Ismail Ugur DUNDAR
Space Engineering, master's level (120 credits) 2018
Luleå University of Technology
Department of Computer Science, Electrical and Space Engineering
IMPROVEMENT OF A SPACE SURVEILLANCE TRACKING ANALYSIS TOOL
SUBMITTED BY
ISMAIL UGUR DUNDAR
Prepared at
Airbus Defence and Space GmbH 88039 Friedrichshafen, Germany
Examiner:
Prof. Dr. Sergio Montenegro, Julius–Maximilians–University Würzburg, Germany Dr. Victoria Barabash, Lulea University of Technology , Kiruna, Sweden
Supervisor:
Dr. Jens Utzmann
MSc. Oscar Rodriguez Fernandez
December 5, 2018
DISCLAIMER
"This project has been funded with support from the European Commission. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein."
i
Abstract
Since the beginning of space exploration, the amount of space debris has increased with the development of new space technologies. In fact, when a collision happens, new space debris are generated. Hence, collision risk between space debris and operational satellites rises. The purpose of a surveillance network system consists of the detection of space objects, their classification and their tracking. To avoid collisions, space debris objects’ orbit must be computed with sufficient accuracy.
The goal of this thesis is the improvement of a pre-existing Space Surveillance and Tracking Analysis Tool. The tool is able to simulate different observation scenarios for radar or optical observer, which can be space-based or ground-based. To enhance the orbit determination, an Extended Square Root Information Filter is implemented and incremented with a Smoother. Smoothers have been implemented for the existing filters as well, such as the Extended Kalman Filter and the Unscented Kalman Filter. A bias estimation method was added as part of the OD for all filter types.
Additionally, different outlier detection methods were implemented for the automatic detection of outliers within the measurement data. To find the optimum orbit determination interval, different scenarios were considered in LEO, MEO and GEO orbits. The methods were implemented and different scenarios for validation will be discussed. A wide discussion on the methods implementation and their validation on different scenarios is presented, together with a comparison of the orbit determination results with the other filters.
All the recently implemented features increase the efficiency of the tool to simulate the different
scenarios and enhance the tracking of space debris objects.
Acknowledgements
I would like to thank my supervisor at Airbus Defence and Space, MSc. Oscar Rodriguez Fernandez, for his patience, enthusiasm, motivation, immense knowledge and continuous support. His guidance helped me through all the research and writing of this thesis. My sincere gratitude also goes to Dr.
Jens Utzmann, who provided me an opportunity to join their team as a master thesis student.
I would also like to thank my university supervisors: Prof. Dr. Sergio Montenegro and Dr. Victoria Barabash, for their support and insightful comments.
iii
Declaration of Authorship
I declare that I completed this thesis on my own and that information which has been directly or indirectly taken from other sources has been noted as such. Neither this nor a similar work has been presented to an examination committee.
Name of Student
Friedrichshafen, December 5, 2018
. . . .
Contents
List of Figures . . . . xii
List of Tables . . . xiv
List of Abbreviations . . . . xv
List of Symbols . . . xvi
1 Introduction 1 1.1 Thesis Objectives . . . . 3
2 Theoretical Background 4 2.1 Coordinate Frames . . . . 4
2.1.1 Geocentric Celestial Coordinate System . . . . 4
2.1.2 International Terrestrial Reference Frame . . . . 5
2.2 Object State . . . . 6
2.2.1 Space-based Objects . . . . 6
2.2.2 Ground-based Observers . . . . 7
2.3 Observation Theory . . . . 7
2.3.1 Optical Observer . . . . 8
2.3.2 Radar Observer . . . . 8
2.4 Dynamic Model - Orbital Perturbation . . . . 9
2.4.1 Non-spherical Gravitational Field . . . . 9
2.4.2 Atmospheric Drag . . . . 10
2.4.3 Solar Radiation Pressure . . . . 10
2.4.4 Third-body Perturbations . . . . 10
2.5 Initial Orbit Determination . . . . 11
2.5.1 Gibbs Algorithm . . . . 11
2.5.2 Herrick-Gibbs Algorithm . . . . 11
2.6 Orbit Determination . . . . 11
2.6.1 The Weighted Least Squares Method . . . . 12
