Design and fabrication of test-bed for testing attitude ...1077742/FULLTEXT02.pdf · Design and fabrication of test-bed for testing attitude determination of spin stabilized spacecraft

Design and fabrication of test-bed for

testing attitude determination of spin

stabilized spacecraft

Venkata Narayana Chowdary Kudaravalli

Space Engineering, masters level (120 credits)

2017

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

Design and fabrication of test-bed for testing attitude determination

of spin stabilized spacecraft.


January 2017

Matrikel Nr: 1952071

Supervisors:

Dr. Rees Fullmer,

Department of Mechanical and Aerospace Engineering

Utah State University, Logan, UT, USA

Dr. Sergio Montenegro,

Informationstechnik für Luft- und Raumfahrt Informatik VIII,

Julius-Maximilians-Universität Würzburg

.Dr. Leonard Felicetti,

Space Technology Department of Computer Science,

Electrical and Space Engineering

Luleå University of Technology.

i Spin stabilized satellite attitude determination

Abstract

Design and fabrication of test-bed for testing attitude determination of spin stabilized

spacecraft.

Main Supervisors:

Professor: Rees Fullmer,

Utah State University, USA.

Professor: Dr. Sergio Montenegro,

JMUW, Germany.

Attitude determination for a spin stabilized satellite is calibrated. A mock spacecraft and a

spin simulation test bed are designed and developed. Spin simulation testbed which provides

position data is used to acquire true position data. The data from the simulator test bed is used as

reference for the sensor data to estimate the error in position of sensor data. Two vector method

attitude solutions are used here for attaining the estimated position. Two vectors used for attitude

determination are magnetic field vector and sun sensor vector. Calibration of accuracy for sensors

is main goal, which is attained by calculating error by comparing the estimated position with true

position.

ii Spin stabilized satellite attitude determination

Acknowledgement

I would like to express my gratitude for Professor. Rees Fullmer for his guidance regarding

the thesis. Without Professor. Rees Fullmer’s support I couldn’t have overcome all the barriers to

produce a gratified extremity. I would also like to thank Professor. Charles Swenson (Director of

Space Engineering department, Utah State University) for suggesting me the changes in the design

which helped to produce a better design. A very special thanks to an important person who

provided his perfection for machining components, which provided me best products for the design

Terry Zollinger (MAE 138, old engineering building, Utah State University). I would also like to

thank Dr. Sergio Montenegro and Dr. Leonard Felicetti for supervising my thesis. A very special

to the guys who have been helping me to overcome the problems with software and matlab coding,

Thank you very much Jorden Luke, Dana Sorensen, Colton Lindstorm, Russell Bab and of course

my colleague Charles Pope. Finally to the person who has been my inspiration and motivation for

every work in my life for each and every second Mohana Rao Kudaravalli ( My Father).


iii Spin stabilized satellite attitude determination

Contents Page no

Abstract……………………………………………………………………………... i

Acknowledgement…………………………………………………………………... ii

List of Tables………………………………………………………………………... vii

Chapter 1. Introduction……………………………………………………………… 1

1.1. Attitude determination and stabilization of spacecraft………………... 1

1.2. Spin stabilization……………………………………………………… 1

1.3. Motivation…………………………………………………………….. 1

1.4. Overview……………………………………………………………… 2

Chapter 2. Electronics and sensors………………………………………………….. 4

2.1. Spacecraft/satellite sensor systems……………………………………. 5

2.1.1. Dowty TAM………………………………………………... 5

2.1.2. IMU………………………………………………………... 6

2.1.3. GPS………………………………………………………… 6

2.1.4. Camera…………………………………………………….. 7

2.2. Motor and encoder system…………………………………………….. 8

2.2.1. Motor………………………………………………………. 8

2.2.2. Encoder…………………………………………………….. 9

2.3. Power and operating systems…………………………………………. 10

2.3.1. Power bank………………………………………………… 10

2.3.2. Raspberry pi 2……………………………………………… 11

2.4. Electronics schematics…………………………………………………. 12

iv Spin stabilized satellite attitude determination

Chapter 3. Mechanical design and analysis………………………………………… 14

3.1. Introduction to CATIA………………………………………………… 15

3.2. Mechanical design and introduction…………………………………... 15

3.3. Design of test-bed……………………………………………………... 16

3.3.1. Motor………………………………………………………. 16

3.3.2. Shaft coupler………………………………………………. 17

3.3.3. Thrust bearing……………………………………………… 18

3.3.4. Spool………………………………………………………. 19

3.3.5. Motor mounting plate……………………………………… 22

3.3.6. Bearing case………………………………………………... 24

3.3.7. Entire test-bed assembly…………………………………… 27

3.4. Spacecraft design……………………………………………………… 28

3.5. Over all testbed-spacecraft setup………………………………………. 32

3.5. Numerical analysis……………………………………………………. 33

Chapter 4. Co-ordinate systems……………………………………………………... 34

4.1. A brief overview………………………………………………………. 34

4.2. CATIA simulations for orientation of sensor in body frame…………… 36

4.3. CATIA results failure for estimation of rotational matrix…………….. 37

Chapter 5. Sensors calibration………………………………………………………. 38

5.1. GPS……………………………………………………………………. 38

5.2. Encoder………………………………………………………………... 38

5.3. TAM…………………………………………………………………... 40

5.4. IMU…………………………………………………………………… 41

v Spin stabilized satellite attitude determination

5.5. Sunsensor…………………………………………………………….... 46

Chapter 6. TRIAD and Q methods overview and results…………………………… 52

6.1. TRIAD method………………………………………………………... 52

6.2. Conversion of rotation matrix to quaternion…………………………... 53

6.3. Wahba’s problem……………………………………………………… 55

6.4. Q-method……………………………………………………………… 56

6.5. Results………………………………………………………………… 58

Chapter 7. Conclusion and future work……………………………………………… 72

Appendix……………………………………………………………………………. 74

vi Spin stabilized satellite attitude determination

List of Tables

Chapter 2. Electronics and sensors used

2.1 Magnetometer specifications……………………………………………………..........

2.2 IMU specifications……………………………………………………………….................

2.3 Camera specifications…………………………………………………………………………

2.4 Motor specifications………………………………………………………………………….

2.5 Encoder specifications……………………………………………………………………….

2.6 Raspberry pi 2 specifications……………………………………………………............

Chapter 3. Mechanical design and analysis

3.1 Shaft coupler specifications……………………………………………………………….

3.2 Thrust bearing specifications………………………………………………………………

Chapter 5. Sensors calibration

5.1 Angular transformations from IMU to body frame…………………………….

5.2 Magnetic vector in inertial co-ordinate system………………………………….

5.3 Angular transformations from sunsensor to body frame……………………

5.4 Sunsensor vector in inertial co-ordinate system…………………………………

Chapter 6. Introduction TRIAD and Q- methods, results

6.1 TRIAD and Q mean error and stadard deviation with velocities……………

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vii Spin stabilized satellite attitude determination

List of figures

Page no:

Chapter 1. Introduction

1.1 Overview of entire setup…………………………………………………………………….

Chapter 2. Electronics and sensors used

2.1 Overview of entire setup (electronics)……………………………………………….

2.2 Dowty magnetometer……………………………………………………………………….

2.3 IMU-3DM-GX-3………………………………………………………………………………....

2.4 GPS FGPMMOPAH………………………………………………………………………………

2.5 ASI 120 mm lens camera…………………………………………………………………….

2.6 Motor GP 32 A 166156………………………………………………………..................

2.7 Encoder HEDS 5540…………………………………………………………………………….

2.8 Power bank…………………………………………………………………………………………

2.9 Raspberry pi 2…………………………………………………………………………………….

2.10 Spacecraft’s sensor schematics…………………………………………………………

2.11 Testbed schematics…………………………………………………………………………..

Chapter 3. Mechanical design and analysis

3.1 Test bed rough sketch…………………………………………………………………………

3.2 Motor modelled using CATIA V5 R18 ………………………………………………….

3.3 Wac 25-8mm-6mm shaft coupler……………………………………………………….

3.4 Thrust bearing…………………………………………………………………………………….

3.5 Views of spool…………………………………………………………………………………….

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viii Spin stabilized satellite attitude determination

Page no:

3.6 Spool dimensions 1…………………………………………………………………………….

3.7 Spool dimensions 2…………………………………………………………………………….

3.8 Spool with shaft coupler and thrust bearing……………………………………….

3.9 views of motor mounting plate……………………………………………….............

