ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 39: Combining State Estimates and Measurement Modeling

Announcements/Reminders No lecture quiz next week. Exam 3 posted today by 5pm In-class Students: Due December 11 by 5pm CAETE Students: Due 11:59pm (Mountain) on 12/13 Final Project Due December 14 by noon Eduardo’s office hours next week: He will be out of town Thursday and Friday, but is available via e-mail. Office hours Thursday (2:30-4:30pm) have been moved to Tuesday (2:30-4:30pm in ECAE 1B44)

Exam 3 Comments Your solutions must be uploaded to D2L as a searchable PDF Same rules as homework apply in regards to format, code appendices, etc. Open-book, open notes You may use a computer, MATLAB, etc. Honor code rules apply Do not give or ask for help from your peers The TA has been instructed to redirect all questions to the instructor I can answer questions to clarify what is being asked, but cannot provide guidance on solutions

Project Q&A

Combining State Estimates

Consider This Scenario A ground station in Maui observed our satellite several times over the past week Generated a filtered solution using their observations A ground station in Florida also observed the satellite several times over the past week Generated a filter solution using their observations What is the best approach to fusing this information?

Solution Setup Treat one solution as the a priori and the other as the observation Does it matter which one is which? For this case, H=I

Combining Solutions Does not require the additional processing of observations

General Combination of Solutions

Modeling Measurements Tapley, Schutz, and Born, Chapter 3 Montenbruck and Gill, Satellite Orbits, Chapter 6

Range Types One-way Range Example: GNSS Signal travels to/from reference from/to satellite

Range Types Two-way Range Examples: SLR, DSN Satellite is a relay for signal

Range Types Multi-way Range Examples: DSN, TDRSS Multiple satellite and/or ground stations used

Ideal Range and Range-Rate We have been using range and range-rate: In the real world, what is wrong with these equations?

Light Time Correction At best, a signal travels at the speed of light We must approximate the signal propagation time δt Approximately 0.06 seconds for GPS signal to reach Earth A LEO spacecraft will have moved approximately 500 meters in that time

Light Time Computation Assume we have estimates of our satellite trajectory and the reference station/satellite We need to solve for δt No analytic solution so we solve for the correction using iteration

Light Time Correction Algorithm Start with δt=0 Compute the distance with the satellite state at time t and the reference state at t-δt Given that distance, compute the light propagation time Δδt Set δt=δt+Δδt Continue until Δδt is sufficiently small

Any other issues? We have taken care of light-time correction assuming the speed of light in a vacuum. Any other things we should account for? Signal does not always propagate through a vacuum Ionosphere Troposphere Charged particle interactions Solar corona etc. Coordinate and time systems This requires a very careful treatment in the filter

Coordinate Systems Not including accurate coordinate system information creates systematic errors. Violates our assumption of random errors! Creates a time-varying bias in the measurement Table courtesy of Bradley, et al., 2011

Doppler as Range-Rate Is it possible to measure range-rate instantaneously? No! (at least not that I am aware of) We have to observe this indirectly Instead, we look at the change in a signal over time to approximate the range-rate

Pulsed Transmission Satellite sends pulse at fixed interval

Repeat Pulse Transmission

Doppler Shift Image Courtesy of WikiCommons The velocity of the spacecraft affects the frequency of any radar signal Requires us to observe the change in frequency over some period of time Known as integrated Doppler shift

Doppler Shift Image Courtesy of WikiCommons The velocity of the spacecraft affects the frequency of any radar signal Requires us to observe the change in frequency over some period of time Known as integrated Doppler shift

Two-Way Doppler Measurements Range-rate model assumes: Linear change in range over integration time Constant transmission frequency over integration time

Doppler Measurements (one model) Do I need to perform any light time correction? Is there anything different about this case when compared to range?

FCQs