NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Visual Scanning Hartmann 

Optical Tester (VSHOT) 

Uncertainty Analysis

A. Gray, A. Lewandowski, and T. Wendelin

National Renewable Energy Laboratory

Milestone Report

NREL/TP-5500-48482

October 2010
NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

National Renewable Energy Laboratory

1617 Cole Boulevard

Golden, Colorado 80401

303-275-3000 • www.nrel.gov

Contract No. DE-AC36-08GO28308

Visual Scanning Hartmann 

Optical Tester (VSHOT) 

Uncertainty Analysis

A. Gray, A. Lewandowski, and T. Wendelin

National Renewable Energy Laboratory

Prepared under Task No. CP09.1001

Milestone Report

NREL/TP-5500-48482

October 2010
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Table of Contents 

Table of Contents .......................................................................................................................................... 1 

Abstract ......................................................................................................................................................... 2 

Introduction .................................................................................................................................................. 3 

VSHOT ........................................................................................................................................................... 3 

Zernike Polynomial ....................................................................................................................................... 5 

Uncertainty Contributions ............................................................................................................................ 9 

Target Tilt .......................................................................................................................................... 12 

Target Face to Laser Scanner Output ................................................................................................ 13 

Instrument Vertical Offset ................................................................................................................ 14 

Scanner Tilt ........................................................................................................................................ 16 

Distance from Target to Test Piece ................................................................................................... 18 

Camera Calibration ............................................................................................................................ 20 

Scanner/Calibration .......................................................................................................................... 25 

Uncertainty Estimate – Slope Error ............................................................................................................ 29 

Uncertainty Estimate – Focal Length and Test Article Tilt .......................................................................... 35 

Summary ..................................................................................................................................................... 36 

Acknowledgements ..................................................................................................................................... 37 

References .................................................................................................................................................. 37 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Abstract 

Concentrating solar power plants are being developed and deployed around the world using 

various concentrator technologies, including parabolic troughs, power towers, and dishes. 

Sunlight is focused onto a receiver that collects the heat to generate steam for a conventional 

power plant or Stirling engine. To concentrate sunlight and achieve the high temperatures needed 

to obtain high cycle efficiency, these technologies typically use reflective optics. Glass mirrors 

are commonly used, but reflective films can also be employed. In either case, the reflective 

surface needs to concentrate the sunlight by reflecting it to a desired location. This is done by 

designing the reflective surface to conform to a certain shape that will optimize the amount of 

light reaching the receiver. Once a reflector has been designed and fabricated, it is important to 

know how close it is to the ideal shape. The Video Scanning Hartmann Optical Tester (VSHOT) 

is a surface slope and contour measuring tool for concentrating solar power (CSP) reflector 

panels used in line- and point-focus technologies. VSHOT was developed by the U.S. 

Department of Energy’s SunLab in the early 1990s in a collaboration between the National 

Renewable Energy Laboratory (NREL) and Sandia National Laboratories (Sandia) to provide 

accurate surface characterization of CSP reflective surfaces. 

The VSHOT is a proven tool that has been used on heliostat, dish, and trough mirror facets to 

provide accurate surface slope deviations that characterize optical quality. These data are used to 

estimate optical performance within the overall system. A study of the uncertainty and sensitivity 

of this instrument was completed in 1997 showed that there was a 0.1 mrad slope uncertainty in a 

full scan. Since then, the hardware and software have been upgraded with new technology. To 

ensure that both industry and laboratory users understand the accuracy of the data provided by 

the VSHOT, we have conducted a new uncertainty analysis. 

This analysis is based primarily on the geometric optics of the system and shows sensitivities to 

various design and operational parameters. We discuss sources of error with measuring devices, 

instrument calibrations, and operator measurements for a parabolic trough test. These help to 

guide the operator in proper setup, and help end users to understand the data they are provided. 

In this report, we include both the systematic (bias) and random (precision) errors for VSHOT 

testing and their contributions to the uncertainty. The contributing factors that we considered in 

this study are the following: target tilt; target face to laser output distance; instrument vertical 

offset; scanner tilt; distance between the tool and the test piece; camera calibration; and 

scanner/calibration. These contributing factors were applied to the calculated slope error, focal 

length, and test article tilt that are generated by the VSHOT data processing. The results shown 

in this work estimate the 2σ slope error uncertainty for a parabolic trough line scan test to be 

from ±0.21 - 0.46 mrad for any given single slope error measurement. The 2σ uncertainty for 

slope errors over a single scan is ±0.33 mrad, ±0.6 mm (±0.03%) for focal length and ±0.2 mrad 

for test article tilt. 

2
 
 

 

 

 

 

 

 

 

 

 

 

 

 

Introduction 

Measurement error is defined as the difference between the true and measured value [1] and 

includes both the systematic and random errors. The measurement uncertainty discussed in this 

report provides an estimate of the 2σ error in the slope of a reflective surface that may be 

expected for VSHOT testing. Errors larger than the stated uncertainty will occur 5% of the time. 

A previous uncertainty analysis [2] used a 16-inch telescope mirror to conduct the experimental 

analysis. The distance between the mirror and the VSHOT ranged from 6.32–8.20 m (249–323”). 

With this setup, the full cone angle for the laser to scan the entire mirror was less than 

0.0698 radians (4o). The VSHOT is often used to test flat and parabolic trough facets that require 

the laser to scan a 1.38-radian (80o) arc to collect data over the entire test piece. The analysis 

presented here is for laser scanning angles of ±0.69 radians (±40o

), or 1.38 radians (80o). The 

case study for this uncertainty analysis is an ideal parabolic trough with a 6-m aperture, a 1.71-m 

focal length, and continuous reflective surface. VSHOT is assumed to have a ±0.69-radian cone 

angle and is 4.928 m (194”) away from the vertex of the collector. This arrangement was 

selected because it represents a majority of the VSHOT tests conducted. 

VSHOT 

The VSHOT is a laser ray-trace system designed to characterize the optical surfaces of solar 

concentrators [2]. Originally designed to test point-focus (dish) concentrators, it was later 

modified to include characterization of line-focus (trough) concentrators and has been used to 

test mirror panels for heliostats. The VSHOT uses computer controlled laser scanner and digital 

camera to provide surface contour data. The laser scans a mirror in a pattern predefined by the 

user. At each scanned position, the laser beam is reflected back to a target and the location is 

imaged using a camera. The surface slope is calculated at each position using the laser output 

angle and return-spot location. A Zernike Polynomial is used to mathematically fit the surface 

using the slope data. 

During setup and before each test, many of the components are checked and measurements are 

taken to insure correct orientation of the VSHOT relative to the test article. This procedure is 

listed below: 

1.	 Level the front of the target. 

2.	 Level the optical rail (Figure 1). 

3.	 Calibrate the camera to the target (Figure 2). Measure the distance between the laser 

output mirror on the scanner head and the target face. 

4.	 Determine the VSHOT vertical location. Check the level of the target and the scanner 

again while another operator checks the location of the laser at the vertex of the collector. 

5.	 Measure the distance between the target face and vertex of the trough. 

6.	 Begin a test using the VSHOT software. 

3
 
4 

Figure 1. Image of the VSHOT TO-GO being set up for a test. The optical rail and target were being 

leveled at the time this picture was taken. “TO-GO” refers to the version of the hardware used for 

field testing. (credit: Jen Crawford, NREL)

Figure 2. Image of the VSHOT “TO-GO” camera calibration grid used to calibrate the camera pixel 

space to the target. (credit: Mark Bernardi, NREL)

Camera support arms

Camera

Pedestal

Operators

Optical rail

Target

Laser

Camera

Camera 

calibration grid
5 

Zernike Polynomial 

A Zernike Polynomial is used to mathematically describe common optical surfaces [3] where k is 

the order of the monomial and the ∆ݔ and ∆ݕ terms compensate for known position offsets of the 

mirror vertex relative to VSHOT coordinates. These last two terms are useful when fitting the 

data from an actual test, but for the purpose of this analysis, we will assume they are equal to 

zero (errors associated with the offset are accounted for in the instrument vertical offset). 

ݖሺݔ െ ∆ݔ, ݕ െ ∆ݕሻ ൌ ෍ ෍ ܤ௜,௝ሺݔ െ ∆ݔሻ௝ሺݕ െ ∆ݕሻ௜ି௝

௜

௝ୀ଴

௞

௜ୀ଴

Equation 1 

Equation 2 is the second-order expansion where k=2. 

