Statistics of state-vector corrections for Apollo onboard computers
Apollo onboard computers tracking data state- vector corrections covariance matrix, considering initial estimate, noise and tracking geometry errors
Pixley, P. T.↗
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Two Reduced Resolution Filter Approaches to Data Assimilation
In this paper we evaluate the performance of two reduced resolution filter approaches to data assimilation. The main distinction between these approaches is in the manner they propagate error covariances. Both account for error covariances in a space with dimension m smaller than the model's state vector dimension n. In the first approach the m dimensional error covariance matrix is interpolated to the n-dimensional space and propagated with the n-dimensional dynamics. In the second approach the low-dimensional error covariance matrix is propagated by a dynamical operator generated in the m-dimensional space. Our experiments indicate that the first approach provides a more reliable simplified scheme for error covariance propagation than the second approach.
Todling, Ricardo↗
Kalman Filtering of Angular-Momentum-Based Attitude Parameters
This paper presents an extended Kalman filter using an attitude parameterization that is advantageous for attitude estimation of spinning spacecraft. The parameters are the angular momentum components in an inertial reference frame, the angular momentum components in the body frame, and a rotation angle. To avoid the singularity of the 7x7 covariance of this state vector arising from the constraint that the magnitude of the angular momentum vector is the same in the inertial and body frames, the Kalman filter employs the nonsingular 6x6 covariance of a reduced error state. Three of the components of this six-component error state are the usual infinitesimal attitude error angles, so the usual 3x3 attitude covariance matrix is a submatrix of the 6x6 covariance. The performance of the resulting filter is compared with that of a quaternion-based filter.
Markley, F. Landis↗
Linear, Kalman, and epoch state filtering techniques Quarterly progress report, 1 Dec. 1970 - 30 Apr. 1971
Numerical analyses of error covariance matrix in linear filtering problem, computational errors in Kalman filtering, and coasting flight equations
Source record↗
ILIAD Testing; and a Kalman Filter for 3-D Pose Estimation
This report presents the results of a two-part project. The first part presents results of performance assessment tests on an Internet Library Information Assembly Data Base (ILIAD). It was found that ILLAD performed best when queries were short (one-to-three keywords), and were made up of rare, unambiguous words. In such cases as many as 64% of the typically 25 returned documents were found to be relevant. It was also found that a query format that was not so rigid with respect to spelling errors and punctuation marks would be more user-friendly. The second part of the report shows the design of a Kalman Filter for estimating motion parameters of a three dimensional object from sequences of noisy data derived from two-dimensional pictures. Given six measured deviation values represendng X, Y, Z, pitch, yaw, and roll, twelve parameters were estimated comprising the six deviations and their time rate of change. Values for the state transiton matrix, the observation matrix, the system noise covariance matrix, and the observation noise covariance matrix were determined. A simple way of initilizing the error covariance matrix was pointed out.
Richardson, A. O.↗
Parameter identification of linear discrete stochastic systems with time delays
An identification algorithm that uses the maximum likelihood technique to identify the unknown time delays, plant parameters, and noise covariances of linear discrete stochastic systems is presented. Cases of additive white noise and colored measurement noises are considered. The likelihood function is evaluated using either a minimum-variance (Kalman) filter or a minimal-order observer. The Kalman filter is used in the identification algorithm to provide minimum-variance estimates. The minimal-order observer is a lower-dimensional and computationally simpler filter, and is advantageous especially for systems with long delays. It provides a less optimal solution to the minimum-mean-square state estimation problem. The colored-noise observer algorithm has the disadvantage of having to compute an extra error covariance matrix of lower order.