2.6.2 The Sequential Batched Least Squares Algorithm . . . . 13
2.6.3 The Extended Kalman Filter . . . . 13
2.6.4 Unscented Kalman Filter . . . . 14
2.7 Orbital Regions . . . . 16
v
3 SPOOK 18
3.1 SPOOK Structure . . . . 18
3.1.1 Simulation Layer . . . . 19
3.1.2 Interface Layer . . . . 20
3.1.3 Analysis Layer . . . . 21
4 Method 25 4.1 Bias Estimation . . . . 25
4.2 Extended Square Root Information Filter . . . . 26
4.2.1 Correction of Non-Positive Definite Matrices due to Numerical Error . . . . . 31
4.3 Smoother . . . . 31
4.3.1 EKF . . . . 32
4.3.2 UKF . . . . 33
4.3.3 ESRIF . . . . 33
4.4 Outlier Detection . . . . 33
4.4.1 Chi-Square Test . . . . 34
4.4.2 Measurement Variance Test . . . . 34
4.4.3 3σ Test . . . . 34
4.4.4 Data Editing . . . . 35
4.5 Orbit Determination Interval . . . . 35
4.5.1 Credibility Test . . . . 36
5 Evaluation and Discussion 38 5.1 Experiment . . . . 38
5.1.1 GEO Orbit - Object Scenario 1 . . . . 38
5.1.2 LEO Orbit - Object Scenario 2 . . . . 39
5.1.3 GEO Orbit - Object Scenario 3 . . . . 40
5.2 Bias estimation . . . . 41
5.3 Extended Square Root Information Filter . . . . 51
5.4 Smoother . . . . 57
5.4.1 EKF Smoother . . . . 58
5.4.2 ESRIF Smoother . . . . 62
5.4.3 UKF Smoother . . . . 66
5.5 Outlier Detection . . . . 70
5.5.1 EKF Outlier . . . . 71
5.5.2 ESRIF Outlier . . . . 74
5.5.3 UKF Outlier . . . . 76
5.6 Orbit Determination Interval . . . . 78
5.6.1 LEO Orbit . . . . 83
5.6.2 MEO Orbit . . . 107
5.6.3 GEO Orbit . . . 110
6 Conclusion 113
7 Future Work 114
A.4 WLS . . . 122
A.5 SBLS . . . 123
B Performance of Filters 125 B.1 Initial Covariance Propagation (ICP) - GEO Object Scenario 1 . . . 125
B.1.1 EKF . . . 125
B.1.2 ESRIF . . . 126
B.1.3 UKF . . . 126
B.2 Initial Covariance Propagation (ICP) - LEO Object Scenario 2 . . . 127
B.2.1 EKF . . . 127
B.2.2 ESRIF . . . 128
B.2.3 UKF . . . 128
B.3 Initial Orbit Determination (IOD) - GEO Object Scenario 1 . . . 129
B.3.1 EKF . . . 129
B.3.2 ESRIF . . . 130
B.3.3 UKF . . . 130
B.4 Initial Orbit Determination (IOD) - LEO Object Scenario 2 . . . 131
B.4.1 EKF . . . 131
B.4.2 ESRIF . . . 132
B.4.3 UKF . . . 132
B.5 Comparison of the Smoother Results with WLS . . . 133
vii
List of Figures
1.1 Distribution of space debris in the Earth orbit . . . . 1
1.2 Simulation of the collision before and during the first three hours . . . . 2
1.3 Number of objects per orbital region included in the TLE catalogue . . . . 2
2.1 Geocentric Celestial Reference Frame . . . . 5
2.2 The International Terrestrial Reference Frame . . . . 5
2.3 Keplerian Classical Orbital Elements . . . . 7
2.4 Angular measurements for optical observer: right ascension α and declination δ . . . 8
2.5 Angular measurement of the radar observer . . . . 9
3.1 SPOOK functional diagram . . . . 19
3.2 SPOOK visualisation tool . . . . 21
3.3 Main structure of SPOOK. . . . . 23
3.4 Structure of Orbit Determination. . . . . 24
4.1 ESRIF Algorithm Structure. . . . 30
4.2 Forward - Backward Filter . . . . 31
4.3 Filter Time Step . . . . 32
5.1 Errors of position and 3σ-sigma uncertainty from covariance of the orbit determination of the object for case 1. . . . . 42
5.2 Residuals of Orbit Determination of the object for case 1. . . . 43
5.3 Errors of position and 3σ-sigma uncertainty from covariance of the orbit determination of the object for case 2. . . . . 44
5.4 Residuals of Orbit Determination of the object for case 2. . . . 45
5.5 Errors of position and 3σ-sigma uncertainty from covariance of the orbit determination of the object for case 3. . . . . 46
5.6 Residuals of Orbit Determination of the object for case 3. . . . 47
5.7 Bias Estimation for case 3. . . . . 48
5.8 Errors of position and 3σ-sigma uncertainty from covariance of the orbit determination of the object for case 4. . . . . 49
5.9 Residuals of Orbit Determination of the object for case 4. . . . 50
5.10 Bias Estimation for case 4. . . . . 51
5.11 Errors of position and uncertainty of the orbit determination - ESRIF. . . . 52
5.12 Residuals of the Orbit Determination - ESRIF. . . . 53
5.13 Errors of position and uncertainty of the orbit determination - EKF. . . . 54
5.14 Residuals of the Orbit Determination - EKF. . . . 55
5.15 Errors of position and uncertainty of the orbit determination - UKF. . . . 56
