Analytical determination of the adjoint vector for optimum space trajectories
Analytic approximation for initial adjoint vector for optimal /minimum propellant/ space trajectories
Hazelrigg, G. A.↗
SEARCH · Search NASA
Results for “VECTOR SPACE”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
NASA Tech Briefs, November 2009
Topics covered include: Cryogenic Chamber for Servo-Hydraulic Materials Testing; Apparatus Measures Thermal Conductance Through a Thin Sample from Cryogenic to Room Temperature; Rover Attitude and Pointing System Simulation Testbed; Desktop Application Program to Simulate Cargo-Air-Drop Tests; Multimodal Friction Ignition Tester; Small-Bolt Torque-Tension Tester; Integrated Spacesuit Audio System Enhances Speech Quality and Reduces Noise; Hardware Implementation of a Bilateral Subtraction Filter; Simple Optoelectronic Feedback in Microwave Oscillators; Small X-Band Oscillator Antennas; Free-Space Optical Interconnect Employing VCSEL Diodes; Discrete Fourier Transform Analysis in a Complex Vector Space; Miniature Scroll Pumps Fabricated by LIGA; Self-Assembling, Flexible, Pre-Ceramic Composite Preforms; Flight-speed Integral Image Analysis Toolkit; Work Coordination Engine; Multi-Mission Automated Task Invocation Subsystem; Autonomously Calibrating a Quadrupole Mass Spectrometer; Determining Spacecraft Reaction Wheel Friction Parameters; Composite Silica Aerogels Opacified with Titania; Multiplexed Colorimetric Solid-Phase Extraction; Detecting Airborne Mercury by Use of Polymer/Carbon Films; Lattice-Matched Semiconductor Layers on Single Crystalline Sapphire Substrate; Pressure-Energized Seal Rings to Better Withstand Flows; Rollerjaw Rock Crusher; Microwave Sterilization and Depyrogenation System; Quantifying Therapeutic and Diagnostic Efficacy in 2D Microvascular Images; NiF2/NaF:CaF2/Ca Solid-State High-Temperature Battery Cells; Critical Coupling Between Optical Fibers and WGM Resonators; Microwave Temperature Profiler Mounted in a Standard Airborne Research Canister; Alternative Determination of Density of the Titan Atmosphere; Solar Rejection Filter for Large Telescopes; Automated CFD for Generation of Airfoil Performance Tables; Progressive Classification Using Support Vector Machines; Active Learning with Irrelevant Examples; A Data Matrix Method for Improving the Quantification of Element Percentages of SEM/EDX Analysis; Deployable Shroud for the International X-Ray Observatory; Improved Model of a Mercury Ring Damper; Optoelectronic pH Meter: Further Details; X-38 Advanced Sublimator; and Solar Simulator Represents the Mars Surface Solar Environment.
Source record↗
A unified development of several techniques for the representation of random vectors and data sets
Linear vector space theory is used to develop a general representation of a set of data vectors or random vectors by linear combinations of orthonormal vectors such that the mean squared error of the representation is minimized. The orthonormal vectors are shown to be the eigenvectors of an operator. The general representation is applied to several specific problems involving the use of the Karhunen-Loeve expansion, principal component analysis, and empirical orthogonal functions; and the common properties of these representations are developed.
Bundick, W. T.↗
Structure of the isotropic transport operators in three independent space variables
Based on the idea of separation of variables, a spectral theory for the three-dimensional, stationary, isotropic transport operator in a vector space of complex-valued Borel functions results in continuous sets of regular and generalized eigenfunctions.
Abu-Shumays, I. K.↗
Reduced conservatism in stability robustness bounds by state transformation
This note addresses the issue of 'conservatism' in the time domain stability robustness bounds obtained by the Liapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Liapunov approach with respect to the basis of the vector space in which the Liapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the 'optimal' transformation. Several examples are presented to illustrate the proposed analysis.
Yedavalli, R. K.↗
Using trees to compute approximate solutions to ordinary differential equations exactly
Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.
