Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “vector fitting”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRSDate not supplied

Application of Support Vector Regression to Derive Crater Depth/Diameter From Satellite Images

Through the study of impact crater shapes, one can draw important conclusions about the nature and evolution of planetary surfaces [e.g., 1-4].In particular, studying the depth (d) to diameter (D)ratio (d/D) of a population of impact craters, in combination with crater count statistics, can yield valuable insights regarding rates of erosion and burial[5]. Motivated by the great abundance of available planetary surface image data, the goal of this project is to develop an efficient way to estimate d/D from satellite images of impact craters for which stereo information is not available [6]. We set out to develop and train a machine learning algorithm to extract d/D from a dataset of synthetic impact crater images for which model d/D is known. The applications of machine learning to planetary science are numerous and diverse [7], including automatic planetary surface mapping [8] and the detection of impact craters [9]. Our algorithm makes use of Support Vector Regression (SVR), which is a type of Support Vector Machine (SVM) [10, 11].SVMs are a branch of supervised machine learning valued for their straightforward implementation and versatility in solving both classification and regression problems. In regression analysis, an SVR algorithm produces a hyperplane function to fit the training data points, as well as an ε-tube that surrounds the hyperplane. Tunable hyperparameters include the width of the ε-tube (ε) and the amount an algorithm is penalized for points which fall outside the ε-tube.

L R Chin↗
NASA NTRS1997

Response properties of pigeon otolith afferents to linear acceleration

In the present study, the sensitivity to sinusoidal linear accelerations in the plane of the utricular macula was tested in afferents. The head orientation relative to the translation axis was varied in order to determine the head position that elicited the maximal and minimal responses for each afferent. The response gain and phase values obtained to 0.5-Hz and 2-Hz linear acceleration stimuli were then plotted as a function of head orientation and a modified cosine function was fit to the data. From the best-fit cosine function, the predicted head orientations that would produce the maximal and minimal response gains were estimated. The estimated maximum response gains to linear acceleration in the utricular plane for the afferents varied between 75 and 1420 spikes s-1 g-1. The mean maximal gains for all afferents to 0.5-Hz and 2-Hz sinusoidal linear acceleration stimuli were 282 and 367 spikes s-1 g-1, respectively. The minimal response gains were essentially zero for most units. The response phases always led linear acceleration and remained constant for each afferent, regardless of head orientation. These response characteristics indicate that otolith afferents are cosine tuned and behave as one-dimensional linear accelerometers. The directions of maximal sensitivity to linear acceleration for the afferents varied throughout the plane of the utricle; however, most vectors were directed out of the opposite ear near the interaural axis. The response dynamics of the afferents were tested using stimulus frequencies ranging between 0.25 Hz and 10 Hz (0.1 g peak acceleration). Across stimulus frequencies, most afferents had increasing gains and constant phase values. These dynamic properties for individual afferents were fit with a simple transfer function that included three parameters: a mechanical time constant, a gain constant, and a fractional order distributed adaptation operator.

Non-NASA Center↗
NASA NTRS2015

Suomi-NPP VIIRS Solar Diffuser Stability Monitor Performance

When illuminated by the Sun, the onboard solar diffuser (SD) panel provides a known spectral radiance source to calibrate the reflective solar bands of the Visible Infrared Imaging Radiometer Suite on the Suomi-NPP satellite. The SD bidirectional reflectance distribution function (BRDF) degrades over time due to solar exposure, and this degradation is measured using the SD stability monitor (SDSM). The SDSM acts as a ratioing radiometer, comparing solar irradiance measurements off the SD panel to those from a direct Sun view. We discuss the design and operations of the SDSM, the SDSM data analysis, including improvements incorporated since launch, and present the results through 1000 days after launch. After 1000 days, the band-dependent H-factors, a quantity describing the relative degradation of the BRDF of the SD panel since launch, range from 0.716 at 412 nanometers to 0.989 at 926 nanometers. The random uncertainty of these H-factors is about 0.1 percent, which is confirmed by the similar standard deviation values computed from the residuals of quadratic exponential fits to the H-factor time trends. The SDSM detector gains have temperature sensitivity of up to about 0.36 percent per kelvin, but this does not affect the derived H-factors. An initial error in the solar vector caused a seasonal bias to the H-factors of up to 0.5 percent. The total exposure of the SD panel to UV light after 1000 orbits is equivalent to about 100 hours of direct sunlight illumination perpendicular to the SD panel surface.

