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NASA NTRS2011

Multi-Maneuver Clohessy-Wiltshire Targeting

Orbital rendezvous involves execution of a sequence of maneuvers by a chaser vehicle to bring the chaser to a desired state relative to a target vehicle while meeting intermediate and final relative constraints. Intermediate and final relative constraints are necessary to meet a multitude of requirements such as to control approach direction, ensure relative position is adequate for operation of space-to-space communication systems and relative sensors, provide fail-safe trajectory features, and provide contingency hold points. The effect of maneuvers on constraints is often coupled, so the maneuvers must be solved for as a set. For example, maneuvers that affect orbital energy change both the chaser's height and downrange position relative to the target vehicle. Rendezvous designers use experience and rules-of-thumb to design a sequence of maneuvers and constraints. A non-iterative method is presented for targeting a rendezvous scenario that includes a sequence of maneuvers and relative constraints. This method is referred to as Multi-Maneuver Clohessy-Wiltshire Targeting (MM_CW_TGT). When a single maneuver is targeted to a single relative position, the classic CW targeting solution is obtained. The MM_CW_TGT method involves manipulation of the CW state transition matrix to form a linear system. As a starting point for forming the algorithm, the effects of a series of impulsive maneuvers on the state are derived. Simple and moderately complex examples are used to demonstrate the pattern of the resulting linear system. The general form of the pattern results in an algorithm for formation of the linear system. The resulting linear system relates the effect of maneuver components and initial conditions on relative constraints specified by the rendezvous designer. Solution of the linear system includes the straight-forward inverse of a square matrix. Inversion of the square matrix is assured if the designer poses a controllable scenario - a scenario where the the constraints can be met by the sequence of maneuvers. Matrices in the linear system are dependent on selection of maneuvers and constraints by the designer, but the matrices are independent of the chaser's initial conditions. For scenarios where the sequence of maneuvers and constraints are fixed, the linear system can be formed and the square matrix inverted prior to real-time operations. Example solutions are presented for several rendezvous scenarios to illustrate the utility of the method. The MM_CW_TGT method has been used during the preliminary design of rendezvous scenarios and is expected to be useful for iterative methods in the generation of an initial guess and corrections.

Dannemiller, David P.↗
NASA NTRS1996

Elevation correction factor for absolute pressure measurements

With the arrival of highly accurate multi-port pressure measurement systems, conditions that previously did not affect overall system accuracy must now be scrutinized closely. Errors caused by elevation differences between pressure sensing elements and model pressure taps can be quantified and corrected. With multi-port pressure measurement systems, the sensing elements are connected to pressure taps that may be many feet away. The measurement system may be at a different elevation than the pressure taps due to laboratory space or test article constraints. This difference produces a pressure gradient that is inversely proportional to height within the interface tube. The pressure at the bottom of the tube will be higher than the pressure at the top due to the weight of the tube's column of air. Tubes with higher pressures will exhibit larger absolute errors due to the higher air density. The above effect is well documented but has generally been taken into account with large elevations only. With error analysis techniques, the loss in accuracy from elevation can be easily quantified. Correction factors can be applied to maintain the high accuracies of new pressure measurement systems.

Panek, Joseph W.↗
NASA NTRS2016

Probing Acceleration and Turbulence at Relativistic Shocks in Blazar Jets

Diffusive shock acceleration (DSA) at relativistic shocks is widely thought to be an important acceleration mechanism in various astrophysical jet sources, including radio-loud active galactic nuclei such as blazars. Such acceleration can produce the non-thermal particles that emit the broad-band continuum radiation that is detected from extragalactic jets. An important recent development for blazar science is the ability of Fermi-Large Area Telescope spectroscopy to pin down the shape of the distribution of the underlying non-thermal particle population. This paper highlights how multiwavelength spectra spanning optical to X-ray to gamma-ray bands can be used to probe diffusive acceleration in relativistic, oblique, magnetohydrodynamic (MHD) shocks in blazar jets. Diagnostics on the MHD turbulence near such shocks are obtained using thermal and non-thermal particle distributions resulting from detailed Monte Carlo simulations of DSA. These probes are afforded by the characteristic property that the synchrotron vF(sub v), peak energy does not appear in the gamma-ray band above 100 MeV. We investigate self-consistently the radiative synchrotron and inverse Compton signatures of the simulated particle distributions. Important constraints on the diffusive mean free paths of electrons, and the level of electromagnetic field turbulence are identified for three different case study blazars, Mrk 501, BL Lacertae and AO 0235+164. The X-ray excess of AO 0235+164 in a flare state can be modelled as the signature of bulk Compton scattering of external radiation fields, thereby tightly constraining the energy-dependence of the diffusion coefficient for electrons. The concomitant interpretations that turbulence levels decline with remoteness from jet shocks, and the probable significant role for non-gyroresonant diffusion, are posited.

