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NASA NTRS1994

A simple suboptimal least-squares algorithm for attitude determination with multiple sensors

Three-axis attitude determination is equivalent to finding a coordinate transformation matrix which transforms a set of reference vectors fixed in inertial space to a set of measurement vectors fixed in the spacecraft. The attitude determination problem can be expressed as a constrained optimization problem. The constraint is that a coordinate transformation matrix must be proper, real, and orthogonal. A transformation matrix can be thought of as optimal in the least-squares sense if it maps the measurement vectors to the reference vectors with minimal 2-norm errors and meets the above constraint. This constrained optimization problem is known as Wahba's problem. Several algorithms which solve Wahba's problem exactly have been developed and used. These algorithms, while steadily improving, are all rather complicated. Furthermore, they involve such numerically unstable or sensitive operations as matrix determinant, matrix adjoint, and Newton-Raphson iterations. This paper describes an algorithm which minimizes Wahba's loss function, but without the constraint. When the constraint is ignored, the problem can be solved by a straightforward, numerically stable least-squares algorithm such as QR decomposition. Even though the algorithm does not explicitly take the constraint into account, it still yields a nearly orthogonal matrix for most practical cases; orthogonality only becomes corrupted when the sensor measurements are very noisy, on the same order of magnitude as the attitude rotations. The algorithm can be simplified if the attitude rotations are small enough so that the approximation sin(theta) approximately equals theta holds. We then compare the computational requirements for several well-known algorithms. For the general large-angle case, the QR least-squares algorithm is competitive with all other know algorithms and faster than most. If attitude rotations are small, the least-squares algorithm can be modified to run faster, and this modified algorithm is faster than all but a similarly specialized version of the QUEST algorithm. We also introduce a novel measurement averaging technique which reduces the n-measurement case to the two measurement case for our particular application, a star tracker and earth sensor mounted on an earth-pointed geosynchronous communications satellite. Using this technique, many n-measurement problems reduce to less than or equal to 3 measurements; this reduces the amount of required calculation without significant degradation in accuracy. Finally, we present the results of some tests which compare the least-squares algorithm with the QUEST and FOAM algorithms in the two-measurement case. For our example case, all three algorithms performed with similar accuracy.

Brozenec, Thomas F.↗

NASA NTRS1994

Semigroup approximation and robust stabilization of distributed parameter systems

Theoretical results that enable rigorous statements of convergence and exponential stability of Galerkin approximations of LQR controls for infinite dimensional, or distributed parameter, systems have proliferated over the past ten years. In addition, extensive progress has been made over the same time period in the derivation of robust control design strategies for finite dimensional systems. However, the study of the convergence of robust finite dimensional controllers to robust controllers for infinite dimensional systems remains an active area of research. We consider a class of soft-constrained differential games evolving in a Hilbert space. Under certain conditions, a saddle point control can be given in feedback form in terms of a solution to a Riccati equation. By considering a related LQR problem, we can show a convergence result for finite dimensional approximations of this differential game. This yields a computational algorithm for the feedback gain that can be derived from similar strategies employed in infinite dimensional LQR control design problems. The approach described in this paper also inherits the additional properties of stability robustness common to game theoretic methods in finite dimensional analysis. These theoretical convergence and stability results are verified in several numerical experiments.

Kurdila, A. J.↗

NASA NTRS1994

A rapid radiative transfer model for reflection of solar radiation

A rapid analytical radiative transfer model for reflection of solar radiation in plane-parallel atmospheres is developed based on the Sobolev approach and the delta function transformation technique. A distinct advantage of this model over alternative two-stream solutions is that in addition to yielding the irradiance components, which turn out to be mathematically equivalent to the delta-Eddington approximation, the radiance field can also be expanded in a mathematically consistent fashion. Tests with the model against a more precise multistream discrete ordinate model over a wide range of input parameters demonstrate that the new approximate method typically produces average radiance differences of less than 5%, with worst average differences of approximately 10%-15%. By the same token, the computational speed of the new model is some tens to thousands times faster than that of the more precise model when its stream resolution is set to generate precise calculations.

