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NASA NTRS1988

Dispositional logic

The applicability of conventional mathematical analysis (based on the combination of two-valued logic and probability theory) to problems in which human judgment, perception, or emotions play significant roles is considered theoretically. It is shown that dispositional logic, a branch of fuzzy logic, has particular relevance to the common-sense reasoning typical of human decision-making. The concepts of dispositionality and usuality are defined analytically, and a dispositional conjunctive rule and dispositional modus ponens are derived.

Le Balleur, J. C.↗

NASA NTRS1991

Analysis Of Lock Detection In Costas Loops

Report presents analysis of detection of phase lock in Costas loops, used in coherent binary-phase-shift-keying communication systems to track both subcarrier and suppressed carrier signals. Detection of phase lock important part of operation and monitoring of operation of Costas or other tracking loop, provides insight into behavior of loop in real time. Focuses on effects of phase jitter and correlation between samples of phase error in all-digital Costas loop, in which lock detection implemented via algorithm. Applicable to both sinusoidal and square-law carrier signals, incorporates new mathematical models of square-law and absolute-value detectors.

Mileant, Alexander↗

NASA NTRS1993

The complex structured singular value

A tutorial introduction to the complex structured singular value (mu) is presented, with an emphasis on the mathematical aspects of mu. The mu-based methods discussed here have been useful for analyzing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using linear fractional transformations which naturally unify the frequency-domain and state space methods.

Packard, A.↗

NASA NTRS1994

Real-time sensor data validation

This report describes the status of an on-going effort to develop software capable of detecting sensor failures on rocket engines in real time. This software could be used in a rocket engine controller to prevent the erroneous shutdown of an engine due to sensor failures which would otherwise be interpreted as engine failures by the control software. The approach taken combines analytical redundancy with Bayesian belief networks to provide a solution which has well defined real-time characteristics and well-defined error rates. Analytical redundancy is a technique in which a sensor's value is predicted by using values from other sensors and known or empirically derived mathematical relations. A set of sensors and a set of relations among them form a network of cross-checks which can be used to periodically validate all of the sensors in the network. Bayesian belief networks provide a method of determining if each of the sensors in the network is valid, given the results of the cross-checks. This approach has been successfully demonstrated on the Technology Test Bed Engine at the NASA Marshall Space Flight Center. Current efforts are focused on extending the system to provide a validation capability for 100 sensors on the Space Shuttle Main Engine.

Bickmore, Timothy W.↗

NASA NTRS2008

A New Bond Albedo for Performing Orbital Debris Brightness to Size Transformations

We have developed a technique for estimating the intrinsic size distribution of orbital debris objects via optical measurements alone. The process is predicated on the empirically observed power-law size distribution of debris (as indicated by radar RCS measurements) and the log-normal probability distribution of optical albedos as ascertained from phase (Lambertian) and range-corrected telescopic brightness measurements. Since the observed distribution of optical brightness is the product integral of the size distribution of the parent [debris] population with the albedo probability distribution, it is a straightforward matter to transform a given distribution of optical brightness back to a size distribution by the appropriate choice of a single albedo value. This is true because the integration of a powerlaw with a log-normal distribution (Fredholm Integral of the First Kind) yields a Gaussian-blurred power-law distribution with identical power-law exponent. Application of a single albedo to this distribution recovers a simple power-law [in size] which is linearly offset from the original distribution by a constant whose value depends on the choice of the albedo. Significantly, there exists a unique Bond albedo which, when applied to an observed brightness distribution, yields zero offset and therefore recovers the original size distribution. For physically realistic powerlaws of negative slope, the proper choice of albedo recovers the parent size distribution by compensating for the observational bias caused by the large number of small objects that appear anomalously large (bright) - and thereby skew the small population upward by rising above the detection threshold - and the lower number of large objects that appear anomalously small (dim). Based on this comprehensive analysis, a value of 0.13 should be applied to all orbital debris albedo-based brightness-to-size transformations regardless of data source. Its prima fascia genesis, derived and constructed from the current RCS to size conversion methodology (SiBAM Size-Based Estimation Model) and optical data reduction standards, assures consistency in application with the prior canonical value of 0.1. Herein we present the empirical and mathematical arguments for this approach and by example apply it to a comprehensive set of photometric data acquired via NASA's Liquid Mirror Telescopes during the 2000-2001 observing season.