3.10 motor mounting plate dimensions……………………………………………………

3.11 Motor mounted to motor mounting plate…………………………………………

3.12 Bearing case………………………………………………………………………………......

3.13 Bearing case dimensions………………………………………………………………….

3.14 Fabricated bearing case……………………………………………………………………

3.15 Test bed assembly…………………………………………………………………………….

3.16 Mock spacecraft without sensors…………………………………………………….

3.17 Mock spacecraft with sensors……………………………………………….............

3.18 Mock spacecraft with sensors working model…………………………………

3.19 Overall experimental setup with sun location……………………………………

Chapter 4. Co-ordinate systems

4.1 Spacecraft with co-ordinate systems………………………………………………….

Chapter 5. Sensor calibration

5.1 Encoder data vs GPS time……………………………………………………………………

5.2 Dowty TAM vs GPS time…………………………………………………………………….

5.3 IMU magnetometer vs GPS time (a)…………………………………………...........

5.4 IMU magnetometer vs GPS time (b)……………………………………………………

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ix Spin stabilized satellite attitude determination

Page no:

5.5 Sun vector vs GPS time (a)………………………………………………………………….

5.6 Sun vector vs GPS time (b)………………………………………………………………….

5.7 Angular position (sun visible) of spacecraft vs GPS time……………….......

Chapter 6. Introduction TRIAD and Q- methods, results

6.1 (a). Angle error TRIAD method …………………………………………………………..

6.1 (b). Angle error TRIAD method with zeros eliminated…………………………

6.2 Angle error Q method…………………………………………………………………………

6.3 Angle error (Q and TRIAD) vs GPS time 1…………………………………………….




6.7 Angle error (Q and TRIAD) vs GPS time 5…………………………………............



6.10 Angle error (Q and TRIAD) vs GPS time 8…………………………………………..

6.11 GPS time vs Angle error…………………………………………………………………….

6.12 GPS time vs Angle error (1)……………………………………………………………….

6.13 GPS time vs Angle error (2)……………………………………………………………….

6.14 Velocity vs Mean angular error…………………………………………………………

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1 Spin stabilized satellite attitude determination

Chapter 1. Introduction

1.1 Attitude determination and stabilization of spacecraft/satellite:

The main aim for stabilizing a spacecraft is to keep its pointing direction as desired. Spin

stabilization and three axis stabilization are used for the orientation of satellites [1]. In spin

stabilization a satellite/spacecraft will be spinning around the axis of maximum moment of inertia

[2]. A Spin stabilized satellite/spacecraft stabilizes itself on the principal of gyroscopic effect [5]

[3][4].

1.2 Spin stabilization:

Spin stabilization involves spinning of the spacecraft/satellite around the axis of maximum

moment of inertia [4]. Usually the axis with maximum moment of inertia is assumed to be aligned

along the principal major axis of spacecraft/satellite. Spinning around principal major axis (axis

with maximum moment of inertia [5]) provides stability which is undisturbed by the other two axes,

this is due to conservation of angular momentum [5]. The angular momentum is the factor that

keeps the satellite/spacecraft to point in desired direction. Spinning satellites are mostly used in

astronomy [4] when scanning is required, here are some of the spin stabilized spacecraft/satellites:

1.3 Motivation:

Design a test-bed to create spinning motion. A motor provided with encoder is used for

providing spinning motion.

A mock spacecraft is designed and equipped with sensors capable of determining the

attitude.

The calibrated data from sensors are compared to encoder data to determine sensor attitude

accuracy.


1.4 Overview:

The current mission is to design a spacecraft/satellite test-bed with spin stabilization, for

Dr. Charles Swenson’s spacecraft program-upper atmospheric weather 3U cubesats. The

requirements here are spinning motion and sensor data. For spinning motion, a DC motor used,

spacecraft/satellite is coupled to the motor to achieve spinning. This motor should be equipped

with an encoder to provide rpm count and position of motor. Magnetometer, accelerometer and

gyroscope are the sensors used for the attitude determination, these sensors are mounted to the

spacecraft. The data from these sensors is compared to encoder data. This provides the accuracy

of the sensors; the rotation is assumed to be around the z-axis with an angular displacement of 𝛩.

Figure 1.1 Overview of entire setup


Let 𝛩𝑒𝑛𝑐𝑜𝑑𝑒𝑟 be the angular displacement from the motor encoder data, and 𝛩𝑠𝑒𝑛𝑠𝑜𝑟 be the

displacement calculated using the data from the sensor data, now

if 𝛩𝑒𝑛𝑐𝑜𝑑𝑒𝑟 = 𝛩𝑠𝑒𝑛𝑠𝑜𝑟 the sensors are accurate but this rarely true

𝐴𝑛𝑔𝑙𝑒 𝐸𝑟𝑟𝑜𝑟 = |𝛩𝑠𝑒𝑛𝑠𝑜𝑟 − 𝛩𝑒𝑛𝑐𝑜𝑑𝑒𝑟|

This error can be same for every reading or may be varying depending upon the sensor

preciseness and accuracy. The overview of each chapter is discussed below as follows.

Chapter 2 provides the details about sensors and electronics used.

Chapter 3 provides a detailed view of mechanical design and construction.

Chapter 4 provides an overview of the coordinate systems.

Chapter 5 provides an overview of sensor calibration.

Chapter 6 provides an overview of calculation of results.

Chapter 7 is the conclusion.


Chapter 2. Electronics and sensors

The aim of the entire thesis is to provide a spin simulation and to use sensors to predict the

position of the spacecraft at required instances. The Electronics are used here to provide spinning

motion, attitude determination and system that power up and control all of the entire electronics.

Requirements of sensors for current design is the first step to know about the electronics used.

Sensors are used in spacecraft and also for the motor for encoder data. The Electronics can be

subdivided in to three subsystems.

2.1. Spacecraft/satellite Sensor systems

2.2. Motor Encoder system (Ground station or Control unit)

2.3. Operating systems

Figure 2.1 Overview of the entire setup(electronics)


2.1 Spacecraft Sensor systems:

The sensor systems are used depending on the requirement. For current mission which is

a spinning spacecraft, accelerometer, magnetometer and rate gyro are required for acquiring

position data and a GPS is also used to provide time stamp for the position data. So the sensors

used here are as follows

2.1.1 Magnetometer:

The magnetometer here is used to estimate the magnetic field strength in all three axis, so

Dowty magnetics TAM 7 MK 2 [6] is used here. This three axis magnetometer works on the

fluxgate principal. The specifications are as follows:

Dynamic range +/- 60,000 nT

Output scaling 24,000 nT/V

Operating temperature -32 0C to +71 0C

Storage temperature -50 0C to +100 0C

Alignment accuracy < 0.20

Table 2.1 Magnetometer specifications

Figure: 2.2 Dowty Magnetometer


2.1.2 IMU:

The IMU (Inertial Measurement Unit) used here is 3DM-GX3-25 [7] Miniature attitude

heading reference system. It is a combination of triaxial accelerometer, triaxial magnetometer

and triaxial gyro.

Figure 2.3 IMU – 3DM-GX-3

Accelerometer Gyroscope Magnetometer

Range 3600 about all axes +/- 5 g +/- 2.5 Gauss

Table 2.2 IMU specifications

2.1.3 GPS:

The GPS is used here to provide a time stamp for the data from the sensors. Two GPS

are used here, one is used to provide a time stamp for the sensors used in spacecraft and the other

is used for providing a timestamp for the data from the motor encoder. FGPMMOPA6H [8] is the

GPS used here.

Figure 2.4 GPS FGPMMOPA6H


2.1.4 Sun sensor (or) Camera:

The purpose to use a camera to track the location of the sun.ASI 120mm [9] is the camera

used here, to trace the sun. The product description is as follows:

Resolution 1.2 Mega pixel

Bit depth 12 bits

Weight 120gma

Type Black and white

Table 2.3 Camera specifications

Figure 2.5 ASI 120mm lens camera


2.2 Motor and the Encoder system:

This motor encoder system is more or less like a ground station which powers up the spacecraft

to provide motion. The motor and the motor encoder used here are from Maxon a Swiss motor

company.

2.2.1 Motor:

The motor used here is a planetary gear motor. The following are the details of the motor

used here, Motor model and part number: GP (planetary gearhead) 32A (outer diameter in mm)

166156 (part number) [10].

Gear type Planetary gear

Gear ration 4.8:1

Max torque (continuous) 0.75Nm

Max axial load 120N

Max continuous speed 6000rpm

Input voltage 3.3

Table 2.4 Motor specifications

Figure 2.6 Motor GP 32A 166156


2.2.2 Encoder:

A twelve-bit encoder; HEDS 5540 [11]. is used here. This is used to acquire the position

data of the motor which is the true position of the spacecraft. This HEDS 5540 [11]. is mounted to

the planetary gear shaft which has a gear ratio of 4.8:1. So when motor has one revolution the gear

head makes 4.8 revolutions. The encoder reads 500 counts for 1 gear head revolution. Encoder is

a quad encoder so counts for 1 motor revolution are equal to 500 ∗ 4.8 ∗ 4 = 9600 counts and

counts for 1 gearhead revolution are equal to 500 ∗ 4 = 2000 counts.

