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# ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones - PowerPoint PPT Presentation

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

• 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

• 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 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.

• 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

• Process observations forward in time:

• If you were to process them backward in time (given everything needed to do that):

• Process observations forward in time:

• If you were to process them backward in time (given everything needed to do that):

• 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.

• 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.

• First, we use

• If Q = 0,

• Hence, in the CKF, we store:

• Optionally, we may smooth the state error covariance matrix

• If we suppose that there is no process noise (Q=0), then the smoothing algorithm reduces to the CKF mapping relationships:

Book p. 283

Book p. 284

• Say there are 100 observations

• We want to construct new estimates using all data, i.e.,

• Say there are 100 observations

• Say there are 100 observations

• Say there are 100 observations

• 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

• 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.)

• Qualitatively, how does the mapped covariance look for the Batch processor?

• 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…

• 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

Image: Bertiger, et al., 2010

• 1 Cycle = approximately 10 days

• Differences on the order of millimeters

• Compare different fit intervals:

• Consider the “abutment test”:

• 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?

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”