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© 2020 Hongrui Zhao

 DEVELOPMENT OF A
LOW-COST MULTI-CAMERA STAR TRACKER
FOR SMALL SATELLITES

BY

HONGRUI ZHAO

THESIS

Submitted in partial fulfillment of the requirements

for the degree of Master of Science in Aerospace Engineering
in the Graduate College of the
University of Illinois at Urbana-Champaign, 2020

Urbana, Illinois

Adviser:

Associate Professor Michael Lembeck

 ABSTRACT

This thesis presents a novel small satellite star tracker that uses an array of low-cost, off

the shelf imaging sensors to achieve high accuracy attitude determination performance. The

theoretical analysis of improvements in star detectability achieved by stacking images from

multiple cameras is presented. An image processing algorithm is developed to combine images

from multiple cameras with arbitrary focal lengths, principal point offsets, distortions, and

misalignments. The star tracker also implements other algorithms including the region growing

algorithm, the intensity weighted centroid algorithm, the geometric voting algorithm for star

identification, and the singular value decomposition algorithm for attitude determination. A star

tracker software simulator is used to test the algorithms by generating star images with sensor

noises, lens defocusing, and lens distortion. A hardware prototype is being assembled for eventual

night sky testing to verify simulated performance levels. Star tracker flight hardware is being

developed in the Laboratory for Advanced Space Systems at Illinois (LASSI) at the University of

Illinois at Urbana Champaign for future CubeSat missions.

ii
 ACKNOWLEDGEMENTS

I want to express my sincere gratitude to Dr.Michael Lemebck for giving me the opportunity

to work in the Laboratory for Advanced Space Systems at Illinois. He entrusts the important tasks

on attitude determination and control to me which I am passionate about. I learned a lot from his

rich industrial experience, his visions, and his great sense of humor. He certainty played an

immense role in my aerospace engineering career.

I want to thank my talented partners working on spacecraft attitude determination and control

systems: Marc Akiki, Eric Alpine and Walker Dimon as well as everyone else who contributed to

our CubeSat missions.

I also want to thank my friends who accompany me during this difficult time of the COVID-

19 pandemic: Antonio Watson, Bethany Garland, Tiffany Dao, Terry Chen, and Pang Ge.

While completing my master’s degree, I was blessed with an inordinate amount of support

from my mother, Qihong Hu, and my father, Ge Zhao. Thank you for all the support despite being

11,671 kilometers away.

iii
To my parents, for their love and guidance. To Space, our final frontier and the boundary of

human imagination.

iv
 TABLE OF CONTENTS
CHAPTER 1: MOTIVATION ........................................................................................................ 1
CHAPTER 2: BACKGROUND ..................................................................................................... 4
2.1 Star Tracker Concept of Operations ................................................................................. 4
2.2 Star Tracker Historical Development ............................................................................... 5
2.3 Camera Arrays and Image Stacking ................................................................................. 9
CHAPTER 3: STAR TRACKER ALGORITHMS ...................................................................... 12
3.1 Reference Frames............................................................................................................ 13
3.2 Imaging Stacking and Star Detection ............................................................................. 14
3.3 Imaging Processing ......................................................................................................... 18
3.4 Centroiding ..................................................................................................................... 25
3.5 Star Identification............................................................................................................ 28
3.6 Attitude Determination ................................................................................................... 32
CHAPTER 4: SOFTWARE SIMULATION AND HARDWARE VERIFICATION ................. 35
4.1 Star Tracker Software Simulator .................................................................................... 35
4.2 Software Simulation Results and Analysis ..................................................................... 42
4.3 Hardware Verification .................................................................................................... 49
CHAPTER 5: CONCLUSION ..................................................................................................... 51
REFERENCES ............................................................................................................................. 53

v
 CHAPTER 1: MOTIVATION

Space missions requiring precise attitude determination are beginning to take advantage of

CubeSat platforms for their low cost and design simplicity [1][2]. Historically, magnetometers and

sun sensors have been used as attitude sensors for CubeSat missions providing low accuracy

attitude information (no better than 0.2° ) [3]. Magnetometers have limited resolution and sun

sensors do not work in eclipse, hindering their utility on high performance missions. Another

approach is required to improve performance and achieve higher accuracies.

One way to improve attitude determination accuracy is to add a star tracker to the attitude

control system sensor suite. The AeroCube-OCSD CubeSat, developed by the Aerospace

Corporation, utilizes a star tracker to provide three-axis determination accuracy better than

0.15° (540 arcseconds) in support of laser communication downlinks [1]. Under ideal

circumstances, star trackers can provide attitude determination with an accuracy better than 0.1°

(360 arcseconds) throughout the orbit. Such accuracy can enable a variety of new satellite

applications including optical communications and Earth observation [1][2].

There are several commercial star trackers that are compatible with the CubeSat format.

Commercial star trackers with high accuracy and robustness tend to be costly, while lower cost

star trackers, some developed by universities, fail to achieve required levels of performance. Table

1.1 compares three university-built and three commercial star trackers compatible with CubeSat

applications.

University-built star trackers have not yet demonstrated high performance metrics. In part,

this may be due to the lack of suitable verification test equipment. The stated accuracy performance

of the star tracker in Ref.[4] is based on software simulation and theoretical analysis. Unfortunately,

1
 night sky testing of this star tracker failed, in part because the noise level in the images was high

and the noise removal method was not accurate enough to remove false stars. Another star tracker

in Ref.[5] utilizes a computer monitor based system to project stars onto the star tracker sensor to

test the star tracker accuracy performance. The monitor is not able to accurately simulate the actual

brightness of stars. Only CubeStar in Ref.[3] has successfully performed a series of night sky tests

to verify its performance metrics.

Table 1.1 Comparisons of available small star trackers for CubeSat applications

Model and Accuracy Radiation Stray Light Cost

Manufacturer (cross boresight Tolerance Tolerance
/roll, arcsecond)
York University [4] 11.448 N/A N/A $695
(theoretical) (camera and lens)
University of Texas Within tens of N/A N/A $1,250
at Austin [5] arcseconds (camera and lens)
CubeStar, 55.4~77.4 / N/A N/A $94
Stellenbosch 218.88 (image sensor and lens)
University [3] (3σ)
ST-200, 30 / 200 9 krad 34° / 45° Sun $30,000 [3]
Berlin Space (3σ) exclusion
Technologies [6]
ST-16, Sinclair 5 / 55 N/A 34° Sun exclusion $120,000 ~ $140,000
Interplanetary [7] (RMS) and Moon-avoidance
MAI-SS, 4 / 27 75 krad 45° Sun exclusion $32,500 [9]

Adcole Maryland

Aerospace [8]

2
 Radiation tolerance is usually not accommodated in university star tracker designs due to low

budgets. Commercial star trackers such as the ST-200 use shielding and hot pixel detection

algorithms to achieve better radiation robustness. Some higher-end star trackers [10][11] use

radiation-hardened glass materials and special radiation-hardened image sensors to improve their

performance.

Stray light is also a factor for university-built, low-cost star trackers. Stray light may be

introduced into the optics chain by the Sun, Moon, Earth, or other bright bodies. A baffle is

commonly used to capture stray light before it reaches the sensor. Stray light measurement

facilities add to the overhead cost of star tracker development and can be cost-prohibitive for small

research groups [12].

With these considerations in mind, a set of objectives and stretch goals has been formulated

for the development of an accurate, low-cost star tracker for small satellite applications. This thesis

focuses on developing the design of a low-cost star tracker prototype with a novel camera array

configuration, providing 0.1° (360 arcseconds) attitude determination accuracy for a cost of less

than $5,000 (US). The availability of such a tracker could be enabling for missions with high-

performance attitude determination requirements.

3
 CHAPTER 2: BACKGROUND

2.1 Star Tracker Concept of Operations

A star tracker (sometimes referred as star sensor or star imager) consists of a camera(s) and a

microcomputer that determines the attitude of a satellite from captured star images. Star trackers

are the most accurate attitude sensor, because fixed stars with accurately mapped positions are

used as references [13]. The concept of operations for a common star tracker is illustrated in Figure

2.1. The light from a distant star is focused by the lens of a star tracker and falls on a pixel area of

the image sensor. The pixels illuminated by the star light are detected and the coordinate (x, y)

representing the centroid of the star light is calculated. A star vector can be constructed using (x,

y) and the focal length of the lens. The attitude of the star tracker can then be calculated using

multiple star vectors.

Figure 2.1 A star tracker takes star light patterns projected through a lens onto an image sensor to

determine the pointing orientation of the satellite it is attached to.

In practice, the attitude determination using a star tracker faces several challenges: 1) An ideal

pinhole camera doesn't exist, and all lens have some degree of distortion. A distorted image results
4
in an inaccurate star centroid. 2) A star can be so dim that its signals collected by the image sensor

are overshadowed by the background noise. This can result in a failed star detection or an

inaccurate star centroid. 3) Non-star objects can illuminate pixels and generate false stars. 4) The

parameters of the camera stored in the star tracker's processor, such as focal length, can be

inaccurate. This can be caused by inaccurate ground calibration. 5) Mechanical alignments

between the sensor and structure, induced by vibration (during launch) and thermal expansion, can

also impact attitude determination accuracy.

2.2 Star Tracker Historical Development

Star trackers have changed dramatically since they were first introduced in the late 1940s and

early 1950s [14]. Several generations of star trackers have been developed. The first-generation

star trackers provided star vector readouts for an external processor to analyze. The second-

generation star trackers introduced autonomous, stand-alone computations. More recently, robust

complementary metal–oxide–semiconductor active-pixel sensor-based (CMOS APS) and smaller

Nano/Pico star trackers have become available.

