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NASA NTRS2004

Deriving Laws from Ordering Relations

The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication among logical statements to degrees of implication, which are manipulated using rules derived from consistency with Boolean algebra. These rules are known as the sum rule, the product rule and Bayes Theorem, and the measure resulting from this generalization is probability. In this paper, I will describe how Cox s technique can be further generalized to include other algebras and hence other problems in science and mathematics. The result is a methodology that can be used to generalize an algebra to a calculus by relying on consistency with order theory to derive the laws of the calculus. My goals are to clear up the mysteries as to why the same basic structure found in probability theory appears in other contexts, to better understand the foundations of probability theory, and to extend these ideas to other areas by developing new mathematics and new physics. The relevance of this methodology will be demonstrated using examples from probability theory, number theory, geometry, information theory, and quantum mechanics.

Knuth, Kevin H.↗
NASA NTRS1984

The semi-Markov unreliability range evaluator program

The SURE program is a design/validation tool for ultrareliable computer system architectures. The system uses simple algebraic formulas to compute accurate upper and lower bounds for the death state probabilities of a large class of semi-Markov models. The mathematical formulas used in the program were derived from a mathematical theorem proven by Allan White under contract to NASA Langley Research Center. This mathematical theorem is discussed along with the user interface to the SURE program.

Butler, R. W.↗
NASA NTRS1985

Space Mathematics: A Resource for Secondary School Teachers

A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.

Kastner, Bernice↗
NASA NTRS1993

Semi-Markov Unreliability Range Evaluator

Semi-Markov Unreliability Range Evaluator, SURE, computer program is software tool for analysis of reconfigurable, fault-tolerant systems. Traditional reliability analyses based on aggregates of fault-handling and fault-occurrence models. SURE provides efficient means for calculating accurate upper and lower bounds for probabilities of death states for large class of semi-Markov mathematical models, and not merely those reduced to critical-pair architectures.

Butler, Ricky W.↗
NASA NTRS2004

The Investigation Of Carbon Contamination And Sputtering Effects Of Xenon Ion Thrusters

The Electro-Physics Branch of the NASA Glenn Research Center investigates the effect of atomic oxygen, environmental durability of high performance power materials and surfaces, and low earth orbit. One of its current projects involves the analysis of ion thrusters. Ion thrusters are devices that initiate a beam of ions to a target area. The type of ion thruster that I have been working with this Summer of 2004 emits positively charged Xenon (Xe(+)) atoms through two grids, the screen grid and the accelerator grid, after it enters an ionization chamber. Insulators are used to mechanically hold and separate these two grids. A propellant isolator, an instrument that closely resembles insulators, is placed in front of the ionization chamber. Both the insulator and isolator are made with a ceramic compound and filled with insulating beads. The main difference between the two devices is that the propellant isolator allows gas to flow through, in this case, the gas is Xe(+) and the insulators do not. In order to avoid carbon deposits and other contaminating chemicals to settle on the insulators and propellant isolator, a metal shadow shield is placed around them. These shadow shields function as a protectant and can be shaped in numerous configurations. Part of my job responsibility this summer is to investigate the effectiveness of different shadow shields that are utilized on three different ion engines: the NSTAR (NASA Solar Electric Propulsion Technology Application Readiness), JIMO (Jupiter Icy Moons Orbiter), and NEXIS (Nuclear Electric Xenon Ion System). Using calculus and other mathematical tactics, I was asked to find the total flux of carbon contamination that was able to pass the protectant shadow shield. I familiarized myself with the software program, MathCad2004, to help perform some mathematical computations such as complex integration. Another method of studying the probability of contamination is by experimental simulation. After attaining the precise parameters of the actual shadow shields, I created replicas of three types of shadow shielding to be used to undergo testing. It will be placed in a machine that produces carbon atoms at a high temperature of 200 C. or beam is aimed at a targeted material. As a result of this collision, atoms and other particles are ejected out of the target surface. Another part of my internship consisted of research on sputter ejection, or the angle distribution of sputtered material. This research entailed finding the past results of sputter ejection investigation as well as creating another type of mock simulation. Other minor projects include calculating the path of Xe(+) gas through the insulating beads of the isolators and assisting my mentor in collecting data for his paper for the Joint Propulsion Conference & Exhibit to be held July 11-14,2004 in Fort Lauderdale, Florida.

