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ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 1: Introduction to Stat OD. Course Outline. Instructor Professor George H. Born < [email protected] > Office: ECNT 316 Office Hour: Wed 2-3 PM - PowerPoint PPT Presentation
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CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder
ASEN 5070Statistical Orbit determination I
Fall 2012
Professor George H. BornProfessor Jeffrey S. Parker
Lecture 1: Introduction to Stat OD
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder 2
Instructor◦ Professor George H. Born <[email protected]>
Office: ECNT 316 Office Hour: Wed 2-3 PM
◦ Professor Jeff Parker <[email protected]> Office: ECNT 418 Office Hours: Mon 2-3 PM, Wed 10-11 AM
Course Assistants◦ Eduardo Villalba <[email protected]
Office: ECNT 414 Office Hours: Tues 11-12 AM
◦ Paul Anderson <[email protected]> Office: ECEE 275 Office Hours: Mon 10-11 AM
Course Outline
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder
George Born: Wrote the book on Stat OD.
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder 4
Jeff Parker◦ Graduated from CU in 2007◦ “Low-Energy Ballistic Lunar
Transfers”
◦ JPL since then Chandrayaan-1 GRAIL MoonRise Team-X
Introductions
CCARColorado Center for
Astrodynamics Research
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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 ;)
Introductions
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder 6
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.
Course Websites
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Homework = 20%◦ 11-12 assignments
Quizzes/Exams = 50%◦ Concept quizzes (before/during class): 10%◦ 2 mid-terms◦ 1 take-home final
Course Project = 30 %
Course Grade
CCARColorado Center for
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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.
Honor Code
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder 9
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.
Homework
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder
Course Textbook
Tapley, B.D., B.E. Schutz, and G.H. Born, Statistical Orbit Determination, Elsevier Academic Press, New York, 2004.
CCARColorado Center for
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University of ColoradoBoulder 11
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
What is Statistical Orbit Determination?
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Use numerous observations of a satellite and estimate its state using a filter.
What is Statistical Orbit Determination?
Required skills:• Astrodynamics, Linear Algebra• Signal Analysis, Awesomeness
CCARColorado Center for
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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
What can you do with Stat OD?
CCARColorado Center for
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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)
Course Topics
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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
Course Topics
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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
Course Topics
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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)
Course Topics
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Any Questions?
(Show syllabus)
(quick break)
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Homework # 1
Problem 1:
Problem 2:
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Homework # 1
Problem 3:
Problem 4:
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Homework # 1
Problem 5:
Problem 6:
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Homework # 1
Problem 7:
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What’s μ?
Review of Astrodynamics
CCARColorado Center for
Astrodynamics Research
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What’s μ?
Review of Astrodynamics
μ
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What’s μ?
μ is the gravitational parameter of a massive body
μ = GM
Review of Astrodynamics
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Astrodynamics Research
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What’s μ?
μ is the gravitational parameter of a massive body
μ = GM
What’s G? What’s M?
Review of Astrodynamics
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder 27
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
Review of Astrodynamics
CCARColorado Center for
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What’s μ? μ = GM G = 6.67384 ± 0.00080 × 10-20 km3/kg/s2
MEarth ~ 5.97219 × 1024 kg ◦ or 5.9736 × 1024 kg◦ 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?
Review of Astrodynamics
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Review of Astrodynamics
Problem of Two Bodies
XYZ is nonrotating, with zero acceleration; an inertial reference frame
µ = G(M1 + M2)
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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)
A. 2B. 4C. 6D. 12
Review of Astrodynamics
CCARColorado Center for
Astrodynamics Research
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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)
A. 2B. 4C. 6D. 12
Review of Astrodynamics
6 for each body:
3 position and 3 velocity X 2
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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
Integrals of Motion
CCARColorado Center for
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University of ColoradoBoulder Topic:
Astrodynamics
Statistical Orbit Determination
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Orbit Plane in Space
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Equations of Motion in the Orbit Plane
The uθ component yields:
which is simply h = constant
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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.
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder Topic:
Astrodynamics
Statistical Orbit Determination
University of Colorado at Boulder
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” fromanalytical 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)
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder Topic:
Astrodynamics
Statistical Orbit Determination
University of Colorado at Boulder
Types of Orbital Motion
CCARColorado Center for
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University of ColoradoBoulder Topic:
Astrodynamics
Statistical Orbit Determination
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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”
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder Topic:
Astrodynamics
Statistical Orbit Determination
University of Colorado at Boulder
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
CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder Topic:
Astrodynamics
Statistical Orbit Determination
University of Colorado at Boulder
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)
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a e i Ω ω ν
M(t-tp)
One more picture of an orbit
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Six orbital elements:◦ a, e, i, Ω, ω, ν
How do we measure μEarth?
Satellite in orbit
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Six orbital elements:◦ a, e, i, Ω, ω, ν
How do we measure μEarth?
Observe orbital period, P
Satellite in orbit
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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.
Satellite in orbit
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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.
End of Lecture 1