Attitude Determination and Control Hardware Acceptance ... · PDF fileAbstract Attitude...

Attitude Determination and Control Hardware AcceptanceTesting for Next Generation Microsatellites

by

Young-Young Shen

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of the Institute for Aerospace StudiesUniversity of Toronto

c© Copyright 2015 by Young-Young Shen

Abstract

Attitude Determination and Control Hardware Acceptance Testing for Next Generation

Microsatellites

Young-Young Shen

Master of Applied Science

Graduate Department of the Institute for Aerospace Studies

University of Toronto

2015

In a spacecraft development cycle, components must undergo a rigorous regime of testing,

known as acceptance testing, to ensure that they can operate in the required environment

and at the required level before being accepted for spaceflight. This thesis discusses the

acceptance testing of attitude determination and control hardware using these procedures

for four satellite missions currently under development at the time of writing in the

Space Flight Laboratory at the University of Toronto Institute for Aerospace Studies.

Focus is placed on improvements made to the acceptance testing procedures and analysis

methods for the hardware. The key contributions of the thesis are improvements to

the magnetometer functional test and calibration, rate sensor calibration, sun sensor

calibration, and reaction wheel long-form functional test. This work demonstrates that

the hardware integrated into the satellites is reliable, improving confidence in mission

success, and in some cases could improve the performance of missions on-orbit.

ii

Acknowledgements

This thesis would not have been possible without the support of some key people along

the way. I would first like to thank Dr. Robert Zee for providing me with the opportunity

to study at SFL where I got a taste of working in the space industry on real satellite

missions. I am certain that the skills I had learned here will continue to prove useful as

I continue with my studies elsewhere. I only wish I could have had more time to fully

explore the many other exciting topics I had encountered in my work.

Next, I would like to acknowledge Karan Sarda, manager of the attitude group at SFL,

with whom I’ve had the pleasure of working closely and who has directly coordinated my

work over the last two years. I appreciate the countless hours you’ve spent answering my

questions and puzzling with me over problems I had run into in my work, and also doing

everything in your power to help me make the most of my experience at SFL.

I am also grateful to the other members of the attitude group, namely Niels Roth,

Marc Fournier, Najmus Ibrahim, and Pawel Lukaszynski, for getting me started on many

of my responsibilities at SFL.

Penultimately, I would like to thank all of my fellow students and the staff at SFL

for being a part of my experience here. I have enjoyed working with you all, and I have

learned a lot from everyone I had worked with.

Last, but certainly not least, I would like to thank my parents for going to sometimes

extreme lengths to support me in my undergraduate and graduate studies at the Univer-

sity of Toronto for the last six years. I absolutely could not have made it this far without

your support.

iii

Contents

1 Introduction 1

2 Acceptance Testing 7

2.1 Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Theory of Operation . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Testing Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Functional Test Updates . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Outdoor Calibration Verification . . . . . . . . . . . . . . . . . . 9

2.2 Angular Rate Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11


2.2.2 Testing Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Reduction of Calibration Orientations . . . . . . . . . . . . . . . . 13

2.2.4 Screw Torque and Bias . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Fine Sun Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


2.3.2 Testing Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.3 Calibration Improvements . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Reaction Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


2.4.2 Testing Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.3 Torque Test Improvements . . . . . . . . . . . . . . . . . . . . . . 32

2.4.4 Timestamp Uniformity . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Conclusion 49

Bibliography 52

iv

List of Tables

2.1 Orientation Sets for Rate Sensor Calibration . . . . . . . . . . . . . . . . 15

2.2 Reaction Wheel Torque Test Speeds and Torques . . . . . . . . . . . . . 32

2.3 Reaction Wheel Torque Test Old Metrics . . . . . . . . . . . . . . . . . . 33

v

List of Figures

1.1 Attitude Determination and Control Architecture for GNB and NEMO

Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 UTIAS-SFL ADCS Hardware Testing Cycle . . . . . . . . . . . . . . . . 3

1.3 UTIAS-SFL Standard Thermal Acceptance Profile [6] . . . . . . . . . . . 5

2.1 Magnetometer Calibration Setup . . . . . . . . . . . . . . . . . . . . . . 10

2.2 UTIAS-SFL Rate Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Misalignment Angle Definitions . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Rate Sensor Calibration Orientations . . . . . . . . . . . . . . . . . . . . 15

2.5 EV9 Scale Factor vs. Orientation Set . . . . . . . . . . . . . . . . . . . . 16

2.6 AISSat-3 Scale Factor vs. Orientation Set . . . . . . . . . . . . . . . . . 17

2.7 GHGSat-D Scale Factor vs. Orientation Set . . . . . . . . . . . . . . . . 17

2.8 EV9 Skew Angle vs. Orientation Set . . . . . . . . . . . . . . . . . . . . 18

2.9 AISSat-3 Skew Angle vs. Orientation Set . . . . . . . . . . . . . . . . . . 18

2.10 GHGSat-D Skew Angle vs. Orientation Set . . . . . . . . . . . . . . . . . 19

2.11 Screw Torque vs. Bias for Y-Axis Sensor . . . . . . . . . . . . . . . . . . 20

2.12 UTIAS-SFL Fine Sun Sensor Theory of Operation . . . . . . . . . . . . . 21

2.13 Fine Sun Sensor Calibration Setup, Side View . . . . . . . . . . . . . . . 23

2.14 Calibration Results, ρrli and ρrsr Hard-Coded . . . . . . . . . . . . . . . . 26

2.15 On Orbit vs. Simulated Sun Sensor Error . . . . . . . . . . . . . . . . . . 27

2.16 Calibration Results, ρrli and ρrsr Estimated . . . . . . . . . . . . . . . . . 29

2.17 Reaction Wheel Control Modes . . . . . . . . . . . . . . . . . . . . . . . 31

2.18 Reaction Wheel Torque Jitter FFT . . . . . . . . . . . . . . . . . . . . . 37

2.19 Torque Jitter FFT Spike on NORSAT-1 Reaction Wheel . . . . . . . . . 44

2.20 Torque Jitter FFT Spike on NEMO-HD Reaction Wheel . . . . . . . . . 46

2.21 Old WheelTerm Time Step Variation, 60 Hz Torque Test . . . . . . . . . 47

2.22 New WheelTerm Time Step Variation, 50 Hz Torque Test . . . . . . . . . 47

vi

List of Abbreviations

AIS Automatic Identification System

AISSat Automatic Identification System Satellite

BRITE Bright Target Explorer

CMOS Complementary Metal-Oxide Semiconductor

CVG Coriolis Vibratory Gyro

EV9 exactView-9

FSS Fine Sun Sensor

FFT Fast Fourier Transform

GHGSat-D Greenhouse Gas Satellite Demonstrator

GNB Generic Nanosatellite Bus

IGRF International Geomagnetic Field Reference

LEO Low Earth Orbit

LFFT Long Form Functional Test

MEMS Micro-Electromechanical System

NEMO Next-Generation Earth Monitoring and Observation

NOAA National Oceanic and Atmospheric Administration

NORSAT-1 Norwegian Satellite 1

OASYS On-board Attitude Subsystem Software

PCB Printed Circuit Board

PSD Power Spectral Density

RMS Root Mean Square

SFFT Short Form Functional Test

T-Shock Temperature Shock

UTIAS-SFLSpace Flight Laboratory at the University of Toronto Institute for

Aerospace StudiesVHF Very High Frequency

vii

List of Notation

F b Reference frame~F b Vectrix of reference frame, i.e. column matrices consisting of basis

vectors of frame F b, e.g. ~F b =[~b1

~b2~b3

]T~(·) Vector

v Column matrices, e.g. v =[v1 v2 . . . vn

]Tbi Vector ~b resolved in frame F i

A Matrix

Cbi Rotation matrix from frame F i to frame F b, i.e. vb = Cbivi

1 Identity matrix

|·| Absolute value (scalar), determinant (matrix)

〈·〉 Expected value˙(·) Time derivative

viii

Chapter 1

Introduction

Attitude determination and control refers to the determination and control of a space-

craft’s orientation in three-dimensional space. This is achieved using sensors to measure

attitude and using actuators to change it. Taken together, these sensors and actuators,

along with the software used to operate them, are referred to as a spacecraft’s attitude

determination and control system (ADCS). The ability of a spacecraft to determine and

control its attitude is often critical to accomplishing key objectives of its mission, and

is typically a defining metric of a mission’s capabilities. Many of the recent missions

developed by the Space Flight Laboratory at the University of Toronto Institute for

Aerospace Studies (UTIAS-SFL) have been enabled by improved pointing accuracy due

to advances in the lab’s technology for small satellite attitude determination and control.

For example, the satellites comprising the Bright Target Explorer (BRITE) mission, a

constellation of five astronomy nanosatellites observing brightness and temperature vari-

ations in bright stars, achieve a pointing accuracy of 12 arc-seconds within one standard

deviation [1], enabling them to direct their telescope payloads with the accuracy required

to meet their science objectives.

The generic ADCS architecture for UTIAS-SFL satellites is shown in Figure 1.1. This

architecture is used in two UTIAS-SFL small satellite platforms: the Generic Nanosatel-

lite Bus (GNB) – a 7 kg nanosatellite platform, and the Next-Generation Earth Moni-

toring and Observation (NEMO) bus – a 15 kg microsatellite platform. For the purposes

of this thesis, a nanosatellite refers to a satellite with mass between 1 kg and 10 kg, and

a “microsatellite” refers to a satellite with mass between 10 kg and 100 kg, in conformity

with the generally accepted industry definitions [2]. In the illustrated ADCS architec-

ture, reaction wheels and magnetic torquers are used as actuators. Fine sun sensors, a

magnetometer, and an angular rate sensor are used as coarse attitude sensors. Some

missions also have a star tracker for fine attitude determination. A discussion of the

1

Chapter 1. Introduction 2

ActuatorsSensors

On-board Attitude Software Subsystem

(OASYS)

Fine Sun SensorsMagnetometer Reaction Wheels

Magnetorquers

Attitude Dynamics

Star Tracker

Disturbance Torques

Commanded Attitude

Angular Rate Sensor

Figure 1.1: Attitude Determination and Control Architecture for GNB and NEMO Plat-forms

theory of operation for these devices is presented later in this thesis. Finally, the On-

board Attitude Subsystem Software (OASYS) processes sensor measurements, estimates

the spacecraft attitude state, and commands the actuators to apply a torque in order to

track a given commanded torque. An extended Kalman filter (EKF) is used to produce

state estimates and a proportional-derivative attitude control law is used as a controller.

The focus of this thesis is acceptance testing of attitude determination and control

hardware. Acceptance testing is a key part of the later stages of most spacecraft develop-

ment cycles [3]. Its purpose is to ensure that pieces of hardware designated for spaceflight

operate within the range of conditions required in the mission design, perform at the level

required by the system design, and are free of defects. This is distinguished from qual-

ification testing, which ensures that hardware of a particular design or model can be

included in the design of the spacecraft, which is usually completed in the earlier stages

of the spacecraft development cycle. The general acceptance testing cycle for ADCS

hardware is shown in Figure 1.2. Note that for some hardware, some of the steps in this

cycle are skipped or do not apply. The specifics of the testing cycle for each type of

hardware is discussed in their respective sections. The testing cycle includes four types

of tests: the functional test, the thermal shock (T-Shock) test, the thermal acceptance

campaign, and calibration.