Wong, E. C.↗
Toward Improving Short-Term Predictions of Fine Particulate Matter Over the United States Via Assimilation of Satellite Aerosol Optical Depth Retrievals
This study develops a new approach to improve simulations of the particulate matter of aerodynamic diameter smaller than 2.5μm (PM2.5) in the Community Multiscale Air Quality (CMAQ) model via assimilation of Moderate Resolution Imaging Spectroradiometer (MODIS) aerosol optical depth(AOD) retrievals using the Grid point Statistical Interpolation (GSI) system. In contrast to previous studies that only consider errors due to transport, our computation of the background error covariance matrix incorporates uncertainties in anthropogenic emissions. To understand the impact of this approach, three experiments (one background and two assimilations) are performed over the contiguous United States(CONUS) from 15 July to 14 August 2014. The background CMAQ experiment significantly underestimates both the MODIS AOD and surface PM2.5levels. MODIS AOD assimilation pushes both the CMAQ AOD and surface PM2.5distributions toward the observed distributions, but CMAQ still underestimates the observations. Averaged over CONUS, the two assimilation experiments with and without including the anthropogenic emission uncertainties improve the correlation coefficient between the model and independent observations of PM2.5 by ~67% and ~48%, respectively, and reduces the mean bias by ~38% and~10%, respectively. The assimilation improves the model performance everywhere over CONUS, except the New York and Wisconsin, where CMAQ overestimates the observed PM2.5during nighttime after assimilation likely because of overcorrection of aerosol mass concentrations by the AOD assimilation. Future work should incorporate uncertainties in other processes (biomass burning and biogenic emissions, deposition, chemistry, transport, and boundary conditions) to further enhance the value of assimilating spaceborne AOD retrievals.
Assimilation↗
Extended Kalman Filter Performance on the Artemis-1 Mission
The Artemis Program is NASA’s campaign to explore the Moon and beyond. Artemis-1, the uncrewed exoLEO test flight of the Orion spacecraft, was completed in 2022. There were four navigation Extended Kalman Filters (EKFs) that are part of the Orion navigation system. The Atmospheric Extended Kalman Filter (ATMEKF) estimates the vehicle position, velocity, and attitude (referred to as the vehicle state) during the ascent and entry phases of flight. Once Orion is outside of Earths atmosphere, the Earth Orbit Extended Kalman Filter (EOEKF) and CisLunar Extended Kalman Filter (CLEKF) estimate the translational states, depending on the phase of flight, while the Attitude Extended Kalman Filter (ATTEKF) estimates the rotational state of the vehicle. The Kalman filters propagate the vehicle state forward in time using a combination of dynamics models and the output data from the Inertial Measurement Unit (IMU). The filters update the vehicle states and associated uncertainties, in the form of the covariance matrix, using pseudorange measurements from GPS (in ATMEKF/EOEKF), optical navigation measurements of the Earth or Moon (in CLEKF), and star tracker measurements (in ATTEKF). Simultaneously, the Kalman filters estimate error sources in the sensors, which are included in the state vectors as Exponentially Correlated Random Variables (ECRVs). This paper will summarize the performance of these filters during the Artemis-1 mission.
Artemis-1↗
An Upper Bound on Orbital Debris Collision Probability When Only One Object has Position Uncertainty Information
Upper bounds on high speed satellite collision probability, P (sub c), have been investigated. Previous methods assume an individual position error covariance matrix is available for each object. The two matrices being combined into a single, relative position error covariance matrix. Components of the combined error covariance are then varied to obtain a maximum P (sub c). If error covariance information for only one of the two objects was available, either some default shape has been used or nothing could be done. An alternative is presented that uses the known covariance information along with a critical value of the missing covariance to obtain an approximate but useful P (sub c) upper bound. There are various avenues along which an upper bound on the high speed satellite collision probability has been pursued. Typically, for the collision plane representation of the high speed collision probability problem, the predicted miss position in the collision plane is assumed fixed. Then the shape (aspect ratio of ellipse), the size (scaling of standard deviations) or the orientation (rotation of ellipse principal axes) of the combined position error ellipse is varied to obtain a maximum P (sub c). Regardless as to the exact details of the approach, previously presented methods all assume that an individual position error covariance matrix is available for each object and the two are combined into a single, relative position error covariance matrix. This combined position error covariance matrix is then modified according to the chosen scheme to arrive at a maximum P (sub c). But what if error covariance information for one of the two objects is not available? When error covariance information for one of the objects is