Showing papers on "Equations of motion published in 1987"

A unified approach for motion and force control of robot manipulators: The operational space formulation

Abstract: A framework for the analysis and control of manipulator systems with respect to the dynamic behavior of their end-effectors is developed. First, issues related to the description of end-effector tasks that involve constrained motion and active force control are discussed. The fundamentals of the operational space formulation are then presented, and the unified approach for motion and force control is developed. The extension of this formulation to redundant manipulator systems is also presented, constructing the end-effector equations of motion and describing their behavior with respect to joint forces. These results are used in the development of a new and systematic approach for dealing with the problems arising at kinematic singularities. At a singular configuration, the manipulator is treated as a mechanism that is redundant with respect to the motion of the end-effector in the subspace of operational space orthogonal to the singular direction.

Frictional–collisional constitutive relations for granular materials, with application to plane shearing

Abstract: Within a granular material stress is transmitted by forces exerted at points of mutual contact between particles. When the particles are close together and deformation of the assembly is slow, contacts are sustained for long times, and these forces consist of normal reactions and the associated tangential forces due to friction. When the particles are widely spaced and deformation is rapid, on the other hand, contacts are brief and may be regarded as collisions, during which momentum is transferred. While constitutive relations are available which model both these situations, in many cases the average contact times lie between the two extremes. The purpose of the present work is to propose constitutive relations and boundary conditions for this intermediate case and to solve the corresponding equations of motion for plane shear of a cohesionless granular material between infinite horizontal plates. It is shown that, in general, not all the material between the plates participates in shearing, and the solutions for the shearing material are coupled to a yield condition for the non-shearing material to give a complete solution of the problem.

Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Final report

Abstract: New numerical algorithms are devised (PSC algorithms) for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be used also for more general Hamilton-Jacobi-type problems. The algorithms are demonstrated by computing the solution to a variety of surface motion problems.

Equations of Motion from a Data Series.

Abstract: Temporal pattern learning, control and prediction, and chaotic data analysis share a common problem: deducing optimal equations of motion from observations of time-dependent behavior. Each desires to obtain models of the physical world from limited information. We describe a method to reconstruct the deterministic portion of the equations of motion directly from a data series. These equations of motion represent a vast reduction of a chaotic data set’s observed complexity to a compact, algorithmic specification. This approach employs an informational measure of model optimality to guide searching through the space of dynamical systems. As corollary results, we indicate how to estimate the minimum embedding dimension, extrinsic noise level, metric entropy, and Lyapunov spectrum. Numerical and experimental applications demonstrate the method’s feasibility and limitations. Extensions to estimating parametrized families of dynamical systems from bifurcation data and to spatial pattern evolution are presented. Applications to predicting chaotic data and the design of forecasting, learning, and control systems, are discussed.

A finite-element approach to control the end-point motion of a single-link flexible robot

Abstract: A structural finite-element technique based on Bernoulli-Euler beam theory is presented which will permit the finding of the torques (or forces) that are necessary to apply at one end of a flexible link to produce a desired motion at the other end. This technique is suitable for the open loop control of the tip motion. It may also provide a good control law for feedback control. The finite-element method is used to discretize the equations of motion. This method has a major advantage in the fact that different material properties and boundary conditions like hubs, tip loads, changes in cross sections, etc., can be handled in a very simple and straightforward manner. The resulting differential equations are integrated via the frequency domain. This allows for the expansion of the desired end motion into its harmonic components and helps to visualize the complex wave propagation nature of the problem. The performance of the proposed technique is illustrated in the solution of a practical example. Results point out the potential that this technique has in the study of the dynamics and control not only of flexible robots, but also of any other flexible mechanisms like those used in biomechanics, where high precision at the tip of very light flexible arms is required.

A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems

Abstract: A recursive formulation of the equations of motion of constrained mechanical systems with closed loops is derived, using tools of variational and vector calculus. Kinematic couplings between pairs of contiguous bodies presented in Part 1 of this paper are generalized. Lagrange multipliers are introduced to account for the effects of joints that are cut to define a tree structure. Constraint Jacobian terms are added to the reduced variational equations derived in Part I. Cut-joint constraint acceleration equations are derived, to complete the reduced equations of motion. Lagrange multipliers associated with each cut-joint are eliminated at the first junction body encountered that permits closing the loop that constraints in cut joint. The inductive algorithm developed in Part I is used to calculate accelerations for the system. A multi-loop compressor is analyzed to illustrate use of the method.

Dynamics of bubbles in general relativity

Abstract: This is a systematic study of the evolution of thin shell bubbles in general relativity. We develop the general thin-wall formalism first elaborated by Israel and apply it to the investigation of the motion of various bubbles arising in the course of phase transitions in the very early Universe including new phase bubbles, old phase remnants, and domains. We consider metric junction conditions and derive constraints both on the decay of metastable states and on the evolution of non- equilibrium scalar field configurations (fluctuations) following from the global geometry of spacetime.

Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit

Abstract: Based on the linearized equations of motion of a spacecraft near a satellite in general Keplerian orbit and the assumptions of a bounded thrust magnitude and constant exhaust velocity, a fixed-duration, fuel-optimal rendezvous problem is formulated and is investigated for the constant-mass case through the solution for the primer vector, which is shown to satisfy the original differential equations that describe the spacecraft motion during unpowered flight. It is shown that there are no singular solutions to this rendezvous problem for noncircular Keplerian orbits and that consequently all optimal solutions for orbit eccentricities greater than zero are constructed from a finite number of intervals of full thrust and coast where the switches are determined from the primer vector.

The equations of motion of a perfect fluid with free boundary are not well posed.

Abstract: (1987). The equations of motion of a perfect fluid with free boundary are not well posed. Communications in Partial Differential Equations: Vol. 12, No. 10, pp. 1175-1201.

Free Massless Fields of Arbitrary Spin in the de Sitter Space and Initial Data for a Higher Spin Superalgebra

Abstract: Linearized curvatures are constructed which allow a completely uniform description of free massless fields of all integer and half-integer spins sgreater than or equal to3/2 on an anti-de Sitter background. The corresponding action functionals and equations of motion are found. The proposed linearized curvatures specify the ''initial data'' for deriving a nonabelian symmetry corresponding to a (hypothetical) nontrivial theory of massless higher-spin fields interacting with gravity.

The Weyssenhoff fluid in Einstein-Cartan theory

Abstract: The variational theory of an ideal spinning fluid is developed. This is described by original Weyssenhoff-Raabe tensors of spin and energy-momentum. Both neutral and charged fluids are considered, and the equations of motion and conservation laws for the Weyssenhoff neutral fluid in the Riemann-Cartan spacetime are discussed. The results are applied to the study of cosmological models with rotation, shear and expansion in the framework of the Einstein-Cartan theory of gravity.

A two‐dimensional theory for high‐frequency vibrations of piezoelectric crystal plates with or without electrodes

Abstract: Two‐dimensional equations of motion of successively higher‐order approximations for piezoelectric crystal plates with triclinic symmetry are deduced from the three‐dimensional equations of linear piezoelectricity by expansion in series of trigonometric functions of the thickness coordinate of the plate. These equations, complemented by two additional relations: one, the usual relation of face tractions to the mass of electrodes, and the other relating face charges to face potentials and face displacements, can accommodate either the traction and charge boundary conditions at the faces of the plate without electrodes or the traction and potential boundary conditions at the faces of the plate with electrodes. Dispersion curves are obtained from the first‐ to fourth‐order approximate plate equations for a rotated 45° Y‐cut lithium tantalate plate without electrodes, and these curves are compared with those from the frequency equation of the three‐dimensional equations with close agreement. Solutions of force...

Bubble dynamics in a compressible liquid. Part 2. Second-order theory

Abstract: The radial dynamics of a spherical bubble in a compressible liquid is studied by means of a rigorous singular-perturbation method to second order in the bubble-wall Mach number. The results of Part 1 (Prosperetti & Lezzi, 1986) are recovered at orders zero and one. At second order the ordinary inner and outer structure of the solution proves inadequate to correctly describe the fields and it is necessary to introduce an intermediate region the characteristic length of which is the geometric mean of the inner and outer lengthscales. The degree of indeterminacy for the radial equation of motion found at first order is significantly increased by going to second order. As in Part 1 we examine several of the possible forms of this equation by comparison with results obtained from the numerical integration of the complete partial-differential-equation formulation. Expressions and results for the pressure and velocity fields in the liquid are also reported.

Resonance absorption of magnetohydrodynamic surface waves Physical discussion

Abstract: It is shown how the phenomenon of MHD surface wave resonance absorption can be described in simple terms, both physically and mathematically, by applying the 'thin flux tube equations' to the finite-thickness transition layer which supports the surface wave. The thin flux tubes support incompressible slow-mode waves that are driven by fluctuations in the total pressure which exist due to the presence of the surface wave. It is shown that the equations for the slow-mode waves can be reduced to a simple equation, equivalent to a driven harmonic oscillator. Certain field lines within the transition layer are equivalent to a harmonic oscillator driven at resonance, and neighboring field lines are effectively driven at resonance as long as a given condition is satisfied. Thus, a layer which secularly extracts energy from the surface wave develops. The analysis indicates that nonlinear effects may destroy the resonance which is crucial to the whole effect.