Functional Test The purpose of a functional test is to operate hardware in a flight-


1. Initial Functional

2. T-Shock3. Post-T-

Shock Functional

4. Thermal Acceptance

5. Calibration6. Final

Functional

Pre-T-Shock Inspection

Post-T-Shock Inspection

Figure 1.2: UTIAS-SFL ADCS Hardware Testing Cycle

representative manner [4]. This ensures key components of the hardware are func-

tioning, but does not usually measure the performance of the hardware. UTIAS-

SFL distinguishes between two types of functional tests. A long form functional test

(LFFT) is a comprehensive test providing high confidence of a device’s function-

ality, and is usually performed before and after major environmental tests such as

a thermal acceptance test. A short form functional test (SFFT) is an abbreviated

test usually consisting only of a subset of the procedures of the LFFT to expedite

testing when only a basic assurance of a device’s functionality is needed. This may

be performed in the middle of an environmental test, or when small changes are

made to the hardware that don’t warrant an LFFT. For some ADCS hardware,

a comprehensive functional test is sufficiently simple to make it unnecessary to

distinguish between long form and short form tests.

Thermal Shock A thermal shock test subjects the hardware to sudden and extreme

changes in temperature in order to test the hardware’s capacity to survive extreme

temperature changes and to check the workmanship of solder joints on printed

circuit boards (PCBs) [5]. The rapid temperature changes cause uneven expansion

and contraction of PCB components, applying stress to them and dislodging ones

that are not properly soldered. At UTIAS-SFL, this test is normally performed by

a third-party contractor based on UTIAS-SFL specifications and is consequently

not part of the work discussed in this thesis. However, inspections of parts before

and after the T-shock are performed in-house.

Thermal Acceptance The thermal acceptance campaign is not a single test, but is

instead a series of tests that operate the test unit at various temperatures as well

as throughout changing temperatures. This ensures that the hardware is able to

tolerate the full range of temperatures required by the mission profile [6]. The

hardware is tested to two categories of temperatures – non-operational temper-

atures and operating temperatures. Non-operational temperatures refer to the


temperature range over which a device must be capable of being powered on while

remaining undamaged, whereas operating temperatures refer to the temperature

range over which the device must be capable of powering on and operating within

specifications. A third category of temperatures, survival temperatures, refers to

the range of temperatures over which a device must be capable of remaining free

from damage while powered off. However, devices are never subjected to these tem-

peratures during testing, and the thermal designs of UTIAS-SFL satellites prevent

devices from reaching these temperatures on orbit. Operating temperatures are

usually more stringent than non-operational temperatures, which in turn are more

stringent than survival temperatures. Non-operational and operating temperature

ranges are specified by the design of a satellite’s mission and are a function of orbit

and attitude profile. The design takes into consideration survival and operational

ranges typically provided by manufacturers of the parts out of which the devices

are constructed. General practice at UTIAS-SFL calls for these ranges to be de-

rated, i.e. applying stricter limits, leading to typical non-operational temperatures

for ADCS hardware of −30◦C to 70◦C and operational temperatures of −20◦C to

60◦C [7]. This practice allows for additional safety margin.

Figure 1.3 shows the UTIAS-SFL standard thermal testing profile. As illustrated,

the thermal testing campaign tests hardware at both their non-operational hot

and cold limits and their operating hot and cold limits, as well as throughout

temperature changes. Note that full SFFTs and LFFTs for ADCS sensors are not

usually possible during a thermal test since physical access to the sensors is required

in order to expose them to a stimulus, for example by moving them. Physical access

to the units is restricted once they are placed in the thermal chamber. Note also

that the temperature extremes shown in the figure are labeled as “survival” hot

and cold temperatures, a typical albeit misleading internal shorthand for a device’s

non-operational temperatures.

Calibration A calibration is a characterization of the performance of a piece of hard-

ware. This is usually performed only for ADCS sensors. In general, a calibration

involves comparing measurements made by a sensor against a set of reference mea-

surements that are known to be accurate in order to calculate the value of correction

factors. These correction factors are applied on orbit when measurements from the

sensor are used. Details of the calibration techniques for some sensors are provided

in later sections.

Note that while satellite components must also in general be tested to survive in the hard

Chapter 1. Introduction 5UTIAS/SFL Thermal Vacuum and Standard Thermal Test Procedure, Issue 1.2, 2009/06/04

UTIAS/SFL, Tel (416) 667-7864, E-mail: [email protected], UTIAS/SFL Confidential Page 4 of 6

Standard Temperature

25 deg C

T_hot, survival

T_cold, survival

T_hot

T_cold

Long-Form Functional

Test (LFFT)

Temperature

Soak, for t_soak

Warm Start + LFFT

Cold Start +

LFFT

Temperature

Soak, for t_soak

Final LFFT

All Slews at

2 deg C / min

Temperature Soak, for t_soak

Temperature

Soak, for t_soak

Temperature Soak, for t_soak

Temperature

Soak, for t_soak

Phase IV – Temperature Cycling – Check Telemetry and Communications at Regular Intervals (short-form test every t_sfft seconds)

Phase I – Initial Check Phase II – Warm Start Phase III – Cold Start Phase V –

Final Check

Temperature Profile for Unit Thermal Vacuum or Standard Thermal Tests

Figure 1.3: UTIAS-SFL Standard Thermal Acceptance Profile [6]

vacuum of space, vacuum testing is only performed at the system, or satellite, level if the

component has already been qualified in thermal vacuum.

The contribution of this thesis was to perform acceptance testing of ADCS hard-

ware for the exactView-9 (EV9), Automatic Identification System Satellite 3 (AISSat-

3), Greenhouse Gas Satellite Demonstrator (GHGSat-D), and Norwegian Satellite 1

(NORSAT-1) missions. EV9 and AISSat-3 are two ship-tracking nanosatellites based

on the Generic Nanosatellite Bus platform [8][9]. The main role of the attitude deter-

mination and control subsystem for these satellites is to point a very high frequency

(VHF) antenna to nadir, or Earthward, in order to receive positioning broadcasts from

ships. GHGSat-D is a mission that will demonstrate space-based monitoring of indus-

trial greenhouse gas emissions using a microsatellite based on the NEMO platform [10].

Attitude determination and control is used to allow the satellite to track fixed ground

targets with high accuracy as it passes over them to enable measuring of greenhouse gas

emissions from a single industrial site. Finally, NORSAT-1 is a science microsatellite also

based on the NEMO platform that will carry a radiometer for studying variations in solar

irradiance and a pair of Langmuir probes to study the plasma environment in low Earth

orbit (LEO) [11]. The satellite will also carry an AIS payload. Attitude determination

and control will be used to point the radiometer payload continuously at the Sun while

keeping the Langmuir probes pointed in the local velocity direction. GHGSat-D has the

most stringent pointing accuracy requirements of the four missions.

Testing for each mission was completed according to project timelines handed down


from the system-level design of the satellites. The goal of the testing was to ensure

not only that all hardware passed each test in its testing cycle, but also to ensure that

existing test procedures and acceptance criteria appropriately reflected system require-

ments. Where necessary, previously established test procedures and acceptance criteria

were adjusted, and new test procedures and criteria were developed. The testing of

ADCS hardware contributed by this thesis will help ensure to a high degree of confidence

that the satellites using the hardware will be capable of achieving the pointing objectives

required by their missions.

Chapter 2

Acceptance Testing

The sections in this chapter treat the acceptance testing of each type of attitude determi-

nation and control hardware involved in this thesis. Each section begins with a discussion

of the theory of operation of the device, followed by a description of its specific accep-

tance testing cycle, and then identifies the contributions of this thesis to the acceptance

testing of that device.

2.1 Magnetometers

2.1.1 Theory of Operation

Magnetometers are used to measure the magnitude and direction of the local magnetic

field. Combined with a model of the geomagnetic field, these can be used to determine

a satellite’s attitude about two axes at any point in its orbit. A single magnetometer

measurement leaves uncertainty in attitude about the local magnetic field vector, and

is usually combined with a measurement of another vector to obtain a full attitude

solution. UTIAS-SFL’s in-house magnetometers use three magneto-inductive sensors

oriented orthogonally to each other to measure the components of the magnetic field

vector. These sensors have the property that their inductance is proportional to the

magnetic field applied along their axes. An interface circuit measures this inductance

and converts it into a magnetic field strength value. In pointing budgets for UTIAS-SFL

missions, the magnetometer is expected to achieve an accuracy on the order of 3◦ [12] to

5◦ [13].

Due to their ability to make measurements at any point in a satellite’s orbit as well as

their relatively low cost, magnetometers have become ubiquitous in UTIAS-SFL missions.

7

Chapter 2. Acceptance Testing 8

2.1.2 Testing Cycle

The testing cycle for magnetometers at UTIAS-SFL follows the general cycle identified

in Figure 1.2. This thesis contributes updates to the functional and calibration tests for

UTIAS-SFL’s magnetometers in addition to performing the acceptance testing for the

EV9, AISSat-3, GHGSat-D, and NORSAT-1 magnetometers.

2.1.3 Functional Test Updates

A functional test of the magnetometer involves a polarity check, noise measurement, and

a rough test of sensor bias [14]. The purpose of the polarity check is to ensure that

measurements qualitatively respond in the right way to a change in the applied magnetic

field. Previously, the polarity check called for placing a permanent magnet close to the

sensor [15]. The change in the observed magnetic field along each axis should be positive

when aligned with the magnet south-to-north and negative when aligned with it north-

to-south. However, this test carried a risk of permanently magnetizing components on

the magnetometer or damaging the magneto-inductive sensors due to the strength of the

permanent magnet. It was observed that the ambient magnetic field within the lab was

sufficiently strong and consistent to be used in place of the permanent magnet in the

polarity check. Specifically, the ambient field seemed always to be the strongest along

the local vertical and in the downward direction, and was strong enough to be greater

than both the typical noise and bias levels of the magnetometer. A new procedure was

developed for the polarity check in which the magnetometer would be polled continuously

while being held in a series of orientations to align each axis with the vertical. Since the

field in the lab is known to be dominated by the vertical component in the downward

direction, the test would check that the field measured by each axis would measure a

positive field when pointed downward, and change to a negative field when flipped to

point upward.

Another update to the functional test procedure was the addition of the rough bias

check. During calibration of the magnetometer for the GHGSat-D mission, it was noticed

that measurements on one axis exhibited an abnormally large bias of approximately

23µT , where typical biases have been observed to be in the low thousands of nano-

Tesla. Additional testing confirmed that the bias was inherent to the sensor and not a

result of experimental error. Despite this bias, the unit was accepted for the mission

as a spare since it was determined that, even with the bias, the maximum expected on-

orbit field measurement on any axis (36µT unbiased) would fall well within the sensor’s

measurement range of ±131µT , and that the unit was functioning properly otherwise.


However, since no attempt at measuring the bias was made prior to calibration, it was

unclear at what point the bias had changed, or if it had always been high. It was therefore

decided to add a bias check to the functional test. In a similar vein to the polarity check,

the bias check consisted of aligning each axis of the sensor with the vertical while polling

for one minute, then inverting the alignment and polling for another minute. The bias for

each axis is estimated by summing the average fields measured in positive and negative

alignment. This bias check revealed through the acceptance testing campaign of the

NORSAT-1 magnetometers that a large bias similar to that seen on the GHGSat-D

sensor only appeared once the magnetometer was integrated into an interface plate that

mounts the sensor to the calibration facility. The cause of this bias was traced to a

magnetic threaded helical insert used in this plate. The helical inserts have since been

replaced with nonmagnetic titanium substitutes. Another benefit of adding this check

to the magnetometer functional test is that it could help detect faulty units with biases

outside of manufacturer specifications early in the acceptance testing cycle. This helps

reduce schedule cost in the event of a faulty sensor since replacement sensors can be

ordered before most of the testing is done rather than after calibration is completed near

the end of the cycle.

