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At least 145 records · Page 8

Methodology for Producing a Uniform Distribution of UO2 in a Tungsten Matrix

Current work at NASA's Marshall Space Flight Center (MSFC) is focused on the development CERMET fuel materials for Nuclear Thermal Propulsion (NTP). The CERMETs consist of uranium dioxide (UO2) fuel particles embedded in a tungsten (W) metal matrix. Initial testing of W-UO2 samples fabricated from fine angular powders performed reasonably well, but suffered from significant fuel loss during repeated thermal cycling due to agglomeration of the UO2 (1). The blended powder mixtures resulted in a non-uniform dispersion of the UO2 particles in the tungsten matrix, which allows rapid vaporization of the interconnected UO2 from the sample edges into the bulk material. Also, the angular powders create areas of stress concentrations due to thermal expansion mismatch, which eventually cracks the tungsten matrix. Evenly coating spherical UO2 particles with chemical vapor deposited (CVD) tungsten prior to consolidation was previously demonstrated to provide improved performance. However, the CVD processing technology is expensive and not currently available. In order to reduce cost and enhance performance, a powder coating process has been developed at MSFC to produce a uniform distribution of the spherical UO2 particles in a tungsten matrix. The method involves utilization of a polyethylene binder during mixing which leads to fine tungsten powders clinging to the larger UO2 spherical particles. This process was developed using HfO2 as a surrogate for UO2. Enough powder was mixed to make 8 discs (2cm diameter x 8mm thickness) using spark plasma sintering. A uniaxial pressure of 50 MPa was used at four different temperatures (2 samples at each temperature). The first two samples were heated to 1400C and 1500C respectively for 5 minutes. Densities for these samples were less than 85% of theoretical, so the time at temperature was increased to 20 minutes for the remaining samples. The highest densities were achieved for the two samples sintered at 1700C (approx. 92% of theoretical). Scanning electron microscopy (SEM) of the mixed powders and the sintered samples along with energy dispersive x-ray analysis was obtained. The SEM of the powders clearly show the fine W powder adhered to the larger HfO2 particles and a uniform distribution of HfO2 particles in a tungsten matrix upon densification. Vicker's Microhardness testing was also performed on all samples using 0.5, 1.0 and 2.0 kg loads. Five indents were made at each load level. All indents were placed in the tungsten matrix to assist as a proxy in measuring densification. The highest hardness value was obtained for the 1700C specimens. The hardness average for these samples was 312.14 MPa. This powder processing method has been applied to W/UO2 powders with the SEM of the powders appearing similar to the W/HfO2 powder images.

Tucker, Dennis S.

Fe and O EELS Studies of Ion Irradiated Murchison CM2 Carbonaceous Chondrite Matrix

Introduction: The physical and chemical response of hydrated carbonaceous chondrite materials to space weathering processes is poorly understood. Improving this understanding is a key part of establishing how regoliths on primitive carbonaceous asteroids respond to space weathering processes, knowledge that supports future sample return missions (Hayabusa 2 and OSIRISREx) that are targeting objects of this type. We previously reported on He+ irradiation of Murchison matrix and showed that the irradiation resulted in amorphization of the matrix phyllosilicates, loss of OH, and surface vesiculation. Here, we report electron energy-loss spectroscopy (EELS) measurements of the irradiated material with emphasis on the Fe and O speciation. Sample and Methods: A polished thin section of the Murchison CM2 carbonaceous chondrite was irradiated with 4 kilovolts He(+) (normal incidence) to a total dose of 1 x 10(exp 18) He(+) per square centimeter. We extracted thin sections from both irradiated and unirradiated regions in matrix using focused ion beam (FIB) techniques with electron beam deposition for the protective carbon strap to minimize surface damage artifacts from the FIB milling. The FIB sections were analyzed using a JEOL 2500SE scanning and transmission electron microscope (STEM) equipped with a Gatan Tridiem imaging filter. EELS spectra were collected from 50 nanometer diameter regions with an energy resolution of 0.7 electronvolts FWHM at the zero loss. EELS spectra were collected at low electron doses to minimize possible artifacts from electron-beam irradiation damage. Results and Discussion: Fe L (sub 2,3) EELS spectra from matrix phyllosilicates in CM chondrites show mixed Fe(2+)/Fe(3+) oxidation states with Fe(3+)/Sigma Fe approximately 0.5. Fe L(sub 2,3) spectra from the irradiated/ amorphized matrix phyllosilicates show higher Fe(2+)/Fe(3+) ratios compared to spectra obtained from pristine material at depths beyond the implantation/amorphization layer. We also obtained O Ka spectra from phyllosilicates in both regions of the sample. The O Ka spectra show a pre-edge feature at approximately 530.5 electronvolts that is related to O 2p states hybridized with Fe 3d states. The intensity ratio of the O Ka pre-edge peak relative to the main part of the O Ka edge (that results from transitions of O 1s to 2p states) is lower in the irradiated layer compared to the pristine material and may reflect the loss of O (as OH) as was observed by IR spectroscopy. Conclusions: In addition to amorphization and OH loss, EELS spectra of He(+) irradiated matrix phyllosilicates in Murchison show that some of the Fe(3+) is reduced to Fe(2+). Spectral deconvolution is underway to extract quantitative ratios from the EELS spectra.

