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University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 29: Observability and Introduction to Process Noise
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University of Colorado Boulder Exam 2 Friday Seminar Friday: 2
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University of Colorado Boulder 3 Givens Transformations – Lingering Question
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University of Colorado Boulder We do not want to add non-zero terms to the previously altered rows, so we use the identity matrix except in the rows of interest: 4
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University of Colorado Boulder Apply a series of rotations such that: 5
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University of Colorado Boulder A commonly used tool in linear algebra is the QR factorization of a matrix: Using the Given rotations, we have: Givens is one way to get the QR solutions where: 7
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University of Colorado Boulder 8 Observability
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University of Colorado Boulder How do I determine what parameters may be successfully estimated in the filter? ◦ Can I use observations of a spacecraft to estimate the height of Folsom Field Stadium? ◦ What about observations of a spacecraft to measure variations in rainfall in the Amazon river basin? ◦ How do I determine if either of these are possible? 9
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University of Colorado Boulder Consider the case of two spacecraft and a ground station with a fixed inertial position ◦ Two-body gravity field (no perturbations) ◦ No modeling error ◦ Infinite precision ◦ Little/no error on range observations 10
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University of Colorado Boulder 12 The two plots look similar (this is not a copy/paste error) Does anyone think there is a problem? Satellite 1Satellite 2
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University of Colorado Boulder Gather more observations? ◦ Unfortunately, No. Gather range-rate to go with the range data? ◦ Nope – we run into the same problem Orthogonal data type, e.g., angles? ◦ Actually that would work, but how do we find out? 14
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University of Colorado Boulder We can use the information matrix: 15
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University of Colorado Boulder In other words, when designing our filter, we should study the information matrix to determine if we can get a solution Let’s say you solve for the information matrix defined by some simulation. ◦ How would you determine if it is positive definite? ◦ Do you need to generate simulated observations? 16
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University of Colorado Boulder What if the condition number of the information matrix is very large (too large for any of the more numerically stable methods to apply)? ◦ Maybe we should reconsider what parameters to estimate? ◦ This can be the case for gravity field estimation with spatially sparse measurements 17
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University of Colorado Boulder ◦ Can I use observations of a spacecraft to estimate the height of Folsom Field? Only if observations of/from a well-known spacecraft are gathered with respect to the top of the stadium ◦ What about observations of a spacecraft to measure variations in rainfall in the Amazon river basin? Actually – you can! Scientific studies of GRACE data do this type of analysis regularly ◦ How do I determine if either of these are possible? You perform an observability study! 18
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University of Colorado Boulder Can we estimate the absolute position of two spacecraft in Earth orbit (two-body dynamics) using relative range and/or range-rate measurements? 19
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University of Colorado Boulder Can we do it if we put one of the spacecraft near the Moon and keep one at Earth? 20 Image Credit: Hill and Born, 2007
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University of Colorado Boulder 21 Process Noise
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University of Colorado Boulder 22 What happened to u (modeling error) ? ◦ This is true process noise… Can we ignore it? How do we account for it?
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University of Colorado Boulder Random process u maps to the state through the matrix B ◦ Consider it a random process for our purposes Usually (for OD), we consider random accelerations: 24
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University of Colorado Boulder For the sake of our discussion, assume: 25
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University of Colorado Boulder This is a non-homogenous differential equation The derivation of the general (continuous) solution to this equation is derived in the book (Section 4.9), and is: 26
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University of Colorado Boulder 27 If we want to instead map between two discrete times:
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University of Colorado Boulder 28 For the case of a noise process with zero mean: The zero-mean noise process does not change the mapping of the mean state
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University of Colorado Boulder What about the covariance matrix? The derivation of the general (continuous) solution to this equation is derived in the book (Section 4.9), and is: 29
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University of Colorado Boulder The previous discussion considered the case where the noise process is continuous, i.e, 30 Things may be simplified if we instead consider a discrete process:
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University of Colorado Boulder Using the discrete noise process, we instead get (for zero mean process): 32
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University of Colorado Boulder This defines, mathematically, how we can select the minimum covariance to prevent saturation ◦ Saturation is typically dominated by dynamic model error ◦ With a stochastic (probabilistic) description of the modeling error, we have our minimum 33
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University of Colorado Boulder The addition of a noise process is better suited for a sequential filter ◦ Must include the process noise transition matrix in the Batch formulation ◦ Changes the mapping of the state (deviation) back to the epoch time, which requires alterations to the H matrix definition ◦ Tapley, Schutz, and Born (p. 229) argue that this is cumbersome and impractical for real-world application Advantage: Kalman, EKF, Potter, and others 35
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