2.1.4 Outdoor Calibration Verification

The magnetometer calibration compares measurements of a series of applied magnetic

fields made by an uncalibrated “flight” magnetometer with those made by a calibrated

“lab” magnetometer in order to calculate a number of calibration parameters. The

magnetometers are mounted side-by-side on an aluminum L-bracket, which is secured

to a platform in the center of three pairs of orthogonal Helmholtz coils that generate

the applied magnetic fields. Figure 2.1 shows the magnetometer calibration setup. The

parameters determined from the calibration are programmed into OASYS, the attitude

determination and control software, to be used for estimating the true magnetic field

based on measured values and include factors that correct for sensor bias, nonlinearity of

the magnetometer’s measurements, and non-orthogonality of its axes, as well as scaling

factors for converting between the magnetometer’s native units, or “counts”, to nano-

Tesla [14].

Previous iterations of the calibration procedure were considered complete once these

parameters had been determined. However, in order to provide an additional measure

of confidence for calibrations performed with a newly commissioned in-house calibration

facility, it was decided to implement a test to verify the parameters extracted from cal-


Lab Magnetometer Flight Magnetometer

Helmholtz Coil (1 of 6)

Aluminum L Bracket

Figure 2.1: Magnetometer Calibration Setup

ibration. In this test, both the lab and flight magnetometers are brought outside of the

magnetically noisy lab environment and into a field far away from buildings and machin-

ery. The two magnetometers are held in a fixed orientation using the same L-bracket

from the calibration setup, and measurements of the ambient geomagnetic field are col-

lected for five minutes from both magnetometers. The calibration parameters are applied

to the flight magnetometer measurements and the corrected average field is compared

against that measured by the lab magnetometer. The error between the corrected flight

magnetometer measurement and the lab magnetometer measurement has been found to

be typically below 0.5◦ in direction and 0.6% in magnitude within 3 standard deviations.

This provides an estimate of the calibration error contributed by variations in the indoor

magnetic field that was applied to the flight and lab magnetometers during calibration.

Other contributions to the calibration error include mounting misalignment between the

lab and flight magnetometers due to manufacturing tolerances in the placement of holes

for screws and the calibration error of the lab magnetometer. The mounting misalignment

contributes up to 1.42◦ within 3 standard deviations. The lab magnetometer calibration

is accurate to 0.5% on any axis [16], which corresponds to 0.29◦ peak error. No confi-

dence interval is provided, but 3 standard deviations is assumed. Treating each source

of calibration error as a zero-mean normally distributed random variable, the total cal-


ibration error is found to be 1.53◦ within three standard deviations, and is dominated

by the misalignment error. Adding on misalignment error from satellite integration of

0.41◦ within 3 standard deviations, these results show that the total magnetometer error

could amount to 1.59◦ within three standard deviations. This falls well below the allotted

magnetometer error of 3.62◦ in the coarse pointing budget for the NEMO-AM mission

[12], which is also used for GHGSat-D and NORSAT-1. The AISSat-1 pointing budget

[13], which is also used for EV9 and AISSat-3, allows 5.4◦ within 3 standard deviations.

The corrected flight magnetometer measurements from the outdoor test are also com-

pared against the local magnetic field vector predicted by the International Geomagnetic

Reference Field (IGRF) model for the day of the test, retrieved from the National Oceanic

and Atmospheric Administration (NOAA) website [17]. The error between the magni-

tude of the corrected field measurement and that of the IGRF predicted field is typically

below 0.5%, suggesting that the location chosen for the verification test is magnetically

clean, as desired. The angle between the measured and predicted fields cannot be com-

pared since the attitude of the lab and flight magnetometers is not accurately measured

during the test.

2.2 Angular Rate Sensors


Angular rate sensors measure the angular velocity, or angular rate, of a spacecraft with

respect to inertial space. This is used to propagate the spacecraft’s attitude based on

previous measurements in order to balance the effect of measurement error in sun sensors,

the magnetometer, or other sensors that measure attitude using the ambient environment.

Propagating attitude based on angular velocity measurements is also useful for tracking

satellite attitude when one of these sensors is not available. An example of this is when

the satellite passes into the shadow of the Earth, a condition known as eclipse, during

which the sun sensors have no view of the Sun. However, as an inertial sensor, which

measures the spacecraft’s motion rather than its position or orientation, a rate sensor is

unreliable on its own for tracking spacecraft attitude. As with any sensor, a rate sensor’s

output is biased and has noise. Although the bias is characterized during calibration, the

bias of an inertial sensor drifts over time and is usually also a function of temperature.

Moreover, even with the bias deducted from the output signal, integrating the noisy rate

measurements to track attitude results in a gradual departure from the true attitude.

This process is known as random walk.


UTIAS-SFL’s in-house rate sensors use three orthogonally mounted micro-electromechanical

system (MEMS) Coriolis vibratory gyros (CVGs) to measure the three components of

the angular velocity vector, as shown in Figure 2.2. The Z sensor is placed on a mother-

board, and the X and Y sensors are placed on two daughter-boards1 A MEMS CVG

measures angular velocity by measuring the Coriolis force on a proof mass vibrating ra-

dially in the measurement plane, the amplitude of which is linearly proportional to the

component of the angular velocity in the measurement plane [18]. This force is then

transduced into a voltage. Rate sensors for UTIAS-SFL missions are usually expected to

be sufficiently accurate to contribute to no more than 1.8◦ of pointing error within one

standard deviation [19].

X Sensor

Y Sensor

Z Sensor

+X

+Y

+Z

Figure 2.2: UTIAS-SFL Rate Sensor

Rate sensors based on MEMS CVGs are relatively inexpensive and for this reason,

UTIAS-SFL’s in house rate sensor has made its way into the designs of most of the lab’s

newer satellites, including each of the four with which this thesis is concerned.

1The axes shown in Figure 2.2 form a right-handed triad, but the assignment of the X and Y CVGsin the actual sensor design is swapped such that the axes of the rate sensor do not correspond to aright-handed triad. This detail is omitted here for simplicity. The handedness of the rate sensor axeswas not considered during its original design because the axes are easily remapped in flight software.However, since this was found to cause confusion during testing, future revisions of the rate sensor designwill use the axes shown here.


2.2.2 Testing Cycle

The testing cycle for angular rate sensors at UTIAS-SFL follows the general cycle identi-

fied in Figure 1.2. This thesis contributes updates to the calibration test for UTIAS-SFL’s

rate sensors, discovered a correlation between sensor bias and screw torque, and evalu-

ated the performance of a new sensor package. Additionally, acceptance testing was

performed for the EV9, AISSat-3, GHGSat-D, and NORSAT-1 rate sensors.

2.2.3 Reduction of Calibration Orientations

The calibration procedure for UTIAS-SFL’s in-house angular rate sensor involves mount-

ing the sensor to a rate table and polling measurements from the sensor while it is rotated

on the table at specified, fixed rates [20]. This is repeated with the sensor mounted in

various orientations in order to exercise each measurement axis of the sensor. The rate

of rotation of the rate table is accurately known, to ±0.1◦ accuracy [21], and forms the

truth data to which the rate sensor measurements are compared to find the calibration

parameters. These are applied to correct the raw angular rate measurements from each

of the three measurement axes using

ωz,i =ωz,i − bz1 + sz

(2.1)

ωx,i = ωz,i tan θy +ωx,i − bx1 + sx

1

cos θy(2.2)

ωy,i = ωx,i tan θz +

(ωy − by1 + sy

1

cos θx− ωz,i tan θx

)1

cos θz(2.3)

where ωx,i, ωy,i, and ωz,i are the corrected rate measurements at time ti, in rad/s and

ωx,i, ωy,i, and ωz,i are the raw rate measurements at time ti in counts. The remaining

variables are the calibration parameters to be solved for. The variables bx, by, and bz

are the biases on each axis, sx, sy, and sz are the scale factors for each axis, and θx, θy,

and θz are misalignment angles between the raw non-orthogonal axes and an orthogonal

set of axes defined in Figure 2.3. Equations 2.1 to 2.3 treat the raw z axis as being

aligned perfectly with the z axis of the desired orthogonal axes and the raw x axis is

taken to lie in the x-z plane of the orthogonal axes, with its misalignment described

by θy. The misalignment of the raw y axis is then described by a rotation in the x-

y plane about the z axis by θz followed by a rotation out of the plane toward the z

axis by θx. The misalignment between the sensor axes and the satellite body axes is

dealt with by including an estimate for the rate sensor misalignment in the mission’s

pointing budget. For AISSat-1, UTIAS-SFL’s first mission flown with a rate sensor,


this misalignment was 0.10◦ within one standard deviation [13]. Note that although the

biases of the sensor are solved for in the calibration, these only reflect the biases at room

temperature. Since CVG biases are a strong function of temperature, these are not used

in flight. Instead, a separate thermal calibration is performed to characterize the bias as

a function of temperature across the operating temperature range. Additionally, the on-

board extended Kalman filter (EKF) is designed to be capable of continuously estimating

and tracking the rate sensor bias to allow for changes throughout the life of the mission,

and uses the measured temperature of the sensor and the thermal calibration results to

generate its initial guess. Details of the thermal calibration procedure are provided in

[22].

Figure 2.3: Misalignment Angle Definitions

A least-squares method is used to estimate the calibration parameters by minimizing

the residual sum of squares given by

Sres =N∑i=1

(|ωT,i| −

√ω2x,i + ω2

y,i + ω2z,i

)2

(2.4)

where ωT,i is the rate table rate at time ti, used as the truth value, ωx,i, ωy,i, and ωz,i

are functions of the calibration variables calculated using Equations 2.1 to 2.3, and N is

the total number of measurements over all rate setpoints and all orientations. A detailed

derivation of the calibration method is found in [23].

Previously, the calibration procedure called for the rate sensor to be mounted in 21

different orientations [20]. These consisted of each axis being placed in alignment with

the rotation axis of the rate table and stepping through 15◦ increments until the axis is

perpendicular to the table rotation axis for a total of 7 orientations per axis. A selection

of these orientations are shown in Figure 2.4. This number of orientations provided


accurate estimation of the calibration parameters, but required an entire work day to

complete.

(a) Orientation 1 (b) Orientation 2 (c) Orientation 5

Figure 2.4: Rate Sensor Calibration Orientations

It was shown that the duration of the test could be significantly reduced while achiev-

ing similar results using fewer orientations. Calibrations were performed using 12, 9, 6,

and 3 orientations by removing intermediate orientations from the original set of 21. The

difference in the resulting scale factors and misalignment (skew) angles for three rate

sensors with serial numbers AUZ, B1A, and B1D are shown in Figures 2.5 to 2.7 and

Figures 2.8 to 2.10. The enumeration of the orientation sets is explained in Table 2.1.

These results indicate that Orientation Sets 3 to 6 achieve a similar level of performance

in estimating the skew angles, with rapidly diminishing return in all cases for increas-

ing the number of calibration orientations past that of Set 3. No significant change is

observed in the accuracy of estimating the scale factors for any orientation set since the

value of the scale factors is typically in the range of 200 counts·s/◦.