The first-generation star trackers were locked onto a known star(s) with the help of other

attitude sensors. Star identification and attitude determination were computed by algorithms

operating on an external processor [13][15][16]. Photomultiplier and dissector tubes were used in

the first-generation star trackers designs [14][17][18]. However, designs incorporating charge-

coupled device (CCD) technology were also pioneered in the early 1970s at NASA's Jet Propulsion

Lab (JPL) [19]. One example of the first generation star tracker is the Canopus star tracker on the

Voyager spacecraft, as shown in Figure 2.2 [18]. The Canopus star tracker is designed to track

Canopus or some other guide stars using an image dissector tube. The star tracker only generates

5
an error signal along its roll axis when the light source focused on the photocathode is not at the

roll center line of the field of view (FOV), thus only two-axis attitude information is provided.

Figure 2.2 Canopus star tracker used by the Voyager spacecraft has been operational for over 43

years in space [18].

Another example is the STELLAR CCD star tracker developed in the 1970s at JPL [14]. The

STELLAR star tracker is designed to acquire and track up to ten stars exceeding a preset signal

threshold. The X- and Y-axis coordinates and magnitude of the acquired stars are measured and

transmitted to the user system for further processing.

In the 1990s, the availability of more powerful computers enabled the development of

autonomous star trackers [16][19]. An autonomous star tracker can independently provide three-

axis attitude quaternions by performing a pattern recognition of the stars in the FOV (at least three

6
stars matches are required) [20]. An internal star catalog is required to perform this task. Compared

with the image dissector, a CCD sensor is superior in tracking multiple stars, and has been widely

used in autonomous star trackers [21]. One example of the second-generation star tracker is the

SED-16 developed by SODERN, as shown in Figure 2.3 [22]. The SED-16 autonomous star

tracker can provide 3-axis attitude measurements with accuracy between 0.01° and 0.02° (3𝜎) at

10 Hz and for body rotation velocities up to 0.3°/𝑠.

Figure 2.3 SED-16 autonomous star tracker is capable of acquiring stars at high body rates [22].

In the 2000s, new design concepts improved the robustness of available star trackers.

Compared to CCD detectors, CMOS APS devices add a direct pixel readout capability that

improves the kinematic robustness of a tracker [23]. Also, CMOS APS detectors have better

radiation tolerance and incorporate simplified electronics [24]. One example of a robust CMOS

APS star tracker is the HYDRA multi-headed star tracker also developed by SODERN, pictured

in Figure 2.4 [10][23]. The HYDRA can conduct lost-in-space attitude acquisition with angular

7
rates up to 6°/𝑠. Thanks to the multi-headed system, the HYDRA is tolerant of Earth and Sun

occultations. Another robust example is the ASTRO APS star tracker developed by Jena-Optronik

GmbH [24]. The APS star tracker utilizes a STAR-100 radiation hardened CMOS image sensor

featuring on-chip fixed pattern noise (FPN) correction. This APS star tracker also has a hardwired

image data pre-processor to filter non-star objects under bad solar flare conditions.

Figure 2.4 HYDRA multi-headed star tracker is a robust design for high reliability missions [23].

Recently, Nano/Pico star trackers have been developed for small satellites, especially

CubeSat, applications. Nano star trackers fit within a 1 U (10 cm X 10 cm) volume and weigh less

than 1 kg [3]. Pico star trackers weigh up to 90 g (without baffle) and fit within 0.5 U volume [25].

Several Nano/Pico star trackers have been developed, such as the ST-16 from Sinclair

Interplanetary and the ST-200 from Berlin Space Technologies. The ST-16 is one of the few

Nano/Pico star trackers that has flight heritage [26]. Unfortunately, the flight performance of the

8
ST-16 was far worse than expected. After comprehensive fault diagnosis, several issues were

found, and improvements were made to the baseline design. In addition, the ST-200 star tracker

was launched onboard the Aalto-1 CubeSat on June 2017, and the CubeStar star tracker was

launched onboard ZA-AeroSat on May 2017 [26][28][29]. However, information regarding their

on-orbit performance is not readily available.

Figure 2.5 The Sinclair Interplanetary ST-16 star tracker has flight heritage, but lacked the desired

performance [26].

2.3 Camera Arrays and Image Stacking

The concept of using a camera array to improve star imaging has been applied to astronomical

telescope systems such as Dragonfly Telephoto Array [30], but has not yet been explored for star

tracker applications. The Dragonfly system utilizes a camera array to increase the sensitivity of the

composite system to detect ultra–low surface brightness objects. The array consists of eight lenses

with 143 mm apertures coupled to eight science-grade commercial CCD cameras. By combining

9
the light from all of the cameras, the effective aperture for N lens is √𝑁 × 143 𝑚𝑚, and the signal-

to-noise (SNR) ratio is significantly improved over a single cameras system.

Image combining/ stacking techniques can be applied to increase the sensitivity of an imaging

system if synchronous images of a moving object, or asynchronous images of a static object are

available. For example, Ref.[31] applies an image stacking algorithm to increase the sensitivity of

fluorescence detection using a low-cost webcam. A 10 ~ 15 seconds video of a static sample chip

is taken by the webcam and an averaging image stacking algorithm is applied. In the stacked image,

the random noise present in each frame is significantly reduced which results in an enhanced SNR.

By using this method, a set of low-cost webcams can achieve a sensitivity similar to a single

expensive CCD camera.

Various image combining/ stacking algorithms are available for increasing SNR and rejecting

bad pixels caused by cosmic ray hits and satellite trails [32][33]. An averaging algorithm simply

divides the sum of pixels by the number of frame exposures taken. The averaging algorithm has

the highest SNR but does not reject bad pixels. Another approach is to use the median algorithm

to sort pixels from highest to lowest values and select the values in the middle. Bad pixel rejection

of the median algorithm is excellent, but its SNR is lower than the averaging algorithm.

The Min/Max clip algorithm excludes the maximum and minimum pixel values before

computing the mean, thereby rejecting most of the bad pixels. A sigma clip algorithm first

computes the mean and compares the deviation of a pixel from the mean with a“clipping sigma."

The "clipping sigma" can be obtained from the standard deviation of the pixels or the sensor noise

model. The Standard Deviation Masking (SDM) algorithm is a combination of the average, median,

and sigma clip methods. The SDM algorithm selects either the median or the average value based

on the standard deviation.

10
 The use of camera arrays and suitable algorithms in a low-cost star tracker implementation,

and the simulated performance obtained, is discussed in the following chapters.

11
 CHAPTER 3: STAR TRACKER ALGORITHMS

The software system for the multi-camera star tracker consists of an image processing

algorithm, a centroiding algorithm, a star identification algorithm, and an attitude determination

algorithm. A flowchart of the algorithms used by the star tracker is illustrated in Figure 3.1.

Figure 3.1 Flowchart of the algorithms used by the star tracker. Multiple images are transformed

and stacked to provide a composite image for attitude determination.

One camera (A) is selected as the reference camera and its image is the reference image.

Images from other cameras (B) are the non-reference images. The non-reference images are

aligned to the reference by the image processing algorithms (C). The image processing algorithms

consists of the projective transformation, the distortion correction, the image interpolation, and the

image stacking. The projective transform is applied on the non-reference images to align them

with the reference image. The lens distortion effects on the images are removed. The image
12
interpolation is used to generate an image with correct intensity values. The images are stacked to

reduce noise. The star centroids of the stacked image are calculated. Star vectors are constructed

by the star identification algorithm using the star centroids and searches for corresponding inertial

star vector in a star catalog. By using the inertial star vectors, an attitude quaternion (D) of the star

tracker is calculated by the attitude determination routine.

3.1 Reference Frames

Several reference frames are used in this thesis: a focal plane frame, a star tracker body frame,

and a J2000 Earth-Centered Inertial (ECI) frame.

Both the focal plane frame and star tracker body frame are defined inside the model star

tracker, as shown in the Figure.3.2. The star tracker body frame XYZ has its origin located at the

vertex of the optical system (the pinhole camera model is used) and its z-axis aligned with the star

tracker's boresight. The focal plane frame UV is a 2D reference frame, originating at either its

center or one of its corners. The transformation for a star vector from the focal plane frame to the

star tracker body frame is given by:

𝑢0 − 𝑢
1
𝑠= [ 𝑣0 − 𝑣 ] (3.1)
2 2
√𝑓 + (𝑢0 − 𝑢) + (𝑣0 − 𝑣) 2
𝑓

where (𝑢, 𝑣) is the star image centroid coordinate under the focal plane frame. Coordinate

(𝑢0 , 𝑣0 ) represents the intersection of the boresight and the focal plane under the focal plane frame.

The focal length of the optics is represented by 𝑓. 𝑠 is the star vector under the star tracker body

frame.

13
 Figure 3.2 Focal plane reference frame and star tracker body frame are used in the development of

the star tracker attitude determination scheme.

3.2 Imaging Stacking and Star Detection

The sensitivity of a star tracker's image sensor determines the star detectability that sets a

threshold for the dimmest stars that the star tracker can detect. Improving the detectability of a star

tracker improves its attitude estimation accuracy and robustness. Combining images from multiple

low-cost, off-the-shelf sensors is proposed to provide a sensitivity similar to a single high-cost,

high-sensitivity camera.