Grossman, Robert↗
Lie theory and control systems defined on spheres
It is shown that in constructing a theory for the most elementary class of control problems defined on spheres, some results from the Lie theory play a natural role. To understand controllability, optimal control, and certain properties of stochastic equations, Lie theoretic ideas are needed. The framework considered here is the most natural departure from the usual linear system/vector space problems which have dominated control systems literature. For this reason results are compared with those previously available for the finite dimensional vector space case.
Brockett, R. W.↗
Hybrid NN/SVM Computational System for Optimizing Designs
A computational method and system based on a hybrid of an artificial neural network (NN) and a support vector machine (SVM) (see figure) has been conceived as a means of maximizing or minimizing an objective function, optionally subject to one or more constraints. Such maximization or minimization could be performed, for example, to optimize solve a data-regression or data-classification problem or to optimize a design associated with a response function. A response function can be considered as a subset of a response surface, which is a surface in a vector space of design and performance parameters. A typical example of a design problem that the method and system can be used to solve is that of an airfoil, for which a response function could be the spatial distribution of pressure over the airfoil. In this example, the response surface would describe the pressure distribution as a function of the operating conditions and the geometric parameters of the airfoil. The use of NNs to analyze physical objects in order to optimize their responses under specified physical conditions is well known. NN analysis is suitable for multidimensional interpolation of data that lack structure and enables the representation and optimization of a succession of numerical solutions of increasing complexity or increasing fidelity to the real world. NN analysis is especially useful in helping to satisfy multiple design objectives. Feedforward NNs can be used to make estimates based on nonlinear mathematical models. One difficulty associated with use of a feedforward NN arises from the need for nonlinear optimization to determine connection weights among input, intermediate, and output variables. It can be very expensive to train an NN in cases in which it is necessary to model large amounts of information. Less widely known (in comparison with NNs) are support vector machines (SVMs), which were originally applied in statistical learning theory. In terms that are necessarily oversimplified to fit the scope of this article, an SVM can be characterized as an algorithm that (1) effects a nonlinear mapping of input vectors into a higher-dimensional feature space and (2) involves a dual formulation of governing equations and constraints. One advantageous feature of the SVM approach is that an objective function (which one seeks to minimize to obtain coefficients that define an SVM mathematical model) is convex, so that unlike in the cases of many NN models, any local minimum of an SVM model is also a global minimum.
Rai, Man Mohan↗
On the convergence of an iterative formulation of the electromagnetic scattering from an infinite grating of thin wires
Contraction theory is applied to an iterative formulation of electromagnetic scattering from periodic structures and a computational method for insuring convergence is developed. A short history of spectral (or k-space) formulation is presented with an emphasis on application to periodic surfaces. The mathematical background for formulating an iterative equation is covered using straightforward single variable examples including an extension to vector spaces. To insure a convergent solution of the iterative equation, a process called the contraction corrector method is developed. Convergence properties of previously presented iterative solutions to one-dimensional problems are examined utilizing contraction theory and the general conditions for achieving a convergent solution are explored. The contraction corrector method is then applied to several scattering problems including an infinite grating of thin wires with the solution data compared to previous works.
Brand, J. C.↗
Investigations on the hierarchy of reference frames in geodesy and geodynamics
Problems related to reference directions were investigated. Space and time variant angular parameters are illustrated in hierarchic structures or towers. Using least squares techniques, model towers of triads are presented which allow the formation of linear observation equations. Translational and rotational degrees of freedom (origin and orientation) are discussed along with and the notion of length and scale degrees of freedom. According to the notion of scale parallelism, scale factors with respect to a unit length are given. Three-dimensional geodesy was constructed from the set of three base vectors (gravity, earth-rotation and the ecliptic normal vector). Space and time variations are given with respect to a polar and singular value decomposition or in terms of changes in translation, rotation, deformation (shear, dilatation or angular and scale distortions).