SetNumber

Number ofOrientations

Description

1 3 One orientation for each axis aligned with table axis2 6 As in Set 1 plus one for each axis aligned 45◦ with table axis.3 6 One for each axis aligned perpendicular with table axis, and

one for each axis aligned 45◦.4 9 As in Set 3 plus one for each axis aligned with table axis.5 12 Each axis oriented 0◦, 30◦, 60◦, and 90◦ from table axis.6 21 Full original set consisting of each axis oriented 0◦, 15◦, 30◦,

45◦, 60◦, 75◦, and 90◦ from table axis.

Table 2.1: Orientation Sets for Rate Sensor Calibration

Though Orientation Set 3 provided good performance, it was decided to use Orienta-

tion Set 4 for future rate sensor calibrations as an added measure of confidence, reducing


the number of mounting orientations to 9 and cutting the calibration time roughly in

half. However, although this analysis has shown that the effect of reducing the number

of orientations has had a small effect on the accuracy with which the calibration pa-

rameters are determined, the resulting impact on the satellite pointing error is not well

understood. Future work would explore the effect of reducing the number of calibration

orientations on pointing error arising from errors in determining the rate sensor’s scale

factors and axis skew angles.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.5

0

0.5

1

1.5

2AUZ Scale Factor Error* vs. Orientation Set

Sca

le F

acto

r E

rror

(co

unt/(

° /s))

Orientation Set *Using 6 (full set) as true scale factor

sx

sy

sz

Figure 2.5: EV9 Scale Factor vs. Orientation Set


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4B1A Scale Factor Error* vs. Orientation Set

Sca

le F

acto

r E

rror

(co

unt/(

° /s))


sx

sy

sz

Figure 2.6: AISSat-3 Scale Factor vs. Orientation Set

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5B1D Scale Factor Error* vs. Orientation Set

Sca

le F

acto

r E

rror

(co

unt/(

° /s))


sx

sy

sz

Figure 2.7: GHGSat-D Scale Factor vs. Orientation Set


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−6

−4

−2

0

2

4

6

Orientation Set Number *Using set 6 (full set) as true skew angle.

Ske

w A

ngle

Err

or (°

)

AUZ Skew Angle Error* vs. Orientation Set

θx

θy

θz

Figure 2.8: EV9 Skew Angle vs. Orientation Set

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−5

−4

−3

−2

−1

0

1

2

3


Ske

w A

ngle

Err

or (°

)

B1A Skew Angle Error* vs. Orientation Set

θx

θy

θz

Figure 2.9: AISSat-3 Skew Angle vs. Orientation Set


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−3

−2

−1

0

1

2

3

4


Ske

w A

ngle

Err

or (°

)

B1D Skew Angle Error* vs. Orientation Set

θx

θy

θz

Figure 2.10: GHGSat-D Skew Angle vs. Orientation Set

2.2.4 Screw Torque and Bias

It was discovered during calibration of EV9’s rate sensor, after the sensor was integrated

into its aluminum enclosure, that the biases on its X- and Y-axis sensors had changed

drastically since the thermal acceptance campaign and prior functional tests completed

before the sensor was integrated into its enclosure. Where the biases were typically in the

range of −200 counts to 200 counts at room temperature, the bias on the Y-axis sensor

was now around 800 counts. A similar level of change in the bias was subsequently

observed in the test campaigns for the AISSat-3 and GHGSat-D rate sensors. Further

investigation revealed that the bias change was related to the torque applied to the

screws used to fix the daughter-boards on which the X- and Y-axis sensors reside to the

enclosure. A plot of bias versus a rough measurement of screw torque is shown in Figure

2.11. The screw torque measurements were made with an adjustable-torque screwdriver

and are accurate to 8.8× 10−3 N m.

Upon consulting with the manufacturer of the rate sensor’s CVG modules, it was

established that the cause behind this relationship between screw torque and bias is that

the stress profile on the sensor package influences the sensor’s measurement of Coriolis

force upon which the rate measurements depend. According to the manufacturer, this

level of change is not unexpected. Since the problem has not been observed to occur


0.08 0.1 0.12 0.14 0.16 0.18 0.2−940

−930

−920

−910

−900

−890

−880

−870

−860

−850

−840Y Axis Angular Rate Bias vs. Torque

Ang

ular

Rat

e ω

(co

unts

)

Screw Torque (Nm)

Figure 2.11: Screw Torque vs. Bias for Y-Axis Sensor

for the Z-axis sensor on the mother-board, whose screws are further away from the

sensor package, it was recommended that the daughter-boards be made larger in future

revisions of the rate sensor design to increase the spacing between the screws and the

sensor package, or to use adhesives to bond the daughter-boards to the enclosure.

There is no guarantee that the present mounting scheme for the rate sensor daughter-

boards does not place the X- and Y-axis CVG modules under unsafe stress levels. How-

ever, the current schedules for the four missions discussed in this thesis do not allow

sufficient time for a redesigned sensor to be implemented for those missions. Further

testing revealed that, though unusually high, the bias was relatively stable once the

sensor was integrated into its enclosure. Furthermore, other aspects of the rate sensor’s

performance appear to be unaffected, and the rate sensor was at no risk of saturating dur-

ing operation since spacecraft body rates were expected to remain well below ±10rad/s,

or approximately ±2200 counts, where the full range of the rate sensor is ±8192 counts.

Moreover, as previously explained, the EKF implemented in the attitude determination

and control flight software is capable of estimating and tracking a slowly changing bias,

and, additionally, is also capable of propagating spacecraft attitude state using actuator

inputs in place of rate sensor measurements. It was therefore decided to accept the rate

sensors for flight despite the high bias, and a recommendation was made to integrate

future sensors into their enclosures before the start of the acceptance test campaign to

ensure consistent performance throughout the campaign.


2.3 Fine Sun Sensors


Sun sensors are used to measure the direction of the Sun relative to the spacecraft. Fine

sun sensors provide a more accurate measurement of the sun direction vector relative to

the spacecraft, while coarse sun sensors provide a much rougher measurement and are

used in some space missions to point fine sun sensors. Pointing budgets for UTIAS-SFL

missions typically require fine sun sensors to be accurate to approximately 3◦ within three

standard deviations or better [12][19]. Coarse sun sensors are no longer used on UTIAS-

SFL missions, but have been implemented in the past by using solar panel currents to

determine the incident angle of the sun on each face of the satellite, which have been

observed to be accurate to about 30◦ or better.

As with the magnetometer, a single fine sun sensor measurement can only determine

a satellite’s attitude about two independent axes, leaving uncertainty about the Sun

vector, and is therefore typically used in combination with a magnetometer to provide

a full three-axis attitude solution. UTIAS-SFL’s in-house fine sun sensors measure the

sun vector using a two-dimensional complementary metal-oxide semiconductor (CMOS)

phototransistor array located underneath a pinhole lens. As light from the Sun passes

through the pinhole lens, a spot is projected onto the CMOS array below. A cross-section

of this is shown in Figure 2.12. In the figure, w is the width of the array. Given known

Incident Sunlight

CMOS Array

Pinhole

Figure 2.12: UTIAS-SFL Fine Sun Sensor Theory of Operation

values for the height of the pinhole h and the two components of the bias of the pinhole

from the center of the CMOS array b = [ bx by ]T , the location of the centroid of the

light spot c = [ cx cy ]T can be used to calculate the direction of the Sun in the sensor


frame, indicated by the subscript s, using

ss =1√

(b− c)T (b− c) + h2

[b− c

h

](2.5)

where ss, the sun vector has been normalized to unit length. Note that in Figure 2.12, ~b

and ~c are the bias and centroid vectors, with components in the sensor frame of b and c

as defined above, respectively. These are confined to the plane of the CMOS array, but

can in general be in any direction in that plane. A neutral density filter is placed on top

of the pinhole to attenuate the intensity of the incoming sunlight and also reduces the

intensity of most sources of stray light to nearly undetectable levels.

One of these sensors is typically placed on each face of a satellite in order to allow

measuring the sun vector in any attitude so long as the Sun is visible from the satellite’s

position in orbit. The low price of CMOS arrays in recent years has allowed sun sensors

of this type to be made relatively cheaply. Various iterations of UTIAS-SFL’s in-house

fine sun sensor have therefore been used on every UTIAS-SFL mission to date.

2.3.2 Testing Cycle

The testing cycle for fine sun sensors at UTIAS-SFL follows the general cycle identified in

Figure 1.2. This thesis contributed improvements to the sun sensor calibration procedure

in addition to performing acceptance testing for the EV9, AISSat-3, GHGSat-D, and

NORSAT-1 sun sensors.

2.3.3 Calibration Improvements

The sun sensor calibration estimates a sensor’s pinhole height and bias [24]. These

are estimated by finding the values of height and bias that minimize the angular error

between the measured sun direction and the true sun direction for a large number of

measurements across the range of the sensor field of view. This is elaborated below.

The sun sensor calibration setup is shown in Figure 2.13. Note that the size of the

rotator has been exaggerated for clarity. The sensor is mounted on a 3-axis rotating

platform and a simulated sun source is used to illuminate the sensor. Mirrors are used

to extend the distance between the light source and the sensor. The sensor is mounted

such that its Z-axis is parallel within misalignment error with the rotator roll axis. In

the rotator’s home position, when all actuator angles are set to 0◦, the rotator roll axis is

pointed toward the front of the room directly at the light source with some misalignment


error, the rotator yaw axis points at the ceiling, and the rotator pitch axis points to the

right side of the room, or out of the page in Figure 2.13.

Three right-handed orthogonal reference frames are defined in the figure: these are

the lab frame F i, the rotator frame F r, and the sensor frame F s. The lab frame

has its origin at the rotator’s center of rotation and remains fixed with respect to the

room. Its three axes[~i1 ~i2 ~i3

]Tcoincide with the rotator’s yaw, pitch, and roll axes,

respectively, at home position. The rotator frame also has its origin at the rotator’s

center of rotation, but remains fixed with respect to the sensor throughout the test.

Its three axes[~r1 ~r2 ~r3

]Tare a transformation of the axes of F i by the rotator’s

rotation angles in a yaw-pitch-roll, or 1-2-3, order. Finally, the sensor frame finds its

origin at the center of the CMOS array and remains fixed with respect to the sun sensor

throughout the test. Its three axes[~s1 ~s2 ~s3

]Tcoincide with the CMOS array’s X, Y,

and Z axes, respectively. Three vectors are also defined: ~ρrs, ~ρrl, and ~ρsl. The vector ~ρrs

extends from the rotator center of rotation to the sensor CMOS and stays fixed relative

to F r throughout the calibration. The vector ~ρrl extends from the rotator center of

rotation to the apparent position of the light source. This stays fixed with respect to

F i. Finally, ~ρsl is the vector from the sensor CMOS to the apparent light source. The

superscript letters r, s, and l refer to the rotator center of rotation, CMOS center, and

light source origin, respectively. Note that the size of the CMOS is considered negligible

at this scale and it is assumed that the rays of light incident across the CMOS are all

parallel to ~ρsl.

Rotator

Sensor

Mirror

Light Source (True)

Light Source (Apparent)

Mirror

Figure 2.13: Fine Sun Sensor Calibration Setup, Side View

Measurements of the vector from the sensor to the sun source ~ρsl are made across the


sensor field of view by holding the sensor at a series of pitch angles between 0◦ and 57◦,

which covers most of a typical sensor’s field of view. For each pitch angle, the sensor is

rotated into a series of roll angles evenly spaced between −180◦ and 180◦. A measurement

of the centroid of the projected light spot on the CMOS is made at each position and

the rotator angles are recorded. The yaw axis is held fixed at 0◦ throughout the test.