The improvement of star detectability through the addition of multiple cameras can be

quantified to determine the optimal camera array size. Detectability and sensitivity analyses start

with the identification of noise components. According to Ref.[4], there are two types of noise that

influence the pixel values of an image sensor: temporal (or time variant) noise, and spatial (or time

invariant) noise. The effects of spatial noise can be largely removed through calibrations. Thus,

temporal noise dominates the sensitivity of a star tracker.

14
 Shot noise, dark current, and readout noise are the principal temporal noise sources, according

to Ref.[16][20][34]. Shot, or photon noise, results from the discrete and random nature of photons.

Shot noise follows a Poisson distribution, but can be modeled as zero mean Gaussian noise. Dark

current represents the accumulation of electrons in the absence of light. Dark current can be

modeled as Gaussian noise with zero mean and a standard deviation varying with integration time

and temperature. Read noise combines noise sources from the image sensor charge transfer and

read-out amplifiers. Read noise is modeled using a zero mean normal distribution with a constant

standard deviation. Considering these temporal noise sources, the expression for SNR is given as:

𝑆
𝑆𝑁𝑅 =
(3.2)
√𝑆 + 𝑁𝑠𝑒𝑛𝑠𝑜𝑟 2

where 𝑆 is the electrons generated in the sensor by photons from a target star; 𝑁𝑠𝑒𝑛𝑠𝑜𝑟 is the

standard deviation of the temporal noise measured in electrons. 𝑁𝑠𝑒𝑛𝑠𝑜𝑟 is given by:

𝑁𝑠𝑒𝑛𝑠𝑜𝑟 = √𝑁𝑑𝑎𝑟𝑘 ∙ 𝑇 + 𝑁𝑟𝑒𝑎𝑑 2 (3.3)

where √𝑁𝑑𝑎𝑟𝑘 ∙ 𝑇 is the standard deviation of the dark current in electrons. T is the exposure

time in seconds. 𝑁𝑟𝑒𝑎𝑑 is the standard deviation of read noise in electrons. By stacking n images

or calculating the mean of n images, SNR increases by a factor of √𝑛:.

𝑆∙𝑛
𝑆𝑁𝑅 =
(1.4)
√(𝑆 + 𝑁𝑠𝑒𝑛𝑠𝑜𝑟 2 ) ∙ 𝑛

To determine the highest star visual magnitude a star tracker can detect based on SNR, it is

necessary to calculate how many photons are generated by a target star. According to Ref.[35], the

photons collected on the focal plane from a magnitude m star is given as:

𝑁𝑝ℎ = 𝐴𝑙 ∙ 𝑇𝑙 ∙ ∆B ∙ Φ𝑚 ∙ 𝑇 (3.5)

15
 𝜋
where 𝐴𝑙 is the area of light collecting surface in 𝑐𝑚2 , 𝐴𝑙 = 𝑑 2 and d is the diameter of the
4

camera's aperture; 𝑇𝑙 is optical transmittance, usually 0.6~0.8. 𝑇𝑙 represents the fraction of light

of a specific wavelength which passes through the lens; ∆B is the bandwidth of the lens, usually

3000-6000 Å (Angstrom); T is the exposure time in seconds; Φ𝑚 is the luminous flux of m visual

𝑝ℎ𝑜𝑡𝑜𝑛𝑠
magnitude ( unit ) , given Ref.[35] as:
𝑐𝑚2 ∙Å∙𝑠

Φ𝑚 = 10(15−2𝑚)/5 (3.6)

Using Eq(3.5), the signal 𝑆 (electrons per pixel) received from a magnitude m star can be

expressed as:

𝑆𝑚 = 𝑁𝑝ℎ ∙ 𝑄𝐸 ∙ 𝐾𝑓𝑖𝑙𝑙 ∙ 𝐾 (3.7)

where QE is the general quantum efficiency, usually ranging from 0.3 to 0.7; 𝐾𝑓𝑖𝑙𝑙 is the fill

factor, the ratio of a pixel's light sensitive area to its total area, ranging from 0.3 to 0.7; and K is

the energy concentrative degree. The value of K is about 0.25~0.4 when the starlight is spread over

a 3x3 pixel grid.

An SNR threshold 𝑉𝑡ℎ is defined to determine if an SNR value is sufficient for detecting stars.

According to Rose criterion explained in Ref.[36], a SNR > 5 can guarantee the certain detection

of a signal. Based on Eq(3.4), SNR should fulfill the expression given below to detect a star with

magnitude m when N images are stacked:

𝑆𝑚 ∙ 𝑛
𝑆𝑁𝑅 = ≥ 𝑉𝑡ℎ
2 (3.8)
√(𝑆𝑚 + 𝑁𝑠𝑒𝑛𝑠𝑜𝑟 ) ∙ 𝑛

Eq(3.8) can be rewritten as:

𝑉𝑡ℎ 2 + √𝑉𝑡ℎ 4 + 4𝑉𝑡ℎ 2 ∙ 𝑁𝑠𝑒𝑛𝑠𝑜𝑟 2 ∙ 𝑛 (3.9)

𝑆𝑚 ≥
2𝑛

16
 Substituting Eqs(3.3)(3.7) into Eq(3.9) gives the highest star visual magnitude a star tracker

can detect:

𝑉𝑡ℎ 2 + √𝑉𝑡ℎ 4 + 4𝑉𝑡ℎ 2 ∙ (𝑁𝑑𝑎𝑟𝑘 ∙ 𝑇 + 𝑁𝑟𝑒𝑎𝑑 2 ) ∙ 𝑛

𝑚𝑥 ≤ 7.5 − 2.5𝑙𝑜𝑔10 (3.10)
2𝑛 ∙ 𝐴𝑙 ∙ 𝑇𝑙 ∙ ∆B ∙ 𝑇 ∙ 𝑄𝐸 ∙ 𝐾𝑓𝑖𝑙𝑙 ∙ 𝐾
( )

Eq(3.10) can be used to plot the star detectability as a function of the number of cameras

added, as shown in Figure 3.3. The parameters are set to represent a MT9V022 image sensor and

a LS-12020 lens used for the hardware prototype, where 𝑁𝑑𝑎𝑟𝑘 = 25 𝑒 − , 𝑁𝑟𝑒𝑎𝑑 = 2.6 𝑒 − , 𝐴𝑙 =
𝜋
𝑐𝑚2, 𝑇𝑙 = 0.88, ∆B = 3000 Å, 𝑇 = 0.1 𝑠, 𝑄𝐸 = 0.4, 𝐾𝑓𝑖𝑙𝑙 = 0.3, 𝐾 = 0.25.
16

The circular field of view (FOV) of a LS-12020 lens paired with a MT9V022 image sensor

is 13.69°. A star visual magnitude higher than five is required to achieve sky coverage above 90%

with this circular FOV, according to Ref.[3]. Thus, there should be more than three cameras in a

star tracker camera array to achieve adequate star detection sensitivity across the entire sky.

However, this analysis does not consider the impacts of errors in image processing algorithms,

which would hinder the improvement of star detectability. Those impacts are examined in Chapter

4.

17
 Figure 3.3 Star detectability as a function of the number of cameras in a star tracker camera array.

There should be more than three cameras in an array to ensure sensitivity for detecting stars in 90% of

the visible sky.

3.3 Imaging Processing

Without calibration, there are misalignments between the arrayed star tracker cameras. Each

camera also may have different intrinsic parameters. As a result, camera array images are

processed and transformed before stacking. One camera is selected as the reference camera and its

star images are defined as the reference images. The images from non-reference cameras are then

mapped to the reference images. The reference images and non-reference images are processed

differently.

Unlike an ideal pin hole camera, every camera lens causes some degree of distortion. As a

result, straight lines in a scene aren't projected as straight lines on an image plane. Two common

distortions, known as the pincushion distortion and the barrel distortion, are illustrated in Figure

18
3.4 [37][38]. In the pincushion distortion, image magnification increases with the distance from

the lens boresight. In the barrel distortion, the image magnification decreases with the distance

from the lens boresight. Lens distortion models can be used to correct a distorted image. To

implement distortion correction, the model distortion coefficients need to be measured by using

calibration tools such as those identified in Ref.[39].

Figure 3.4 Illustration of two common distortions [38]. In the pincushion distortion, image

magnification increases with the distance from the lens boresight. In the barrel distortion, the image

magnification decreases with the distance from the lens boresight.

For the reference images, the distortion correction is applied. According to Ref.[40], a

mapping from the undistorted coordinates to the distorted position is required to obtain the image

intensity at the corrected position, which follows:

(1) A raw distorted image from the reference camera is obtained and saved into a matrix A.

(2) An empty matrix B is created to represent the undistorted image.

(3) The coordinate (𝑢𝑐 , 𝑣𝑐 ) of a pixel from j row k column of the matrix B relative to the

principal point is calculated.

(4) The UV frame coordinate (𝑢𝑑 , 𝑣𝑑 ) of the corresponding pixel on the matrix A is

calculated using the tangential and radial distortions model.

19
 (5) The value for the j row k column pixel on the matrix B is computed by interpolation of

pixels on the matrix A.