5.23 ESRIF - Forward Filter. . . . . 62
5.24 ESRIF - Backward Filter. . . . . 63
5.25 ESRIF - Forward Filter and Backward Filter Together. . . . . 64
5.26 ESRIF - Difference of the Forward Filter and Backward Filter. . . . 64
5.27 ESRIF - Forward Filter Residual. . . . . 65
5.28 ESRIF - Backward Filter Residual. . . . . 66
5.29 UKF - Forward Filter of the Orbit Determination Error. . . . . 66
5.30 UKF - Backward Filter of the Orbit Determination Error. . . . . 67
5.31 UKF - Forward and Backward Filter Together. . . . 68
5.32 UKF - Difference of the Forward Filter and Backward Filter. . . . 68
5.33 UKF - Forward Filter of the Residual. . . . . 69
5.34 UKF - Forward Filter of the Residual with large scale. . . . . 69
5.35 UKF - Backward Filter of the Residual. . . . 70
5.36 EKF - Outlier Detection Algorithm not Active. . . . 71
5.37 EKF - Outlier Detection Method 1 Active . . . . 72
5.38 EKF - Outlier Detection Method 2 Active. . . . 72
5.39 EKF - Outlier Detection Method 3 Active. . . . 73
5.40 ESRIF - Outlier Detection Method 1 Active . . . . 74
5.41 ESRIF - Outlier Detection Method 2 Active. . . . . 75
5.42 ESRIF - Outlier Detection Method 3 Active. . . . . 76
5.43 UKF - Outlier Detection not Active. . . . 76
5.44 UKF - Outlier Detection Method 1 Active. . . . . 77
5.45 UKF - Outlier Detection Method 2 Active. . . . . 77
5.46 UKF - Outlier Detection Method 3 Active. . . . . 78
5.47 Orbit Determination Interval Scenario result part 1. . . . 79
5.48 Orbit Determination Interval Scenario result part 2. . . . 79
5.49 Orbit Determination - Object Number 1. . . . 80
5.50 Residual - Object Number 1. . . . . 80
5.51 Orbit Determination - Object Number 4. . . . 81
5.52 Residual - Object Number 4. . . . . 81
5.53 Orbit Determination - Object Number 13. . . . 82
5.54 Residual - Object Number 13. . . . 82
5.55 Sensor Type 1 - Consistency result in Region 1. . . . 85
5.56 Sensor Type 1 - WRMS result in Region 1. . . . 85
5.57 Sensor Type 1 - Consistency result in Region 2. . . . 86
5.58 Sensor Type 1 - WRMS result in Region 2. . . . 86
5.59 Sensor Type 1 - Consistency result in Region 3. . . . 87
5.60 Sensor Type 1 - WRMS result in Region 3. . . . 87
5.61 Sensor Type 1 - Consistency result in Region 4. . . . 88
5.62 Sensor Type 1 - WRMS result in Region 4. . . . 88
ix
5.63 Sensor Type 1 - Consistency result in Region 7. . . . 89
5.64 Sensor Type 1 - WRMS result in Region 7. . . . 89
5.65 Sensor Type 1 - Consistency result in Region 7. . . . 90
5.66 Sensor Type 1 - WRMS result in Region 7. . . . 90
5.67 Sensor Type 2 - Consistency result in Region 1. . . . 91
5.68 Sensor Type 2 - WRMS result in Region 1. . . . 91
5.69 Sensor Type 2 - Consistency result in Region 2. . . . 92
5.70 Sensor Type 2 - WRMS result in Region 2. . . . 92
5.71 Sensor Type 2 - Consistency result in Region 3. . . . 93
5.72 Sensor Type 2 - WRMS result in Region 3. . . . 93
5.73 Sensor Type 2 - Consistency result in Region 4. . . . 94
5.74 Sensor Type 2 - WRMS result in Region 4. . . . 94
5.75 Sensor Type 2 - Consistency result in Region 7. . . . 95
5.76 Sensor Type 2 - WRMS result in Region 7. . . . 95
5.77 Sensor Type 3 - Consistency result in Region 1. . . . 96
5.78 Sensor Type 3 - WRMS result in Region 1. . . . 96
5.79 Sensor Type 3 - Consistency result in Region 2. . . . 97
5.80 Sensor Type 3 - WRMS result in Region 2. . . . 97
5.81 Sensor Type 3 - Consistency result in Region 3. . . . 98
5.82 Sensor Type 3 - WRMS result in Region 3. . . . 98
5.83 Sensor Type 3 - Consistency result in Region 4. . . . 99
5.84 Sensor Type 3 - WRMS result in Region 4. . . . 99
5.85 Sensor Type 3 - Consistency result in Region 7. . . 100
5.86 Sensor Type 3 - WRMS result in Region 7. . . 100
5.87 Sensor Type 5 - Consistency result for Leo Orbit. . . 101
5.88 Sensor Type 5 - WRMS result for LEO Orbit. . . 101
5.89 Sensor Type 7 - Consistency result in Region 1. . . 102
5.90 Sensor Type 7 - WRMS result in Region 1. . . 102
5.91 Sensor Type 7 - Consistency result in Region 2. . . 103
5.92 Sensor Type 7 - WRMS result in Region 2. . . 103
5.93 Sensor Type 7 - Consistency result in Region 3. . . 104
5.94 Sensor Type 7 - WRMS result in Region 3. . . 104
5.95 Sensor Type 7 - Consistency result in Region 4. . . 105
5.96 Sensor Type 7 - WRMS result in Region 4. . . 105
5.97 Sensor Type 7 - Consistency result in Region 7. . . 106
