470 likes | 666 Views
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 < georgeb@colorado.edu > Office: ECNT 316 Office Hour: Wed 2-3 PM
E N D
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 <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
Introductions • 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 ;)
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.
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 %
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.
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.
Course Textbook Tapley, B.D., B.E. Schutz, and G.H. Born, Statistical Orbit Determination, Elsevier Academic Press, New York, 2004.
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
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
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
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)
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
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)
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:
Review of Astrodynamics • What’s μ?
Review of Astrodynamics • What’s μ? μ
Review of Astrodynamics • What’s μ? • μ is the gravitational parameter of a massive body • μ = GM
Review of Astrodynamics • What’s μ? • μ is the gravitational parameter of a massive body • μ = GM • What’s G? • What’s M?
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
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?
Review of Astrodynamics Problem of Two Bodies µ = G(M1 + M2) XYZ is nonrotating, with zero acceleration; an inertial reference frame
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
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
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
Orbit Plane in Space Statistical Orbit Determination University of Colorado at Boulder
Equations of Motion in the Orbit Plane The uθcomponent 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.
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
Types of Orbital Motion Statistical Orbit Determination University of Colorado at Boulder
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
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
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
One more picture of an orbit • a • e • i • Ω • ω • νM(t-tp)
Satellite in 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.
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.