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NASA NTRS1990

Structure/control synthesis with nonnegligible actuator mass

The simultaneous design of a structure/active control system is addressed in which the mass of the actuators required to implement the active control is considered. An algorithm for estimating the required actuator mass given the control law and the desired maneuver is presented. A nonlinear optimization algorithm is used to direct the design process. Results are given for the design of a L shaped structure where it is desired to minimize the line of sight pointing error after a worst case slew maneuver.

Zimmerman, David C.↗

NASA NTRS1990

A physical model of the bidirectional reflectance of vegetation canopies. I - Theory. II - Inversion and validation

A new physically based analytical model of the bidirectional reflectance of vegetation canopies is derived. The model expresses the bidirectional reflectance field of a semiinfinite canopy as a combination of functions describing (1) the optical properties of the leaves through their single-scattering albedo and their phase function, (2) the average distribution of leaf orientations, and (3) the architecture of the canopy. The model is validated against laboratory and ground-based measurements in the visible and IR spectral regions, taken over two vegetation covers. The intrinsic optical properties of leaves and the information on the geometrical canopy arrangements in space were obtained using an inversion procedure based on a nonlinear optimization technique. Model predictions of bidirectional reflectances obtained using the inversion procedure compare well with actual observations.

Verstraete, Michel M.↗

NASA NTRS1992

Frequency domain state-space system identification

An algorithm for identifying state-space models from frequency response data of linear systems is presented. A matrix-fraction description of the transfer function is employed to curve-fit the frequency response data, using the least-squares method. The parameters of the matrix-fraction representation are then used to construct the Markov parameters of the system. Finally, state-space models are obtained through the Eigensystem Realization Algorithm using Markov parameters. The main advantage of this approach is that the curve-fitting and the Markov parameter construction are linear problems which avoid the difficulties of nonlinear optimization of other approaches. Another advantage is that it avoids windowing distortions associated with other frequency domain methods.

Chen, Chung-Wen↗

NASA NTRS1992

Assigning Damping Parameters To Component Dynamical Modes

Damping parameters found by solving nonlinear optimization problem. Paper discusses assignment of matrices of damping parameters to mathematical models of vibrations and rotations of interconnected flexible components of large, flexible structure. Such models truncated at component level to reduce amount of computation necessary to simulate dynamics of structure numerically at both component and system levels.

Lee, Allan Y.↗

NASA NTRS1992

Analytical formulation of optimum rotor interdisciplinary design with a three-dimensional wake

An analytical formulation of optimum rotor interdisciplinary design is presented. A finite-state aeroelastic rotor model, coupling generalized dynamic wake with blade finite elements, is applied to perform the optimum rotor blade design for improved aerodynamic performance and vehicle vibration, while a feasible direction nonlinear optimizer, CONMIN, provides the optimization algorithm. The approach features a systematic rotor aeroelastic model which offers an efficient analytical tool, and retains necessary aerodynamic and blade dynamic building blocks for a sufficient rotor dynamic response analysis. The formulation is well suited for an efficient design sensitivity computation without resorting to finite difference, and thus provides a practical design tool. The results show improved rotor aerodynamic performance and reduced hub vibratory loads for the optimized blade as compared to the advanced rotor of reference design.

He, Chengjian↗

NASA NTRS1991

Path planning algorithms for assembly sequence planning

Planning for manipulation in complex environments often requires reasoning about the geometric and mechanical constraints which are posed by the task. In planning assembly operations, the automatic generation of operations sequences depends on the geometric feasibility of paths which permit parts to be joined into subassemblies. Feasible locations and collision-free paths must be present for part motions, robot and grasping motions, and fixtures. This paper describes an approach to reasoning about the feasibility of straight-line paths among three-dimensional polyhedral parts using an algebra of polyhedral cones. A second method recasts the feasibility conditions as constraints in a nonlinear optimization framework. Both algorithms have been implemented and results are presented.