Baring, Matthew G.↗
NASA NTRS1993

Visual display aid for orbital maneuvering - Design considerations

This paper describes the development of an interactive proximity operations planning system that allows on-site planning of fuel-efficient multiburn maneuvers in a potential multispacecraft environment. Although this display system most directly assists planning by providing visual feedback to aid visualization of the trajectories and constraints, its most significant features include: (1) the use of an 'inverse dynamics' algorithm that removes control nonlinearities facing the operator, and (2) a trajectory planning technique that separates, through a 'geometric spreadsheet', the normally coupled complex problems of planning orbital maneuvers and allows solution by an iterative sequence of simple independent actions. The visual feedback of trajectory shapes and operational constraints, provided by user-transparent and continuously active background computations, allows the operator to make fast, iterative design changes that rapidly converge to fuel-efficient solutions. The planning tool provides an example of operator-assisted optimization of nonlinear cost functions.

Grunwald, Arthur J.↗
NASA NTRS2022

Employing Relaxed Smoothness Constraints on Imaginary Part of Refractive Index in AERONET Aerosol Retrieval Algorithm

In the Aerosol Robotic Network (AERONET) retrieval algorithm, smoothness constraints on the imaginary part of the refractive index provide control of retrieved spectral dependence of aerosol absorption by preventing the inversion code from fitting the noise in optical measurements and thus avoiding unrealistic oscillations of retrievals with wavelength. The history of implementation of the smoothness constraints in the AERONET retrieval algorithm is discussed. It is shown that the latest version of the smoothness constraints on the imaginary part of refractive index, termed standard and employed by Version 3 of the retrieval algorithm, should be modified to account for strong variability of light absorption by brown-carbon-containing aerosols in UV through mid-visible parts of the solar spectrum. In Version 3 strong spectral constraints were imposed at high values of the Ångström exponent (440–870 nm) since black carbon was assumed to be the primary absorber, while the constraints became increasingly relaxed as aerosol exponent deceased to allow for wavelength dependence of absorption for dust aerosols. The new version of the smoothness constraints on the imaginary part of the refractive index assigns different weights to different pairs of wavelengths, which are the same for all values of the Ångström exponent. For example, in the case of four-wavelength input, the weights assigned to short-wavelength pairs (440–675, 675–870 nm) are small so that smoothness constraints do not suppress natural spectral variability of the imaginary part of the refractive index. At longer wavelengths (870–1020 nm), however, the weight is 10 times higher to provide additional constraints on the imaginary part of refractive index retrievals of aerosols with a high Ångström exponent due to low sensitivity to aerosol absorption for longer channels at relatively low aerosol optical depths. The effect of applying the new version of smoothness constraints, termed relaxed, on retrievals of single-scattering albedo is analyzed for case studies of different aerosol types: black- and brown-carbon-containing fine mode aerosols, mineral dust coarse mode aerosols, and urban industrial fine mode aerosol. It is shown that for brown-carbon-containing aerosols employing the relaxed smoothness constraints resulted in significant reduction in retrieved single-scattering albedo and spectral residual errors (compared to standard) at the short wavelengths. For example, biomass burning smoke cases showed a reduction in single-scattering albedo and spectral residual error at 380 nm of ∼ 0.033 and ∼ 17 %, respectively, for the Rexburg site and ∼ 0.04 and ∼ 12.7 % for the Rimrock site, both AERONET sites in Idaho, USA. For a site with very high levels of black-carbon-containing aerosols (Mongu, Zambia), the effect of modification in the smoothness constraints was minor. For mineral dust aerosols at small Ångström exponent values (Mezaira site, UAE), the spectral constraint on the imaginary part of the refractive index was already relaxed in Version 3; therefore the new relaxed constraint results in minimal change. In the case of weakly absorbing urban industrial aerosols at the GSFC site, there are significant changes in retrieved single-scattering albedo using relaxed assumption, especially reductions at longer wavelengths: ∼ 0.016 and ∼ 0.02 at 875 and 1020 nm, respectively, for 440 nm aerosol optical depth (AOD) ∼ 0.3. The modification of smoothness constraints on the imaginary part of the refractive index has a minor effect on retrievals of other aerosol parameters such as the real part of the refractive index and parameters of the aerosol size distribution. The implementation of the relaxed smoothness constraints on the imaginary part of the refractive index in the next version of the AERONET inversion algorithm will produce significant impacts at some sites in short wavelength channels (380 and 440 nm) for some biomass burning smoke cases with significant brown carbon content and possibly in mid-visible channels (500 and 675 nm) to near-infrared channels (870 to 1020 nm) for some urban industrial aerosol types. However, most differences in single-scattering albedo retrievals between those applying the new relaxed constraint and the standard constraint will be within the uncertainty of the single-scattering albedo retrievals, depending on the level of aerosol optical depth, Ångström exponent, brown carbon content and wavelength.

