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NASA NTRS1989

An analytic development of orbit determination for a distant, planetary orbiter

With the advent of the Mariner '71 Mission, NASA has been sending spacecraft to orbit various distant bodies within the solar system. At present, there is still no adequate theory describing the inherent state estimation accuracy, based on two-way, coherent range-rate data. It is the purpose of this article to lay the groundwork for a general elliptic theory, and in addition to provide an analytic solution for the special case of circular orbits. It is shown that circular orbits about distant planets may suffer singularities in over-all position error estimation. These singularities are due to orbit inclination, placement of the line-of-nodes, and insignificant cross-velocity at the start and end of retrograde motion when orbiting a superior planet. Even though these conclusions appear to yield poor state estimation, one should not be unduly alarmed inasmuch as the stated conditions for singularity are not maintained for extended periods during typical mission scenarios. However, mission analysts should be aware of these potential pitfalls and realize that spuriously large results for circular orbiters can be obtained and are not the result of incorrect assumptions or faulty software. The general elliptic problem appears so involved that analytic inversion at this time is just not feasible, and in any case the resulting expression for the position error would likely be so lengthy that any understanding would be lost in the maze.

Russell, R. K.↗

NASA NTRS1989

Modern CACSD using the Robust-Control Toolbox

The Robust-Control Toolbox is a collection of 40 M-files which extend the capability of PC/PRO-MATLAB to do modern multivariable robust control system design. Included are robust analysis tools like singular values and structured singular values, robust synthesis tools like continuous/discrete H(exp 2)/H infinity synthesis and Linear Quadratic Gaussian Loop Transfer Recovery methods and a variety of robust model reduction tools such as Hankel approximation, balanced truncation and balanced stochastic truncation, etc. The capabilities of the toolbox are described and illustated with examples to show how easily they can be used in practice. Examples include structured singular value analysis, H infinity loop-shaping and large space structure model reduction.

Chiang, Richard Y.↗

NASA NTRS1989

Robotic control of the seven-degree-of-freedom NASA laboratory telerobotic manipulator

A computationally efficient robotic control scheme for the NASA Laboratory Telerobotic Manipulator (LTM) is presented. This scheme utilizes the redundancy of the seven-degree-of-freedom LTM to avoid joint limits and singularities. An analysis to determine singular configurations is presented. Performance criteria are determined based on the joint limits and singularity analysis. The control scheme is developed in the framework of resolved rate control using the gradient projection method, and it does not require the generalized inverse of the Jacobian. An efficient formulation for determining the joint velocities of the LTM is obtained. This control scheme is well suited for real-time implementation, which is essential if the end-effector trajectory is continuously modified based on sensory feedback. Implementation of this scheme on a Motorola 68020 VME bus-based controller of the LTM is in progress. Simulation results demonstrating the redundancy utilization in the robotic mode are presented.

Dubey, R. V.↗

NASA NTRS1990

On the Lagrangian description of unsteady boundary-layer separation. I - General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Van Dommelen, Leon L.↗

NASA NTRS1991

Fractal dimension in nonhyperbolic chaotic scattering

In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.

Lau, Yun-Tung↗

NASA NTRS1990

A scheme for theoretical and experimental evaluation of multivariable system stability robustness

A scheme for theoretical and experimental evaluation of multivariable system stability robustness measures is described. The experimental scheme is based on the estimation of frequency responses from power-spectra analyses of a set of time responses due to sine-sweep excitation at each input and then computation of the appropriate singular values. The procedure can also be used on an open-loop stable system to predict the closed-loop stability before closing the loop. Classical Nyquist diagrams can also be constructed to determine one-loop-at-a-time gain or phase margins. The scheme has been implemented in a wind tunnel test for singular-value evaluation of digital flutter suppression control laws and compared with theory. The singular values at the plant input and output of the actual system and the theoretical model are qualitatively similar but have some discrepancies.

Mukhopadhyay, Vivek↗

NASA NTRS1991

On the nonlinear development of the most unstable Goertler vortex mode

The nonlinear development of the most unstable Gortler vortex mode in boundary layer flows over curved walls is investigated. The most unstable Gortler mode is confined to a viscous wall layer of thickness O(G -1/5) and has spanwise wavelength O(G 11/5); it is, of course, most relevant to flow situations where the Gortler number G is much greater than 1. The nonlinear equations covering the evolution of this mode over an O(G -3/5) streamwise lengthscale are derived and are found to be of a fully nonparallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Gortler equations valid for O(1) Gortler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion, whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flowfield as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.