The coordinate of an undistorted pixel from j row k column of the matrix B relative to the

principal point is calculated by:

(𝑢𝑐 , 𝑣𝑐 ) = (𝑘 + 0.5, 𝑗 + 0,5) ∙ 𝑝 − (𝑢0 , 𝑣0 ) (3.11)

where p is the pixel width/length of the image sensor, (𝑢0 , 𝑣0 ) is the principal point of the

reference camera. (𝑢𝑐 , 𝑣𝑐 ) is the coordinate of an undistorted pixel. The UV frame coordinate of

the corresponding distorted pixel is calculated by applying radial and tangential distortion model

from Ref.[41]:

(𝑢𝑑 , 𝑣𝑑 ) = (𝑢𝑐 , 𝑣𝑐 ) + (𝛿𝑢𝑟 , 𝛿𝑣 𝑟 ) + (𝛿𝑢𝑡 , 𝛿𝑣 𝑡 ) + (𝑢0 , 𝑣0 ) (3.12)

The radial distortion term is given by:

(𝛿𝑢𝑟 , 𝛿𝑣 𝑟 ) = (𝑢𝑐 , 𝑣𝑐 ) ∙ (𝑘1 ∙ 𝑟 2 + 𝑘2 ∙ 𝑟 4 ) (3.13)

where 𝑘1 and 𝑘2 are the coefficients for radial distortion, and 𝑟 = √𝑢𝑐2 + 𝑣𝑐2 . The tangential

distortion term is given by:

𝛿𝑢𝑡 = 2 ∙ 𝑝1 ∙ 𝑢𝑐 ∙ 𝑣𝑐 + 𝑝2 ∙ (𝑟 2 + 2 ∙ 𝑢𝑐2 ) (3.14)

𝛿𝑣 𝑡 = 2 ∙ 𝑝2 ∙ 𝑢𝑐 ∙ 𝑣𝑐 + 𝑝1 ∙ (𝑟 2 + 2 ∙ 𝑣𝑐2 ) (3.15)

where 𝑝1 and 𝑝2 are the coefficients for tangential distortion

The distortion computation usually results in a non-integral pixel position lying between

discrete pixels of the distorted image (the matrix A) [40][42]. It is necessary to estimate the

unknown pixel amplitude from its neighbors using image interpolation [40][42]. The nearest

neighbor interpolation is the most basic interpolation method, where each pixel is given the value

of the sample closest to it [43]. The nearest neighbor interpolation requires the least processing

time and preserves some high frequency signals, but cannot preserve subpixel image relations,

20
according to Ref.[43][44]. Bilinear interpolation considers the closest 2x2 neighborhood of known

pixel values surrounding the unknown pixel, and then takes a weighted average of these four pixels

to arrive at its final interpolated value[45]. This results in much smoother looking images than the

nearest neighbor method, but also results in strong attenuation of high frequencies [44][45].

Ref.[46] has implemented the Gaussian interpolation, which can achieve sub-pixel precision, to

process the images from particle image velocimetry and particle tracking velocimetry. Star signals

usually spread over a small pixel area and have much higher pixel values in comparison to the

background noises. Thus, they are considered high frequency signals because the pixel values are

rapidly changing in space.

Algorithm processing speed is a critical factor for a star tracker's performance at high rotation

rates, with star images being smeared on the sensor by relative motion. As a result, the nearest

neighbor interpolation is selected for implementation here. After all pixels in the matrix B are

assigned with values through interpolation, the image processing of the reference image is

complete, and the matrix B is used to stack with other processed images.

Misalignments between the reference camera and the non-reference cameras must also be

accommodated. As a result, in addition to the distortion correction, the projective transformation

is applied for the non-reference images, as follows:

(1) A raw distorted and misaligned image from a non-reference camera i is obtained and saved

into a matrix 𝐶𝑖 .

(2) An empty matrix 𝐷𝑖 is created to represent the aligned and undistorted image.

(3) The UV frame coordinate (𝑢1 , 𝑣1 ) of a pixel from j row k column of the matrix 𝐷𝑖 is

calculated.

21
 (4) The coordinate (𝑢2 , 𝑣2 ) of the corresponding misaligned but undistorted pixel relative to

the principal point is calculated using the projective transformation.

(5) The UV frame coordinate (𝑢𝑑 , 𝑣𝑑 ) of the corresponding misaligned and distorted pixel on

the matrix 𝐶𝑖 is calculated using the tangential and radial distortions model.

(6) The value for the j row k column pixel on the matrix 𝐶𝑖 is computed by interpolation of

pixels on the matrix 𝐷𝑖

The UV frame coordinate (𝑢1 , 𝑣1 ) of a pixel from j row k column of the matrix 𝐷𝑖 is

calculated by:

(𝑢1 , 𝑣1 ) = (𝑘 + 0.5, 𝑗 + 0,5) ∙ 𝑝 (3.16)

Since all stars can be considered infinitely far away from the star tracker, translations between

reference camera and non-reference cameras can be ignored. When only the rotations between the

cameras are considered, point correspondences between the images are related by an explicit one-

to-one projective transformation given by [47]:

𝑟11 (𝑢0 − 𝑢1 ) + 𝑟12 (𝑣0 − 𝑣1 ) + 𝑟13 ∙ 𝑓1 (3.17)

𝑢2 = −𝑓2
𝑟31 (𝑢0 − 𝑢1 ) + 𝑟32 (𝑣0 − 𝑣1 ) + 𝑟33 ∙ 𝑓1

𝑟21 (𝑢0 − 𝑢1 ) + 𝑟22 (𝑣0 − 𝑣1 ) + 𝑟23 ∙ 𝑓1

𝑣2 = −𝑓2
𝑟31 (𝑢0 − 𝑢1 ) + 𝑟32 (𝑣0 − 𝑣1 ) + 𝑟33 ∙ 𝑓1
(3.18)

where 𝑓2 is the focal length of the non-reference camera, 𝑓1 is the focal length of the reference

camera, (𝑢0 , 𝑣0 ) is the principal point of the reference camera. The (𝑢2 , 𝑣2 ) coordinate is

determined relative to the principal point of the non-reference camera. The rotation matrix from

the reference camera to the non-reference camera is given by:

𝑟11 𝑟12 𝑟13 (3.19)
𝑟
𝑅 = [ 21 𝑟22 𝑟23 ]
𝑟31 𝑟32 𝑟33

22
 After the projective transformation, (𝑢𝑑 , 𝑣𝑑 ) can be calculated by replacing (𝑢𝑐 , 𝑣𝑐 ) in

Eq(3.12)~(3.15) with (𝑢2 , 𝑣2 ) . When calculating Eq(3.12) for a non-reference camera, the

principal point of the non-reference should be used instead of (𝑢0 , 𝑣0 ). After all pixels in the

matrix 𝐷𝑖 are assigned with values through interpolation, the image processing of the non-

reference image is complete. The image stacking is simply averaging the pixel values from matrix

B and 𝐷𝑖 (i = 1, 2, 3, …).

For illustrative purposes, a simulated example of the image processing results for a two-

camera array is shown in Figure 3.5. The same process is applied to higher dimensional arrays.

From the top to the bottom, the first image is a raw image taken by the reference camera, the

second image is a raw image taken by a non-reference camera, and the third image is the final

stacked image. The X and Y axes are the pixel coordinates, and the grey scale bars represent the

pixel values. All images are generated by a star image simulator, which will be discussed in

Chapter 4. Stars A, B, C, D, E can be seen at the different locations in the reference and non-

reference images due to misalignments and different intrinsic camera parameters. After the images

are processed, the two images are aligned and combined into a single stacked image.

23
 Figure 3.5 A reference image (top), a non-reference image (middle), and a stacked image (bottom).

Stars A, B, C, D, E appear at the different locations in the reference image and the non-reference image,

but they are eventually aligned in the stacked image.

24
3.4 Centroiding
After the camera array images are stacked, the centroiding process is applied to find the

location of individual stars. The optics of the star trackers are often defocused to spread incoming

star light over several pixels, as shown in Figure 3.6. This facilitates the calculation of the star

centroid on the imaging sensor with subpixel accuracy [16].

In Ref.[16], a pixel is considered as illuminated if its readout value is above a given threshold.

A region of interest (ROI) window is aligned with the detected pixel in the center. The average

pixel value on the border of the ROI is calculated and subtracted from all pixels within the ROI.

A center of mass calculation is then applied to the subtracted pixels in the ROI to obtain the star's

centroid. Ref.[16] also pointed out that centroiding accuracy is proportional to the square root of

the number of photoelectrons generated by the star. Accuracy is also dependent on the sensor noise

level.

Figure 3.6 A defocused star image. There are eleven contiguously lit pixels with different intensity

values. The centroid can be calculated with subpixel accuracy by using the distribution of light intensity.
25
 Based on the ROI method from Ref.[16], Ref.[48] proposed a faster centroid algorithm. This

faster algorithm utilizes the centroids from the previous two images to predict the location of the

stars on the current image. Then a windowed centroid code uses the predicted locations to obtain

the new centroids in the current image. After analyzing test data from an air bearing validation,

Ref.[48] concluded that this method is about ten times faster than the conventional full frame

method for an ROI with 80~100 pixels.

In addition to the ROI method, Ref.[3][49] applied the region growing algorithm before the

center of mass calculation. The region growing algorithm scans over the entire image searching

for pixels brighter than a pre-determined threshold. Once a pixel is found, the algorithm determines

the area connected to this given pixel by checking its eight neighboring pixels for values higher

than the threshold. The same region growing process is also applied to the neighboring pixels.