Aliaksandr Siniuk↗
NASA NTRS2002

Site distribution and aliasing effects in the inversion for load coefficients and geocenter motion from GPS data

Precise GPS measurements of elastic relative site displacements due to surface mass loading offer important constraints on global surface mass transport. We investigate effects of site distribution and aliasing by higher- degree (n ≥ 2) loading terms on inversion of GPS data for n = 1 load coefficients and geocenter motion. Covariance and simulation analyses are conducted to assess the sensitivity of the inversion to aliasing and mismodeling errors and possible uncertainties in the n = 1 load coefficient determination. We found that the use of center-of-figure approximation in the inverse formulation could cause 10– 15% errors in the inverted load coefficients. n = 1 load estimates may be contaminated significantly by unknown higher-degree terms, depending on the load scenario and the GPS site distribution. The uncertainty in n = 1 zonal load estimate is at the level of 80–95% for two load scenarios.

Webb, Frank H.↗
NASA NTRS2012

Parametric Deformation of Discrete Geometry for Aerodynamic Shape Design

We present a versatile discrete geometry manipulation platform for aerospace vehicle shape optimization. The platform is based on the geometry kernel of an open-source modeling tool called Blender and offers access to four parametric deformation techniques: lattice, cage-based, skeletal, and direct manipulation. Custom deformation methods are implemented as plugins, and the kernel is controlled through a scripting interface. Surface sensitivities are provided to support gradient-based optimization. The platform architecture allows the use of geometry pipelines, where multiple modelers are used in sequence, enabling manipulation difficult or impossible to achieve with a constructive modeler or deformer alone. We implement an intuitive custom deformation method in which a set of surface points serve as the design variables and user-specified constraints are intrinsically satisfied. We test our geometry platform on several design examples using an aerodynamic design framework based on Cartesian grids. We examine inverse airfoil design and shape matching and perform lift-constrained drag minimization on an airfoil with thickness constraints. A transport wing-fuselage integration problem demonstrates the approach in 3D. In a final example, our platform is pipelined with a constructive modeler to parabolically sweep a wingtip while applying a 1-G loading deformation across the wingspan. This work is an important first step towards the larger goal of leveraging the investment of the graphics industry to improve the state-of-the-art in aerospace geometry tools.

Anderson, George R.↗
NASA NTRS2002

Site Distribution and Aliasing Effects in the Inversion for Load Coefficients and Geocenter Motion from GPS Data

Precise GPS measurements of elastic relative site displacements due to surface mass loading offer important constraints on global surface mass transport. We investigate effects of site distribution and aliasing by higher-degree (n greater than or equal 2) loading terms on inversion of GPS data for n = 1 load coefficients and geocenter motion. Covariance and simulation analyses are conducted to assess the sensitivity of the inversion to aliasing and mismodeling errors and possible uncertainties in the n = 1 load coefficient determination. We found that the use of center-of-figure approximation in the inverse formulation could cause 10- 15% errors in the inverted load coefficients. n = 1 load estimates may be contaminated significantly by unknown higher-degree terms, depending on the load scenario and the GPS site distribution. The uncertainty in n = 1 zonal load estimate is at the level of 80 - 95% for two load scenarios.

global positioning system↗
NASA NTRS1985

Algorithm Solves Constrained and Unconstrained Optimization Problems

Is quasi-Newton iteration utilizing Broyden/Fletcher/Goldfarb/Shanno update on inverse Hessian matrix. Capable of solving constrained optimization unconstrained optimization and constraints only problems with one to five independent variables from one to five constraint functions and one dependent function optimized.