5.98 Sensor Type 7 - WRMS result in Region 7. . . 106
5.99 Sensor Type 1 - Consistency result in Region 7 and 8. . . 107
5.100Sensor Type 1 - WRMS result in Region 7 and 8. . . 107
5.101Sensor Type 1 - Consistency result in Region 10. . . 108
5.102Sensor Type 1 - WRMS result in Region 10. . . 108
5.103Sensor Type 1 - Consistency result in Region 12. . . 109
5.104Sensor Type 1 - WRMS result in Region 12. . . 109
5.105Optical Observer - Consistency result for GEO Orbit. . . . 110
5.106Optical Observer - WRMS result for GEO Orbit. . . 110
5.107Sensor Type 1 - Consistency result in Region 12 and 13. . . 111
A.5 EKF - Result of the OD using Initial Orbit Determination. . . . 119
A.6 Result of the Bias Estimation - Right Ascension (0") and Declination (0") of bias values.119 A.7 EKF - Result of the OD using Initial Orbit Determination. . . . 120
A.8 Result of the Bias Estimation - Right Ascension (1") and Declination (1") of bias values.120 A.9 ESRIF - Result of the OD using Initial Covariance Propagation. . . . 120
A.10 Result of the Bias Estimation - Right Ascension (0") and Declination (0") of bias values.120 A.11 ESRIF - Result of the OD using Initial Covariance Propagation. . . . 121
A.12 Result of the Bias Estimation - Right Ascension (1") and Declination (1") of bias values.121 A.13 UKF - Result of the OD using Initial Covariance Propagation. . . . 121
A.14 Result of the Bias Estimation - Right Ascension (0") and Declination (0") of bias values.121 A.15 UKF - Result of the OD using Initial Covariance Propagation. . . . 122
A.16 Result of the Bias Estimation - Right Ascension (1") and Declination (1") of bias values.122 A.17 WLS - Result of the OD. . . . 122
A.18 Result of the Bias Estimation - Right Ascension (0") and Declination (0") of bias values.122 A.19 WLS - Result of the OD . . . 123
A.20 Result of the Bias Estimation - Right Ascension (1") and Declination (1") of bias values.123 A.21 SBLS - Result of the OD. . . 123
A.22 Result of the Bias Estimation - Right Ascension (0") and Declination (0") of bias values.123 A.23 SBLS - Result of the OD. . . 124
A.24 Result of the Bias Estimation - Range(0.5m), Azimuth (1"), Elevation (1") and Range-Rate (0") of bias values. . . 124
B.1 EKF - Result of the OD. . . 125
B.2 EKF - Result of the residual. . . 125
B.3 ESRIF - Result of the OD. . . . 126
B.4 ESRIF - Result of the residual. . . 126
B.5 UKF - Result of the OD. . . . 126
B.6 UKF - Result of the residual. . . 126
B.7 EKF - Result of the OD. . . 127
B.8 EKF - Result of the residual. . . 127
B.9 ESRIF - Result of the OD. . . . 128
B.10 ESRIF - Result of the residual. . . 128
B.11 UKF - Result of the OD. . . . 128
B.12 UKF - Result of the residual. . . 128
B.13 EKF - Result of the OD. . . 129
B.14 EKF - Result of the residual. . . 129
B.15 ESRIF - Result of the OD . . . 130
B.16 ESRIF - Result of the residual. . . 130
B.17 UKF - Result of the OD. . . . 130
B.18 UKF - Result of the residual. . . 130
B.19 EKF - Result of the OD. . . 131
xi
B.20 EKF - Result of the residual. . . 131
B.21 ESRIF - Result of the OD . . . 132
B.22 ESRIF - Result of the residual. . . 132
B.23 UKF - Result of the OD. . . . 132
B.24 UKF - Result of the residual. . . 132
B.25 WLS - Orbit Determination Result. . . . 133
B.26 WLS - Residual of the Measurements. . . 133
List of Tables
2.1 Definition of Classical Keplerian Orbital Elements . . . . 6
2.2 Orbital Region Classification . . . . 16
2.3 Augmented Orbital Region Classification for Orbit Determination Interval . . . . 17
3.1 Observer Types. . . . . 20
4.1 One of the combinations of the metrics parameter . . . . 36
4.2 Parameters of Metrics . . . . 36
5.1 Scenario 1 - Target Object and Observer . . . . 38
5.2 Scenario 1 Parameters . . . . 39
5.3 LEO Orbit - Target Object . . . . 39
5.4 Ground Radar Observer Location . . . . 39
5.5 Scenario 2 Parameters . . . . 40
5.6 Scenario 3 Parameters . . . . 40
5.7 Ground Optical Observer Location . . . . 40
5.8 Standard Deviation and Mean values for case 1. . . . 43
5.9 Standard Deviation and Mean values for case 2. . . . 45
5.10 Standard Deviation and Mean values for case 3. . . . 48
5.11 Standard Deviation and Mean values for case 4. . . . 50
5.12 Standard Deviation and Mean values. . . . . 57
5.13 RMS Error of OD and RMS Error of the Smoother OD. . . . 70