Prak, Moline K.↗
NASA NTRS2012

Model-Based Method for Sensor Validation

Fault detection, diagnosis, and prognosis are essential tasks in the operation of autonomous spacecraft, instruments, and in situ platforms. One of NASA s key mission requirements is robust state estimation. Sensing, using a wide range of sensors and sensor fusion approaches, plays a central role in robust state estimation, and there is a need to diagnose sensor failure as well as component failure. Sensor validation can be considered to be part of the larger effort of improving reliability and safety. The standard methods for solving the sensor validation problem are based on probabilistic analysis of the system, from which the method based on Bayesian networks is most popular. Therefore, these methods can only predict the most probable faulty sensors, which are subject to the initial probabilities defined for the failures. The method developed in this work is based on a model-based approach and provides the faulty sensors (if any), which can be logically inferred from the model of the system and the sensor readings (observations). The method is also more suitable for the systems when it is hard, or even impossible, to find the probability functions of the system. The method starts by a new mathematical description of the problem and develops a very efficient and systematic algorithm for its solution. The method builds on the concepts of analytical redundant relations (ARRs).

Vatan, Farrokh↗
NASA NTRS2024

Safety Assessment of Conformance Monitoring for Situational Awareness in UTM Operations

This report presents a systematic approach to evaluating the safety benefit of utilizing strategic deconfliction and Conformance Monitoring for Situational Awareness to manage the traffic that is comprised of both conforming aircraft and contingent aircraft. First, we developed a Monte-Carlo-based simulation platform that generates a range of nominal flight paths, a set of flight paths associated with contingent aircraft, encounters between conforming and contingent aircraft, and simulates aircraft's reported positions within conformance bounds. Next, we derived the mathematical equations for quantifying the level of safety, represented by the probability of mid-air collision per flight hour, using the flight data from the simulation platform. Finally, we compared the safety benefit of CMSA services in scenarios generated at different traffic densities and rates of contingent aircraft. This study suggests that strategic deconfliction and CMSA services can collaboratively handle traffic mixed with conforming and contingent aircraft more safely than strategic deconfliction alone.

CMSA↗
NASA NTRS2003

Self-Supervised Dynamical Systems

Some progress has been made in a continuing effort to develop mathematical models of the behaviors of multi-agent systems known in biology, economics, and sociology (e.g., systems ranging from single or a few biomolecules to many interacting higher organisms). Living systems can be characterized by nonlinear evolution of probability distributions over different possible choices of the next steps in their motions. One of the main challenges in mathematical modeling of living systems is to distinguish between random walks of purely physical origin (for instance, Brownian motions) and those of biological origin. Following a line of reasoning from prior research, it has been assumed, in the present development, that a biological random walk can be represented by a nonlinear mathematical model that represents coupled mental and motor dynamics incorporating the psychological concept of reflection or self-image. The nonlinear dynamics impart the lifelike ability to behave in ways and to exhibit patterns that depart from thermodynamic equilibrium. Reflection or self-image has traditionally been recognized as a basic element of intelligence. The nonlinear mathematical models of the present development are denoted self-supervised dynamical systems. They include (1) equations of classical dynamics, including random components caused by uncertainties in initial conditions and by Langevin forces, coupled with (2) the corresponding Liouville or Fokker-Planck equations that describe the evolutions of probability densities that represent the uncertainties. The coupling is effected by fictitious information-based forces, denoted supervising forces, composed of probability densities and functionals thereof. The equations of classical mechanics represent motor dynamics that is, dynamics in the traditional sense, signifying Newton s equations of motion. The evolution of the probability densities represents mental dynamics or self-image. Then the interaction between the physical and metal aspects of a monad is implemented by feedback from mental to motor dynamics, as represented by the aforementioned fictitious forces. This feedback is what makes the evolution of probability densities nonlinear. The deviation from linear evolution can be characterized, in a sense, as an expression of free will. It has been demonstrated that probability densities can approach prescribed attractors while exhibiting such patterns as shock waves, solitons, and chaos in probability space. The concept of self-supervised dynamical systems has been considered for application to diverse phenomena, including information-based neural networks, cooperation, competition, deception, games, and control of chaos. In addition, a formal similarity between the mathematical structures of self-supervised dynamical systems and of quantum-mechanical systems has been investigated.

