Results for “Covariance Matrix” — Page 8

NASA NTRSDate not supplied

Structured Covariance Gaussian Networks for Orion Crew Module Aerodynamic Uncertainty Quantification

In this paper we propose a new approach for nonlinear regression and uncertainty quantification. The method is based on a pair of neural networks which parameterize mean and dense covariance functions of a multivariate Gaussian process, trained together to maximize the log-likelihood of observing the given data. The covariance matrix is made positive definite at every input by construction. We also propose a sampling approach that produces viable surrogate function realizations from the Gaussian process. We call the proposed model a Structured Covariance Gaussian Network (SCGN). We illustrate the use of SCGNs for learning an aerodynamic response surface with built-in uncertainty for the Orion crew module. We find that SCGN provides an efficient and systematic way to learn nonlinear functional relationships and dense covariances. We compare results to a baseline Gaussian process regressor and observe that the SCGN provides comparable uncertainty descriptions with improved scalability to dataset size. The sample functions generated by SCGN are fast to evaluate online and are therefore convenient for use in trajectory simulations. These results suggest that SCGN may be a viable computational method for aerodynamic uncertainty quantification.

machine learning↗

NASA NTRSDate not supplied

Structured Covariance Gaussian Networks for Orion Crew Module Aerodynamic Uncertainty Quantification

In this paper we propose a new approach for nonlinear regression and uncertainty quantification. The method is based on a pair of neural networks which parameterize mean and dense covariance functions of a multivariate Gaussian process, trained together to maximize the log-likelihood of observing the given data. The covariance matrix is made positive definite at every input by construction. We also propose a sampling approach that produces viable surrogate function realizations from the Gaussian process. We call the proposed model a Structured Covariance Gaussian Network (SCGN). We illustrate the use of SCGNs for learning an aerodynamic response surface with built-in uncertainty for the Orion crew module. We find that SCGN provides an efficient and systematic way to learn nonlinear functional relationships and dense covariances. We compare results to a baseline Gaussian process regressor and observe that the SCGN provides comparable uncertainty descriptions with improved scalability to dataset size. The sample functions generated by SCGN are fast to evaluate online and are therefore convenient for use in trajectory simulations. These results suggest that SCGN may be a viable computational method for aerodynamic uncertainty quantification.

machine learning↗

NASA NTRS1991

External calibration of polarimetric radars using point and distributed targets

Polarimetric calibration algorithms using combinations of point targets and reciprocal distributed targets are developed. From the reciprocity relations of distributed targets, and equivalent point target response is derived. Then the problem of polarimetric calibration using two point targets and one distributed target reduces to that using three point targets, which has been previously solved. For calibration using one point target and one reciprocal distributed target, two cases are analyzed with the point target being a trihedral reflector or a polarimetric active radar calibrator (PARC). For both cases, the general solutions of the system distortion matrices are written as a product of a particular solution and a matrix with one free parameter. For the trihedral-reflector case, this free parameter is determined by assuming azimuthal symmetry for the distributed target. For the PARC case, knowledge of one ratio of two covariance matrix elements of the distributed target is required to solve for the free parameter. Numerical results are simulated to demonstrate the usefulness of the developed algorithms.

Yueh, S. H.↗

NASA NTRS1998

A low cost approach to simultaneous orbit, attitude, and rate estimation using an extended Kalman filter

An innovative approach to autonomous attitude and trajectory estimation is available using only magnetic field data and rate data. The estimation is performed simultaneously using an Extended Kalman Filter (EKF), a well known algorithm used extensively in onboard applications. The magnetic field is measured on a satellite by a magnetometer, an inexpensive and reliable sensor flown on virtually all satellites in low earth orbit. Rate data is provided by a gyro, which can be costly. This system has been developed and successfully tested in a post-processing mode using magnetometer and gyro data from 4 satellites supported by the Flight Dynamics Division at Goddard. In order for this system to be truly low cost, an alternative source for rate data must be utilized. An independent system which estimates spacecraft rate has been successfully developed and tested using only magnetometer data or a combination of magnetometer data and sun sensor data, which is less costly than a gyro. This system also uses an EKF. Merging the two systems will provide an extremely low cost, autonomous approach to attitude and trajectory estimation. In this work we provide the theoretical background of the combined system. The measurement matrix is developed by combining the measurement matrix of the orbit and attitude estimation EKF with the measurement matrix of the rate estimation EKF, which is composed of a pseudo-measurement which makes the effective measurement a function of the angular velocity. Associated with this is the development of the noise covariance matrix associated with the original measurement combined with the new pseudo-measurement. In addition, the combination of the dynamics from the two systems is presented along with preliminary test results.

