GEONSMS_R3_0_NASA-TP-20240004259.pdf

March 2024

NASA/TP-20240004259

Goddard Enhanced Onboard Navigation System (GEONS) Mathematical Specifications

Anne C. Long Goddard Space Flight Center, Greenbelt, Maryland

NASA STI Program Report Series

Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA scientific and technical information (STI) program plays a key part in helping NASA maintain this important role.

The NASA STI program operates under the auspices of the Agency Chief Information Officer. It collects, organizes, provides for archiving, and disseminates NASA’s STI. The NASA STI program provides access to the NTRS Registered and its public interface, the NASA Technical Reports Server, thus providing one of the largest collections of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, which includes the following report types:

• TECHNICAL PUBLICATION. Reports of completed research or a major significant phase of research that present the results of NASA Programs and include extensive data or theoretical analysis. Includes compilations of significant scientific and technical data and information deemed to be of continuing reference value.

NASA counterpart of peer-reviewed formal professional papers but has less stringent limitations on manuscript length and extent of graphic presentations.

• TECHNICAL MEMORANDUM.

Scientific and technical findings that are preliminary or of specialized interest, e.g., quick release reports, working papers, and bibliographies that contain minimal annotation. Does not contain extensive analysis.

• CONTRACTOR REPORT. Scientific and technical findings by NASA-sponsored contractors and grantees.

• CONFERENCE PUBLICATION.

Collected papers from scientific and technical conferences, symposia, seminars, or other meetings sponsored or co-sponsored by NASA.

• SPECIAL PUBLICATION. Scientific, technical, or historical information from NASA programs, projects, and missions, often concerned with subjects having substantial public interest.

• TECHNICAL TRANSLATION.

English-language translations of foreign scientific and technical material pertinent to NASA’s mission.

Specialized services also include organizing and publishing research results, distributing specialized research announcements and feeds, providing information desk and personal search support, and enabling data exchange services.

For more information about the NASA STI program, see the following:

• Access the NASA STI program home page at http://www.sti nasa.gov

National Aeronautics and Space Administration

Goddard Space Flight Center Greenbelt, Maryland

NASA/TP-20240004259

Anne C. Long Goddard Space Flight Center, Greenbelt, Maryland

Available from:

NASA STI Program / Mail Stop 050 NASA Langley Research Center

Hampton, VA 23681-2199

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics and Space Administration.

GEONS-MS-2024

REV 1.0

Release 3.0

Prepared by:

Anne C. Long Date

a. i. solutions, Inc, 4500 Forbes Blvd.

Lanham, MD 20706

Approved by:

David Gaylor Date Code 595 Goddard Spaceflight Center

Nathan Stacey Date Code 595

Preface

This document defines the mathematical specifications for the Goddard Enhanced Onboard Navigation System (GEONS), which was previously known as the GPS Enhanced Onboard Navigation System. This document is a revision of the document FDSS-23-0035, issued November 28, 2012 (Reference 4). It has been updated to include all capabilities implemented in GEONS Flight Software Release 3.0.

Proposed changes to this document should be submitted to the signatories along with supportive material justifying the proposed change. Changes to this document shall be made by complete revision.

Comments or questions concerning this document and proposed changes shall be addressed to:

David Gaylor Code 595 v

Change Information Page

List of Effective Pages Page Number Release Version

Title ii through xvi 1-1 through 1-6 2-1 through 2-42 3-1 through 3-58 4-1 through 4-88 5-1 through 5-154 6-1 through 6-4 7-1 through 7-7 8-1 through 8-10 9-1 through 9-10 10-1 through 10-10 11-1 through 11-20 12-1 through 12-17 A-1 and A-2 R-1 through R-5

Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0 Release 3.0

Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original

Document History Document Number Release/Issue Issue Date

CSC-5506-02R0UD0

CSC-5506-02R0UD1

CSC-5506-02R1UD0

CSC-5547-02R0UD0

CSC-5547-09R0UD0

CSC-5547-09R1UD0

CSC-5547-09R1UD1

CSC-5570-01R0UD0

CSC-5570-01R0UD1

CSC-5570-14R0UD0

FDF-59-001

FDF-59-001

FDF-59-002

FDF-59-002

FDF-59-003

FDF-59-005

FDF-59-009

FDF-59-010

Original Release 1.1, Update 1

Release 1.2 Release 1.2, Update 1

Release 1.3 Release 1.4

Release 1.4, Update 1 Release 2.0

Release 2.0, Update 1 Release 2.1 Release 2.2

Release 2.2, Revision 1 Release 2.3

Release 2.3, Revision 1 Release 2.4 Release 2.5 Release 2.6 Release 2.7

August 2001 November 2001 February 2002

July 2002 November 2002 December 2002 January 2003 February 2003

June 2003 December 2003

March 2004 August 2004 June 2004 April 2005 May 2005

December 2005 February 2006

September 2006

Document Number Release/Issue Issue Date vi

FDF-59-012

FDF-59-014

FDF-59-016

FDF-59-017

FDF-59-018

FDF-59-019

FDSS-14-0002

FDSS-14-0006

FDSS-14-0010

FDSS-23-0035

GEONS-MS-2024

Release 2.8 Release 2.9 Release 2.10 Release 2.11 Release 2.12 Release 2.13 Release 2.14 Release 2.15 Release 2.16 Release 2.17 Release 3.0

January 2007 September 2007 December 2007 September 2008 February 2009

November 2009 November 2010

April 2011 November 2011 November 2012 vii

Contents Preface .......................................................................................................................... iii