Keller, L. P.

An Improved Plastically Dilatant Unified Viscoplastic Constitutive Formulation for Multiscale Analysis of Polymer Matrix Composites Under High Strain Rate Loading

Polymer matrix composites are commonly used to fabricate energy-absorbing structures expected to experience impact loading. As such, a detailed understanding of the dynamic response of the constituent materials is necessary. Since the rate, temperature, and pressure dependence of carbon fiber reinforced polymer matrix composites are primarily manifestations of the rate, temperature, and pressure dependence of the polymer matrix, it is crucial that the constitutive behavior of the matrix be accurately characterized. In this work, an existing unified viscoplastic constitutive formulation is extended to ensure thermodynamic consistency and to more accurately account for the tension-compression asymmetry observed in the response of polymeric materials. A new plastic potential function is proposed, and elementary loading conditions are utilized to determine relations between model constants to ensure nonnegative plastic dissipation, a necessary thermodynamic requirement. Expressions for plastic Poisson’s ratios are derived and are bounded by enforcing nonnegative plastic dissipation. The model is calibrated against available experimental data from tests conducted over a range of strain rates, temperatures, and loading cases on a representative thermoset epoxy; good correlation between simulations and experimental data is obtained. Temperature rises due to the conversion of plastic work to heat are computed via the adiabatic heat energy equation. The viscoplastic polymer model is then used as a constitutive model in the generalized method of cells micromechanics theory to investigate the effects of matrix adiabatic heating on the high strain rate response of a unidirectional composite. The thermodynamic consistency of the model ensures plastic dissipation can only cause an increase in temperature. Simulation results indicate that significant thermal softening due to the conversion of plastic work to heat is observed in the composite for matrix dominated deformation modes.

Impact loading

Effect of Phenolic Matrix Microcracking on the Structural Response of a 3-D Woven Thermal Protection System

The effect of microcracking in the phenolic matrix of a three-dimensional woven thermal protection system (TPS) and the resulting material stiffness reduction was studied via a comparison of finite element analysis results from a linear analysis and an iterative linear analysis. A TPS is necessary to protect space vehicles from the aerodynamic heating of planetary entry. The Heatshield for Extreme Entry Environment Technology (HEEET) project has developed a TPS for use in high heat-flux and pressure missions. The material is a dual-layer continuous dry weave, which is then infiltrated with a low-density phenolic resin matrix to form a rigid ablator. The phenolic resin matrix does not have structural load transfer requirements, and testing has shown that the phenolic resin can fully satisfy thermal requirements when the matrix contains microcracks. Due to high stresses in the through-the-thickness direction of the material, phenolic microcracks may form in the matrix material, which would result in a reduction of stiffness. An exploratory study was conducted to determine if reduction in material stiffness would change the load paths and/or decrease the structural margins. A comparison was performed between a linear finite element analysis that did not take into account phenolic microcracking and an iterative linear finite element analysis that accounted for propagation of phenolic microcracking. Four subcases using varying assumptions were analyzed and the results indicate that the assumed strength at which the phenolic microcracking propagates was the critical parameter for determining the extent of microcracking in the phenolic matrix. Phenolic microcracking does not have an adverse effect on the structural response of the test article and is not a critical failure.