not available the analyst has commonly defaulted to the situation in which only the relative miss position and velocity are known without any corresponding state error covariance information. The various usual methods of finding a maximum P (sub c) do no good because the analyst defaults to no knowledge of the combined, relative position error covariance matrix. It is reasonable to think, given an assumption of no covariance information, an analyst might still attempt to determine the error covariance matrix that results in an upper bound on the P (sub c). Without some guidance on limits to the shape, size and orientation of the unknown covariance matrix, the limiting case is a degenerate ellipse lying along the relative miss vector in the collision plane. Unless the miss position is exceptionally large or the at-risk object is exceptionally small, this method results in a maximum P (sub c) too large to be of practical use. For example, assuming that the miss distance is equal to the current ISS alert volume along-track (+ or -) distance of 25 kilometers and that the at-risk area has a 70 meter radius. The maximum (degenerate ellipse) P (sub c) is about 0.00136. At 40 kilometers, the maximum P (sub c) would be 0.00085 which is still almost an order of magnitude larger than the ISS maneuver threshold of 0.0001. In fact, a miss distance of almost 340 kilometers is necessary to reduce the maximum P (sub c) associated with this degenerate ellipse to the ISS maneuver threshold value. Such a result is frequently of no practical value to the analyst. Some improvement may be made with respect to this problem by realizing that while the position error covariance matrix of one of the objects (usually the debris object) may not be known the position error covariance matrix of the other object (usually the asset) is almost always available. Making use of the position error covariance information for the one object provides an improvement in finding a maximum P (sub c) which, in some cases, may offer real utility. The equations to be used are presented and their use discussed.
Frisbee, Joseph H., Jr.↗
Relative Navigation In Elliptical Orbits Using An Iterative Nonlinear Filter
The two step filter is applied to process intersatellite radar measurements to determine the motion of one satellite relative to another in close elliptical orbits. This filter breaks a nonlinear estimation problem into two state vectors. The 'first step' state is chosen so as to have a linear measurement equation. This is nonlinearly related to the 'second step' state which describes the dynamics. Two different forms are used. In one, the first step state is the second step state vector augmented by the measurement equation. In the other, the first step and second step state vectors are of equal dimension. The two step filter is compared against an iterated extended Kalman filter and a Kalman filter using a change of variables. Analytical differences between the two step estimator and these conventional filters are highlighted. Special concerns for initializing the first step state covariance matrix and handling the possibility of numerically rank deficient covariance matrices are addressed. Numerical simulations are performed which show that the Two Step estimator produces a lower estimation bias under two circumstances; large apriori initial error; and small dimension observation vectors which require a longer arc of measurements to generate observability of the state.
Garrison, James L.↗
True covariance simulation of the EUVE (extreme ultra violet explorer) update filter
This paper presents a covariance analysis of the performance and sensitivity of the attitude determination Extended Kalman Filter (EKF) used by the onboard computer (OBC) of the Extreme Ultra Violet Explorer (EUVE) spacecraft. The linearized dynamics and measurement equations of the error states are used in formulating the 'truth model' describing the order of the systems involved. The 'design model' used by the OBC EKF is then obtained by reducing the order of the truth model. The covariance matrix of the EKF which uses the reduced order model is not the correct covariance of the EKF estimation error. A 'true covariance analysis' has to be carried out in order to evaluate the correct accuracy of the OBC generated estimates. The results of such analysis are presented which indicate both the performance and the sensitivity of the OBC EKF.
Bar-Itzhack, I. Y.↗
True covariance simulation of the EUVE update filter
A covariance analysis of the performance and sensitivity of the attitude determination Extended Kalman Filter (EKF) used by the On Board Computer (OBC) of the Extreme Ultra Violet Explorer (EUVE) spacecraft is presented. The linearized dynamics and measurement equations of the error states are derived which constitute the truth model describing the real behavior of the systems involved. The design model used by the OBC EKF is then obtained by reducing the order of the truth model. The covariance matrix of the EKF which uses the reduced order model is not the correct covariance of the EKF estimation error. A true covariance analysis has to be carried out in order to evaluate the correct accuracy of the OBC generated estimates. The results of such analysis are presented which indicate both the performance and the sensitivity of the OBC EKF.