Krishnan, S. S.↗

NASA NTRS1994

Dynamically consistent hydrography and absolute velocity in the eastern North Atlantic Ocean

The problem of mapping a dynamically consistent hydrographic field and associated absolute geostrophic flow in the eastern North Atlantic between 24 deg and 36 deg N is related directly to the solution of the so-called thermocline equations. A nonlinear optimization problem involving Needler's P equation is solved to find the hydrography and resulting flow that minimizes the vertical mixing above about 1500 m in the ocean and is simultaneously consistent with the observations. A sharp minimum (at least in some dimensions) is found, apparently corresponding to a solution nearly conserving potential vorticity and with vertical eddy coefficient less than about 10(exp -5) sq m/s. Estimates of `residual' quantities such as eddy coefficients are extremely sensitive to slight modifications to the observed fields. Boundary conditions, vertical velocities, etc., are a product of the optimization and produce estimates differing quantitatively from prior ones relying directly upon observed hydrography. The results are generally insensitive to particular elements of the solution methodology, but many questions remain concerning the extent to which different synoptic sections can be asserted to represent the same ocean. The method can be regarded as a practical generalization of the beta spiral and geostrophic balance inverses for the estimate of absolute geostrophic flows. Numerous improvements to the methodology used in this preliminary attempt are possible.

Wunsch, Carl↗

NASA NTRS1998

Incorporating Endmember Variability into Spectral Mixture Analysis Through Endmember Bundles

Variation in canopy structure and biochemistry induces a concomitant variation in the top-of-canopy spectral reflectance of a vegetation type. Hence, the use of a single endmember spectrum to track the fractional abundance of a given vegetation cover in a hyperspectral image may result in fractions with considerable error. One solution to the problem of endmember variability is to increase the number of endmembers used in a spectral mixture analysis of the image. For example, there could be several tree endmembers in the analysis because of differences in leaf area index (LAI) and multiple scatterings between leaves and stems. However, it is often difficult in terms of computer or human interaction time to select more than six or seven endmembers and any non-removable noise, as well as the number of uncorrelated bands in the image, limits the number of endmembers that can be discriminated. Moreover, as endmembers proliferate, their interpretation becomes increasingly difficult and often applications simply need the aerial fractions of a few land cover components which comprise most of the scene. In order to incorporate endmember variability into spectral mixture analysis, we propose representing a landscape component type not with one endmember spectrum but with a set or bundle of spectra, each of which is feasible as the spectrum of an instance of the component (e.g., in the case of a tree component, each spectrum could reasonably be the spectral reflectance of a tree canopy). These endmember bundles can be used with nonlinear optimization algorithms to find upper and lower bounds on endmember fractions. This approach to endmember variability naturally evolved from previous work in deriving endmembers from the data itself by fitting a triangle, tetrahedron or, more generally, a simplex to the data cloud reduced in dimension by a principal component analysis. Conceptually, endmember variability could make it difficult to find a simplex that both surrounds the data cloud and has vertices that are realistic endmember spectra with reflectances between 0 and 1. In this paper, we create endmember bundles and bounding fraction images for an AVIRIS subscene simulated with a plant canopy radiative transfer model. The simulated subscene is spatially patterned after a subscene from the AVIRIS image acquired August, 1993 over La Copita, Texas. In addition, for comparison, we performed a traditional unmixing with image endmembers.

Bateson, C. Ann↗

NASA NTRS2000

Analytical and Computational Properties of Distributed Approaches to MDO

Historical evolution of engineering disciplines and the complexity of the MDO problem suggest that disciplinary autonomy is a desirable goal in formulating and solving MDO problems. We examine the notion of disciplinary autonomy and discuss the analytical properties of three approaches to formulating and solving MDO problems that achieve varying degrees of autonomy by distributing the problem along disciplinary lines. Two of the approaches-Optimization by Linear Decomposition and Collaborative Optimization-are based on bi-level optimization and reflect what we call a structural perspective. The third approach, Distributed Analysis Optimization, is a single-level approach that arises from what we call an algorithmic perspective. The main conclusion of the paper is that disciplinary autonomy may come at a price: in the bi-level approaches, the system-level constraints introduced to relax the interdisciplinary coupling and enable disciplinary autonomy can cause analytical and computational difficulties for optimization algorithms. The single-level alternative we discuss affords a more limited degree of autonomy than that of the bi-level approaches, but without the computational difficulties of the bi-level methods. Key Words: Autonomy, bi-level optimization, distributed optimization, multidisciplinary optimization, multilevel optimization, nonlinear programming, problem integration, system synthesis

