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All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

Vibrating Systems with Singular Mass-Inertia Matrices

Vibrating systems with singular mass-inertia matrices arise in recent continuum models of Smart Structures (beams with PZT strips) in assessing the damping attainable with rate feedback. While they do not quite yield 'distributed' controls, we show that they can provide a fixed nonzero lower bound for the damping coefficient at all mode frequencies. The mathematical machinery for modelling the motion involves the theory of Semigroups of Operators. We consider a Timoshenko model for torsion only, a 'smart string,' where the damping coefficient turns out to be a constant at all frequencies. We also observe that the damping increases initially with the feedback gain but decreases to zero eventually as the gain increases without limit.

Balakrishnan, A. V.↗
NASA NTRS1999

Singular Plasma Disturbance in the Low-Latitude F Region

We describe here a new phenomenon characterized by unusual patterns of ion drifts inside ion density depletion regions observed by the AE-E satellite in the low-latitude F region. In about 30 depletions, vertical ion drift relative to the background was upward on the western sides, downward on the eastern sides, and zero near the middle where the density depletion was greatest. These drift characteristics are distinct from those observed in plasma bubble depletions. The structures reported here were observed on circular orbits below 300 km altitude and had density depletions of up to 2 orders of magnitude or more below the ambient ion density. The upward and downward drift excursions were up to 200 m/s relative to the background. Almost all these structures were observed over oceans or near coasts and largely between +/- 10 deg and +/- 30 deg clip latitude. The structures were observed mostly as isolated, single depletion regions with the majority of them about 250 km wide in the east-west direction. They occurred during quiet magnetic conditions with near-equal occurrence frequencies in the premidnight and postmidnight periods. The characteristic density and drift signatures indicate westward propagating disturbances in which the bottomside F layer is first lifted and then returned back to its original position, leaving the ionosphere undisturbed after the disturbance passes by. The estimated speed of these disturbances is of the order of 200 m/s. These unique solitary plasma disturbances, which we designate as singular plasma disturbances, are associated with a propagating source of E x B drift, not driven by neutral perturbations at the altitude of observation.

Singh, Sardul↗
NASA NTRS2000

Singular Isothermal Disks and the Formation of Multiple Stars

A crucial missing ingredient in previous theoretical studies of fragmentation is the inclusion of dynamically important levels of magnetic fields. As a minimal model for a candidate presursor to the formation of binary and multiple stars, we therefore consider the equilibrium configuration of isopedically magnetized, scale-free, singular isothermal disks, without the assumption of axial symmetry. We find that lopsided (M = 1) configurations exist at any dimensionless rotation rate, including zero. Multiple-lobed (M = 2, 3, 4, ...) configurations bifurcate from an underlying axisymmetric sequence at progressively higher dimensionless rates of rotation, but such nonaxisymmetric sequences always terminate in shockwaves before they have a chance to fission into separate bodies. We advance the hypothesis that binary and multiple star-formation from smooth (i.e., not highly turbulent) starting states that are supercritical but in unstable mechanical balance requires the rapid (i.e., dynamical) loss of magnetic flux at some stage of the ensuing gravitational collapse.

Galli, Daniele↗
NASA NTRS2000

Singular Isothermal Disks: Nonaxiymmetric Bifurcations and Equilibria - Paper 2

We review the difficulties of the classical fission and fragmentation hypotheses for the formation of binary and multiple stars. A crucial missing ingredient in previous theoretical studies is the inclusion of dynamically important levels of magnetic fields. As a minimal model for a candidate presursor to the formation of binary and multiple stars, we therefore formulate and solve the problem of the equilibria of isopedically magnetized, singular isothermal disks, without the assumption of axial symmetry. Considerable analytical progress can be made if we restrict our attention to models that are scale-free, i.e., that have surface densities that vary inversely with distance omega from the rotation axis of the system. In agreement with earlier analysis by Syer and Tremaine, we find that lopsided (M = 1) configurations exist at any dimensionless rotation rate, including zero. Multiple-lobed (M = 2, 3, 4, ...) configurations bifurcate from an underlying axisymmetric sequence at progressively higher dimensionless rates of rotation, but such nonaxisymmetric sequences always terminate in shockwaves before they have a chance to fission into M = 2, 3, 4, ... separate bodies. On the basis of our experience in this paper, we advance the hypothesis that binary and multiple star-formation from smooth (i.e., not highly turbulent) starting states that are supercritical but in unstable mechanical balance requires the rapid (i.e., dynamical) loss of magnetic flux at some stage of the ensuing gravitational collapse.