Mulrooney, Mark K.↗

NASA NTRS2010

Spatial Standard Observer

The present invention relates to devices and methods for the measurement and/or for the specification of the perceptual intensity of a visual image. or the perceptual distance between a pair of images. Grayscale test and reference images are processed to produce test and reference luminance images. A luminance filter function is convolved with the reference luminance image to produce a local mean luminance reference image . Test and reference contrast images are produced from the local mean luminance reference image and the test and reference luminance images respectively, followed by application of a contrast sensitivity filter. The resulting images are combined according to mathematical prescriptions to produce a Just Noticeable Difference, JND value, indicative of a Spatial Standard Observer. SSO. Some embodiments include masking functions. window functions. special treatment for images lying on or near border and pre-processing of test images.

Watson, Andrw B.↗

NASA NTRS2012

Spatial Standard Observer

The present invention relates to devices and methods for the measurement and/or for the specification of the perceptual intensity of a visual image, or the perceptual distance between a pair of images. Grayscale test and reference images are processed to produce test and reference luminance images. A luminance filter function is convolved with the reference luminance image to produce a local mean luminance reference image. Test and reference contrast images are produced from the local mean luminance reference image and the test and reference luminance images respectively, followed by application of a contrast sensitivity filter. The resulting images are combined according to mathematical prescriptions to produce a Just Noticeable Difference, JND value, indicative of a Spatial Standard Observer, SSO. Some embodiments include masking functions, window functions, special treatment for images lying on or near borders and pre-processing of test images.

Watson, Andrew B.↗

NASA NTRS1983

Multivalued-Velocity-Field Model of Turbulence

Report presents multi-valued-velocity model used in calculations of turbulence formation. Developed as part of continuing study of turbulent fluid motion, model expected to evolve into comprehensive mathematical tool to explain origin and effects of turbulence. Model has great theoretical and practical value in such fields as aerodynamics, meteorology, and combustion.

Zak, M. A.↗

NASA NTRS2003

Analytic Solution to the Problem of Aircraft Electric Field Mill Calibration

It is by no means a simple task to retrieve storm electric fields from an aircraft instrumented with electric field mill sensors. The presence of the aircraft distorts the ambient field in a complicated way. Before retrievals of the storm field can be made, the field mill measurement system must be "calibrated". In other words, a relationship between impressed (i.e., ambient) electric field and mill output must be established. If this relationship can be determined, it is mathematically inverted so that ambient field can be inferred from the mill outputs. Previous studies have primarily focused on linear theories where the relationship between ambient field and mill output is described by a "calibration matrix" M. Each element of the matrix describes how a particular component of the ambient field is enhanced by the aircraft. For example the product M(sub ix), E(sub x), is the contribution of the E(sub x) field to the i(th) mill output. Similarly, net aircraft charge (described by a "charge field component" E(sub q)) contributes an amount M(sub iq)E(sub q) to the output of the i(th) sensor. The central difficulty in obtaining M stems from the fact that the impressed field (E(sub x), E(sub y), E(sub z), E(sub q) is not known but is instead estimated. Typically, the aircraft is flown through a series of roll and pitch maneuvers in fair weather, and the values of the fair weather field and aircraft charge are estimated at each point along the aircraft trajectory. These initial estimates are often highly inadequate, but several investigators have improved the estimates by implementing various (ad hoc) iterative methods. Unfortunately, none of the iterative methods guarantee absolute convergence to correct values (i.e., absolute convergence to correct values has not been rigorously proven). In this work, the mathematical problem is solved directly by analytic means. For m mills installed on an arbitrary aircraft, it is shown that it is possible to solve for a single 2m-vector that provides all other needed variables (i.e., the unknown fair weather field, the unknown aircraft charge, and the unknown matrix M). Numerical tests of the solution, effects of measurement errors, and studies of solution non-uniqueness are ongoing as of this writing.

Koshak, William↗

NASA NTRS1972

An innovative approach to compensator design

The primary goal is to present for a control system a computer-aided-compensator design technique from a frequency domain point of view. The thesis for developing this technique is to describe the open loop frequency response by n discrete frequency points which result in n functions of the compensator coefficients. Several of these functions are chosen so that the system specifications are properly portrayed; then mathematical programming is used to improve all of these functions which have values below minimum standards. In order to do this several definitions in regard to measuring the performance of a system in the frequency domain are given. Next, theorems which govern the number of compensator coefficients necessary to make improvements in a certain number of functions are proved. After this a mathematical programming tool for aiding in the solution of the problem is developed. Then for applying the constraint improvement algorithm generalized gradients for the constraints are derived. Finally, the necessary theory is incorporated in a computer program called CIP (compensator improvement program).

