ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 1:

1. ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 1: Introduction to Stat OD

2. Course Outline • Instructor • Professor George H. Born <georgeb@colorado.edu> • Office: ECNT 316 • Office Hour: Wed 2-3 PM • Professor Jeff Parker <parkerjs@colorado.edu> • Office: ECNT 418 • Office Hours: Mon 2-3 PM, Wed 10-11 AM • Course Assistants • Eduardo Villalba <eduardo.villalba@colorado.edu • Office: ECNT 414 • Office Hours: Tues 11-12 AM • Paul Anderson <paul.anderson@colorado.edu> • Office: ECEE 275 • Office Hours: Mon 10-11 AM

3. George Born: Wrote the book on Stat OD.

4. Introductions • Jeff Parker • Graduated from CU in 2007 • “Low-Energy Ballistic Lunar Transfers” • JPL since then • Chandrayaan-1 • GRAIL • MoonRise • Team-X

5. Introductions • Eduardo and Paul • Everyone else! • Name • Where are you from? Or really, where do you want people to think you’re from? • An interesting hobby or tidbit. • Who are we going to know the best by the end of the semester? • The ones who come to office hours the most ;)

6. Course Websites • Course website: ccar.colorado.edu/asen5070 • Homework, project, and reference materials • Desire2Learn website is brand new • Forums, Dropbox, Links, Quizzes, News, etc. • Short quizzes before each lecture. • They become available at midnight before the lecture • They are due at 1:00pm before the lecture. • CAETE students can access them longer • If you attempt the quiz, you get 50% - any correct answers add to the score (max 100%). • Be honest: if you don’t know an answer, we’ll review the subject in lectures.

7. Course Grade • Homework = 20% • 11-12 assignments • Quizzes/Exams = 50% • Concept quizzes (before/during class): 10% • 2 mid-terms • 1 take-home final • Course Project = 30 %

8. Honor Code • You are expected to follow the Honor Code • We will treat you as an engineer in the field as practice for your career. • This course teaches you to navigate spacecraft. Spacecraft are worth many $Millions. Don’t crash them. • You can work together, but give each other credit when credit is due. We use software to detect plagiarism. The Honor Code will be enforced. • If you’re concerned about your grade, please come talk to us rather than cheating.

9. Homework • Homework Policy • Assigned on a Tuesday • Due 9 days later (a week from Thursday) • You are encouraged to work with others. • Turn in your own work. • If you work with others, give them credit – this is totally fine for most things! • Behave according to the Honor Code • Turn in a searchable PDF to the D2L Dropbox • There are free PDF converters if you need it. • Encouraged to use LaTex / pdflatex • Late policy • It should be on-time (practice for careers in engineering!). But it’s better correct and late than incorrect and on-time for this course.

10. Course Textbook Tapley, B.D., B.E. Schutz, and G.H. Born, Statistical Orbit Determination, Elsevier Academic Press, New York, 2004.

11. What is Statistical Orbit Determination? • It is the process of estimating the state/orbit of a satellite using a collection of observations. • We never know where a satellite is. • Launch errors • Modeling errors • Spacecraft performance errors • maneuvers, electromagnetic interactions with the environment, etc • Track a satellite • Observation errors • Locations of tracking stations • Atmosphere • Hardware modeling • Geometry issues

12. What is Statistical Orbit Determination? • Use numerous observations of a satellite and estimate its state using a filter. • Required skills: • Astrodynamics, Linear Algebra • Signal Analysis, Awesomeness

13. What can you do with Stat OD? • Navigate satellites and spacecraft! • A huge portion of the population of people in the world who navigate satellites learned their skills from Born, Tapley and Schutz. • Commercial: • GEO communication sats • Human spaceflight • Defense: • Spy satellites • Interplanetary: • JPL, Goddard, APL • Human Exploration: • ISS, Orion • Missions to LEO, Moon, NEOs, Mars

14. Course Topics • Introduction • Overview, Background, Notation, References • Review of Astrodynamics • Review of Matrix Theory (App. B in Text) • Uniform Gravity Field Problem (1.2) • The Orbit Determination (OD) Problem • The Observation – State Relationship • Linearization of the OD Process (1.2.4, 4.2) • Transformation to a Common Epoch – The State Transition Matrix (1.2.5, 4.2, 4.2.3)

15. Course Topics • Solution Methods • Least Squares (4.3) • Weighted Least Squares (4.3.3) • Minimum Norm (4.3.1) • Least Squares with a prioriinformation (4.3.3, 4.4.2) • Review of Probability and Statistics (App. A in Text) • Density/Distribution Functions • Moment Generating Functions • Bivariate Density Functions • Properties of Covariance and Correlation

16. Course Topics • Review of Probability and Statistics (App. A in Text) • Central Limit Theorem • Bayes Theorem • Stochastic Processes • Statistical Interpretation of Least Squares • Computational Algorithms • Cholesky(5.2) • Square Root Free Cholesky (5.2.2) • Givens Algorithm (Orthogonal Transformations 5.3, 5.4

17. Course Topics • The Sequential Estimation Algorithm (4.7) • The Extended Sequential Estimation Algorithm • Numerical Problems with the Kalman Filter Algorithm • Square Root Filter Algorithms • Potter Algorithm • State Noise Compensation Algorithms • Information Filters • Smoothing Algorithms • Gauss-Markoff Theorem • The Probability Ellipsoid (4.16) • Combining Estimates (4.17)

