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Performance Evaluation of a Cartesian Move Algorithm for the LSMS Family of Cable-Driven Cranes

The Lightweight Surface Manipulation System, or LSMS, is a family of cable-actuated cranes, designed for surface operations on the Moon and Mars. Its structural design focuses on three fundamental traits for space environments: a high payload-to-mass ratio to reduce launch costs, compactness for space travel in reduced cargo bays, and self-erecting capabilities for operations before astronauts arrive. The resulting design confers nonlinear and switched dynamics to the LSMS, making these cranes an interesting challenge for autonomous and teleoperated applications. One of the protocols required for these applications is a Cartesian move algorithm. This algorithm is responsible for placing the end effector of the crane at a desired location, specified in Cartesian coordinates. This paper evaluates the performance of a Cartesian move algorithm developed for the LSMS at the NASA Langley Research Center. To this end, several payload manipulation tasks were performed with LSMS-L35, the smallest version within the LSMS family. The evaluated algorithm can place the end effector with accuracy, as shown in the hardware and simulation tests.

cable-actuated↗

NASA NTRS1976

Polyatomic molecule vibrations

Polyatomic molecule vibrations are analyzed as harmonic vibrations along normal coordinates. The energy eigenvalues are found for linear and nonlinear symmetric triatomic molecules for valence bond models of the potential function with arbitrary coupling coefficients; such models can usually be fitted to observed energy levels with reasonably good accuracy. Approximate normal coordinates for the H2O molecule are discussed. Degenerate vibrational modes such as occur in CO2 are analyzed and expressions for Fermi resonance between close-lying states of the same symmetry are developed. The bending modes of linear triatomic molecules are expressed in terms of Laguerre polynomials in cylindrical coordinates as well as in terms of Hermite polynomials in Cartesian coordinates. The effects of large-amplitude bending such as occur in the C3 molecule are analyzed, along with anharmonic effects, which split the usually degenerate bending mode energy levels. Finally, the vibrational frequencies, degeneracies, and symmetry properties of XY3, X2Y2, and XY4 type molecules are discussed.

Source record↗

NASA NTRS1979

Non-force-free solar magnetic fields in magnetohydrostatic equilibrium

The objective of the paper is to examine a class of non-force-free fields analytically. Specifically, magnetic fields in magnetohydrostatic equilibrium with a plasma in a gravitational field are treated in an approximation of two independent variables but three vector components. Spherical coordinates are emphasized, although the formal results for cylindrical and Cartesian coordinates are presented in the appendices. Formal solutions for the magnetic-field components are obtained in terms of the plasma variables, and field line equations are derived. A final equation governing the plasma variables is then obtained. Procedures are developed for analyzing this equation and obtaining sets of self-consistent particular solutions to the governing equations; a number of such sets of solutions are presented. As an example, one solution set is examined, illustrating the application of the results to the analysis of solar observational data.

Comfort, R. H.↗

NASA NTRS1993

Parametric motion control of robotic arms: A biologically based approach using neural networks

A neural network based system is presented which is able to generate point-to-point movements of robotic manipulators. The foundation of this approach is the use of prototypical control torque signals which are defined by a set of parameters. The parameter set is used for scaling and shaping of these prototypical torque signals to effect a desired outcome of the system. This approach is based on neurophysiological findings that the central nervous system stores generalized cognitive representations of movements called synergies, schemas, or motor programs. It has been proposed that these motor programs may be stored as torque-time functions in central pattern generators which can be scaled with appropriate time and magnitude parameters. The central pattern generators use these parameters to generate stereotypical torque-time profiles, which are then sent to the joint actuators. Hence, only a small number of parameters need to be determined for each point-to-point movement instead of the entire torque-time trajectory. This same principle is implemented for controlling the joint torques of robotic manipulators where a neural network is used to identify the relationship between the task requirements and the torque parameters. Movements are specified by the initial robot position in joint coordinates and the desired final end-effector position in Cartesian coordinates. This information is provided to the neural network which calculates six torque parameters for a two-link system. The prototypical torque profiles (one per joint) are then scaled by those parameters. After appropriate training of the network, our parametric control design allowed the reproduction of a trained set of movements with relatively high accuracy, and the production of previously untrained movements with comparable accuracy. We conclude that our approach was successful in discriminating between trained movements and in generalizing to untrained movements.

