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NASA NTRS1991

Method and apparatus for configuration control of redundant robots

A method and apparatus to control a robot or manipulator configuration over the entire motion based on augmentation of the manipulator forward kinematics is disclosed. A set of kinematic functions is defined in Cartesian or joint space to reflect the desirable configuration that will be achieved in addition to the specified end-effector motion. The user-defined kinematic functions and the end-effector Cartesian coordinates are combined to form a set of task-related configuration variables as generalized coordinates for the manipulator. A task-based adaptive scheme is then utilized to directly control the configuration variables so as to achieve tracking of some desired reference trajectories throughout the robot motion. This accomplishes the basic task of desired end-effector motion, while utilizing the redundancy to achieve any additional task through the desired time variation of the kinematic functions. The present invention can also be used for optimization of any kinematic objective function, or for satisfaction of a set of kinematic inequality constraints, as in an obstacle avoidance problem. In contrast to pseudoinverse-based methods, the configuration control scheme ensures cyclic motion of the manipulator, which is an essential requirement for repetitive operations. The control law is simple and computationally very fast, and does not require either the complex manipulator dynamic model or the complicated inverse kinematic transformation. The configuration control scheme can alternatively be implemented in joint space.

Seraji, Homayoun↗

NASA NTRS1991

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. As an alternative to the positive real lemma and to the s-domain inequalities, these conditions provide a recursive algorithm for testing positive realness which result in a set of simple algebraic conditions. By relating the positive real property to the associated variational problem, a unified derivation of necessary and sufficient conditions for optimality of both singular and nonsingular problems is derived.

Wang, Q.↗

NASA NTRS1990

System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. As an alternative to the positive real lemma and to the s-domain inequalities, these conditions provide a recursive algorithm for testing positive realness which results in a set of simple algebraic conditions. By relating the positive real property to the associated variational problem, a unified derivation of necessary and sufficient conditions for optimality of both singular and nonsingular problems is derived.

Wang, Q.↗

NASA NTRS1993

State-constrained booster trajectory solutions via finite elements and shooting

This paper presents an extension of a FEM formulation based on variational principles. A general formulation for handling internal boundary conditions and discontinuities in the state equations is presented, and the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions which only touch the state constraint and solutions which have a boundary arc of finite length are considered. Suitable shape and test functions are chosen for a FEM discretization. All element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form. The final form of the algebraic equations is then derived. A simple state-constrained problem is solved. Then, for a practical application of the use of the FEM formulation, a launch vehicle subject to a dynamic pressure constraint (a first-order state inequality constraint) is solved. The results presented for the launch-vehicle trajectory have some interesting features, including a touch-point solution.

Bless, Robert R.↗

NASA NTRS1979

The near-wake structure of the Atmosphere Explorer C /AE-C/ satellite - A parametric investigation

Measurements of ion current, electron temperature, and values of space potential obtained from the cylindrical electrostatic probe on board the Atmosphere Explorer C (AE-C) satellite were used to examine, in a parametric manner, the angular distribution of charge around the satellite. Interest is focused on nighttime equatorial data in the altitude range 275-620 km, which yields a wide range for the parameter R sub D (the radius of the satellite divided by the ambient value of the Debye lengths), including R sub D greater than 100, which is of practical significance to large space platforms. The variations of normalized ion current in the wake zone of the AE-C satellite appear to display an exponential dependence on R sub D for 'constant' values of other relevant parameters. The angular variations of electron temperature and space potential in the close vicinity of the satellite's surface were examined and compared with results from the Explorer 31 satellite. The variation of the ratio of measured to computed space potential with electron temperature was examined using data from both the AE-C and Explorer 31 satellites. It was found that the ratio is greater than unity. Possible causes for the above inequality are discussed.