Grafarend, E. W.↗
Automation of POST Cases via External Optimizer and "Artificial p2" Calculation
During early conceptual design of complex systems, speed and accuracy are often at odds with one another. While many characteristics of the design are fluctuating rapidly during this phase there is nonetheless a need to acquire accurate data from which to down-select designs as these decisions will have a large impact upon program life-cycle cost. Therefore enabling the conceptual designer to produce accurate data in a timely manner is tantamount to program viability. For conceptual design of launch vehicles, trajectory analysis and optimization is a large hurdle. Tools such as the industry standard Program to Optimize Simulated Trajectories (POST) have traditionally required an expert in the loop for setting up inputs, running the program, and analyzing the output. The solution space for trajectory analysis is in general non-linear and multi-modal requiring an experienced analyst to weed out sub-optimal designs in pursuit of the global optimum. While an experienced analyst presented with a vehicle similar to one which they have already worked on can likely produce optimal performance figures in a timely manner, as soon as the "experienced" or "similar" adjectives are invalid the process can become lengthy. In addition, an experienced analyst working on a similar vehicle may go into the analysis with preconceived ideas about what the vehicle's trajectory should look like which can result in sub-optimal performance being recorded. Thus, in any case but the ideal either time or accuracy can be sacrificed. In the authors' previous work a tool called multiPOST was created which captures the heuristics of a human analyst over the process of executing trajectory analysis with POST. However without the instincts of a human in the loop, this method relied upon Monte Carlo simulation to find successful trajectories. Overall the method has mixed results, and in the context of optimizing multiple vehicles it is inefficient in comparison to the method presented POST's internal optimizer functions like any other gradient-based optimizer. It has a specified variable to optimize whose value is represented as optval, a set of dependent constraints to meet with associated forms and tolerances whose value is represented as p2, and a set of independent variables known as the u-vector to modify in pursuit of optimality. Each of these quantities are calculated or manipulated at a certain phase within the trajectory. The optimizer is further constrained by the requirement that the input u-vector must result in a trajectory which proceeds through each of the prescribed events in the input file. For example, if the input u-vector causes the vehicle to crash before it can achieve the orbital parameters required for a parking orbit, then the run will fail without engaging the optimizer, and a p2 value of exactly zero is returned. This poses a problem, as this "non-connecting" region of the u-vector space is far larger than the "connecting" region which returns a non-zero value of p2 and can be worked on by the internal optimizer. Finding this connecting region and more specifically the global optimum within this region has traditionally required the use of an expert analyst.
Dees, Patrick D.↗
Acceleration hodograph analysis techniques for powered orbital trajectories, phase 3 Final report
Hodographic equations of motion for powered orbital trajectories in acceleration vector space
Altman, S. P.↗
Geometric Theory of Functional Differential Equations
Functional differential equations for determining n-dimemensional vector spaces - Liapunov function asymptotic theory of linear systems, and Floquet theory
Hale, J. K.↗
A Generalized Unimodality
Generalization of unimodality for random objects taking values in finite dimensional vector space
Olshen, R. A.↗
Principles of error detection and error correction codes
Report is reviewed which considers theoretical basis of groups, rings, fields, and vector spaces, and their relationship to algebraic coding theory. Report serves as summary for engineers and scientists involved in data handling and processing systems.
Perlman, M.↗
Attitude determination and sensor alignment via weighted least squares affine transformations
Vehicle attitude determination and guidance sensor orientation by vector space matrix method, minimizing errors by weighted least squares affine transformation technique
Davenport, P. B.↗
A linear programming manual
Computer solutions of linear programming problems are outlined. Information covers vector spaces, convex sets, and matrix algebra elements for solving simultaneous linear equations. Dual problems, reduced cost analysis, ranges, and error analysis are illustrated.
Tuey, R. C.↗
Structure in the nonlinear saturation spectrum of parametric instabilities.
The nonlinear saturation spectrum of the decay instability is obtained in the limit of small spontaneous emission, for comparable ion and electron temperatures, from numerical solutions of a kinetic equation based on an accurate expression for the nonlinearity. The spectral energy occupies several pairs of isolated saucer-shaped regions in wave-vector space. The regions increase in thickness, angular diameter, and number as the pump power is increased. The theory thus predicts the generation of waves propagating in directions which can differ substantially from the direction of the pump field. Ionospheric observations confirm this prediction; they were difficult to reconcile with the predictions of previous theories based on an approximated expression for the nonlinearity. The present work also corrects the results of previous one-dimensional theories that used an accurate expression for the nonlinearity and predicted 'spectral lines' in the limit of vanishing spontaneous emission.
Fejer, J. A.↗