Section 1. Introduction

1.1 Purpose and Scope ................................................................................ 1-1

1.2 Document Organization .......................................................................... 1-1

1.3 Overview of the GEONS Navigation Algorithms ..................................... 1-2

Section 2. State Estimation Algorithms

2.1 Estimation State Vector .......................................................................... 2-1

2.2 State Error Covariance Matrix ................................................................. 2-7

2.2.1 Covariance Factorization ....................................................... 2-10

2.2.2 Semimajor Axis Variance ....................................................... 2-12

2.3 Extended Kalman Filter Algorithms ....................................................... 2-14

2.3.1 Time Update Process ............................................................ 2-15

2.3.2 Measurement Update Process .............................................. 2-18

2.3.3 Fault Detection Tests ............................................................. 2-34

2.3.4 GEONS Operational Modes ................................................... 2-37

2.4 Hierarchical Relative Navigation Capability .......................................... 2-39

Section 3. Coordinate Systems, Transformations, and Time Systems

3.1 Coordinate Systems................................................................................ 3-1

3.1.1 Planet-Centered Inertial Systems ............................................ 3-1

3.1.2 Earth-Centered True Equator and Equinox of Date (TOD)

Coordinate Systems................................................................. 3-3

3.1.3 Earth-Centered Earth-Fixed (ECEF) Coordinate System......... 3-4

3.1.4 RIC Orbit Frame Coordinate System ....................................... 3-5

3.1.5 Rotating Libration Point Coordinate System ............................ 3-7

3.1.6 Libration Point RIC Coordinate System ................................... 3-9

3.1.7 Lunar and Planet Body-Fixed Systems .................................. 3-11

3.1.8 VBN Coordinate System ........................................................ 3-11

3.2 Coordinate Transformations ................................................................. 3-13

3.2.1 IAU-76/FK5 ECI-to-Mean-of-Date Coordinate Transformation

............................................................................................... 3-13

3.2.2 Earth Mean-of-Date-to-True-of-Date Coordinate

Transformation ....................................................................... 3-16

3.2.3 Earth True-of-Date-to-ECEF Coordinate Transformation....... 3-23 viii

3.2.4 Central Body Inertial-to-RIC Coordinate Transformation ....... 3-27

3.2.5 Central Body Inertial-to-RIC Covariance Transformation ....... 3-28

3.2.6 Inertial Cartesian to Rotating Libration Point Coordinate

Transformation ....................................................................... 3-28

3.2.7 Rotating Libration Point to Libration-Point RIC Coordinate

Transformation ....................................................................... 3-29

3.2.8 Spacecraft Antenna Offset Transformations .......................... 3-29

3.2.9 ECI to Lunar Principal Axis Frame Coordinate

Transformation ....................................................................... 3-33

3.2.10 VBN to CBI Coordinate Transformation ................................. 3-37

3.2.11 ECI to Non-Earth CBI Coordinate Transformation ................. 3-37

3.2.12 Non-Earth CBI to PBF Coordinate Transformation ................ 3-38

3.2.13 Ecliptic to Mean of J2000 Coordinate Transformation ........... 3-39

3.2.14 J2 Mean Semimajor Axis Error .............................................. 3-39

3.2.15 IAU-2000 ECI/GCRF to ECEF/ITRF Coordinate

Transformation ....................................................................... 3-41

3.3 Time Systems ....................................................................................... 3-43

3.3.1 Conversion From GPS System Time to UTC ......................... 3-43

3.3.2 Conversion From GPS SV Clock Time (tsv) to GPS System

Time ...................................................................................... 3-45

3.3.3 Conversion From Julian Date to Calendar Date .................... 3-47

3.3.4 Conversion From Calendar Date to Julian Date .................... 3-48

3.3.5 Conversion From UTC to TAI and Terrestrial Time ............... 3-49

3.3.6 Relativistic Clock Corrections ................................................ 3-49

3.3.7 Conversion From Galileo System Time (GST) to UTC .......... 3-53

3.3.8 Conversion From Galileo System Time (GST) to GPS Time

(GPST) ................................................................................... 3-55

3.3.9 Conversion From Galileo Satellite Clock Time to Galileo System

Time (GST) ........................................................................... 3-56

Section 4. State Propagation

4.1 Spacecraft Equations of Motion .............................................................. 4-1

4.1.1 Earth, Solar, Lunar, and Planetary Point-Mass Accelerations . 4-2

4.1.2 Nonspherical Gravitational Acceleration ................................ 4-14

4.1.3 Atmospheric Drag Acceleration ............................................. 4-23

4.1.4 Solar Radiation Pressure Acceleration Model ........................ 4-31

4.1.5 Externally Measured Accelerations ........................................ 4-36

4.1.6 Unmodeled Accelerations ...................................................... 4-37

4.2 Runge-Kutta Integration Algorithm ........................................................ 4-39

4.3 Nonspacecraft State Vector Propagation .............................................. 4-41 ix

4.3.1 Random Walk Processes ....................................................... 4-44

4.3.2 First-Order Gauss-Markov Processes.................................... 4-47

4.3.3 Second-Order Gauss-Markov Processes .............................. 4-51

4.4 State Error Covariance Propagation ..................................................... 4-53

4.4.1 State Transition Matrix Computation ...................................... 4-53

4.4.2 Process Noise Covariance Matrix .......................................... 4-60

4.4.3 Time Propagation .................................................................. 4-74

4.5 Spacecraft Maneuver Covariance and Clock Variance Increments ...... 4-76

4.5.1 Maneuver Process Noise Models Based on Maneuver Variances and Variance Rates .............................................................. 4-76