Xiang, X.↗

NASA NTRS1995

A conservative staggered-grid Chebyshev multidomain method for compressible flows

We present a new multidomain spectral collocation method that uses staggered grids for the solution of compressible flow problems. The solution unknowns are defined at the nodes of a Gauss quadrature rule. The fluxes are evaluated at the nodes of a Gauss-Lobatto rule. The method is conservative, free-stream preserving, and exponentially accurate. A significant advantage of the method is that subdomain corners are not included in the approximation, making solutions in complex geometries easier to compute.

Kopriva, David A.↗

NASA NTRS1999

Some Continuum Aspects of Data Assimilation

A long-sought goal in data assimilation is to build more fully the time dimension into the data assimilation process. Truly four-dimensional data assimilation requires evolving second-moment information, namely the estimation error covariance, along with the state estimate itself. The continuum evolution equations for the estimation error covariance constitute a system of partial differential equations in six space dimensions, forced by a model error covariance and by the observations themselves. The high dimensionality of this system poses a difficult computational problem. Numerous methods have been proposed for approximate, discrete solution of this system of equations. Four-dimensional variational schemes solve these equations indirectly, while schemes based on Kalman filtering solve the equations more directly, usually on a small subspace of the full six-dimensional space. In both cases, most proposed solution methods are not derived from the continuum covariance evolution equations themselves. Instead, most methods simply inherit the discretization used for evolving the state estimate, either in the form of a discrete tangent linear model or, in the case of ensemble schemes, in the form of a fully nonlinear discrete model. In this lecture we show that solutions of the continuum covariance evolution equations possess simple properties that are not readily satisfied upon discretization of the equations. we give numerical examples illustrating that "inherited" discretizations sometimes fail to produce meaningfully accurate solutions. Finally, we suggest alternative discretization methods that may yield more faithful approximate solutions of the continuum problem.

Cohn, Stephen E.↗

NASA NTRS1996

Cloud Properties Derived from Surface-Based Near-Infrared Spectral Transmission

Surface based near-infrared cloud spectral transmission measurements from a recent precipitation/cloud physics field study are used to determine cloud physical properties and relate them to other remote sensing and in situ measurements. Asymptotic formulae provide an effective means of closely approximating the qualitative and quantitative behavior of transmission computed by more laborious detailed methods. Relationships derived from asymptotic formulae are applied to measured transmission spectra to test objectively the internal consistency of data sets acquired during the field program and they confirmed the quality of the measurements. These relationships appear to be very useful in themselves, not merely as a quality control measure, but also a potentially valuable remote-sensing technique in its own right. Additional benefits from this analysis have been the separation of condensed water (cloud) transmission and water vapor transmission and the development of a method to derive cloud liquid water content.

Pilewskie, Peter↗

NASA NTRS2002

Learning Assumptions for Compositional Verification

Compositional verification is a promising approach to addressing the state explosion problem associated with model checking. One compositional technique advocates proving properties of a system by checking properties of its components in an assume-guarantee style. However, the application of this technique is difficult because it involves non-trivial human input. This paper presents a novel framework for performing assume-guarantee reasoning in an incremental and fully automated fashion. To check a component against a property, our approach generates assumptions that the environment needs to satisfy for the property to hold. These assumptions are then discharged on the rest of the system. Assumptions are computed by a learning algorithm. They are initially approximate, but become gradually more precise by means of counterexamples obtained by model checking the component and its environment, alternately. This iterative process may at any stage conclude that the property is either true or false in the system. We have implemented our approach in the LTSA tool and applied it to the analysis of a NASA system.