Alexandrov, Natalia M.↗

NASA NTRS1995

Profiles of Gamma-Ray Bursts and Their Component Pulses

One physically informative regularity of their otherwise heterogeneous ensemble, is that many Gamma-Ray Bursts consist of well defined pulses. To objectively quantify the temporal structure of BATSE bursts, we have developed an automatic modeling procedure that separates overlapping pulses and determines the energy-dependence of the pulse-shape parameters. No binning of photon arrival times is needed, so when applied to time-tagged events (TTE) the procedure captures variability information down to the shortest time scales present in the raw data. Maximizing the Bayesian likelihood function Pr(data/model) yields estimates of the model parameters, including the number of pulses present, and allows intercomparison of models of different forms. As with any nonlinear optimization, good initial guesses are crucial to avoid convergence to undesirable local minima. We find excellent initial pulse decompositions by wavelet-denoising a cumulative distribution of the raw photon arrival data; differentiation then gives a time profile mostly free of the systematic effects of degraded resolution (as in ordinary Fourier smoothing) and binning. We present statistical information on pulse rise-time, decay-time, peakedness, and amplitudes, plus their energy dependences - both within a single burst and for a large ensemble of bursts.

Scargle, Jeff D.↗

NASA NTRS2003

Binary-Phase Fourier Gratings for Nonuniform Array Generation

We describe a design method for a binary-phase Fourier grating that generates an array of spots with nonuniform, user-defined intensities symmetric about the zeroth order. Like the Dammann fanout grating approach, the binary-phase Fourier grating uses only two phase levels in its grating surface profile to generate the final spot array. Unlike the Dammann fanout grating approach, this method allows for the generation of nonuniform, user-defined intensities within the final fanout pattern. Restrictions governing the specification and realization of the array's individual spot intensities are discussed. Design methods used to realize the grating employ both simulated annealing and nonlinear optimization approaches to locate optimal solutions to the grating design problem. The end-use application driving this development operates in the near- to mid-infrared spectrum - allowing for higher resolution in grating specification and fabrication with respect to wavelength than may be available in visible spectrum applications. Fabrication of a grating generating a user-defined nine spot pattern is accomplished in GaAs for the near-infrared. Characterization of the grating is provided through the measurement of individual spot intensities, array uniformity, and overall efficiency. Final measurements are compared to calculated values with a discussion of the results.