4.5.2 Maneuver Process Noise Models Using Maneuver Acceleration

............................................................................................... 4-79

4.5.3 Addition of the Maneuver Covariance to the State Covariance

............................................................................................... 4-81

4.5.4 Addition of the Clock Covariance to the State Covariance..... 4-81

4.6 Ground Receiver State Propagation ..................................................... 4-83

4.6.1 Ground Receiver State Vector Prediction .............................. 4-83

4.6.2 State Transition Matrix ........................................................... 4-84

4.6.3 Process Noise Matrix ............................................................. 4-85

4.7 Moon-Based Receiver State Propagation ............................................. 4-86

4.7.1 Moon-Based Receiver State Vector Prediction ...................... 4-86

4.7.2 State Transition Matrix ........................................................... 4-87

4.7.3 Process Noise Matrix ............................................................. 4-88

Section 5. Measurement Models

5.1 Measurement Selection and Processing Overview ................................. 5-1

5.2 GPS/Galileo Space Vehicle/WAAS GEO Ephemeris Computation ........ 5-5

5.2.1 GPS/Galileo Broadcast Ephemeris Parameters ...................... 5-6

5.2.2 User Algorithm for GPS/Galileo SV Antenna Phase Center

Position Computation ............................................................... 5-7

5.2.3 GPS/Galileo Antenna Phase Center Velocity Computation ... 5-10

5.2.4 TASS GPS Differential Corrections ....................................... 5-12

5.2.5 GPS Improved Clock and Ephemeris (ICE) Differential

Correction Parameters ........................................................... 5-13

5.3 GNSS Pseudorange, Doppler, and Carrier Phase Measurement Models ..

.............................................................................................................. 5-15

5.3.1 Pseudorange Measurement Preprocessing ........................... 5-15

5.3.2 GPS/WAAS Pseudorange Measurement Model and Associated

Partial Derivatives .................................................................. 5-21 x

5.3.3 GPS/WAAS Doppler Instantaneous Measurement Model and

Associated Partial Derivatives ............................................... 5-27

5.3.4 GPS/WAAS Integrated Carrier Phase Measurement Model and Associated Partial Derivatives ............................................... 5-32

5.3.5 Ionospheric Refraction Delay Using GPS Ionospheric

Parameters ............................................................................ 5-34

5.3.6 Galileo Pseudorange Measurement Model and Associated

Partial Derivatives .................................................................. 5-42

5.3.7 Galileo Instantaneous Doppler Measurement Model and

Associated Partial Derivatives ............................................... 5-45

5.3.8 Galileo Integrated Carrier Phase Measurement Model and

Associated Partial Derivatives ............................................... 5-47

5.4 Singly Differenced GPS/WAAS Measurement Models ......................... 5-51

5.4.1 Singly Differenced Measurement Preprocessing ................... 5-51

5.4.2 Singly Differenced GPS/WAAS Pseudorange Measurement

Model and Associated Partial Derivatives .............................. 5-52

5.4.3 Singly Differenced GPS/WAAS Doppler Measurement Model

and Associated Partial Derivatives ........................................ 5-55

5.4.4 Singly Differenced GPS/WAAS Carrier Phase Measurement

Model and Associated Partial Derivatives .............................. 5-57

5.5 Cross-Link Measurement Models ......................................................... 5-60

5.5.1 One-Way Cross-Link Pseudorange Measurement Model and

Associated Partial Derivatives ............................................... 5-60

5.5.2 One-Way Cross-Link Averaged Doppler Measurement Model

and Associated Partial Derivatives ........................................ 5-68

5.5.3 Two-Way Cross-Link Range Measurement Model and

Associated Partial Derivatives ............................................... 5-74

5.5.4 Two-Way Cross-Link Averaged Doppler Measurement Model

and Associated Partial Derivatives ........................................ 5-79

5.6 GS Measurement Models ..................................................................... 5-83

5.6.1 Raw GS Doppler Measurement Processing .......................... 5-83

5.6.2 GS Pseudorange and Associated Partial Derivatives ............ 5-86

5.6.3 GS One-Way Forward Averaged Doppler Measurement Model

and Associated Partial Derivatives ........................................ 5-90

5.6.4 GS Round-Trip Range-Rate Measurement Preprocessing .... 5-95

5.7 Geometric Point Solution Measurement Models ................................... 5-97

5.7.1 Geometric Point Solution Measurement Processing .............. 5-97

5.7.2 Geometric Point Solution Measurement Model and

Associated Partial Derivatives ............................................... 5-97

5.7.3 Measurement Update for Geometric Point Solution

Processing ............................................................................. 5-99 xi

5.8 Celestial Object Measurement Models ............................................... 5-101

5.8.1 Spacecraft Models ............................................................... 5-102

5.8.2 Celestial Object and Intersatellite Measurements for a Three-

Axis-Stabilized Satellite........................................................ 5-106

5.8.3 Planetary Sensor Measurements for a Spin-Stabilized

Satellite ................................................................................ 5-120

5.8.4 Pseudoangle Measurements ............................................... 5-131

5.9 TDRSS Measurement Models ............................................................ 5-135

5.9.1 Raw TDRSS Doppler Measurement Preprocessing ............ 5-135

5.9.2 TDRSS Pseudorange and Associated Partial Derivatives ... 5-136

5.9.3 TDRSS Forward-Link Averaged Doppler Measurement Model

and Associated Partial Derivatives ...................................... 5-138

5.9.4 TDRSS Differenced One-Way Doppler (DOWD) Measurement

Model and Associated Partial Derivatives ............................ 5-143

5.10 X-Ray Pulsar Measurement Models ................................................... 5-147

5.10.1 X-Ray Pulsar Navigation Measurement Preprocessing ....... 5-147

5.10.2 X-Ray Pulsar Navigation Measurement Model and Associated

Partial Derivatives ................................................................ 5-148