Langston, Sarah L.

Tensile Creep Behavior of SiC(f)/SiC Ceramic Matrix Minicomposites

Single fiber-tow minicomposites represent the major load-bearing element of woven and laminate ceramic matrix composites (CMCs). To understand the effects of fiber type, fiber content, and matrix cracking on tensile creep in SiCf/SiC CMCs, single-tow SiCf/SiC minicomposites with different fiber types and contents were investigated. The minicomposites studied contained either Hi-Nicalon™ or Hi-Nicalon™ Type S SiC fibers with a boron nitride (BN) interphase and a chemical-vapor-infiltrated-silicon-carbide (CVI-SiC) matrix. Tensile creep was performed at 1200 °C in air. A bottom-up creep modeling approach was applied where creep parameters of the fibers and matrix were obtained separately at 1200 °C. Next, a theoretical model based on the rule of mixtures was derived to model the fiber and matrix creep-time-dependent stress redistribution. Fiber and matrix creep parameters, load transfer model results, and numerical modeling were used to construct a creep strain model to predict creep damage evolution of minicomposites with different fiber types and contents.

SiC/SiC Minicomposites Creep

Tracing Presolar Grain Populations Through Matrix Alteration Histories

Primitive meteorites preserve remnants of evolved stars as presolar grains, from asymptotic and red giant branch stars, novae and supernovae. These grains retain records of the formation mechanisms of the stellar environment from which they condensed, as well as physical and chemical processes in the interstellar medium (ISM), solar nebula and the parent bodies in which they are found. Consequently, their study can refine our understanding of the solar system and external stellar and planetary systems. However, presolar grains and their records can be altered, re-equilibrated with the surrounding matrix, and eventually lost through aqueous and thermal processing on the meteorite parent body. NanoSIMS is the most rapid and effective method for in situ identification of presolar grains via their highly anomalous isotopic ratios. Previous studies have measured variations in presolar grain populations between different matrix regions within the same meteorite thin section. However, most of these studies do not identify if there are any ties between these variations in presolar grain populations and the chemical and mineralogical differences in measured matrix regions. Therefore, the goal of this study is to determine if there is measurable variations in the degree of alteration of different matrix regions within the same thin section, and if mineralogical and chemical changes in matrix regions can be used as indicators for presolar grain preservation. Such a correlation could improve the efficiency of presolar grain identification during searches across large surface areas. To do this, we carried out coordinated scanning electron microscope (SEM) and NanoSIMS analyses, using a new SEM method which measures the ppm concentration of 100 nm to 5 μm sized Mg-, S-, Al-, C- and Ca-rich matrix grains.

presolar grains

Randomized Algorithms for Symmetric Nonnegative Matrix Factorization

Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a matrix with a product of a nonnegative, low-rank matrix and it transpose. To design faster and more scalable algorithms for SymNMF we develop two randomized algorithms for its computation. The first method uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to uses this as a low-rank input to rapidly compute a SymNMF. The second methods uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. Here, we prove theoretically that leverage score sampling can approximately solve constrained least squares problems to e-accuracy. Finally we demonstrate both methods work in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.

97 MATHEMATICS AND COMPUTING

Collagen in the spicule organic matrix of the gorgonian Leptogorgia virgulata

Decalcification of the calcareous spicules from the gorgonian Leptogorgia virgulata reveals an organic matrix that may be divided into water insoluble and soluble fractions. The insoluble fraction displays characteristics typical of collagen, which is an unusual component of an invertebrate calcium carbonate structure. This matrix fraction exhibits a collagenous amino acid profile and behavior upon SDS-PAGE. Furthermore, the reducible crosslink, dihydroxylysinonorleucine (DHLNL), is detected in this fraction. The composition of the matrix varies seasonally; i.e., the collagenous composition is most prevalent in the summer. These results indicate that the insoluble matrix is a dynamic structure. Potential roles of this matrix in spicule calcification are discussed.