After a region is identified, the number of contiguously lit pixels in this region is calculated. The

number needs to be higher than corresponding thresholds to filter out hot pixels (pixels producing

higher readout values than others in the sensor for the same illumination levels) and cosmic ray

hits (pixels energized by relativistic particles). After all individual star regions are identified, the

centroid of each region is calculated by the center of mass method given by [49]:

(∑𝑛𝑖=1 𝑥𝑖 × 𝐼𝑖 , ∑𝑛𝑖=1 𝑦𝑖 × 𝐼𝑖 ) (3.20)

(𝑥, 𝑦) =
∑𝑛𝑖=1 𝐼𝑖

where (𝑥, 𝑦) is the coordinate of the star centroid, (𝑥𝑖 , 𝑦𝑖 ) is the coordinate of the i th pixel in

the region, 𝐼𝑖 is the intensity of the i th pixel in the region, and n is the number of lit pixels in this

region.

The use of a focused optic is an alternative approach to simplify star centroiding. Ref.[50]

explained that, despite providing subpixel accuracy, there are two disadvantages of using a

defocused optic: First, more computational power is required and second, spreading a star over
26
several pixels results in a lower SNR. One example is the ST-200 star tracker [50]. In each sensor

row of the ST-200, only the brightest pixel is counted and saved into a table. This method

significantly speeds up the star centroiding process and reduces the required computational power

by a factor of 1000. To correct hot pixels/cosmic ray hits, the ST-200 regularly saves a standard

sensor image as a reference image and subtracts it from all the following images. The centroiding

is conducted on the subtracted images.

The detection and accurate centroiding of dim stars is challenging. The ST-16 star tracker

struggled with these issues [26] . The ST-16 utilizes a relatively complicated star detection logic

that includes a moving average filter used to estimate the local background level. It also sets a

threshold to define the minimum intensity for an illuminated pixel, a blob size that defines the

minimum number of contiguous bright pixels that each candidate star must possess, and an

integrated candidate star intensity threshold.

Still, the detection routines failed to detect many dim stars that were distinguishable by eye.

The developers of the ST-16 were forced to retune the detection parameters by using downloaded

images. For centroiding, the ST-16 was utilizing a 4-connectivity method to identify pixels in

candidate star region in an attempt to reduce the contributions from image noise sources. However,

for dim stars detected under non-ideal conditions, only an asymmetric portion of the actual star

was identified, which negatively impacted centroiding accuracy. Ultimately, the ST-16 was

modified to compute the centroid using all pixels within a circular window centered on the initial

centroid estimate.

The major focus of star detection/centroiding research has been to tune detection parameters

for more reliable detections, improve centroiding accuracy, filter outliers, and to generally speed

up the required computations. Many novel approaches are proposed to tackle these problems.

27
 Ref.[51] discussed the systematic errors of the conventional center of mass centroiding

method and proposed a compensation for these errors. Ref.[52] proposed a weighted iterative

threshold algorithm to find the optimal threshold for star detection. Ref.[53] proposed a novel

architecture for star detection and centroid calculation that would speed up the process and enable

a higher update rate.

Here, the region growing algorithm is implemented for its robustness. The center of mass

method given by Eq.(3.20) is also selected for centroid calculation. The detection threshold is

calculated by [16]:

1 (3.21)
𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 = 𝐴𝑝𝑖𝑥𝑒𝑙 + 5 ∙ 𝜎𝑝𝑖𝑥𝑒𝑙 ∙ −𝑥 2 +𝑦 2
1 1 1
∫0 ∫0 2 ∙ 𝜋 ∙ 𝜎 𝑒 2∙𝜎𝑃𝑆𝐹 𝑑𝑥𝑑𝑦
𝑃𝑆𝐹

where 𝐴𝑝𝑖𝑥𝑒𝑙 is the mean pixel values of the stacked image, 𝜎𝑝𝑖𝑥𝑒𝑙 is the standard deviation

of the pixel values in a dark frame, and 𝜎𝑃𝑆𝐹 is the point spread function (PSF) radius in pixels

(assuming a Gaussian PSF).

3.5 Star Identification

A star identification algorithm constructs body-frame vectors from star centroids by using

Eq.(3.1) and finds corresponding inertial-frame vectors from a star pattern database that is

constructed from a star catalog, such as the Hipparcos catalog [54]. To match the body-frame

vectors to the inertial-frame vectors in a star pattern database, certain features must be extracted

from the body-frame vectors [55]. Depending on how the features are extracted, two star

identification methods are available from the literature. These methods are the subgraph

isomorphism-based feature extraction method and the pattern-based feature extraction method [56].

28
 The first method treats stars as vertices in a subgraph, utilizing the radial distances, angular

separations, area, and other parameters as the features for comparison and matching. To identify

the stars, an isomorphic subgraph has to be found in a star pattern database. The second method

assigns each star to a pattern based on its relative positioning to neighboring stars and tries to find

the closest match to the measured pattern in a star pattern base.

Star identification algorithms can also be categorized into either "lost-in-space" algorithms

or recursive algorithms [55]. The lost-in-space algorithms identify stars with no information

regarding the attitude of the spacecraft, while the prior attitude information is available for the

recursive algorithms. The lost-in-space operation mode of the star tracker is more critical, and

some star trackers, such as the ST-16RTS star tracker, only utilize the lost-in-space algorithms and

generate full lost-in-space solutions for each frame [7]. Accordingly, this section focuses on the

lost-in-space algorithms.

As mentioned in Ref.[55], development of star identification techniques tends to focus on

speeding up computation time. Ref.[57] was the first to implement a binary search tree to search

in the star pattern database. A "k-vector" method approach is proposed by Ref.[58] for the "Search-

Less Algorithm," which allows it to search the database in an amount of time independent of the

size of the database. As pointed out by Ref.[56], the best performance is achieved by deep learning

solutions because a search is eliminated and the complexity remains constant regardless of the size

of the problem. However, the neural network itself takes up a significant amount of memory space

in a star tracker.

The triangle algorithm is a well-known subgraph isomorphism-based feature extraction

algorithm [59], as illustrated in Figure 3.7. It uses the two stars closet to a given star to form a

pattern. The angular separations from the given stars to the two closet stars and the angle between

29
them are used as the search parameters. The pyramid algorithm is an improvement based on the

triangle technique, which utilizes the three closest neighbors to a given guide star for feature

extraction [60]. The pyramid algorithm can robustly reject false stars and has been successfully

tested using an image containing five real stars and 63 false stars.

Figure 3.7 The pattern used by the triangle algorithm. Two inter-star angles (A and B), and one

interior angle (C) are used.

However, both methods rely on the nearest stars to form a geometric feature. In non-ideal,

low SNR application environments, of which low-cost star trackers designed for small satellites

tend to be, these nearest neighbor stars could be missed and not captured in the image [61]. Also,

inaccurate camera calibrations can cause variations in the inter-star angles [55]. This makes it

difficult to implement the above-mentioned algorithms on low-cost star trackers that do not have

a budget for fine temperature control systems and high-end calibration equipment. To solve this

30
problem, a non-dimensional star identification algorithm is proposed by Ref.[62]. The non-

dimensional star identification method can be used to identify the stars captured by uncalibrated

cameras or when dealing with inaccurate parameter estimations. The stars' triangle interior angles

are used in this method and they are, to first order, independent of both the focal length and optical

axis offsets.

The majority of research to date has been focused on the pattern-based feature extraction

methods [56]. The singular value method, the modified grid algorithms, the log-polar transform

method, the adaptive ant colony algorithm, the image-based method, the deep learning method,

and many other methods have been proposed to improve the algorithms' performance in terms of

dealing with magnitude and positional noise [56]. To deal with low SNR application environments,

some algorithms also utilize more complex solutions involving iterative validation. Some of these

novel approaches are restricted by computing resources, which would require the application of

more powerful space avionic to resolve.

The geometric voting algorithm, which is a type of subgraph isomorphism-based feature

extraction algorithm, is implemented here for its robustness against false stars and low SNR

application environments [61][63]. The angular distances between all pairs of stars from the

Hipparcos star catalog are calculated and saved into a star pattern database. The database is sorted

by the angular distance value, and every row in the database contains an angular distance 𝑑𝑝𝑞 and

the identities of the related star pair. For each star centroid on the focal plane, a body-frame star

vector 𝑠𝑖 can be constructed using Eq.(3.1). The angular distance for a pair of body-frame vectors

can be computed by:

𝑑𝑖𝑗 = 𝑠𝑖𝑇 𝑠𝑗 (3.22)

31
 Given a threshold ϵ, a binary search is carried to find every pair of stars p and q in the star

pattern database whose angular distance satisfies:

𝑑𝑖𝑗 − 𝜖 ≤ 𝑑𝑝𝑞 ≤ 𝑑𝑖𝑗 + 𝜖 (3.23)

Both image stars i and j will receive votes from database stars p and q as possible identities

for them. Once all pairs of image stars have been processed, each image star is assigned to the

database star with the highest votes. After the voting is over, an extra verification step is conducted.

Image stars i and j are matched to database stars with inertial-frame (J2000 ECI) vectors 𝑣𝑖 and 𝑣𝑗 ,

whose angular distance can be found by:

𝑟𝑖𝑗 = 𝑣𝑖𝑇 𝑣𝑗 (3.24)

We then check whether:

𝑑𝑖𝑗 − 𝜖 ≤ 𝑟𝑖𝑗 ≤ 𝑑𝑖𝑗 + 𝜖 (3.25)

which represents whether the angular distance between the image stars is close to the angular

distance between their matched database stars. When the distance is within the threshold, the two

matched database stars get votes because they are likely to be correctly identified. If the number

of votes for a matched star is close to the maximal number of votes among all matched stars, the

star identification for this matched star is assumed to be correct.