5.11 Two-Leg GPS Pseudorange Measurement Model ............................. 5-151

Section 6. Real-Time State Propagation

6.1 Real-Time Propagation Algorithm ........................................................... 6-1

6.2 Real-Time Propagation Acceleration Model ............................................ 6-1

Section 7. Doppler Compensation Prediction

7.1 GS Visibility Test ..................................................................................... 7-1

7.2 GS FCW Computation ............................................................................ 7-2

7.3 TDRSS FCW Computation ..................................................................... 7-4

7.4 TDRS Visibility Test ................................................................................ 7-5

Section 8. Averaged Orbital Element Ephemeris

8.1 Computation of Averaged State Vector................................................... 8-1

8.2 Computation of Reference Averaged Equinoctial Elements and Rates .. 8-2

8.3 Equinoctial Element Transformations ..................................................... 8-5

8.3.1 Definition of Equinoctial Elements ........................................... 8-5

8.3.2 Transformation From Equinoctial Elements to Cartesian

Coordinates ............................................................................. 8-7

8.3.3 Transformation From Cartesian Coordinates to Equinoctial

Elements .................................................................................. 8-8 xii

8.3.4 Transformation From Keplerian Elements to Equinoctial

Elements ................................................................................ 8-10

Section 9. Orbit Control Algorithms

9.1 Generic Maneuver Targeting Algorithm .................................................. 9-1

9.1.1 Solution to Lambert’s Problem ................................................. 9-4

9.1.2 Computation of f and g Functions ............................................ 9-5

9.1.3 Central Body Collision Detection Algorithm ............................. 9-8

9.2 Formation Control Algorithms ............................................................... 9-10

Section 10. Cold Start Initialization Algorithm

10.1 Algorithm Overview ............................................................................... 10-1

10.2 Initialization Algorithm Summary ........................................................... 10-3

10.3 Coarse Search Algorithm ...................................................................... 10-3

10.4 Golden Section Search for Minimum .................................................... 10-6

Section 11. Attitude Estimation Algorithms

11.1 Attitude Estimation State Vector ........................................................... 11-1

11.2 Attitude State Error Covariance Matrix.................................................. 11-3

11.3 Attitude Estimation Algorithm ................................................................ 11-4

11.3.1 Attitude State Initialization ...................................................... 11-4

11.3.2 Time Update Process ............................................................ 11-4

11.3.3 Measurement Update ........................................................... 11-5

11.4 Attitude State and Covariance Propagation Algorithms ........................ 11-8

11.5 Attitude Measurement Models ............................................................ 11-12

11.5.1 SNR Measurement Model and Partial Derivatives ............... 11-13

11.5.2 Self Calibration Using SNR Measurements ......................... 11-14

11.5.3 GPS Double-Difference Carrier Phase Measurements and

Partial Derivatives ................................................................ 11-15

11.5.4 Double-Difference Carrier Phase Integer Resolution and

Residual Check .................................................................... 11-18

11.6 Attitude State Initialization ................................................................... 11-19

Section 12. GPS/Galileo Measurement Simulation

12.1 GPS and Galileo Measurement Simulation Models .............................. 12-2

12.1.1 GNSS Pseudorange Simulation ............................................. 12-2

12.1.2 GNSS Instantaneous Doppler Simulation .............................. 12-3

12.1.3 GNSS Carrier Phase Simulation ............................................ 12-3

12.2 Receiver Clock Error Model Simulation ................................................ 12-4 xiii

12.3 Transmitter Ephemeris and Clock Error Model ..................................... 12-8

12.4 Measurement Noise Model ................................................................... 12-9

12.4.1 Step-Function Measurement Noise Model ............................. 12-9

12.4.2 Pseudorange Measurement Noise Model Based on Thermal

Noise Theory ....................................................................... 12-10

12.5 Measurement Validity Editing ............................................................. 12-11

12.5.1 Radial Distance Editing ........................................................ 12-11

12.5.2 HORP and Earth Occultation Editing ................................... 12-11

12.5.3 Antenna Limit Tests ............................................................. 12-11

12.5.4 Receiving Antenna Signal-to-Noise Ratio Calculation ......... 12-12

12.5.5 Antenna Gain Limit Tests ..................................................... 12-13

12.5.6 GPS Signal Acquisition Probability Calculation and Tracking

Test ...................................................................................... 12-14

12.5.7 Time Interval Editing ............................................................ 12-15

12.5.8 Transmitter Antenna Models ................................................ 12-15

12.5.9 Receiving Antenna Models .................................................. 12-16

Abbreviations and Acronyms

References

List of Figures 1-1 GEONS Software Library .................................................................................. 1-1

1-2 GEONS Navigation Processing Scenario .......................................................... 1-6

2-1 HORP Geometry ............................................................................................. 2-21

2-2 Atmospheric Editing for GS Measurements ..................................................... 2-23

2-3 GEONS Mode State Transition Diagram ......................................................... 2-38

3-1 IAU 1976 FK5 ECI Coordinate System .............................................................. 3-2

3-2 Earth-Centered Earth-Fixed Coordinate System ............................................... 3-5

3-3 Radial/In-Track/Cross-Track (RIC) Orbit Frame Coordinate System ................. 3-6

3-4 Rotating Libration Point Coordinate System ...................................................... 3-7

