An application of least squares method to the solution of the inverse problem of heat conduction.
Heat input to a body surface, using least squares method
THERMAL CONDUCTOR↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Iterative methods for solving nonlinear least squares problems
Iterative methods for solving nonlinear least squares problems
ITERATIVE SOLUTION↗
ORACLS: A system for linear-quadratic-Gaussian control law design
A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.
Armstrong, E. S.↗
A least-squares procedure for the solution of the lifting-line integral equation
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LIFT DISTRIBUTION, WING-SPAN↗
Shuttle onboard IMU alignment methods
The current approach to the shuttle IMU alignment is based solely on the Apollo Deterministic Method. This method is simple, fast, reliable and provides an accurate estimate for the present cluster to mean of 1,950 transformation matrix. If four or more star sightings are available, the application of least squares analysis can be utilized. The least squares method offers the next level of sophistication to the IMU alignment solution. The least squares method studied shows that a more accurate estimate for the misalignment angles is computed, and the IMU drift rates are a free by-product of the analysis. Core storage requirements are considerably more; estimated 20 to 30 times the core required for the Apollo Deterministic Method. The least squares method offers an intermediate solution utilizing as much data that is available without a complete statistical analysis as in Kalman filtering.
Henderson, D. M.↗
The covariance matrix for the solution vector of an equality-constrained least-squares problem
Methods are given for computing the covariance matrix for the solution vector of an equality-constrained least squares problem. The methods are matched to the solution algorithms given in the book, 'Solving Least Squares Problems.'
Lawson, C. L.↗
Solution of an integral equation encountered in studies on radiative transfer in completely absorbing leaf canopies
The leaf normal distribution function is related to the extinction coefficient of direct solar radiation by a Fredholm integral equation of the first kind. The integral equation is solved by the method of constrained least squares. The solutions are stable and correct only in extreme cases and, in general, it is not possible to recover an arbitrary leaf normal distribution function.
Myneni, Ranga B.↗
Applications of a mathematical programming technique to finite thrust rocket trajectory optimization
Iterative solution of least squares programming problem for finite thrust rocket trajectory optimization
Armstrong, E. S.↗
Weighted least squares formulas for multiple Doppler radar analysis
An algorithm is developed for the analysis of multiple Doppler radar data in meteorological applications. The algorithm is based on the least squares linear solutions for the variable parameters of a local particle velocity field. The computational sequence for the algorithm is described, and some factors influencing the covariance of Doppler measurement errors are discussed.
Wilson, F. W., Jr.↗
Determination of relative site motions in the western United States using Mark III very long baseline interferometry
Four years of very long baseline interferometry (VLBI) measurements involving fixed and mobile stations in the western U.S. are analyzed using weighted least squares. Baseline solution time series are used to estimate rates of change of the vector baseline between pairs of VLBI stations. Horizontal vector motions are determined for 18 sites with respect to a site in the Mojave desert. Six sites west of and two sites just to the east of the San Andreas fault show northwestward motion closely parallel to the San Andreas fault. Their motions range from 18 to 43 mm/yr and are significant at confidence levels greater than 99 percent. Motions in California relative to interior North American plate sites are used to estimate an integrated contribution from Basin and Range spreading to the total relative Pacific plate motion as 9-10 mm/yr. A maximum relative velocity of 50-51 mm/yr is found between the interior North American sites and the western California sites, in close agreement with the NUVEL-1 plate motion model.
Clark, T. A.↗
An analysis of the Voyager 2 Ultraviolet Spectrometer occultation data at Uranus - Inferring heat sources and model atmospheres
Heat source information is derived here from the Voyager 2 Ultraviolet Spectrometer occultation data of Uranus. Analytic functions for the local heat dependence on altitude are used to obtain a temperature profile by solving the heat equation. The stellar entrance and exit occultation and a solar occultation are used to infer the thermal and density structure of the atmosphere. The least squares fit solution to the solar occultation data gives one source located at 1.8 x 10 exp -5 microbar with a strength of 0.056 +/- 0.01 erg/sq cm/s. Latitudinal temperature gradients are obtained.