Zak, Michail↗
NASA NTRS1999

Structural Reliability Analysis and Optimization: Use of Approximations

This report is intended for the demonstration of function approximation concepts and their applicability in reliability analysis and design. Particularly, approximations in the calculation of the safety index, failure probability and structural optimization (modification of design variables) are developed. With this scope in mind, extensive details on probability theory are avoided. Definitions relevant to the stated objectives have been taken from standard text books. The idea of function approximations is to minimize the repetitive use of computationally intensive calculations by replacing them with simpler closed-form equations, which could be nonlinear. Typically, the approximations provide good accuracy around the points where they are constructed, and they need to be periodically updated to extend their utility. There are approximations in calculating the failure probability of a limit state function. The first one, which is most commonly discussed, is how the limit state is approximated at the design point. Most of the time this could be a first-order Taylor series expansion, also known as the First Order Reliability Method (FORM), or a second-order Taylor series expansion (paraboloid), also known as the Second Order Reliability Method (SORM). From the computational procedure point of view, this step comes after the design point identification; however, the order of approximation for the probability of failure calculation is discussed first, and it is denoted by either FORM or SORM. The other approximation of interest is how the design point, or the most probable failure point (MPP), is identified. For iteratively finding this point, again the limit state is approximated. The accuracy and efficiency of the approximations make the search process quite practical for analysis intensive approaches such as the finite element methods; therefore, the crux of this research is to develop excellent approximations for MPP identification and also different approximations including the higher-order reliability methods (HORM) for representing the failure surface. This report is divided into several parts to emphasize different segments of the structural reliability analysis and design. Broadly, it consists of mathematical foundations, methods and applications. Chapter I discusses the fundamental definitions of the probability theory, which are mostly available in standard text books. Probability density function descriptions relevant to this work are addressed. In Chapter 2, the concept and utility of function approximation are discussed for a general application in engineering analysis. Various forms of function representations and the latest developments in nonlinear adaptive approximations are presented with comparison studies. Research work accomplished in reliability analysis is presented in Chapter 3. First, the definition of safety index and most probable point of failure are introduced. Efficient ways of computing the safety index with a fewer number of iterations is emphasized. In chapter 4, the probability of failure prediction is presented using first-order, second-order and higher-order methods. System reliability methods are discussed in chapter 5. Chapter 6 presents optimization techniques for the modification and redistribution of structural sizes for improving the structural reliability. The report also contains several appendices on probability parameters.

Grandhi, Ramana V.↗
NASA NTRS1977

Cost reduction potential of the DSN data base

The cost of the DSN data base can be reduced by computerizing and unifying the current multiplicity of separate manual, computer, and hybrid data bases. Savings would accrue from eliminating all manual system costs, increasing efficiency in data base implementation and maintenance-and-operation, and increasing data accuracy. By applying a simple mathematical savings model to current data base costs, this study estimates the probable range of net ten-year savings. The minimum net savings, under the assumptions of the study, are calculated as $7.5 million.

Mckenzie, M.↗
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 NTRS1993

The potential of space exploration for the fine arts

Art provides an integrating function between the 'upper' and 'lower' centers of the human psyche. The nature of this function can be made more specific through the triune model of the brain. The evolution of the fine arts - painting, drawing, architecture, sculpture, literature, music, dance, and drama, plus cinema and mathematics-as-a-fine-art - are examined in the context of their probable stimulations by space exploration: near term and long term.