Resolution 2000 counts for 1 gear head revolution/9600

counts for 1 motor revolution

Operational velocity 30,000 rpm

Operational acceleration 250,000 rad/sec2

Table 2.5 Encoder specifications

Figure 2.7 Encoder HEDS 5540


2.3. Power and Operating systems:

2.3.1 Power:

To power up all these sensors a power bank [12] of 10,000 mAh is used. This is capable of

powering up all the sensors, used for powering up the sensors and raspberry pi 2 which access the

data from the sensors. The power bank weighs around 300 grams. It has a volume of 202050 mm3.

Figure 2.8 Power bank


2.3.2 Operating system:

Raspberry pi 2 [13] is used here to acquire data from all the sensors; this raspberry pi 2

is used as system to monitor, control the sensors and acquire the data from them. Two raspberry

pi 2s are used here; one for the sensors along with the spacecraft and other to control motor and

encoder. The specifications of the raspberry pi 2:

Clock speed 900MHz

Ram 1GB

Processor BCM2836 ARMv7

Type 32 bit quad core processor

Table 2.6 Raspberry pi2 specifications

Figure 2.9 Raspberry pi2


Electronics Schematics:

The electronics are integrated as two systems.

Spacecraft’s sensor schematics.

Testbed schematics.

These schematics are drawn using Eagle Cad software with the help of Jorden Luke,

Dana Sorensen and Colton Lindstorm.

Figure 2.10 Spacecraft’s sensor schematics

Raspberry pi 2 is connected to 10,000 mAh battery. This battery provides pi to power up

with a voltage of 5V. This pi will power up all the sensors. All the sensors included in the

spacecraft are estimated to have a common input and output voltage of 5V, except GPS and TAM.

GPS has an input and output voltage of 3.3V. The GPS is directly connected to GPIO pins

which supply a voltage of 3.3 V. A level shifter is used to convert the output of the GPS to

5V.

TAM has an input voltage of 28V. A Boost converter is used for the TAM to convert the

5V to 28V. Output of the TAM varies from 0-5V, so a level shifter is used to convert the

output to 5V.


Figure 2.11 Testbed schematics

Raspberry pi is connected to an electric source which continuously provides the pi a voltage of

5V.

GPS is connected to GPIO pins here also and a level shifter is used for the output here as

well.

A control board is used for the motor which controls it by altering the transmission

voltage.

Encoder data is sent through the level shifter to keep the output voltage at 5V.


Chapter:3 Mechanical Design and its analysis

3.1 Introduction to CATIA:

Here CATIA (Computer Aided Three-dimensional Interactive Application) V5 R18 [14]

is used as designing and analysis tool for the test-bed. This is a tool to design objects in three-

dimensions, it helps an individual to design each part and assemble them later using assembly

feature. The parts can be designed individually using the part design feature. This software also

allows to analyze for the moment of inertia and center of gravity for the assembled model. The

main purpose of this software is to allow an individual to design, assemble individual designs

and analyze for structural analysis.

3.2 Mechanical design introduction:

The mechanical design is sketched concerning the requirements in motivation. The

requirements as stated in the motivation are Spacecraft mounted to the spinning test-bed. The

spacecraft is basically the design that holds the sensors and their control unit. The spinning test-

bed is a platform to provide the spacecraft spinning motion. The test-bed is provided with a

motor to which the platform or spinning spool is coupled using a shaft coupler. The motor is

attached rigidly to a table; the shaft of the motor is coupled to the spool. The spacecraft is fixed

rigidly to the spool for spinning simulation. So from the above concerns the requirements here

are

Mounting the motor rigidly to table

Coupling shafts of the motor and the spool, which provides frictionless motion

Design of spacecraft and mounting it to spool rigidly

So the main design is subdivided as

1. Design of test-bed

2. Design of spacecraft-location of sensors

3. Overall testbed-spacecraft design

4. Numerical analysis for the mechanical design


Figure 3.1 Test bed with spacecraft rough sketch


3.3 Design of test-bed:

Test-bed is a setup which is used to provide spinning motion using a motor. Components

used here are Motor, shaft coupler, spool and thrust bearing. So now in this section we will

describe the physical structure and assembly of test-bed. So now we will initiate with

components used and fabricated for the test-bed design assembly,

1. Motor

2. Shaft coupler

3. Thrust bearing

4. Spool

5. Motor mounting plate

6. Bearing case

7. Overview of assembly of test-bed

3.3.1. Motor:

Technical data regarding the motor is explained in a briefly in earlier chapter 2.

Schematic structure of the motor is drawn using the help of CATIA V5 R18 is shown here

Figure 3.2 Motor modelled using CATIA V5 R18


3.3.2 Shaft coupler:

Shaft coupler is used to couple the shafts of spool and the motor. Here wac25-8mm-

6mm [15] shaft coupler which has a tolerance of 5deg angular misalignment and 0.25 mm of

axial misalignment. This tolerance will allow spintable (or) spinning spool to have perfect

rotation without any wobbling even if there is a misalignment either in axial or angular ways.

The following describe more about the shaft couple.,

Figure 3.3 Wac25-8mm-6mm shaft coupler

Outside Diameter 25 mm

Length 30 mm

Major bore diameter 8 mm

Minor bore diameter 6 mm

Momentary Dynamic Torque 2.6 N

Speed 10000 RPM

Tolerate level of angular misalignment 5 deg

Tolerate level of parallel misalignment 0.25 mm

Tolerate level of axial misalignment 0.25 mm

Table 3.1 Shaft coupler specifications


3.3.3 Thrust bearing:

A thrust bearing is used to reduce the axial load on motor shaft and also for providing a

frictionless motion for the spool. Thrust bearing here used should support for axial load. So

angular-contact ball bearing, Trade No. 7204 [16], Single Row, for 20 mm Shaft Diameter is used

to provide the support for motor shaft. This thrust bearing is bought from mcmaster.com. The

product details are as follows.

Radial load capacity Dynamic 2990 lbs

Radial load capacity static 1715 lbs

Maximum speed 18,000 rpm

Table 3.2 Thrust bearing specifications

Figure 3.4 Thrust bearing


3.3.4 Spool:

The Spool is designed as a platform for the spacecraft which can be coupled to shaft of the

motor to provide spinning action. Spool is machined according to the CATIA design. Machining

of spool is done by Terrry Zollinger, 138 ME, Old Engineering building, USU. Raw material used

is a solid aluminum shaft. The design of the spool is as follows.

Figure 3.5 Views of spool

Dimensions of all the parts are represented in mm. Components are designed according to

SI system. So all dimensions which will be mentioned in the entire document will be denoted in

millimeters(mm). This spool will be mounted between spacecraft and the motor.


Figure 3.6 Spool dimensions 1

Figure 3.7 Spool dimensions 2


Figure 3.8 Spool with shaft coupler and thrust bearing


3.3.5 Motor mounting plate:

The Motor mounting plate is the mount to hold motor firmly under the table. The design

of this mounting plate is as follows.

Figure 3.9 Views of motor mounting plate

As show in the above figure, we notice four bigger holes used for mounting the plate to

the table. There is a small clearance at the center; this is for the motor screw heads which will

not affect the mount of the plate. The small holes within the clearance are used for mounting

the motor. As said earlier the design specifications in the entire report will be denoted in mm

(millimeters).

Figure 3.10 Motor mounting plate dimensions


Figure 3.11 Motor mounted to motor mounting plate


3.3.6 Bearing case:

This is designed to provide a rigid support to the bearing when it is fixed to the rotating

spool. This bearing case is machined using aluminum with reference to the design in CATIA.

This bearing case is provided with four holes near each corner which will match with the motor

mounting plate exterior holes. This bearing case is machined as two parts instead of one part so

it can provide the clearance to the shaft coupler during assembly. So the design is done

considering all the assembly limitations,

Figure 3.12 bearing case

As shown in the above figure bearing case is machined as two parts, this makes it

easier to couple the spool shaft and motor shaft during assembly. Two parts are designed in

such a way that one part is a mirror design of the other.


Figure 3.13 Bearing case dimensions


Figure 3.14 Fabricated bearing case

SPIN-AXIS ATTITUDE DETERMINATION


3.3.7 Entire test-bed assembly:

All parts are mounted together for assembly of test-bed. As shown in figure the parts are

labelled in the CATIA assembly, because each and every part is clearly visible. This visibility in

CATIA is due to a feature called transparency for objects.

Figure 3.15 Test-bed assembly


3.4 Spacecraft design:

The design of the spacecraft is initiated using the specific dimensions of each sensor and

positioning them. Positioning the sensors is dependent upon the sensitivity of each sensor, the

sensors are as following:

GPS:

GPS is located on the top of spacecraft above all sensors. The reason for positioning GPS

on top is it should not have any obstacles that will affect its sensitivity.

IMU and TAM:

TAM is a triaxial magnetometer which is sensitive to the magnetic field created when the

motor spins; this is the same with the triaxial magnetometer of IMU. Magnetic field generated by

the motor can affect these sensors, so these sensors are placed at distance of 180 mm from the

motor. The IMU is placed in such a way that it is just right above the TAM.