Fulbright, Jon↗
NASA NTRS2008

Visual Data Analysis for Satellites

The Visual Data Analysis Package is a collection of programs and scripts that facilitate visual analysis of data available from NASA and NOAA satellites, as well as dropsonde, buoy, and conventional in-situ observations. The package features utilities for data extraction, data quality control, statistical analysis, and data visualization. The Hierarchical Data Format (HDF) satellite data extraction routines from NASA's Jet Propulsion Laboratory were customized for specific spatial coverage and file input/output. Statistical analysis includes the calculation of the relative error, the absolute error, and the root mean square error. Other capabilities include curve fitting through the data points to fill in missing data points between satellite passes or where clouds obscure satellite data. For data visualization, the software provides customizable Generic Mapping Tool (GMT) scripts to generate difference maps, scatter plots, line plots, vector plots, histograms, timeseries, and color fill images.

Lau, Yee↗
NASA NTRS2012

Comparison of PARASOL Observations with Polarized Reflectances Simulated Using Different Ice Habit Mixtures

Insufficient knowledge of the habit distribution and the degree of surface roughness of ice crystals within ice clouds is a source of uncertainty in the forward light scattering and radiative transfer simulations required in downstream applications involving these clouds. The widely used MODerate Resolution Imaging Spectroradiometer (MODIS) Collection 5 ice microphysical model assumes a mixture of various ice crystal shapes with smooth-facets except aggregates of columns for which a moderately rough condition is assumed. When compared with PARASOL (Polarization and Anisotropy of Reflectances for Atmospheric Sciences coupled with Observations from a Lidar) polarized reflection data, simulations of polarized reflectance using smooth particles show a poor fit to the measurements, whereas very rough-faceted particles provide an improved fit to the polarized reflectance. In this study a new microphysical model based on a mixture of 9 different ice crystal habits with severely roughened facets is developed. Simulated polarized reflectance using the new ice habit distribution is calculated using a vector adding-doubling radiative transfer model, and the simulations closely agree with the polarized reflectance observed by PARASOL. The new general habit mixture is also tested using a spherical albedo differences analysis, and surface roughening is found to improve the consistency of multi-angular observations. It is suggested that an ice model incorporating an ensemble of different habits with severely roughened surfaces would potentially be an adequate choice for global ice cloud retrievals.

Cole, Benjamin H.↗
NASA NTRS2016

Modeling in the State Flow Environment to Support Launch Vehicle Verification Testing for Mission and Fault Management Algorithms in the NASA Space Launch System