Keys, Andrew S.↗

NASA NTRS2007

Iterative-Transform Phase Retrieval Using Adaptive Diversity

A phase-diverse iterative-transform phase-retrieval algorithm enables high spatial-frequency, high-dynamic-range, image-based wavefront sensing. [The terms phase-diverse, phase retrieval, image-based, and wavefront sensing are defined in the first of the two immediately preceding articles, Broadband Phase Retrieval for Image-Based Wavefront Sensing (GSC-14899-1).] As described below, no prior phase-retrieval algorithm has offered both high dynamic range and the capability to recover high spatial-frequency components. Each of the previously developed image-based phase-retrieval techniques can be classified into one of two categories: iterative transform or parametric. Among the modifications of the original iterative-transform approach has been the introduction of a defocus diversity function (also defined in the cited companion article). Modifications of the original parametric approach have included minimizing alternative objective functions as well as implementing a variety of nonlinear optimization methods. The iterative-transform approach offers the advantage of ability to recover low, middle, and high spatial frequencies, but has disadvantage of having a limited dynamic range to one wavelength or less. In contrast, parametric phase retrieval offers the advantage of high dynamic range, but is poorly suited for recovering higher spatial frequency aberrations. The present phase-diverse iterative transform phase-retrieval algorithm offers both the high-spatial-frequency capability of the iterative-transform approach and the high dynamic range of parametric phase-recovery techniques. In implementation, this is a focus-diverse iterative-transform phaseretrieval algorithm that incorporates an adaptive diversity function, which makes it possible to avoid phase unwrapping while preserving high-spatial-frequency recovery. The algorithm includes an inner and an outer loop (see figure). An initial estimate of phase is used to start the algorithm on the inner loop, wherein multiple intensity images are processed, each using a different defocus value. The processing is done by an iterative-transform method, yielding individual phase estimates corresponding to each image of the defocus-diversity data set. These individual phase estimates are combined in a weighted average to form a new phase estimate, which serves as the initial phase estimate for either the next iteration of the iterative-transform method or, if the maximum number of iterations has been reached, for the next several steps, which constitute the outerloop portion of the algorithm. The details of the next several steps must be omitted here for the sake of brevity. The overall effect of these steps is to adaptively update the diversity defocus values according to recovery of global defocus in the phase estimate. Aberration recovery varies with differing amounts as the amount of diversity defocus is updated in each image; thus, feedback is incorporated into the recovery process. This process is iterated until the global defocus error is driven to zero during the recovery process. The amplitude of aberration may far exceed one wavelength after completion of the inner-loop portion of the algorithm, and the classical iterative transform method does not, by itself, enable recovery of multi-wavelength aberrations. Hence, in the absence of a means of off-loading the multi-wavelength portion of the aberration, the algorithm would produce a wrapped phase map. However, a special aberration-fitting procedure can be applied to the wrapped phase data to transfer at least some portion of the multi-wavelength aberration to the diversity function, wherein the data are treated as known phase values. In this way, a multiwavelength aberration can be recovered incrementally by successively applying the aberration-fitting procedure to intermediate wrapped phase maps. During recovery, as more of the aberration is transferred to the diversity function following successive iterations around the ter loop, the estimated phase ceases to wrap in places where the aberration values become incorporated as part of the diversity function. As a result, as the aberration content is transferred to the diversity function, the phase estimate resembles that of a reference flat.

Dean, Bruce H.↗

NASA NTRS2011

Broadband Achromatic Phase Shifter for a Nulling Interferometer

Nulling interferometry is a technique for imaging exoplanets in which light from the parent star is suppressed using destructive interference. Light from the star is divided into two beams and a phase shift of radians is introduced into one of the beams. When the beams are recombined, they destructively interfere to produce a deep null. For monochromatic light, this is implemented by introducing an optical path difference (OPD) between the two beams equal to lambda/2, where lambda is the wavelength of the light. For broadband light, however, a different phase shift will be introduced at each wavelength and the two beams will not effectively null when recombined. Various techniques have been devised to introduce an achromatic phase shift a phase shift that is uniform across a particular bandwidth. One popular technique is to use a series of dispersive elements to introduce a wavelength-dependent optical path in one or both of the arms of the interferometer. By intelligently choosing the number, material and thickness of a series of glass plates, a nearly uniform, arbitrary phase shift can be introduced between two arms of an interferometer. There are several constraints that make choosing the number, type, and thickness of materials a difficult problem, such as the size of the bandwidth to be nulled. Several solutions have been found for bandwidths on the order of 20 to 30 percent (Delta(lambda)/lambda(sub c)) in the mid-infrared region. However, uniform phase shifts over a larger bandwidth in the visible regime between 480 to 960 nm (67 percent) remain difficult to obtain at the tolerances necessary for exoplanet detection. A configuration of 10 dispersive glass plates was developed to be used as an achromatic phase shifter in nulling interferometry. Five glass plates were placed in each arm of the interferometer and an additional vacuum distance was also included in the second arm of the interferometer. This configuration creates a phase shift of pi radians with an average error of 5.97 x 10(exp -8) radians and standard deviation of 3.07 x 10(exp -4) radians. To reduce ghost reflections and interference effects from neighboring elements, the glass plates are tilted such that the beam does not strike each plate at normal incidence. Reflections will therefore walk out of the system and not contribute to the intensity when the beams are recombined. Tilting the glass plates, however, introduces several other problems that must be mitigated: (1) the polarization of a beam changes when refracted at an interface at non-normal incidence; (2) the beam experiences lateral chromatic spread as it traverses multiple glass plates; (3) at each surface, wavelength- dependent intensity losses will occur due to reflection. For a fixed angle of incidence, each of these effects must be balanced between each arm of the interferometer in order to ensure a deep null. The solution was found using a nonlinear optimization routine that minimized an objective function relating phase shift, intensity difference, chromatic beam spread, and polarization difference to the desired parameters: glass plate material and thickness. In addition to providing a uniform, broadband phase shift, the configuration achieves an average difference in intensity transmission between the two arms of the interferometer of 0.016 percent with a standard deviation of 3.64 x 10(exp -4) percent, an average difference in polarization between the two arms of the interferometer of 5.47 x 10(exp -5) percent with a standard deviation of 1.57 x 10(exp -6) percent, and an average chromatic beam shift between the two arms of the interferometer of -47.53 microns with a wavelength-by-wavelength spread of 0.389 microns.