 2 Equation ଶݔଶ,ଶܤ ൅ ݕݔଵ,ଶܤ ൅ ଶݕ଴,ଶܤ൅ݔଵ,ଵܤ൅ݕ଴,ଵܤ ൅ ଴,଴ܤ ൌ ሻݕ ,ݔሺݖ

Most surfaces tested by VSHOT are parabolic, making the relationship between a second-order 

Zernike Polynomial equation and the designed surface correlation simple (dish and heliostat 

mirror panel focal lengths are often long enough such that a sphere and parabola are essentially 

identical). An ideal parabola can be mathematically described in three dimensions as 

ଶݔ ൌ ሻݕ ,ݔሺݖ

4݂௫

ଶݕ ൅

4݂௬

Equation 3 

with coefficient fx equal to the focal length in the x direction and fy equal to the focal length in 

the y direction. If we relate Equation 3 to the Zernike monomial, ܤଶ,଴ ൌ ଵ

ସ௙೤

 and ܤଶ,ଶ ൌ ଵ

ସ௙ೣ

଴,ଵܤ .

and ܤଵ,ଵ coefficients describe the test piece tilt relative to the instrument. Under ideal conditions, 

the tilt terms equal zero. Usually, this is not the case and it is common for these tilt terms to be in 

the mrad range. 

ݐ݈݅ݐ௫ ൌ atan ሺܤଵ,଴ሻ Equation 4 

ݐ݈݅ݐ௬ ൌ atan ሺܤଵ,ଵሻ Equation 5 

The ܤ଴, ଴coefficient is the piston term (offset along the z axis). ܤଶ,ଵ is the cross term and is used 

with the focal length coefficients to determine if there is any astigmatism. If ܤଶ,଴ ് ܤଶ,ଶ and 

ܤଶ,ଵ ൌ 0, there is an astigmatism error on the horizontal or vertical axis. If ܤଶ,ଵ ് 0 and 

ܤଶ,଴ ് ܤଶ,ଶ, there is an astigmatism error with an arbitrary axis orientation [3]. For a perfect 

point-focus collector, ܤଶ,଴=ܤଶ,ଶ ൌ 

ଵ

ସ௙ and ܤଶ,ଵ ൌ 0; for a perfect line-focus collector 

ܤଶ,଴ ൌ ଵ

ସ௙೤

 ,ܤଶ,ଶ,=0, and ܤଶ,ଵ ൌ 0. Table 1 lists the coefficients for a point-focus and parabola 

collector. 
Table 1. Zernike coefficients for a perfect point-focus and line-focus collector with a 6-m aperture 

and 1.71-m focal length and no tilt. 

Coefficient Point Focus Line Focus 

B0,0 0 0 

B1,0 0 0 

B1,1 0 0 

B2,0 0.1462 0.1462 

B2,1 0 0 

B2,2 0.1462 0 

In this study, a two-dimensional analysis was completed for a parabolic trough collector. Data 

were generated in a single-column profile because this is representative of the majority of 

VSHOT field testing. In two dimensions, the Zernike Polynomial can be simplified so that z is a 

ଶܤ ൅ݕ

ଵ and ܤଵ ସ௙ ೤

function of y only. Assuming a second-order surface (k=2), the equation becomes: 

ଵܤ ൅ ଴ܤ ൌ ሻݕሺݖ

ൌ tan ሺݐ ሻ ݅ ݐ

ݕଶ Equation 6 

with ܤଶ ൌ ݈ . For VSHOT testing of troughs in the field, only the slope 

errors in the transverse, y, direction (along the curvature) are typically collected (depicted in 

Figure 3). The laser is scanned in a vertical direction (with the trough facing the horizon, i.e., 

with a vertical aperture). VSHOT records the output angle of the laser, αy, and the return-spot 

location on the target, Hy, then uses this information to calculate the slope. The ideal return-spot 

location is compared to the actual to calculate slope error. 

6 
 

 

 

 

 

 

 

 

 

Figure 3. Schematic of VSHOT used with a trough collector. VSHOT target and laser are on the left 

and a two-dimensional representation of a parabolic collector is on the right. The red line is the 

ideal laser output angle as it scans the test piece with its ideal reflection onto the target. The 

green line is actual simulated reflection caused by the surface slope error at that point on the 

parabola. This figure is for a 6-m aperture with a 1.71-m focal length, laser output angle of -0.531 

radian (30.45o

). The distance from the target to the vertex of the collector is 4.93 m. The laser hits 

the collector at yp=-2.40 m, zp=5.77 m. Ideally, the laser return spot on the target is Hyideal=0.948 m 

and the actual return spot for this example is Hyactual= 0.75 m. With no other setup errors, the slope 

error at this point is +15 mrad. 

VSHOT results are presented as a slope error, Ry, for each data point collected, and as an overall 

root-mean-square (RMS) of the slope errors for either a single scan or test. The RMS slope error 

is also defined as the sample standard deviation, σRMS (Equation 7) for a distribution with zero 

7
 
 

mean. The slope errors can be provided relative to a best-fit focal length and/or the design focal 

length. 

ଶ

ଵ௜ୀ ܴ௬, ∑ே

ൌ ඨ ߪோெௌ ܰെ1

The slope at each point on the test piece is the slope of the tangent line at that point. For the 

perfect parabolic case, the slope should be linear across the entire surface and can be described 

௜ Equation 7 

mathematically by Equation 8. The effect of focal length on slope is shown in Figure 4. 

݂௬

ݕ

ݕ݀ ൌ 2

ݖ݀

The derivative of the two-dimensional Zernike Polynomial is shown in Equation 9. If there is a 

tilt in the test piece, ܤ will not be equal to zero and the line will translate in the vertical 

ܤଵ. An example of this is shown in Figure 5. The slope of the line gives 

ଵ

direction by the value

the “best fit’ focal length through the ܤଶ ଵ ଶ term. ܤ and ܤ

Equation 8 

are independent and thus can be 

uniquely determined for a test that provides slope (and slope error) as a function of position 

along the aperture. 

ݕ݀ ܤ2 ൅ ܤ ൌ

ݖ݀

ଵ ଶݕ 
Equation 9 

2.21 m 1.71 m 1.21 m 

1.5 

Slope (radians) 

1.0 

0.5 

0.0 

-0.5 

-1.0 

-1.5 

dz/dy = 0.2924y 

dz/dy = 0.2262y 

dz/dy = 0.4132y 

-3 -2 -1 0 1 2 

Position on Aperture (m) 

Figure 4. This figure shows how the focal length affects the slope, dz/dy, along the aperture. The 

surface slope along a parabola with three different focal lengths is plotted, 2.21 m, 1.71 m, and 

1.21 m. As the focal length decreases, the slope increases. 

8 

3 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+0.245 radian tilt No tilt -0.245 radian tilt 

Slope (radians) 

1.5 

1.0 

0.5 

0.0 

-0.5 

-1.0 

-1.5 

dz/dy= 0.2924y + 0.25 

dz/dy = 0.2924y 

dz/dy = 0.2924y - 0.25 

-3 -2 -1 0 1 2 

Position on Aperture (m) 

Figure 5. This figure shows how tilt affects the slope along the aperture. Tilt does not affect the 

calculated focal length. 

Uncertainty Contributions 

The measurement uncertainty in this paper is applied to the calculated value of the reflective 

surface slope, focal length, and test-article tilt. The slope (or slope error) is of primary interest, so 

we primarily address the impacts of measurement error on it and then summarize the impacts on 

focal length and test-article tilt in a later section. We consider six random error sources and 

seven systematic error sources in this uncertainty analysis which are listed in Table 2. Random 

error uncertainty comes from hardware vendor data on repeatability or precision where we have 

not explicitly distinguished between those two terms. All but the camera calibration random 

errors will vary over the short time while measurements are being taken. Systematic uncertainty 

can be difficult to quantify and our assumption is that over a large number of tests the biases 

imposed by the operator in measuring setup parameters will be randomly distributed. We 

determined those values based on experience, judgment or some approach specific to each 

uncertainty. Initially, we show the impact of all the errors considered (except camera calibration 

and scanner/calibration), then address the individual errors in separate subsections. 

The affect on the uncertainty in the calculated slope varies at different laser output angles for 

each of the uncertainties listed in Table 2. A computer program was written to simulate the test 

geometry and calculate the impact of the uncertainties listed in Table 2. Figure 6 (random 

uncertainty) and Figure 7 (systematic uncertainty) show the slope error results for each of the 

uncertainties listed in Table 2 as a function of laser output angle; -0.69 radians corresponds to ­

40o (lower edge of the aperture). 

The target tilt is measured with a level that has precision of ±0.208 mrad. It is assumed that the 

operator will cause a systematic error when taking this measurement of 0.416 mrad (twice the 

random error). The target face to laser output is measured with calipers that have an accuracy of 

0.0127 mm. We estimate the operator can measure this to within 0.508 mm of the true value. 

Overall, these two measurements cause the smallest slope error uncertainties, with less than 

9 

3 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.035 mrad over the aperture. The minimum for both of these contributors is at a laser output 

angle of 0 radians (vertex of the collector). 