Cobleigh, Jamieson M.↗

NASA NTRS2010

Automated Desalting Apparatus

Because salt and metals can mask the signature of a variety of organic molecules (like amino acids) in any given sample, an automated system to purify complex field samples has been created for the analytical techniques of electrospray ionization/ mass spectroscopy (ESI/MS), capillary electrophoresis (CE), and biological assays where unique identification requires at least some processing of complex samples. This development allows for automated sample preparation in the laboratory and analysis of complex samples in the field with multiple types of analytical instruments. Rather than using tedious, exacting protocols for desalting samples by hand, this innovation, called the Automated Sample Processing System (ASPS), takes analytes that have been extracted through high-temperature solvent extraction and introduces them into the desalting column. After 20 minutes, the eluent is produced. This clear liquid can then be directly analyzed by the techniques listed above. The current apparatus including the computer and power supplies is sturdy, has an approximate mass of 10 kg, and a volume of about 20 20 20 cm, and is undergoing further miniaturization. This system currently targets amino acids. For these molecules, a slurry of 1 g cation exchange resin in deionized water is packed into a column of the apparatus. Initial generation of the resin is done by flowing sequentially 2.3 bed volumes of 2N NaOH and 2N HCl (1 mL each) to rinse the resin, followed by .5 mL of deionized water. This makes the pH of the resin near neutral, and eliminates cross sample contamination. Afterward, 2.3 mL of extracted sample is then loaded into the column onto the top of the resin bed. Because the column is packed tightly, the sample can be applied without disturbing the resin bed. This is a vital step needed to ensure that the analytes adhere to the resin. After the sample is drained, oxalic acid (1 mL, pH 1.6-1.8, adjusted with NH4OH) is pumped into the column. Oxalic acid works as a chelating reagent to bring out metal ions, such as calcium and iron, which would otherwise interfere with amino acid analysis. After oxalic acid, 1 mL 0.01 N HCl and 1 mL deionized water is used to sequentially rinse the resin. Finally, the amino acids attached to the resin, and the analytes are eluted using 2.5 M NH4OH (1 mL), and the NH4OH eluent is collected in a vial for analysis.

Spencer, Maegan K.↗

NASA NTRS2011

Ares I-X In-Flight Modal Identification

Operational modal analysis is a procedure that allows the extraction of modal parameters of a structure in its operating environment. It is based on the idealized premise that input to the structure is white noise. In some cases, when free decay responses are corrupted by unmeasured random disturbances, the response data can be processed into cross-correlation functions that approximate free decay responses. Modal parameters can be computed from these functions by time domain identification methods such as the Eigenvalue Realization Algorithm (ERA). The extracted modal parameters have the same characteristics as impulse response functions of the original system. Operational modal analysis is performed on Ares I-X in-flight data. Since the dynamic system is not stationary due to propellant mass loss, modal identification is only possible by analyzing the system as a series of linearized models over short periods of time via a sliding time-window of short time intervals. A time-domain zooming technique was also employed to enhance the modal parameter extraction. Results of this study demonstrate that free-decay time domain modal identification methods can be successfully employed for in-flight launch vehicle modal extraction.

Bartkowicz, Theodore J.↗

NASA NTRS2006

Rotating-Pump Design Code

Pump Design (PUMPDES) is a computer program for designing a rotating pump for liquid hydrogen, liquid oxygen, liquid nitrogen, water, methane, or ethane. Using realistic properties of these fluids provided by another program called GASPAK, this code performs a station-by-station, mean-line analysis along the pump flow path, obtaining thermodynamic properties of the pumped fluid at each station and evaluating hydraulic losses along the flow path. The variables at each station are obtained under constraints that are consistent with the underlying physical principles. The code evaluates the performance of each stage and the overall pump. In addition, by judiciously choosing the givens and the unknowns, the code can perform a geometric inverse design function: that is, it can compute a pump geometry that yields a closest approximation of given design point. The code contains two major parts: one for an axial-rotor/inducer and one for a multistage centrifugal pump. The inducer and the centrifugal pump are functionally integrated. The code can be used in designing and/or evaluating the inducer/centrifugal-pump combination or the centrifugal pump alone. The code is written in standard Fortran 77.