Analysis methods and testing processes are essential activities in the engineering development and verification of the National Aeronautics and Space Administration's (NASA) new Space Launch System (SLS). Central to mission success is reliable verification of the Mission and Fault Management (M&FM) algorithms for the SLS launch vehicle (LV) flight software. This is particularly difficult because M&FM algorithms integrate and operate LV subsystems, which consist of diverse forms of hardware and software themselves, with equally diverse integration from the engineering disciplines of LV subsystems. M&FM operation of SLS requires a changing mix of LV automation. During pre-launch the LV is primarily operated by the Kennedy Space Center (KSC) Ground Systems Development and Operations (GSDO) organization with some LV automation of time-critical functions, and much more autonomous LV operations during ascent that have crucial interactions with the Orion crew capsule, its astronauts, and with mission controllers at the Johnson Space Center. M&FM algorithms must perform all nominal mission commanding via the flight computer to control LV states from pre-launch through disposal and also address failure conditions by initiating autonomous or commanded aborts (crew capsule escape from the failing LV), redundancy management of failing subsystems and components, and safing actions to reduce or prevent threats to ground systems and crew. To address the criticality of the verification testing of these algorithms, the NASA M&FM team has utilized the State Flow environment6 (SFE) with its existing Vehicle Management End-to-End Testbed (VMET) platform which also hosts vendor-supplied physics-based LV subsystem models. The human-derived M&FM algorithms are designed and vetted in Integrated Development Teams composed of design and development disciplines such as Systems Engineering, Flight Software (FSW), Safety and Mission Assurance (S&MA) and major subsystems and vehicle elements such as Main Propulsion Systems (MPS), boosters, avionics, Guidance, Navigation, and Control (GN&C), Thrust Vector Control (TVC), liquid engines, and the astronaut crew office. Since the algorithms are realized using model-based engineering (MBE) methods from a hybrid of the Unified Modeling Language (UML) and Systems Modeling Language (SysML), SFE methods are a natural fit to provide an in depth analysis of the interactive behavior of these algorithms with the SLS LV subsystem models. For this, the M&FM algorithms and the SLS LV subsystem models are modeled using constructs provided by Matlab which also enables modeling of the accompanying interfaces providing greater flexibility for integrated testing and analysis, which helps forecast expected behavior in forward VMET integrated testing activities. In VMET, the M&FM algorithms are prototyped and implemented using the same C++ programming language and similar state machine architectural concepts used by the FSW group. Due to the interactive complexity of the algorithms, VMET testing thus far has verified all the individual M&FM subsystem algorithms with select subsystem vendor models but is steadily progressing to assessing the interactive behavior of these algorithms with LV subsystems, as represented by subsystem models. The novel SFE applications has proven to be useful for quick look analysis into early integrated system behavior and assessment of the M&FM algorithms with the modeled LV subsystems. This early MBE analysis generates vital insight into the integrated system behaviors, algorithm sensitivities, design issues, and has aided in the debugging of the M&FM algorithms well before full testing can begin in more expensive, higher fidelity but more arduous environments such as VMET, FSW testing, and the Systems Integration Lab7 (SIL). SFE has exhibited both expected and unexpected behaviors in nominal and off nominal test cases prior to full VMET testing. In many findings, these behavioral characteristics were used to correct the M&FM algorithms, enable better test coverage, and develop more effective test cases for each of the LV subsystems. This has improved the fidelity of testing and planning for the next generation of M&FM algorithms as the SLS program evolves from non-crewed to crewed flight, impacting subsystem configurations and the M&FM algorithms that control them. SFE analysis has improved robustness and reliability of the M&FM algorithms by revealing implementation errors and documentation inconsistencies. It is also improving planning efficiency for future VMET testing of the M&FM algorithms hosted in the LV flight computers, further reducing risk for the SLS launch infrastructure, the SLS LV, and most importantly the crew.

Trevino, Luis↗
NASA NTRS1994

Root-sum-square structural strength verification approach

Utilizing a proposed fixture design or some variation thereof, this report presents a verification approach to strength test space flight payload components, electronics boxes, mechanisms, lines, fittings, etc., which traditionally do not lend themselves to classical static loading. The fixture, through use of ordered Euler rotation angles derived herein, can be mounted on existing vibration shakers and can provide an innovative method of applying single axis flight load vectors. The versatile fixture effectively loads protoflight or prototype components in all three axes simultaneously by use of a sinusoidal burst of desired magnitude at less than one-third the first resonant frequency. Cost savings along with improved hardware confidence are shown. The end product is an efficient way to verify experiment hardware for both random vibration and strength.

Lee, Henry M.↗
NASA NTRS1997

Kinematic principles of primate rotational vestibulo-ocular reflex. II. Gravity-dependent modulation of primary eye position