3.6 Attitude Determination

The attitude determination algorithm calculates an attitude quaternion using the identified star

vectors given by the star identification algorithm. The attitude quaternion represents a rotation

from the star tracker body frame to the J2000 ECI frame. The attitude determination methods fall

into two categories: the point-by-point deterministic methods and the recursive methods. The

deterministic methods require at least two vector observations at a single point in time to calculate

32
the attitude quaternion. The recursive methods utilize observations obtained over a short period of

time. The deterministic solutions are simpler and straight forward to implement and test [64].

The deterministic solutions are especially useful for star trackers because a tracker computes

a quaternion from multiple star vectors. The quaternion output from the star tacker can be used as

an input to a recursive method such as the extended Kalman filter (EKF) [65]. Among the

deterministic methods available, a quaternion estimator (QUEST), a fast optimal attitude matrix

(FOAM), and estimators of the optimal quaternion (ESOQ and ESOQ2) are referred to as fast

attitude estimation algorithms [66].

Of the deterministic methods, Davenport's original q-method and Markley's singular value

decomposition (SVD) algorithm are numerically more robust [66]. The SVD method is selected

for its robustness and simplicity in this thesis [67]. In order to find the transformation matrix A

from the star tracker body frame to the J200 ECI frame, we first compute matrix B given by:
𝑛
(3.26)
𝐵 = ∑ 𝑏𝑖 𝑟𝑖𝑇
𝑖=1

where bi is a body-frame vector constructed by Eq(3.1), and riT is the transpose of a database

inertial-frame vector assigned to this body-frame vector. The SVD of B is given by:

𝐵 = 𝑈𝑆𝑉 𝑇 (3.27)

Now we can obtain the transformation matrix A given as:

1 0 0 𝐴11 𝐴12 𝐴13 (3.28)

𝑇
𝐴 = 𝑈 [0 1 0 ] 𝑉 = [𝐴21 𝐴22 𝐴23 ]
0 0 (det𝑈)(det𝑉) 𝐴31 𝐴32 𝐴33

If 𝐴33 < 0 and 𝐴11 > 𝐴22 , then an attitude quaternion q can be calculated as:

𝑞 = [1 + 𝐴11 − 𝐴22 − 𝐴33 𝐴12 + 𝐴21 𝐴31 + 𝐴13 𝐴23 − 𝐴32 ]T (3.29)

If 𝐴33 < 0 and 𝐴11 ≤ 𝐴22 , an attitude quaternion q can be calculated as:

33
 𝑞 = [𝐴12 + 𝐴21 1 − 𝐴11 + 𝐴22 − 𝐴33 𝐴23 + 𝐴32 𝐴31 − 𝐴13 ]T (3.30)

If 𝐴33 ≥ 0 and 𝐴11 < −𝐴22 , an attitude quaternion q can be calculated as:

𝑞 = [𝐴31 + 𝐴13 𝐴23 + 𝐴32 1 − 𝐴11 − 𝐴22 + 𝐴33 𝐴12 − 𝐴21 ]T (3.31)

If 𝐴33 ≥ 0 and 𝐴11 ≥ −𝐴22 , an attitude quaternion q can be calculated as:

𝑞 = [𝐴23 + 𝐴32 𝐴31 − 𝐴13 𝐴12 − 𝐴21 1 + 𝐴11 + 𝐴22 + 𝐴33 ]T (3.32)

Having derived the algorithms required for image processing, star detection, centroiding, and

attitude determination, the following chapter provides a demonstration of the simulated software

system and its attitude determination performance.

34
 CHAPTER 4: SOFTWARE SIMULATION AND HARDWARE
VERIFICATION

4.1 Star Tracker Software Simulator

A star tracker software simulator was developed to test the algorithms by generating star

images for a simulated array of sensors. The flowchart of the star tracker simulator is provided in

Figure 4.1. Given a quaternion input, the star tracker's boresight vector is rotated into the J2000

ECI frame and compared with star vectors. Star vectors within the FOV are projected onto the

image plane. The star light is defocused to form star images. Several layers of noises are added.

The distortion is applied. The sensor pixel values are quantized by an analog-to-digital converter

(ADC) and a simulated star image is generated. The following section describes the details of the

star tracker software simulation.

Figure 4.1 Flowchart of the star tracker simulator. Star vectors within the tracker's FOV are

projected onto the sensor image plane. Defocusing, noises, distortion, and quantization are applied to

generate a simulated star image.

35
 Given a quaternion input, an inertial star tracker boresight vector 𝑧𝑖 is calculated. As shown

in Figure 3.2, the Z axis of the star tracker body frame is aligned with the boresight of the star

tracker. An attitude quaternion 𝑞 = [𝑞1 𝑞2 𝑞3 𝑞4 ]𝑇 representing the rotation from the star

tracker body frame to the J2000 ECI frame is given as an input to the simulator (𝑞1 is the scalar

part). The inertial boresight vector 𝑧𝑖 is obtained by:

0 (4.1)
𝑧𝑖 = 𝐴𝑖𝑏 ∙ [0]
1

where 𝐴𝑖𝑏 is the rotation matrix from the star tracker body frame to the J2000 ECI frame:

𝑞12 + 𝑞22 − 𝑞32 − 𝑞42 2 ∙ (𝑞2 𝑞3 − 𝑞1 𝑞4 ) 2 ∙ (𝑞2 𝑞4 + 𝑞1 𝑞3 ) (4.2)

𝐴𝑖𝑏 = [ 2 ∙ (𝑞2 𝑞3 + 𝑞1 𝑞4 ) 𝑞12 − 𝑞22 + 𝑞32 − 𝑞42 2 ∙ (𝑞3 𝑞4 − 𝑞1 𝑞2 ) ]
2 ∙ (𝑞2 𝑞4 − 𝑞1 𝑞3 ) 2 ∙ (𝑞3 𝑞4 + 𝑞1 𝑞2 ) 𝑞12 − 𝑞22 − 𝑞32 + 𝑞42

A star vector 𝑠𝑖 under the J2000 ECI frame is picked from the star catalog. The Hipparcos

catalog from the European Space Agency (ESA) is used [54]. The angle between 𝑠𝑖 and 𝑧𝑖 is

calculated from the dot product of the two vectors. If the calculated angle is larger than a

predetermined FOV angle of the star tracker, the star is considered to be out of the FOV, and

another star vector is picked from the catalog. Otherwise, the star is seen by image sensor.

Based on Eq(3.1), The real centroid coordinate of a star on the sensor focal plane can be

calculated as:

𝑥𝑓
𝑢 𝑢 0 −
[ ]= [ 𝑧]
𝑣 𝑦𝑓 (4.3)
𝑣0 −
𝑧

where x, y, and z are three components of a star vector s under the star tracker body frame

that can be obtained by 𝑠 = 𝐴𝑇𝑖𝑏 ∙ 𝑠𝑖 . The photons collected on the focal plane from a magnitude m

star can be calculated using Eq(3.5). A star's image is defocused by the optics and spreads over

36
several pixels. A point spread function (PSF) is selected to model the defocusing of the star light.

The PSF of an ideal lens can be described by an Airy Disk [68]. Alternatively, a Gaussian

distribution can also be used as a good approximation of the actual PSF [69]. There are two typical

types of PSF models: integrated PSF (IPSF) and simple PSF (SPSF) [68]. The SPSF is usually

preferred for its simplified computation. The SPSF used here is given by [70].

1 (𝑖 − 𝑥𝑐 )2 + (𝑗 − 𝑦𝑐 )2 (4.4)
𝑔(𝑖, 𝑗) = 𝑒𝑥𝑝 [− ]
2𝜋𝜎 2 2𝜎 2

where (𝑖, 𝑗) is the coordinate of an arbitrary pixel, 𝑔(𝑖, 𝑗) is the probability that a photon will

fall onto pixel (𝑖, 𝑗), (𝑥𝑐 , 𝑦𝑐 ) is the coordinate of the center of a star image, 𝜎 is the Gaussian radius

describing the size of the defocused image.

Therefore, the photons collected on an arbitrary pixel coordinate (𝑖, 𝑗) is given by:

𝑃ℎ𝑜𝑡𝑜𝑛𝑠 = 𝑁𝑝ℎ ∙ 𝑔(𝑖, 𝑗) (4.5)

Then, shot noise, as described in Section 3.1, is added to this raw photon counts:

𝑃ℎ𝑜𝑡𝑜𝑛𝑠 = 𝑃ℎ𝑜𝑡𝑜𝑛𝑠 + √𝑃ℎ𝑜𝑡𝑜𝑛𝑠 ∙ 𝑛𝑢𝑚𝑝𝑦. 𝑟𝑎𝑛𝑑𝑜𝑚. 𝑛𝑜𝑟𝑚𝑎𝑙(0,1) (4.6)

where 𝑛𝑢𝑚𝑝𝑦. 𝑟𝑎𝑛𝑑𝑜𝑚. 𝑛𝑜𝑟𝑚𝑎𝑙(0,1) is a Python function that generates random numbers

from a Gaussian distribution with zero mean and a standard deviation equal to one. After shot

noise is added, the electrons collected on this pixel is calculated based on Eq(3.7):

𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 = 𝑃ℎ𝑜𝑡𝑜𝑛𝑠 ∙ 𝑄𝐸 ∙ 𝐾𝑓𝑖𝑙𝑙 ∙ 𝐾 (4.7)

The next step is to apply the lens distortion to the undistorted image. As describe in Section

3.3, the forward distortion model maps an undistorted pixel to a distorted pixel. To generate

distortions for simulated images, the corresponding inverse distortion mapping is required. The

inverse distortion maps a distorted pixel to an undistorted pixel. If both radial and tangential

distortion components are considered, there is no analytical solution to the inverse mapping [41].