Samir, U.↗

NASA NTRS1988

On the convergence of difference approximations to scalar conservation laws

A unified treatment is given for time-explicit, two-level, second-order-resolution (SOR), total-variation-diminishing (TVD) approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced to obtain results in terms of the latter. The existence of a cell entropy inequality is discussed, and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first-order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Osher, Stanley↗

NASA NTRS1984

Global variation of the para hydrogen fraction in Jupiter's atmosphere and implications for dynamics on the outer planets

A detailed analysis of the Voyager infrared spectrometer measurements on Jupiter's atmosphere is presented, and possible implications of para hydrogen disequilibrium for the energetics and dynamics of that atmosphere are examined. The method of data analysis is described, and results for the large scale latitude variation of the para hydrogen fraction are presented. The Jovian results show pronounced latitude variation, and are compared with other parameters including wind fields, thermal structure, and various indicators of atmospheric clouds. The problem of equilibration rate is reexamined, and it is concluded that on Jupiter the equilibration time is longer than the radiative time constant at the level of emission to space, but that this inequality reverses at greater depths. A model for the interaction of fluid motions with the ortho-para conversion process is presented, and a consistent mixing length theory for the reacting ortho-para mixture is developed. Several implications of the Jovian data for atmospheric energetics and stability on the outer planets are presented.

Conrath, B. J.↗

NASA NTRS1971

The influence of the great inequality on the secular disturbing function of the planetary system.

This paper derives the contribution by the great inequality to the secular disturbing function of the principal planets. Andoyer's expansion of the planetary disturbing function and von Zeipel's method of eliminating the periodic terms is employed; thereby, the corrected secular disturbing function for the planetary system is derived. The conclusion is drawn that the canonicity of the equations for the secular variation of the heliocentric elements can be preserved if there be retained, in the secular disturbing function, terms only of the second and fourth order relative to the eccentricity and inclinations. The Krylov-Bogoliubov method is suggested for eliminating periodic terms, if it is desired to include the secular perturbations of the fifth and higher order in the heliocentric elements. The additional part of the secular disturbing function derived in this paper can be included in existing theories of the secular effects of principal planets.

Musen, P.↗

NASA NTRS1972

N-burn optimal analytic trajectories.

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

Jezewski, D. J.↗

NASA NTRS1978

Expansion and broadening of coronal loop transients - A theoretical explanation

Consequences are examined of the assumption that an observed coronal loop transient is a twisted rope of magnetic-field lines expanding and broadening in the background coronal plasma and magnetic field. It is shown that the expansion can be accounted for by the azimuthal component of the field; the observed broadening of the loop as it moves outward can be accounted for by the longitudinal component of the field. In order to have a net outward force and at the same time avoid a classical pinch (sausage) instability, the two components of the field must satisfy a certain inequality. It is predicted that, as the loop rises, the width (h) of its top portion should vary proportionally with distance (R) from the sun's center. This is in good agreement with measurements that show h is proportional to the 0.8 power of R. The prediction that the radius of curvature of the top portion of the loop should be proportional to R differs from the measured variation. The difference could be accounted for by a drag due to the background coronal field that flattens the loop's top.

Mouschovias, T. CH.↗

NASA NTRS1989

Flutter suppression using eigenspace freedoms to meet requirements

A constrained optimization methodology has been developed which allows specific use of eigensystem freedoms to meet design requirements. A subset of the available eigenvector freedoms was employed. The eigenvector freedoms associated with a particular closed-loop eigenvalue are coefficients of basis vectors which span the subspace in which that closed-loop vector must lie. Design requirements are included as a vector of inequality constraints. The procedure was successfully applied to develop an unscheduled controller which stabilizes symmetric flutter of an aeroelastic vehicle to a dynamic pressure 44 percent above the open-loop flutter point. The design process proceeded from full-state feedback to the inclusion of a full-order observer to the selection of an eighth-order controller which preserved the full-state sensitivity characteristics. Only a subset of the design freedoms was utilized (i.e., assuming full-state feedback only four out of 26 eigenvectors were used, and no variations were made in the closed-loop eigenvalues). Utilization of additional eigensystem freedoms could further improve the controller.