Stevens, Michael H.↗
An improved gravity model for Mars: Goddard Mars Model-1 (GMM-1)
Doppler tracking data of three orbiting spacecraft have been reanalyzed to develop a new gravitational field model for the planet Mars, GMM-1 (Goddard Mars Model-1). This model employs nearly all available data, consisting of approximately 1100 days of S-bank tracking data collected by NASA's Deep Space Network from the Mariner 9, and Viking 1 and Viking 2 spacecraft, in seven different orbits, between 1971 and 1979. GMM-1 is complete to spherical harmonic degree and order 50, which corresponds to a half-wavelength spatial resolution of 200-300 km where the data permit. GMM-1 represents satellite orbits with considerably better accuracy than previous Mars gravity models and shows greater resolution of identifiable geological structures. The notable improvement in GMM-1 over previous models is a consequence of several factors: improved computational capabilities, the use of optimum weighting and least-squares collocation solution techniques which stabilized the behavior of the solution at high degree and order, and the use of longer satellite arcs than employed in previous solutions that were made possible by improved force and measurement models. The inclusion of X-band tracking data from the 379-km altitude, near-polar orbiting Mars Observer spacecraft should provide a significant improvement over GMM-1, particularly at high latitudes where current data poorly resolves the gravitational signature of the planet.
Smith, D. E.↗
An improved gravity model for Mars: Goddard Mars Model 1
Doppler tracking data of three orbiting spacecraft have been reanalyzed to develop a new gravitational field model for the planet Mars, Goddard Mars Model 1 (GMM-1). This model employs nearly all available data, consisting of approximately 1100 days of S band tracking data collected by NASA's Deep Space Network from the Mariner 9 and Viking 1 and Viking 2 spacecraft, in seven different orbits, between 1971 and 1979. GMM-1 is complete to spherical harmonic degree and order 50, which corresponds to a half-wavelength spatial resolution of 200-300 km where the data permit. GMM-1 represents satellite orbits with considerably better accuracy than previous Mars gravity models and shows greater resolution of identifiable geological structures. The notable improvement in GMM-1 over previous models is a consequence of several factors: improved computational capabilities, the use of otpimum weighting and least squares collocation solution techniques which stabilized the behavior of the solution at high degree and order, and the use of longer satellite arcs than employed in previous solutions that were made possible by improved force and measurement models. The inclusion of X band tracking data from the 379-km altitude, nnear-polar orbiting Mars Observer spacecraft should provide a significant improvement over GMM-1, particularly at high latitudes where current data poorly resolve the gravitational signature of the planet.
Smith, D. E.↗
Geopotential Model Improvement Using POCM_4B Dynamic Ocean Topography Information: PGM2000A
The two-year mean (1993-1994) Dynamic Ocean Topography (DOT) field implied by the POCM_4B circulation model was used to develop normal equations for DOT, in a surface spherical harmonic representation. These normal equations were combined with normal equations from satellite tracking data, surface gravity data, and altimeter data from TOPEX/Poseidon and ERS-1. Several least-squares combination solutions were developed in this fashion, by varying parameters such as the maximum degree of the estimated DOT and the relative weights of the different data. The solutions were evaluated in terms of orbit fit residuals, GPS/Leveling-derived undulations, and independent DOT information from in situ WOCE hydrographic data. An optimal solution was developed in this fashion which was originally presented at the 1998 EGS meeting in Nice, France. This model, designated here PGM2000A, maintains the orbit and land geoid modeling performance of EGM96, while improving its marine geoid modeling capability. In addition, PGM2000A's error spectrum is considerably more realistic than those of other contemporary gravitational models and agrees well with the error spectrum of EGM96. We will present the development and evaluation of PGM2000A, with particular emphasis on the weighting of the DOT information implied by POCM_4B. We will also present an inter-comparison of PGM2000A with the GRIM5-C1 and TEG-4 models. Directions for future work and problematic areas will be identified.
Pavlis, N. K.↗
The Lunar GNSS Receiver Experiment (LuGRE)
The Lunar GNSS Receiver Experiment (LuGRE) is a joint NASA-Italian Space Agency (ASI) payload on the Firefly Blue Ghost Mission 1 (BGM1) with the goal to demonstrate GNSS-based positioning, navigation, and timing at the Moon. LuGRE was chosen by the NASA Commercial Lunar Payload Services (CLPS) program as one of ten payloads on its “19D” task order for delivery to the lunar surface in 2023. The LuGRE payload consists of a weak-signal GNSS receiver, a high-gain L-band patch antenna, a low-noise amplifier, and an RF filter. The receiver will track GPS L1 C/A and L5, and Galileo E1 and E5a signals and will return pseudorange, carrier phase, and Doppler measurements to the ground. It will also calculate least-squares point solutions and Kalman-filter based navigation solutions onboard. In addition, the receiver features the capability to record raw I/Q baseband samples for downlink and ground processing. LuGRE will build on the legacy of prior missions in the Space Service Volume (SSV) including the initial experiments by AMSAT-OSCAR 40 and others, the GOES-R series of geostationary weather satellites, and the NASA Magnetospheric Multiscale (MMS) mission currently operating on GPS-based navigation at nearly 50% of lunar distance. Further, LuGRE will be one of the very first demonstrations of GNSS signal reception and navigation in the lunar environment and on the lunar surface, paving the way for operational use by future lunar missions such as Orion, Gateway, robotic and human landers, and surface rovers. Ultimately, all LuGRE science data will be released to a public data archive for the benefit of the GNSS and space communities. This paper provides a detailed overview of the LuGRE payload, including its design, concept of operations, and its predicted ability to meet its core science objectives. The baseline science investigations and priorities are outlined. Simulated performance results are shown based on the latest calibrated models including signal strength, signal availability, onboard navigation performance and convergence properties, and ground-based post-processed navigation performance.