Mclaughlin, William I.↗
NASA NTRS2017

Sensitivity Analysis of the Bone Fracture Risk Model

Introduction: The probability of bone fracture during and after spaceflight is quantified to aid in mission planning, to determine required astronaut fitness standards and training requirements and to inform countermeasure research and design. Probability is quantified with a probabilistic modeling approach where distributions of model parameter values, instead of single deterministic values, capture the parameter variability within the astronaut population and fracture predictions are probability distributions with a mean value and an associated uncertainty. Because of this uncertainty, the model in its current state cannot discern an effect of countermeasures on fracture probability, for example between use and non-use of bisphosphonates or between spaceflight exercise performed with the Advanced Resistive Exercise Device (ARED) or on devices prior to installation of ARED on the International Space Station. This is thought to be due to the inability to measure key contributors to bone strength, for example, geometry and volumetric distributions of bone mass, with areal bone mineral density (BMD) measurement techniques. To further the applicability of model, we performed a parameter sensitivity study aimed at identifying those parameter uncertainties that most effect the model forecasts in order to determine what areas of the model needed enhancements for reducing uncertainty. Methods: The bone fracture risk model (BFxRM), originally published in (Nelson et al) is a probabilistic model that can assess the risk of astronaut bone fracture. This is accomplished by utilizing biomechanical models to assess the applied loads; utilizing models of spaceflight BMD loss in at-risk skeletal locations; quantifying bone strength through a relationship between areal BMD and bone failure load; and relating fracture risk index (FRI), the ratio of applied load to bone strength, to fracture probability. There are many factors associated with these calculations including environmental factors, factors associated with the fall event, mass and anthropometric values of the astronaut, BMD characteristics, characteristics of the relationship between BMD and bone strength and bone fracture characteristics. The uncertainty in these factors is captured through the use of parameter distributions and the fracture predictions are probability distributions with a mean value and an associated uncertainty. To determine parameter sensitivity, a correlation coefficient is found between the sample set of each model parameter and the calculated fracture probabilities. Each parameters contribution to the variance is found by squaring the correlation coefficients, dividing by the sum of the squared correlation coefficients, and multiplying by 100. Results: Sensitivity analyses of BFxRM simulations of preflight, 0 days post-flight and 365 days post-flight falls onto the hip revealed a subset of the twelve factors within the model which cause the most variation in the fracture predictions. These factors include the spring constant used in the hip biomechanical model, the midpoint FRI parameter within the equation used to convert FRI to fracture probability and preflight BMD values. Future work: Plans are underway to update the BFxRM by incorporating bone strength information from finite element models (FEM) into the bone strength portion of the BFxRM. Also, FEM bone strength information along with fracture outcome data will be incorporated into the FRI to fracture probability.

mathematical models↗
NASA NTRS1989

The Fault Tree Compiler (FTC): Program and mathematics

The Fault Tree Compiler Program is a new reliability tool used to predict the top-event probability for a fault tree. Five different gate types are allowed in the fault tree: AND, OR, EXCLUSIVE OR, INVERT, AND m OF n gates. The high-level input language is easy to understand and use when describing the system tree. In addition, the use of the hierarchical fault tree capability can simplify the tree description and decrease program execution time. The current solution technique provides an answer precisely (within the limits of double precision floating point arithmetic) within a user specified number of digits accuracy. The user may vary one failure rate or failure probability over a range of values and plot the results for sensitivity analyses. The solution technique is implemented in FORTRAN; the remaining program code is implemented in Pascal. The program is written to run on a Digital Equipment Corporation (DEC) VAX computer with the VMS operation system.