Adams, William M., Jr.↗

NASA NTRS2021

Spatial Variation of Fine Particulate Matter Levels in Nairobi Before and During the COVID-19 Curfew: Implications for Environmental Justice

The temporary decrease of fine particulate matter (PM(sub 2.5)) concentrations in many parts of the world due to the COVID-19 lockdown spurred discussions on urban air pollution and health. However there has been little focus on sub-Saharan Africa, as few African cities have air quality monitors and if they do, these data are often not publicly available. Spatial differentials of changes in PM(sub 2.5) concentrations as a result of COVID also remain largely unstudied. To address this gap, we use a serendipitous mobile air quality monitoring deployment of eight Sensirion SPS 30 sensors on motorbikes in the city of Nairobi starting on 16 March 2020, before a COVID-19 curfew was imposed on 25 March and continuing until 5 May 2020. We developed a random-forest model to estimate PM(sub 2.5) surfaces for the entire city of Nairobi before and during the COVID-19 curfew. The highest PM(sub 2.5) concentrations during both periods were observed in the poor neighborhoods of Kariobangi, Mathare, Umoja, and Dandora, located to the east of the city center. Changes in PM(sub 2.5) were heterogeneous over space. PM(sub 2.5) concentrations increased during the curfew in rapidly urbanizing, the lower-middle-class neighborhoods of Kahawa, Kasarani, and Ruaraka, likely because residents switched from LPG to biomass fuels due to loss of income. Our results indicate that COVID-19 and policies to address it may have exacerbated existing air pollution inequalities in the city of Nairobi. The quantitative results are preliminary, due to sampling limitations and measurement uncertainties, as the available data came exclusively from low-cost sensors. This research serves to highlight that spatial data that is essential for understanding structural inequalities reflected in uneven air pollution burdens and differential impacts of events like the COVID pandemic. With the help of carefully deployed low cost sensors with improved spatial sampling and at least one reference-quality monitor for calibration, we can collect data that is critical for developing targeted interventions that address environmental injustice in the African context.

Environmental justice↗

NASA NTRS2023

Chemical Thermodynamics and the Mathematical Integration of Reaction Kinetics

Key advances in the development of numerical methods for non-reacting compressible flows have been enabled by translating physical requirements into concrete numerical guidelines, such as the satisfaction of entropy inequalities for shock-capturing techniques [Lax, Contributions to Nonlinear Functional Analysis (1971) 603-634]. In the present work, we present nonlinear numerical analysis tools that draw from Chemical Thermodynamics , the branch of Nonequilibrium Thermodynamics that deals with chemical reactions. Through Gibbs formalism, chemical thermodynamics provides a well-known theoretical expression for the chemical equilibrium constant of a reaction in terms of reduced chemical potentials. A less-known, yet extremely valuable result, due to [Krambeck, Arch. Ration. Mech. Anal. , 38 (1970) 317], states that when this expression is implemented, mass-action kinetic models are consistent with the dynamical prescriptions of the 2nd law of thermodynamics. For fixed-temperature ordinary differential equations modeling constant-volume reacting gas mixtures, this leads to a decreasing Helmholtz free energy. If the temperature is allowed to vary in accordance with conservation of energy (1st law), this leads to the statement of increasing entropy. These nonlinear prescriptions can, and should be, used to further develop temporal integration techniques for reaction kinetics. We demonstrate that Krambeck's result holds even when the equilibrium constants are approximated from data. We prove this result by constructing the implicit free energy and the implicit entropy inherent to a given approximation. This is first done for a 5-species, 17-reaction model problem for air. With this structure established, elements of discrete entropy-stability theory [Tadmor, Acta Numer. , 12 (2003) 451] are leveraged to examine the consistency of time-integration schemes with these prescriptions. Using chemical potentials, one can compute the respective contributions of the kinetics model and of the temporal scheme to free energy/entropy variations. We introduce a nonlinear-stable version of the Discontinuous-Galerkin (DG) scheme in time which shows robustness improvements over the standard linearly-stable version. Most notably, the maximum timestep that can be resolved with the nonlinearly-stable variant tends to grow with polynomial order, in contrast to the linearly-stable variant. We generalize our constructions to arbitrary systems of reversible chemical reactions, ultimately showing that the compressible reacting Euler system admits the opposite of the implicitly constructed thermodynamic entropy as a mathematical entropy . This lays important theoretical foundations towards robust scheme development [Harten, J. Comput. Phys. 49 (1983) 151-164].

STMD↗