Mitchell, J. R.↗

NASA NTRS1973

An innovative approach to compensator design

The design is considered of a computer-aided-compensator for a control system from a frequency domain point of view. The design technique developed is based on describing the open loop frequency response by n discrete frequency points which result in n functions of the compensator coefficients. Several of these functions are chosen so that the system specifications are properly portrayed; then mathematical programming is used to improve all of these functions which have values below minimum standards. To do this, several definitions in regard to measuring the performance of a system in the frequency domain are given, e.g., relative stability, relative attenuation, proper phasing, etc. Next, theorems which govern the number of compensator coefficients necessary to make improvements in a certain number of functions are proved. After this a mathematical programming tool for aiding in the solution of the problem is developed. This tool is called the constraint improvement algorithm. Then for applying the constraint improvement algorithm generalized, gradients for the constraints are derived. Finally, the necessary theory is incorporated in a Computer program called CIP (compensator Improvement Program). The practical usefulness of CIP is demonstrated by two large system examples.

Mitchell, J. R.↗

NASA NTRS1972

Aerodynamic parameters of the Navion airplane extracted from flight

An iterative method, which is characterized as a maximum-likelihood minimum-variance technique, was used to extract the aerodynamic parameters of a Navion airplane from flight data. The purposes were to compare the results with parameters obtained from wind-tunnel tests and with results obtained by analog matching the same data, and to develop techniques for application of the parameter extraction program. Results from the study showed that the parameter-extraction program can produce aerodynamic parameters which will permit close estimation of the aircraft time histories used in the extraction process. The program determined an estimate of the standard deviations of the states and parameters. These estimates were used to indicate how well the calculated states fit the flight data and the confidence in the values of the estimated parameters. The study also showed that the values of the parameters were affected by the data and mathematical model used during the extraction process. Because of the lack of confidence in the parameters extracted by use of some of the sets of data, several parameters were estimated by other methods. By using a combination of methods, a set of parameters which gave a fit to the data was obtained.

Suit, W. T.↗

NASA NTRS1943

Periodic Heat Transfer at Small Pressure Fluctuations

The effect of cyclic gas pressure variations on the periodic heat transfer at a flat wall is theoretically analyzed and the differential equation describing the process and its solution for relatively. Small pressure fluctuations developed, thus explaining the periodic heat cycle between gas and wall surface. The processes for pure harmonic pressure and temperature oscillations, respectively, in the gas space are described by means of a constant heat transfer coefficient and the equally constant phase angle between the appearance of the maximum values of the pressure and heat flow most conveniently expressed mathematically in the form of a complex heat transfer coefficient. Any cyclic pressure oscillations, can be reduced by Fourier analysis to harmonic oscillations, which result in specific, mutual relationships of heat-transfer coefficients and phase angles for the different harmonics.

Pfriem, H.↗

NASA NTRS2010

Inductive Non-Contact Position Sensor

Optical hardware has been developed to measure the depth of defects in the Space Shuttle Orbiter's windows. In this hardware, a mirror is translated such that its position corresponds to the defect's depth, so the depth measurement problem is transferred to a mirror-position measurement problem. This is preferable because the mirror is internal to the optical system and thus accessible. Based on requirements supplied by the window inspectors, the depth of the defects needs to be measured over a range of 200 microns with a resolution of about 100 nm and an accuracy of about 400 nm. These same requirements then apply to measuring the position of the mirror, and in addition, since this is a scanning system, a response time of about 10 ms is needed. A market search was conducted and no sensor that met these requirements that also fit into the available housing volume (less than one cubic inch) was found, so a novel sensor configuration was constructed to meet the requirements. This new sensor generates a nearly linearly varying magnetic field over a small region of space, which can easily be sampled, resulting in a voltage proportional to position. Experiments were done with a range of inductor values, drive voltages, drive frequencies, and inductor shapes. A rough mathematical model was developed for the device that, in most aspects, describes how it operates and what electrical parameters should be chosen for best performance. The final configuration met all the requirements, yielding a small rugged sensor that was easy to use and had nanometer resolution over more than the 200-micron range required. The inductive position sensor is a compact device (potentially as small as 2 cubic centimeters), which offers nanometer-position resolution over a demonstrated range of nearly 1 mm. One of its advantages is the simplicity of its electrical design. Also, the sensor resolution is nearly uniform across its operational range, which is in contrast to eddy current and capacitive sensors whose sensitivity is dependent upon position.