This generates concentric annuli of measured centroids spanning the field of view. The

spacing between the pitch and roll angles are chosen to get a total of 2000 roughly evenly

distributed measurement orientations.

The measurements made in each orientation throughout the test allow for the direction

of the sun vector ~ρslk to be calculated in two different ways. The subscript k refers to the

kth measurement orientation. The measured sun vector is calculated and normalized to

unit length using Equation 2.5, which naturally resolves the vector in the sensor frame.

Adding indices, Equation 2.5 can be rewritten as

ss,k =1√

(b− ck)T (b− ck) + h2

[b− ck

h

](2.6)

where (·) is used to indicate that this is the measured sun direction vector. This is similar

to how the sun direction vector is calculated on orbit. The true sun vector is calculated

by observing that ~ρsl = ~ρrl − ~ρrs, and using

ρslr,k = Cri,kρrli − ρrsr (2.7)

where ρrli and ρrsr are ~ρrl and ~ρrs resolved in the lab frame and rotator frame, respectively,

which are the frames in which the two vectors can be measured. ρslr,t is the true sun vector

resolved in the rotator frame, and Cri is the rotation matrix from the lab frame to the

rotator frame. In order to compare the measured and true sun vectors, the two must be

expressed in the same frame. The true sun vector can be expressed in the sensor frame by

applying an additional rotation from the rotator frame to the sensor frame to Equation

2.7 to get

ρsls,k = Csr

(Cri,kρ

rli + ρrsr

)(2.8)

This is then normalized to get the unit vector for the true sun direction

ss,k =ρsls,kρsls,k

(2.9)

Cri,k is obtained by composing three principal rotations in the order of the rotator’s order


of rotation

Cri,k =

cos θ3,k sin θ3,k 0

− sin θ3,k cos θ3,k 0

0 0 1

cos θ2,k 0 − sin θ2,k

0 1 0

sin θ2,k 0 cos θ2,k

1 0 0

0 cos θ1,k sin θ1,k

0 − sin θ1,k cos θ1,k

(2.10)

where θ1,k, θ2,k, and θ3,k correspond to the rotator yaw, pitch, and roll angles recorded

during the test, respectively. Csr can similarly be expressed as the composition of three

principal rotations in 3-2-1 order

Csr =

1 0 0

0 cosα − sinα

0 sinα cosα

cos β 0 sin β

0 1 0

− sin β 0 cos β

cos γ − sin γ 0

sin γ cos γ 0

0 0 1

(2.11)

where α, β, and γ are respectively referred to as the tip, tilt, and clock angles, which

characterize the misalignment between the sensor frame and the rotator frame and remain

constant throughout the test.

The angle between the measured and true sun vectors is obtained by taking

δk = cos−1 sTs,kss,k (2.12)

The calibration variables are then determined by finding the values that minimize the

root-mean-square (RMS) of δk

δRMS =

√√√√ 1

N

N∑k=1

δ2k (2.13)

where N is the total number of measurement orientations. A complete derivation of the

UTIAS-SFL fine sun sensor calibration method can be found in [25].

Since only the actuator angles and centroid locations are recorded during each cal-

ibration test, the pinhole height h and bias b are not the only parameters estimated

in post-processing of the calibration data. Previously, the sun sensor calibration script

also estimated the three sensor misalignment angles α, β, and γ, while ρrli and ρrsr were

hard-coded to values that reflect an estimate of these quantities made in the past [25].

The calibration, stated as an optimization problem, is therefore

{c∗, h∗, α∗, β∗, γ∗} = arg minc,h,α,β,γ

δRMS. (2.14)


A sample plot of the output from this calibration script from the NORSAT-1 sun sensor

calibration campaign is shown in Figure 2.14. The color map shows the residual sun

direction measurement error across the sensor field of view resulting from the optimal

pinhole height and bias values found by the script, while the red points indicate the

centroid locations measured throughout the test. The pinhole height and bias found

by the script are shown above the plot, and fall within the expected range for these

parameters. The tip, tilt, and clock angles are not shown, but were found to be α =

−0.19◦, β = −0.13◦, and γ = 266.1◦. These are also within the expected ranges, with

the tip and tilt angles being small and the clock angle corresponding roughly to a 3/4

rotation about the rotator frame roll axis, which is how the sensor was mounted. The

full accuracy and inner 90% accuracy displayed above the plot refer to the root mean

square (RMS) error of the whole field of view and the inner 90% of the field of view,

respectively. These results suggest that the RMS accuracy of the sensor is 0.35◦ within

the inner 90% of the field of view.

Figure 2.14: Calibration Results, ρrli and ρrsr Hard-Coded

However, on-orbit performance of the sun sensors calibrated using this procedure has


shown that this accuracy is not being achieved, and that the actual accuracy is typically

around 2◦. This is shown using on-orbit data from one of the BRITE satellites in Figure

2.15. The angle between the sun direction measured by a sun sensor and that predicted

based on the star tracker attitude, which is considered much more accurate than the

sun sensor, are shown from both on-orbit data and a simulation. One suggested reason

Figure 2.15: On Orbit vs. Simulated Sun Sensor Error

for this discrepancy was that the values of ρrli and ρrsr implemented in the script were

wrong, possibly due to changes in the setup geometry since the values were measured.

It was therefore decided to investigate the possibility of estimating ρrli and ρrsr together

with the pinhole height and bias and sensor misalignment angles using the same data.

If successful, this would improve the accuracy of sun sensors flying on satellites that are

already on-orbit without recalibrating.

Estimating ρrli and ρrsr amounts to estimating the geometry of the triangle formed

by ~ρsl, ~ρrl, and ~ρrs. In general, this would be defined by 6 independent parameters.

However, since the calibration works with unit vectors for the sun direction, the true

lengths of the sides of this triangle are ignored by the calibration routine, which removes


one independent parameter. It was therefore decided to enforce ~ρrl to be unit length.

This constraint was enforced by defining a new frame F l and a new rotation matrix

Cil , ~F i · ~FT

l . F l shares an origin with F i at the rotator center of rotation and its axes

are such that ~l3 is colinear with ~ρrl, ~l2 lies in the plane defined by ~ρrl and ~i3, between

~ρrl and ~i3, and ~l1 completes the right-handed orthogonal triad. The components of ~ρrl

in F l are then simply ρrll =[

0 0 1]T

. The rotation between F l and F i can then be

expressed as a composition of two principal rotations, first about ~l1, and then about the

third axis of the intermediate frame so that

Cil =

cosφ − sinφ 0

sinφ cosφ 0

0 0 1

1 0 0

0 cosλ − sinλ

0 sinλ cosλ

(2.15)

This guarantees that the resulting vector estimated in F i is of unit length. The newly

added parameters to be estimated are therefore φ, λ, and the three components of ρrsr ,

corresponding to the 5 independent parameters that define the triangle described earlier.

The optimization problem for the calibration thus becomes

{c∗, h∗, α∗, β∗, γ∗, φ∗, λ∗,ρrsr∗} = arg min

c,h,α,β,γ,φ,λ,ρrsrδRMS. (2.16)

The output from the modified calibration routine is shown in Figure 2.16. The plot

shows that the residual error has been highly reduced, with the RMS error for the

inner 90% of the field of view roughly halved. Moreover, the error in the higher er-

ror regions near the edge of the field of view has been drastically reduced. However,

the pinhole height falls far outside of the expected range of 1 mm ± 0.1 mm that has

been observed for all past calibrations of identical sensors, including those performed

with an older setup that produced good results. The calibration also estimated that

ρrsr =[

0.00077 −0.0050 −0.28]T

, which suggests that ~ρrs is nearly one third of the

length of ~ρrl, where in reality this fraction should be an order of magnitude smaller. The

true length of ~ρrl is on the order of a few meters, whereas the true length of ~ρrs is just

a few centimeters. However, λ and φ were found to be 1.53◦ and 44.2◦, both believable

values. There was no appreciable change in the tip, tilt, and clock angles.

Given the unrealistic estimates of the pinhole height and the rotator-sensor vector,

these results are not expected to provide accurate measurement of the sun direction vector

on-orbit despite what the residual error plot might suggest, showing that there is still a an

error in the updated calibration method. It was found through further experimentation

that constraining the third component of ρrsr to fall within 0±0.1 resulted in an estimate


Figure 2.16: Calibration Results, ρrli and ρrsr Estimated

of −0.1, but the RMS accuracy would be degraded, with the higher error regions around

the edge of the field of view becoming visible once again. The estimated pinhole height

would also be reduced. Fixing the third component of ρrsr to a value closer to 0 would

result in a further reduction in the RMS accuracy and a proportional increase in the

pinhole height. There was no appreciable difference observed in the other parameters.

It appears, therefore, that the degree of freedom introduced by adding ρrsr allows the

optimizer to falsely compensate for an unmodeled effect at the edge of the field of view,

with an accompanying change in the pinhole height.

More work is needed in order to understand and address this problem. One possible

approach is to measure the true values of ρrsr and ρrli and hard-code them into the cali-

bration script. This is, as was the previous version of the calibration routine, susceptible

to changes in the test setup and measurement error. Another approach, more preferable,

is to run the optimization on data collected for a smaller portion of the field of view. The

challenge with this approach is in deciding how much of the field of view to exclude, and

would require further study of the problem to understand the nature of the unmodeled


effect. In either case, validation of the calibration can be performed by comparing to

results from an on-orbit calibration of the fine sun sensors on the BRITE mission.

2.4 Reaction Wheels


Reaction wheels are used to control a spacecraft’s attitude by transferring angular mo-

mentum between the wheels and the rest of the spacecraft. A reaction wheel consists of

a rotor of known moment of inertia spun by a motor about a single axis. Three reaction

wheels are mounted orthogonally to the satellite bus for UTIAS-SFL satellites in order

to provide control on three independent axes. The dynamics of a spacecraft with three

reaction wheels expressed in some reference frame F b fixed to the spacecraft body and

centered at the center of mass is governed by [26]

Ibωb + ω×b Ibωb + IsAbωs + ωb×IsAbωs = Gb, (2.17)

where Ib is the satellite moment of inertia, ωb is the satellite angular velocity, Is is the

reaction wheel moment of inertia, and Gb is the sum of the remaining torques on the

satellite. The elements of ωs =[ωs,1 ωs,2 ωs,3

]Tare the rates of rotation, or wheel

speeds, of each reaction wheel, with each column ab,j ofAb containing the rotation axis of

the corresponding wheel. The term Isωs,j is sometimes referred to as the wheel “torque”.

Note, though, that this does not correspond to the total torque applied by the wheel to

the satellite since it ignores the gyric term ωb×Isωs,jab,j. On UTIAS-SFL satellites, the

wheel speed is measured by differentiating output from a Hall effect encoder. Two sizes

of wheel are used. The smaller model can sustain a maximum Isωs = 30 mN m s and

has a moment of inertia of Is = 5.12× 10−5 kg m2. These are standard on the Generic

Nanosatellite Bus platform and are used by AISSat-3 and EV9. The larger model can

sustain Isωs = 60 mN m s and has a moment of inertia Is = 8.78× 10−5 kg m2. These are

standard on the NEMO bus platform and are used by GHGSat-D and NORSAT-1. The

reaction wheel axes are usually aligned with the satellite axes, such that Ab is usually

treated as simply the 3 × 3 identity matrix, though in reality there is also a mounting

misalignment, which is budgeted for in the mission pointing budget.