Raspberry Pi 2:

Raspberry pi 2 is placed quite below the TAM and it is positioned in an inverted way, so

that it is available for every sensor to be connected with this control board. An additional board is

used along with this raspberry pi 2 to power up the sensors.

CMOS camera:

Camera is not much affected by motor; the main requirement is sun should be visible

enough to the camera so that it can trace the sun while spinning. The camera is located quite above

the power bank.

Power Bank:

Power bank weighs around 300 grams, so this is preferred to be placed at the bottom of

spacecraft. Power bank is located in such a way that the center of mass of this battery is aligned

with the spinning axis of the spacecraft.


Figure 3.16 Mock spacecraft without sensors


Figure 3.17 Mock spacecraft with sensors


Figure 3.18 Mock spacecraft with sensors working model


3.5 Over all testbed-spacecraft setup:

The spacecraft is mounted to the spool using the five screw holes which are available on

both spool and spacecraft. These five holes in spool are threaded with a screw pitch of 0.05

inch/thread (or) 1.27 mm/thread. These holes in spacecraft have allocated spaces for the head of

the bolts to sit inside. After mounting together sun (LED) is place at a distance of 30-50 centimeters

to be visible for the camera to track the sun. The overall setup is as shown in the following figure.

Figure 3.19 Overall experimental setup with sun location.


3.6 Numerical analysis:

The spacecraft spins around Z-axis. There are no deployments, or any extensions so this

doesn’t let to choose movable masses. So the concept of movable masses is omitted here. The

design is initiated as if location of sensors will try to maintain the center of gravity within the

spinning axis; this will provide a balance to the spacecraft.

The center of gravity is simulated for the spacecraft the results are:

𝐺 = (0, 0, −87.3)

The spacecraft spins around z-axis which means motor should be able to produce enough

torque, so that this torque is enough to overcome torque [17] around z-axis for the spacecraft and

spool mount.

𝜏 = 𝐼𝑍 ∗ 𝛼

Where Iz is the moment of inertia around Z-axis or XY-plane for the spacecraft and

spool mount. The reason for simulating moment of inertia for the spacecraft and the spool mount

is motor should be capable of spinning this mounting section. α is the angular acceleration. The

maximum continuous torque for the motor.

The maximum continuous torque of the motor is 0.75 N-m from the motor table

𝐼𝑍 is moment of inertia along Z-axis of the entire spacecraft 0.05 kg-m2

Angular acceleration [18] can be calculated using the formula.

𝛼 =𝜏

𝐼𝑍

𝛼 = 15 𝑟𝑎𝑑/𝑠𝑒𝑐2

This angular acceleration is relatively more than required so the motor is fine enough to

rotate the spacecraft. The motor has limit regarding axial load; as of here load on the motor is acted

axially through shaft coupler which holds the spool shaft and the motor shaft firmly. So axial load

is analyzed using CATIA V5 R 18. Weight of the spacecraft when all the masses are equipped is

1.84795 kilograms almost equal to 2 kilograms. Axial weight (𝐴𝑊𝑠) acted by spacecraft is

𝐴𝑊𝑠 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 ∗ 𝑚𝑎𝑠𝑠

𝐴𝑊𝑠 = 9.81 ∗ 2 = 19.62 Axial weight can be accepted by the motor from table (2.1) is

120 N. so from the above calculations we can say that the motor is capable of rotating the spacecraft

with its axial weight. Entire assembly looks like.


Chapter. 4 Co-ordinate systems

4.1 A brief overview:

Attitude of spacecraft is defined as rotational orientation of the spacecraft body frame to

an inertial fixed frame. So the axes definition are important.

Figure 4.1 Spacecraft with co-ordinate systems


Inertial Co-ordinate system:

This lies exactly along with the body co-ordinate system. This is not stationary along

with the body co-ordinate system [19] [1]. This is used as a reference for body frame to count the

rotations. When the body co-ordinate frame is at home position then inertial co-ordinate system

perfectly coincides with inertial co-ordinate system.

Body Co-ordinate system:

The body co-ordinate frame coincides with the inertial frame when the motor or the

spacecraft is at home position [1] [19]. The alignment of this body co-ordinate system coincides with

the Z-axis of the inertial co-ordinate frame.

OB =OI at home position

Sensor Co-ordinate system:

This is defined as the coordinate system that represents the orientation of each induvidual

sensor. The sensor frame is used as reference to estimate the position of each sensor with body and

inertial coordinate frames. These are nothing but coordinates frames that use geometric centre of

sensor as orgin of the the coordinate system.

IMU Co-ordinate system:

The IMU is located quite below the GPS. The geometric centre of this IMU coincides with

the centre of gravity. The origin for the IMU co-ordinate system is estimated using simulations in

CATIA V5 R 18. This simulation for the IMU is done in such a way that it provides centre of

gravity for the IMU when mounted on the spacecraft with reference to spacecraft body co-ordinate

frame.

CMOS camera or Sun sensor co-ordinate system:

Focal point for every camera lens is the place of reference for its pictures. This point lies

exactly at the centre of the camera lens. Here in the current spacecraft/satellite the focal point of

this camera is assumed to be at the chip or the image sensor and which lies inside the camera and

exacly aligned with the lens of the camera.


4.2 CATIA simulations for orientation of sensor in body frame:

A rotational matrix which represents the sensor orientation in body coordinate frame is

calculated. For this we try to relate the origin of the sensor coordinate frame to body coordinate

frame. In other words, we represent the sensor coordinate system origin in body coordinate system.

The geometric center of sensor which is assumed to be the origin of the co-ordinate system is

found using CATIA. This is equated to the origin of the entire spacecraft which is spinning as a

product of rotation matrix [1] [19] [20] 𝑅𝐵𝑜𝑑𝑦𝑆𝑒𝑛𝑠𝑜𝑟 .

Now the equation looks like

𝑂𝑆𝑒𝑛𝑠𝑜𝑟 = (𝑥, 𝑦, 𝑧) in body coordinate system

𝑂𝐵 = 𝑅 ∗ 𝑂𝑆𝑒𝑛𝑠𝑜𝑟𝐵𝑜𝑑𝑦𝑆𝑒𝑛𝑠𝑜𝑟 Equation (4.1)

Where 𝑂𝐵 is origin of the body co-ordinate system

𝑂𝑆𝑒𝑛𝑠𝑜𝑟 is origin of the sensor co-ordinate system

𝑅𝐵𝑜𝑑𝑦𝑆𝑒𝑛𝑠𝑜𝑟 rotation matrix from sensor frame to body frame

IMU:

𝑂𝐵 = 𝑅 ∗ 𝑂𝐼𝑀𝑈𝐵𝑜𝑑𝑦𝐼𝑀𝑈 Equation (4.2)

𝑂𝐵 = 𝑅(𝛷, 𝜃,𝛹) ∗ 𝑂𝐼𝑀𝑈 Equation (4.3)

Where Φ, θ and Ψ are Euler angles [1] [19] [20] representing the rotation sequences around x,

y and z axes

After solving the equation

𝛷 = 0, 𝜃 = 0, 𝛹=0

Sun sensor:

𝑂𝐵 = 𝑅 ∗ 𝑂𝑆𝑢𝑛 𝑆𝑒𝑛𝑠𝑜𝑟𝐵𝑜𝑑𝑦𝑆𝑢𝑛 𝑆𝑒𝑛𝑠𝑜𝑟 Equation (4.4)


𝑂𝐵 = 𝑅(𝛷, 𝜃,𝛹) ∗ 𝑂𝑆𝑢𝑛 𝑆𝑒𝑛𝑠𝑜𝑟

Equation (4.5)

Results after solving equation (4.5)

𝛷 = 90, 𝜃 = 0, 𝛹= -90

4.3 CATIA results failure for estimation of rotational matrix:

Reasons:

The simulations are done on the model designed in CATIA. The CATIA design or

the model which is designed in CATIA is done perfectly during its design because its creation or

the design is virtual. There is a mismatch between virtual design and real fabricated design. The

sensor location is perfect without any misalignments. The sensors are virtually assembled in

CATIA without any alignment errors, because of this the CATIA simulations are not satisfactory.


Chapter 5. Sensors and Calibration.

Sensors:

The data is extracted from all the sensors and formatted in such a way that the data from

every sensor have same sampled frequency. When we have all the data in same frequency it will

provide an ease in determination of attitude. Here this section will provide a view of data of

different sensors and the data used for the attitude determination.

5.1 GPS:

GPS is used only for the timestamp reference and not for determination of attitude here.

This time stamp is used to compare the data of each different sensor.

5.2 Encoder:

Encoder provides the true angular rotation or the true attitude. The data from the encoder

is shown as follows using GPS time stamp as reference

Figure 5.1 Encoder data vs GPS time


The position is varied in the form of a cyclic rotation as shown in the above plot. The

encoder data is provided in degrees here which is clearly indicated on the measuring bar of y-axis.

From the above data it is clear that the encoder provides data regarding position of the spinning

spacecraft for a particular period of time.

