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Bierman, G. J.

Publications and source records attributed to Bierman, G. J..

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

A decentralized square root information filter/smoother

A number of developments has recently led to a considerable interest in the decentralization of linear least squares estimators. The developments are partly related to the impending emergence of VLSI technology, the realization of parallel processing, and the need for algorithmic ways to speed the solution of dynamically decoupled, high dimensional estimation problems. A new method is presented for combining Square Root Information Filters (SRIF) estimates obtained from independent data sets. The new method involves an orthogonal transformation, and an information matrix filter 'homework' problem discussed by Schweppe (1973) is generalized. The employed SRIF orthogonal transformation methodology has been described by Bierman (1977).

Bierman, G. J.↗
NASA NTRS1983

Voyager orbit determination at Jupiter

This paper summarizes the Voyager 1 and Voyager 2 orbit determination activity extending from encounter minus 60 days to the Jupiter encounter, and includes quantitative results and conclusions derived from mission experiences. The major topics covered include an identification and quantification of the major orbit determination error sources and a review of salient orbit determination results from encounter, with emphasis on the Jupiter approach phase orbit determination. Special attention is paid to the use of combined spacecraft-based optical observations and earth-based radiometric observations to achieve accurate orbit determination during the Jupiter encounter approach phase.

Campbell, J. K.↗
NASA NTRS1980

Seasat orbit refinement for altimetry application

This paper describes the use of stochastic differential correction models in refining the Seasat orbit based on post-flight analysis of tracking data. The objective is to obtain orbital-height precision that is commensurate with the inherent Seasat altimetry data precision level of 10 cms. Local corrections to a mean ballistic arc, perturbed principally by atmospheric drag variations and local gravitational anomalies, are obtained by the introduction of stochastic dynamical models in conjunction with optimal estimation/smoothing techniques. Assessment of the resulting orbit with 'ground truth' provided by Seasat altimetry data shows that the orbital height precision is improved by 32% when compared to a conventional least-squares solution using the same data set. The orbital height precision realized by employing stochastic differential correction models is in the range of 73 cms to 208 cms rms.

Mohan, S. N.↗
NASA NTRS1980

UDU/T/ covariance factorization for Kalman filtering

There has been strong motivation to produce numerically stable formulations of the Kalman filter algorithms because it has long been known that the original discrete-time Kalman formulas are numerically unreliable. Numerical instability can be avoided by propagating certain factors of the estimate error covariance matrix rather than the covariance matrix itself. This paper documents filter algorithms that correspond to the covariance factorization P = UDU(T), where U is a unit upper triangular matrix and D is diagonal. Emphasis is on computational efficiency and numerical stability, since these properties are of key importance in real-time filter applications. The history of square-root and U-D covariance filters is reviewed. Simple examples are given to illustrate the numerical inadequacy of the Kalman covariance filter algorithms; these examples show how factorization techniques can give improved computational reliability.

Thornton, C. L.↗
NASA NTRS1979

Modern estimation techniques applied to microwave sensing of the marine boundary layer

Previous efforts in the area of satellite microwave sensing of the marine boundary layer have relied upon linear regression techniques to extract geophysical parameters from the microwave measurement data. The approach used in the present paper shifts emphasis away from the generation of regression weighting matrices which implicitly assume that the data are linear in the parameters to be determined and that the problem is statistically stationary. The idea is simply to employ modern computational estimation techniques to obtain parameter estimates from nonlinear noisy measurements. The approach is limited only to the region of validity of Grody's (1976) model. Attention is focused on documenting how estimation techniques, in particular the square root information filter (SRIF), are used to solve a nonlinear function optimization problem.

Bierman, G. J.↗
NASA NTRS1979

A subroutine package for discrete estimation problems

In this paper we describe a well documented, compactly coded, storage efficient, thoroughly tested, and easy to use set of FORTRAN IV subroutines for use in Kalman filter or least-squares applications. The package contains both the UDU covariance factorization and the square root information filter algorithms developed at the Jet Propulsion Laboratory. Numerical reliability of the algorithms is a key feature of the package.

Bierman, G. J.↗
NASA NTRS1978

A parameter estimation subroutine package

Linear least squares estimation and regression analyses continue to play a major role in orbit determination and related areas. A library of FORTRAN subroutines were developed to facilitate analyses of a variety of estimation problems. An easy to use, multi-purpose set of algorithms that are reasonably efficient and which use a minimal amount of computer storage are presented. Subroutine inputs, outputs, usage and listings are given, along with examples of how these routines can be used. The routines are compact and efficient and are far superior to the normal equation and Kalman filter data processing algorithms that are often used for least squares analyses.

Bierman, G. J.↗
NASA NTRS1978

Filtering and error analysis via the UDU super T covariance factorization

Kalman filter algorithms based on the UDU super T covariance factorization are discussed, with special attention given to algorithm implementation efficiency. A U-D-factored covariance error-analysis algorithm is formulated, and its efficiency and numerical stability are demonstrated in a representative orbit determination problem. The numerical results are compared with those obtained using covariance error-analysis formulas, and the comparison highlights the numerical superiority of the present algorithm. A byproduct of the U-D analysis is a highly efficient algorithm mechanization of the arbitrary gain covariance update formula.