The kinematic constraints of three-dimensional eye positions were investigated in rhesus monkeys during passive head and body rotations relative to gravity. We studied fast and slow phase components of the vestibulo-ocular reflex (VOR) elicited by constant-velocity yaw rotations and sinusoidal oscillations about an earth-horizontal axis. We found that the spatial orientation of both fast and slow phase eye positions could be described locally by a planar surface with torsional variation of <2.0 +/- 0.4 degrees (displacement planes) that systematically rotated and/or shifted relative to Listing's plane. In supine/prone positions, displacement planes pitched forward/backward; in left/right ear-down positions, displacement planes were parallel shifted along the positive/negative torsional axis. Dynamically changing primary eye positions were computed from displacement planes. Torsional and vertical components of primary eye position modulated as a sinusoidal function of head orientation in space. The torsional component was maximal in ear-down positions and approximately zero in supine/prone orientations. The opposite was observed for the vertical component. Modulation of the horizontal component of primary eye position exhibited a more complex dependence. In contrast to the torsional component, which was relatively independent of rotational speed, modulation of the vertical and horizontal components of primary position depended strongly on the speed of head rotation (i.e., on the frequency of oscillation of the gravity vector component): the faster the head rotated relative to gravity, the larger was the modulation. Corresponding results were obtained when a model based on a sinusoidal dependence of instantaneous displacement planes (and primary eye position) on head orientation relative to gravity was fitted to VOR fast phase positions. When VOR fast phase positions were expressed relative to primary eye position estimated from the model fits, they were confined approximately to a single plane with a small torsional standard deviation ( approximately 1.4-2.6 degrees). This reduced torsional variation was in contrast to the large torsional spread (well >10-15 degrees ) of fast phase positions when expressed relative to Listing's plane. We conclude that primary eye position depends dynamically on head orientation relative to space rather than being fixed to the head. It defines a gravity-dependent coordinate system relative to which the torsional variability of eye positions is minimized even when the head is moved passively and vestibulo-ocular reflexes are evoked. In this general sense, Listing's law is preserved with respect to an otolith-controlled reference system that is defined dynamically by gravity.

NASA Discipline Neuroscience↗
NASA NTRS1973

Coordinated observations of chemical releases from the ground and from aircraft at high latitudes

The ground observations of the Na-Li trail released from a Nike-Apache rocket obtained by the Geophysical Institute are discussed. By using the nominal trajectory for a 60 pound payload and the particular rocket, a best fit trajectory was determined based on the Ester Dome photographic data, launch time and earth-sun geometrical shadow height. From these calculations, the height of obvious features along the trail were determined and their velocity estimated. A clockwise rotation of the velocity vector with increasing height was observed. Velocities deduced at various altitudes were then compared to meter radar data also obtained during this period. The comparisons of these two neutral wind measurements techniques are satisfactory.

Romick, G. J.↗
NASA NTRS1996

The alpha Centauri Line of Sight: D/H Ratio, Physical Properties of Local Interstellar Gas, and Measurement of Heated Hydrogen (The 'Hydrogen Wall') Near the Heliopause

We analyze high-resolution spectra of the nearby (1.34 pc) stars alpha Cen A (G2 V) and alpha Cen B (K1 V), which were obtained with the Goddard High Resolution Spectrograph on the Hubble Space Telescope. The observations consist of echelle spectra of the Mg II 2800 A and Fe II 2599 A resonance lines and the Lyman-alpha lines of hydrogen and deuterium. The interstellar gas has a velocity (v = - 18.0 +/- 0.2 km/s) consistent with the local flow vector proposed for this line of sight by Lailement & Berlin (1992). The temperature and nonthermal velocity inferred from the Fe II, Mg II, and D I line profiles are T = 5400 +/- 500 K and xi = 1.20 +/- 0.25 km/s, respectively. However, single-component fits to the H I Lyman-alpha lines yield a Doppler parameter (b(sub HI) = 11.80 km/s) that implies a significantly warmer temperature of 8350 K, and the velocity of the H I absorption (v = - 15.8 +/- 0.2 km/s) is redshifted by about 2.2 km/s with respect to the Fe II, Mg II, and D I lines. The one-component model of the interstellar gas suggests natural logarithm N base HI = 18.03 +/- 0.01 and D/H = (5.7 +/- 0.2) x 10(exp -6) . These parameters lead to a good fit to the observed spectra, but this model does not explain the higher temperature and redshift of H I relative to the other interstellar lines. The most sensible way to resolve the discrepancy between H(I) and the other lines is to add a second absorption component to the H(I) lines. This second component is hotter (T approx. equals 30,000 K), is redshifted relative to the primary component by 2-4 km/s, and has a column density too low to be detected in the Fe(II), Mg(II), and D(I) lines. We propose that the gas responsible for this component is located near the heliopause, consisting of the heated H I gas from the interstellar medium that is compressed by the solar wind. This so-called 'hydrogen wall' is predicted by recent multifluid gasdynamical models of the interstellar gas and solar wind interaction. Our data provide the first measurements of the temperature and column density of H(I) in the hydrogen wall. After considering the effects that a corresponding hydrogen wall around alpha Cen would have on our analysis, our best estimates for the parameters of the solar hydrogen wall are natural log N(sup (2))(H(I)) = 14.74 +/- 0.24, b(sup (2))(H(I)) = 21.9 +/- 1.7 km/s (corresponding to T = 29,000 +/- 5000 K), and v(sup (2))(H(I)) greater than -16km/s. Unfortunately, the existence of this heated H(I) reduces our ability to compute the H(I) column density of the interstellar medium accurately because, with slight alterations to our assumed stellar Lyman-alpha profiles, we discovered that acceptable two-component fits also exist with natural log N(H(I))approx. 17.6. We, therefore, quote large error bars for the H I column density along the alpha Cen line of sight, natural log N(H(I)) = 17.80 +/- 0.30. For this range in N(H(I)), n(H(I)) = 0.15 /cu.cm (+/- a factor of 2) and D/H = (0.5-1.9) x 10(exp -5). This is the first direct measurement of the H(I) density in a local cloud and allows us to predict the distance from the Sun to the edge of the local cloud along various lines of sight. This range in D/H is consistent with the value D/H = 1.6 x 10(exp -5) previously derived for the Capella and Procyon lines of sight. We cannot tell whether D/H ratio varies or is constant in the local interstellar medium, but we do find that the D(I)/Mg(II) ratio for the alpha Cen line of sight is about 4 times smaller than for the Capella and Procyon lines of sight. Therefore, either D/H or the Mg depletion varies significantly over distance scales of only a few parsecs.

