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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
- 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

- 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:

An example: 4-41 and 4-42

Book p. 283

An example: 4-41 and 4-42

Book p. 284

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”

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