Walker, James F.↗

NASA NTRS2003

Instrument for Measuring Temperature of Water

A pseudo-Brewster-angle infrared radiometer has been proposed for use in noncontact measurement of the surface temperature of a large body of water (e.g., a lake or ocean). This radiometer could be situated on a waterborne, airborne, or spaceborne platform. The design of the pseudo-Brewster-angle radiometer would exploit the spectral-emissivity and polarization characteristics of water to minimize errors attributable to the emissivity of water and to the reflection of downwelling (e.g., Solar and cloud-reflected) infrared radiation. The relevant emissivity and polarization characteristics are the following: . The Brewster angle is the angle at which light polarized parallel to the plane of incidence on a purely dielectric material is not reflected. The pseudo-Brewster angle, defined for a lossy dielectric (somewhat electrically conductive) material, is the angle for which the reflectivity for parallel-polarized light is minimized. For pure water, the reflectivity for parallel-polarized light is only 2.2 x 10(exp -4) at its pseudo- Brewster angle of 51deg. The reflectivity remains near zero, several degrees off from the 51deg optimum, allowing this angle of incidence requirement to be easily achieved. . The wavelength range of interest for measuring water temperatures is 8 to 12 microns. The emissivity of water for parallel- polarized light at the pseudo-Brewster angle is greater than 0.999 in this wavelength range. The radiometer would be sensitive in the wavelength range of 8 to 12 microns, would be equipped with a polarizer to discriminate against infrared light polarized perpendicular to the plane of incidence, and would be aimed toward a body of water at the pseudo- Brewster angle (see figure). Because the infrared radiation entering the radiometer would be polarized parallel to the plane of incidence and because very little downwelling parallel-polarized radiation would be reflected into the radiometer on account of the pseudo-Brewster arrangement, the radiation received by the radiometer would consist almost entirely of thermal emission from the surface of the water. Because the emissivity of the water would be very close to 1, the water could be regarded as a close approximation of a blackbody for the purpose of computing its surface temperature from the radiometer measurements by use of the Planck radiation law.

Ryan, Robert↗

NASA NTRS2012

Plume-Free Stream Interaction Heating Effects During Orion Crew Module Reentry

During reentry of the Orion Crew Module (CM), vehicle attitude control will be performed by firing reaction control system (RCS) thrusters. Simulation of RCS plumes and their interaction with the oncoming flow has been difficult for the analysis community due to the large scarf angles of the RCS thrusters and the unsteady nature of the Orion capsule backshell environments. The model for the aerothermal database has thus relied on wind tunnel test data to capture the heating effects of thruster plume interactions with the freestream. These data are only valid for the continuum flow regime of the reentry trajectory. A Direct Simulation Monte Carlo (DSMC) analysis was performed to study the vehicle heating effects that result from the RCS thruster plume interaction with the oncoming freestream flow at high altitudes during Orion CM reentry. The study was performed with the DSMC Analysis Code (DAC). The inflow boundary conditions for the jets were obtained from Data Parallel Line Relaxation (DPLR) computational fluid dynamics (CFD) solutions. Simulations were performed for the roll, yaw, pitch-up and pitch-down jets at altitudes of 105 km, 125 km and 160 km as well as vacuum conditions. For comparison purposes (see Figure 1), the freestream conditions were based on previous DAC simulations performed without active RCS to populate the aerodynamic database for the Orion CM. Other inputs to the analysis included a constant Orbital reentry velocity of 7.5 km/s and angle of attack of 160 degrees. The results of the study showed that the interaction effects decrease quickly with increasing altitude. Also, jets with highly scarfed nozzles cause more severe heating compared to the nozzles with lower scarf angles. The difficulty of performing these simulations was based on the maximum number density and the ratio of number densities between the freestream and the plume for each simulation. The lowest altitude solutions required a substantial amount of computational resources (up to 1800 processors) to simulate approximately 2 billion molecules for the refined (adapted) solutions.

Marichalar, J.↗

NASA NTRS2017

High-Resolution OH and CH2O Visualization in a Premixed Cavity-Anchored Ethylene-Air Flame in a M = 0.6 Flowfield

OH and CH2O were imaged in a premixed, cavity-anchored, ethylene-air turbulent flame using a high resolution planar laser-induced fluorescence (PLIF) system. The electrically-heated, continuous flow facility (UVa Supersonic Combustion Facility, Configuration E) consisted of a Mach 2 nozzle, an isolator with fuel injectors, a test section with a cavity flame holder and optical access, and an extender. Standard test conditions comprised total temperature 1200 K, total pressure 300 kPa, local equivalence ratio near 0.4, and local Mach number near 0.6. OH PLIF data was also collected for a case with reduced total temperature and another with reduced equivalence ratio. OH and CH2O were excited in separate experiments with light sheets at 283.55 nm and 352.48 nm, respectively. A light sheet of approximate thickness 25 ?m illuminated the stream-wise midplane. This plane was imaged for 120 mm downstream of the backward-facing step. The intensified camera system imaged OH with magnification 1.97, a square 6.67 mm field of view, and in-plane resolution of 39 ?m. The smallest observed OH structures observed were approximately 100 ?m wide. The CH2O PLIF image signal was much weaker; the smallest observed structures were approximately 200 ?m wide. Composite fluorescence images were computed for the observed area.