Table 2. List of uncertainties considered in this study. All of the random and systematic errors are 

assumed to have a 95% confidence (2σ). 

Description Measuring Device Random Systematic 

Target tilt Bubble level ±0.208 mrad (0.012o

) ±0.416 mrad (0.024o) 

Target face to laser 

scanner output Calipers ±0.0127 mm (0.0005“) ±0.5 mm (0.02”) 

Instrument vertical 

offset Human eye N/A ±1.59 mm (0.0625”, 1

/16”) 

Scanner tilt Inclinometer ±0.087 mrad (0.005o

) ±0.523 mrad (0.03o) 

Distance from 

target to test piece Laser range finder ±1.27 mm ( 0.05”) ±0.751 mm (0.030”) 

Camera calibration Prosilica GE2040 GigE 

Camera 

1.49 ± 0.084 mm/pixel 

(0.0587 ± 0.0034”) ±0.374 mm/pixel (0.014”) 

Cambridge Technology 

Scanner/calibration closed-loop galvanometer ±8 μrad ±0.62 mrad (0.36o

) 

model 6220 

* A visual description for some of these measurements is shown in Figure 3. 

The laser, at 0 radians, should be in line with the vertex of the collector as shown in Figure 3. 

The location of the laser is verified by visually inspecting the location of the laser on the 

collector. The uncertainty of this measurement is assumed to be 1.59 mm (1/16”). Only a 

systematic error is considered for this error source. Although random error may exist in this 

measurement between scans, it is not quantified because this results in a systematic error for each 

scan. This uncertainty is highest at the vertex and decreases as the laser angle increases. 

The laser scanner is leveled before each test with an inclinometer that has an uncertainty of 

0.087 mrad. The operator levels the scanner to 0.523 mrad or less before each test. This 

uncertainty peaks at the vertex and at the outer portions of the aperture and follows the trend in 

the return-spot location on the target. Figure 8 shows the return-spot location trend on the target. 

The step change in the laser output angle is constant, but the distance between the return-spot 

locations is not. The minimum change in distance is where the return spot changes direction 

(circled in black). The plot on the right is the laser return-spot locations with respect to the output 

angle. The negative output angles for the laser have positive return-spot locations on the target, 

and positive laser output angles have negative return-spot locations. The distance from the target 

to the vertex of the collector is measured with a laser range finder that has an accuracy of 

±1.27 mm. The slope error uncertainty trend for this is linear and has a positive slope if the 

distance is below the true value or negative if the distance is above the true value. 

10
 
 

 

 

 

 

 

 

 

 

 

 

 

 

Target Tilt (-0.208 mrads) Target Tilt (+0.208 mrads) 

Target Face to Scanner Output (-0.013 mm) Target Face to Scanner Output (+0.013 mm) 

Scanner Tilt (-0.087 mrad) Scanner Tilt (+0.087 mrad) 

Distance (-1.27 mm) Distance (+1.27 mm) 

-0.2 

-0.1 

0.0 

0.1 

0.2 

Slope Error Uncertainty (mrads) 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Laser Output Angle (radians) 

Figure 6. Results from random slope error uncertainties listed in Table. The only uncertainties not 

shown are the camera and scanner. Resolution errors in the MATLAB program used to generate 

these results are visible for slope error differences less than 0.01 mrad. The resolution errors 

cause the slope error trend to appear jagged rather than smooth. 

Instrument Vertical Location (-1.59 mm) Instrument Vertical Location (+1.59 mm) 

Scanner Tilt (-0.526 mrad) Scanner Tilt (+0.526 mrad) 

Distance (-0.751 mm) Distance ((+0.751 mm)
 Target Face to Scanner Output (-0.5 mm)
 Target Facet to Scanner Output (+0.5 mm) 

Target Tilt (-0.416 mrad) Target Tilt (+0.416 mrad) 

0.4 

Slope Error Uncertainty (mrad) 0.2 

0.0 

-0.2 

-0.4 

Laser Output Angle (radians) 

Figure 7. Results from systematic slope error uncertainties listed in Table 2. The only 

uncertainties not shown in the plot are those of the camera and scanner. 

11 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
 
 

-1.5 

-1.0 

-0.5 

0.0 

0.5 

1.0 

1.5 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Return Spot Location on Target (m) 

Scanner Output Angle (radians) 

+40o output angle 

-40o output angle 

Figure 8. The figure on the left shows ray paths while scanning a collector. The laser output rays 

are shown in magenta and the reflected laser rays in blue. The areas circled in black are for laser 

output angles of ±0.36–0.50 radian and shows where the laser return spot changes directions on 

the target. 

Target Tilt 

During setup and at the beginning of each test, the target is leveled in both the vertical and 

horizontal directions. This is done by an operator with a bubble level that has a random 

uncertainty of 0.127 mm (0.005”) over a length of 0.61 m (24”), providing a random uncertainty 

in the level of the target of 0.208 mrad. This random uncertainty causes slope errors ranging 

from ±0.000205 mrad. Figure 9 shows the calculated slope for a perfect parabola as a function of 

output laser position along the aperture assuming a target tilt angle of ±0.208 mrad. The linear 

, with the -6 regression for this random slope error uncertainty data yields /݀ݕ݀ݖ= 0.2924y ± 8*10

sum of the squared residuals (R2

) equal to 1.00. The slope of the line is 0.2924, resulting in a 

focal length of 1.71 m. This shows that the random errors in the target tilt have a negligible effect 

on the calculated focal length. The tilt term is 8*10-6 radians, or 8 μrad and can be considered a 

negligible contribution (within the noise of the instrument). 

12 
Targe t Tilt (-0.208 mrad ) Target Tilt (+0.208 mrad) 

0.8 0.8 

0.6 0.6 

0.4 0.4 

dz/dy = 0.2924y - 8E-06 

R2 = 1.00 

dz/dy = 0.2924y + 8E-06 

R2 = 1.00 

Slope (radians) 0.2 

0.0 

-0.2

Slope (radians) 0.2
 

0.0
 

-0.2
 

-0.4 -0.4

-0.6
 -0.6

-0.8 -0.8

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2

Position on Aperture (m) Position on Aperture (m) 

Figure 9. The calculated slopes along the aperture of a parabola with the target tilted at ±0.208 

mrad. 

The systematic error for this bubble-level measurement is arbitrarily estimated to be two times 

the random error or 0.416 mrad. This systematic error is caused by the operator’s 

handling/placement of the instrument, flatness of the target or other sources. The slope errors 

caused by this systematic uncertainty ranges from 0.0002 to 0.031 mrad. A linear regression of 

, with the sum of the squared -5 this systematic slope error yields /݀ݕ݀ݖ = 0.2924y ± 2*10 

residuals (R2

) equal to 1.00. The slope of the line is 0.2924, resulting in a focal length of 1.71 m. 

The tilt term is 2*10-5radians, or 0.02 mrad. 

Target Face to Laser Scanner Output 

The distance between the front of the VSHOT target and the laser scanner output is measured 

during the setup. This measurement is highlighted in Figure 10 with a thick white line. Calipers 

are used to measure this distance, and they have a random uncertainty of ±0.127 mm. The slope 

errors caused by this random uncertainty are relatively small, ranging from -0.0235 to 

+0.0235 mrad across the aperture. A linear regression of the random slope error uncertainty in 

)

2 , with the sum of the squared residuals (R -7 this measurement yields /݀ݕ݀ݖ = 0.2924y ± 7*10 

equal to 1.00. The slope of the line is 0.2924, resulting in a focal length of 1.71m. The tilt term is 

7*10-7 radians, or 0.0007 mrad, and can be considered negligible (within the noise of the 

instrument). 

This measurement has a systematic error of ±0.5 mm based on observed consistency among 

operators making the measurement. The resulting systematic slope error uncertainty is relatively 

small, ranging between ±0.0355 mrad with the absolute minimum at a laser output angle of 0o

. 

Figure 11 shows the calculated slope for a perfect parabola with systematic measurement error of 

±0.508 mm. A linear regression of these data was completed to determine the effect on the 

13 

3 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

calculated focal length and tilt. Based on this regression, the focal length matches the design at 

1.71 m with a tilt of 7*10-7radians, or 0. 7 μrad, and can be considered negligible. 