The attitude control system for UTIAS-SFL satellites is, in theory, a zero-momentum

system, meaning that the wheels are nominally non-spinning when the satellite is held

fixed relative to inertial space. However, in practice, each wheel is set to track 50 rad/s


while idling in order to avoid operating in the zero-crossing range where wheel speed

measurement accuracy is poor due to less frequent encoder output. A spacecraft will

also gradually accumulate angular momentum in orbit due to the action of disturbance

torques. This angular momentum must be absorbed by the wheels and eventually re-

moved from the spacecraft to avoid saturating the wheels. For this reason, UTIAS-SFL

satellites are augmented with magnetic torquers, or magnetorquers, as secondary actua-

tors to continuously dump angular momentum as it is absorbed.

The reaction wheel controller can be configured in “speed” mode to track a com-

manded wheel speed, or “torque” mode to track a commanded wheel torque. The torque

mode overlies the speed mode in that the commanded torque is achieved by commanding

a profile of speeds to the wheel. The speed mode itself overlies another control loop, the

current controller, which tracks a given current commanded by the speed control mode.

This is illustrated in Figure 2.17.

-Wheel

DynamicsMotor Drive

Stages

Current Control LoopCurrent Control Loop

+-+- +

Speed Control LoopSpeed Control Loop

Torque Control LoopTorque Control Loop

TrueWheelTorque

TrueWheelSpeed

True Current

CommandedCurrent

CommandedWheel SpeedCommanded

WheelTorque Current

ControllerSpeed

ControllerTorque

Controller

Figure 2.17: Reaction Wheel Control Modes

Reaction wheels are the most complex and, consequently, the most expensive piece of

ADCS hardware used on UTIAS-SFL satellites, but enable a satellite to achieve highly

accurate pointing and allow it to slew to a pointing target relatively quickly. Though they

are used on each of the four satellites discussed in this thesis, some missions developed

by the lab forgo the advantages provided by reaction wheels when requirements allow in

order to reduce cost and complexity, and opt to use magnetorquers or magnetic hysteresis

rods as primary actuators instead.

2.4.2 Testing Cycle

Reaction wheels undergo a reduced set of tests compared to the standard test cycle.

The typical acceptance testing campaign for a reaction wheel consists of an initial long


form functional test (LFFT), a thermal acceptance campaign, and a closeout LFFT. The

LFFT for a reaction wheel consists of a built-in short form functional test (SFFT), a

torque tracking error test, a noise over speed test, and a power consumption over speed

test. The wheels are delivered by the manufacturer as an assembled unit, and T-shock

testing is handled by the manufacturer.

This thesis contributed improvements to the torque error test, investigated issues with

timestamping in wheel test results, and assessed the effect of a current sensor bias on the

EV9 and AISSat-3 wheels.

2.4.3 Torque Test Improvements

The purpose of the torque test is to measure a reaction wheel’s performance in tracking

torque commands at various speeds. The reaction wheel is commanded to hold a certain

speed, and then issued a torque command. Wheel speed measurements are collected for

one minute while the torque is being applied. This is repeated for other combinations

of wheel speeds and torques. The particular speeds and torques are notionally chosen to

reflect mission requirements, but in practice are standardized for the 30 mNms wheels

and the 60 mNms wheels. In either case, the speeds consist of a low speed and a high

speed, and the torques similarly consist of a low torque and a high torque. The low

torques are chosen to be roughly representative of the torque required for nadir tracking,

while the high torque is chosen to be reflective of the torque needed for slewing. The

low speed is chosen as the idle speed of the wheels, reflecting the wheel speed at the

beginning of a maneuver, and the high speed is chosen as the highest speed expected

for the wheels during a slew, reflecting the wheel speed mid-maneuver. Previously, the

low and high torques were applied in both the negative and positive directions, but

only positive speeds were tested, giving a total of eight speed and torque combinations.

However, it was decided that both negative and positive speeds should be included for

completeness to give a new total of 16 cases. Table 2.2 gives the values of low and high

speed and torque used for the 30 mNms wheels and the 60 mNms wheels.

30 mNms 60 mNms

Low Speed (rad/s) 100 50High Speed (rad/s) 300 300Low Torque (Nm) 10−6 10−5

High Torque (Nm) 4× 10−4 6× 10−5

Table 2.2: Reaction Wheel Torque Test Speeds and Torques


For each segment of the test, a regression line is fit to the collected speed measurements

ωk, the slope of which is referred to as the nominal acceleration α, where the subscript k

refers to the kth measurement. Multiplying this by the wheel moment of inertia gives the

nominal torque τn = Isα. This should be close to the commanded torque τc. The wheel

speeds obtained by evaluating the best fit line at the sample times are correspondingly

referred to as the nominal wheel speeds ωn,k. The difference between the measured speeds

and the nominal speeds is referred to as the wheel speed error ωe,k = ωk − ωn,k. Torque

jitter is then defined as the derivative of the wheel speed error multiplied by the wheel

moment of inertia. This is obtained by numerically differentiating using a first difference

method

τe,k =Is

∆tk(ωe,k+1 − ωe,k) . (2.18)

Several metrics are calculated using these quantities. Previously, these included the

torque bias, the torque jitter mean, the torque jitter standard deviation, the equivalent

one second torque jitter standard deviation, the torque jitter fast Fourier transform (FFT)

peak, and the FFT peak frequency. These are defined in Table 2.3. In the table, fs is

the sampling frequency of the wheel speed measurements.

Metric Symbol Formula

Torque Bias τb τn − τc

Torque Jitter Mean τe1

N

N∑k=1

τe,k

Torque Jitter Std. Dev. στ,e

√√√√ 1

N − 1

N∑k=1

(τe,k − τe)2

Equiv. 1s Torque Jitter Std. Dev. στ,1 See Equation 2.26

Torque Jitter FFT Peak Te,max maxn=1...floorN

2

∣∣∣∣∣N∑k=1

τe,ke−j2π(k−1)(n−1)

N

∣∣∣∣∣Torque Jitter FFT Peak Frequency fp

fsN· arg max

n=1...floorN2

∣∣∣∣∣N∑k=1

τe,ke−j2π(k−1)(n−1)

N

∣∣∣∣∣Table 2.3: Reaction Wheel Torque Test Old Metrics

This thesis made updates to the torque jitter mean, equivalent one second standard

deviation, and the two FFT metrics.


Torque Jitter Mean

The purpose of the torque jitter mean metric was to measure the mean deviation of

the applied torque from the nominal torque [27]. The intent was to detect long-period

drifts from the nominal torque. However, it was discovered that this metric was only

reflective of the difference between the first wheel speed measurement and final wheel

speed measurement as well as the total measurement duration in each test segment, and

was therefore not a meaningful metric. This result can be shown by first substituting

Equation 2.18 into the formula for torque jitter mean in Table 2.3

τe =IsN

N∑k=1

1

∆tk(ωe,k+1 − ωe,k) (2.19)

Usually, ∆tk = ∆tl = ∆t ∀k, l, i.e. the sampling interval is uniform. Substituting in the

definition of ωe,k gives

τe =IsN∆t

N∑k=1

(ωk+1 − ωn,k+1 − ωk + ωn,k) (2.20)

=IsN∆t

N∑k=1

(ωk+1 − ωk)−IsN∆t

N∑k=1

(ωn,k+1 − ωn,k) . (2.21)

Working with the second term, in Equation 2.21, we find

IsN∆t

N∑k=1

(ωn,k+1 − ωn,k) =IsN

N∑k=1

α = Isα = τn. (2.22)

Now working with the first term in Equation 2.21, the middle terms in the summation

cancel and the summation collapses into

N∑k=1

(ωk+1 − ωk) = ωN+1 − ω1. (2.23)

Finally, substituting Equations 2.22 and 2.23 into Equation 2.21 gives

τe =Is (ωN+1 − ω1)

N∆t− τn, (2.24)

which contains no information about the data between the first and last wheel speed

measurements.

It was decided to replace this metric with the coefficient of determination, or R-


squared value, of the regression line fitted through the wheel speed measurements. This

serves the original intent of the torque jitter mean to detect long-period drifts, since

these drifts would also be apparent in the wheel speed measurements and be reflected

by a lower R-squared value, indicating a poor fit. Further work is needed to validate the

efficacy of this metric in detecting real long-period drifts.

One-Second Equivalent Torque Jitter Standard Deviation

The one second equivalent standard deviation metric is used in Mirage, UTIAS-SFL’s in-

house satellite attitude simulator, as the standard deviation of an additive white Gaussian

source of torque noise. Using this standard deviation, the noise source should, when

sampled at the simulation update frequency, generate torque noise that has a similar

effect on spacecraft pointing as the torque jitter measured in the torque test. For Mirage,

the simulation update frequency is 1 Hz. The usual torque jitter standard deviation

στ,e could not be used to define the noise source because aliasing results in larger-than-

expected pointing error on the next cycle.

Previously, the approach used to calculate this metric was to first partition the torque

jitter into M one-second segments, where M =⌊N∆t

1s

⌋and calculate the means of each

segment

τe,l =1

Nl+1 −Nl

Nl+1−1∑k=Nl

τe,k (2.25)

where l indicates the lth segment and Nl refers to the index of the first point in the lth

segment. The one-second equivalent standard deviation is then the standard deviation

of these means

στ,1 =

√√√√ 1

M − 1

M∑l=1

(τe,l −

1

M

M∑l=1

τe,l

)2

. (2.26)

στ,1 usually has values on the order of 10−6, which is much lower than στ,e, which typically

has values on the order of 10−4, as desired. This seems intuitively to be a valid approach,

but it was never shown that this would result in a noise source that would accurately

reflect the measured torque jitter, and it was suspected that this approach produced

results that are too optimistic. A more rigorous approach to this metric was desired.

One approach that was considered was to apply an anti-aliasing filter to the signal and

calculate the standard deviation of the filtered signal. The anti-aliasing filter is simply

an ideal low-pass filter with a cutoff frequency of 0.5 Hz, or one half the sampling rate.

The filtered signal is calculated by taking its convolution with the anti-aliasing filter. In


the frequency domain, this is a multiplication given by

Te,k = H0.5Te,k (2.27)

where Te,k is the discrete Fourier transform of τe,k obtained by taking the fast Fourier

transform of the latter, and H0.5 is the anti-aliasing low-pass filter defined by

H0.5 =

{1, −0.5 Hz < fk < 0.5 Hz

0, otherwise(2.28)

Taking the inverse discrete Fourier transform of Te,k to convert it back to the time domain,

and assuming that the torque jitter is an ergodic process it is possible to estimate the

standard deviation as desired using

στ,1 =

√√√√ 1

N − 1

N∑l=1

(τe,l −

1

N

N∑l=1

τe,l

)2

. (2.29)

The drawback to this approach is that much of the content of the noise is deleted by

the anti-aliasing filter. Figure 2.18 shows a typical plot of the FFT magnitude of the

torque jitter. The plot shows that most of the power of the noise is contained in frequen-

cies above 0.5 Hz. Using this method of calculating the one second equivalent torque

jitter standard deviation would mean that the effect of the noise at higher frequencies

would not be captured in the simulation, i.e. the noise is undersampled. Ideally, this

would be taken into consideration when choosing the simulator update frequency. How-

ever, changing the simulator update frequency would require substantial code changes in

addition to bogging down the simulation.