Bolcar, Matthew R.↗

NASA NTRS2012

On Heatshield Shapes for Mars Entry Capsules

The 70deg sphere-cone - the standard geometry for all US Mars entry missions - is thoroughly examined via flow field simulations at a select few peak heating points along candidate flight trajectories. Emphasis is placed on turbulent heating based on the Baldwin- Lomax turbulence model. It is shown that increased leeward turbulent heating for a 70 sphere-cone flying at angle of attack is primarily due to the discontinuity in curvature between the spherical nose cap and the conical frustum - the attachment of the sonic line at this sphere-cone junction leads to a supersonic edge Mach number over the leeward acreage. In an attempt to mitigate this problem of elevated turbulent heating, alternate geometries, without any curvature discontinuities in the acreage, are developed. Two approaches, one based on nonlinear optimization with constraints, and one based on the use of non-uniform rational B-splines, are considered. All configurations examined remain axisymmetric. The aerothermal performance of alternate geometries is shown to be superior to that of the 70 sphere-cone.

Prabhu, DInesh K.↗

NASA NTRS2012

Uncertainty Quantification for Polynomial Systems via Bernstein Expansions

This paper presents a unifying framework to uncertainty quantification for systems having polynomial response metrics that depend on both aleatory and epistemic uncertainties. The approach proposed, which is based on the Bernstein expansions of polynomials, enables bounding the range of moments and failure probabilities of response metrics as well as finding supersets of the extreme epistemic realizations where the limits of such ranges occur. These bounds and supersets, whose analytical structure renders them free of approximation error, can be made arbitrarily tight with additional computational effort. Furthermore, this framework enables determining the importance of particular uncertain parameters according to the extent to which they affect the first two moments of response metrics and failure probabilities. This analysis enables determining the parameters that should be considered uncertain as well as those that can be assumed to be constants without incurring significant error. The analytical nature of the approach eliminates the numerical error that characterizes the sampling-based techniques commonly used to propagate aleatory uncertainties as well as the possibility of under predicting the range of the statistic of interest that may result from searching for the best- and worstcase epistemic values via nonlinear optimization or sampling.

Crespo, Luis G.↗

NASA NTRS2012

Cascade Storage and Delivery System for a Multi Mission Space Exploration Vehicle (MMSEV)