3-5 Libration-Point Centered Radial/In-Track/Cross-Track (LRIC) Orbit Frame Coordinate System .......................................................................................... 3-10

3-6 Precession Angles ........................................................................................... 3-14

3-7 Nutation Angles ............................................................................................... 3-16 xiv

3-8 Definition of Rotating Body Frame ................................................................... 3-32

3-9 Relativistic Effects on a GPS SV Clock ........................................................... 3-51

3-10 Relativistic Effects on a HEO Satellite Clock ................................................... 3-52

5-1 Measurement Selection/Processing Algorithm .................................................. 5-3

5-2 Special Cases for Computing the Pseudorange Ambiguity ............................. 5-17

5-3 Signal Path Geometry ..................................................................................... 5-36

5-4 Ground Station Forward-Link Signal Path ....................................................... 5-84

5-5a Definition of Sun-Pointing 3-Axis Stabilized Absolute Reference Frame ....... 5-102

5-5b Definition of True Ecliptic Coordinate Frame ................................................. 5-102

5-6 Definition of Spin-Stabilized Attitude Reference Frame ................................. 5-105

5-7 SOHO Fine Sun Sensor Measurements ........................................................ 5-107

5-8 Sun Angle Measurement Geometry .............................................................. 5-121

5-9 Earth-Out Crossing Measurement Geometry ................................................ 5-123

5-10 Earth/Out Measurement Geometry ................................................................ 5-125

5-11 TDRSS Forward-Link Signal Path ................................................................. 5-135

6-1 GEONS Real-Time Interface Sequence ............................................................ 6-3

7-1 Line-of-Sight Visibility Cases ............................................................................. 7-2

7-2 TDRS Visibility ................................................................................................... 7-7

8-1 Equinoctial Coordinate System .......................................................................... 8-6

9-1 Geometry of Rendezvous Targeting Problem ................................................... 9-1

9-2 Overview of Lambert Targeting Algorithm ......................................................... 9-2

9-3 First Iteration of the Targeting Problem ............................................................. 9-4

12-1 Measurement Simulation Algorithm ................................................................. 12-1

12-2 Parameter Fits to Typical Oscillator Performance Data ................................... 12-8

12-3 Typical Simulated GPS Ephemeris and Clock Errors ...................................... 12-9

12-4 Comparison of EIRP based on GPS ACE Patterns and Conservative Gal E1 Antenna Gain Model. Left plot shows per-SV patterns averaged over azimuth.

Right plot shows all azimuth cuts ................................................................... 12-16 xv

List of Tables

1-1 Summary of GEONS Flight Software Algorithms ............................................... 1-4

3-1a Coefficients for the Series for Nutation in Longitude ∆Ψ and Obliquity ∆ε, With T Measured in Julian Centuries From Epoch J2000.0 ........................................ 3-18

3-1b Coefficients for the Truncated Series for Nutation in Longitude ∆Ψ and Obliquity ∆ε, With T Measured in Julian Centuries From Epoch J2000.0 ....................... 3-21

3-2 GPS ICE Earth Orientation Parameters .......................................................... 3-26

3-3 Values for the Direction of the North Pole of Rotation and Prime Meridian of the Sun and Planets .............................................................................................. 3-38

3-4 Standard Epochs ............................................................................................. 3-43

3-5 GPS-to-UTC Time Conversion Parameters Included in Page 18 of Subframe ........................................................................................................ 3-44

3-6 Standard Epochs ............................................................................................. 3-53

3-7 GPS-to-UTC Time Conversion Parameters (from Table 65 in Reference 63) . 3-54

3-8 GGTO Parameters for the GPS Time to GST Offset Computation (from Table 66 in Reference 63) ...................................................................... 3-56

3-9 Galileo Clock Correction Parameters (from Table 60 in Reference 63) ........... 3-57

4-1 Trigonometric Series for the Sun ....................................................................... 4-7

4-2 Trigonometric Series for the Moon .................................................................... 4-8

4-3 Best-Fit Parameters for the Harris-Priester Minimum Atmospheric Density, ρmin ..................................................................................................... 4-27

4-4 Best-Fit Parameters for the Harris-Priester Maximum Atmospheric Density, ρmax .................................................................................................... 4-29

4-5 Coefficients for the Eight-Order Runge-Kutta Method ..................................... 4-40

5-1 Measurement Selection/Processing Algorithm .................................................. 5-3

5-2 Ephemeris Parameters Contained in the LNAV Broadcast Navigation Message ............................................................................................................ 5-6

5-3 Clock and Ephemeris Differential Correction Parameters Contained in the Broadcast Navigation Message Types 34 or 13 and 14 .................................. 5-13

5-4 Relationship Between Iamb and Nc ................................................................... 5-18

6-1 Real-Time State Propagation Sequence ........................................................... 6-4

12-1 Hadamard Deviations Used in Clock Models................................................... 12-7

1-1

Section 1. Introduction

The National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) has developed the capability to provide high-accuracy attitude, orbit, and time autonomously onboard NASA spacecraft. The GSFC Mission Engineering and Systems Analysis Division has implemented NASA-developed navigation algorithms for high-accuracy real-time onboard orbit determination in the Global Positioning System (GPS) Enhanced Orbit Determination (GEODE) flight software. The Goddard Enhanced Onboard Navigation System (GEONS) extends the capabilities of the GEODE flight software to include additional measurement types and additional navigation algorithms.

1.1 Purpose and Scope

This document presents the mathematical algorithms implemented in the GEONS Software Library (or GEONS). Reference 1 defines the GEONS flight software requirements. These algorithms were initially developed based on prototype flight software developed to support the Explorer Platform (EP)/Extreme Ultraviolet Explorer (EUVE) Tracking and Data Relay Satellite System (TDRSS) Onboard Navigation System (TONS) experiment (Reference 2) and the TONS flight software implemented for the Earth Observing System (EOS) Terra mission (Reference 3).

