Transcript Lecture 37
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 37: Solution Characterization and IOD University of Colorado Boulder Homework 11 due on Friday Lecture quiz due by 5pm on Friday Exam 3 Posted On Friday ◦ In-class Students: Due December 11 by 5pm ◦ CAETE Students: Due 11:59pm (Mountain) on 12/13 Final Project Due December 14 by noon University of Colorado Boulder 2 Project Q&A University of Colorado Boulder 3 Solution Characterization University of Colorado Boulder 4 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 University of Colorado Boulder 5 Image: Bertiger, et al., 2010 1 Cycle = approximately 10 days Differences on the order of millimeters University of Colorado Boulder 6 Compare different fit intervals: University of Colorado Boulder 7 Consider the “abutment test”: University of Colorado Boulder 8 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? University of Colorado Boulder 9 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” University of Colorado Boulder 10 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? University of Colorado Boulder 11 Image: Bertiger, et al., 2010 Results imply that the GPS-based radial error is on the order of millimeters Why is the DORIS/SLR/GPS solution better here? University of Colorado Boulder 12 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? University of Colorado Boulder 13 Initial Orbit Determination (IOD) University of Colorado Boulder 14 Up to now, we’ve assumed a given a priori state for the spacecraft ◦ Required to establish a reference trajectory and a priori information ◦ For the statistical filters, we need a probabilistic description of the trajectory Mean Covariance How do we get this in the real world? University of Colorado Boulder 15 You have a GNSS (e.g., GPS) receiver on your spacecraft and are in communication with the vehicle ◦ Every 10 seconds, the GPSR provides 8 to 12 (pseudo)range measurements relative to known satellites ◦ How could you get an initial estimate to use in an orbit determination filter? ◦ What about the covariance matrix? ◦ How did they do it before GNSS? University of Colorado Boulder 16 Without sufficient observations, we have an underdetermined problem ◦ Range measurements provide one component of position ◦ Angle measurements provide no range data University of Colorado Boulder 17 Different methods for different measurement types ◦ Angles-only IOD Gauss’s Method Double r-iteration ◦ Range-only Homotopy Continuation ◦ Range and Range-Rate Trilateration ◦ Position Vectors Gibbs Method Herrick-Gibbs Lambert’s Problem University of Colorado Boulder 18 Each method has different strengths and weaknesses ◦ Method selected is typically based on the observations available and not a trade-off between methods Those mentioned previously do not provide a PDF ◦ All assume deterministic ◦ Exception: recent research into probabilistic Gauss’ method Vallado provides a good summary of the classic methods in his book ◦ More detail on classic methods in ASEN 5050 University of Colorado Boulder 19 IOD via the Admissible Region University of Colorado Boulder 20 Introduced in early-2000s to address the Too-Short Arc (TSA) problem for asteroid surveys Leverage a 4D observation vector for a new target In the unobserved directions, leverage constraints on the system to restrict the space of possible solutions ◦ Dubbed an “attributable vector” ◦ Given a time series of angles observations, approximate the angle and angle-rates at a given time ◦ For example, a maximum value for range given semimajor axis ◦ Bounds on range-rate given energy constraints Refine knowledge of the orbit with follow-up tracking University of Colorado Boulder 21 For optical observations: Given a time series of right ascension and declination measurements, how could we get the angle rates? University of Colorado Boulder 22 What are some reasonable constraints on an orbit? University of Colorado Boulder 23 We can include a constraint based on upper/lower limits of the semimajor axis University of Colorado Boulder 24 We can include a constraint based on an upper limits in the eccentricity University of Colorado Boulder 25 We can combine them to further constrain the space of solutions University of Colorado Boulder 26 “Virtual Asteroids” Image Credit: Milani and Knežević, 2005 ◦ Milani, et al. University of Colorado Boulder 27 “Direct Bayesian” Method ◦ Fujimoto and Scheeres ◦ Following slides from Kohei Fujimoto as part of ASEN 6519: Orbital Debris (Fall 2012) University of Colorado Boulder 28 Adm. Region Direct Bayesian approach ASEN 6519 - Orbital Debris 29 Adm. Region Direct Bayesian approach ASEN 6519 - Orbital Debris 30 “Direct Bayesian” Method ◦ Fujimoto and Scheeres Allows for hypothesis-free correlation of two tracks ◦ From there, a 6D state vector for a possible new target is available ◦ Allows for correlation of tracks over relatively large time spans University of Colorado Boulder 31 Constrained Admissible Region (CAR) ◦ DeMars and Jah, 2013 Image Credit: Jones, et al., 2014 University of Colorado Boulder 32 Probabilistic Admissible Region (PAR) ◦ Hussein, et al. (2014, 2015) ◦ Attempts to map probabilistic description for SMA and eccentricity to range/range-rate PDF Image Credit: Jones, et al., in preparation University of Colorado Boulder 33 Relatively new way to approach the IOD problem Has its own set of advantages and disadvantages ◦ Advantages: Only one observation arc required to create track hypotheses May be use to generate an initial-state PDF ◦ Disadvantages: Still need follow-up observation to refine orbit (not always easy) Make assumptions on the class of orbit based on constraints Needs some pre-processing to get 4-D observation vector University of Colorado Boulder 34