Let this angle be 𝛹𝑇𝑅𝑈𝐸 , which means the true angular position

Rotation matrix [21] for body to inertial from using 𝛹𝑇𝑅𝑈𝐸

𝑅𝑇𝑅𝑈𝐸 = [

𝐶𝑜𝑠 (𝛹𝑇𝑅𝑈𝐸) 𝑆𝑖𝑛 (𝛹𝑇𝑅𝑈𝐸) 0

−𝑆𝑖𝑛 (𝛹𝑇𝑅𝑈𝐸) 𝐶𝑜𝑠 (𝛹𝑇𝑅𝑈𝐸) 00 0 1

]

Equation (5.1)

Now this is converted or represented in the form of Quaternion [22]

[

1 1 −11 −1 111

−11

−11

−1−111

] ∗

[ 𝑞0𝑇𝑟𝑢𝑒

2

𝑞1𝑇𝑟𝑢𝑒2


𝑞3𝑇𝑟𝑢𝑒2 ]

= [

𝑟11

𝑟22𝑟33

1

] = [

𝐶𝑜𝑠 (𝛹𝑇𝑅𝑈𝐸)𝐶𝑜𝑠 (𝛹𝑇𝑅𝑈𝐸)

11

]

Where 𝑄𝑇𝑟𝑢𝑒= [

𝑞0𝑇𝑟𝑢𝑒

𝑞1𝑇𝑟𝑢𝑒𝑞2𝑇𝑟𝑢𝑒


]

Equation (5.2)

[ 𝑞0𝑇𝑟𝑢𝑒

2



𝑞3𝑇𝑟𝑢𝑒2 ]

= 1

4∗ [

1 1 11 −1 −1

−1−1

1−1

−11

1111

] ∗ [

𝐶𝑜𝑠 (𝛹𝑇𝑅𝑈𝐸)𝐶𝑜𝑠 (𝛹𝑇𝑅𝑈𝐸)

11

]


5.3 TAM:

Although the Dowty TAM magnetometer data is not used as one of the vectors in the

attitude solution, the data can be used as reference for the magnetometer data from IMU. The data

from the Dowty TAM magnetometer is plotted using matlab with reference to GPS time stamp

data.

Figure 5.2 Dowty TAM vs GPS time

The reason for not using the data from Dowty TAM is due to amount of error in angle is

high compared to IMU. The Dowty TAM data vector Q and TRIAD methods results for first three

data sets can be found in appendix page numbers: 85 and 83.So IMU TAM data is preferred over

Dowty TAM. This due to the influence of camera which perturbs TAM data when spacecraft is not

in motion (or) when motor has no power.


5.4 IMU:

The IMU has three sensors in it- accelerometer, gyroscope and magnetometer. The current

thesis uses the magnetometer data from the IMU as a vector for attitude determination. IMU data

is plotted with reference to timestamp of GPS as shown

Figure 5.3 IMU magnetometer vs GPS time (a)

Data plotted in the above plot is in IMU co-ordinate system. Trying to convert this data

from IMU co-ordinate frame to Body frame results in z component of the magnetometer to be

constant. This is because the spacecraft is spinning around z-axis which means the magnetic field

varies in xy-plane. The IMU magnetic vector in body frame [1] [19] [25] is

𝐵1 = [𝑉𝐵 ] = [ 𝑅𝐵

𝐼𝑀𝑈 ] ∗ [𝑉𝐼𝑀𝑈 ]

Equation (5.3)


Here 𝑉𝐵 vector matrix in body co-ordinate system, 𝑅𝐵

𝐼𝑀𝑈 is rotation matrix from IMU co-

ordinate frame to body frame and 𝑉𝐼𝑀𝑈 is vector in IMU co-ordinate system.

[

𝑉𝐵𝑥

𝑉𝐵𝑦

𝑉𝐵𝑧

] = [𝑅(𝛷, 𝜃, 𝛹)] ∗ [

𝑉𝐼𝑀𝑈𝑥

𝑉𝐼𝑀𝑈𝑦

𝑉𝐼𝑀𝑈𝑧

]

Equation (5.4)

Euler’s rotation matrix is equal to

𝑅(𝛷, 𝜃,𝛹) = [𝐶𝜃 ∗ 𝐶𝛹 𝑆𝜃 ∗ 𝑆𝛷 ∗ 𝐶𝛹 − 𝐶𝛷 ∗ 𝑆𝛹 𝑆𝜃 ∗ 𝐶𝛷 ∗ 𝐶𝛹 + 𝑆𝛷 ∗ 𝑆𝛹𝐶𝜃 ∗ 𝑆𝛹 𝑆𝜃 ∗ 𝑆𝛷 ∗ 𝑆𝛹 + 𝐶𝛷 ∗ 𝐶𝛹 𝑆𝜃 ∗ 𝐶𝛷 ∗ 𝑆𝛹 − 𝑆𝛷 ∗ 𝐶𝛹

−𝑆𝜃 𝐶𝜃 ∗ 𝑆𝛷 𝐶𝜃 ∗ 𝐶𝛷]

Equation

(5.5)

𝐶𝜃 = 𝐶𝑜𝑠(𝜃) and 𝑆𝜃 = 𝑆𝑖𝑛(𝜃), Considering 𝛹 = 0

[

𝑉𝐵𝑥

𝑉𝐵𝑦

𝑉𝐵𝑧

] = [𝐶𝜃 𝑆𝜃 ∗ 𝑆𝛷 𝑆𝜃 ∗ 𝐶𝛷0 𝐶𝛷 −𝑆𝛷

−𝑆𝜃 𝐶𝜃 ∗ 𝑆𝛷 𝐶𝜃 ∗ 𝐶𝛷] ∗ [

𝑉𝐼𝑀𝑈𝑥

𝑉𝐼𝑀𝑈𝑦

𝑉𝐼𝑀𝑈𝑧

]

Equation (5.6)

After the matrix multiplication the equations are

𝑉𝐵𝑥 = 𝐶𝜃 ∗ 𝑉𝐼𝑀𝑈𝑥

+ 𝑆𝜃 ∗ 𝑆𝛷 ∗ 𝑉𝐼𝑀𝑈𝑦 + 𝑆𝜃 ∗ 𝐶𝛷 ∗ 𝑉𝐼𝑀𝑈𝑧

Equation (5.7)

𝑉𝐵𝑦 = 𝐶𝛷 ∗ 𝑉𝐼𝑀𝑈𝑦

− 𝑆𝛷𝑉𝐼𝑀𝑈𝑧

Equation (5.8)

𝑉𝐵𝑧 = −𝑆𝜃 ∗ 𝑉𝐼𝑀𝑈𝑥

+ 𝐶𝜃 ∗ 𝑆𝛷 ∗ 𝑉𝐼𝑀𝑈𝑦 + 𝐶𝜃 ∗ 𝐶𝛷 ∗ 𝑉𝐼𝑀𝑈𝑧

Equation (5.9)


To attain the rotational transformation to convert the vector in IMU frame to body co-

ordinate frame, the system is run several times to attain the sensor data. This data is calibrated

using least squares method and z component of magnetometer (𝑉𝐼𝑀𝑈𝑧 ) is made constant. This

provide transformation rotations to attain a vector in body coordinate system is made almost

constant. From these tests the angular transformations to convert vector in IMU frame to body

frame are calculated using the above stated equations for several tests and their mean is used as

the angular transformations of the IMU [1] [19] [25]. The angular transformations to convert vector in

IMU frame to body frame (courtesy of Charles Pope Space master round 10)

Test number 𝛷 (degrees) 𝜃 (degrees)

1 -0.09 0.37

2 -0.08 0.4

3 -0.07 0.4

4 -0.06 0.38

5 -0.06 0.4

6 -0.05 0.4

7 -0.06 0.4

8 -0.06 0.4

Table 5.1: Angular transformations from IMU to body frame

𝛷 = 𝑚𝑒𝑎𝑛 𝑜𝑓 𝛷

𝜃 = 𝑚𝑒𝑎𝑛 𝑜𝑓 𝜃

𝛹 = 0

Result is

𝛷 = −0.07 𝑑𝑒𝑔

𝜃 = 0.40 𝑑𝑒𝑔

Standard deviations are calculated for the above data

Standard deviations for 𝛷 = 0.01 and 𝜃 =0.01


Now the rotation matrix is used to convert the data from IMU co-ordinate system to the

body co-ordinate system. This converted data is plotted again using Matlab with GPS time stamp,

the plot is as shown.

Figure 5.4 IMU magnetometer in body co-ordinates vs GPS time (b)

As shown in the above plot after converting the data to body frame, the z-component of

the magnetic field is almost constant. The magnetometer data is considered to be the first reference

vector in body co-ordinate system which 𝐵1 and now the corresponding inertial frame vector is

calibrated in the same way as for the misalignment calibration for 𝛷 and 𝜃.