5.14 Outliers’ time amount in the measurement. . . . 71
5.15 Ground Observer Location. . . . 83
5.16 Sensor Type 1 - Combination Parameters. . . . 84
5.17 Sensor Type 1 - Summary of the ODI. . . . 90
5.18 Sensor Type 2 - Combination Parameters. . . . 91
5.19 Sensor Type 2 - Summary of the ODI. . . . 95
5.20 Sensor Type 3 - Combination Parameters. . . . 96
5.21 Sensor Type 3 - Summary of the ODI. . . 100
5.22 Sensor Type 5 - Combination Parameters. . . 101
5.23 Sensor Type 7 - Combination Parameters. . . 102
5.24 Sensor Type 7 - Summary of the ODI. . . 106
5.25 Orbit Determination Interval for Leo Orbit. . . 107
B.1 RMS Error of OD and Mean value of the Residual using ICP for Geo orbit object Scenario. . . 127
B.2 RMS Error of OD and Mean value of the Residual using ICP for Leo orbit object Scenario. . . 129
xiii
B.3 RMS Error of OD and Mean value of the Residual using IOD for Geo orbit object Scenario. . . 131 B.4 RMS Error of OD and Mean value of the Residual using IOD for Leo orbit object
Scenario. . . 133
B.5 RMS Error of OD and RMS Error of the Smoother OD. . . 134
List of Abbreviations
BL Backward Filter
BC Ballistic Coefficient
COPUOS Committee on the Peaceful Uses of Outer Space
ESA European Space Agency EKF Extended Kalman Filter
ESRIF Extended Square Root Information Fil- ter
FoV Field of View
FL Forward Filter
GCRF Geocentric Celestial Coordinate System GEO Geostationary Earth Orbit
GPS Global Positioning System IOD Initial Covariance Propagation IOD Initial Orbit Determination
ICRF International Celestial Reference Frame IERS International Earth Rotation Service ISS International Space Station
ITRS International Terrestrial Reference Frame
LEO Low Earth Orbit
MSIS Mass Spectrometer Incoherent Scatter
MEO Medium Earth Orbit
NEES Normalized Estimation Error Squared NIS Normalized Innovation Squared ODI Orbit Determination Interval OD Orbit Determination
RTN Radial Tangential Normal RTS Rauch Tung Striebel RMS Root Mean Square
SBLS Sequential Batch Least Square SRP Solar Radiation Pressure
SPOOK Space Object Observations and Kalman Filtering
SST Space Surveillance Tracking SRIF Square Root Information Filter TLE Two Line Element
UKF Unscented Kalman Filter
USKF Unscented Schmidt Kalman Filter WLS Weighted Least Square
WRMS Weighted Root Mean Square WGS84 World Geodetic System 1984
xv
List of Symbols
A D Object’s cross-sectional area
" arcsec
α Semi-major Axis
β Azimuth
χ Sigma Points
χ 2 Chi-Square Distribution δ Declination
˙
ρ Slant Range-Rate λ Longitude e Eccentricity i Inclination v True Anomaly
w Argument of periapsis µ Gravitational Parameter
Ω Right Ascension of the Ascending Node φ Latitude
Φ State Transition Matrix ρ Slant Range
σ Standard Deviation τ Truncation Error α Right Ascension bj ~ Residual
~ r Position Vector
~ v Velocity Vector
~a Drag Atmospheric Drag Acceleration
~
v t,winds Thermospheric Wind
~
w ⊕ Earth’s Rotational Speed A j Partial Derivative Matrix C D Drag Coefficient
C R Reflectivity Coefficient D Smoother Kalman Gain
el Elevation
F Partial Derivative of the state rates matrix
h Altitude
H j Observation Matrix
K Kalman Gain MAtrix
k Time Step
m obj Mass of Object P Covariance Matrix p SRP Solar Radiation Pressure Q Process Noise Matrix r Spacecraft’s Position R Measurement Noise Matrix R Information Matrix S k Eigen Vector U Gravity Field W Weighted Matrix
y Measurement
Chapter 1
Introduction
Since the beginning of Space activities, man-made space debris have been created as a result of more than 5400 rocket launches and more than 500 on-orbit fragmentation events [1]. Space activities have grown substantially, and space exploration will increase further in the future. Because of lack of the space laws, several organizations have emerged, which are the United Nations Committee on the Peaceful Uses of Outer Space (COPUOS) and the Inter-Agency Space Debris Coordination Committee. The main purpose of these committees is to develop guidelines to decrease the amount of new space debris, upper stage rockets and defunct satellites [2]. Space debris is defined by IASDCC as follows:
"Space debris are all man made objects, including fragments and elements thereof, in Earth orbit or reentering the atmosphere, that are nonfunctional"[3]
Figure 1.1: Distribution of space debris in the Earth orbit [4].