Scanner 

Target center hole 

Target 

Optical rail Inclinometers used to level the optical rail 

Figure 10. The image on the left is the back of the VSHOT TO-GO system. The optical rail is 

mounted on the back of the target. The image on the right is of the front of the VSHOT TO-GO. The 

distance between the front of the VSHOT target and the laser scanner output is measured with a 

caliper and is shown as a thick white line in the center of the left image. This measurement is 

taken through the center hole in the front of the target to the front of the scanner head behind the 

target. Design drawings provide the additional distance from the front of the scanner head to the 

center of the output mirror. (Left credit: PIX 17379) (Right credit: Jen Crawford, NREL) 

Instrument Vertical Offset 

Before each test, the VSHOT is aligned to the vertex of the optic being tested. The optical rail 

and target are leveled using the inclinometers shown in Figure 10. The operator does a visual 

inspection to make sure the laser (set to zero angle in both x and y) is directed to the vertex of the 

optic. The vertex point is usually specified by the manufacturer because there is usually no welldefined physical feature at the vertex. Only a systematic error is considered for this 

measurement. Although a random error may exist between scans, it is not quantified because of 

the difficulty in the operator’s ability to measure the centroid of the laser. 

The laser beam is about 6 mm in diameter. We believe that the operator can estimate the center 

of the laser relative to the vertex of the optic to within 2 mm. This results in a systematic 

uncertainty in slope error ranging from about 0.02 to 0.30 mrad, with the peak at a laser output 

angle of zero. The calculated slopes for a perfect parabola with the VSHOT vertical offset at 

±1.59 mm are plotted in Figure 12. A linear regression calculates the effect of this uncertainty on 

14
 
 

 

 

 

 

 

 

 

 

 

 

 

 

the best-fit focal length and tilt term. The best-fit focal length is 1.71 m and the tilt term is 

2*10-4 radians, or 0.2 mrad. 

Target face to laser scanner output (-0.5 mm ) Target face to laser scanner output (+0.5 mm ) 

0.8 0.8 

0.6 0.6 

0.4 0.4 

0.2 0.2 

dz/dy = 0.2924y + 7E-07 

R2 = 1 

dz/dy = 0.2924y + 7E-07 

Slope (radians) 

R2 = 1 

-0.2 -0.2 

-0.4 -0.4 

-0.6 -0.6 

-0.8 -0.8 Slope (radians) 

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 

Position on Aperture (m) Position on Aperture (m) 

Figure 11. Calculated slope along the aperture of a perfect parabola with target face to laser 

scanner output mirror distance error of ±0.508 mm. 

Target Vertical Location (-1.59 mm) Target Vertical Location (+1.59 mm) 

Slope Error (-1.59 mm) Slope Error (+1.59 mm) 

0.0
 0.0
 

-0.45 

-0.30 

-0.15 

0.00 

0.15 

0.30 

0.45 

-6 

-4 

-2 

0 

2 

4 

6 

Slope Error Uncertainty Caused by the

Incorrect Return Spot Location (mrad) 

Difference in Return Spot location from

uncertainties (ideal-uncertainty) (mm) 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Laser Output Angle (radians) 

Figure 12. Systematic error caused by the vertical offset of the VSHOT relative to the collector 

vertex. The vertical axis on the left is the difference in return-spot location on the target caused by 

the uncertainty at each laser output angle. The vertical axis on the right is the calculated slope 

error. 

15 

3 
 

 

 

 

 

 

 

 

 

 

 

 

Instrument Vertical Offset (-1.59 mm) Instrument Vertical Offset (+1.59 mm) 

0.2 

dz/dy = 0.2924y - 0.0002 

R2 = 1.00 

-0.8 

0.8 0.8 

0.6 0.6 

0.4 0.4 

dz/dy = 0.29

R2 = 

24y + 

1.00 

Slope (radians) 

0.0002 

0.2 

Slope (radians)

0.0
 0.0
 

-0.2
 -0.2
 

-0.4
 -0.4
 

-0.6
 -0.6
 

-0.8 

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 

Position on Aperture (m) Position on Aperture (m) 

Figure 13. Calculated slopes along the aperture of a perfect parabola with the VSHOT vertical 

offset of ±1.59 mm are plotted in the charts. 

Scanner Tilt 

The scanner is securely fixed to an optical rail which is fixed to the back of the target. The rail 

also holds the laser in a rigid position and orientation. The inclinometers used to level the 

scanner and rail assembly have a random uncertainty of 0.009 mrad (0.005o). The random 

uncertainty associated with the scanner tilt peaks at the vertex and at the outer portions of the 

aperture. This uncertainty follows the trend in the change of return-spot location on the target. 

The minimum is where the return-spot locations are close together (turn-around points). The 

maximum and minimum slope errors for this random uncertainty range from ±0.04 mrad, 

regression of these data yielded 

depending on the laser output angle. A linear regression was completed on the slope data to 

estimate the effect of this random uncertainty on the best-fit focal length and tilt term. The linear 

ݖ݀ݕ/݀

and the tilt term is 4*10-6 radians, or 0.004 mrad. 

 = 0.2924y ± 4*10-6. The best-fit focal length is 1.71 m 

The scanner is leveled to ±0.523 mrad (0.03o) or less before each test using two inclinometers 

located on the optical rail (shown in Figure 10). The systematic uncertainty is plotted in Figure 

14. The maximum and minimum slope errors for this systematic uncertainty range between 

±0.22 mrad, depending on the laser output angle. Figure 15 shows two plots of the calculated 

slopes assuming a laser output angle error of ±0.524 mrad. The linear regression performed on 

these data shows that this uncertainty has a negligible impact on the best-fit focal length and a 

small impact on the tilt term, 0.03 mrad. 

16
 
 

 

 

 

 

 

 

 

 

 

 

 

 

Laser Tilt (-0.523 mrad) Laser Tilt (+0.523 mrad) 

-0.30 

-0.20 

-0.10 

0.00 

0.10 

0.20 

0.30 

-6 

-4 

-2 

0 

2 

4 

6 

Difference in Return Spot location from

uncertainties (ideal

Slope Error Uncertainty Caused by the

Incorrect Return Spot Location (mrad) 

-uncertainty) (mm) 

Slope Error (-0.523 mrad) Slope Error (+0.523 mrad) 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Laser Output Angle (radians) 

Figure 14. Scanner tilt systematic error for tilt values of ±0.523 mrad. The vertical axis on the left is 

the difference in return-spot location on the target caused by the uncertainty in the laser tilt. The 

vertical axis on the right is the calculated slope error caused by the error in the return-spot 

location. 

Laser output angle (-0.523 mrad) Laser output angle (+0.523 mrad) 

0.8 0.8 

0.6 0.6 

dz/dy = 0.2924x - 3E-05 

R2 = 1.00 

dz/dy = 0.2924y + 3E-05 

R2 = 1.00 

0.4 0.4
 
 

Slope (radians) -0.2 

-0.4 

Slope (radians) 0.2
 

0.0
 

-0.2
 

0.2
 

0.0
 

-0.4
 

-0.6
 -0.6
 

-0.8 

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 

-0.8 

Position on Aperture (m) Position on Aperture (m) 

Figure 15. Calculated slopes for a perfect parabola with scanner tilt of ±0.523 mrad are plotted. 

17
 
 

 

 

 

 

 

Distance from Target to Test Piece 

The distance between the VSHOT target and the vertex of the test article is measured before 

every test. This measurement is taken by an operator with a laser distance range finder that has 

an accuracy of ±1.27 mm (0.05”). This random uncertainty has a linear effect on the return-spot 

location on the target and leads to a linear effect on the slope errors between ±0.17 mrad, as 

shown in Figure 16. Figure 17 is a plot of the calculated slopes for a perfect parabola with 

measured distance errors between the target and the test piece of ±1.27 mm. A linear 

ݖ݀ݕ/݀

0.6 mm variation in the focal length (1.71 ±0.0006 m). This random uncertainty had a negligible 

impact on the tilt. 

regression of this positive and negative random uncertainty slope error data yields

 =(0.2924±0.0001) y ± 7*10-7

. Using Equation 9 to solve for the focal length results in a 

Distance (-1.27mm) Distance (+1.27mm) 

Slope Error (-1.27mm) Slope Error (+1.27mm) 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Laser Output Angle (radians) 

-0.3 

-0.2 

-0.1 

0.0 

0.1 

0.2 

0.3 

-6 

-4 

-2 

0 

2 

4 

6 

Slope 

Error 

Unc

ert

ainty 

C

aus

ed by the

Incorre

ct R

eturn 

Spot Loc

ation (mrad) 

Difference in 

R

eturn 

Spot loc

ation from

u

ncertai

nties (ideal-unc

ert

aint

y) (m

m) 

Figure 16. Random errors caused by the laser range finder used to measure the distance between 

the VSHOT and the vertex. The vertical axis on the left is the difference in return-spot location on 

the target caused by the uncertainty over the collector. The vertical axis on the right is the 

calculated slope error associated with the error caused by the error in the return-spot location. 