System is run several times to attain the sensor data and the inertial vectors are calibrated

whenever the mock spacecraft reaches its home position, as the inertial coordinate frame coincides

with the body coordinate frame at this position. The magnetic vectors in inertial frame (courtesy

of Charles Pope Space master round 10)

Test Number X-component(Gauss) Y-component(Gauss) Z-component(Gauss)

1 -0.04 -0.06 0.64

2 -0.04 -0.06 0.64

3 -0.04 -0.06 0.64

4 -0.04 -0.06 0.64

5 -0.04 -0.06 0.64

6 -0.04 -0.06 0.64

7 -0.04 -0.06 0.64

8 -0.04 -0.06 0.64

Table 5.2: Magnetic vector in inertial co-ordinate system

𝐼1 = [−0.04 − 0.06 0.64] (Gauss)

𝐼1 = [−4 ∗ 10−6 − 6 ∗ 10−6 64 ∗ 10−6] (Tesla)


5.5 Sun sensor:

The data from the sun sensor is in the form of binary output. These numbers build up a

matrix which represent each pixel of image. Every image used here from the camera is 320x320

pixels, and each pixel consists of a 320x320 matrix. So for the values are acquired from the cosmos

and these values are formatted and represented in the form of vectors in each axis. These vectors

are calculated and used for calibrating the sun sensor. Now this matrix is calibrated, threshold is

set as50, usually the camera frame has its pixel value varying from an intensity of 0-225, so the

pixels which have intensity under 50 are considered to be zeros. Now this data is calibrated and

weighted centroid intensities are normalized. Which means the value of the highest intensity pixel

centroid which is 225 is assumed to be equal to 1. So now the data is in form of each frame having

the value of its weighted centroid in all the three axis.

Figure 5.5 Sun vector vs GPS time (a)


𝐵2 = �� 𝐵𝑜𝑑𝑦 = [ 𝑅𝐵𝑜𝑑𝑦𝑆𝑢𝑛𝑠𝑒𝑛𝑜𝑠𝑜𝑟 ] ∗ [�� 𝑆𝑆]

Equation (5.10)

Where �� 𝐵𝑜𝑑𝑦 and �� 𝑆𝑆 [1] [19] [25] represent the matrices of vectors of body and sunsensor,

𝑅𝐵𝑜𝑑𝑦𝑆𝑢𝑛𝑠𝑒𝑛𝑜𝑠𝑜𝑟 is the rotation matrix to transform a vector in sun sensor frame to body frame

[

�� 𝐵𝑜𝑑𝑦_𝑥

�� 𝐵𝑜𝑑𝑦_𝑦

�� 𝐵𝑜𝑑𝑦_𝑧

] = [𝑅(𝛷, 𝜃,𝛹)] ∗ [

�� 𝑆𝑆_𝑥

�� 𝑆𝑆_𝑦

�� 𝑆𝑆_𝑧

]

Equation (5.11)

𝛷, 𝜃, 𝛹 are the Euler angles used in the transformation of Body frame to sun sensor

frame.

[

�� 𝐵𝑜𝑑𝑦_𝑥

�� 𝐵𝑜𝑑𝑦_𝑦

�� 𝐵𝑜𝑑𝑦_𝑧

] = [𝐶𝜃 ∗ 𝐶𝛹 𝑆𝜃 ∗ 𝑆𝛷 ∗ 𝐶𝛹 − 𝐶𝛷 ∗ 𝑆𝛹 𝑆𝜃 ∗ 𝐶𝛷 ∗ 𝐶𝛹 + 𝑆𝛷 ∗ 𝑆𝛹𝐶𝜃 ∗ 𝑆𝛹 𝑆𝜃 ∗ 𝑆𝛷 ∗ 𝑆𝛹 + 𝐶𝛷 ∗ 𝐶𝛹 𝑆𝜃 ∗ 𝐶𝛷 ∗ 𝑆𝛹 − 𝑆𝛷 ∗ 𝐶𝛹

−𝑆𝜃 𝐶𝜃 ∗ 𝑆𝛷 𝐶𝜃 ∗ 𝐶𝛷] ∗ [

�� 𝑆𝑆_𝑥

�� 𝑆𝑆_𝑦

�� 𝑆𝑆_𝑧

]

Equation (5.12)

Multiplying the matrices and trying to calculate each vector induvidually

�� 𝐵𝑜𝑑𝑦𝑥= 𝐶𝜃 ∗ 𝐶𝛹 ∗ �� 𝑆𝑆𝑥

+ (𝑆𝜃 ∗ 𝑆𝛷 ∗ 𝐶𝛹 − 𝐶𝛷 ∗ 𝑆𝛹) ∗ �� 𝑆𝑆𝑦+ (𝑆𝜃 ∗ 𝐶𝛷 ∗ 𝐶𝛹 + 𝑆𝛷 ∗ 𝑆𝛹) ∗ �� 𝑆𝑆_𝑧

Equation (5.13)

�� 𝐵𝑜𝑑𝑦_𝑦 = 𝐶𝜃 ∗ 𝑆𝛹 ∗ �� 𝑆𝑆𝑥+ (𝑆𝜃 ∗ 𝑆𝛷 ∗ 𝑆𝛹 + 𝐶𝛷 ∗ 𝐶𝛹) ∗ �� 𝑆𝑆𝑦

+ (𝑆𝜃 ∗ 𝐶𝛷 ∗ 𝑆𝛹 − 𝑆𝛷 ∗ 𝐶𝛹) ∗ �� 𝑆𝑆_𝑧

Equation (5.14)

�� 𝐵𝑜𝑑𝑦_𝑧 = −𝑆𝜃 ∗ �� 𝑆𝑆𝑥+ 𝐶𝜃 ∗ 𝑆𝛷 ∗ �� 𝑆𝑆𝑦

+ 𝐶𝜃 ∗ 𝐶𝛷 ∗ �� 𝑆𝑆_𝑧

Equation (5.15)

To attain the rotational transformation to convert the vector in sunsensor frame to body co-

ordinate frame, the system is run several times to attain the sensor data. This data is calibrated

using least squares method and z component of sunsensor (𝑉𝑆𝑆_𝑧 ) is made constant. This provide

transformation rotations to attain a vector in body coordinate system is made almost constant.

From these tests the angular transformations to convert vector in sunsensor frame to body frame

are calculated using the above stated equations for several tests and their mean is used as the


angular transformations of the sunsensor [1] [19] [25]. The angular transformations to convert vector

in sunsensor frame to body frame (courtesy of Charles Pope Space master round 10)

Test number 𝛷 (degrees) 𝜃 (degrees) 𝛹 (degrees)

1 90.31 -0.25 -90.20

2 90.36 -0.25 -89.95

3 90.32 -0.27 -89.49

4 90.34 -0.25 -90.44

5 90.34 -0.25 -90.64

6 90.33 -0.24 -90.05

7 90.39 -0.26 -89.86

8 90.37 -0.26 -90.34

Table 5.3 Angular transformations from sunsensor to body frame

𝛷 = 𝑚𝑒𝑎𝑛 𝑜𝑓 𝛷

𝜃 = 𝑚𝑒𝑎𝑛 𝑜𝑓 θ

𝛹 = 𝑚𝑒𝑎𝑛 𝑜𝑓 𝛹

Result:

𝛷 = 89.83 𝑑𝑒𝑔

𝜃 = −0.25 𝑑𝑒𝑔

𝛹 = −90.02 𝑑𝑒𝑔

Standard deviations are calculated for the above data

Standard deviations of 𝛷 = 0.03 , 𝜃 = 0.01 and 𝛹 =0.34


Figure 5.6 Sun vector vs GPS time (b)

In the above figure it is clear that Z-component of the sun vector is made constant Sun is

not visible for camera in all positions, so these positions where sun cannot be traced have no

solutions for the attitude determination. The locations where sun is visible are plotted in matlab.

The plot shows the encoder data location of position where is the sun is visible to the camera.


Figure 5.7 Angular position (sun visible) of spacecraft vs GPS time

By observing the above plot the field view of sun for the camera is 0-40 degrees and 320-

360 degrees, which is a total of 80 degrees from -40 degrees to +40 degrees. The inertial coordinate

frame sun vector is achieved by following the same process used for Inertial frame magnetic vector.


System is run several times to attain the sensor data and the inertial vectors are calibrated

whenever the mock spacecraft reaches its home position, as the inertial coordinate frame coincides

with the body coordinate frame at this position. The sunsensor vectors in inertial frame (courtesy

of Charles Pope Space master round 10).

Test Number X-component Y-component Z-component

1 0.07 0.27 0.95

2 0.07 0.27 0.95

3 0.07 0.27 0.95

4 0.07 0.27 0.95

5 0.07 0.27 0.95

6 0.07 0.27 0.95

7 0.07 0.27 0.95

8 0.07 0.27 0.95

Table 5.4 Sunsensor vector in inertial co-ordinate system

𝐼2 = [0.08 0.03 0.96]


Chapter 6. Introduction to TRIAD and Q-methods, Results

6.1 TRIAD method:

Triad algorithm was introduced by Harold D Black in 1964; this algorithm is very simple

and elegant. This algorithm is used for spacecraft attitude determination for two decades. Here in

the current case as per the requirement and calculation ease TRIAD[1] I is opted. Consider two

independent vectors B1 and B2. I1 and I2 are reference vectors in Inertial Co-ordinate system.

