Catherine L. Thornton

Reduced-dynamic technique for precise orbit determination of low earth satellites

A reduced-dynamic technique for precise orbit determination of low earth satellites is described. This technique optimally combines the conventional dynamic technique with the nondynamic technique which uses differential GPS continuous carrier phase to define the state transition. A Kalman filter formulation for this reduced-dynamic technique is given. A covariance analysis shows that when neither the dynamic nor the nondynamic technique is clearly superior, the reduced-dynamic technique appreciably improves the orbit accuracy. Guidelines for selecting a near-optimum weighting for the combination are given. Sensitivity to suboptimal weighting is assessed.

Paper: 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.

Precise tracking of remote sensing satellites with the Global Positioning System

The Global Positioning System (GPS) can be applied in a number of ways to track remote sensing satellites at altitudes below 3000 km with accuracies of better than 10 cm. All techniques use a precise global network of GPS ground receivers operating in concert with a receiver aboard the user satellite, and all estimate the user orbit, GPS orbits, and selected ground locations simultaneously. The GPS orbit solutions are always dynamic, relying on the laws of motion, while the user orbit solution can range from purely dynamic to purely kinematic (geometric). Two variations show considerable promise. The first one features an optimal synthesis of dynamics and kinematics in the user solution, while the second introduces a novel gravity model adjustment technique to exploit data from repeat ground tracks. These techniques, to be demonstrated on the TOPEX/Poseidon mission in 1992, will offer subdecimeter tracking accuracy for dynamically unpredictable satellites down to the lowest orbital altitudes. >

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 = UDUT, 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 colored process noise parameters increases 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.

UDUT 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.

Constraints on Pacific Plate Kinematics and Dynamics with Global Positioning System Measurements

Global positioning system (GPS) receivers are capable of generating precise geodetic data that can yield important constraints on both plate kinematics and dynamics. Geodetic measurements can determine plate motion rates over time scales for which little data are currently available. High-quality geodetic data may also allow the investigation of plate driving forces because changes in the intraplate stress field can potentially be inferred from measurement of the resulting crustal strain. A measurement program designed to investigate kinematic and dynamic aspects of plate tectonics should be concentrated in the Pacific region. This area contains the largest and fastest moving plate, up to 17 cm/year. Furthermore, subduction zones, which play a quantitatively important role in plate driving forces, are largely restricted to the Pacific region. We summarize accuracy studies showing that for short (< 100 km) baselines, centimeter-level accuracy can be expected using only mobile stations. For longer baselines, uncertainty in the orbit ephemeris of the GPS satellites is a major error source. Performing simultaneous observations at widely separated (~3000 km) fiducial stations near the region of interest, however, should allow centimeter-level accuracy for baselines up to several thousand kilometers in length. This performance level is predicated upon the assumption that fiducial baselines are known a priori to the centimeter level, for example, from very-long-baseline interferometry, and that corrections for tropospheric path delay are accurate to the centimeter level. The fiducial network location is flexible, limited mainly by the requirement for mutual satellite visibility.

Numerical comparison of discrete 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 UD 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 UD filter is further demonstrated by its insensitivity to variations in the a priori statistics.