LuGRE↗
Least squares collocation applied to local gravimetric solutions from satellite gravity gradiometry data
An autonomous spaceborne gravity gradiometer mission is being considered as a post Geopotential Research Mission project. The introduction of satellite diometry data to geodesy is expected to improve solid earth gravity models. The possibility of utilizing gradiometer data for the determination of pertinent gravimetric quantities on a local basis is explored. The analytical technique of least squares collocation is investigated for its usefulness in local solutions of this type. It is assumed, in the error analysis, that the vertical gravity gradient component of the gradient tensor is used as the raw data signal from which the corresponding reference gradients are removed to create the centered observations required in the collocation solution. The reference gradients are computed from a high degree and order geopotential model. The solution can be made in terms of mean or point gravity anomalies, height anomalies, or other useful gravimetric quantities depending on the choice of covariance types. Selected for this study were 30 x 30 foot mean gravity and height anomalies. Existing software and new software are utilized to implement the collocation technique. It was determined that satellite gradiometry data at an altitude of 200 km can be used successfully for the determination of 30 x 30 foot mean gravity anomalies to an accuracy of 9.2 mgal from this algorithm. It is shown that the resulting accuracy estimates are sensitive to gravity model coefficient uncertainties, data reduction assumptions and satellite mission parameters.
Robbins, J. W.↗
Improved solution accuracy for TDRSS-based TOPEX/Poseidon orbit determination
Orbit determination results are obtained by the Goddard Space Flight Center (GSFC) Flight Dynamics Division (FDD) using a batch-least-squares estimator available in the Goddard Trajectory Determination System (GTDS) and an extended Kalman filter estimation system to process Tracking and Data Relay Satellite (TDRS) System (TDRSS) measurements. GTDS is the operational orbit determination system used by the FDD in support of the Ocean Topography Experiment (TOPEX)/Poseidon spacecraft navigation and health and safety operations. The extended Kalman filter was implemented in an orbit determination analysis prototype system, closely related to the Real-Time Orbit Determination System/Enhanced (RTOD/E) system. In addition, the Precision Orbit Determination (POD) team within the GSFC Space Geodesy Branch generated an independent set of high-accuracy trajectories to support the TOPEX/Poseidon scientific data. These latter solutions use the geodynamics (GEODYN) orbit determination system with laser ranging and Doppler Orbitography and Radiopositioning integrated by satellite (DORIS) tracking measurements. The TOPEX/Poseidon trajectories were estimated for November 7 through November 11, 1992, the timeframe under study. Independent assessments were made of the consistencies of solutions produced by the batch and sequential methods. The batch-least-squares solutions were assessed based on the solution residuals, while the sequential solutions were assessed based on primarily the estimated covariances. The batch-least-squares and sequential orbit solutions were compared with the definitive POD orbit solutions. The solution differences were generally less than 2 meters for the batch-least-squares and less than 13 meters for the sequential estimation solutions. After the sequential estimation solutions were processed with a smoother algorithm, position differences with POD orbit solutions of less than 7 meters were obtained. The differences among the POD, GTDS, and filter/smoother solutions can be traced to differences in modeling and tracking data types, which are being analyzed in detail.
Doll, C. E.↗
Exact spacecraft detumbling and reorientation maneuvers with gimbaled thrusters and reaction wheels
The equations of rotational motion for a spacecraft equipped with external jets and internal reaction wheels are shown to be feedback-equivalent to those of a linear system in attitude parameter space. Reorientation maneuvers are thereby formulated as linear optimal control problems with least mean square acceleration in attitude parameter space, solved in closed form and implementable either with internal or external torque commands, the choice depending on power and throttling requirements. For prior detumbling, an alternative solution with least mean square torque by angular momentum feedback is also given, that is implementable with gimbaled pairs of thrusters at constant throttle. Such a detumbling maneuver may then be followed by an acceleration-commanded rest-to-rest maneuver by means of the reaction wheels.
Dwyer, T. A. W., III↗