Galli, Danielle↗
NASA NTRS2000

Robust Fault Detection for Aircraft Using Mixed Structured Singular Value Theory and Fuzzy Logic

The purpose of fault detection is to identify when a fault or failure has occurred in a system such as an aircraft or expendable launch vehicle. The faults may occur in sensors, actuators, structural components, etc. One of the primary approaches to model-based fault detection relies on analytical redundancy. That is the output of a computer-based model (actually a state estimator) is compared with the sensor measurements of the actual system to determine when a fault has occurred. Unfortunately, the state estimator is based on an idealized mathematical description of the underlying plant that is never totally accurate. As a result of these modeling errors, false alarms can occur. This research uses mixed structured singular value theory, a relatively recent and powerful robustness analysis tool, to develop robust estimators and demonstrates the use of these estimators in fault detection. To allow qualitative human experience to be effectively incorporated into the detection process fuzzy logic is used to predict the seriousness of the fault that has occurred.

Collins, Emmanuel G.↗
NASA NTRS2002

On the Formulation of Weakly Singular Displacement/Traction Integral Equations; and Their Solution by the MLPG Method

In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.

Atluri, Satya N.↗
NASA NTRS2002

On the Singularity in the Estimation of the Quaternion-of-Rotation

It has been claimed in the archival literature that the covariance matrix of a Kalman filter, which is designed to estimate the quaternion-of-rotation, is necessarily rank, deficient because the normality constraint of the quaternion produces dependence between the quaternion elements. In reality, though, this phenomenon does not occur. The covariance matrix is not singular, and the filter is well behaved. Several simple examples are presented th at demonstrate the regularity of the covariance matrix. First, a Kalman filter is designed to estimate variables subject to a functional relationship. Then the particular problem of quaternion estimation is analyzed. It is shown that the discrepancy stems from the fact that the functional relationship exists between the elements of the quaternion but not between its estimate elements.

Bar-Itzhack, Itzhack Y.↗
NASA NTRS1998

Planning Paths Through Singularities in the Center of Mass Space

The center of mass space is a convenient space for planning motions that minimize reaction forces at the robot's base or optimize the stability of a mechanism. A unique problem associated with path planning in the center of mass space is the potential existence of multiple center of mass images for a single Cartesian obstacle, since a single center of mass location can correspond to multiple robot joint configurations. The existence of multiple images results in a need to either maintain multiple center of mass obstacle maps or to update obstacle locations when the robot passes through a singularity, such as when it moves from an elbow-up to an elbow-down configuration. To illustrate the concepts presented in this paper, a path is planned for an example task requiring motion through multiple center of mass space maps. The object of the path planning algorithm is to locate the bang- bang acceleration profile that minimizes the robot's base reactions in the presence of a single Cartesian obstacle. To simplify the presentation, only non-redundant robots are considered and joint non-linearities are neglected.

Doggett, William R.↗
NASA NTRS2005

A Survey of Singular Value Decomposition Methods and Performance Comparison of Some Available Serial Codes

This contractor report describes a performance comparison of available alternative complete Singular Value Decomposition (SVD) methods and implementations which are suitable for incorporation into point spread function deconvolution algorithms. The report also presents a survey of alternative algorithms, including partial SVD's special case SVD's, and others developed for concurrent processing systems.

Plassman, Gerald E.↗
NASA NTRS2009

Optimal Control Modification for Robust Adaptation of Singularly Perturbed Systems with Slow Actuators

Recently a new optimal control modification has been introduced that can achieve robust adaptation with a large adaptive gain without incurring high-frequency oscillations as with the standard model-reference adaptive control. This modification is based on an optimal control formulation to minimize the L2 norm of the tracking error. The optimal control modification adaptive law results in a stable adaptation in the presence of a large adaptive gain. This study examines the optimal control modification adaptive law in the context of a system with a time scale separation resulting from a fast plant with a slow actuator. A singular perturbation analysis is performed to derive a modification to the adaptive law by transforming the original system into a reduced-order system in slow time. The model matching conditions in the transformed time coordinate results in increase in the feedback gain and modification of the adaptive law.

Nguyen, Nhan T.↗
NASA NTRS2015

Robust Stability Analysis of the Space Launch System Control Design: A Singular Value Approach

Classical stability analysis consists of breaking the feedback loops one at a time and determining separately how much gain or phase variations would destabilize the stable nominal feedback system. For typical launch vehicle control design, classical control techniques are generally employed. In addition to stability margins, frequency domain Monte Carlo methods are used to evaluate the robustness of the design. However, such techniques were developed for Single-Input-Single-Output (SISO) systems and do not take into consideration the off-diagonal terms in the transfer function matrix of Multi-Input-Multi-Output (MIMO) systems. Robust stability analysis techniques such as H(sub infinity) and mu are applicable to MIMO systems but have not been adopted as standard practices within the launch vehicle controls community. This paper took advantage of a simple singular-value-based MIMO stability margin evaluation method based on work done by Mukhopadhyay and Newsom and applied it to the SLS high-fidelity dynamics model. The method computes a simultaneous multi-loop gain and phase margin that could be related back to classical margins. The results presented in this paper suggest that for the SLS system, traditional SISO stability margins are similar to the MIMO margins. This additional level of verification provides confidence in the robustness of the control design.