A time-domain approach was also conceived for calculating this metric. The idea

behind this approach was to calculate a crude estimate of the maximum satellite pointing

jitter standard deviation, and then solve for the value of the torque jitter metric that

would achieve the computed pointing jitter in the simulator. To begin, a one-axis model

of the spacecraft is used to simplify the attitude dynamics. This has the form

Iθ = τ, (2.30)

where I is the spacecraft moment of inertia, θ is the spacecraft attitude, and τ is the wheel

torque. Note that this model includes torque and pointing jitter. This is a reasonable

approximation of the satellite dynamics when performing target tracking slew maneuvers


0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

−5Torque Jitter Single−Sided Amplitude Spectrum of Wheel: 60−171 at 48HzParsing of commanded torque 1×10−5 Nm at 300 rad/s completed.

Frequency (Hz)

Tor

que

Jitte

r A

mpl

itude

(N

m/√H

z)

Figure 2.18: Reaction Wheel Torque Jitter FFT

during which the satellite only slews about one axis. The nominal jitter free dynamics

are given by

Iθn = τn, (2.31)

where θn and τn are the nominal spacecraft attitude and wheel torque, respectively.

Subtracting Equation 2.31 from Equation 2.30 gives the jitter dynamics

I(θ − θn

)= τ − τn (2.32)

Iθe = τe. (2.33)

Here, θe and τe are the pointing and torque jitter, respectively. Integrating Equation 2.33

twice over one-second segments and dividing by the moment of inertia gives the pointing

jitter over each segment

θe,l =1

I

∫ tl+1

tl

∫ t′

tl

τedtdt′. (2.34)

This can be related to the one-second equivalent torque jitter τe,1,l by observing that the

latter is treated by the simulator as a zero-order hold, represented here by τe,1 as it is


propagated through the attitude dynamics. It is desired that the values of τe,1 satisfy

θe,l =1

I

∫ tl+1

tl

∫ t′

tl

τe,1dtdt′. (2.35)

Since τe,1 is a zero-order hold of a 1 Hz discrete-time signal, it is constant over one-second

intervals. Thus,

θe,l =τe,1,1I

∫ tl+1

tl

∫ t′

tl

dtdt′ (2.36)

=τe,1∆t2

2I(2.37)

where ∆t = tk+1 − tk.Taking the standard deviation of θe,l gives

σθ,1,l =√⟨

θ2e,l

⟩− 〈θe,l〉2 (2.38)

where 〈·〉 is the expected value operator. Assuming the pointing jitter is ergodic, σθ,1,l =

σθ,1,m = σθ,1 ∀l,m. Substituting in Equation 2.37 gives

σθ,1 =

√√√√⟨(τe,1,l∆t22I

)2⟩−⟨τe,1,l∆t2

2I

⟩2

(2.39)

=

√∆t4

4I2

(⟨τ 2e,1,l

⟩− 〈τe,1,l〉2

)(2.40)

=∆t2

2I

√⟨τ 2e,1,l

⟩− 〈τe,1,l〉2 (2.41)

=∆t2

2Iστ,1 (2.42)

which can be rearranged to get

στ,1 =2I

∆t2σθ,1. (2.43)

In practice, using discrete data and again assuming the pointing jitter is ergodic, θe,l

is calculated from the wheel torque measurements by trapezoidal integration

θe,l =1

I

kl+1−1∑k′=kl

∆t′k2

(k′+1∑k=kl

∆tk (τe,k+1 + τe,k)

2+

k′∑k=kl

∆tk (τe,k+1 + τe,k)

2

)(2.44)


and σθ,1 is estimated using

σθ,1 =

√√√√ 1

M − 1

M∑l=1

(θe,l −

1

M

M∑l=1

θe,l

)2

. (2.45)

where M is the total number of one-second intervals. The axial moments of inertia of

the spacecraft are used for I in order to produce an estimate of the metric for each axis.

This method is similar to the old method of Equation 2.26, the key difference being

that the torque signal is integrated twice rather than just once. The advantage of this

method is that it attempts to capture the effect of the torque jitter on the spacecraft

pointing jitter. Additionally, by substituting the allowable pointing jitter from the mis-

sion requirements as σθ,1 into Equation 2.43, a rough estimate of the required one-second

equivalent torque jitter of the wheels can be calculated and compared to the value cal-

culated from measurements as a quick check of a wheel’s performance. However, this

method relies on approximate numerical integration of the torque jitter to estimate the

pointing jitter, and also neglects the effects of gyric torques since it relies on a one-axis

model.

A third method conceived for calculating the metric arrives at a new model for gen-

erating torque noise in Mirage rather than a formula for calculating the existing metric.

This method approaches the conversion between the standard deviation of the measured

torque jitter to an equivalent one-second torque jitter standard deviation as a discretiza-

tion problem. As with the previous approach, this method uses a single-axis model for

spacecraft attitude dynamics but with variables defined differently. Here, the model is

constructed with the form

Iθ = u+ w (2.46)

where I is again the spacecraft moment of inertia, θ is the spacecraft attitude, u is the

applied wheel torque, and w is the wheel torque jitter. w is assumed to be an additive

white Gaussian noise source, i.e.

w ∼ N (0, στ,e) , 〈w(t)w(t′)〉 = 0 ∀t 6= t′ (2.47)

where στ,e is the usual torque jitter standard deviation as defined in Table 2.3.

Defining a state x =[x1 x2

]so that x1 , θ and x2 , θ and output y = θ,


Equation 2.46 can be written in state-space form

˙[x1

x2

]=

[0 1

0 0

]︸︷︷︸

A

[x1

x2

]+

[01I

]︸︷︷︸B

u+

[01I

]︸︷︷︸E

w (2.48)

y =[

1 0]

︸︷︷︸C

x (2.49)

Multiplying both sides of Equation 2.48 by e−At gives

e−Atx = e−AtAx+ e−AtBu+ e−AtEw (2.50)

Now, moving the first term on the right side of Equation 2.50 to the left side, then

integrating over the sample period T ,∫ tk+T

tk

(e−Atx− e−AtAx

)dt =

∫ tk+T

tk

e−At (Bu+Ew) dt (2.51)

e−A(tk+T )xk+1 = e−A(tk)xk +

∫ tk+T

tk

e−At (Bu+Ew) dt (2.52)

xk+1 = eATxk +

∫ tk+T

tk

eA(tk+T−t) (Bu+Ew) dt (2.53)

where xk = x(tk). Now making a change of variables so that

τ = tk + T − t, dτ = −dt, (2.54)

Equation 2.53 becomes

xk+1 = eATxk +

∫ T

0

eAτ (Bu+Ew) dτ. (2.55)

The commanded torque u remains constant over each control cycle. The control cycle

also has a period of one second, which means that u remains constant over the integration

period and may be factored out of the integral

xk+1 = eATxk +

∫ T

0

eAτdτBuk +

∫ T

0

eAτEwdτ (2.56)


where uk = u(tk). Now,

eAT =

[1 T

0 1

](2.57)

and ∫ T

0

eAτdτ =

∫ T

0

[1 τ

0 1

]dτ =

[T T 2

2

0 T

](2.58)

Dealing with the second integral term in Equation 2.56 is more involved due to the

integration of a random variable. However, this can be simplified by treating the entire

term as a random variable wk with covariance

QT =

⟨(∫ T

0

eAτEwdτ

)(∫ T

0

eAτEwdτ

)T⟩(2.59)

Combining the two integrals gives

QT =

⟨∫ T

0

∫ T

0

eAτEw(τ)w(τ ′)ET eAT τdτdτ ′

⟩(2.60)

The order of the integral and the expectation can be switched since the expectation

operator is an ensemble average, whereas the integration is over the time domain, getting

QT =

∫ T

0

∫ T

0

eAτE 〈w(τ)w(τ ′)〉ET eAT τ ′dτdτ ′ (2.61)

However,

〈w(τ)w(τ ′)〉 =

{0, τ 6= τ ′

σ2τ,e, τ = τ ′

(2.62)

Equation 2.61 thus becomes

QT = σ2τ,e

∫ T

0

eAτEET eAT τdτ (2.63)

= σ2τ,e

∫ T

0

[1 τ

0 1

][0 0

0 1I2

][1 0

τ 1

]dτ (2.64)

= σ2τ,e

[T 3

3I2T 2

2I2

T 2

2I2TI2

](2.65)

Substituting Equations 2.57, 2.58, and 2.65 into Equation 2.56, it is possible to write the


discretized state space equations[x1,k+1

x2,k+1

]=

[1 T

0 1

][x1,k

x2,k

]+

[T T 2

2

0 T

][01I

]uk +

[w1,k

w2,k

](2.66)

=

[1 T

0 1

]︸︷︷︸

Ad

[x1,k

x2,k

]+

[T 2

2ITI

]︸︷︷︸Bd

uk +

[w1,k

w2,k

](2.67)

yk =[

1 0]

︸︷︷︸Cd

xk (2.68)

where [w1,k

w2,k

]∼ N

(0, σ2

τ,e

[T 3

3I2T 2

2I2

T 2

2I2TI2

])(2.69)

Note that, under the one-axis model for spacecraft attitude dynamics, Equations 2.67

and 2.68 are the exact discretized forms of Equations 2.48 and 2.49, i.e. the solution of

Equations 2.67 and 2.68 satisfy xk = x (tk) ∀k.

In order to express the equivalent one-second wheel torque jitter in a form that can

be used in Mirage, Equations 2.67 and 2.68 must be converted into a single difference

equation. To do this, first take z transforms of both sides to get

zX (z) = AdX (z) +BdX (z) +W (z) (2.70)

Y (z) = CdX (z) . (2.71)

Rearranging and solving for Y as a function of U and W ,

(z1−Ad)X (z) = BdU (z) +W (z) (2.72)

X (z) = (z1−Ad)−1BdU (z) + (z1−Ad)

−1W (z) (2.73)

Y (z) = Cd (z1−Ad)−1BdU (z) +Cd (z1−Ad)

−1W (z) (2.74)

where 1 is the identity matrix. Now,

z1−Ad =

[z − 1 −T

0 z − 1

](2.75)


(z1−Ad)−1 =

[z − 1 T

0 z − 1

]∣∣∣∣∣ z − 1 −T

0 z − 1

∣∣∣∣∣=

1

(z − 1)2

[z − 1 T

0 z − 1

](2.76)

Substituting Equation 2.76 back into Equation 2.74,

Y (z) =1

(z − 1)2

([1 0

] [ z − 1 T

0 z − 1

][T 2

2ITI

]U (z)

+[

1 0] [ z − 1 T

0 z − 1

]W (z)

) (2.77)

Rearranging eventually leads to

z2Y (z)− 2zY (z) + Y (z) =T 2

2IzU (z) +

T 2

2IU (z) + zW1 (z)−W1 (z) + TW2 (z) (2.78)

Taking the inverse z transform gives

yk+2 − 2yk+1 + yk =T 2

2Iuk+1 +

T 2

2Iuk + w1,k+1 − w1,k + Tw2,k, (2.79)

and rearranging to convert back into units of torque, as well as substituting back θ in

place of y,

I

T 2(θk+2 − 2θk+1 + θk) =

1

2uk+1 +

1

2uk + w1,k+1 − w1,k +

w2,k

T, (2.80)

where wk = IT 2wk with [

w1,k

w2,k

]∼ N

(0, σ2

τ,e

[1

3T1

2T 2

12T 2

1T 3

])(2.81)

The last three terms of Equation 2.80 can be added to the nominal wheel torque in

Mirage to generate torque jitter, and constitute the terms of the new torque model. The

covariances of these terms are calculated using the standard deviation of the measured

torque jitter and scaled by the sampling time. This model suggests that an accurate

discrete torque jitter model requires two statistically dependent Gaussian white noise

sources rather than a single noise source.