Bock, O.↗

NASA NTRS1998

An Alternative Lunar Ephemeris Model for On-Board Flight Software Use

In calculating the position vector of the Moon in on-board flight software, one often begins by using a series expansion to calculate the ecliptic latitude and longitude of the Moon, referred to the mean ecliptic and equinox of date. One then performs a reduction for precession, followed by a rotation of the position vector from the ecliptic plane to the equator, and a transformation from spherical to Cartesian coordinates before finally arriving at the desired result: equatorial J2000 Cartesian components of the lunar position vector. An alternative method is developed here in which the equatorial J2000 Cartesian components of the lunar position vector are calculated directly by a series expansion, saving valuable on-board computer resources.

Simpson, David G.↗

NASA NTRS1999

An Alternative Lunar Ephemeris Model for On-Board Flight Software Use

Simpson, David G.↗

NASA NTRS1980

Nonlinear finite element analysis: An alternative formulation

A geometrical nonlinear analysis based on an alternative definition of strain is presented. Expressions for strain are obtained by computing the change in length of the base vectors in the curvilinear element coordinate system. The isoparametric element formulation is assumed in the global Cartesian coordinate system. The approach is based on the minimization of the strain energy, and the resulting nonlinear equations are solved by the modified Newton method. Integration of the first and second variation of the strain energy is performed numerically in the case of two and three dimensional elements. Application is made to a simple long cantilever beam.

Merazzi, S.↗

NASA NTRS1991

A solution-adaptive mesh algorithm for dynamic/static refinement of two and three dimensional grids

An adaptive grid algorithm has been developed in two and three dimensions that can be used dynamically with a solver or as part of a grid refinement process. The algorithm employs a transformation from the Cartesian coordinate system to a general coordinate space, which is defined as a parallelepiped in three dimensions. A weighting function, independent for each coordinate direction, is developed that will provide the desired refinement criteria in regions of high solution gradient. The adaptation is performed in the general coordinate space and the new grid locations are returned to the Cartesian space via a simple, one-step inverse mapping. The algorithm for relocation of the mesh points in the parametric space is based on the center of mass for distributed weights. Dynamic solution-adaptive results are presented for laminar flows in two and three dimensions.

Benson, Rusty A.↗

NASA NTRS2002

A Lightning Channel Retrieval Algorithm for the North Alabama Lightning Mapping Array (LMA)

A new multi-station VHF time-of-arrival (TOA) antenna network is, at the time of this writing, coming on-line in Northern Alabama. The network, called the Lightning Mapping Array (LMA), employs GPS timing and detects VHF radiation from discrete segments (effectively point emitters) that comprise the channel of lightning strokes within cloud and ground flashes. The network will support on-going ground validation activities of the low Earth orbiting Lightning Imaging Sensor (LIS) satellite developed at NASA Marshall Space Flight Center (MSFC) in Huntsville, Alabama. It will also provide for many interesting and detailed studies of the distribution and evolution of thunderstorms and lightning in the Tennessee Valley, and will offer many interesting comparisons with other meteorological/geophysical wets associated with lightning and thunderstorms. In order to take full advantage of these benefits, it is essential that the LMA channel mapping accuracy (in both space and time) be fully characterized and optimized. In this study, a new revised channel mapping retrieval algorithm is introduced. The algorithm is an extension of earlier work provided in Koshak and Solakiewicz (1996) in the analysis of the NASA Kennedy Space Center (KSC) Lightning Detection and Ranging (LDAR) system. As in the 1996 study, direct algebraic solutions are obtained by inverting a simple linear system of equations, thereby making computer searches through a multi-dimensional parameter domain of a Chi-Squared function unnecessary. However, the new algorithm is developed completely in spherical Earth-centered coordinates (longitude, latitude, altitude), rather than in the (x, y, z) cartesian coordinates employed in the 1996 study. Hence, no mathematical transformations from (x, y, z) into spherical coordinates are required (such transformations involve more numerical error propagation, more computer program coding, and slightly more CPU computing time). The new algorithm also has a more realistic definition of source altitude that accounts for Earth oblateness (this can become important for sources that are hundreds of kilometers away from the network). In addition, the new algorithm is being applied to analyze computer simulated LMA datasets in order to obtain detailed location/time retrieval error maps for sources in and around the LMA network. These maps will provide a more comprehensive analysis of retrieval errors for LMA than the 1996 study did of LDAR retrieval errors. Finally, we note that the new algorithm can be applied to LDAR, and essentially any other multi-station TWA network that depends on direct line-of-site antenna excitation.