Deutschmann, Julie↗

NASA NTRS1998

A Low Cost Approach to Simultaneous Orbit, Attitude, and Rate Estimation Using an Extended Kalman Filter

An innovative approach to autonomous attitude and trajectory estimation is available using only magnetic field data and rate data. The estimation is performed simultaneously using an Extended Kalman Filter, a well known algorithm used extensively in onboard applications. The magnetic field is measured on a satellite by a magnetometer, an inexpensive and reliable sensor flown on virtually all satellites in low earth orbit. Rate data is provided by a gyro, which can be costly. This system has been developed and successfully tested in a post-processing mode using magnetometer and gyro data from 4 satellites supported by the Flight Dynamics Division at Goddard. In order for this system to be truly low cost, an alternative source for rate data must be utilized. An independent system which estimate spacecraft rate has been successfully developed and tested using only magnetometer data or a combination of magnetometer data and sun sensor data, which is less costly than a gyro. This system also uses an Extended Kalman Filter. Merging the two systems will provide an extremely low cost, autonomous approach to attitude and trajectory estimation. In this work we provide the theoretical background of the combined system. The measurement matrix is developed by combining the measurement matrix of the orbit and attitude estimation EKF with the measurement matrix of the rate estimation EKF, which is composed of a pseudo-measurement which makes the effective measurement a function of the angular velocity. Associated with this is the development of the noise covariance matrix associated with the original measurement combined with the new pseudo-measurement. In addition, the combination of the dynamics from the two systems is presented along with preliminary test results.

Deutschmann, Julie↗

NASA NTRS1971

Applications of generalized inverses to estimation in dynamic systems.

Generalized matrix inverses are used to obtain an estimation procedure for estimation of the state vector of a dynamic system. This sequential procedure is studied analytically with respect to the choice of an arbitrary vector. The covariance matrix of the estimator is determined and compared to the optimal Kalman type procedure. A numerical example illustrates the procedure and compares it to the optimal one.

Prochaska, B. J.↗

NASA NTRS1991

Constant covariance in local vertical coordinates for near-circular orbits

A method is presented for devising a covariance matrix that either remains constant or grows in keeping with the presence of a period error in a rotating local-vertical coordinate system. The solution presented may prove useful in the initialization of simulation covariance matrices for near-circular-orbit problems. Use is made of the Clohessy-Wiltshire equations and the travelling-ellipse formulation.

Shepperd, Stanley W.↗

NASA NTRS1993

Attitude determination using vector observations: A fast optimal matrix algorithm

The attitude matrix minimizing Wahba's loss function is computed directly by a method that is competitive with the fastest known algorithm for finding this optimal estimate. The method also provides an estimate of the attitude error covariance matrix. Analysis of the special case of two vector observations identifies those cases for which the TRIAD or algebraic method minimizes Wahba's loss function.

Markley, F. Landis↗

NASA NTRS1993

Attitude determination using vector observations - A fast optimal matrix algorithm

The attitude matrix minimizing Wahba's loss function is computed directly by a method that is competitive with the fastest known algorithm for finding this optimal estimate. The method also provides an estimate of the attitude error covariance matrix. Analysis of the special case of two vector observations identifies those cases for which the TRIAD or algebraic method minimizes Wahba's loss function.