Bar-Itzhack, Itzhack Y.↗
True covariance simulation of the EUVE update filter
This paper presents a covariance analysis of the performance and sensitivity of the attitude determination Extended Kalman Filter (EKF) used by the On Board Computer (OBC) of the Extreme Ultra Violet Explorer (EUVE) spacecraft. The linearized dynamics and measurement equations of the error states are used in formulating the 'truth model' describing the real behavior of the systems involved. The 'design model' used by the OBC EKF is then obtained by reducing the order of the truth model. The covariance matrix of the EKF which uses the reduced order model is not the correct covariance of the EKF estimation error. A 'true covariance analysis' has to be carried out in order to evaluate the correct accuracy of the OBC generated estimates. The results of such analysis are presented which indicate both the performance and the sensitivity of the OBC EKF.
Bar-Itzhack, I. Y.↗
A numerical algorithm to propagate navigation error covariance matrices associated with generalized strapdown inertial measurement units
The design and operation of a Strapdown Navigation Analysis Program (SNAP) developed to perform covariance analysis on spacecraft inertial-measurement-unit (IMU) navigation errors are described and demonstrated. Consideration is given to the IMU modeling subroutine (with user-specified sensor characteristics), the data input procedures, state updates and the simulation of instrument failures, the determination of the nominal trajectory, the mapping-matrix and Monte Carlo covariance-matrix propagation methods, and aided-navigation simulation. Numerical results are presented in tables for sample applications involving (1) the Galileo/IUS spacecraft from its deployment from the Space Shuttle to a point 10 to the 8th ft from the center of the earth and (2) the TDRS-C/IUS spacecraft from Space Shuttle liftoff to a point about 2 h before IUS deployment. SNAP is shown to give reliable results for both cases, with good general agreement between the mapping-matrix and Monte Carlo predictions.
Weir, Kent A.↗
Improved Stratospheric Temperature Retrievals for Climate Reanalysis
The Data Assimilation Office (DAO) is embarking on plans to generate a twenty year reanalysis data set of climatic atmospheric variables. One of the focus points will be in the evaluation of the dynamics of the stratosphere. The Stratospheric Sounding Unit (SSU), flown as part of the TIROS Operational Vertical Sounder (TOVS), is one of the primary stratospheric temperature sensors flown consistently throughout the reanalysis period. Seven unique sensors made the measurements over time, with individual instrument characteristics that need to be addressed. The stratospheric temperatures being assimilated across satellite platforms will profoundly impact the reanalysis dynamical fields. To attempt to quantify aspects of instrument and retrieval bias we are carefully collecting and analyzing all available information on the sensors, their instrument anomalies, forward model errors and retrieval biases. For the retrieval of stratospheric temperatures, we adapted the minimum variance approach of Jazwinski (1970) and Rodgers (1976) and applied it to the SSU soundings. In our algorithm, the state vector contains an initial guess of temperature from a model six hour forecast provided by the Goddard EOS Data Assimilation System (GEOS/DAS). This is combined with an a priori covariance matrix, a forward model parameterization, and specifications of instrument noise characteristics. A quasi-Newtonian iteration is used to obtain convergence of the retrieved state to the measurement vector. This algorithm also enables us to analyze and address the systematic errors associated with the unique characteristics of the cell pressures on the individual SSU instruments and the resolving power of the instruments to vertical gradients in the stratosphere. The preliminary results of the improved retrievals and their assimilation as well as baseline calculations of bias and rms error between the NESDIS operational product and col-located ground measurements will be presented.
