Covariance expressions for eigenvalue and eigenvector problems

There are a number of important scientific and engineering problems whose solutions take the form of an eigenvalue--eigenvector problem. Some notable examples include solutions to linear systems of ordinary differential equations, controllability of linear systems, finite element analysis, chemical kinetics, fitting ellipses to noisy data, and optimal estimation of attitude from unit vectors. In many of these problems, having knowledge of the eigenvalue and eigenvector Jacobians is either necessary or is nearly as important as having the solution itself. For instance, Jacobians are necessary to find the uncertainty in a computed eigenvalue or eigenvector estimate. This uncertainty, which is usually represented as a covariance matrix, has been well studied for problems similar to the eigenvalue and eigenvector problem, such as singular value decomposition. There has been substantially less research on the covariance of an optimal estimate originating from an eigenvalue-eigenvector problem.;In this thesis we develop two general expressions for the Jacobians of eigenvalues and eigenvectors with respect to the elements of their parent matrix. The expressions developed make use of only the parent matrix and the eigenvalue and eigenvector pair under consideration. In addition, they are applicable to any general matrix (including complex valued matrices, eigenvalues, and eigenvectors) as long as the eigenvalues are simple. Alongside this, we develop expressions that determine the uncertainty in a vector estimate obtained from an eigenvalue-eigenvector problem given the uncertainty of the terms of the matrix. The Jacobian expressions developed are numerically validated with forward finite, differencing and the covariance expressions are validated using Monte Carlo analysis. Finally, the results from this work are used to determine covariance expressions for a variety of estimation problem examples and are also applied to the design of a dynamical system

Optical Navigation Algorithm Performance

There is a wide variety of optical navigation (OpNav) techniques that can be used to extract observables from images of natural bodies. Each of these techniques has a number of strengths and weaknesses and domains where they are most applicable. In this paper, we compare the performance of some of the most commonly used OpNav techniques across a variety of orbital regimes and a variety of body types through the use of synthetic images. Specifically, we consider the techniques of analytic model fitting, phase corrected moment estimation, limb-scanning, ellipsoid matching, and cross correlation using synthetic images of a tri-axial ellipsoid, the asteroid Bennu, and the comet 67P/Churyumov-Gerasimenko. For each technique, regime, and body, we examine the overall accuracy and the type of information available. The resulting information provides a useful tool for understanding which techniques are best suited for a given image, as well as for understanding the relative performance of each technique

Optical Navigation Simulation and Performance Analysis for Osiris-Rex Proximity Operations

The OSIRIS-REx mission timeline with OpNav milestones is presented in Figure 1. The first three proximity operations (ProxOps) mission phases focus on Navigation. During these phases, OSIRIS-REx approaches Bennu, conducts equatorial and polar flybys in Preliminary Survey, and inserts into the first mission orbit: Orbit A. During these phases, the OpNav techniques evolve from point-source to resolved-body centroiding to landmark tracking

Serendipitous Geodesy from Bennu's Short-Lived Moonlets

The Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer (OSIRIS-REx; or OREx) spacecraft arrived at its target, near-Earth asteroid (101955) Bennu, on December 3, 2018. The OSIRIS-REx spacecraft has since collected a wealth of scientific information in order to select a suitable site for sampling. Shortly after insertion into orbit on December 31, 2018, particles were identified in starfield images taken by the navigation camera (NavCam 1). Several groups within the OSlRlS-REx team analyzed the particle data in an effort to better understand this newfound activity of Bennu and to investigate the potential sensitivity of the particles to Bennu's geophysical parameters. A number of particles were identified through automatic and manual methods in multiple images, which could be turned into short sequences of optical tracking observations. Here, we discuss the precision orbit determination (OD) effort focused on these particles at NASA GSFC, which involved members of the Independent Navigation Team (INT) in particular. The particle data are combined with other OSIRIS-REx tracking data (radiometric from OSN and optical landmark data) using the NASA GSFC GEODYN orbit determination and geodetic parameter estimation software. We present the results of our study, particularly those pertaining to the gravity field of Bennu. We describe the force modeling improvements made to GEODYN specifically for this work, e.g., with a raytracing-based modeling of solar radiation pressure. The short-lived, low-flying moonlets enable us to determine a gravity field model up to a relatively high degree and order: at least degree 6 without constraints, and up to degree 10 when applying Kaula-like regularization. We can backward- and forward-integrate the trajectory of these particles to the ejection and landing sites on Bennu. We assess the recovered field by its impact on the OSIRIS-REx trajectory reconstruction and prediction quality in the various mission phases (e.g., Orbital A, Detailed Survey, and Orbital B)

Autonomous Detection of Particles and Tracks in Optical Images

During its initial orbital phase in early 2019, the Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer (OSIRIS-REx) asteroid sample return mission detected small particles apparently emanating from the surface of the near-Earth asteroid (101955) Bennu in optical navigation images. Identification and characterization of the physical and dynamical properties of these objects became a mission priority in terms of both spacecraft safety and scientific investigation. Traditional techniques for particle identification and tracking typically rely on manual inspection and are often time-consuming. The large number of particles associated with the Bennu events and the mission criticality rendered manual inspection techniques infeasible for long-term operational support. In this work, we present techniques for autonomously detecting potential particles in monocular images and providing initial correspondences between observations in sequential images, as implemented for the OSIRIS-REx mission.Comment: 23 pages, 10 figure