Denier, James P.↗

NASA NTRS1992

Solution of two-body relativistic bound state equations with confining plus Coulomb interactions

Studies of meson spectroscopy have often employed a nonrelativistic Coulomb plus Linear Confining potential in position space. However, because the quarks in mesons move at an appreciable fraction of the speed of light, it is necessary to use a relativistic treatment of the bound state problem. Such a treatment is most easily carried out in momentum space. However, the position space Linear and Coulomb potentials lead to singular kernels in momentum space. Using a subtraction procedure we show how to remove these singularities exactly and thereby solve the Schroedinger equation in momentum space for all partial waves. Furthermore, we generalize the Linear and Coulomb potentials to relativistic kernels in four dimensional momentum space. Again we use a subtraction procedure to remove the relativistic singularities exactly for all partial waves. This enables us to solve three dimensional reductions of the Bethe-Salpeter equation. We solve six such equations for Coulomb plus Confining interactions for all partial waves.

Maung, Khin Maung↗

NASA NTRS1992

Kinematic equations for control of the redundant eight-degree-of-freedom advanced research manipulator 2

The forward position and velocity kinematics for the redundant eight-degree-of-freedom Advanced Research Manipulator 2 (ARM2) are presented. Inverse position and velocity kinematic solutions are also presented. The approach in this paper is to specify two of the unknowns and solve for the remaining six unknowns. Two unknowns can be specified with two restrictions. First, the elbow joint angle and rate cannot be specified because they are known from the end-effector position and velocity. Second, one unknown must be specified from the four-jointed wrist, and the second from joints that translate the wrist, elbow joint excluded. There are eight solutions to the inverse position problem. The inverse velocity solution is unique, assuming the Jacobian matrix is not singular. A discussion of singularities is based on specifying two joint rates and analyzing the reduced Jacobian matrix. When this matrix is singular, the generalized inverse may be used as an alternate solution. Computer simulations were developed to verify the equations. Examples demonstrate agreement between forward and inverse solutions.

Williams, Robert L., II↗

NASA NTRS1993

On the nonlinear development of the most unstable Goertler vortex mode TLSP - Final report

The nonlinear development of the most unstable Goertler vortex mode in boundary layer flows over curved walls is investigated. The most unstable Goertler mode is confined to a viscous wall layer of thickness O(G -1/5) and has spanwise wavelength O(G 11/5); it is, of course, most relevant to flow situations where the Goertler number G is much greater than 1. The nonlinear equations covering the evolution of this mode over an O(G -3/5) streamwise lengthscale are derived and are found to be of a fully nonparallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Goertler equations valid for O(1) Goertler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion, whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flowfield as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.

Denier, James P.↗

NASA NTRS1992

Minimum fuel spacecraft reorientation

Fuel optimal solutions for the reorientation of an inertially symmetric rigid spacecraft with independent 3-axes controls are investigated. All possible optimal control strategies are identified. These include bangbang solutions, finite order singular arcs and infinite order singular arcs. Higher order necessary conditions for optimality of finite order singular arcs are presented. Numerical examples of fuel optimal solutions with fixed maneuver time are presented involving all of the theoretically possible control logics.

Seywald, H.↗

NASA NTRS1994

Robust inverse kinematics using damped least squares with dynamic weighting

This paper presents a general method for calculating the inverse kinematics with singularity and joint limit robustness for both redundant and non-redundant serial-link manipulators. Damped least squares inverse of the Jacobian is used with dynamic weighting matrices in approximating the solution. This reduces specific joint differential vectors. The algorithm gives an exact solution away from the singularities and joint limits, and an approximate solution at or near the singularities and/or joint limits. The procedure is here implemented for a six d.o.f. teleoperator and a well behaved slave manipulator resulted under teleoperational control.

Schinstock, D. E.↗

NASA NTRS1993

Tension fracture of laminates for transport fuselage. Part 2: Large notches

Tests were conducted on over 200 center-crack specimens to evaluate: (a) the tension-fracture performance of candidate materials and laminates for commercial fuselage applications; and (b) the accuracy of several failure criteria in predicting response. Crack lengths of up to 12 inches were considered. Other variables included fiber/matrix combination, layup, lamination manufacturing process, and intraply hybridization. Laminates fabricated using the automated tow-placement process provided significantly higher tension-fracture strengths than nominally identical tape laminates. This confirmed earlier findings for other layups, and possibly relates to a reduced stress concentration resulting from a larger scale of repeatable material inhomogeneity in the tow-placed laminates. Changes in material and layup result in a trade-off between small-notch and large-notch strengths. Toughened resins and 0 deg-dominate layups result in higher small-notch strengths but lower large-notch strengths than brittle resins, 90 deg and 45 deg dominated layups, and intraply S2-glass hybrid material forms. Test results indicate that strength-prediction methods that allow for a reduced order singularity of the crack-tip stress field are more successful at predicting failure over a range of notch sizes than those relying on the classical square-root singularity. The order of singularity required to accurately predict large-notch strength from small-notch data was affected by both material and layup. Measured crack-tip strain distributions were generally higher than those predicted using classical methods. Traditional methods of correcting for finite specimen width were found to be lacking, confirming earlier findings with other specimen geometries. Fracture tests of two stiffened panels, identical except for differing materials, with severed central stiffeners resulted in nearly identical damage progression and failure sequences. Strain-softening laws implemented within finite element models appear attractive to account for load redistribution in configured structure due to damage-induced crack tip softening

Walker, Tom H.↗

NASA NTRS1996

Balanced Placement Of Vibration Sensors And Actuators

Mathematical approach that involves Hankel singular values leads to method for estimation of optimal placement of relatively few vibration sensors and actuators in multivariable flexible structure. Approach extends and complements previous developments regarding approximate decomposition, to Hankel singular values, of singular values of controllability and observability grammian matrices of multivariable flexible structure. Mathematical constructs also described in "Designing An Approximately Balanced LQG Compensator" (NPO-19000).