NASA is developing a Multi Mission Space Exploration Vehicle (MMSEV) for missions beyond Low Earth Orbit (LEO). The MMSEV is a pressurized vehicle used to extend the human exploration envelope for Lunar, Near Earth Object (NEO), and Deep Space missions. The Johnson Space Center is developing the Environmental Control and Life Support System (ECLSS) for the MMSEV. The MMSEV s intended use is to support longer sortie lengths with multiple Extra Vehicular Activities (EVAs) on a higher magnitude than any previous vehicle. This paper presents an analysis of a high pressure oxygen cascade storage and delivery system that will accommodate the crew during long duration Intra Vehicular Activity (IVA) and capable of multiple high pressure oxygen fills to the Portable Life Support System (PLSS) worn by the crew during EVAs. A cascade is a high pressure gas cylinder system used for the refilling of smaller compressed gas cylinders. Each of the large cylinders are filled by a compressor, but the cascade system allows small cylinders to be filled without the need of a compressor. In addition, the cascade system is useful as a "reservoir" to accommodate low pressure needs. A regression model was developed to provide the mechanism to size the cascade systems subject to constraints such as number of crew, extravehicular activity duration and frequency, and ullage gas requirements under contingency scenarios. The sizing routine employed a numerical integration scheme to determine gas compressibility changes during depressurization and compressibility effects were captured using the Soave-Redlich-Kwong (SRK) equation of state. A multi-dimensional nonlinear optimization routine was used to find the minimum cascade tank system mass that meets the mission requirements. The sizing algorithms developed in this analysis provide a powerful framework to assess cascade filling, compressor, and hybrid systems to design long duration vehicle ECLSS architecture. 1

Yagoda, Evan↗

NASA NTRS2016

Phase-Retrieval Uncertainty Estimation and Algorithm Comparison for the JWST-ISIM Test Campaign

Phase retrieval, the process of determining the exitpupil wavefront of an optical instrument from image-plane intensity measurements, is the baseline methodology for characterizing the wavefront for the suite of science instruments (SIs) in the Integrated Science Instrument Module (ISIM) for the James Webb Space Telescope (JWST). JWST is a large, infrared space telescope with a 6.5-meter diameter primary mirror. JWST is currently NASA's flagship mission and will be the premier space observatory of the next decade. ISIM contains four optical benches with nine unique instruments, including redundancies. ISIM was characterized at the Goddard Space Flight Center (GSFC) in Greenbelt, MD in a series of cryogenic vacuum tests using a telescope simulator. During these tests, phase-retrieval algorithms were used to characterize the instruments. The objective of this paper is to describe the Monte-Carlo simulations that were used to establish uncertainties (i.e., error bars) for the wavefronts of the various instruments in ISIM. Multiple retrieval algorithms were used in the analysis of ISIM phase-retrieval focus-sweep data, including an iterativetransform algorithm and a nonlinear optimization algorithm. These algorithms emphasize the recovery of numerous optical parameters, including low-order wavefront composition described by Zernike polynomial terms and high-order wavefront described by a point-by-point map, location of instrument best focus, focal ratio, exit-pupil amplitude, the morphology of any extended object, and optical jitter. The secondary objective of this paper is to report on the relative accuracies of these algorithms for the ISIM instrument tests, and a comparison of their computational complexity and their performance on central and graphical processing unit clusters. From a phase-retrieval perspective, the ISIM test campaign includes a variety of source illumination bandwidths, various image-plane sampling criteria above and below the Nyquist- Shannon critical sampling value, various extended object sizes, and several other impactful effects.

Design Analysis↗

NASA NTRS2019

Retrieval of Vegetation Water Content Using Brightness Temperatures from the Soil Moisture Active Passive (SMAP) Mission

In this paper, we explore a time series approach to using the tau-omega (τ-ω) model to retrieve vegetation water content (kg/m2) with minimal use of ancillary data. Analytically, this approach calls for nonlinear optimization in two steps. First, multiple days of co-located brightness temperature observations are used to retrieve the effective vegetation opacity, which incorporates the combined radiometric and polarization effects of surface roughness and vegetation opacity. The resulting effective vegetation opacity is then used to retrieve vegetation water content to within a gain factor α and an offset factor β. By using a climatological vegetation water content ancillary database as the one adopted in the development of the SMAP standard and enhanced soil moisture products, α and β can be determined globally using the annual minimum and annual maximum of vegetation water content. The resulting values of α and β can then be used to reconstruct the retrieved vegetation water content. Formulation, assumptions, and limitations of this approach are presented alongside the preliminary global retrieval of vegetation water content using one year (2016) ofSMAP brightness temperature observations.

radiative transfer↗