Denson, M. A.↗
NASA NTRS1990

The shape of Eros

Monte Carlo simulations are presently used to optimize estimation, ascertain associated errors, and guide bias-correction procedures, for the Eros polar silhouette convex hull that has been estimated from radar echo spectra. This hull is trapezoidal; this nonaxisymmetric shape may account for odd harmonics in Eros' echo spectral signature as a function of rotation phase. Additional constraints have been obtained for the figure of Eros through the inversion of the optical lightcurve to estimate the asteroid's two-dimensional average of the three-dimensional shape. This 'mean cross-section' and the polar silhouette exhibit similar elongations.

Ostro, S. J.↗
NASA NTRS2002

Turbomachinery Airfoil Design Optimization Using Differential Evolution

An aerodynamic design optimization procedure that is based on a evolutionary algorithm known at Differential Evolution is described. Differential Evolution is a simple, fast, and robust evolutionary strategy that has been proven effective in determining the global optimum for several difficult optimization problems, including highly nonlinear systems with discontinuities and multiple local optima. The method is combined with a Navier-Stokes solver that evaluates the various intermediate designs and provides inputs to the optimization procedure. An efficient constraint handling mechanism is also incorporated. Results are presented for the inverse design of a turbine airfoil from a modern jet engine and compared to earlier methods. The capability of the method to search large design spaces and obtain the optimal airfoils in an automatic fashion is demonstrated. Substantial reductions in the overall computing time requirements are achieved by using the algorithm in conjunction with neural networks.

Madavan, Nateri K.↗
NASA NTRS2002

Turbomachinery Airfoil Design Optimization Using Differential Evolution

An aerodynamic design optimization procedure that is based on a evolutionary algorithm known at Differential Evolution is described. Differential Evolution is a simple, fast, and robust evolutionary strategy that has been proven effective in determining the global optimum for several difficult optimization problems, including highly nonlinear systems with discontinuities and multiple local optima. The method is combined with a Navier-Stokes solver that evaluates the various intermediate designs and provides inputs to the optimization procedure. An efficient constraint handling mechanism is also incorporated. Results are presented for the inverse design of a turbine airfoil from a modern jet engine. The capability of the method to search large design spaces and obtain the optimal airfoils in an automatic fashion is demonstrated. Substantial reductions in the overall computing time requirements are achieved by using the algorithm in conjunction with neural networks.

Madavan, Nateri K.↗
NASA NTRS1974

Structure of the Jovian envelope from Pioneer 10 gravity data

Measurement of Jupiter's zonal harmonics J2 and J4 by the celestial mechanics experiment on Pioneer 10 may be used to obtain a constraint on the structure of the outer envelope of Jupiter, using an inversion technique which is insensitive to the structure of the deep interior for a plausible class of planetary models. The derived structure is consistent with an adiabatic, solar-composition envelope with a starting temperature of 250 plus or minus 40 K at 1 bar pressure.

Anderson, J. D.↗
NASA NTRS1985

Observations of the J = 10 manifold of the pure rotational band of phosphine on Saturn

Saturn was observed in the vicinity of the J = 10 manifold of the pure rotational band of phosphine on 1984 July 10 and 12 from NASA's Kuiper Airborne Observatory with the facility far-infrared cooled grating spectrometer. On each night observations of the full disk plus rings were made at 4 to 6 discrete wavelengths which selectively sampled the manifold and the adjacent continuum. The previously reported detection of this manifold is confirmed. After subtraction of the flux due to the rings, the data are compared with disk-averaged models of Saturn. It is found that PH3 must be strongly depleted above the thermal inversion (approx. 70 mbar). The best fitting models consistent with other observational constraints indicate that PH3 is significantly depleted at even deeper atmospheric levels (or = 500 mbar), implying an eddy diffusion coefficient for Saturn of 10 to the 4 cm sq/sec.

Haas, M. R.↗
NASA NTRS1989

Stress analysis method for clearance-fit joints with bearing-bypass loads

A simple direct stress analysis method is proposed for a laminate with a clearance-fit fastener subjected to combined bearing and bypass loads in tension or compression, including bearing and bypass loads on bolt-hole contact and local stresses. The approach uses a linear-elastic finite element analysis with an inverse formulation; conditions along the bolt-hole interface are specified by constraint equations that limit nodal displacements to a circular arc corresponding to the bolt diameter. The application of the method to the analysis of the effects of bearing-bypass loading on bolt-hole contact angles and local stresses, with the plate material properties representing a quasi-isotropic T300/5208 graphite/epoxy laminate, is discussed.

