1. ASEN 5050 SPACEFLIGHT DYNAMICS Spacecraft Navigation Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 36: Navigation 1

2. Announcements FCQs today! End of class. STK Lab 3 due riiiiiight now! STK Lab 4 due 12/12 Final Exam on 12/12, due 12/18 –Take-home, open book open notes Final project and exam due 12/18 Lecture 36: Navigation 2

3. Schedule from here out Lecture 36: Navigation 3 12/5: Spacecraft Navigation 12/8: Final Review, part 1 12/10: Final Review, part 2 12/12: Small Body Missions

4. Space News Orion’s Exploration Flight Test 1: Friday 12/5 at 7:05 am Eastern Time (5:05 am Mountain!). Duration: 4.5 hours. Lecture 36: Navigation 4

5. ASEN 5050 SPACEFLIGHT DYNAMICS Navigation Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 36: Navigation 5

6. Spacecraft Navigation Mission Design –Define trajectories, error tolerances, navigation strategies, tracking strategies, science requirements, etc. Launch –Launch targets, guidance, tracking, performance estimation Hand-off to the spacecraft operators –Best estimate of the spacecraft’s state and associated uncertainty Tracking and orbit determination –Ground station tracking and maybe other sources of tracking (GNSS, TDRSS, LiAISON, SoOp, etc) –Satellite telemetry –Estimate the spacecraft’s state and associated uncertainty Maneuver design and execution –Maneuver guidance (attitude, timing, propulsion, execution) –Maneuver execution estimation Lecture 36: Navigation 6

7. Spacecraft State The “State” ( X ) describes anything that the satellite operators wish to estimate, with associated uncertainties. Example state parameters: –Satellite’s position and velocity (6 parameters) –Satellite’s mass, Cd, Cr, and other similar parameters –Maneuver execution parameters –Science Atmospheric density profile (GOCE, DANDE, etc) Gravity signatures (GRACE) –Others as needed Lecture 36: Navigation 7

8. Spacecraft State Uncertainty The state is estimated and an uncertainty is associated with that estimate. –State = X –Uncertainty = P, the variance-covariance matrix Lecture 36: Navigation 8 Useful variation, showing only the correlations in the lower-left Symmetric matrix

9. Orbit Determination The art of estimating the state of a satellite in the presence of a great deal of uncertainty. We start with either no information or an estimate of the state. We model the dynamics (gravity, SRP, drag, etc) We acquire tracking data, including noise and systematic errors Filter the data and generate an estimate of the state. Take ASEN 5070 for a lot more information. Lecture 36: Navigation 9

10. Tracking Data Ground stations –1-way, 2-way, 3-way –Range, Doppler, RA, Dec –Delta-DOR –Other beacons, Laser reflectors Satellite tracking –TDRSS, GNSS, LiAISON Telemetry information –Gyros, accelerometers, propulsion system data Other –OpNav, SoOps, X-Ray Pulsar, Radar, Lidar, etc. Lecture 36: Navigation 10

11. Lecture 36: Navigation 11 Methods of Initial Orbit Determination Observations of range, azimuth, and elevation Angles-only observations –Laplace’s Method –Gauss’s Technique –Double r-iteration Mixed Observations –Range and Range-Rate Processing –Range-only Processing Three Position Vectors and Time –Gibbs Method –Herrick-Gibbs Two Position Vectors and Time  Lambert’s Problem –Minimum Energy –Gauss’s Solution –Universal Variables –Battin Method

12. Lecture 36: Navigation 12 Range, Azimuth, Elevation

13. Lecture 36: Navigation 13 Range, Azimuth, and Elevation

14. Angles-Only Observations Lecture 36: Navigation 14

15. Lecture 36: Navigation 15 Three Positions and Time Problem Statement: Given three time-sequential position vectors, determine the velocity vector corresponding to the 2 nd position vector. Note that determining the position and velocity vectors at the 2 nd time essentially completely determines the orbit, and they may be converted into a complete set of orbital elements.

16. Lecture 36: Navigation 16 Gibbs Method We have covered a lot of equations here, so lets summarize the Gibbs Method: Given: Determine: Algorithm: Check if coplanar: Check if too close together:

17. Lecture 36: Navigation 17 Gibbs Method Compute velocity:

18. Lecture 36: Navigation 18 Lambert’s Problem Given two positions and the time-of-flight between them, determine the orbit between the two positions.

19. Orbit Determination If your state has 6 parameters and you get 6 pieces of information, you can estimate each parameter – but you are very vulnerable to modeling and measurement errors! Standard OD methods involve collecting as much data as possible and using a least-squares estimate to fit the state to the data. –i.e., 106 pieces of information, including noise and errors! –Batch processor –Kalman Filters Lecture 36: Navigation 19

20. Lecture 36: Navigation 20 Orbit Determination

21. Things to Note Not all observations are created equal. –Range and Doppler data only provides line-of-sight information. –Angular data – or data over a long arc – may be very useful to improve the observability of the state. Clean data should be weighted higher than noisy data. The satellite is never where you think it is. Pay attention to the covariance matrix! Lecture 36: Navigation 21

22. Maneuver Planning Once you have an estimate of your state, you may require a maneuver downstream in order to achieve the mission’s requirements. What are your acceptable error corridors for each orbital element / state parameter? Lecture 36: Navigation 22

23. Maneuver Execution Error Maneuvers are never performed perfectly –Though they’re getting better and better. There is typically a minimum expected execution error – you can’t perform better without switching to a different system / nozzle / flow rate. There is also typically a proportional error that grows with the maneuver’s duration / magnitude. Lecture 36: Navigation 23

24. Maneuver Navigation Gates Model, describing maneuver execution errors. –C. R. Gates, “A Simplified Model of Midcourse Maneuver Execution Errors,” Tech. Rep. 32-504, JPL, Pasadena, CA, October 15, 1963. Error in pointing of the burn Error in magnitude direction of burn Sample these errors and apply them to Monte Carlo simulations. Lecture 36: Navigation 24

25. Gates Model Error in pointing of the burn Error in magnitude direction of burn Lecture 36: Navigation 25

26. Gates Model Now, define a coordinate frame, where z is aligned with the commanded Delta-V direction: Lecture 36: Navigation 26

27. Gates Model Sample a simple Gaussian distribution three times to generate Execution errors, in maneuver frame: Finally, rotate them into any useful frame. Lecture 36: Navigation 27

28. Navigation Lecture 36: Navigation 28 Example Gates Model inputs into a navigation study:

29. Good practices in spaceflight navigation –Practice! Tune your filter on all kinds of data. Be ready for a wide variety of data. –Perform lots of solutions and compare them. Solutions over different arcs of time, long and short Solutions with different data types Solutions with different parameters, consider parameters, process noise, etc. –Present the information in a manager-friendly format, but understand the details. B-Plane, error ellipses, covariance, residuals –Look at that covariance in different ways. –Don’t neglect the numerical details. 29 Flight Operations

30. All of them. 30 Missions that require navigation

31. All spacecraft require OD –For communication Most spacecraft require OD for other reasons –For science –For mission planning –For maneuver design 31 Spacecraft missions

32. ASEN 5050 SPACEFLIGHT DYNAMICS FCQs Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 36: Navigation 32