ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 36: Smoothing and State Accuracy Estimation. Announcements. Homework 11 due on Friday Sample solutions posted online Lecture quiz due by 5pm on Wednesday

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ASEN 5070: Statistical Orbit Determination I

Fall 2013

Professor Brandon A. Jones

Professor George H. Born

Lecture 36: Smoothing and State

Accuracy Estimation

Announcements
• Homework 11 due on Friday
• Sample solutions posted online
• Lecture quiz due by 5pm on Wednesday
• Final Exam Posted On Friday
• Due December 16 by noon
• By 11:59pm for CAETE Students
• Final Project Due December 16 by noon
• By 11:59pm for CAETE Students
Motivation
• The batch processor provides an estimate based on a full span of data
• When including process noise, we lose this equivalence between the batch and any of the sequential processors
• Is there some way to update the estimated state using information gained from future observations?
Smoothing
• Smoothing is a method by which a state estimate (and optionally, the covariance) may be constructed using observations before and after the epoch.
• Step 1. Process all observations using a CKF with process noise (SNC, DMC, etc.).
• Step 2. Start with the last observation processed and smooth back through the observations.
Notation
• As presented in the book, the most common source of confusion for the smoothing algorithm is the notation

Based on observations up to and including

Value/vector/matrix

Time of current estimate

Smoothing visualization
• Process observations forward in time:
• If you were to process them backward in time (given everything needed to do that):
Smoothing visualization
• Process observations forward in time:
• If you were to process them backward in time (given everything needed to do that):
Smoothing visualization
• Smoothing does not actually combine them, but you can think about it in order to conceptualize what smoothing does.
• Smoothing results in a much more consistent solution over time. And it results in an optimal estimate using all observations.
Smoothing
• Caveats:
• If you use process noise or some other way to increase the covariance, the result is that the optimal estimate at any time really only pays attention to observations nearby.
• While this is good, it also means smoothing doesn’t always have a big effect.
• Smoothing shouldn’t remove the white noise found on the signals.
• It’s not a “cleaning” function, it’s a “use all the data for your estimate” function.
Smoothing of State Estimate
• First, we use
• If Q = 0,
Smoothing of State Estimate
• Hence, in the CKF, we store:
Smoothing of Covariance
• Optionally, we may smooth the state error covariance matrix
Smoothing
• If we suppose that there is no process noise (Q=0), then the smoothing algorithm reduces to the CKF mapping relationships:
Smoothing
• Say there are 100 observations
• We want to construct new estimates using all data, i.e.,
Smoothing
• Say there are 100 observations
Smoothing
• Say there are 100 observations
Smoothing
• Say there are 100 observations
Factors Influencing Filter Accuracy
• Truncation error (linearization)
• Round-off error (fixed precision arithmetic)
• Mathematical model simplifications (dynamics and measurement model)
• Errors in input parameters (e.g., J2)
• Amount, type, and accuracy of tracking data
How do we characterize our accuracy?
• For the Jason-2 / OSTM mission, the OD fits are quoted to have errors less than centimeter (in radial)
• How do they get an approximation accuracy?
• Residuals?
• Depends on how much we trust the data
• Provides information on fit to data, but solution accuracy?
• Covariance Matrix?
• How realistic is the output covariance matrix?
• (Actually, I can make the output matrix whatever I want through process noise or other means.)
Preliminary Discussion – Batch Processor Covariance
• Qualitatively, how does the mapped covariance look for the Batch processor?
Solution Characterization
• Characterization requires a comparison to an independent solution
• Different solution methods, models, etc.
• Different observations data sets:
• Global Navigation Satellite Systems (GNSS) (e.g., GPS)
• Doppler Orbitography and Radio-positioning Integrated by Satellite (DORIS)
• Satellite Laser Ranging (SLR)
• Deep Space Network (DSN)
• Delta-DOR
• Others…
Compare to Independent Solution
• Jason-2 / OSTM positions solutions generated by/at:
• JPL – GPS only
• GSFC – SLR, DORIS, and GPS
• CNES – SLR, DORIS, and GPS
• Algorithms/tools differ by team:
• Different filters
• Different dynamic/stochastic models
Comparison of Jason-2 / OSTM Solutions

Image: Bertiger, et al., 2010

• 1 Cycle = approximately 10 days
• Differences on the order of millimeters
Orbit Overlap Studies
• Compare different fit intervals:
Orbit Overlap Studies
• Consider the “abutment test”:
Example: Jason-2 / OSTM
• Each data fit at JPL uses 30 hrs of data, centered at noon
• This means that each data fit overlaps with the previous/next fit by six hours
• Compare the solutions over the middle four hours
• Why?
Example: Jason-2 / OSTM

Image: Bertiger, et al., 2010

• Histogram of daily overlaps for almost one year
• Imply solution consistency of ~1.7 mm
• This an example of why it is called “precise orbit determination” instead of “accurate orbit determination”