Youngquist, Robert↗

NASA NTRS1974

Evaluation of importance of lateral acceleration derivatives in extraction of lateral-directional derivatives at high angles of attack

A theoretical investigation was conducted to determine the importance of the lateral acceleration (beta) derivatives in the extraction of lateral-directional stability derivatives for swept wing airplanes at high angles of attack. Representative values of lateral acceleration derivatives in yaw and roll (Cn beta and Cl beta) were used in a computer program to generate representative flight motions at several angles of attack and altitudes. The computer-generated motions were then subjected to a parameter identification process based on a modified Newton-Raphson method. Two identification techniques were evaluated, one which included the beta derivatives and one which neglected them. The results of the study indicate that omission of the beta derivatives from mathematical models used in the derivative-extraction techniques can produce erroneous values for the lateral-directional stability derivatives particularly at high angles of attack, where the beta derivatives are large. The largest errors occur in the dynamic derivatives, but large errors may also occur in the static derivatives for cases in which the beta derivatives have large effects on the flight motions of the airplane. In addition, the resulting identified mathematical models provide poor motion prediction as well as erroneous predictions of dynamic modal characteristics. These results strongly indicate that the effects of beta derivatives should be considered in any attempt to extract lateral-directional aerodynamic parameters at high angles of attack.

Nguyen, L. T.↗

NASA NTRS1986

Evaluation of a nonlinear parameter extraction mathematical model including the term C(subm(sub delta e squared))

Shuttle flight test data were used to determine values for the short-period parameters. The best identified, as judged by its estimated standard deviation, was the elevon effectiveness parameter C (sub m (sub sigma e squared)). However, the scatter about the preflight prediction of C (sub m (sub sigma e squared)) was large. Other investigators have suggested that adding nonlinear terms to the mathematical model used to identify C (sub m (sub sigma e)) could reduce the scatter. The results of this investigation show that C (sub m (sub sigma e squared)) is the only identifiable nonlinear parameter applicable and that the changes in C (sub m (sub sigma e)) values when C (sub m (sub sigma e squared)) is included are in the order of ten percent for the data estimated.

Suit, W. T.↗

NASA NTRS1998

Towards a Theory for Integration of Mathematical Verification and Empirical Testing

From the viewpoint of a project manager responsible for the V&V (verification and validation) of a software system, mathematical verification techniques provide a possibly useful orthogonal dimension to otherwise standard empirical testing. However, the value they add to an empirical testing regime both in terms of coverage and in fault detection has been difficult to quantify. Furthermore, potential cost savings from replacing testing with mathematical verification techniques cannot be realized until the tradeoffs and synergies can be formulated. Integration of formal verification with empirical testing is also difficult because the idealized view of mathematical verification providing a correctness proof with total coverage is unrealistic and does not reflect the limitations imposed by computational complexity of mathematical techniques. This paper first describes a framework based on software reliability and formalized fault models for a theory of software design fault detection - and hence the utility of various tools for debugging. It then describes a utility model for integrating mathematical and empirical techniques with respect to fault detection and coverage analysis. It then considers the optimal combination of black-box testing, white-box (structural) testing, and formal methods in V&V of a software system. Using case studies from NASA software systems, it then demonstrates how this utility model can be used in practice.

Lowry, Michael↗

NASA NTRS1988

Two dimensional convolute integers for machine vision and image recognition

Machine vision and image recognition require sophisticated image processing prior to the application of Artificial Intelligence. Two Dimensional Convolute Integer Technology is an innovative mathematical approach for addressing machine vision and image recognition. This new technology generates a family of digital operators for addressing optical images and related two dimensional data sets. The operators are regression generated, integer valued, zero phase shifting, convoluting, frequency sensitive, two dimensional low pass, high pass and band pass filters that are mathematically equivalent to surface fitted partial derivatives. These operators are applied non-recursively either as classical convolutions (replacement point values), interstitial point generators (bandwidth broadening or resolution enhancement), or as missing value calculators (compensation for dead array element values). These operators show frequency sensitive feature selection scale invariant properties. Such tasks as boundary/edge enhancement and noise or small size pixel disturbance removal can readily be accomplished. For feature selection tight band pass operators are essential. Results from test cases are given.

Edwards, Thomas R.↗