Koshak, William↗

NASA NTRS1995

Easing The Calculation Of Bolt-Circle Coordinates

Bolt Circle Calculation (BOLT-CALC) computer program used to reduce significant time consumed in manually computing trigonometry of rectangular Cartesian coordinates of holes in bolt circle as shown on blueprint or drawing. Eliminates risk of computational errors, particularly in cases involving many holes or in cases in which coordinates expressed to many significant digits. Program assists in many practical situations arising in machine shops. Written in BASIC. Also successfully compiled and implemented by use of Microsoft's QuickBasic v4.0.

Burley, Richard K.↗

NASA NTRS1974

Conservation equations of gasdynamics in curvilinear coordinate systems

Description of a new method of writing the conservation equations of gasdynamics in curvilinear coordinates which eliminates undifferentiated terms. It is thus possible to readily apply difference schemes derived for Cartesian coordinates which conserve mass, momentum, and energy in the total flow field. The method is derived for orthogonal coordinates, and then extended to cover the most general class of coordinate transformations, using general tensor analysis. Several special features of the equations are discussed.

Vinokur, M.↗

NASA NTRS1992

Minimization of deviations of gear real tooth surfaces determined by coordinate measurements

The deviations of a gear's real tooth surface from the theoretical surface are determined by coordinate measurements at the grid of the surface. A method was developed to transform the deviations from Cartesian coordinates to those along the normal at the measurement locations. Equations are derived that relate the first order deviations with the adjustment to the manufacturing machine-tool settings. The deviations of the entire surface are minimized. The minimization is achieved by application of the least-square method for an overdetermined system of linear equations. The proposed method is illustrated with a numerical example for hypoid gear and pinion.

Litvin, F. L.↗

NASA NTRS1972

Coordinate transformation by minimizing correlations between parameters

This investigation was to determine the transformation parameters (three rotations, three translations and a scale factor) between two Cartesian coordinate systems from sets of coordinates given in both systems. The objective was the determination of well separated transformation parameters with reduced correlations between each other, a problem especially relevant when the sets of coordinates are not well distributed. The above objective is achieved by preliminarily determining the three rotational parameters and the scale factor from the respective direction cosines and chord distances (these being independent of the translation parameters) between the common points, and then computing all the seven parameters from a solution in which the rotations and the scale factor are entered as weighted constraints according to their variances and covariances obtained in the preliminary solutions. Numerical tests involving two geodetic reference systems were performed to evaluate the effectiveness of this approach.

Kumar, M.↗

NASA NTRS1977

National Geodetic Satellite Program, Part II: Ohio State University

Data analysis was carried out to obtain an improved global network combining all participating tracking stations in a single worldwide coordinate system. Mathematical formulations were derived, programmed, and tested. Observational data were used to determine the relative positions of tracking stations in an arbitrary Cartesian coordinate system. The position of this coordinate system was estimated with respect to an absolute system. A primary network was established in which station positions were known to an internal consistency of 10 meters or better for the following purposes: (1) to establish the relative relationships between the various geodetic datums in use around the world; and (2) to connect isolated tracking stations, islands, navigational beacons, and other points of interest. Results are presented.

Mueller, I. I.↗

NASA NTRS2004

Rapid Structured Volume Grid Smoothing and Adaption Technique

A rapid, structured volume grid smoothing and adaption technique, based on signal processing methods, was developed and applied to the Shuttle Orbiter at hypervelocity flight conditions in support of the Columbia Accident Investigation. Because of the fast pace of the investigation, computational aerothermodynamicists, applying hypersonic viscous flow solving computational fluid dynamic (CFD) codes, refined and enhanced a grid for an undamaged baseline vehicle to assess a variety of damage scenarios. Of the many methods available to modify a structured grid, most are time-consuming and require significant user interaction. By casting the grid data into different coordinate systems, specifically two computational coordinates with arclength as the third coordinate, signal processing methods are used for filtering the data [Taubin, CG v/29 1995]. Using a reverse transformation, the processed data are used to smooth the Cartesian coordinates of the structured grids. By coupling the signal processing method with existing grid operations within the Volume Grid Manipulator tool, problems related to grid smoothing are solved efficiently and with minimal user interaction. Examples of these smoothing operations are illustrated for reduction in grid stretching and volume grid adaptation. In each of these examples, other techniques existed at the time of the Columbia accident, but the incorporation of signal processing techniques reduced the time to perform the corrections by nearly 60%. This reduction in time to perform the corrections therefore enabled the assessment of approximately twice the number of damage scenarios than previously possible during the allocated investigation time.

