asen 5070 statistical orbit determination i fall 2012 professor george h. born
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ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 7: Spaceflight Ops and Statistics. Announcements. Homework 1 Graded Comments included on D2L Any questions, talk with us this week Homework 2 CAETE due Thursday - PowerPoint PPT PresentationTRANSCRIPT
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ASEN 5070Statistical Orbit determination I
Fall 2012
Professor George H. BornProfessor Jeffrey S. Parker
Lecture 7: Spaceflight Ops and Statistics
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Homework 1 Graded◦ Comments included on D2L◦ Any questions, talk with us this week
Homework 2 CAETE due Thursday◦ Graded soon after
Homework 3 due Thursday
Homework 4 out today
Announcements
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Review Homework and Quizzes
Finish Mars Odyssey
Review Statistics
Start on some serious Stat OD!
Lecture Plan
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Questions yet?
Homework 3
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Homework 4
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Homework 4
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Homework 4
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Mars Odyssey
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Mars Odyssey
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• Requirement– Must predict Periapsis Time to within 225 sec– Must predict Periapsis Altitude to within 1.5 km
• Capability– Altitude requirement easily met with MGS gravity field (Nav Plan)– Timing requirement uncertainty dominated by assumption on future drag pass
atmospheric uncertainty
• Atmospheric Variability– Total Orbit-to-Orbit Atmospheric variability: 80% (MGS: 90%)
• Periapsis timing prediction– To first order, the expected change in orbit period per drag pass will indicate how
well future periapses can be predicted– This simplifying assumption is supported by OD covariance analysis
Aerobraking Nav Prediction Accuracy
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• Example– Total expected Period change for a given drag pass is 1000 seconds– Atmosphere could change density by 80%– Resulting Period change could be off by 80% = 800 sec– If orbit Period is different by 800 seconds, then the time of the next periapsis will be
different by 800 seconds– This fails to meet the 225 sec requirement
• Large Period Orbits– Period change per rev is large– Therefore can never predict more than 1 periapsis ahead within the 225 sec
requirement with any confidence
• Small Period Orbits– Period change per rev is small (for example 30 seconds)– Therefore can predict several periapses in the future to within the 225 second
requirement– Example: 80% uncertainty (24 sec) will allow a 9 rev predict
Nav Predict Capability
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Aerobraking Navigation Process
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Aerobraking Navigation Process
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• A Baseline set of Navigation solution strategies were identified– Varied data arcs, data types, data weights, parameter estimates, a-
prioris
• These solutions were regularly performed and trended– Built a time history of trajectory solutions– Trended evolution of parameter estimates and encounter conditions– Lessons learned from MCO and MPL
• Regularly demonstrate consistency to Project and NAG– Weekly Status Reports– Daily Status after TCM-4 (MOI-12 days) “Daily Show”
• Shadow navigators– Independent solutions run by Sec312 personnel (Bhaskaran, Portock)
What contributed to ODY success?
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Questions
Quick Break
Next up:◦ Statistics◦ Stat OD
Mars Odyssey
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The Variance-Covariance Matrix
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Quiz #5 Results
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Quiz #5 Results
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The Variance-Covariance Matrix
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Quiz #5 Results
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Quiz #5 Results
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The Variance-Covariance Matrix
If two parameters are perfectly correlated then
Say we have and all other correlations are 0.
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Quiz #6 Results
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Quiz #6 Results
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Quiz #6 Results
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Quiz #6 Results
The probability of sampling a distribution and getting something is equal to 100%
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Quiz #6 Results
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Quiz #6 Results
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Quiz #6 Results
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Quiz #6 Results
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Example Problems in Statistics
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Example Problems in Statistics
f = joint density function
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Example Problems in Statistics
f = joint density function
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Example Problems in Statistics
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Example Problems in Statistics
What is the marginal density function of x? What does that mean?
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Example Problems in Statistics
The marginal density function of x is the probability density function of x in the presence of any y.
What is the marginal density function of x? What does that mean?
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Example Problems in Statistics
We’ll work this one
You’ll work this one
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Example Problems in Statistics
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Example Problems in Statistics
???
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
also,
and,
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problems in Statistics
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Example Problem continuedDetermine the Variance-Covariance matrix for the example problem
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Example Problem continuedDetermine the Variance-Covariance matrix for the example problem
We have shown that the marginal density functions are given by
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Example Problem continuedThe elements of the variance-covariance matrix are computed below
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Example Problem continuedCont.
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Example Problem continuedThe variance-covariance matrix, P, is given by
The correlation coefficient for random variables x and y is given by
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Example Problem continuedThe variance-covariance matrix, P, is given by
The correlation coefficient for random variables x and y is given by
A common OD expression for the variance-covariance matrix has variances on the diagonal, covariance's in the upper triangle and correlation coefficients in the lower triangle.
Hence,
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Questions?
One last quick subject:◦ Distributions with multiple variables that are
Gaussian in any combination of those variables
Statistical Problems
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Bivariate Normal Distribution
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Bivariate Normal Distribution
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Marginal Density Function
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Conditional Density Function
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Conditional Density Function
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The Multivariate Normal Distribution
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The Multivariate Normal Distribution
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CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder
Conditional Distribution for Multivariate Normal Variables
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CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder
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CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder 78
Homework 1 Graded◦ Comments included on D2L◦ Any questions, talk with us this week
Homework 2 CAETE due Thursday◦ Graded soon after
Homework 3 due Thursday
Homework 4 out today
Quiz available tomorrow at 1pm
Final Statements