Linsky, Jeffrey L.↗
NASA NTRS1999

Novel, Miniature Multi-Hole Probes and High-Accuracy Calibration Algorithms for their use in Compressible Flowfields

Two new calibration algorithms were developed for the calibration of non-nulling multi-hole probes in compressible, subsonic flowfields. The reduction algorithms are robust and able to reduce data from any multi-hole probe inserted into any subsonic flowfield to generate very accurate predictions of the velocity vector, flow direction, total pressure and static pressure. One of the algorithms PROBENET is based on the theory of neural networks, while the other is of a more conventional nature (polynomial approximation technique) and introduces a novel idea of local least-squares fits. Both algorithms have been developed to complete, user-friendly software packages. New technology was developed for the fabrication of miniature multi-hole probes, with probe tip diameters all the way down to 0.035". Several miniature 5- and 7-hole probes, with different probe tip geometries (hemispherical, conical, faceted) and different overall shapes (straight, cobra, elbow probes) were fabricated, calibrated and tested. Emphasis was placed on the development of four stainless-steel conical 7-hole probes, 1/16" in diameter calibrated at NASA Langley for the entire subsonic regime. The developed calibration algorithms were extensively tested with these probes demonstrating excellent prediction capabilities. The probes were used in the "trap wing" wind tunnel tests in the 14'x22' wind tunnel at NASA Langley, providing valuable information on the flowfield over the wing. This report is organized in the following fashion. It consists of a "Technical Achievements" section that summarizes the major achievements, followed by an assembly of journal articles that were produced from this project and ends with two manuals for the two probe calibration algorithms developed.

Rediniotis, Othon K.↗
NASA NTRS1978

Numerical solution of the Navier-Stokes equations for arbitrary blunt bodies in supersonic flows

A time-dependent, two-dimensional Navier-Stokes code employing the body-fitted coordinate technique has been developed for supersonic flows past blunt bodies of arbitrary shape. The computer program is based on the finite-difference approximation of the compressible Navier-Stokes equations transformed to nonorthogonal curvilinear coordinates with the contravariant components of the velocity vector as dependent variables. The bow shock ahead of the body is obtained as part of the solution, by 'shock capturing'. Numerical solutions of the complete equations are presented in detail for free-stream Mach number 4.6, Reynolds number 10,000, and an isothermal wall temperature of 556 K for a circular cylinder with the free-stream outer boundaries forming a hyperbola in the front and a circular arc in the back.

Warsi, Z. U. A.↗
NASA NTRS1980

Design of a recursive vector processor using polynomial splines

The problem of obtaining smoothed estimates of function values, particularly their derivatives, from a finite set of inaccurate measurements is considered. A recursive two-dimensional vector processor is introduced as an approximation to the nonrecursive constrained least-squares estimation. Here, piecewise bicubic Hermite polynomials are extensively used as approximating functions, and the smoothing integral is converted to a discrete quadratic form. This makes it possible to convert the problem of fitting an approximating function to one of estimating the function values and derivatives at the nodes.