Thornton, C. L.↗
NASA NTRS1978

A parameter estimation subroutine package

Linear least squares estimation and regression analyses continue to play a major role in orbit determination and related areas. In this report we document a library of FORTRAN subroutines that have been developed to facilitate analyses of a variety of estimation problems. Our purpose is to present an easy to use, multi-purpose set of algorithms that are reasonably efficient and which use a minimal amount of computer storage. Subroutine inputs, outputs, usage and listings are given along with examples of how these routines can be used. The following outline indicates the scope of this report: Section (1) introduction with reference to background material; Section (2) examples and applications; Section (3) subroutine directory summary; Section (4) the subroutine directory user description with input, output, and usage explained; and Section (5) subroutine FORTRAN listings. The routines are compact and efficient and are far superior to the normal equation and Kalman filter data processing algorithms that are often used for least squares analyses.

Bierman, G. J.↗
NASA NTRS1978

An application of the square-root information filter to large scale linear interconnected systems

The paper considers the use of numerically stable square-root information filter (SRIF) algorithms to reduce the computation and storage requirements of a certain class of large-scale linear interconnected systems (multistation satellite tracking is examined as an example). The reductions are in comparison with conventional sequential covariance type formulations. To illustrate the SRIF algorithm: a 40 subsystem, 10 state problem, for example, has its storage requirements reduced by a full order of magnitude (from 84255 to 8100).

Bierman, G. J.↗
NASA NTRS1977

Integration-free interval doubling for Riccati equation solutions

Starting with certain identities obtained by Reid (1972) and Redheffer (1962) for general matrix Riccati equations (RE's), we give various algorithms for the case of constant coefficients. The algorithms are based on two ideas - first, relate the RE solution with general initial conditions to anchored RE solutions; and second, when the coefficients are constant, the anchored solutions have a basic shift-invariance property. These ideas are used to construct an integration-free, superlinearly convergent iterative solution to the algebraic RE. Preliminary numerical experiments show that our algorithms, arranged in square-root form, provide a method that is numerically stable and appears to be competitive with other methods of solving the algebraic RE.

Sidhu, G. S.↗
NASA NTRS1977

A parameter estimation subroutine package

Linear least squares estimation and regression analyses continue to play a major role in orbit determination and related areas. FORTRAN subroutines have been developed to facilitate analyses of a variety of parameter estimation problems. Easy to use multipurpose sets of algorithms are reported that are reasonably efficient and which use a minimal amount of computer storage. Subroutine inputs, outputs, usage and listings are given, along with examples of how these routines can be used.

Bierman, G. J.↗
NASA NTRS1977

Gram-Schmidt algorithms for covariance propagation

This paper addresses the time propagation of triangular covariance factors. Attention is focused on the square-root free factorization, P = UD(transpose of U), where U is unit upper triangular and D is diagonal. An efficient and reliable algorithm for U-D propagation is derived which employs Gram-Schmidt orthogonalization. Partitioning the state vector to distinguish bias and coloured process noise parameters increase mapping efficiency. Cost comparisons of the U-D, Schmidt square-root covariance and conventional covariance propagation methods are made using weighted arithmetic operation counts. The U-D time update is shown to be less costly than the Schmidt method; and, except in unusual circumstances, it is within 20% of the cost of conventional propagation.

Thornton, C. L.↗
NASA NTRS1977

Numerical comparison of Kalman filter algorithms - Orbit determination case study

Numerical characteristics of various Kalman filter algorithms are illustrated with a realistic orbit determination study. The case study of this paper highlights the numerical deficiencies of the conventional and stabilized Kalman algorithms. Computational errors associated with these algorithms are found to be so large as to obscure important mismodeling effects and thus cause misleading estimates of filter accuracy. The positive result of this study is that the U-D covariance factorization algorithm has excellent numerical properties and is computationally efficient, having CPU costs that differ negligibly from the conventional Kalman costs. Accuracies of the U-D filter using single precision arithmetic consistently match the double precision reference results. Numerical stability of the U-D filter is further demonstrated by its insensitivity to variations in the a priori statistics.

Bierman, G. J.↗
NASA NTRS1977

Application of Kalman filtering to spacecraft range residual prediction

One function of the Deep Space Network is validation of the range data that they receive. In this paper we present an automated online sequential range predictor which shows promise of significantly reducing computational and manpower expenditures. The proposed algorithm, a U-D covariance factored Kalman filter, is demonstrated by processing a four-month record of Viking spacecraft data taken enroute to Mars.

Madrid, G. A.↗
NASA NTRS1977

Applications of modern estimation techniques to aircraft navigation

Kalman filter design and analysis are discussed, and application of the analysis to multidimensional aircraft navigation problems is considered. The application of perturbation matrices, U-D factors, and filter variance component error source percentages to avionics covariance analysis is suggested. Filter design is divided into four phases, and the procedures applied in each phase are examined.

Bierman, G. J.↗
NASA NTRS1977

An application of the square root information filter to large scale linear interconnected systems

It is demonstrated that use of the square root information filter (SRIF) can reduce the storage and computation required for estimation of certain classes of large-scale interconnected systems. The SRIF uses an information array that is related to the Kalman filter covariance and estimate. The SRIF algorithm, which is optimal, is a direct application of matrix partitioning to some optimal filtering algorithms described in the literature. The SRIF algorithm is able to reduce the storage requirements of a 40-subsystem 10-state problem by a full order of magnitude.

Bierman, G. J.↗