The GEONS Software Design Document (Reference 76) describes the software architecture and design of the GEONS Computer Software Configuration Items (CSCIs). The GEONS Software Library consists of four CSCIs: GEONS Flight, GEONS Ground Tools, GEONS Analysis Tools, and GEONS Beta, shown in Figure 1-1. This document includes mathematical specifications for all capabilities implemented in Release 3.0 of the GEONS Software Library. In addition, an indication is provided for algorithms that are implemented in the Beta and Ground CSCIs or planned for implementation in future versions of GEONS.

Figure 1-1 GEONS Software Library

1.2 Document Organization

Section 1.3 provides an overview of the algorithms implemented in GEONS. Sections 2 through 12 define the explicit algorithms implemented and provide associated mathematical background useful in understanding these algorithms. To distinguish the flight software

1-2 algorithms from the background discussion, each equation number associated with an explicit flight software algorithm is followed by an asterisk (*).

Section 2 provides a high-level description of the orbit estimation algorithms. The estimation state is defined, and the extended Kalman filter (EKF) processing flow is described. Section 3 describes the coordinate systems, transformations, and time systems used in GEONS. In addition, the algorithm used in the calculation of the Greenwich hour angle is provided.

Section 4 describes the single-step Runge-Kutta numerical integration algorithm and force models used in the propagation of the spacecraft equations of motion. Algorithms for propagation of nonspacecraft state variables and ground-based receivers are given. State error covariance propagation algorithms are also presented.

Section 5 defines the Global Navigation Satellite Systems (GNSS)/Wide Area Augmentation System (WAAS) and cross-link measurement models and measurement partial derivatives.

Algorithms for computing the GNSS space vehicle (SV) position and velocity are provided. This section also includes the point solution, ground-station-to–satellite range and Doppler, celestial object, and TDRSS forward-link Doppler measurement models and measurement partial derivatives.

Section 6 provides the algorithms used to propagate the user state vector to real time. Section 7 provides an algorithm for Doppler compensation prediction. Section 8 provides a backup ephemeris computation algorithm. Section 9 describes the generic maneuver targeting algorithm based on Lambert’s method. Section 10 provides a cold start initialization algorithm that uses range and Doppler measurements. Section 11 provides attitude estimation algorithms that may be implemented in a future version of GEONS. Section 12 provides algorithms that are used to simulate GNSS measurements.

A list defining the abbreviations and acronyms used and a list of the references cited throughout this document follows Section 12.

1.3 Overview of the GEONS Navigation Algorithms

Table 1-1 summarizes the GEONS navigation algorithms defined in this document. Algorithms supported in Release 3.0 of the GEONS Flight CSCI (referred to as GEONS 3.0 throughout this document) are indicated as well as algorithms currently included in the GEONS Beta or Ground CSCI or future if not currently available in a GEONS CSCI. A ground receiver propagation model is also provided to support prelaunch testing of spacecraft receivers and vehicles on the surface of the Moon, planet, or asteroid. These algorithms are defined in detail in Sections 2 through 12 of this document.

GEONS navigation processing consists of two primary activities:

• State vector estimation based on the processing of GNSS SV, WAAS GEO, cross-link range and Doppler, cross-link line-of-sight, Ground-Station-to-satellite range and Doppler, celestial object, TDRSS forward-link Doppler, and/or point solution position measurements

• Real-time state vector propagation

1-3

In addition, Doppler compensation prediction, backup ephemeris computation, maneuver targeting, and cold start initialization are also implemented in Beta versions of GEONS. Figure 1- 2 provides an overview of the GEONS navigation processing scenario. In this figure, dt is the time interval between state vector updates by the estimator, nominally equal to 30 to 60 seconds.

During the same time intervals, measurement, GEONS state estimation, and real-time propagation processing are performed for different time periods. For example, the GPS/WAAS measurement process and real-time propagation could occur once per second. GEONS state vector estimation could be performed at regular intervals, e.g., every 30 or 60 seconds, with intermediate propagation of the filter state vector if required to maintain prediction accuracy.

2-1

Section 2. State Estimation Algorithms

The state estimation algorithm for GEONS consists of an extended Kalman filter (EKF) that uses physically connected process noise covariance models to account for force model and measurement errors.

This section provides a high-level description of the state estimation algorithms. Section 2.1 defines the estimation state, Section 2.2 defines the state error covariance, and Section 2.3 describes the state estimation processing flow, with appropriate references made to the more detailed algorithms provided in Sections 3 through 5.

2.1 Estimation State Vector

The GEONS estimation algorithm estimates a state vector, X , which consists of one or more satellite state vectors, X i , a vector of tracking-system-dependent measurement biases, B , and the ionospheric delay scale factor, Iγ . Each satellite state vector consists of the satellite’s position and velocity vectors and optionally the receiver time bias, the receiver time bias drift, the receiver time bias acceleration, the atmospheric drag coefficient correction, the solar radiation pressure coefficient correction, and/or acceleration biases. The estimator models the drag coefficient correction, solar radiation pressure coefficient correction, the receiver time bias corrections, acceleration biases, accelerometer sensor biases, and tracking-sensor-dependent measurement biases as random variables. Alternatively, the state vector can include the position and velocity vectors of a surface receiver located on the Earth, Moon or planet. In the case of a surface receiver, the drag coefficient correction and solar radiation pressure coefficient correction cannot be estimated.