Kim, C. S.↗
NASA NTRS1983

Intensity dependence of the pulse profile and polarization of the Vela pulsar

Polarization measurements of the Vela pulsar PSR 0833-45 at 2295 MHz revealed that the radio pulse profile is composed of four components. The four component configuration was found to fit the observed dependence of the pulse profile on the intensity of the pulsar, and the dependence of the modulation index and power spectrum of the intensity fluctuations on the pulse-longitude. Deviations of the observed polarization position angles from those predicted by a rotating vector model were explained by the emission regions for each component being located at different distances from the star. The components are affected by retardation, aberration, and magnetic field curvature effects, all of which contribute to arrival of the components at different longitudes even though the components actually carry the position angle signature predicted by the rotating vector model.

Krishnamohan, S.↗
NASA NTRS1985

Development of an explicit multigrid algorithm for quasi-three-dimensional viscous flows in turbo-machinery

A rapid quasi three-dimensional analysis was developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations which are solved by a two-stage Runge-Kutta scheme made efficient by use of vectorization, a variable time-step, and a flux-based multigrid scheme, are described. A stability analysis is presented for the two-stage scheme. Results for a flat-plate model problem show the applicability of the method to axial, radial, and rotating geometries. Results for a centrifugal impeller and a radial diffuser show that the quasi three-dimensional viscous analysis can be a practical design tool.

Chima, R. V.↗
NASA NTRS1986

Development of an explicit multigrid algorithm for quasi-three-dimensional viscous flows in turbomachinery

A rapid quasi three-dimensional analysis was developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations which are solved b a two-stage Runge-Kutta scheme made efficient by use of vectorization, a variable time-step, and a flux-based multigrid scheme, are described. A stability analysis is presented for the two-stage. Results for a flat-plate model problem show the applicability of the method to axial, radial, and rotating geometries. Results for a centrifugal impeller and a radial diffuser show that the quasi three-dimensional viscous analysis can be a practical design tool.

Chima, R. V.↗
NASA NTRS1996

Evaluation of IGS Orbits with Satellite Laser Ranging

The accuracy with which orbits for the Global Positioning System (GPS) spacecraft, can be computed directly affects the accuracy of the resulting site coordinates and polar motion. Several groups routinely analyze GPS ground tracking data to compute precise orbits and terrestrial reference frame solutions. In this paper, we infer the accuracy of the orbits of two of the GPS satellites by comparing to independent laser ranges of subcentimeter accuracy obtained by a small but reasonably well distributed network of tracking sites. We find that all seven International GPS Service for Geodynamics (IGS) analysis centers achieve range residual root mean square (rms) errors at or below the 100 mm level. The best orbit solutions, from JPL, CODE, and the IGS combined product, yield a residual rms of about 50 mm. These residuals are consistent with three dimensional orbit errors of less than 150 mm. Estimating yaw rates for the spacecraft during shadow events, and using these estimates to compute the laser residual, significantly improves the fit. A small mean residual value of -15 to -30 mm seems to exist for most centers and laser sites which is not fully explained at present, but may be due to uncertainties in the corrections to the laser data, such as the reflector to spacecraft center of mass vector or small reference frame differences between the SLR sites and the GPS orbits.

Watkins, M. M.↗
NASA NTRS1975

Time-controlled descent guidance in uncertain winds

A procedure has been developed for constructing a statistical model of the altitude-dependent mean wind profile from the historical record of wind measurements at particular locations. The model is constructed by fitting a Markov process, with altitude as the stage variable, to the historical wind data. The wind model, together with the aircraft dynamics and the error characteristics of the navigation system, are incorporated in the design of a state estimator, which gives the minimum variance estimate of the aircraft state and the wind vector. The state and wind estimates are used as inputs to a linear feedback law for guiding the aircraft along the nominal trajectory. An example design of a time-constrained (4D RNAV) descent guidance system is presented, showing tracking accuracy, control activity, and probability of arrival time with and without the wind estimator.

Menga, G.↗