Butler, Ricky W.↗
NASA NTRSDate not supplied

Multihierarchy Gaussian Process Models for Probabilistic Aerodynamic Databases using Uncertain Nominal and Off-Nominal Configuration Data

Probabilistic aerodynamic databases are a crucial component of the development lifecycle for aerospace vehicles. A key challenge when building aerodynamic databases is that most data used to construct them represent various simplifications of the real flight vehicle. For example, wind tunnel models often simplify the vehicle geometry and surface roughness characteristics, while CFD computations often make simplifications to the physics being modeled, such as fully laminar or turbulent calculations. Multifidelity data fusion models rely on a user being able to define a hierarchy of fidelity levels anchored to some "truth" data. This approach is unsatisfactory when no data can be considered to accurately reflect real flight conditions. In this work, we provide an alternative approach by presenting a consistent mathematical framework for building probabilistic aerodynamic databases in the form of a conditional probability distribution described by an ensemble of multifidelity Gaussian Processes. Instead of relying on a single hierarchy of data fidelity levels, the presented framework identifies a "nominal" configuration and potential corrections to the nominal which represent specific physical phenomena not represented in the nominal data. The nominal and correction functions themselves are constructed as multifidelity Gaussian Processes and linearly combined to form an ensemble model which fuses the uncertainties associated nominal and correction models. Results obtained using the proposed framework on a simplified Orion Crew Module wind tunnel dataset demonstrate the predictive capability of the multihierarchy framework. We further demonstrate the benefits of such a probabilistic aerodynamic database approach through function sampling and computing the conditional distributions of derived quantities, such as the trim angle of attack and aerodynamic coefficients at trim.

Gaussian Processes↗
NASA NTRSDate not supplied

Bias in Planning Algorithms

Does bias exist in planning algorithms? If so, how does bias manifest, and how important is this bias? Answering this question requires a formal, mathematical definition of bias. We formally define bias as the distance between the probability distributions of solutions returned by various algorithms, and the uniform distribution over solutions. We show in this paper that deterministic algorithms are inherently biased, as they don’t return all solutions, and that this property holds even when algorithms return a set of plans instead of just one plan. Exceptions are problem instances or problem classes for which only a single solution exists. We then discuss changing the definition of bias to compare the probability distributions of properties of sets of plans instead of individual plans. We show the property bias is smaller than the bias of actual plans. Finally, we show that entropy is a proxy for the more complex and more expensive distance measurement between pairs of probability distributions. We then describe a roadmap for future investigations of bias in planning.

Planning Scheduling Algorithms↗
NASA NTRS2003

Automated Monitoring with a BCP Fault-Decision Test

The Bayesian conditional probability (BCP) technique is a statistical fault-decision technique that is suitable as the mathematical basis of the fault-manager module in the automated-monitoring system and method described in the immediately preceding article. Within the automated-monitoring system, the fault-manager module operates in conjunction with the fault-detector module, which can be based on any one of several fault-detection techniques; examples include a threshold-limit-comparison technique or the BSP or SPRT technique mentioned in the preceding article. The present BCP technique is used to evaluate a series of one or more fault-detection events for the purpose of filtering out occasional false alarms produced by many types of statistical fault-detection procedures. The BCP technique increases the probability that an automated monitoring system produces a correct decision regarding the presence or absence of a fault. Because occasional false alarms are an inevitable consequence of the SPRT, BSP, or any other statistically based fault-detection test, there is a need for a logical procedure to distinguish between true and false alarms. Heretofore, it has been common practice to make a fault decision on an ad hoc basis for example by following a multiple-observation voting strategy in which a signal is declared to be indicative of a fault if m of the last n observations produced a fault-detection alarm. The BCP technique was developed to obtain results more reliable than those afforded by a voting strategy. The BCP technique involves a test in which one applies Bayesian inference techniques to a series of one or more single-observation alarms produced by a fault-detection test. One considers the last n decisions generated by a fault-detection test in order to evaluate the conditional probability that a failure is indicated (see figure). Each new decision reached by a fault-detection test is treated as a new piece of evidence about the state of the monitored asset, and the conditional probability of failure for the system is updated on the basis of this new evidence. The conditional probability of failure is compared with a predefined limit. For a probability below the limit, the asset is declared to be healthy. For a probability above the limit, the asset is declared to be faulty.

Bickford, Randall L.↗