18
 
 

 

 

 

 

 

 

 

 

 

 

 

Distance from target to test piece (-1.27 mm) Distance from target to test piece (+1.27 mm) 

0.8 0.8 

0.6 0.6 

0.4 0.4 

0.2 

dz/dy = 0.2923x + 7E-07 

R2 = 1.00 

-0.8 

dz/dy = 0.2925x + 7E-07 

R2 = 1.00 

Slope (radians) 

-0.2 

-0.4 -0.4 Slope (radians) 0.2
 

0.0
 0.0
 

-0.2
 

-0.6 -0.6
 
 

-0.8 

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 

Position on Aperture (m) Position on Aperture (m) 

Figure 17. Calculated slope at multiple points across a collector assuming a ±1.27-mm variation in 

the distance between the target and test piece. 

This distance measurement is made by an operator before each test. Depending on how the 

operator makes this measurement, a systematic error can be introduced into the data. The 

operator takes this measurement by placing the laser range finder in front of the target. It is easy 

for the operator to slightly angle the laser range finder and measure a distance longer than the 

actual distance. It is estimated that the operator could angle the laser range finder up to 

0.0175 radian (1o), potentially causing a ±0.751mm (0.0300”) systematic error in this 

measurement. Figure 18 is a top view of the VSHOT relative to a collector. The operator takes 

the measurement as close to the center of the target as possible. 

Measurement 

taken up to 8 

cm away 

from center 

hole 

Top of VSHOT target 

Target center hole 

Distance from target face to vertex (4.93 m) 

Collector 

vertex 

Operator measurement 

Figure 18. Top view of the VSHOT and a collector. The dashed lines between the target and the 

collector show two ways an operator can measure the distance between the target and the 

collector vertex. 

19 

3 
The errors caused by the systematic uncertainty follow the same trend as the random 

uncertainties except the errors are smaller. The slope error trend is linear, with a maximum at 

+0.11 mrad and a minimum of -0.11 mrad. A linear regression was performed on the systematic 

slope error data to determine its effect on the focal length and tilt. A linear regression of these 

) equal to 1.00. 2 , with the sum of the squared residuals (R -7 data yielded /݀ݕ݀ݖ = 0.2924y ± 7*10 

Equation 9 was used to calculate a focal length of 1.71 m. The error in focal length at this level 

of systematic error is negligible. 

Camera Calibration 

The camera must be calibrated so that pixel space can be related to actual xy space on the target. 

This is done during setup once the camera location, orientation, lens focal length and focus are 

fixed. The camera is calibrated using the calibration grid shown in Figure 2. The calibration grid 

has 46 dots in the vertical direction and 7 dots in the horizontal. The diameter of the dots is 

63.5 mm (2.50”) and they are spaced 76.2 mm (3.00”) apart. Twelve camera-calibration files 

taken over recent VSHOT testing history were used to estimate the uncertainty in the typical 

camera calibrations. 

To estimate the random errors in the camera, we evaluated the number of pixels between the 

center of each circle. The number of pixels between circle centers was divided by the known 

distance, 76.2 mm (3.00”). This provided an estimate for the variation in pixel response over 

76.2 mm. The area of the target used for testing can vary slightly depending on the aperture and 

focal length of the collector being tested, lens focal length and camera position. For the twelve 

camera calibrations used in this study, the calibration region ranged from 35 to 43 spots in the 

vertical direction and 5 spots in the horizontal. The regions of the target that are typically used 

are shown in Figure 19. As the calibration region increases, the pixels per area decrease. This is 

expected because the number of pixels in the camera is fixed. The number of pixels/mm ranged 

from 0.54 to 0.68. The data sets from each calibration file were normalized by scaling the 

dataset with respect to their mean values. The normalized number of pixels/mm was 0.673±0.038 

(2σ). One pixel thus corresponds to 1.49 70.084 mm on the target. Figure 20 is a plot of this 

uncertainty. Errors in return-spot location range from ±0.7 mm. Slope errors range from ±0.07 

mrad. 

The positive random uncertainty for the camera uses less pixels per area than the negative. This 

reduces the camera’s ability to image the calibration spots (and the laser during testing), 

increasing the uncertainty. This appears to have a random effect on the slope error trend over the 

aperture. This is expected because determining the laser location on the target is limited to 1.41 

and 1.57 mm/pixel. An example of this effect is shown in Figure 21. 

20 
 

 

 

 

 

 

Figure 19. Drawing of the camera calibration target. The dashed lines represent two common 

camera calibration regions that are used for trough testing. 

Camera (-0.084mm) Camera (+0.084 mm) 

Slope error (-0.084 mm) Slope error (+0.084 mm) 

-0.10 

-0.05 

0.00 

0.05 

0.10 

-1.0 

-0.5 

0.0 

0.5 

1.0 

Difference in Return Spot location from

uncertainties (ideal

Slope Error Uncertainty Caused by the

Incorrect Return Spot Location (mrad) 

-uncertainty) (mm) 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Laser Output Angle (radians) 

Figure 20. Plot of the random error in the camera’s ability to centroid. The vertical axis on the left 

is the difference in return-spot location on the target caused by the uncertainty over the collector. 

The vertical axis on the right is the calculated slope error associated with the error caused by the 

error in the return-spot location. 

21 
22 

A linear regression was performed on the calculated slopes across a perfect collector assuming a 

camera random uncertainty of 1.41 and 1.57 mm. This data set (not shown in a figure) yields 

dz/dy = 0.2924 y + 5*10-7 and dz/dy = 0.2924 y + 7*10-7, with the sum of the squared residuals 

(R2

) equal to 1.00. The slope of the line is 0.2924 and using Equation 9 to calculate the focal 

length results in a 1.71-m focal length. The tilt term is 5*10-7radian, or 0.5 μrad, and 

7*10-7radian, or 0.7 μrad, and both can be considered negligible. 

Figure 21. Drawing of camera to target response in pixel/mm (boxes correspond to a camera 

pixel). Errors in the camera’s ability to locate the laser increase as the number of mm increase per 

pixel. ∆ yerror is the error in locating the laser on the target. The larger (red) dot is where the laser 

is striking the target. When the camera images this, it assumes the laser is in the center, not the 

lower edge, causing an error in the measured return-spot location. This effect causes the random 

error effects shown in Figure 20 and 22. 

The same twelve camera-calibration files were used to estimate the systematic uncertainty. 

Based on previous VSHOT experience the standard deviation in the camera calibration errors for 

x and y directions should be less than 0.381 mm (0.015”). The standard deviation of each file is 

listed in Table 3. Each file has a standard deviation for each direction. To get an estimate for 

standard deviation over all the data sets, the individual standard deviations were pooled. Using 

Equation 10, with vi equal to the degrees of freedom in the calibration. Sx,i and SY,i are the 

standard deviation in the x and y direction associated with the data set. Systematic errors in this 

study have a 95% confidence, or 2σ. The camera calibration systematic error is 0.374 mm (2σ) 

from Table 3. 

 
Table 3. Camera calibration files used to estimate the systematic uncertainty. The degrees of 

freedom, ࢏࢜, and standard deviation in the x and y direction are listed. 

ܵ (x-direction) ܵ , (y-direction) ௜ File Number ݒ௑ ,௒ ௜,௜

mm in mm in 

1 204 0.076 0.003 0.102 0.004 

2 184 0.051 0.002 0.076 0.003 

3 174 0.076 0.003 0.076 0.003 

4 174 0.025 0.001 0.076 0.003 

5 184 0.152 0.006 0.127 0.005 

6 184 0.076 0.003 0.127 0.005 

7 184 0.051 0.002 0.152 0.006 

8 185 0.330 0.013 0.305 0.012 

9 174 0.076 0.003 0.152 0.006 

10 184 0.076 0.003 0.127 0.005 

11 194 0.051 0.002 0.102 0.004 

12 300 0.102 0.004 0.127 0.005 

Pooled Standard Deviation (σ), includes both SX,i and SY,i 0.187 0.007 

൅ ൫ܵ௒,

ܵ௣௢௢௟௘ௗ ൌ

ଶ ଵ/ଶ ൨

൪

∑ ݒ൤ ௜൫ܵ௑, ൯

∑ே

௜ୀ

௜

ଶ ൯ ே

௜ୀଵ ௜

 ௜ݒ ൦

Equation 10 

ଵ 

The slope error trend over the aperture resembles a scatter plot. This is expected because 

determining the laser location on the target is limited to 0.374 mm (2σ) and this effect is 

illustrated in Figure 21. Figure 22 is a plot of this uncertainty. Errors in return-spot location 

range from ±0.2 mm. Slope errors range from ±0.02 mrad. Linear regression of this slope data is 

 with the sum of the squared residuals -7 plotted in Figure 23 and yields ݀ݖ/݀ݕ = 0.2924y ± 7*10 

(R2

) equal to 1.00. Equation 9 was used to calculate a focal length of 1.71 m. The tilt term is 

0.0007 radians, or 0.7 μrad. 

