ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey...

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



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




George Born: Wrote the book on Stat OD.




Jeff Parker◦ Graduated from CU in 2007◦ “Low-Energy Ballistic Lunar

Transfers”

◦ JPL since then Chandrayaan-1 GRAIL MoonRise Team-X

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 ;)

Introductions




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




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




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




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




Course Textbook

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




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?




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




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?




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




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




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




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




Any Questions?

(Show syllabus)

(quick break)




Homework # 1

Problem 1:

Problem 2:




Homework # 1

Problem 3:

Problem 4:




Homework # 1

Problem 5:

Problem 6:




Homework # 1

Problem 7:




What’s μ?

Review of Astrodynamics




What’s μ?


μ




What’s μ?

μ is the gravitational parameter of a massive body

μ = GM





What’s μ?


μ = GM

What’s G? What’s M?





What’s μ?


μ = GM

What’s G? Universal Gravitational Constant What’s M? The mass of the body





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?






Problem of Two Bodies

XYZ is nonrotating, with zero acceleration; an inertial reference frame

µ = G(M1 + M2)




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





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


6 for each body:

3 position and 3 velocity X 2




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



University of ColoradoBoulder Topic:

Astrodynamics

Statistical Orbit Determination

University of Colorado at Boulder

Orbit Plane in Space




Equations of Motion in the Orbit Plane

The uθ component yields:

which is simply h = constant




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.




Astrodynamics



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)




Astrodynamics



Types of Orbital Motion




Astrodynamics



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”




Astrodynamics



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




Astrodynamics



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)




a e i Ω ω ν

M(t-tp)

One more picture of an orbit




Six orbital elements:◦ a, e, i, Ω, ω, ν

How do we measure μEarth?

Satellite in orbit




Six orbital elements:◦ a, e, i, Ω, ω, ν

How do we measure μEarth?

Observe orbital period, P

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.

Satellite in orbit




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

Date post:	26-Feb-2016
Category:	Documents
Upload:	palila
View:	40 times
Download:	3 times

ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey...

Documents