𝐵1 = 𝐴 ∗ 𝐼1

𝐵2 = 𝐴 ∗ 𝐼2

𝐵1 × 𝐵2 = 𝐴 ∗ (𝐼1 × 𝐼2) Equation (6.1)

The body frame vectors used in the current experiment are 𝐵1(Equation (5.3)) is the

magnetometer vector in body frame and 𝐵2(Equation (5.10)) is sun vector in body frame. 𝐼1

(chapter 5 page no: 51) and 𝐼2(chapter 5 page no: 57) are reference inertial vectors. A is the

transformation matrix from Inertial to body co-ordinates. These are defined in chapter 5.

Unit vectors from the vectors in inertial frame and body frame

U, V, W and S be assumed as unit vectors here then

𝑈 = 𝐵1

‖𝐵1‖ 𝑉 =

𝐼1

‖𝐼1‖ 𝑊 =

𝐵1×𝐵2

‖𝐵1×𝐵2‖ 𝑆 =

𝐼1×𝐼2

‖𝐼1×𝐼2‖

Now the transformation matrix in terms of these unit vector is as follows

𝑈 = 𝐴 ∗ 𝑉

𝑊 = 𝐴 ∗ 𝑆

𝑈 × 𝑊 = 𝐴 ∗ (𝑉 × 𝑆)

Now the attitude matrix A can be found out using these unit vectors

[𝑈 𝑊 𝑈 × 𝑊] = 𝐴 ∗ [ 𝑉 𝑆 𝑉 × 𝑆]

𝐴𝑇𝑅𝐼𝐴𝐷 = [𝑈 𝑊 𝑈 × 𝑊] ∗ [𝑉 𝑆 𝑉 × 𝑆]𝑇 Equation (6.2)


6.2 Conversion of rotation matrix to quaternion:

A quaternion represents two sets of data regarding the rotation axis and the amount of

rotation around the axis. The quaternion [22] is shown as a 1 × 4 matrix as denoted

𝑄𝑇𝑅𝐼𝐴𝐷 = [𝑞1𝑇𝑅𝐼𝐴𝐷 𝑞2𝑇𝑅𝐼𝐴𝐷 𝑞3𝑇𝑅𝐼𝐴𝐷 𝑞4𝑇𝑅𝐼𝐴𝐷] Equation (6.3)

The first three elements q1, q2 and q3 provide the data regarding the rotation axis, q4

provide the data regarding amount of rotation. In the current thesis every attitude matrix is

represented in quaternion for ease in calculation

Consider attitude matrix 𝐴𝑇𝑅𝐼𝐴𝐷 (Equation 6.2)

𝐴𝑇𝑅𝐼𝐴𝐷 = [

𝐴11 𝐴12 𝐴13

𝐴21 𝐴22 𝐴23

𝐴31 𝐴32 𝐴33

]

Now trying to convert this attitude matrix to quaternion, the equation of relation of

quaternion and rotation matrix.

[

1 1 −11 −1 111

−11

−11

−1−111

] ∗

[ 𝑞0𝑇𝑅𝐼𝐴𝐷

2

𝑞1𝑇𝑅𝐼𝐴𝐷2


𝑞3𝑇𝑅𝐼𝐴𝐷2]

= [

𝐴11

𝐴22

𝐴33

1

]

The quaternion matrix will be found using the above equation


2




= [

1 1 −11 −1 111

−11

−11

−1−111

]

−1

∗ [

𝐴11

𝐴22

𝐴33

1

]


2




= 1

4∗ [

1 1 11 −1 −1

−1−1

1−1

−11

1111

] ∗ [

𝐴11

𝐴22

𝐴33

1

]

The Norm of a quaternion is equal to unity which means

|𝑄𝑇𝑅𝐼𝐴𝐷| = 1

|𝑄𝑇𝑅𝐼𝐴𝐷| = √𝑞0𝑇𝑅𝐼𝐴𝐷2 + 𝑞1𝑇𝑅𝐼𝐴𝐷

2 + 𝑞2𝑇𝑅𝐼𝐴𝐷2 + 𝑞3𝑇𝑅𝐼𝐴𝐷

2 = 1


The angle error is estimated using TRIAD, using the quaternions. This angle error is

estimated by comparing the true position data and the TRIAD data. The plot is as shown.

Figure 6.1(a) Angle error TRIAD method

The zeros in the plot indicate the positions where the sun is not visible to camera. This means there

is no solution at these positions. Eliminating these positions where there is no solution the plot is

replotted again.


Figure 6.1(b) Angle error TRIAD method with zeros eliminated.

6.3 Wahba’s Problem:

The main aim of Wahba’s problem [1] is to improve TRIAD method, which can put up the

TRIAD solution for more than two vectors. This problem was introduced by Grace Wahba in

1965. This is to find the attitude matrix with minimum loss function.

𝐿(𝐴𝑄) = 1

2∗ ∑[𝑎𝑖 ∗ ‖𝐵𝑖 − 𝐴𝑄𝐼𝑖‖

2]

𝑛

𝑖=1

‖𝐵𝑖 − 𝐴𝐼𝑖‖2 = 2 − 2𝑡𝑟(𝐴𝑄𝐼𝑖𝐵𝑖

𝑇)

𝐿(𝐴) = ∑ 𝑎𝑖[1 − 𝑡𝑟(𝐴𝑄𝐼𝑖𝐵𝑖𝑇)]𝑛

𝑖=1

Equation (6.4)

This equation show that the loss function can be minimized by increasing 𝑡𝑟(𝐴𝐼𝑖𝐵𝑖𝑇).

The solutions are solved using two procedures using Wahba’s problem. It can directly provide

The value on n = 1-2 cause here we use only two vectors and n represents number of vectors.


The attitude matrix or the equation can be further mathematically solved to attain the

quaternion representation.

6.4 Q method:

The equations in Wahba’s problem are represented in quaternion [1][26][27] form. This will

provide a robust algorithm for attitude estimation rather than attitude matrix using Wahba’s

problem. This Q method is introduced by Paul Davenport, which provided a breakthrough for

Wahba’s problem.

∑ 𝑎𝑖(𝐵𝑖𝐼𝑖𝑇)]𝑛

𝑖=1 = 𝑈𝑆𝑉𝑇

Equation (6.5)

Where 𝐾(𝑎𝑖,𝐵𝑖, 𝐼𝑖) is symmetric traceless matrix. S is diagonal, U and V are orthogonal

matrices

𝐾(𝑎𝑖,𝐵𝑖, 𝐼𝑖) =

[ ∑[𝑎𝑖𝐵𝑖𝐼𝑖

𝑇]

𝑛

𝑖=1

+ ∑[𝑎𝑖𝐵𝑖𝑇𝐼𝑖]

𝑛

𝑖=1

− 𝑡𝑟(∑[𝑎𝑖𝐵𝑖𝐼𝑖𝑇]

𝑛

𝑖=1

) 𝐸

𝐸𝑇 𝑡𝑟(∑[𝑎𝑖𝐵𝑖𝐼𝑖𝑇]

𝑛

𝑖=1

)]

Equation (6.6)

𝐸 = ∑ 𝑎𝑖(𝐵𝑖 × 𝐼𝑖)𝑛𝑖=1

Equation (6.7)

𝐴 𝑄𝑀 = 𝑈 𝑑𝑖𝑎𝑔([1 1 𝑑𝑒𝑡 𝑈 𝑑𝑒𝑡 𝑉]) 𝑉𝑇 .

Equation (6.8)

Where 𝐴 𝑄𝑀 [1][26][27] is the attitude matrix using quaternion method. Now this attitude

matrix is converted to quaternion [22] form using the following equation.

Consider attitude matrix 𝐴𝑇𝑅𝐼𝐴𝐷 (Equation 6.8)

𝐴𝑄𝑀 = [

𝐴11 𝐴12 𝐴13

𝐴21 𝐴22 𝐴23

𝐴31 𝐴32 𝐴33

]


[ 𝑞0𝑄𝑀

2

𝑞1𝑄𝑀2

𝑞2𝑄𝑀2

𝑞3𝑄𝑀2]

= 1

4∗ [

1 1 11 −1 −1

−1−1

1−1

−11

1111

] ∗ [

𝐴11

𝐴22

𝐴33

1

]

𝑄𝑄𝑀 = [𝑞0𝑄𝑀 𝑞1𝑄𝑀 𝑞2𝑄𝑀 𝑞3𝑄𝑀]𝑇

Equation (6.9)

The angle error is estimated here as well using the quaternion data from the Q-method

and comparing it with quaternion from the true position data.