Geipel, Clayton M.↗

NASA NTRSDate not supplied

High Energy Dynamic Impact Testing of APC AS4/PEKK-FC and TC1225 LMPAEK T700G Thermoplastic Composite Materials

High energy dynamic impact tests were conducted on Solvay APC AS4D/PEKK-FC and Toray TC1225 LMPAEK T700Gthermoplastic composite laminates. The primary objective was to provide data for evaluation of computational composite impact models. Empirical results indicate an approximately 50% increase in penetration threshold velocity and over 100% increase in penetration energy threshold for the TC1225 LMPAEK T700Gcomposite compared to the APC AS4D/PEKK-FC composite. The velocity required to produce NDE-detectable delamination in the TC1225 LMPAEK T700Gcomposite was roughly twice that for the APC AS4D/PEKK-FC composite. Comparison of the normalized energy absorption showed similar results for the APC AS4D/PEKK-FC system compared to a previously studied IM7/8552 composite material. The TC1225 LMPAEK T700Gsystem absorbed approximately twice that of the other two systems when normalized by areal weight.

Impact↗

NASA NTRS1979

A simplified computer program for the prediction of the linear stability behavior of liquid propellant combustors

A program for predicting the linear stability of liquid propellant rocket engines is presented. The underlying model assumptions and analytical steps necessary for understanding the program and its input and output are also given. The rocket engine is modeled as a right circular cylinder with an injector with a concentrated combustion zone, a nozzle, finite mean flow, and an acoustic admittance, or the sensitive time lag theory. The resulting partial differential equations are combined into two governing integral equations by the use of the Green's function method. These equations are solved using a successive approximation technique for the small amplitude (linear) case. The computational method used as well as the various user options available are discussed. Finally, a flow diagram, sample input and output for a typical application and a complete program listing for program MODULE are presented.

Mitchell, C. E.↗

NASA NTRS1992

A three-parameter approximation for radiative transfer in nonhomogeneous atmospheres - Application to the O3 9.6-micron band

A three-parameter scaling approximation is developed in conjunction with random models for the computation of transmittance along nonhomogeneous atmospheres. In addition to the scaled path length and pressure, a third parameter is introduced to account for intermediate absorption. With reference to the results derived from line-by-line calculations, the accuracy of this scaling approximation is assessed in different atmospheric conditions. For the O3 9.6-micron band where the Curtis-Godson and other conventional scaling methods usually produce large errors, the present model can achieve relative accuracies within about 1 percent in transmittance and the downward surface flux. Accuracy of about 0.15 K/d can be obtained in cooling rate calculations. The present three-parameter approximation is well suited for radiative transfer parameterizations in which the demand for high accuracy is required in the transmittance and flux calculations.

Fu, Qiang↗

NASA NTRS2018

Quantifying Shortcomings in the 2-D Pc Calculation

The probability of collision, Pc, between two Earth-orbiting satellites can often, but not always, be approximated adequately using the "2D-Pc" formulation. The objective is to find a set of "boundary conditions" that ensure the 2D-Pc approximation be sufficiently accurate, so that it may be determined when computationally-intensive Brute Force Monte Carlo1 (BFMC) Pc estimates are required. We critically examine the assumptions used in the formulation of the 2D-Pc approximation and then formulate 2D-Pc boundary condition tests to check if these assumptions are satisfied adequately.

Analysis↗

NASA NTRS1994

A PC program to optimize system configuration for desired reliability at minimum cost

High reliability is desired in all engineered systems. One way to improve system reliability is to use redundant components. When redundant components are used, the problem becomes one of allocating them to achieve the best reliability without exceeding other design constraints such as cost, weight, or volume. Systems with few components can be optimized by simply examining every possible combination but the number of combinations for most systems is prohibitive. A computerized iteration of the process is possible but anything short of a super computer requires too much time to be practical. Many researchers have derived mathematical formulations for calculating the optimum configuration directly. However, most of the derivations are based on continuous functions whereas the real system is composed of discrete entities. Therefore, these techniques are approximations of the true optimum solution. This paper describes a computer program that will determine the optimum configuration of a system of multiple redundancy of both standard and optional components. The algorithm is a pair-wise comparative progression technique which can derive the true optimum by calculating only a small fraction of the total number of combinations. A designer can quickly analyze a system with this program on a personal computer.

Hills, Steven W.↗