37
Ref.[71] proposed an iterative approach to approximate the inverse mapping. Ref.[41] used a

polynomial approximation method to solve the back-projection problem.

The residual errors of both methods are calculated and shown in Figure 4.2. The color bars

represent the residual errors in pixels. The residual error of the iterative method is so small that no

gradient can be seen in the image. However, the polynomial method can be computed more quickly,

and its maximum residual error is less than 0.1 pixel. As a result, the polynomial method is selected

here to simulate lens distortion. Similar to the image processing algorithm application discussed

in Section 3.2, the nearest neighbor interpolation is also applied here.

Figure 4.2 Residual error of the iterative method (top) and the polynomial method (bottom). The

residual error of the iterative method is significantly smaller than that of the polynomial method resulting

in a flat error field.

38
 Next, the dark current and readout noise, as explained in Section 3.1, are added to every pixel

in the distorted image:

𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 = 𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 + √𝑁𝑑𝑎𝑟𝑘 ∙ 𝑇 ∙ 𝑛𝑢𝑚𝑝𝑦. 𝑟𝑎𝑛𝑑𝑜𝑚. 𝑟𝑎𝑛𝑑(1)

(4.8)
+ 𝑁𝑟𝑒𝑎𝑑 ∙ 𝑛𝑢𝑚𝑝𝑦. 𝑟𝑎𝑛𝑑𝑜𝑚. 𝑟𝑎𝑛𝑑(1)

where 𝑛𝑢𝑚𝑝𝑦. 𝑟𝑎𝑛𝑑𝑜𝑚. 𝑟𝑎𝑛𝑑(1) is a Python function that generates random numbers from

a uniform distribution over (0,1). Since any analog-to-digital converter (ADC) has a finite number

of bits, quantization needs to be applied to the electron counts [34]:

2𝑁𝑏𝑖𝑡𝑠 − 1 𝐹𝑊𝐶
𝑏𝑖𝑡𝑠 = 𝑚𝑎𝑡ℎ. 𝑓𝑙𝑜𝑜𝑟(𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 ∙ )∙ 𝑁 (4.9)
𝐹𝑊𝐶 2 𝑏𝑖𝑡𝑠 − 1

where 𝑚𝑎𝑡ℎ. 𝑓𝑙𝑜𝑜𝑟() is a Python function that rounds a number into an integral; 𝑁𝑏𝑖𝑡𝑠 is the

resolution of the ADC; 𝐹𝑊𝐶 is the full well capacity of the image sensor in electrons.

The effects of defocusing and distortion are shown in Figure 4.3. The image at the top is a

simulated image after the "Electrons Calculation" process. In this image, every star is spread over

several pixels due to the optics defocusing. The shot noise is added, and photon counts are

converted into electron counts. The image in the middle is a simulated image after the "distortion"

process is applied. Star A is stretched in the bottom image due to distortion effects. Star group B

is "pushed away" from the center of the image by distortion and disappears in the bottom image.

The effects of adding noise and quantization are shown in Figure 4.4. The image at the top is

a simulated image after the distortion process is applied. The image at the bottom is a simulated

image after quantization. The dark current and readout noise are added to the image at the bottom.

Also, the image at the bottom is quantized into 0~255 bit counts to mimic an 8-bit ADC. As a

result, the star group A is barely distinguishable from the background noise after noise adding and

quantization.

39
 Figure 4.3 A simulated image after "Electrons Calculation" (top) and a simulated image after

"Distortion" (bottom). The star A is stretched in the bottom image due to distortion effects. The star group

B is "pushed away" from the center of the image by distortion and disappears in the bottom image.

40
 Figure 4.4 Simulated image after "distortion" (top) and a simulated image after quantization

(bottom). The star group A is barely distinguishable from the background noise after quantization.
41
4.2 Software Simulation Results and Analysis
Using the star tracker simulator described in the previous section, software simulation is

conducted to verify the proposed algorithms. The simulation is run for a one-camera star tracker,

a two-camera star tracker, a three-camera star tracker, and a four-camera star tracker. Each

configuration shares some of the intrinsic parameters, as listed in Table 4.1.

Table 4.1 Intrinsic camera parameters are the same for all cameras.

pixel sizer (MT9V022) = 6 μm 𝑁𝑑𝑎𝑟𝑘 = 500 𝑒 −

height (MT9V022) = 480 pixels 𝑁𝑟𝑒𝑎𝑑 = 500 𝑒 −

width (MT9V022) = 640 pixels 𝑇 =1s

𝐴𝑙 (LS12020) = 1.33 cm2 𝐹𝑊𝐶 = 8500 𝑒 −

𝑇𝑙 = 0.88 𝑁𝑏𝑖𝑡𝑠 = 8

∆B = 3000 Å QE = 30%

𝜎 = 0.85 pixels

First, parameters are defined for all cameras used in the simulations with the same focal length,

principal point offset, and lens distortion (𝑘1 , 𝑘2 , 𝑝1 , 𝑝2 ), as listed in Table 4.2. All cameras are

assumed to be perfectly aligned. The simulation is designed to verify the performance of the

algorithms under ideal conditions where no projective transformation is needed, and all images

have the same distortion.

Table 4.2 Camera parameters used in the simulation under ideal conditions

𝑓 (LS12020) = 12.0 mm principal point offset (U) = 0.9 pixels

pprincipal point offset (V) = −2.0 pixels 𝑘1 = 0.01

𝑘2 = 0.01 𝑝1 = 0.00

𝑝2 = 0.00

42
 The standard deviation (SD) of the stacked images (in bit counts) and the root mean square

(RMS) centroid errors for U axis and V axis (in pixels) for star trackers with different camera array

sizes are listed in Table 4.3. SDs of the raw images are 4.6582, 4.6739, 4.6653, and 4.6742 in bit

counts. RMS centroid errors are calculated from 30~40 star centroiding results.

The distortion correction process increases the noise level of the image, as shown in the SD

result for the one-camera array star tracker. The SD of the stacked image decreases as more

cameras are added to the camera array, proving that image stacking can reduce the noise level of

the image.

Results of centroid errors are more complicated. RMS centroid error is obviously not a linear

function of the number of cameras in the array. Although the cameras in this simulation are

perfectly aligned and share the same intrinsic parameters, image processing steps, such as image

interpolation and distortion correction (since inverse mapping is an approximation, as discussed in

Section 4.1), can bring unexpected impacts to the centroiding accuracy. On the U axis, the four-

camera configuration has the lowest RMS centroid error. On the V axis, the conventional one-

camera configuration has the lowest RMS centroid error. In general, the four-camera configuration

has the best centroiding accuracy performance.

Table 4.3 Stacked images SD and RMS centroid errors of the simulation under ideal conditions

SD of Image RMS Centroid RMS Centroid Mean Centroid

(bit counts) Error U (pixels) Error V (pixels) Error (pixels)

One-Camera 5.1395 0.5293 0.4724 0.5009

Two-Camera 4.4639 0.5216 0.4828 0.5022

Three-Camera 4.2541 0.5239 0.5121 0.5180

Four-Camera 4.1990 0.5047 0.4918 0.4983

43
 The noise equivalent angle (NEA) is the star tracker's capability to reproduce the same

attitude given the same optical setting [16]. It is possible to estimate the cross-boresight NEA of a

star tracker by [16]:

𝐴 ∙ 𝐸centroid
𝐸cross−boresight = (4.10)
𝑁pixel ∙ √𝑁star

where 𝐴 is the circular FOV of the star tracker (13.6855° using f from Table 4.2). 𝐸centroid

is the average centroiding accuracy in pixels. 𝑁pixel is the number of pixels across the focal plane

(480 by Table 4.1). 𝑁star is the average number of stars detected by the star tracker. The roll NEA

can be calculated by [16]:

𝐸centroid 1
𝐸roll = atan ( )∙ (4.10)
0.3825 ∙ 𝑁pixel √𝑁star

𝐸cross−boresight and 𝐸roll can be calculated and compare with the results from a dynamic

simulation under ideal conditions (Table 4.2). The dynamic simulation generates star images at

thirty different attitude orientations. 𝐸cross−boresight , 𝐸roll , the average attitude estimation errors,

and the average numbers of star detected are listed in Table 4.4. The star detectability is improved

by adding more cameras to the array. This agrees with the theoretical analysis presented in Section

3.1.

For 𝐸cross−boresight and 𝐸roll , although the centroiding accuracy could be negatively affected,

the improvement of the star detectability by image stacking results in reducing NEAs. The attitude

44
estimation error shows little deviation after adding cameras to the array. In conclusion, the four-

camera configuration demonstrates much improved star detectability and sufficiently high

accuracy (the stated accuracy objective is 0.1° or 360 arcseconds) in this simulation for use on a

CubeSat.

