 2008.
32

X Position Change- runs 411 to 471

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 10 20 30 40 50 60 70

Position, m Difference, mm

Runs 442 & 443 (9/18/08)

Run 471 (11/5/08)

Figure 19. Runs 411 to 471 over 89 days: 8 August 2008 to 5 November 2008. 
33

X Position Change- runs 472 to 493 

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0 10 20 30 40 50 60 70

Position, m Difference from run 471, mm

Figure 20. Runs 472 to 493 over 25 days: 11 November 2008 to 1 December 2008.
34

X Position Change- runs 555 to 678

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0 10 20 30 40 50 60 70

Position, m Difference from run 554, mm

Figure 21. Runs 555 to 678 over 191 days: 21 September 2009 to 31 March 2010
35

Ball Range Length by Date

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1/3/06 7/4/06 1/2/07 7/3/07 1/1/08 7/1/08 12/30/08 6/30/09 12/29/09 6/29/10

Date, 2006 to 2010 Deviation from 65951.2, mm

Note: unusually dry 

summer in 2007

Note: tape 

tunnel 

temperature

out of control

during this 

time

Figure 22. Change in the measured value of the 66 m length from 24 January 2006 to 6

October 2010. Error bars are k = 2 expanded uncertainty.