More than 23.000 objects are currently catalogued by the U.S Space Surveillance Network [5].
Nevertheless, the total real object number can only be estimated by a scientific model. According to the European Space Agency (ESA) scientific model, an estimation of space debris in 2013, 29.000 objects are larger than 10 cm in size, 670.000 objects are between 1 cm to 10 cm and 170 million objects are between 1 mm to 1 cm. These sizes of objects are a problem for an operational spacecraft.
1. INTRODUCTION 1
For instance, in case of collision with a 10-cm object, a catastrophic fragmentation of a spacecraft will be unavoidable [6]. Larger than 10 cm in size might cause catastrophic breakup of a spacecraft.
Objects with diameter of 1 cm to 10 cm could create significant damage to a spacecraft and possible mission failure. Objects with diameter of 1 mm to 1 cm could cause component damage, craters, spallations and degradation od spacecraft surfaces [7]. Micrometeoroids have an average velocity and density of 20 km/sec and 0.5 g/cm 3 respectively whereas space objects average velocity and density are 9 to 10 km/sec and 2.8 g/cm 3 depending on the altitude [8]. Micrometeoroids could impact penetration of the spacecraft, RF transients, sudden electrical disturbances and gyro-destabilizations [7].
Two main events in history lead to the creation of a large number of new space debris in Earth orbit. One of them is when China destroyed Fengyun-1C in an anti-satellite weapon test in 2007 with a ballistic missile. The destruction created more than 3000 new objects [9]. The second is when a collision happened between Cosmos 2251 and Iridium 33 in 2009. These satellites are an inactive Russian communication satellite and an active satellite operated by the U.S. This collision created almost 2,000 space debris objects [10].
Figure 1.2: Simulation of the collision before and during the first three hours [10].
Figure 1.2 shows the two satellites at the time of the collision (left), the position of two satellites
10 minutes after the collision (center) and space debris clouds three hours later (right) [10].
laser observation from space or ground. Space objects’ orbital elements can be estimated from the observation measurements. This is known as orbit determination and it will be the major subject of this thesis.
1.1 Thesis Objectives
Space Object Observations and Kalman Filtering (SPOOK) is a software tool that has been being developed by Airbus Defence and Space. This is a highly configurable and versatile tool for simula- tion, detection as well as producing orbit determination of Earth-orbiting objects using either real or synthetic data. This tool has the ability to predict and detect the movement of the Earth-orbiting objects. It has the ability to use different observers (ground-based and space-based observers) and different sensors (radar and optical sensors). In order to perform orbit determination for space objects, several techniques such as Extended Kalman Filter, Weighted Least Square, Sequential Batch Least Square, Unscented Kalman Filter and Unscented Schmidt Kalman Filter can be used.
The objective of this thesis is to enhance this tool, focusing on the Orbit Determination aspect. The thesis works are defined as follows:
• Bias Estimation in Orbit Determination
• Implementation and validation of an ESRIF (Extended Square Root Information Filter)
• Implementation and validation of filter Smoother (EKF, UKF, ESRIF)
• Implementation and validation of Outlier Detection Algorithms
• Selection of optimum data interval for OD in LEO, MEO and GEO orbits
• Comparison of performance between different OD methods
In Chapter 2, the theoretical background is provided which is based on the orbit determina- tion problem. In Chapter 3, SPOOK characteristics and its situation of the beginning of the project are described. The methods to enhance the orbit determination are described in Chapter 4. Subsequently, the simulation parameters are defined, and the methods’ results are evaluated and discussed with different observer and orbit simulation in Chapter 5. Afterwards, Chapter 6 summarizes the achievements accomplished during the master thesis. Finally, Chapter 7 gives an aspect for eventual future enhancements of the SPOOK.
1. INTRODUCTION 3
Chapter 2
Theoretical Background
This section introduces the theoretical background, which is necessary to understand the Orbit Determination process. Some brief descriptions of astrodynamics fundamentals will be given in this chapter.
2.1 Coordinate Frames
There are two main reference system which are used: The Geocentric Celestial Coordinate Reference Frame (GCRF) and the International Terrestrial Reference Frame (ITRF). These frames are introduced briefly in this chapter.
2.1.1 Geocentric Celestial Coordinate System
The Geocentric Celestial Coordinate System is one of the classical inertial coordinate system for the Earth. It is the geocentric counterpart of the International Celestial Reference Frame (ICRF). This coordinate frame has been the accepted celestial reference frame for the International Earth Rotation Service (IERS) since January 1, 1997 [12]. This reference frame can be visualized as the first f 1
axis aligned with vernal equinox and the third f 3 axis aligned with Earth’s north pole (alignment
is exact only for the date 1 January 2000) [13]. Figure 2.2 indicates the GCRF and an arbitrary
position vector to indicate right ascension and declination [13]. GCRF is defined using the J2000
date (Julian date for 1 January 2000 at 12:00). In SPOOK, the GCRF is the main reference frame.
Figure 2.1: Geocentric Celestial Reference Frame [13].