Photometry of Particles Ejected From Active Asteroid (101955) Bennu

AbstractNear‐Earth asteroid (101955) Bennu is an active asteroid experiencing mass loss in the form of ejection events emitting up to hundreds of millimeter‐ to centimeter‐scale particles. The close proximity of the Origins, Spectral Interpretations, Resource Identification, and Security–Regolith Explorer spacecraft enabled monitoring of particles for a 10‐month period encompassing Bennu's perihelion and aphelion. We found 18 multiparticle ejection events, with masses ranging from near zero to hundreds of grams (or thousands with uncertainties) and translational kinetic energies ranging from near zero to tens of millijoules (or hundreds with uncertainties). We estimate that Bennu ejects ~104 g per orbit. The largest event took place on 6 January 2019 and consisted of ~200 particles. The observed mass and translational kinetic energy of the event were between 459 and 528 g and 62 and 77 mJ, respectively. Hundreds of particles not associated with the multiparticle ejections were also observed. Photometry of the best‐observed particles, measured at phase angles between ~70° and 120°, was used to derive a linear phase coefficient of 0.013 ± 0.005 magnitudes per degree of phase angle. Ground‐based data back to 1999 show no evidence of past activity for Bennu; however, the currently observed activity is orders of magnitude lower than observed at other active asteroids and too low be observed remotely. There appears to be a gentle decrease in activity with distance from the Sun, suggestive of ejection processes such as meteoroid impacts and thermal fracturing, although observational bias may be a factor

Template Matching Used for Small Body Optical Navigation with Poorly Detailed Objects

Object and template matching becomes difficult when an image lacks detail. This is particularly worrisome when typical matching techniques, cross-correlation, log-polar mapping, and key point matching fail. Work herein describes a formulation that identifies objects of interest, estimates the affine transformation between a template object and scene using Principal Component Analysis (PCA), and provides a fit value for the objects and template incorporating Hu's Moments. The algorithm presented is tested on synthetic images and images obtained from the OSIRIS-REx mission while the spacecraft was approaching its target, Bennu. Results for the current formulation show that, with the presence of large-scale variations and rotation, the fitting scheme performs well when compared with other techniques

Parametric Covariance Model for Horizon-Based Optical Navigation

This Note presents an entirely parametric version of the covariance for horizon-based optical navigation measurements. The covariance can be written as a function of only the spacecraft position, two sensor design parameters, the illumination direction, the size of the observed planet, the size of the lit arc to be used, and the total number of observed horizon points. As a result, one may now more clearly understand the sensitivity of horizon-based optical navigation performance as a function of these key design parameters, which is insight that was obscured in previous (and nonparametric) versions of the covariance. Finally, the new parametric covariance is shown to agree with both the nonparametric analytic covariance and results from a Monte Carlo analysis

Observations on the Computation of Eigenvalue and Eigenvector Jacobians

Many scientific and engineering problems benefit from analytic expressions for eigenvalue and eigenvector derivatives with respect to the elements of the parent matrix. While there exists extensive literature on the calculation of these derivatives, which take the form of Jacobian matrices, there are a variety of deficiencies that have yet to be addressed—including the need for both left and right eigenvectors, limitations on the matrix structure, and issues with complex eigenvalues and eigenvectors. This work addresses these deficiencies by proposing a new analytic solution for the eigenvalue and eigenvector derivatives. The resulting analytic Jacobian matrices are numerically efficient to compute and are valid for the general complex case. It is further shown that this new general result collapses to previously known relations for the special cases of real symmetric matrices and real diagonal matrices. Finally, the new Jacobian expressions are validated using forward finite differencing and performance is compared with another technique

Reconstruction of Bennu Particle Events From Sparse Data

OSIRIS-REx began observing particle ejection events shortly after entering orbit around near-Earth asteroid (101955) Bennu in January 2019. For some of these events, the only observations of the ejected particles come from the first two images taken immediately after the event by OSIRIS-REx's NavCam 1 imager. Without three or more observations of each particle, traditional orbit determination is not possible. However, by assuming that the particles all ejected at the same time and location for a given event, and approximating that their velocities remained constant after ejection (a reasonable approximation for fast-moving particles, i.e., with velocities on the order of 10 cm/s or greater, given Bennu's weak gravity), we show that it is possible to estimate the particles' states from only two observations each. We applied this newly developed technique to reconstruct the particle ejection events observed by the OSIRIS-REx spacecraft during orbit about Bennu. Particles were estimated to have ejected with inertial velocities ranging from 7 cm/s to 3.3 m/s, leading to a variety of trajectory types. Most (>80%) of the analyzed events were estimated to have originated from midlatitude regions and to have occurred after noon (local solar time), between 12:44 and 18:52. Comparison with higher-fidelity orbit determination solutions for the events with sufficient observations demonstrates the validity of our approach and also sheds light on its biases. Our technique offers the capacity to meaningfully constrain the properties of particle ejection events from limited data.Open access articleThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]