Figure 6.2 Angle Error Q method


6.5 Results:

Quaternions are estimated from the attitude matrices of each method and are used to

calculate the angle error using the true position quaternion. Quaternions we have are from true

position data, TRIAD method and Q-method. True position data quaternion [1] [19] [22] [24] (𝑄𝑇𝑟𝑢𝑒-

Equation (5.2)) is used as a reference to estimate the angle error from TRIAD method quaternion

(𝑄𝑇𝑅𝐼𝐴𝐷-Equation (6.3)) and Q-method method quaternion (𝑄𝑄𝑀 − 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (6.9))

𝛹𝑒𝑇𝑅𝐼𝐴𝐷 = 2 ∗ arccos(𝑛0𝑇𝑅𝐼𝐴𝐷) Equation (6.10)

Let 𝑞𝑚(xy) be a function that indicates quaternion multiplication of two quaternions x

and y.

𝑄𝑞𝑚𝑇𝑅𝐼𝐴𝐷 = 𝑞𝑚(𝑄𝑇𝑟𝑢𝑒𝑄𝑇𝑅𝐼𝐴𝐷) Equation (6.11)

[

𝑛0𝑇𝑅𝐼𝐴𝐷

𝑛1𝑇𝑅𝐼𝐴𝐷𝑛2𝑇𝑅𝐼𝐴𝐷

𝑛3𝑇𝑅𝐼𝐴𝐷

] = 𝑞𝑚 ([




] [

𝑞0𝑇𝑅𝐼𝐴𝐷

𝑞1𝑇𝑅𝐼𝐴𝐷𝑞2𝑇𝑅𝐼𝐴𝐷

𝑞3𝑇𝑅𝐼𝐴𝐷

])

Where 𝑛0𝑇𝑅𝐼𝐴𝐷 = (𝑞0𝑇𝑅𝐼𝐴𝐷𝑞1𝑇𝑟𝑢𝑒 − 𝑞1𝑇𝑅𝐼𝐴𝐷𝑞1𝑇𝑟𝑢𝑒 − 𝑞2𝑇𝑅𝐼𝐴𝐷𝑞2𝑇𝑟𝑢𝑒 − 𝑞3𝑇𝑅𝐼𝐴𝐷𝑞3𝑇𝑟𝑢𝑒)

Where 𝛹𝑒𝑇𝑅𝐼𝐴𝐷 is the angular position error in TRIAD method.

This way the angle error is calculated in TRIAD method and a similar way is applied to

estimate the angle error is calculated for Q-method.

𝛹𝑒𝑄𝑀 = 2 ∗ arccos(𝑛0𝑄𝑀)

𝑄𝑞𝑚𝑄𝑀 = 𝑞𝑚(𝑄𝑇𝑟𝑢𝑒𝑄𝑄𝑀)

[

𝑛0𝑄𝑀

𝑛1𝑄𝑀

𝑛2𝑄𝑀

𝑛3𝑄𝑀

] = 𝑞𝑚 ([




] [

𝑞0𝑄𝑀

𝑞1𝑄𝑀

𝑞2𝑄𝑀

𝑞3𝑄𝑀

])

𝑛0𝑄𝑀 = (𝑞0𝑄𝑀𝑞1𝑇𝑟𝑢𝑒 − 𝑞1𝑄𝑀𝑞1𝑇𝑟𝑢𝑒 − 𝑞2𝑄𝑀𝑞2𝑇𝑟𝑢𝑒 − 𝑞3𝑄𝑀𝑞3𝑇𝑟𝑢𝑒)

Where 𝛹𝑒𝑄𝑀 is the angular position error in Q method.


The angle error is plotted using matlab with 8 different sets of data from sensors, as shown.

DATE AND TIME OF FILE: 2016/07/28 (YYYY/MM/DD)

21:34:41 (Hours:Minutes:Seconds)

Figure 6.3 Angle error (Q and TRIAD) vs GPS time 1






























The above plots represent angular error estimated from TRIAD and Q-methods. Some of

the data plotted have few deviations initially so these plots are replotted. This is because of

startup or end velocities which fluctuate the error. So such errors are eliminated for the data sets

21:53:35, 22:10:26 and 22:15:01. After elimination of such error the plots are replotted.

Standard deviation and mean are calculated again for these particular data sets. Plots are as

follows:



Figure 6.11 GPS time vs Angle error (for data set with eliminated error)




Figure 6.12 GPS time vs Angle error (for data set with eliminated error (1))




Figure 6.13 GPS time vs Angle error (for data set with eliminated error (2))


The above data sets are observed regarding the variation of mean angle error and

assumption is made initially for reason as each data set has different angular velocities and then

the data is compared with velocities in tabular format as shown below.

Time

Hr:min:sec

Average

Velocity

(degrees/sec)

Mean Error for

Triad method

angle error

(degrees)

Standard

Deviation for

Triad method

angle error

(degrees)

Mean Error for

Q method

angle error

(degrees)

Standard

Deviation for

Q method

angle error

(degrees)

21:34:41 85.23 1.32 0.21 1.34 0.22

21:53:35 84.04 2.32 0.37 2.35 0.37

22:00:21 28.22 0.49 0.07 0.48 0.07

22:10:26 22.93 0.46 0.07 0.45 0.08

22:15:01 19.28 0.32 0.04 0.29 0.05

22:19:05 2.39 0.22 0.05 0.16 0.04

22:22:27 84.93 1.08 0.08 1.08 0.08

22:25:41 28.43 0.49 0.06 0.49 0.06

Table 6.1 TRIAD and Q mean error and stadard deviation with velocities

Using the above data from the table in matlab a plot is plotted between velocity and

angular error. This plot is to show that error change is dependent on velocity.


Figure 6.14 Velocity vs Mean angular error

Summary:

The sensor accuracy is dependent upon velocity and varies with it. The reason can be

assumed as encoder data is dependent upon motor rpm and it is independent of time. The rate of

encoder output is purely dependent on velocity and it varies along with it. Sensor are dependent

on time and provide a constant output rate with time. The sensors have constant latency but the

size of the latency varies depending upon the speed. So the size of latency increment causes an

increase in angular position error.


Chapter 7. Conclusion and future work

This chapter is about conclusion of results and future possible developments that can be made

with this test-bed. The experimental results have produced some errors and reduction of the errors

to minimum will help in precise attitude estimation of satellite or spacecraft. This spin attitude

determination test-bed can be further more modified to produce results with more accuracy.

Conclusion:

Successfully designed and developed a system for testing of spin stabilized spacecraft.

Attitude determination using Q and TRIAD methods is done, angular error is calculated.

It is proved experimentally that COTS (commercial off the shelf) sensors are not

precisely accurate in positioning.

Few factors which affect the position error are estimated.

Possible future developments:

Modification of algorithms:

Variety of algorithms like Kalmanfilter [1] [19] [25] can be used for reducing the attitude

error. This reduction of error improves positioning accuracy. The errors are being analyzed and

further analysis is done for improvement of attitude determination accuracy.

Sensors:

Sensors can be replaced with more precise ones to provide very accurate data. The

sensors preciseness is vital during the selection. A new version of spintable is being developed at

Utah State University which is in other words spintable version 2.0. This is expected to be

equipped with more accurate sensors.


Mechanical design:

Mechanical design could be modified to reduce vibrations and provide a stabilized spin,

although there are not many vibrations but they can be minimized. Design can be modified for

three axis attitude determination and this three axis determination is more useful than spin

stabilized one. A new version of spintable i.e., spintable version 2.0 is being developed at Utah

state university at department of space engineering with reduction in all possible mechanical

vibrations. Although in version 1 the vibrations are less dominant, there is noticeable wobbling in

the spin which will be eliminated in version 2.0. This wobbling didn’t affect the data from the

sensors much as it was so minute.

A spinning test bed rotates or simulates the spinning satellite rotation around z-axis or principal

axis. Consider a point on a spin satellite test-bed which is located at a distance of ‘r’ from the axis

of rotation. The trajectory of the point in a spinning test bed will be on the circumference of a

circle which is equal to 2𝛱𝑟.

Now assume the same point for a satellite on three axis testbed. This point is located at a distance

of ‘r’ from the origin of the spin axes. Here the spin is attained in all the three axis individually

and also together. As the satellite spins in all the axes the trajectory of the point will be on the

surface area of a sphere is equal to 4𝛱𝑟2.


APPENDIX

Dowty TAM plots for data.




Encoder plots for data





IMU magnetometer plots for data:





Q and TRIAD results for first three data sets when Dowty TAM data is used in place of IMU



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https://creativecommons.org/licenses/by-nc-sa/4.0/

http://www.cg.info.hiroshima-cu.ac.jp/~miyazaki/knowledge/teche52.html

http://www.cg.info.hiroshima-cu.ac.jp/~miyazaki/knowledge/teche52.html

Documents

Design and fabrication of test-bed for testing attitude ...1077742/FULLTEXT02.pdf · Design and fabrication of test-bed for testing attitude determination of spin stabilized spacecraft