Alter, Stephen J.↗

NASA NTRS2006

Rapid Structured Volume Grid Smoothing and Adaption Technique

A rapid, structured volume grid smoothing and adaption technique, based on signal processing methods, was developed and applied to the Shuttle Orbiter at hypervelocity flight conditions in support of the Columbia Accident Investigation. Because of the fast pace of the investigation, computational aerothermodynamicists, applying hypersonic viscous flow solving computational fluid dynamic (CFD) codes, refined and enhanced a grid for an undamaged baseline vehicle to assess a variety of damage scenarios. Of the many methods available to modify a structured grid, most are time-consuming and require significant user interaction. By casting the grid data into different coordinate systems, specifically two computational coordinates with arclength as the third coordinate, signal processing methods are used for filtering the data [Taubin, CG v/29 1995]. Using a reverse transformation, the processed data are used to smooth the Cartesian coordinates of the structured grids. By coupling the signal processing method with existing grid operations within the Volume Grid Manipulator tool, problems related to grid smoothing are solved efficiently and with minimal user interaction. Examples of these smoothing operations are illustrated for reductions in grid stretching and volume grid adaptation. In each of these examples, other techniques existed at the time of the Columbia accident, but the incorporation of signal processing techniques reduced the time to perform the corrections by nearly 60%. This reduction in time to perform the corrections therefore enabled the assessment of approximately twice the number of damage scenarios than previously possible during the allocated investigation time.

Alter, Stephen J.↗

NASA NTRS1984

An efficient approximate factorization implicit scheme for the equations of gasdynamics

An efficient implicit finite-difference algorithm for the gas dynamic equations utilizing matrix reduction techniques is presented. A significant reduction in arithmetic operations is achieved while maintaining the same favorable stability characteristics and generality found in the Beam and Warming approximate factorization algorithm. Steady-state solutions to the conservative Euler equations in generalized coordinates are obtained for transonic flows about a NACA 0012 airfoil. The theoretical extension of the matrix reduction technique to the full Navier-Stokes equations in Cartesian coordinates is presented in detail. Linear stability, using a Fourier stability analysis, is demonstrated and discussed for the one-dimensional Euler equations. It is shown that the method offers advantages over the conventional Beam and Warming scheme and can retrofit existing Beam and Warming codes with minimal effort.

Barth, T. J.↗

NASA NTRS1995

A FFT Method for the Quasiclassical Selection of Initial Ro-Vibrational States of Triatomic Molecules

This paper describes the use of an exact fast Fourier transform (FFT) method to prepare specified vibrational-rotational states of triatomic molecules. The method determines the Fourier coefficients needed to describe the coordinates and momenta of a vibrating-rotating triatomic molecule. Once the Fourier coefficients of a particular state are determined, it is possible to easily generate as many random sets of initial cartesian coordinates and momenta as desired. All the members of each set will correspond to the particular vibrational-rotational state selected. For example, in the case of the ground vibrational state of a non-rotating water molecule, the calculated actions of 100 sets of initial conditions produced actions within 0.001 h(bar) of the specified quantization values and energies within 5 cm(sup -1) of the semiclassical eigenvalue. The numerical procedure is straightforward for states in which all the fundamental frequencies are independent. However for states for which the fundamental frequencies become commensurate (resonance states), there are additional complications. In these cases it is necessary to determine a new set of "fundamental" frequencies and to modify the quantization conditions. Once these adjustments are made, good results are obtained for resonance states. The major problems are in labelling the large number of Fourier coefficients and the presence of regions of chaotic motion. Results are presented for the vibrational states of H2O and HCN and the ro-vibrational states of H2O.

Eaker, Charles W.↗