23 
Camera (0.374 mm) Slope Error

Difference in Return Spot location from

uncertainties (ideal-uncertainty) (mm) 

0.3 

0.2 

0.1 

0.0 

-0.1 

-0.2 

-0.3 

0.03 

0.02 

0.01 

0.00 

-0.01 

-0.02 

-0.03 

Slope Error Uncertainty Caused by the 

Incorrect Return Spot Location (mrad)

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Laser Output Angle (radians) 

Figure 22. Systematic uncertainty in the camera calibration. The vertical axis on the left is the 

difference in return-spot location on the target caused by the uncertainty in the laser tilt. The 

vertical axis on the right is the calculated slope error caused by the error in the return-spot 

location. 

Slope (radians) 

0.8 

0.6 

0.4 

0.2 

0.0 

-0.2 

-0.4 

-0.6 

-0.8 

dz/dy = 0.2924y + 7E-07 

R2 = 1.00 

-3 -2 -1 0 1 2 

Position on Aperture (m) 

Figure 23. Calculated slope at multiple points across an ideal parabola with a 6-m aperture and 

1.71-m focal length accounting for a systematic uncertainty of 0.374 mm is plotted. 

24 

3 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

When using VSHOT to test outdoors, the system setup is completed during the day and testing is 

done at night. The target is made of aluminum honeycomb. The calibration spots are printed on a 

plastic foam core that has been bonded to the front side of the target. The target could potentially 

expand or contract from the time of setup to testing. This expansion is expected to be less than 

our ability to measure it; therefore, it is not expected to contribute to the uncertainty. However, 

before the camera is calibrated, the distance between calibration spots on the target is measured 

with calipers in a few selected areas. This is a precautionary measure to make sure nothing has 

happened to the target since the last time it was used. After this measurement is completed, the 

camera is calibrated. 

A white target is then placed over the calibration grid to make the contrast between the target and 

laser easily distinguishable for both the camera and operator. The aperture on the camera is 

reduced so that it no longer images any stray light on the target and can only image the laser. If 

the dimensions of the target change during testing at night because of thermal expansion or 

contraction, it will not have any effect on the camera’s ability to locate the return-spot location. 

This is because during testing the camera does not use the target as a reference for relating pixel 

space to xy space. The location of the return spot in xy space is based on the camera calibration 

that was performed immediately after the calibration grid was inspected. There could be other 

temperature related impacts on the VSHOT assembly (e.g. thermal expansion on the camera 

support arms) that are not accounted for in this analysis. 

Scanner/Calibration 

The scanner is used to direct the laser beam to the different angles across a test piece. The laser is 

mounted to a fixed location and orientation on the optical rail (Figure 24). The laser beam is 

directed to the x mirror that reflects the laser to the y mirror. At the y mirror, the laser is reflected 

in the direction of the test piece. The maximum output range of the scanner is 1.38 radian (80o

). 

Figure 24. The image on the left is the optical rail mounted on the target. The laser is mounted to 

the top of the rail in a fixed location perpendicular to the target. The scanner is on the top right 

side of the rail. The drawing on the right shows the basic geometry and hardware for the laser 

scanner [4]. (PIX 17379) 

25 
 

 

 

 

 

 

 

 

 

 

 

 

 

The laser scanner has two 16-bit closed-loop galvanometers. One rotates to change the laser 

output angle in the x direction and the other is for the y. According to the manufacturer, the 

random error in the scanner is 8 μrad [4]. This random error has an extremely small effect on the 

slope errors across the aperture ranging from ±4 μrad. This has a negligible effect on the 

calculated focal length and tilt term. 

The scanner must be calibrated so that the galvanometer encoder counts can be correlated to 

angular space when it is mounted on the optical rail. Multiple components are attached to the 

optical rail and each one may be mounted in a slightly different position than ideal, causing 

systematic errors in the setup. To minimize this systematic error, the scanner is calibrated using a 

calibration grid (Figure 25). The scanner calibration grid has a checkerboard pattern on it to 

assist the operator to determine the laser calibration points. Each square is 0.241 m (9.50”) long 

on each side, and each corner represents a laser calibration point. A 12-by12 grid of points is 

used to calibrate the scanner. The calibration is performed periodically in the laboratory with the 

assumption that there are no changes to the calibration with setup and operation in the field. 

Optical rail 

Scanner calibration grid 

Computer 

Tripod 

Figure 25. Image of the scanner being calibrated. The scanner calibration target is mounted on a 

wall and leveled. The optical rail is mounted on a tripod and centered in front of the target. (credit: 

Mark Bernardi, NREL) 

Five calibration files were used to estimate the systematic errors in the scanner calibration. 

Table 4 lists the standard deviation errors for the scanner calibration in the x and y directions. 

Testing should not be done with errors higher than 1 mrad in either direction because this can 

cause slope error uncertainties larger than 0.35 mrad. The error in the x direction of file 4 is 

26
 
significantly higher than any of the other files, with standard deviation of 5.41 mrad. The error in 

the x direction for file 3 is also too high, with a standard deviation of 1.30 mrad. If errors this 

high occur in the scanner calibration, the setup should be checked and the calibration redone. 

To estimate the standard deviation over all the data sets, the individual standard deviations were 

௜ݒ

ܵܵ

pooled. Equation 10 was used to pool the standard deviations, with =144, as degrees of

freedom in the calibration, and ௜ and ௜ ௑, ௒, as the standard deviation in the x and y direction. 

Systematic error associated with the scanner calibration does not include files 3 and 4. The 

standard deviation, σ, in the scanner calibration is 0.31 mrad. The systematic error in the scanner 

calibration with 95% confidence (2σ) is 0.62 mrad. 

Table 4. Standard deviation errors in the scanner calibration. The degrees of freedom, ࢏࢜, for each 

scanner calibration is 144 (12 x 12 grid). 

ܵ௑, ( x-direction) ܵ ௜, (y-direction) ௜ File Number ௒,

X (mrad) Y (mrad) 

1 0.442 0.509 

2 0.141 0.183 

3 1.30 0.413 

4 5.41 0.561 

5 0.191 0.194 

Pooled Standard Deviation (all files) 1.79 

Pooled Standard Deviation (σ) 

(excluding files 3 and 4) 

0.31 

Figure 26 is a plot of the systematic errors in the scanner calibration. Errors in return spot 

location range from ±3.3 mm. Slope errors range from ±0.25 mrad. A linear regression was done 

on the calculated slopes across the aperture to determine its effect on the focal length and tilt. 

) equal to 2 , with the sum of the squared residuals (R -5 These data yielded /݀ݕ ݀ݖ = 0.2924y ± 3*10

1.00. The tilt term is of 3*10-5 radians, or 0.03 mrad. 

27 
 

 

 

 

 

 

 

 

 

 

 

 

 

Scanner Cal (-0.31 mrad) Scanner cal (+.031 mrad) 

-0.45 

-0.30 

-0.15 

0.00 

0.15 

0.30 

0.45 

-6 

-4 

-2 

0 

2 

4 

6 

Difference in Return Spot location from

uncertainties (ideal

Slope Error Uncertainty Caused by the

Incorrect Return Spot Location (mrad) 

-uncertainty) (mm) 

Slope Error (-0.31 mrad) Slope Error (+0.31 mrad) 

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 

Laser Output Angle (radians) 

Figure 26. Scanner calibration systematic errors. The vertical axis on the left is the difference in 

return-spot location on the target caused by the uncertainty in the laser tilt. The vertical axis on 

the right is the calculated slope error caused by the error in the return-spot location. 

scanner cal ( scanner cal (+0.62 mrad) -0.62 mrad) 

0.8 

0.6 

0.4 

Slope (radians) 

-0.8 

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 

-0.4 

-0.2 

Slope (radians) 

dz/dy = 0.2924x - 3E-05 

R2 = 1.00 

-0.8 

-0.6 

-0.4 

-0.2 

0.0 

0.2 

0.4 

0.6 

0.8 

Position on Aperture (m) Position on Aperture (m) 

Figure 27. Calculated slopes for a perfect parabola with scanner calibration error of ±0.62 mrad 

are plotted in the charts. 

28 

dz/dy = 0.2924y + 3E-05 

R2 = 1.00 

0.2
 

0.0
 

-0.6 
The performance of the scanner as a function of temperature is unknown to the manufacturer and 

is not considered in this study. The manufacturer guarantees the repeatability to be 8 μrad from 

0o

–50oC and does not recommend operation outside of this range. For this study, we assumed 

that the scanner is only being operated within this temperature range because no testing has ever 

been done outside this range. No information is known regarding the effect of relative humidity 

on the scanner performance. However, the manufacturer does not expect the performance to 

change and thus it is not considered in this study. 

Uncertainty Estimate – Slope Error 

All of the uncertainties considered in this study are for 2σ or 95% confidence around the true 

measurement. Each of the slope error uncertainties and their relative percentage contribution are 

listed in Table 5 and 6. The slope uncertainty over the laser cone angle varies depending on the 

output angle of the laser, intersection location on the collector, and return-spot location on the 

target. 

