Transcript Lecture 37
ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 37: Solution Characterization and IOD
University of Colorado
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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
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Project Q&A
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Solution Characterization
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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
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Image: Bertiger, et al., 2010
1 Cycle = approximately 10 days
Differences on the order of millimeters
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Compare different fit intervals:
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Consider the “abutment test”:
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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?
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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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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?
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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?
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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?
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Initial Orbit Determination (IOD)
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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?
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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?
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Without sufficient observations, we have an
underdetermined problem
◦ Range measurements provide one component of
position
◦ Angle measurements provide no range data
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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
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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
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IOD via the Admissible Region
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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
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For optical observations:
Given a time series of right ascension and
declination measurements, how could we get
the angle rates?
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What are some reasonable constraints on an
orbit?
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We can include a constraint based on
upper/lower limits of the semimajor axis
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We can include a constraint based on an
upper limits in the eccentricity
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We can combine them to further constrain the
space of solutions
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“Virtual Asteroids”
Image Credit: Milani and Knežević, 2005
◦ Milani, et al.
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“Direct Bayesian” Method
◦ Fujimoto and Scheeres
◦ Following slides from Kohei Fujimoto as part of
ASEN 6519: Orbital Debris (Fall 2012)
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Adm. Region
Direct Bayesian approach
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Adm. Region
Direct Bayesian approach
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“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
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Constrained Admissible Region (CAR)
◦ DeMars and Jah, 2013
Image Credit: Jones, et al., 2014
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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
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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
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