2.1.2 International Terrestrial Reference Frame
A geocentric coordinate frame fixed to the rotating Earth. This coordinate system’s origin is at the center of the Earth [12]. The X-axis directs to the reference meridian, Z-axis directs to the reference pole and Y-axis completes the right-handed system.
More detail about transformation ITRF to GCRF can be found in [12].
Figure 2.2: The International Terrestrial Reference Frame [14].
2. THEORETICAL BACKGROUND 5
2.2 Object State
The goal of the orbit determination is to determine the state of an object at every time step. There are different ways to represent the object state. Space-based (space debris and observers) and ground objects (ground-based observers) state will describe in next section.
2.2.1 Space-based Objects
The object state is described in two ways, which are:
• Object Coordinates
• Classical Orbital Elements
To represent the object state using Cartesian Coordinates frame needs three positions ˆ r and three velocities ˆ v. For example, The International Space Station’s state vector is represented on the 2nd of September 2018 at midday [15]
X = ~ ~r
~ v
=
r I r J r K v I
v J
v K
=
5247.52821 Km
−2515.71764 Km
−3499.83983 Km 4.848498831 Km.s −1 3.822398159 Km.s −1 4.530854374 Km.s −1
(2.1)
The object state can also be described by the 6 classical Keplerian orbital elements which are described in table 2.1
Table 2.1: Definition of Classical Keplerian Orbital Elements [16].
Symbol Name Description
α Semi-major axis Describes the size of the orbit.
e Eccentricity Describes the shape of the orbit.
i Inclination Describes the orientation of the orbit with respect to the Earth’s equator.
Ω Right Ascension of the Ascending Node
Describes where the low point, perigee, of the orbit is with respect to the Earth’s surface.
w Argument of periapsis Describes the location of the ascending and descending orbit loca- tions with respect to the Earth’s equatorial plane.
v True Anomaly Describes where the satellite is within the orbit with respect to
perigee.
Figure 2.3: Keplerian Classical Orbital Elements [12].
The same state vector for ISS in Equation 2.1 is also represented by classical orbital elements as follow:
X = ~
α e i Ω w v
=
6784.93557Km 0.0010405 51.66595 o 2.78131 o 104.38744 o 214.53843 o
(2.2)
2.2.2 Ground-based Observers
The geodetic coordinates latitude φ, longitude λ and altitude h are used for a ground-based observer.
WGS84 datum is used in SPOOK, which is a reference ellipsoid.
2.3 Observation Theory
In order to estimate the space object’s position, orbit determination algorithms use external measurements taken by observers. The observers can be ground-based observers from Earth or from a satellite which is orbiting around the Earth (space-based observers). Different observations methods have already existed for space objects, but the most common are passive optical and radar-based observers. The next sections will describe these two observers’ operational principles.
2. THEORETICAL BACKGROUND 7
2.3.1 Optical Observer
The working principle of an optical observer is based on the light reflection from the observed object. The reflected light can be translated into observer measurements which are right ascension α and declination δ angles. A space-based telescope was first used for space objects in 1996 [17].
Space-based sensors have advantages compared to ground-based ones such as not being affected by atmospheric conditions or geographic restrictions.
Figure 2.4: Angular measurements for optical observer: right ascension α and declination δ [12].
2.3.2 Radar Observer
Radar observers are ground base observers which actively send radio waves that are reflected from
the object and detected by the observers. They are two-way signals. Radar sensor can produce
angles (elevation el and azimuth β) plus distance measurements or slant range. In addition to the
angular angles and slant range ρ, Doppler measurements can measure the alteration of distance over
time or slant range-rate ˙ ρ.
Figure 2.5: Angular measurement of the radar observer [12].
2.4 Dynamic Model - Orbital Perturbation
The dynamic model plays an important role in calculating the object’s orbit. In order to calculate a successful orbit determination, a correct model of the space object must be included. All relevant forces which are acting upon the objects (e.g. space debris) must be taken into account in the orbit determination. This part of the thesis describes the forces which were already implemented in SPOOK before this thesis.
2.4.1 Non-spherical Gravitational Field
If the Earth’s mass concentration is assumed to be in a single point, the expectation will be for a spherical gravitational field. This case is possible only for a simple propagation, but in reality, the Earth’s mass distribution is not homogeneous, and the Earth shape is not spherical. Therefore, a non-spherical gravity field U for the earth is represented in Equation 2.3. More detail about the derivation of U can be found in [12].
U = µ r
"
1 +
∞
X
l=2 l
X
m=0
R ⊕ r
l
P l,m [sin(φ gC
sat)] {C l,m cos(mλ sat ) + S l,m sin(mλ sat )}
#
(2.3)
• µ - the gravitational parameter
• r - the spacecraft’s position
2. THEORETICAL BACKGROUND 9
• l, m - the degree and order of the gravity field
• R ⊕ - Earth’s radius
• P l,m - Legendre functions
• φ gC
satand λ sat - the geocentric latitude and longitude of the spacecraft
• C l,m and S l,m are the geopotential coefficients that define the Earth’s mass distribution [18].