18. Any Questions? • (Show syllabus) • (quick break)

19. Homework # 1 • Problem 1: • Problem 2:

20. Homework # 1 • Problem 3: • Problem 4:

21. Homework # 1 • Problem 5: • Problem 6:

22. Homework # 1 • Problem 7:

23. Review of Astrodynamics • What’s μ?

24. Review of Astrodynamics • What’s μ? μ

25. Review of Astrodynamics • What’s μ? • μ is the gravitational parameter of a massive body • μ = GM

26. Review of Astrodynamics • What’s μ? • μ is the gravitational parameter of a massive body • μ = GM • What’s G? • What’s M?

27. Review of Astrodynamics • What’s μ? • μ is the gravitational parameter of a massive body • μ = GM • What’s G? Universal Gravitational Constant • What’s M? The mass of the body

28. Review of Astrodynamics • What’s μ? • μ = GM • G = 6.67384 ± 0.00080 × 10-20 km3/kg/s2 • MEarth ~ 5.97219 × 1024 kg • or 5.9736 × 1024kg • or 5.9726 × 1024 kg • Use a value and cite where you found it! • μEarth = 398,600.4415 ± 0.0008 km3/s2(Tapley, Schutz, and Born, 2004) • How do we measure the value of μEarth?

29. Review of Astrodynamics Problem of Two Bodies µ = G(M1 + M2) XYZ is nonrotating, with zero acceleration; an inertial reference frame

30. Review of Astrodynamics • How many degrees of freedom are present to fit the orbits of 2 bodies in mutual gravitation (known masses, no SRP, no drag, no perturbations) • 2 • 4 • 6 • 12

31. Review of Astrodynamics • How many degrees of freedom are present to fit the orbits of 2 bodies in mutual gravitation (known masses, no SRP, no drag, no perturbations) • 2 • 4 • 6 • 12 6 for each body: 3 position and 3 velocity X 2

32. Integrals of Motion • Center of mass of two bodies moves in straight line with constant velocity • Angular momentum per unit mass (h) is constant, h = r x V = constant, where V is velocity of M2 with respect to M1, V= dr/dt • Consequence: motion is planar • Energy per unit mass (scalar) is constant

33. Orbit Plane in Space Statistical Orbit Determination University of Colorado at Boulder

34. Equations of Motion in the Orbit Plane The uθcomponent yields: which is simply h = constant

35. Solution of ur Equations of Motion • The solution of the ur equation is (as function of θ instead of t): where e and ω are constants of integration.

36. The Conic Equation • Constants of integration: e and ω • e = ( 1 + 2 ξ h2/µ2 )1/2 • ω corresponds to θ where r is minima • Let f = θ – ω, then r = p/(1 + e cos f) which is “conic equation” from analytical geometry (e is conic “eccentricity”, p is “semi-latus rectum” or “semi-parameter”, and f is the “true anomaly”) • Conclude that motion of M2 with respect to M1 is a “conic section” • Circle (e=0), ellipse (0<e<1), parabola (e=1), hyperbola (e>1) Statistical Orbit Determination University of Colorado at Boulder

37. Types of Orbital Motion Statistical Orbit Determination University of Colorado at Boulder

38. The Orbit and Time • If angle f is known, r can be determined from conic equation • Time is preferred independent variable instead of f • Introduce E, “eccentric anomaly” related to time t by Kepler’s Equation: E – e sin E = M = n (t – tp) where M is “mean anomaly” Statistical Orbit Determination University of Colorado at Boulder

39. Orbit in Space • h = constant • Components of h: • hX, hY, hZ • Inclination, i (angle between Z-axis and h), 0 ≤ i≤ 180° • Line of nodes is line of intersection between orbit plane and (X,Y) plane • Ascending node (AN) is point where M2 crosses (X,Y) plane from –Z to +Z • Ω is angle from X-axis to AN Statistical Orbit Determination University of Colorado at Boulder

40. Six Orbit Elements • The six orbit elements (or Kepler elements) are constant in the problem of two bodies (two gravitationally attracting spheres, or point masses) • Define shape of the orbit • a: semimajoraxis • e: eccentricity • Define the orientation of the orbit in space • i: inclination • Ω: angle defining location of ascending node (AN) • : angle from AN to perifocus; argument of perifocus • Reference time: • tp: time of perifocus(or mean anomaly at specified time) Statistical Orbit Determination University of Colorado at Boulder

41. One more picture of an orbit • a • e • i • Ω • ω • νM(t-tp)

42. Satellite in orbit • Six orbital elements: • a, e, i, Ω, ω, ν • How do we measure μEarth?

43. Satellite in orbit • Six orbital elements: • a, e, i, Ω, ω, ν • How do we measure μEarth? • Observe orbital period, P

44. Satellite in orbit • μ=GM • How do we measure G and M? • We can’t in this way! • Only one is observable using Statistical Orbit Determination • This is why μ is very well known, but G is not.

45. End of Lecture 1 • This is a good place to stop for today • Any questions? • Notes. Quiz 1 is already available. • HW 1 is on the websites and will be due Thursday, 9/6/2012.