Non-NASA Center

Efficient Implementation of the Invariant Imbedding T-Matrix Method and the Separation of Variables Method Applied to Large Nonspherical Inhomogeneous Particles

Three terms, ''Waterman's T-matrix method'', ''extended boundary condition method (EBCM)'', and ''null field method'', have been interchangeable in the literature to indicate a method based on surface integral equations to calculate the T-matrix. Unlike the previous method, the invariant imbedding method (IIM) calculates the T-matrix by the use of a volume integral equation. In addition, the standard separation of variables method (SOV) can be applied to compute the T-matrix of a sphere centered at the origin of the coordinate system and having a maximal radius such that the sphere remains inscribed within a nonspherical particle. This study explores the feasibility of a numerical combination of the IIM and the SOV, hereafter referred to as the IIMþSOV method, for computing the single-scattering properties of nonspherical dielectric particles, which are, in general, inhomogeneous. The IIMþSOV method is shown to be capable of solving light-scattering problems for large nonspherical particles where the standard EBCM fails to converge. The IIMþSOV method is flexible and applicable to inhomogeneous particles and aggregated nonspherical particles (overlapped circumscribed spheres) representing a challenge to the standard superposition T-matrix method. The IIMþSOV computational program, developed in this study, is validated against EBCM simulated spheroid and cylinder cases with excellent numerical agreement (up to four decimal places). In addition, solutions for cylinders with large aspect ratios, inhomogeneous particles, and two-particle systems are compared with results from discrete dipole approximation (DDA) computations, and comparisons with the improved geometric-optics method (IGOM) are found to be quite encouraging.

Invariant imbedding method

Temperature Dependence of Electrical Resistance of Woven Melt-Infiltrated SiCf/SiC Ceramic Matrix Composites

Recent studies have successfully shown the use of electrical resistance (ER)measurements to monitor room temperature damage accumulation in SiC fiber reinforced SiC matrix composites (SiCf/SiC) Ceramic Matrix Composites (CMCs). In order to determine the feasibility of resistance monitoring at elevated temperatures, the present work investigates the temperature dependent electrical response of various MI (Melt Infiltrated)-CVI (Chemical Vapor Infiltrated) SiC/SiC composites containing Hi-Nicalon Type S, Tyranno ZMI and SA reinforcing fibers. Test were conducted using a commercially available isothermal testing apparatus as well as a novel, laser-based heating approach developed to more accurately simulate thermomechanical testing of CMCs. Secondly, a post-test inspection technique is demonstrated to show the effect of high-temperature exposure on electrical properties. Analysis was performed to determine the respective contribution of the fiber and matrix to the overall composite conductivity at elevated temperatures. It was concluded that because the silicon-rich matrix material dominates the electrical response at high temperature, ER monitoring would continue to be a feasible method for monitoring stress dependent matrix cracking of melt-infiltrated SiC/SiC composites under high temperature mechanical testing conditions. Finally, the effect of thermal gradients generated during localized heating of tensile coupons on overall electrical response of the composite is determined.

Ceramic Matrix Composites

Effects of Adiabatic Heating on the High Strain Rate Deformation of Polymer Matrix Composites

Polymer matrix composites (PMCs) are increasingly being used in aerospace structures that are expected to experience complex dynamic loading conditions throughout their lifetime. As such, a detailed understanding of the high strain rate behavior of the constituents, particularly the strain rate, temperature, and pressure dependent polymer matrix, is paramount. In this paper, preliminary efforts in modeling experimentally observed temperature rises due to plastic deformation in PMCs subjected to dynamic loading are presented. To this end, an existing isothermal viscoplastic polymer constitutive formulation is extended to model adiabatic conditions by incorporating temperature dependent elastic properties and modifying the components of the inelastic strain rate tensor to explicitly depend on temperature. It is demonstrated that the modified polymer constitutive model is capable of capturing strain rate and temperature dependent yield as well as thermal softening associated with the conversion of plastic work to heat at high rates of strain. The modified constitutive model is then embedded within a strength of materials based micromechanics framework to investigate the manifestation of matrix thermal softening, due to the conversion of plastic work to heat, on the high strain rate response of a T700Epon 862 (T700E862) unidirectional composite. Adiabatic model predictions for high strain rate composite longitudinal tensile, transverse tensile, and in-plane shear loading are presented. Results show a substantial deviation from isothermal conditions; significant thermal softening is observed for matrix dominated deformation modes (transverse tension and in-plane shear), highlighting the importance of accounting for the conversion of plastic work to heat in the polymer matrix in the high strain rate analysis of PMC structures.