The uncertainties are assumed to be independent of each other and have Gaussian distributions. 

All of the uncertainties, except for the camera calibration, are assumed to be symmetric about the 

true value. The systematic standard uncertainty, US, is used to estimate the combined effect of 

systematic errors (US,i) on slope error test results (Equation 11). The random standard 

uncertainty, ܷோ, is used to estimate the combined effect of random errors ( UR,i) on slope error test

results (Equation 12). The effects on the slope error from these random and systematic 

uncertainties are shown in Figures 28, 29, and 30.

ଵ

ெ ଶ

ଶ ܷௌ ൌ ቌ෍ ܷௌ,௜ቍ Equation 11 

௝ୀଵ

ே

ଶ ܷோ ൌ ൭෍ ܷோ,௜

௜ୀଵ

൱

ଵ

ଶ

Equation 12 

Six random uncertainties were considered in this study (N=6): the measurement of the tilt in the 

target, distance between the target face and scanner, scanner tilt, distance between the target face 

and collector vertex, camera calibration, and scanner/calibration. The measured distance 

between the target face to the test piece had the largest effect on the positive and negative 

random error, ~70% for both the positive and negative (Table 5), contributing up to 0.17 mrad of 

slope uncertainty. The camera calibration was the second largest contributor on the positive and 

negative random uncertainty, causing up to 0.05 mrad of slope error uncertainty at some angles 

(Figure 28 and Figure 29). On average, the random camera calibration errors contribute 14%– 

20% of the total error, with the magnitude depending on whether the uncertainty is positive or 

negative. Note in Figure 28 and Figure 29 that there are some deterministic effects that depend 

29 
on specific output angles where the return position falls within either the upper or lower portion 

of a camera pixel (see the earlier section on Camera Calibration and Figure 21 for more detail). 

Table 5. Random error contributions to the random slope error uncertainty. The random error in 

the camera calibration is not symmetrical, so negative and positive random errors for both were 

calculated. 

Minimum Average Contribution Maximum Contribution Contribution to the 

Description to the Uncertainty (%) to the Uncertainty (%) Uncertainty (%) 

 +࢏,ࡾࢁ -࢏,ࡾࢁ +࢏,ࡾࢁ -࢏,ࡾࢁ +࢏,ࡾࢁ -࢏,ࡾࢁ

Target tilt 0.5 0.6 1.9 2.4 0 0 

Target face to laser 1.7 1.4 3.7 3.0 0.1 0.1

scanner output 

Scanner tilt 9.8 7.6 58.8 59.9 0 0 

Distance from target to 73.1 70 97.9 95.3 1.2 3.2 test piece 

Camera calibration 14.7 20.2 70.8 69.1 0 0.1 

Scanner/calibration 0.2 0 1.3 0.5 0 0 

To decrease the impact of random errors, the measured distance between the target face and 

collector vertex and the camera resolution must be improved. The distance between the target 

face and collector vertex is measured with a laser range finder that has an accuracy of 1.27 mm. 

A laser range finder with an accuracy of less than 1 mm would be needed to reduce the errors 

caused by this measurement. A camera with greater resolution would be needed to reduce the 

camera random errors. This would increase the number of pixels per area and improve the 

software’s ability to accurately calculate the centroid of the return laser on the target. 

Figure 28. Negative random error ࡾࢁ-. The contributing random errors to the negative random 

slope error uncertainty, ࡾࢁ,࢏-, were the tilt in the target, distance between the target face and 

scanner, scanner tilt, distance between the target face and collector vertex, camera calibration, 

and scanner/calibration. 

30 
Figure 29. Positive random error, ࡾࢁ+. The contributing random errors to the positive random 

slope error uncertainty, ࡾࢁ,࢏+, were the tilt in the target, distance between the target face and 

scanner, scanner tilt, distance between the target face and collector vertex, camera calibration, 

and scanner/calibration. 

Seven systematic uncertainties were considered in this study (M=7). These included 

measurement of the tilt in the target, distance between the target face and scanner, instrument 

vertical offset, scanner tilt, distance between the target face and collector vertex, camera 

calibration, and the scanner. The instrument vertical offset had the largest effect on the 

systematic uncertainty, with an average contribution of 42% or 0.12 mrad of error across the 

aperture (Table 6). The uncertainty in this measurement peaks at a laser output angle of zero 

radians, with a slope error of 0.19 mrad (Figure 30). The scanner calibration error was the second 

largest contributor to the systematic error, averaging 27% of the total error (Table 6). The 

uncertainty in the scanner calibration contributed up to 0.18 mrad of slope error. The laser output 

angle was the third largest contributor to the systematic uncertainty, averaging 19% of the total 

error (Table 6). This measurement’s average contribution to the slope error uncertainty was 

0.06 mrad and peaked at 0.12 mrad (Figure 30). 

To decrease the impact of the systematic uncertainty errors, the methods for determining the 

instrument vertical offset, scanner calibration process, and scanner tilt must be improved. 

Currently, the instrument vertical offset is measured using the human eye. If this measurement 

could be made with a mechanical device and the vertex could be accurately determined, the 

uncertainty would be greatly decreased. The scanner calibration is currently done by two 

operators—one controlling the scanner and the other checking the location of the laser on the 

scanner calibration grid. The likely cause for most of the error in this calibration is from the 

operator’s visual inspection of the laser location. If this process was automated, or if a camera 

were used to determine the laser location, the uncertainty in the scanner calibration would be 

reduced. To reduce the uncertainty in the laser output angle, the operator would need to level the 

laser to less than 0.523 mrad (0.03o

) before each test. The inclinometers used to level the laser 

31 
have an accuracy of ±0.087 mrad (0.005o

). Improved inclinometer accuracy would reduce this 

uncertainty contribution. 

Table 6. Percentage contributions to the systematic slope error uncertainty. 

Minimum Average Contribution Maximum Contribution Contribution to the 

Description to the Uncertainty (%) to the Uncertainty (%) Uncertainty (%) 

US,i US,i US,i 

Target tilt 0.8 3.2 0 

Target face to laser 1.3 3.5 0 

scanner output 

Instrument vertical 42.3 81.5 0.6 

offset 

Scanner tilt 19.1 36.9 0 

Distance from target to 9.3 24.7 0 

test piece 

Camera calibration 0.2 0.7 0 

Scanner calibration 27.0 52.1 0 

)LJXUH6\VWHPDWLFVORSHHUURUV Us. 

The systematic and random uncertainties must be combined to give the overall measurement 

uncertainty. There are over 30 degrees of freedom in this analysis, so the Root Sum Square, URSS, 

were assumed to have a 95% confidence. ܷௌ uncertainty model was used (Equation 13). ܷோ, and

ଵ

ൌ േ ଶ ܷோௌௌ ሾ ଶ ൅ ܷ ଶሿ ܷ ௌோ Equation 13 

32 
 

The positive and negative URSS uncertainty results are plotted in Figure 31. URSS - has an average 

uncertainty of 0.32 mrad, with a maximum of 0.47 mrad and a minimum of 0.21 mrad. URSS +

 has 

an average uncertainty of 0.32 mrad, with a maximum of 0.47 mrad and a minimum of 

0.21 mrad. The RMS uncertainty was 0.33 mrad. The positive and negative results are essentially 

the same because the camera contribution (the only one with different positive and negative 

values) is relatively small. Thus for any given test we would expect that 95% of the time the 

RMS error would be less than 0.33 mrad. 

Figure 31. Positive and negative URSS slope error uncertainty at the different laser output angles. - The average uncertainty for URSS and URSS

+ is 0.32 mrad. 

Each uncertainty contributes differently (i.e., randomly) to the total uncertainty over a set of 

tests. An error propagation analysis was completed to estimate the combined effect of these 

different uncertainties using the Monte Carlo method [1]. This was programmed in Excel to 

generate 20,000 trials where at each laser output angle a standard deviation was assigned 

according to the functional relationship for the average URSS shown in Fig. 31. A random slope 

error was then generated using the Box-Muller transformation [5] to achieve a normal (Gaussian) 

distribution. This is shown in Equation 14 below where a1 and a2 are random numbers between 0 

and 1; then b is normally distributed with a mean of 0 and standard deviation of σ. 

ሻ Equation 14 σܾ ൌ ඥെ2 ln ܽଵ ሺcos 2πܽଶ

As expected there are a small fraction of cases (~5%) where the uncertainty is greater than 2σ. 

The gap in the center at zero output angle represents the hole in the target where no return beams 

are captured. A distribution of these slope errors is shown in Fig. 32. The RMS of this 

distribution (with zero mean) is equivalent to the standard deviation, in this case 0.164 mrad. 