Optionally, any parameter in the estimation state vector except for the satellite position and velocity can be processed as a consider parameter. However, GEONS 3.0 only supports the inclusion of the atmospheric drag coefficient correction and sensor and global biases as consider parameters. The value and covariance of a consider parameter is not updated by the filter measurement update process but the effects of consider parameter uncertainties are included in the calculation of corrections to and covariance of estimated parameters and in the error cross-covariance of estimated and consider parameters. In addition, GEONS provides the capability to configure the estimation state vector to estimate a subset of the total estimation state vector. This capability can be used to reduce the state vector when different state vector components are estimated for multiple vehicles.

The total estimation state vector with dimension )1(

B

N n n S Nn

S

, where sn is the size of each receiver’s state vector, sN is the number of satellite/receiver state vectors being estimated, and

BN is the number of measurement biases being estimated, has the following form:

2-2 γ

I

N

B X

X

X S

(2.1-1a)*

Each receiver state vector with dimension ≤11 has the following form:

n

U

R

R

Dn a b C C R R

(2.1-1b)* where

R = satellite/receiver position vector in Earth-centered J2000.0 coordinates (meters)

R = satellite/receiver velocity vector in Earth-centered J2000.0 coordinates (meters per second)

∆ ∆C CD R, = atmospheric drag and solar radiation pressure coefficient corrections (unitless)

Rb = vector of receiver-dependent bias parameters, given by

R

R

R d d b b

(2.1-2a)*

Rb = receiver time bias from onboard reference time (defined in Section 4.3) (meters)

Rd = receiver time bias rate (defined in Section 4.3) (meters per second)

Rd = optional receiver time bias acceleration (defined in Section 4.3) (meters per second2)

2-3 and Ua = unmodeled acceleration biases in radial-intrack-crosstrack (RIC), velocity-binormal-normal (VBN), or spacecraft body frame (defined in Section 4.3) (meters per second2)

Optionally, the state vector can include the relative state vector of each nonlocal satellite with respect to the local satellite (n=1) can be estimated, rather than its absolute state vector:

I

Nrel

B X

X X

X S rel

(2.1-1c)

1;

1 ≠

=−= n a b C C R R aa bb

CC

CC

RR

RR

XXX

rel n rel n rel n rel n n n

U n

U

R n

R n n n n nn

U

R

R

D rel rel

RR

DD

(2.1-1d)*

The BN –dimension vector of tracking-sensor-dependent biases can include 32 GPS + 36 Galileo + 8 WAAS GEO pseudorange biases, 32 GPS + 36 Galileo + 8 WAAS GEO Doppler biases, sN cross-link pseudorange and Doppler biases, measurement biases for each celestial object sensor on each of the sN satellite/receivers being estimated, GSn Ground Station (GS) range and Doppler biases, TDRSn Tracking and Data Relay Satellite System (TDRSS) Doppler biases, 40 GPS/WAAS + 36 Galileo carrier phase biases for each of the sN satellite/receivers being estimated, 40 GPS/WAAS + 36 Galileo singly differenced carrier phase biases between the local satellite/receiver and each of the sN -1 remote satellite/receivers being estimated, and 3 accelerometer sensor measurement biases for each accelerometer:

2-4

A

IMU

WG

WG

TDRS

d

GS

d

GS

CO

CL

d

CL

WG

d

WG

b b b b b b b b b b b

B φ φ ρ ρ ρ

(2.1-2b) where

WGb / ρ = vector of 32 GPS + 36 Galileo + 8 WAAS GEO pseudorange biases (defined in

Section 4.3) (meters)

WG

db / = vector of 32 GPS + 36 Galileo + 8 WAAS GEO Doppler biases (defined in

Section 4.3) (Hertz)

CLbρ = vector of sN cross-link pseudorange biases (defined in Section 4.3) (meters)

CL

db = vector of sN cross-link Doppler biases (defined in Section 4.3) (Hertz)

COb = vector of biases for sn celestial object sensors on sN satellites (defined in Section 4.3)

GSbρ = vector of GSn GS range biases (defined in Section 4.3) (meters)

GS

db = vector of GSn GS Doppler biases (defined in Section 4.3) (Hertz)

TDRS

db = vector of TDRSn TDRS Doppler biases (defined in Section 4.3) (Hertz)

WGb / φ = vector of carrier phase biases between each of 32 GPS + 36 Galileo + 8 WAAS

GEO transmitters and the receiver on each of the sN satellites being estimated (defined in Section 4.3) (meters)

2-5

WGb / φ∆ = vector of singly differenced carrier phase biases for each of 32 GPS + 36 Galileo

+ 8 WAAS GEO transmitters between the local satellite/receiver and each of the sN -1 remote satellite/receivers being estimated (defined in Section 4.3) (meters)

A IMUb = vector of accelerometer sensor measurement biases in the IMU frame (defined in Section 4.3) (meters/second2).