Gawronski, Wodek↗

NASA NTRS1995

Fully-Explicit and Self-Consistent Algebraic Reynolds Stress Models

A fully-explicit, self-consistent algebraic expression for the Reynolds stress, which is the exact solution to the Reynolds stress transport equation in the 'weak equilibrium' limit for two-dimensional mean flows for all linear and some quasi-linear pressure-strain models, is derived. Current explicit algebraic Reynolds stress models derived by employing the 'weak equilibrium' assumption treat the production-to-dissipation (P/epsilon) ratio implicitly, resulting in an effective viscosity that can be singular away from the equilibrium limit. In the present paper, the set of simultaneous algebraic Reynolds stress equations are solved in the full non-linear form and the eddy viscosity is found to be non-singular. Preliminary tests indicate that the model performs adequately, even for three dimensional mean flow cases. Due to the explicit and non-singular nature of the effective viscosity, this model should mitigate many of the difficulties encountered in computing complex turbulent flows with the algebraic Reynolds stress models.

Girimaji, Sharath S.↗

NASA NTRS1993

Linearization of Attitude-Control Error Dynamics

Direction cosines and quaternions are useful for representing rigid-body attitude because they exhibit no kinematic singularities. Each utilizes more variables than the minimum three degrees of freedom required to specify attitude. Therefore, application of a nonlinear inversion procedure to either formulation introduces singularities. Furthermore, in designing an attitude-control system, it is not appropriate to express attitude error as a difference of direction cosines (or quaternions). One should employ a measure of attitude error that not only is minimal but preserves orthogonal rotation properties as well. This note applies an inversion procedure to an appropriate measure of attitude error, so that the singularity occurs when the error reaches +/- 180 deg. This approach leads to the realization of a new model-follower attitude-control system that exhibits exact linear attitude-error dynamics.

Bach, Ralph↗

NASA NTRS1998

Nonlinear Interaction of Detuned Instability Waves in Boundary-Layer Transition: Amplitude Equations

The non-equilibrium critical-layer analysis of a system of frequency-detuned resonant-triads is presented. In this part of the analysis, the system of partial differential critical-layer equations derived in Part I is solved analytically to yield the amplitude equations which are analyzed using a combination of asymptotic and numerical methods. Numerical solutions of the inviscid non-equilibrium oblique-mode amplitude equations show that the frequency-detuned self-interaction enhances the growth of the lower-frequency oblique modes more than the higher-frequency ones. All amplitudes become singular at the same finite downstream position. The frequency detuning delays the occurrence of the singularity. The spanwise-periodic mean-flow distortion and low-frequency nonlinear modes are generated by the critical-layer interaction between frequency-detuned oblique modes. The nonlinear mean flow and higher harmonics as well as the primary instabilities become as large as the base mean flow in the inviscid wall layer in the downstream region where the distance from the singularity is of the order of the wavelength scale.

Lee, Sang Soo↗

NASA NTRS1998

A Framework for a Supervisory Expert System for Robotic Manipulators with Joint-Position Limits and Joint-Rate Limits

This report addresses the problem of path planning and control of robotic manipulators which have joint-position limits and joint-rate limits. The manipulators move autonomously and carry out variable tasks in a dynamic, unstructured and cluttered environment. The issue considered is whether the robotic manipulator can achieve all its tasks, and if it cannot, the objective is to identify the closest achievable goal. This problem is formalized and systematically solved for generic manipulators by using inverse kinematics and forward kinematics. Inverse kinematics are employed to define the subspace, workspace and constrained workspace, which are then used to identify when a task is not achievable. The closest achievable goal is obtained by determining weights for an optimal control redistribution scheme. These weights are quantified by using forward kinematics. Conditions leading to joint rate limits are identified, in particular it is established that all generic manipulators have singularities at the boundary of their workspace, while some have loci of singularities inside their workspace. Once the manipulator singularity is identified the command redistribution scheme is used to compute the closest achievable Cartesian velocities. Two examples are used to illustrate the use of the algorithm: A three link planar manipulator and the Unimation Puma 560. Implementation of the derived algorithm is effected by using a supervisory expert system to check whether the desired goal lies in the constrained workspace and if not, to evoke the redistribution scheme which determines the constraint relaxation between end effector position and orientation, and then computes optimal gains.

Mutambara, Arthur G. O.↗