Rokke, L.↗
The Importance of Semi-Major Axis Knowledge in the Determination of Near-Circular Orbits
Modem orbit determination has mostly been accomplished using Cartesian coordinates. This usage has carried over in recent years to the use of GPS for satellite orbit determination. The unprecedented positioning accuracy of GPS has tended to focus attention more on the system's capability to locate the spacecraft's location at a particular epoch than on its accuracy in determination of the orbit, per se. As is well-known, the latter depends on a coordinated knowledge of position, velocity, and the correlation between their errors. Failure to determine a properly coordinated position/velocity state vector at a given epoch can lead to an epoch state that does not propagate well, and/or may not be usable for the execution of orbit adjustment maneuvers. For the quite common case of near-circular orbits, the degree to which position and velocity estimates are properly coordinated is largely captured by the error in semi-major axis (SMA) they jointly produce. Figure 1 depicts the relationships among radius error, speed error, and their correlation which exist for a typical low altitude Earth orbit. Two familiar consequences are the relationship Figure 1 shows are the following: (1) downrange position error grows at the per orbit rate of 3(pi) times the SMA error; (2) a velocity change imparted to the orbit will have an error of (pi) divided by the orbit period times the SMA error. A less familiar consequence occurs in the problem of initializing the covariance matrix for a sequential orbit determination filter. An initial covariance consistent with orbital dynamics should be used if the covariance is to propagate well. Properly accounting for the SMA error of the initial state in the construction of the initial covariance accomplishes half of this objective, by specifying the partition of the covariance corresponding to down-track position and radial velocity errors. The remainder of the in-plane covariance partition may be specified in terms of the flight path angle error of the initial state. Figure 2 illustrates the effect of properly and not properly initializing a covariance. This figure was produced by propagating the covariance shown on the plot, without process noise, in a circular low Earth orbit whose period is 5828.5 seconds. The upper subplot, in which the proper relationships among position, velocity, and their correlation has been used, shows overall error growth, in terms of the standard deviations of the inertial position coordinates, of about half of the lower subplot, whose initial covariance was based on other considerations.
Carpenter, J. Russell↗
Triangular covariance factorizations for
An improved computational form of the discrete Kalman filter is derived using an upper triangular factorization of the error covariance matrix. The covariance P is factored such that P = UDUT where U is unit upper triangular and D is diagonal. Recursions are developed for propagating the U-D covariance factors together with the corresponding state estimate. The resulting algorithm, referred to as the U-D filter, combines the superior numerical precision of square root filtering techniques with an efficiency comparable to that of Kalman's original formula. Moreover, this method is easily implemented and involves no more computer storage than the Kalman algorithm. These characteristics make the U-D method an attractive realtime filtering technique. A new covariance error analysis technique is obtained from an extension of the U-D filter equations. This evaluation method is flexible and efficient and may provide significantly improved numerical results. Cost comparisons show that for a large class of problems the U-D evaluation algorithm is noticeably less expensive than conventional error analysis methods.
Thornton, C. L.↗
On the Possibility of Ill-Conditioned Covariance Matrices in the First-Order Two-Step Estimator
The first-order two-step nonlinear estimator, when applied to a problem of orbital navigation, is found to occasionally produce first step covariance matrices with very low eigenvalues at certain trajectory points. This anomaly is the result of the linear approximation to the first step covariance propagation. The study of this anomaly begins with expressing the propagation of the first and second step covariance matrices in terms of a single matrix. This matrix is shown to have a rank equal to the difference between the number of first step states and the number of second step states. Furthermore, under some simplifying assumptions, it is found that the basis of the column space of this matrix remains fixed once the filter has removed the large initial state error. A test matrix containing the basis of this column space and the partial derivative matrix relating first and second step states is derived. This square test matrix, which has dimensions equal to the number of first step states, numerically drops rank at the same locations that the first step covariance does. It is formulated in terms of a set of constant vectors (the basis) and a matrix which can be computed from a reference trajectory (the partial derivative matrix). A simple example problem involving dynamics which are described by two states and a range measurement illustrate the cause of this anomaly and the application of the aforementioned numerical test in more detail.
Garrison, James L.↗