The system equations are given by

I

N tB tX tX

)(1

(2.1-3) a b

C

C

R

R n

U

R

R

Dn tw tw tw tw tw tw a b C C R R

X

D

(2.1-3a)

∆ A

IMU

WG

WG

TDRS

d

GS

d

GS

CO

CL

d

CL

WG

d

WG

A

IMU

WG

WG

TDRS

d

GS

d

GS

CO

CL

d

CL

WG

d

WG

w w w w w w w w w w w b b b b b b b b b b b

B φ φ ρ ρ ρ φ φ ρ ρ ρ

(2.1-4)

2-6

II w=γ (2.1-4a)

( )w tR = noise process that models random disturbance on R

= noise process that models random disturbances on R

( )w tCD = white noise process that models random disturbances on ∆CD

( )w tCR = white noise process that models random disturbances on ∆CR

( )w tb = noise process that models random disturbance on bR ( )twa = noise process that models random disturbance on Ua ( )tw WG / ρ = noise process that models random disturbance on WGb / ρ

( )tw WG d

/ = noise process that models random disturbance on WG db /

( )tw CL ρ = noise process that models random disturbance on CLbρ

( )tw CL d = noise process that models random disturbance on CL db ( )tw CO = noise process that models random disturbance on COb ( )tw GS ρ = noise process that models random disturbance on GSbρ ( )tw GS d = noise process that models random disturbance on GS db

( )twTDRS d = noise process that models random disturbance on TDRS db ( )tw WG / φ = noise process that models random disturbance on WGb / φ

( )tw WG / φ∆ = noise process that models random disturbance on WGb / φ∆

( )tw A IMU = noise process that models random disturbance on A

IMUb ( )twI = noise process that models random disturbance on Iγ

The quantity ( )w tb is a column vector given by w w w twb (2.1-5) where w1, w2, and w3 are white noise processes that model random disturbances on Rb , Rd , and

Rd respectively.

The satellite acceleration equations, , are provided in Section 4.1. The derivatives of ∆CD, ∆CR , bR , Ua , WGb / ρ , WG db / , CLbρ , CL db , COb , GSbρ , GS db , TDRS db , WGb / φ , WGb / φ∆ , A

IMUb , and Iγ are defined in Section 4.3. The ground-based receiver propagation equations are given in Section 4.6.

( )w tR

( )R t

2-7

2.2 State Error Covariance Matrix

The state error covariance at time tk is defined as the expectation value of the square of the deviation of the estimated state ( )X k at time tk from the true (unknown) state at time tk, i.e., [ ][ ]P E X X X Xk k k k k

T

(2.2-1)

In the equations below, a plus sign in parentheses (+) denotes the value of a quantity that has been corrected to include the effects of a measurement; a minus sign in parentheses (–) denotes the value of a quantity before this correction has been implemented.

If the effects of the kth measurement have been included in the state error estimate, then

( ) ( )[ ] ( )[ ]P E X X X Xk k k

(2.2-2)

If they have not been included, then

( ) ( )[ ] ( )[ ]P E X X X Xk k k

(2.2-3)

The state error covariance matrix represents the filter uncertainty in the estimated state vector. It also accounts for error correlations between estimated state vector elements. For the filter to be accurate and stable, the covariance matrix must represent the actual errors in the estimated state vector. The state error covariance matrix is defined below:

[ ]P

C C C

C

N N

N N N

1 2 1 2 1 1

2 1 1 2 2

1 1 σ σ σ σ σ σ σ σ σ σ σ

(2.2-4)

( )X k

2-8

[P] = [N × N] state error covariance matrix, where N equals )1( ++× Bss NNn σi = standard deviation in estimate of state vector element i σ i 2 = variance in estimate of state vector element i

Ci,j = Cj,i = correlation coefficient for elements i and j, absolute value < 1

The state error covariance is initialized or reinitialized using command parameters. If the initial state error variances σ i

2 are provided in Mean of J2000.0 XYZ coordinates, they are used directly in Equation 2.2-4 to form [ ]XYZP with Ci,i = 1 and off-diagonal Ci,j = 0.

If the state error variances σ i 2 are provided in instantaneous radial, in-track, cross-track (RIC) coordinates, they are used as follows to form the receiver position and velocity state vector covariance submatrices [ ]n

RICP , for each of the sN satellites/receivers. In this case Ci,i = 1 and all off-diagonal Ci,j = 0 except for the radial velocity/in-track position correlations, C4,2 and C2,4, and the radial position/in-track velocity correlations, C1,5 and C5,1, which are uplinked parameters, nominally equal to –0.95.

σ σσσ σσσ σ σσσ σσσ

1,5

2,4

4,2

5,1

00000 00000

C

IIR

RIR

C

IRI

IRR

n

RIC

C C

C C

P

(2.2-4b)

Each of the resulting RIC covariance submatrices is then transformed to the Earth-centered Mean of J2000.0 frame as defined in Equation (3.2-53) in Section 3.2.5. The initial covariance matrix is then constructed as follows:

γσ

)(0000 0)(000

I

NN

B P

P

SS

(2.2-4c)* where:

2-9

∆ 2

2/

2/

2/

2/

000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000

A

IMU

WG

WG

TDRS

d

GS

d

GS

CO

CL

d

CL

WG

d

WG

B σ σ σ σ σ σ σ σ σ σ σ σ φ φ ρ ρ ρ

(2.2-4d)*

The covariance for individual state vector elements can be reinitialized by resetting the associated diagonal elements in the full covariance matrix to their initial values and the associated off-diagonal elements to zero. Whenever the state error covariance is initialized or reinitialized, it is factored as discussed below.

Similarly, when the relative state vectors of satellite n with respect to satellite 1 are estimated, the initial covariance matrix is given by γσ

)(0000 0)(000

I

NN

rel

B P

P

SS

(2.2-4e)* k n k n k n k nnn rel tXtXtXtXEP relrelrelrel

)()(ˆ)()(ˆ (2.2-4f)

The relative covariance matrices are related to the absolute covariance matrices as follows:

11 1 1nn nn n n rel abs abs abs absP P P P P = + − − (2.2-4g) and similarly

[ ] [ ] [ ] [ ] [ ] n rel n rel nn relabs nn abs PPPPP 1111

+++…