2.4.2 Atmospheric Drag
In Leo Earth Orbit (LEO), the strongest perturbation is the atmospheric drag. For precise orbit modeling, this force model is one of the big challenges because of some parameters such as the drag coefficient, object orientation (e.g. space debris) are difficult to determine. The MSIS-00 model used in SPOOK for the computation of the atmospheric drag. This model is developed by the US Naval Research Laboratory [17]. According to [12], the drag acceleration can be computed as follow:
~a Drag = − 1 2 C D
A D
m obj ρv 2 rel ~ v rel
|~v rel | (2.4)
where C D is the drag coefficient, which describes the spacecraft surface and the interaction of the atmosphere, A D is the object’s cross-sectional area, m obj is the mass of the object. The atmospheric density is represented by ρ which is related to the altitude. ~ v rel is the object velocity which is between the atmosphere and the object. To calculate the relative velocity, the Earth’s rotational speed ( ~ w ⊕ ) must be considered so that this is called thermospheric winds.
~ v rel = ~ v − ~ w ⊕ × ~r − ~v t,winds (2.5) where ~ v and ~ r are the object’s respective velocity and position. ~ v t,winds is the thermospheric wind velocity which affects in the upper part of the atmosphere.
2.4.3 Solar Radiation Pressure
If an object is exposed to solar radiation, the absorption and reflection of the photons cause an additional acceleration on the object. This phenomenon is called Solar Radiation Pressure. This perturbation force has bigger effects in higher orbits. The acceleration can be calculated as follows:
~a SRP = −p SRP C R
A R
m obj
~ r
r (2.6)
where A R is the object’s cross-sectional area, C R is the object’s reflectivity coefficient, m obj is the object’s mass, r is the relative position of Sun to the object position, and p SRP is the solar radiation pressure value as 4.57 · 10 −7 N.m −2 .
2.4.4 Third-body Perturbations
As with the Earth’s gravity field effect on the object, other planets similarly affect the object in an
Earth orbit. There are two main perturbations in the Solar System which are Sun and Moon. To
where ~ S is the relative position between the body and the Earth, and µ body is the gravitational constant of the body.
2.5 Initial Orbit Determination
To compute Orbit Determination (OD) successfully the Initial Orbit must be known. In some cases, initial orbit values can be defined by a user. However, in some cases there is no initial information about the orbit. For that reason, it is important to imply the initial orbit determination algorithm for Orbit Determination using a few measurements from observers. In 1881, Carl Friedrich Gauss used declination and right ascension to determine Dwarf Planet Cere’s orbit. This method is called the Gauss algorithm. This method is much simpler if the data consists of the time of the flight and two position vectors [12]. This algorithm is developed by Gibbs and Herrick- Gibbs. These methods have already been implemented in SPOOK.
2.5.1 Gibbs Algorithm
In 1889, Josiah Gibbs introduced an improvement of the Gauss algorithm. This method uses three position vectors to derive the velocity at the middle time incident. The derivation can be found in [12] and it is implemented by [17] in SPOOK [19].
2.5.2 Herrick-Gibbs Algorithm
The previous approach (Gibbs algorithm) fails when the observations are separated by small angles.
Distinct from the Gibbs algorithm, the Herrick-Gibbs algorithm uses a Taylor-series expansion to obtain an expression for the middle velocity vector [12]. The mathematical expression can be found in [17].
2.6 Orbit Determination
In SPOOK, five different orbit methods were already implemented, and an additional OD method is implemented in this project, giving six OD methods. All filters have a different way to estimate position and velocity, but the greatest difference between the filters are how the filters use the data.
OD methods applied in SPOOK are Weighted Least Squares (WLS), Sequential Batch Least Squares (SBLS), Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Extended Square Root Information Filter (ESRIF). The WLS uses all the available measurements at each iteration, while filter-based algorithms use a single measurement at each update step. The SBLS filter is similar to the WLS filter, but this filter uses smaller batches of data at each step. The EKF, UKF, USKF, and ESRIF filters work in real time. For these filters, initial state and initial covariance should be determined before filtering. These algorithms will be described shortly in the next sections.
2. THEORETICAL BACKGROUND 11
2.6.1 The Weighted Least Squares Method
The WLS filter is a batch filter. The initial state vector ~ X 0 is propagated when all measurements are obtained by an observer:
X ~ j = f ~ X 0 , t j
(2.8) After calculating the predicted step, the predicted measurements are calculated:
~ y p,j = h( ~ X j ) (2.9)
In order to find the residuals, the real measurements (~ y r,j ) and the predicted measurements are compared so that the residual ~b j can be found:
~b j = ~ y r,j − ~y p,j (2.10)
To calculate the covariance matrix, the algorithm needs to find the partial derivative matrix and the weighted matrix. The covariance matrix is calculated as follows:
P o =
n
X A T j W A j
! −1
(2.11) where W is a weighting matrix that provides all measurements, which are considered with their standard deviation, and where n is the total number of measurements from the observer:
W =
1
σ
10 0
0 . . . 0 0 0 σ 1
m