Pei, Jing↗
NASA NTRS2018

A Singular Value Decomposition Framework for Retrievals with Vertical Distribution Information from Greenhouse Gas Column Absorption Spectroscopy Measurements

We describe a variation of the Optimal Estimation (OE) method for greenhouse gas remote sensing retrievals using a singular value decomposition (SVD) and an uninformative prior. The SVD method is capable of discerning vertical information in column integrated absorption measurements. While traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We discuss the method, illustrate its use on an integrated path differential absorption CO2 lidar measurement model, and compare it to traditional optimal estimation using numerical simulations. In the absence of forward model error, the mathematics behind the SVD method guarantee it to be bias-free, which is confirmed by the numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. While the SVD approach can be used for most trace gas retrievals, it is particularly useful for situations where prior knowledge of the trace gas profile is poor. The SVD analysis is also useful in illustrating how vertical information is treated by the traditional OE approach.

Ramanathan, Anand K.↗
NASA NTRS2018

A Singular Value Decomposition Framework for Retrievals with Vertical Distribution Information from Greenhouse Gas Column Absorption Spectroscopy Measurements

We review the singular value decomposition (SVD) framework and use it for quantifying and discerning vertical information in greenhouse gas retrievals from column integrated absorption measurements. While the commonly used traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We review the SVD method, explicitly recognize the use of an uninformative prior and show it to incur no bias from the choice of the prior. We also make the connection between the SVD method and the pseudo-inverse, which makes it more intuitive and easy to understand. We illustrate the use of the SVD method on an integrated path differential absorption CO2 lidar measurement model and verify our derivations and bias-free properties versus optimal estimation using numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. Hence, the SVD method is particularly useful for situations in which knowledge of the prior mean and prior covariance of the true state (e.g., greenhouse gas profiles) is inadequate.

Ramanathan, Anand K.↗
NASA NTRS2019

Efficient On-Orbit Singularity-Free Geopotential Estimation

The complexity of the geopotential model can heavily impact the navigation error in satellites and spacecraft. Geopotential models of the accuracy needed for spaceflight are too complicated for flight computers to run at the rate needed by the navigation system. There are methods to make the geopotential model more efficient while maintaining the needed accuracy, which include: using an efficient method for the full model, propagating to avoid singularities, and running the full model at a low rate and propagating to the needed rate. These methods can decrease the computational requirement enough to be run by the flight computer at the rate required of the navigation system.

Amert, Joel↗
NASA NTRS2019

Efficient On-Orbit Singularity-Free Geopotential Estimation

For vehicles in orbital cruise, either in orbit around a body or in a transfer orbit, gravity is the primary external force; therefore, the complexity of the geopotential model heavily influences the accuracy of the navigation state. As this type of spacecraft requires a highly accurate geopotential model and performs the computation numerous times a second, this calculation needs to be efficient. This calculation needs to include the first and second derivatives of the geopotential because navigation systems can require both the gravitational acceleration and gradient for inertial integration and state filtering. The most efficient method for calculating the geopotential, the forward column method, is discussed in detail, as well as a method to avoid the singularities that exist when using this method. This is shown to decrease the computation time of the geopotential compared to other popular methods. In addition, methods for first and second order propagation are discussed, which decrease the rate at which the full geopotential model needs to be calculated while maintaining the accuracy required of navigation systems. These estimation methods are shown to decrease the necessary computation of the gravity model by multiple orders of magnitude. This method decreases the computation time of the Exploration Upper Stage navigation system geopotential model by potentially an order of magnitude compared with the previous model without affecting the navigation error. A properly implemented geopotential model can have the accuracy of an 8 by 8 model while approaching the computational requirement as using only the J234 coefficients.

Amert, Joel↗
NASA NTRSDate not supplied

Efficient and Singularity-Aware Inverse Kinematics for 6-DOF and 7-DOF Revolute Manipulators

Inverse kinematics (IK) for 6-DOF and 7-DOF revolute robot manipulators may be decomposed into canonical subproblems which find the angles on circles where they intersect with other geometric objects. We present new algebraic solutions and geometric interpretations for six subproblems. We demonstrate significant performance improvements over existing IK methods. 7-DOF manipulators have one redundant DOF to parameterize, so we introduce the general SEW angle, which generalizes the conventional SEW angle but with an arbitrary reference direction function. One special choice, the stereographic SEW angle, has a singularity when the wrist is on a half-line instead of a full line.

Computational Geometry↗