Markley, F. L.↗

NASA NTRS1994

Progress in navigation filter estimate fusion and its application to spacecraft rendezvous

A new derivation of an algorithm which fuses the outputs of two Kalman filters is presented within the context of previous research in this field. Unlike other works, this derivation clearly shows the combination of estimates to be optimal, minimizing the trace of the fused covariance matrix. The algorithm assumes that the filters use identical models, and are stable and operating optimally with respect to their own local measurements. Evidence is presented which indicates that the error ellipsoid derived from the covariance of the optimally fused estimate is contained within the intersections of the error ellipsoids of the two filters being fused. Modifications which reduce the algorithm's data transmission requirements are also presented, including a scalar gain approximation, a cross-covariance update formula which employs only the two contributing filters' autocovariances, and a form of the algorithm which can be used to reinitialize the two Kalman filters. A sufficient condition for using the optimally fused estimates to periodically reinitialize the Kalman filters in this fashion is presented and proved as a theorem. When these results are applied to an optimal spacecraft rendezvous problem, simulated performance results indicate that the use of optimally fused data leads to significantly improved robustness to initial target vehicle state errors. The following applications of estimate fusion methods to spacecraft rendezvous are also described: state vector differencing, and redundancy management.

Carpenter, J. Russell↗

NASA NTRS2022

Continuum Covariance Propagation for Understanding Variance Loss in Advective Systems

Motivated by the spurious variance loss encountered during covariance propagation in atmospheric and other large-scale data assimilation systems, we consider the problem for state dynamics governed by the continuity and related hyperbolic partial differential equations. This loss of variance has been attributed to reduced-rank representations of the covariance matrix, as in ensemble methods for example, or else to the use of dissipative numerical methods. Through a combination of analytical work and numerical experiments, we demonstrate that significant variance loss, as well as gain, typically occurs during covariance propagation, even at full rank. The cause of this unusual behavior is a discontinuous change in the continuum covariance dynamics as correlation lengths become small, for instance in the vicinity of sharp gradients in the velocity field. This discontinuity in the covariance dynamics arises from hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Our numerical experiments demonstrate that standard numerical methods for evolving the state are not adequate for propagating the covariance, because16they do not capture the discontinuity in the continuum covariance dynamics as correlations lengths tend to zero. Our analytical and numerical results show that this leads to significant, spurious variance loss in certain regions, and gain in others. The results suggest that developing local covariance propagation methods designed specifically to capture covariance evolution near the diagonal may prove a useful alternative to current methods of covariance propagation.

covariance propagation↗

NASA NTRS2019

Multivariate Normality of Cartesian-Framed Covariances: Evaluation and Operational Significance

Collision avoidance relies on representative Cartesian uncertainty volumes in order to calculate probabilities of collision. Among the potential shortcomings of a covariance matrix representation of state errors, the most worrisome is the coordinate mismatch between the Cartesian framework in which these matrices are distributed and the curvilinear path that satellite orbits actually follow. The present study compares curvilinear-based and Cartesian covariance representations for ~50,000 conjunctions to determine the frequency in which significant deviations from Gaussianity are observed, then compares the 2-D Pc result from the Cartesian covariance to a Monte Carlo Pc conducted in element space to assess operational significance.

Lechtenberg, Travis F.↗

NASA NTRS2019

Multivariate Normality of Cartesian Frame Covariances: Evaluation and Operational Significance

Collision avoidance relies on representative Cartesian uncertainty volumes in order to calculate probabilities of collision. Among the potential shortcomings of a covariance matrix representation of state errors, the most worrisome is the coordinate mismatch between the Cartesian framework in which these matrices are distributed and the curvilinear path that satellite orbits actually follow. The present study compares curvilinear-based and Cartesian covariance representations for ~50,000 conjunctions to determine the frequency in which significant deviations from Gaussianity are observed, then compares the 2-D Pc result from the Cartesian covariance to a Monte Carlo Pc conducted in element space to assess operational significance.

Lechtenberg, Travis F.↗

NASA NTRS1986

Theory of optimal weighting of data to detect climatic change

A search for climatic change predicted by climate models can easily yield unconvincing results because of 'climatic noise,' the inherent, unpredictable variability of time-average atmospheric data. A weighted average of data that maximizes the probability of detecting predicted climatic change is presented. To obtain the optimal weights, an estimate of the covariance matrix of the data from a prior data set is needed. This introduces additional sampling error into the method. This is presently taken into account. A form of the weighted average is found whose probability distribution is independent of the true (but unknown) covariance statistics of the data and of the climate model prediction.