Naik, R. A.↗
NASA NTRS1991

A theory of photometric stereo for a class of diffuse non-Lambertian surfaces

A theory of photometric stereo is proposed for a large class of non-Lambertian reflectance maps. The authors review the different reflectance maps proposed in the literature for modeling reflection from real-world surfaces. From this, they obtain a mathematical class of reflectance maps to which the maps belong. They show that three lights can be sufficient for a unique inversion of the photometric stereo equation for the entire class of reflectance maps. They also obtain a constraint on the positions of light sources for obtaining this solution. They investigate the sufficiency of three light sources to estimate the surface normal and the illuminant strength. The issue of completeness of reconstruction is addressed. They shown that if k lights are sufficient for a unique inversion, 2k lights are necessary for a complete inversion.

Tagare, Hemant D.↗
NASA NTRS2012

GRAIL Refinements to Lunar Seismic Structure

The present ]day internal structure of the Moon provides insight not only into its own formation and evolution, but also that of all rocky planetary bodies. The most direct way to probe a planet fs interior structure is through seismology. As part of the Apollo lunar missions, four seismometers were deployed on the nearside surface of the Moon between the years 1969 and 1972. These instruments operated continuously until 1977, forming the only substantial extraterrestrial seismic data set in existence. These data have been used to constrain various aspects of the seismic velocity and density structure of the Moon. Typical 1-D models recognize a 30-60 km thick crust overlying a nearly constant ]velocity mantle, and extend to a depth of approximately 1000 km, below which the lack of penetrating moonquake ray ]paths precludes the seismic determination of deeper structure. Previously, the lack of observed moonquakes from the far side of the Moon has been used to infer the presence of a highly attenuating (possibly molten) core. Indirect geophysical measurements such as moment of inertia, magnetic induction, lunar laser ranging, and elemental abundances of mare basalts also place varying constraints on core size and state. In combination with seismic studies, these indirect measurements have been used to arrive at a commonly accepted model of the Moon's deepest interior that includes a solid inner and fluid outer core, overlain by a partial melt boundary layer. We recently applied modern array seismology techniques to the Apollo data and revealed detailed core structure, including the first direct confirmation of the presence of a solid inner core. Our study focused on the identification of core ]reflected phases in deep moonquake seismograms. The resulting model of the Moon fs innermost structure was found to be consistent with the commonly accepted model. However, the modeled layer radii may vary by tens of kilometers, as is expected when accounting for uncertainties such as moonquake location, timing errors, and potential seismic heterogeneities. In addition, the modeled velocities may vary with a 1-to-1 trade ]off with the modeled reflector depth. The GRAIL (Gravity Recovery and Interior Laboratory) mission, launched in Sept. 2011, placed two nearly identical spacecraft in lunar orbit. The two satellites make extremely high-resolution measurements of the lunar gravity field, which can be used to constrain the interior structure of the Moon using a "crust to core" approach. GRAIL fs constraints on crustal thickness, mantle structure, core radius and stratification, and core state (solid vs. molten) will complement seismic investigations in several ways. Here we present a progress report on our efforts to advance our knowledge of the Moon fs internal structure using joint gravity and seismic analyses. We will focus on methodology, including 1) refinements to the seismic core constraint accomplished through array processing of Apollo seismic data, made by applying a set of travel time corrections based on GRAIL structure estimates local to each Apollo seismic station; 2) modeling deep lunar structure through synthetic seismograms, to test whether the seismic core model can reproduce the core reflections observed in the Apollo seismograms; and 3) a joint seismic and gravity inversion in which we attempt to fit a family of seismic structure models with the gravity constraints from GRAIL, resulting in maps of seismic velocities and densities that vary from a nominal model both laterally and with depth.

Weber, Renee↗
NASA NTRS1972

A finite-difference method for transonic airfoil design.

This paper describes an inverse method for designing transonic airfoil sections or for modifying existing profiles. Mixed finite-difference procedures are applied to the equations of transonic small disturbance theory to determine the airfoil shape corresponding to a given surface pressure distribution. The equations are solved for the velocity components in the physical domain and flows with embedded shock waves can be calculated. To facilitate airfoil design, the method allows alternating between inverse and direct calculations to obtain a profile shape that satisfies given geometric constraints. Examples are shown of the application of the technique to improve the performance of several lifting airfoil sections. The extension of the method to three dimensions for designing supercritical wings is also indicated.

Steger, J. L.↗