33 
 

 

 

 

Thus we would expect that 95% of the time, VSHOT test results for RMS error would be 

0.33 mrad or less. The 2σ slope error uncertainty will vary for a given test from about 0.21 to 

0.46 mrad depending on the laser output angle and the resulting position along the aperture. 

Figure 32. Slope error uncertainties at different laser output angles. The 2σ uncertainty is shown 

for reference. 

Figure 33. RMS slope error uncertainty for a VSHOT scan. This is a histogram of the RMS 

uncertainty for 20,000 tests. 

34 
 

 

Uncertainty Estimate – Focal Length and Test Article Tilt 

Each of the individual uncertainties and their contribution to estimated focal length and test 

article tilt were calculated to determine if these effects were significant. For each simulated scan 

across the aperture, the slope error versus y position was fit to the linear Zernike (Eq. 9) to yield 

the focal length and test article tilt. Only the random error in the distance measurement from the 

target to the test piece had any appreciable effect on the focal length, ±0.6 mm. This is the only 

measurement that has a linear effect on the slope across a scan causing an error in the focal 

length term, B2. All of the uncertainties have a relatively small effect on the calculated tilt term, 

ranging from 0.0007 mrad to 0.2 mrad as shown in Table 7. For field testing, a practically 

achievable tilt error of 1 mrad is considered low and all of the tilt terms listed in Table 7 are 

significantly lower. We did not consider the directl impact of test-article tilt error as a specific 

uncertainty error in this analysis because this value is one of the outputs of the data analysis. It is 

also very difficult to accurately determine this tilt in the field. 

The same type error propagation analysis was used to estimate the combined effect of these 

different uncertainties on the focal length and tilt terms using the Monte Carlo method. A 

computer program similar to the one used to determine the slope error was written to calculate

the effect on focal length and tilt. The program was run 20,000 times to determine the 

uncertainty in these terms with a 95% confidence (2σ). The focal length uncertainty for VSHOT 

is ±0.6 mm (2σ), and the tilt uncertainty is ±0.2 mrad (2σ). 

Table 7. Uncertainty impacts on the focal length and test-article tilt. 

Focal Length Uncertainty Tilt Uncertainty

Description UR,i US,i UR,i US,i 

Target tilt negligible negligible ±8 μmrad ±0.02 mrad 

Target face to laser scanner negligible negligible ±0.7 μmrad ±0.7 μmrad

output 

Instrument vertical offset negligible ±0.2 mrad 

Laser output angle negligible negligible ±4 μmrad ±0.03 mrad 

Distance from target to test ±0.6 mm ±0.2 mm negligible negligible piece 

ோത ି =0.5 μmrad Camera calibration negligible negligible 0.7 μmrad

ோത

,௜

,௜

ା =0.7 μmrad 

Scanner calibration negligible negligible negligible ±0.03 mrad 

ݏ

ݏ

35 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summary 

VSHOT has been used to characterize heliostat, dish, and trough reflector panels providing 

accurate surface slope information to determine the optical quality and to estimate optical 

performance. A 1997 uncertainty analysis showed that VSHOT had a RMS slope error 

uncertainty of about 0.1 mrad. The results in this analysis yield a higher 2σ uncertainty, 

~ 0.33 mrad. The 2σ uncertainty in the focal length is ±0.6 mm (0.03%), and the tilt 

2σ uncertainty is ±0.2 mrad. For a single test the expected 2σ slope error will vary between 0.21 

and 0.46 mrad depending on the laser output angle. The focal length uncertainty determined in 

the previous study was 0.8% and is slightly higher than this one.. While there is no direct 

explanation for this there are many differences between these two studies. The original study was 

an experimental analysis that used a 16-inch-diameter telescope mirror and the current study is 

strictly analytical. The original study only used 0.0698 radian (4o

) of the scanner cone angle and 

this one looks at the full 1.38 radians (80o). In addition, the original study did not attempt to 

separate the errors. The original study only looked at the RMS variation between tests, not the 

slope uncertainty at each output angle. 

Out of the six random error sources considered, the measured distance between the target to test 

piece and camera calibration had the largest effects. The measured distance between the target 

face and collector vertex has the largest effect on the positive and negative random error, ~70%, 

and the accuracy of this measurement is 1.27 mm. An accuracy of less than 1 mm would be 

needed to reduce the errors caused by this measurement. The camera was the second largest 

contributor on the positive and negative random uncertainty, contributing 14%–20% of the total 

error. A camera with more pixels would be needed to reduce this random error. This would 

increase the number of pixels per area and improve the software’s ability to calculate the 

centroid of the return laser on the target, thus reducing the slope error uncertainty. 

Seven systematic uncertainties were considered in this study. If the systematic uncertainty errors 

need to be decreased, the methods for determining the instrument vertical offset, scanner 

calibration process, and laser tilt will need to be improved. Currently, the instrument vertical 

offset is measured using the human eye and this has the largest effect on the systematic 

uncertainty, with an average contribution of 42%. The scanner/calibration error was the secondlargest contributor to the systematic error, averaging 27% of the total error. The scanner 

calibration is currently done by two operators—one controlling the scanner and the other 

checking the location of the laser on the scanner calibration’s grid. If this process was 

automated—or if a camera were used to determine the laser location—the uncertainty in the 

scanner calibration could be reduced. The laser output angle was the third-largest contributor to 

the systematic uncertainty, averaging 19% of the total error. To reduce the uncertainty in the 

laser output angle, the operator would need to level the laser to less than 0.523 mrad (0.03o

) 

before each test. 

The estimated uncertainty for slope error is relatively small compared to even the best parabolic 

trough mirror panel (2–3 mrad RMS slope error), and thus, there is currently little incentive to 

implement any of the improvements noted in this report. The estimated uncertainty in focal 

length and test-article tilt are extremely small compared to nominal values; thus, there is no 

36
 
 

 

 

 

incentive to improve individual uncertainty contributions. Overall, it is safe to conclude that the 

uncertainty reported here is well within acceptable levels for the testing of parabolic troughs and 

very likely acceptable for dish or heliostat panels. Future work could extend this analysis to 

point-focus panels or to full panels where both x- and y-direction errors would be accounted for. 

Acknowledgements 

This work was supported by the U.S. Department of Energy’s Concentrating Power Program 

under Task Number CP09.1001. 

References 

1.	 R. Dieck, Measurement Uncertainty Methods and Applications, 4th Edition. Research 

Triangle Park: The Instrumentation, Systems, and Automations Society, 2007. 

2.	 S.A. Jones, J.K. Gruetzner, R.M. Houser, R.M. Edgar, and T.J. Wendelin, VSHOT 

Measurement Uncertainty and Experimental Sensitivity Study. IECEC-97: Proceedings 

of the Thirty-Second Intersociety Energy Conversion Engineering Conference, 27 July–1 

August 1997, Honolulu, Hawaii. Volume 3: Energy Systems, Renewable Energy 

Resources, Environmental Impact and Policy Impacts on Energy, American Institute of 

Chemical Engineers, New York, NY, vol. 3, 1997, pp. 1877–1882. 

3.	 J. Ojeda-Castaneda, Optical Shop Testing, 2nd Edition, edited by Malacara, 1992. 

4.	 Model 6220H Galvanometer Optical Scanner Instruction Manual. Cambridge
 

Technology, Inc., Revision 3.0, February 18, 2007.
 

5.	 G. Box and M. Muller "A Note on the Generation of Random Normal Deviates." Ann. 

Math. Stat. 29, 610-611, 1958. 

37
 
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Visual Scanning Hartmann Optical Tester (VSHOT) Uncertainty

Analysis

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14. ABSTRACT (Maximum 200 Words)

In 1997, an uncertainty analysis was conducted of the Video Scanning Hartmann Optical Tester (VSHOT). In 2010, 

we have completed a new analysis, based primarily on the geometric optics of the system, and it shows sensitivities 

to various design and operational parameters. We discuss sources of error with measuring devices, instrument 

calibrations, and operator measurements for a parabolic trough mirror panel test. These help to guide the operator in 

proper setup, and help end-users to understand the data they are provided. We include both the systematic (bias) 

and random (precision) errors for VSHOT testing and their contributions to the uncertainty. The contributing factors 

we considered in this study are: target tilt; target face to laser output distance; instrument vertical offset; laser output

angle; distance between the tool and the test piece; camera calibration; and laser scanner. These contributing factors 

were applied to the calculated slope error, focal length, and test article tilt that are generated by the VSHOT data 

processing. Results show the estimated 2-sigma uncertainty in slope error for a parabolic trough line scan test to be 

±0.2 milliradians; uncertainty in the focal length is ±0.1 mm, and the uncertainty in test article tilt is ±0.04 milliradians.

15. SUBJECT TERMS

CSP; uncertainty analysis; VSHOT; parabolic trough; systematic error; random error; sensitivity

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