Table 4.4 NEAs and the results from the dynamic simulation under ideal conditions

Average Attitude
𝑬𝐜𝐫𝐨𝐬𝐬−𝐛𝐨𝐫𝐞𝐬𝐢𝐠𝐡𝐭 𝑬𝐫𝐨𝐥𝐥 Average Number of
Estimation Error
(arcsecond) (arcsecond) Stars Detected
(arcsecond)

One-Camera 11.1915 122.4948 72.2269 21.1

Two-Camera 10.0322 109.8060 72.6635 26.4

Three-Camera
10.2512 112.2031 72.4357 26.9

Four-Camera 9.7345 106.5477 72.3475 27.6

Then, another "worst case" simulation is run for cameras with different focal lengths,

principal point offsets, lens distortion coefficients, and misalignments, as listed in Table 4.5. The

misalignment is a quaternion representing the rotation from a non-reference star tracker body

frame to the reference star tracker body frame. Camera One is selected as the reference camera.

The simulation is designed to verify the performance of the algorithms under non-ideal conditions

where projective transformation is required for each non-reference camera, and all camera have

different distortions.

45
 Table 4.5 Camera parameters for the simulation under non-ideal conditions

Camera One Camera Two Camera Three Camera Four

𝒇 (𝐦𝐦) 12.0 12.1 12.2 11.9

𝐏𝐫𝐢𝐧𝐜𝐢𝐩𝐚𝐥 𝐏𝐨𝐢𝐧𝐭
1.0 0.9 -1.1 -0.8
𝐎𝐟𝐟𝐬𝐞𝐭 𝐔 (𝐩𝐢𝐱𝐞𝐥𝐬)

𝐏𝐫𝐢𝐧𝐜𝐢𝐩𝐚𝐥 𝐏𝐨𝐢𝐧𝐭
2.0 -2.0 1.2 0.9
𝐎𝐟𝐟𝐬𝐞𝐭 𝐕 (𝐩𝐢𝐱𝐞𝐥𝐬)

𝒌𝟏 0.0100 0.0090 0.0200 0.0120

𝒌𝟐 0.0100 0.0150 0.0110 0.0080

𝒑𝟏 0.0000 0.0001 0.0002 0.0001

𝒑𝟐 0.0000 0.0001 0.0001 0.0000

0.99954136, 0.99954136, 0.99988555,

0.01755567, 0.01755567, 0.00875214,

Misalignment
0.01714017, 0.01714017, 0.00864942,

0.01775037 0.01775037 0.00880187

The SD of the stacked images (in bit counts) and the RMS centroid errors for U axis and

V axis (in pixels) for star trackers with different camera array size are listed in Table 4.6. SDs of

the raw images are 4.6609, 4.6986, 4.6737, and 4.6499 in bit counts. RMS centroid errors are

calculated from 30~40 star centroiding results. Compared with Table 4.3, although the noise level

of the raw image is still reduced by image stacking, the improvement is negatively affected by the

more complicated image processing. The centroiding accuracy decreases when there are more than

46
two cameras in the array. Compared with Table 4.3, the average centroid error increases by 7.88%

for the three-camera configuration and by 13.97% for the four-camera configuration.

Table 4.6 Stacked images SD and RMS centroid errors of the simulation under non-ideal conditions

SD of Image RMS Centroid RMS Centroid Mean Centroid

(bit counts) Error U (pixels) Error V (pixels) Error (pixels)

One-Camera 5.1503 0.5486 0.5150 0.5318

Two-Camera 4.6532 0.4797 0.5443 0.5120

Three-Camera 4.5153 0.5128 0.6048 0.5588

Four-Camera 4.3445 0.5398 0.5959 0.5679

The dynamic simulation under non-ideal conditions (Table 4.5) generates star images at

thirty different attitude orientations. 𝐸cross−boresight , 𝐸roll , the average attitude estimation errors,

and the average numbers of star detected are listed in Table 4.7. Compared with Table 4.4, the

average number of stars detected still increases from stacking more images, but the improvement

is smaller due to more complicated image processing. Although centroiding accuracy decreases

when there are more than two cameras in the array according to Table 4.6, results in Table 4.7

show that the centroid error is mitigated by the improvement of star detectability. Both NEA

analysis and attitude estimation errors from the dynamic simulation show that having four cameras

in the array can improve the star detectability as well as the attitude estimation accuracy even under

non-ideal conditions.

47
 Table 4.7 NEAs and the results from the dynamic simulation under non-ideal conditions

Average Attitude
𝑬𝐜𝐫𝐨𝐬𝐬−𝐛𝐨𝐫𝐞𝐬𝐢𝐠𝐡𝐭 𝑬𝐫𝐨𝐥𝐥 Average Number of
Estimation Error
(arcsecond) (arcsecond) Stars Detected
(arcsecond)

One-Camera 11.5331 126.2338 71.9893 22.4

Two-Camera 10.6828 116.9267 73.4630 24.2

Three-Camera 11.4942 125.8079 72.3631 24.9

Four-Camera 11.4537 125.3530 70.1363 25.9

A visualization of the simulation results is shown in Figure 4.5. The four-camera

configuration star tracker demonstrates the best star detectability under both conditions. It has a

sufficient attitude estimation accuracy (72.3475 arcseconds) under ideal conditions, and the best

attitude estimation accuracy (70.1363 arcseconds) under non-ideal conditions.

The numerical simulation confirms the hypothesis described in Section 3.2 that star

detectability increases as more cameras are added. The numerical simulation also shows that the

multi-camera star tracker design can provide accurate attitude determination. 72.3475 arcseconds

and 70.1363 arcseconds are both significantly lower that the stated accuracy objective 0.1° (360

arcseconds). The accuracy performance of the multi-camera star tracker is also comparable to the

higher cost miniature star trackers such as the star tracker used for the Aerospace Corporation's

AeroCube-OCSD CubeSat whose determination accuracy is 0.02° (72 arcseconds) [72].

48
 Figure 4.5 A visualization the of the camera array star tracker simulation results. The four-camera

configuration has the overall best star detectability and accuracy performance.

4.3 Hardware Verification

A hardware prototype is being assembled to verify the performance of the multi-camera star

tracker in a more realistic setting, as shown in Figure 4.6. Four MT9V022 monochrome image

sensors will be used to prove the concept of stacking. The MT9V022 image sensor was also

selected for the star tracker of the AeroCube CubeSat mission [1]. Its performance has been

verified on orbit, and it is available off the shelf for 50 USD. The LS-12020 lens is also used,

without an IR filter to improve star detection. A Raspberry Pi 3 (RPi 3) will be used to process

images from the cameras and run the star tracker algorithms. The RPi 3 is remotely controlled

through a laptop. A DXL360S two-axis digital inclinometer will be used to point the star tracker

towards zenith for initial calibrations.

49
 Figure 4.6 Schema of all the components that will be integrated into a future hardware protype

Star tracker performance verification can be challenging on the ground. Arcsecond level

accuracy test equipment is needed for indoor testing, which is costly and requires clean room

environments for successful implementation. To control verification costs, night sky testing

provides the most cost-effective environment for estimating the relative accuracy of the star tracker.

While subject to weather constraints, another challenge of any night sky test is to verify the

attitude output obtained during the test. Ref.[73] used the estimated attitude from a star tracker to

estimate the local coordinates of the camera on the Earth, which can be used to compare with

latitude and longitude from a GPS to qualitatively verify the performance of the star tracker. The

attitude output of the star tracker can also be transformed into celestial coordinates (i.e., declination,

roll angle, and right ascension). It is possible to estimate the relative accuracy of a star tracker as

the statistical fluctuations of the declination and roll angle or the right ascension when the sidereal

rate has been subtracted [16]. Future night sky tests should verify the simulated accuracy

performance of the star tracker demonstrated herein.

50
 CHAPTER 5: CONCLUSION

The concept of a multi-camera star tracker was proposed to improve star detection sensitivity

while providing high accuracy attitude determination and significantly reducing the integrated

system assembly and test costs. A hypothesis was made that star detectability increases as more

cameras are added to the array. This has been verified through theoretical analysis. An image

processing algorithm was developed to correct distortion, align images, and combine them into

one composite image. Centroiding, star identification, and attitude determination algorithms were

selected from the literature and implemented in a star tracker software simulator. Sensor noises,

lens defocusing, lens distortion, and ADC quantization were also considered. Numerical

simulations were run to verify the performance of the proposed method. With a verified simulation

in hand, a hardware prototype is now under development.

In conclusion, the four-camera configuration star tracker has the best star detectability and

sufficient attitude determination accuracy (72.3475 arcseconds for ideal conditions and 70.1363

arcseconds for non-ideal conditions). The accuracy performance of the four-camera star tracker is

significantly better than the stated accuracy objective of 0.1° (360 arcseconds). This performance

is also comparable to the existing miniature star trackers such as that used for the AeroCube-OCSD

CubeSat whose determination accuracy was 0.02° (72 arcseconds) [72].

For the future work, additional analysis and hardware testing is planned to complete the

verification of the low-cost multi-camera star tracker concept. The impacts of the image processing

algorithm on the attitude determination accuracy and centroiding accuracy need to be further

explored in real hardware. A new centroiding method that can find the optimal detection threshold

should be developed. The implementation of a non-dimensional star identification method could

also be explored since the low-cost star trackers usually come with inaccurate calibrations. The
51
hardware prototype will be used to qualitatively and quantitatively verify the proposed multi-

camera star tracker concept. A flight qualified system will be assembled and flown on a future

CubeSat mission to complete the verification program. If successful, CubeSats will be able to take

advantage of a low-cost sensor for achieving higher levels of pointing accuracy in future payload

applications.

52
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