Collapse and revivals in a two-photon absorption process

Abstract: The two-photon absorption process is studied by considering the interaction of a quantized single-mode field with an effective two-level atom through intermediate states. The equations of motion of the coupled atom–field probability amplitudes for the effective two-level atom are obtained after adiabatically eliminating the intermediate states. These equations are then solved in the rotating-wave approximation. The time constants associated with the Rabi oscillation, collapse, and revival of the atomic inversion are derived, and the photon statistics are discussed with special reference to photon antibunching.

Non-linear dynamics of an elastic cable under planar excitation

Abstract: The phenomena of the finite forced dynamics of a suspended cable associated with the quadratic and cubic non-linearities in the equations of motion are studied. A high-order perturbation analysis for the primary resonance is accomplished and numerical results are presented for the frequency-response equation and the region of instability of the steady-state solutions. Multivaluedness of the response curves is shown to occur with different characteristics depending on the cable and forcing parameters. The dependence of the response on the initial conditions is examined by means of the trajectories of the unsteady-state motions.

Equations of motion for maneuvering flexible spacecraft

Abstract: This paper is concerned with the derivation of the equations of motion for maneuvering flexible spacecraft both in orbit and in an earth-based laboratory. The structure is assumed to undergo large rigid-body maneuvers and small elastic deformations. A perturbation approach is presented in which the quantities defining the rigid-body maneuver are regarded as the unperturbed motion and the elastic motions and deviations from the rigid-body motions are regarded as the perturbed motion. The perturbation equations are linear, non-self-adjoint, and with time-dependent coefficients. A maneuver force distribution exciting the least amount of elastic deformation of the spacecraft is developed. Numerical results highlight the vibration caused by rotational maneuvers.

Multiple-quantum dynamics in NMR: A directed walk through Liouville space

Abstract: An approach to spin dynamics in systems with many degrees of freedom, based on a recognition of the constraints common to all large systems, is developed and used to study the excitation of multiple‐quantum coherence under a nonsecular dipolar Hamiltonian. The exact equation of motion is replaced by a set of coupled rate equations whose exponential solutions reflect the severe damping expected when many closely spaced frequency components are superposed. In this model the evolution of multiple‐quantum coherence under any bilinear Hamiltonian is treated as a succession of discrete hops in Liouville space, with each hop taking the system from a K‐spin/n‐quantum mode to a K’‐spin/n’‐quantum mode. In particular, for a pure double‐quantum Hamiltonian the selection rules are ΔK=±1 and Δn=±2. The rate for each move depends on the number of Liouville states at the origin and destination, and on the total number of spins present. All rates are scaled uniformly by a factor dependent on the properties of the material, such as the dipolar linewidth, but otherwise the behavior predicted is universal for all sufficiently complicated systems. Results derived by this generic approach are compared to existing multiple‐quantum data obtained from solids and liquid crystals.

On the reflection of a train of finite-amplitude internal waves from a uniform slope

Abstract: The reflection of a train of two-dimensional finite-amplitude internal waves propagating at an angle β to the horizontal in an inviscid fluid of constant buoyancy frequency and incident on a uniform slope of inclination α is examined, specifically when β > α. Expressions for the stream function and density perturbation are derived to third order by a standard iterative process. Exact solutions of the equations of motion are chosen for the incident and reflected first-order waves. Whilst these individually generate no harmonics, their interaction does force additional components. In addition to the singularity at α = β when the reflected wave propagates in a direction parallel to the slope, singularities occur for values of α and β at which the incident-wave and reflected-wave components are in resonance; strong nonlinearity is found at adjacent values of α and β. When the waves are travelling in a vertical plane normal to the slope, resonance is possible at second order only for α < 8.4° and β < 30°. At third order the incident wave is itself modified by interaction with reflected components. Third-order resonances are only possible for α < 11.8° and, at a given α, the width of the β-domain in which nonlinearities connected to these resonances is important is much less than at second order. The effect of nonlinearity is to reduce the steepness of the incident wave at which the vertical density gradient in the wave field first becomes zero, and to promote local regions of low static stability remote from the slope. The importance of nonlinearity in the boundary reflection of oceanic internal waves is discussed.In an Appendix some results of an experimental study of internal waves are described. The boundary layer on the slope is found to have a three-dimensional structure.

Ideal MHD equations in terms of compressive Elsässer variables

Abstract: By the help of the so-called Elsasser variables, if adopted to the situation of a compressible plasma, the equations of ideal magnetohydrodynamics can be rewritten in a new form. This appears to be particularly suitable to describe the situation of compressive MHD turbulence. After having derived the basic equations of motion, the conservation equations for energy and cross-helicity are rephrased in terms of Elsasser variables. The sinks and sources of cross-helicity associated with the compressibility of the plasma are elucidated. A standard normal mode analysis, performed in terms of Elsasser variables, emphasizes their usefulness in treating hydrodynamic waves. In the conclusions some possible future, more sophisticated applications are outlined.