Bell, T. L.↗

NASA NTRS1991

The behavior of forecast error covariances for a Kalman filter in two dimensions

The behavior of forecast error covariances in a fairly realistic setting is demonstrated via a Kalman filter algorithm. It is used to assimilate simulated data from the existing radiosonde network, from the demonstration network of 31 Doppler wind profilers in the central U.S., and from hypothetical radiometers located at five of the profiler sites. Some theoretical justification of the hypothesis advanced by Phillips (1982), and the hypothesis is used to formulate the model error covariance matrix required by the Kalman filter. The results show that assimilating the profiler wind data leads to a large reduction of forecast/analysis error in heights as well as in winds, over the profiler region and also downstream, when compared with the results of assimilating the radiosonde data alone. The forecast error covariance matrices that the Kalman filter calculates to obtain this error reduction differ considerably from those prescribed by the optimal interpolation schemes that are employed for data assimilation at operational centers.

Cohn, Stephen E.↗

NASA NTRS1987

SMI adaptive antenna arrays for weak interfering signals

The performance of adaptive antenna arrays is studied when a sample matrix inversion (SMI) algorithm is used to control array weights. It is shown that conventional SMI adaptive antennas, like other adaptive antennas, are unable to suppress weak interfering signals (below thermal noise) encountered in broadcasting satellite communication systems. To overcome this problem, the SMI algorithm is modified. In the modified algorithm, the covariance matrix is modified such that the effect of thermal noise on the weights of the adaptive array is reduced. Thus, the weights are dictated by relatively weak coherent signals. It is shown that the modified algorithm provides the desired interference protection. The use of defocused feeds as auxiliary elements of an SMI adaptive array is also discussed.

Gupta, I. J.↗

NASA NTRS1995

Using dark current data to estimate AVIRIS noise covariance and improve spectral analyses

Starting in 1994, all AVIRIS data distributions include a new product useful for quantification and modeling of the noise in the reported radiance data. The 'postcal' file contains approximately 100 lines of dark current data collected at the end of each data acquisition run. In essence this is a regular spectral-image cube, with 614 samples, 100 lines and 224 channels, collected with a closed shutter. Since there is no incident radiance signal, the recorded DN measure only the DC signal level and the noise in the system. Similar dark current measurements, made at the end of each line are used, with a 100 line moving average, to remove the DC signal offset. Therefore, the pixel-by-pixel fluctuations about the mean of this dark current image provide an excellent model for the additive noise that is present in AVIRIS reported radiance data. The 61,400 dark current spectra can be used to calculate the noise levels in each channel and the noise covariance matrix. Both of these noise parameters should be used to improve spectral processing techniques. Some processing techniques, such as spectral curve fitting, will benefit from a robust estimate of the channel-dependent noise levels. Other techniques, such as automated unmixing and classification, will be improved by the stable and scene-independence noise covariance estimate. Future imaging spectrometry systems should have a similar ability to record dark current data, permitting this noise characterization and modeling.

Boardman, Joseph W.↗

NASA NTRS1994

Comparison of Kalman filter and optimal smoother estimates of spacecraft attitude

Given a valid system model and adequate observability, a Kalman filter will converge toward the true system state with error statistics given by the estimated error covariance matrix. The errors generally do not continue to decrease. Rather, a balance is reached between the gain of information from new measurements and the loss of information during propagation. The errors can be further reduced, however, by a second pass through the data with an optimal smoother. This algorithm obtains the optimally weighted average of forward and backward propagating Kalman filters. It roughly halves the error covariance by including future as well as past measurements in each estimate. This paper investigates whether such benefits actually accrue in the application of an optimal smoother to spacecraft attitude determination. Tests are performed both with actual spacecraft data from the Extreme Ultraviolet Explorer (EUVE) and with simulated data for which the true state vector and noise statistics are exactly known.

Sedlak, J.↗