1. Statistical Orbit determination I Fall 2012 Professor George H. Born
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 Course Assistants
Professor George H. Born
Office: ECNT 316
Office Hour: Wed 2-3 PM
Professor Jeff Parker
Office: ECNT 418
Office Hours: Mon 2-3 PM, Wed AM
Course Assistants
Eduardo Villalba
Office: ECNT 414
Office Hours: Tues AM
Paul Anderson
Office: ECEE 275
Office Hours: Mon 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% Quizzes/Exams = 50% Course Project = 30 %
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 The Orbit Determination (OD) Problem
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 priori information (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 = ± × km3/kg/s2
MEarth ~ × 1024 kg
or × 1024 kg
or × 1024 kg
Use a value and cite where you found it!
μEarth = 398, ± 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
Topic: Astrodynamics

34. Equations of Motion in the Orbit Plane
The uθ component 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 = ( 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
Topic: Astrodynamics

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

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
Topic: Astrodynamics

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
Topic: Astrodynamics

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: semimajor axis
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
Topic: Astrodynamics

41. One more picture of an orbitΩ ω
ν M (t-tp)

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

43. Satellite in orbit Six orbital elements: How do we measure μEarth?
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.