University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 37: SNC Example and Solution Characterization

University of Colorado Boulder  Homework 11 due on Friday ◦ Sample solutions will be posted online  Lecture quiz due by 5pm on Friday  Exam 3 Posted On Friday ◦ In-class Students: Due December 12 by 5pm ◦ CAETE Students: Due 11:59pm (Mountain) on 12/14  Final Project Due December 15 by noon 2

University of Colorado Boulder 3 Homework 11

University of Colorado Boulder  Leverage code from HW10 ◦ New data set generated with a different force model ◦ Otherwise, same format, data noise, etc.  Process observations in existing filter ◦ Do not add J 3 to your filter model! ◦ Observe the effects of such errors on OD ◦ Add process noise to improve state estimation accuracy 4

University of Colorado Boulder 5

University of Colorado Boulder 6 3σ

University of Colorado Boulder 7 Application of SNC to Ballistic Trajectory

University of Colorado Boulder 8  Ballistic trajectory with unknown start/stop  Red band indicates time with available observations Start of filter Obs. Stations

University of Colorado Boulder  Object in ballistic trajectory under the influence of drag and gravity 9  Nonlinear observation model ◦ Two observations stations

University of Colorado Boulder 10

University of Colorado Boulder  Now use an EKF  We will vary the truth model to study the benefits of SNC ◦ Look at two cases:  Run each with and without a process noise model  Error in gravity (g = 9.8 m/s vs. 9.9 m/s)  Error in drag (b = 1e-4 vs. 1.1e-4) 11

University of Colorado Boulder 12  Blue – Range  Green – Range-Rate Station 1 Station 2

University of Colorado Boulder 13

University of Colorado Boulder  Added SNC to the filter: 14  Why is the term for x-acceleration smaller?

University of Colorado Boulder 15  Blue – Range  Green – Range-Rate Station 1 Station 2

University of Colorado Boulder 16  vs. 0.8 meters RMS

University of Colorado Boulder 17  Blue – Range  Green – Range-Rate Station 1 Station 2

University of Colorado Boulder 18

University of Colorado Boulder  Added SNC to the filter: 19

University of Colorado Boulder 20  Blue – Range  Green – Range-Rate Station 1 Station 2

University of Colorado Boulder 21  27.6 vs meters RMS

University of Colorado Boulder  Mitigation of the gravity acceleration error yielded better results than the drag error case. Why could that be? 22

University of Colorado Boulder 23 Solution Characterization

University of Colorado Boulder  Truncation error (linearization)  Round-off error (fixed precision arithmetic)  Mathematical model simplifications (dynamics and measurement model)  Errors in input parameters (e.g., J 2 )  Amount, type, and accuracy of tracking data 24

University of Colorado Boulder  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.) 25

University of Colorado Boulder  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…  Provides a measure based on solution precision 26

University of Colorado Boulder  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 ◦ Different measurement models 27

University of Colorado Boulder  1 Cycle = approximately 10 days  Differences on the order of millimeters 28 Image: Bertiger, et al., 2010

University of Colorado Boulder  Compare different fit intervals: 29

University of Colorado Boulder  Consider the “abutment test”: 30

University of Colorado Boulder  Each data fit at JPL uses 30 hrs of data, centered at noon 31  This means that each data fit overlaps with the previous/next fit by six hours  Compare the solutions over the middle four hours ◦ Why?

University of Colorado Boulder  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” 32 Image: Bertiger, et al., 2010

University of Colorado Boulder  In some case, we can leverage observations (ideally not included in the data fit) to estimate accuracy  How might we use SLR to characterize radial accuracy of a GNSS-based solution? 33

University of Colorado Boulder  Results imply that the GPS-based radial error is on the order of millimeters  Why is the DORIS/SLR/GPS solution better here? 34 Image: Bertiger, et al., 2010

University of Colorado Boulder  Must consider independent state estimates and/or observations  Not an easy problem, and the method of characterization is often problem dependent ◦ How do you think they do it for interplanetary missions? 35