This approach of modeling the torque jitter is the most rigorous of the three discussed

in this section. The only simplification made in deriving the model is the use of one-axis


attitude dynamics. However, further work is needed to validate this model and compare

its performance against the others, as well as against the previous method of averaging

the torque jitter. This would involve implementing each of these methods in Mirage

and comparing the simulation results against real on-orbit data collected under the same

pointing conditions. Unfortunately, the time available for this thesis was insufficient to

complete this work due to more mission-critical hardware testing taking higher priority.

Fast-Fourier Transform Area

During the acceptance testing campaigns for the NORSAT-1 and GHGSat-D reaction

wheels, a low-frequency spike was observed in the torque jitter FFT magnitude plots

for the low-speed torque test cases. An example of this is shown in Figure 2.19. Such

low-frequency spikes indicate the presence of a low-frequency oscillation in the reaction

wheel torque jitter, which could have a noticeable impact on the satellite’s pointing

performance.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8x 10

−6Torque Jitter Single−Sided Amplitude Spectrum of Wheel: 60−173 at 97HzParsing of commanded torque 1×10−5 Nm at 50 rad/s completed.

Frequency (Hz)

Tor

que

Jitte

r A

mpl

itude

(N

m/√H

z)

Figure 2.19: Torque Jitter FFT Spike on NORSAT-1 Reaction Wheel

These spikes were only detected upon reviewing the plots generated by the test anal-


ysis script. Due to the large number of plots generated by the script, reviewing them is

a labor-intensive process. In order to help automate the detection of these torque jitter

FFT spikes, it was decided that an estimate of the area under the FFT plot up to 10 Hz

would be added to the torque test analysis. Since the magnitude of the Fourier transform

is the square root of the power spectral density (PSD) of a signal, this gives an estimate

of the power of the torque jitter up to 10 Hz. The cutoff frequency of 10 Hz was chosen

because the spacecraft attitude dynamics act as a low-pass filter, reducing the impact of

torque jitter at higher frequencies.

The area under the FFT magnitude plot is calculated by applying a trapezoidal

integration rule to the torque jitter FFT magnitude, i.e.

A =kc−1∑k=1

∆fk(Te,k+1 − Te,k)

2(2.82)

where fkc = 10 Hz. This calculation was validated by comparing reaction wheel test data

from the earlier NEMO-AM and NEMO-HD missions, which showed even larger spikes,

to GHGSat-D and NORSAT-1 data. A plot of the torque jitter FFT magnitude from a

NEMO-HD reaction wheel is shown in Figure 2.20. The FFT area for NEMO-HD and

NEMO-AM wheels was found to be typically on the order of 3×10−5 whereas that for the

NORSAT-1 and GHGSat-D wheels was found to be around 8× 10−6. Using GHGSat-D

mass properties, analysis has shown that the corresponding overall contributions of these

spikes to the pointing error are 2.5 × 10−3 ◦ and 7.5 × 10−4 ◦, respectively, which are

negligible compared to the overall pointing error of 0.14◦ allocated to the reaction wheels

in the pointing budget [12]. However, further work is needed in order to determine an

acceptance criterion for this metric, since the relationship between the FFT area and the

impact on pointing error is not straightforward. As an interim measure, future wheels

tested with this metric will be compared to results obtained with past wheels, with further

analysis being performed as necessary if it is found that any future wheels perform worse.

2.4.4 Timestamp Uniformity

The reaction wheel torque test was previously run twice in each LFFT: once with a

polling rate of 20 Hz and once with a polling rate of 60 Hz. The purpose of this was to

determine if any anomalies seen in the results were artifacts of poor timestamping. It

has been observed that the 60 Hz test often produces results with inaccurate or duplicate

timestamps, which substantially degrades the results. Although the 20 Hz test usually


0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3x 10

−5Torque Jitter Single−Sided Amplitude Spectrum of Wheel: 60−141 at 64Hz

Parsing of commanded torque 6×10−5 Nm at 50 rad/s completed.

Frequency (Hz)

Tor

que

Jitte

r A

mpl

itude

(N

m/√H

z)

Figure 2.20: Torque Jitter FFT Spike on NEMO-HD Reaction Wheel

produces relatively uniform timestamps, the low sampling rate results in aliasing that

makes it impossible to distinguish between low-frequency torque jitter anomalies that

could affect the satellite’s pointing performance and high-frequency anomalies that would

be filtered out by the spacecraft’s attitude dynamics. A plot of the delta times, or time

steps, between each timestamp for one segment of a 60 Hz torque test is shown in Figure

2.21.

Updates to UTIAS-SFL’s wheel interface software, WheelTerm, were attempted,

which substantially improved the consistency of the timestamps for high poll rates. Figure

2.22 shows a plot of the time steps for a 50 Hz torque test using the updated WheelTerm.

However, it was found that some time stamp jitter still occurs, and, furthermore, that

the results showed significant rates of packet loss. It was therefore decided to turn to

addressing this issue by improving the way post-processing scripts handled timestamps.

Previously, the scripts would either read the timestamps for the test from the dataset,

or hardcoded the timestamps as either 20 Hz or 60 Hz. The problem with the hardcoding

approach was that if a substantial number of data points were lost due to packet losses,

the total duration of the test segment would be substantially reduced. This would skew


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 500 1000 1500 2000 2500 3000 3500 4000

Tim

e S

tep

(s)

Measurement Number

Reaction Wheel Torque Test Time Step vs. Measurement Number

Figure 2.21: Old WheelTerm Time Step Variation, 60 Hz Torque Test

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 500 1000 1500 2000 2500 3000 3500

Tim

e S

tep

(s)

Measurement Number

Reaction Wheel Torque Test Time Step vs. Measurement Number

Figure 2.22: New WheelTerm Time Step Variation, 50 Hz Torque Test

the calculation of torque test metrics, in particular the torque bias. A new option was

therefore added to the analysis scripts to generate timestamps with uniform time steps


between them the entire duration of the test divided by the number of data points.

This change helped reveal a previously unknown phenomenon – that the torque bias

is proportional to the applied torque. Specifically, the torque bias has been observed

to consistently equal approximately 0.55% of the applied torque. This behavior will be

captured in future iterations of Mirage, the attitude simulator, to determine its impact on

pointing performance, though the impact is expected to be small if it is at all noticeable.

Chapter 3

Conclusion

This thesis has contributed acceptance testing of attitude determination and control

system (ADCS) hardware for the AISSat-3, EV9, GHGSat-D, and NORSAT-1 missions

being developed by the Space Flight Laboratory at the University of Toronto Institute

for Aerospace Studies (UTIAS-SFL), as well as improvements to these tests and related

development and analysis work. In particular, acceptance testing was performed for

magnetometers, angular rate sensors, sun sensors, reaction wheels, magnetorquers, and

the GHGSat-D star tracker, and key contributions were made to the acceptance testing

procedures of magnetometers, angular rate sensors, sun sensors, and reaction wheels.

Updates were made to the magnetometer functional test, including the elimination of

the permanent magnet from the polarity check and the addition of a rough bias check.

The former eliminated a source of risk for the missions on which the magnetometers will

fly by reducing the chance that the magnetometer would be damaged or permanently

magnetized during testing. The latter allows the magnetometer bias to be checked more

frequently throughout the acceptance testing campaign, which facilitates troubleshooting

of anomalies encountered during testing and reduces the risk of delays in the acceptance

testing campaign by detecting faulty sensors early in the test cycle. An outdoor ver-

ification of the magnetometer calibration was also added. This test verified that the

calibration error of the magnetometers for each of the missions discussed in this thesis

was well below the alloted error in their pointing budgets.

The rate sensor calibration procedure was simplified and expedited with a reduction in

the number of calibration orientations from 21 to 9. It was found that this reduced num-

ber of orientations provides a comparable level of accuracy to the original 21 orientations

in determining the calibration parameters, and cuts the time required for calibrating the

rate sensor roughly in half. Additionally, a correlation was discovered between the screw

torque in the rate sensor daughter boards and the bias on the X- and Y-axis sensors,

49

Chapter 3. Conclusion 50

which reside on the daughter boards. The cause of this was determined to be stress ap-

plied by the screws to the sensor packages, affecting their measurement of Coriolis force.

It was determined that this issue was of minimum risk to the missions on which the rate

sensors will be flying, and that it would be impractical to design a replacement for the

upcoming missions. Recommendations for addressing this problem in future revisions of

the UTIAS-SFL rate sensor were to either increase the size of the daughter boards, or to

use adhesives to bond the daughter boards to their enclosures.

Corrections were made to the fine sun sensor calibration procedure in order to improve

the accuracy of the calibration. This included adding a sun source misalignment angle and

the vector between the sensor and rotator center of rotation to the calibration variables

estimated by the post-processing script. However, it was found that this extra freedom

was being exploited by the script to incorrectly compensate for unmodeled effects at the

edge of the sun sensor field of view, and was therefore producing unrealistic, unusable

results. Future work would test the calibration on only measurements collected within

a subset of the field of view in order to avoid including regions where the calibration

model breaks down, and explore the possibility of measuring and hardcoding the offset

between the sensor and rotator center of rotation. The results of these approaches would

be validated against on-orbit calibration results, and is expected to help eliminate a 2◦

calibration error observed in on-orbit operation of the sun sensors.

Finally, many improvements were contributed to the torque test in the reaction wheel

long-form functional test (LFFT). These included the replacement of the torque jitter

mean metric with the more meaningful angular velocity coefficient of determination, de-

veloping three new approaches to modeling the reaction wheel torque jitter as a simulated

noise source, adding an estimate of the area under the torque jitter fast Fourier transform

(FFT) magnitude plot as a new metric, and improving the uniformity of timestamps in

the data collected for the test. Examining the angular velocity coefficient of determi-

nation in the reaction wheel torque test helps to detect long-term deviations in applied

torque when the wheel is commanded to apply a constant torque, the occurrence of

which would indicate a faulty wheel or an error in the test setup. The three methods of

modeling torque jitter represent more rigorous approaches to simulating reaction wheel

performance in Mirage, UTIAS-SFL’s in-house attitude simulator, than what was previ-

ously implemented, providing better confidence in the representativeness of the models

to the wheels’ actual performance. The FFT area metric would help detect low-frequency

torque jitter anomalies, which could have an impact on satellite pointing performance,

though it was shown that the particular anomalies detected for the wheels tested in this

thesis did not. Finally, improving the uniformity of the test data timestamps has helped

Chapter 3. Conclusion 51

to improve the accuracy of the analysis performed on the data. Future work would im-

plement the three approaches to modeling torque jitter in Mirage and validating each

against on-orbit results in order to determine which approach provides the most accurate

simulation. Assessment of the effectiveness of the FFT magnitude plot area metric is

also pending.

The contributions of this thesis will contribute in large part to ensuring that the

hardware tested in this thesis and in the future will function and perform as demanded

by the mission and system requirements of the missions on which they will be flying, and

that the test procedures used to test the hardware are efficient and thorough.

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