polymer matrix composites

Analytical Simulation of Effects of Local Mechanisms and Microstructure on Creep Response of Unidirectional Ceramic Matrix Composites

A micromechanics-based method has been developed to analyze the creep response of uncoated ceramic matrix minicomposites. Although the global stress level at which the creep response is analyzed is lower than the composite proportional limit, local stresses are assumed to be high enough that localized damage is present in the composite in the form of matrix microcracks. To model the composite creep response, including the effects of matrix cracking, a fiber shear lag-based methodology is employed. In this approach, stresses are assumed to vary in the fiber as a function of time and distance from the crack plane. The varying stresses are then used to compute the overall creep strain for the composite. Various assumptions regarding the level of matrix microcracking in the composite and the level of creep in the fiber and matrix in various portions of the composite unit cell are also examined. The creep response of the fiber is modeled using a linear Burgers model. The model is applied to a SiCf/SiC unidirectional minicomposite system. The computed creep results are compared to experimentally obtained values. The effects of the local fiber volume fraction on the overall creep response of the composite are also studied. This work will allow increased understanding of the key material damage mechanisms and load sharing that take place during creep conditions and can be expanded to provide improved analysis methods for full macrocomposites.

Micromechanics

Analytical Simulation of the Effects of Local Mechanisms and Microstructure on the Creep Response of Unidirectional Ceramic Matrix Composites

A micromechanics-based method has been developed to analyze the creep response of uncoated ceramic matrix minicomposites. Although the global stress level at which the creep response is analyzed is lower than the composite proportional limit, local stresses are assumed to be high enough that localized damage is present in the composite in the form of matrix microcracks. To model the composite creep response including the effects of matrix cracking, a fiber shear lag-based methodology is employed. In this approach, stresses are assumed to vary in the fiber as a function of time and distance from the crack plane. The varying stresses are then used to compute the overall creep strain for the composite. Various assumptions regarding the level of matrix microcracking in the composite, and the level of creep in the fiber and matrix in various portions of the composite unit cell, are also examined. The creep response of the fiber is modeled using a linear Burgers model. The model is applied to a SiC f /SiC unidirectional minicomposite system. The computed creep results are compared to experimentally obtained values. The effects of the local fiber volume fraction on the overall creep response of the composite are also studied. This work will allow increased understanding of the key material damage mechanisms and load sharing that take place during creep conditions and can be expanded to provide improved analysis methods for full macrocomposites.

Micromechanics

Randomized Algorithms for Low-Rank Matrix and Tensor Decompositions

This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality reduction, such as the singular value decomposition (SVD) or interpolative (ID) and CUR decompositions. Recent advances in randomized dimensionality reduction are discussed, including new methods of fast matrix sketching and sampling techniques, which are incorporated into classical matrix algorithms for fast low-rank matrix approximations. The extension of randomized matrix algorithms to tensors is then explored for several low-rank tensor decompositions in the CP and Tucker formats, including the higher-order SVD, ID, and CUR decomposition.

Pearce, Katherine J. [The University of Texas at A

Is the Matrix Completion of Reduced Density Matrices Unique?

Reduced density matrices are central to describing observables in many-body quantum systems. In electronic structure theory, the two-particle reduced density matrix (2-RDM) suffices to determine the energy and other key properties. Recent work has used matrix completion, leveraging the low-rank structure of RDMs and approximate theoretical models, to reconstruct the 2-RDM from partial data and thus reduce the computational cost. However, matrix completion is, in general, an under-determined problem. Revisiting Rosina’s theorem (Rosina, M. Queen’s Papers on Pure and Applied Mathematics , 1968, No. 11, 369), we here show that the matrix completion is unique under certain conditions, identifying the subset of 2-RDM elements that enables its exact reconstruction from incomplete information. Building on this, we introduce a hybrid quantum–stochastic algorithm that achieves exact matrix completion, demonstrated through applications to the Fermi–Hubbard model.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH

The DYRKP1 kinase regulates cell wall degradation in Chlamydomonas by inducing matrix metalloproteinase expression

Abstract The cell wall of plants and algae is an important cell structure that protects cells from changes in the external physical and chemical environment. This extracellular matrix, composed of polysaccharides and glycoproteins, must be constantly remodeled throughout the life cycle. However, compared to matrix polysaccharides, little is known about the mechanisms regulating the formation and degradation of matrix glycoproteins. We report here that a plant kinase belonging to the dual-specificity tyrosine phosphorylation-regulated kinase (DYRKP1) family present in all eukaryotes regulates cell wall degradation after mitosis of Chlamydomonas reinhardtii by inducing the expression of matrix metalloproteinases. Without DYRKP1, daughter cells cannot disassemble parental cell walls and remain trapped inside for more than 10 days. On the other hand, the dual-specificity tyrosine phosphorylation-regulated kinase complementation lines show normal degradation of the parental cell wall. Transcriptomic and proteomic analyses indicate a marked downregulation of MMP gene expression and accumulation, respectively, in the dyrkp1 mutants. The mutants deficient in matrix metalloproteinases retain palmelloid structures for a longer time than the background strain, like dyrkp1 mutants. Our findings show that dual-specificity tyrosine phosphorylation-regulated kinase, by ensuring timely MMP expression, enables the successful execution of the cell cycle. Altogether, this study provides insight into the life cycle regulation in plants and algae.

Kim, Minjae (ORCID:0000000223561295)

Microstructure and bonding between calcium aluminate cement‐containing gahnite–alumina matrix and refractory aggregates

Calcium aluminate cement enhances the thermomechanical properties of refractory castables through the formation of acicular calcium hexaluminate (CaO·6Al 2 O 3 ), Ca 2 Mg 2 Al 28 O 46 (CAM-I), and CaMg 2 Al 16 O 27 (CAM-II) phases in MgO- or MgAl 2 O 4 -containing castables. The compatibility of CA 6 with gahnite (ZnAl 2 O 4 ), and acicular Ca 2 Zn 2 Al 28 O 46 (CAZ-I) and CaZn 2 Al 16 O 27 (CAZ-II) phases formation have been previously reported. Here, in this work, the interaction between a CAC binder containing ZnAl 2 O 4 -Al 2 O 3 matrix with commonly used refractory aggregates such as tabular alumina (TA), alumina-rich (AR90, AR78) and stoichiometric (SM72) MgAl 2 O 4 spinels, and fused and dead-burned magnesia (FM, DBM, respectively) were investigated at 1650°C for 5 h. Microstructural analysis, using digital microscopy, scanning electron microscopy, and energy dispersive spectroscopy, revealed the formation of acicular CaZn 0.18 Al 11.82 O 18.91 , CAZ-I and CAZ-II grains, and strong interfacial bonding between the matrix and TA and spinel aggregates. FM and DBM were found to debond from the matrix. Thick interface layers were observed between the matrix and all the aggregates but TA. Null hypothesis significance testing (NHST) shows that the difference in the number of acicular grains between the interface zone and the bulk matrix (Z) is statistically significant for AR90/Z, SM72/Z, FM/Z, and DBM/Z interfaces, but not significant for TA/Z and AR78/Z. The role of the aggregates’ chemistry on the interfacial bonding and microstructure evolution is discussed.

Ramteke, Rajat Durgesh [Univ. of Alabama, Birmingh

Space shuttle guidance, navigation and control equation document no. 4: Precision state and filter weighting matrix extrapolation

The Precision State and Filter Weighting Matrix Extrapolation Routine is described which provides the capability to extrapolate any spacecraft geocentric state vector either backwards or forwards in time through a force field consisting of the earth's primary central-force gravitational attraction and a superimposed perturbing acceleration. The routine also provides the capability of extrapolating the filter-weighting matrix along the precision trajectory. This matrix is a square root form of the error covariance matrix and contains statistical information relative to the accuracies of the state vectors and certain other optionally estimated quantities. The routine is a cooled algorithm for the numerical solution of modified forms of the basic differential equations which are satisfied by the geocentric state vector of the spacecraft's center of mass and by the filter-weighting matrix.

Robertson, W. M.