Showing papers on "Configuration space published in 1985"

A subdivision algorithm in configuration space for findpath with rotation

Abstract: A recursive cellular representation for configuration space is presented along with an algorithm for searching that space for collision-free paths. The details of the algorithm are presented for polygonal obstacles and a moving object with two translational and one rotational degrees of freedom.

Using backprojections for fine motion planning with uncertainty

Abstract: This paper outlines a method for planning motions in the presence of uncertainty. Tasks are modelled as geometrical goals in configuration space. The planning process consists of determining regions from which particular motions are guaranteed to successfully reach a desired goal. An algorithm is presented for backprojecting from desired goal states. The backprojection regions are computed by erecting constraints that geometrically capture the uncertainty in motion.

Low-Energy Scattering of Non-Abelian Magnetic Monopoles [and Discussion]

Abstract: The Bogomolny equations represent static configurations of magnetic monopoles, for some non-Abelian group such as SU(2). Geodesic motion on this configuration space represents slowly moving interacting monopoles. Geometric information on this space can then be used to investigate the scattering of monopoles. The results show surprising features, including a 90° scattering and the conversion of angular momentum into electric charge.

A Monte Carlo method for determining free‐energy differences and transition state theory rate constants

Abstract: We present a new Monte Carlo procedure for determining the Helmholtz free‐energy difference between two systems that are separated in configuration space. Unlike most standard approaches, no integration over intermediate potentials is required. A Metropolis walk is performed for each system, and the average Metropolis acceptance probability for a hypothetical step along a probe vector into the other system is accumulated. Either classical or quantum free energies may be computed, and the procedure is also ideally suited for evaluating generalized transition state theory rate constants. As an application we determine the relative free energies of three configurations of a tungsten dimer on the W(110) surface.

A drop of ink falls from my pen...It comes to earth, I know not when

Abstract: The authors obtain, for a Brownian particle in a uniform force field, the mean and asymptotic first-passage times as functions of the particle's initial position and velocity, with the recurrence times given as a special case. They discuss the region of phase space for which the diffusion model of Brownian motion provides an adequate approximation, and conclude that there is no possibility of obtaining the recurrence times within that model. They find that the nature of the boundary-value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space, because the time-development operator is then parabolic rather than elliptic. They argue that such a change in the treatment of Brownian motion places it within the sphere of transport theory rather than diffusion theory, and that, consequently, results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport.

A Voronoi method for the piano-movers problem

Abstract: We describe work in progress toward a polynomial-time algorithm for the classical movers' problem With 6 degrees of freedom. Rotations are represented as a certain 3-dimensional subset of the space of quaternions. This representation gives rise to algebraic constraints on the allowable configurations of the moving object. We define a generalized Voronoi diagram in the configuration space and show that a subset of the diagram, consisting of zero and one dimensional manifolds, can be constructed in polynomial time, and paths found along it. While this subset of the diagram is not always sufficient for path planning, the triangulation of 1- dimensional manifolds can be used as the basis of a complete path planning algorithm by adding other 1-manifolds.

An efficient and simple motion planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers (extended abstract)

Abstract: We present a relatively simple algorithm which runs in time O(n2log n) for the above mentioned problem. The algorithm is an optimized variant of the decomposition technique of the configuration space of the ladder, due to Schwartz and Sharir. The algorithm is based on some ideas which may be exploited to improve the efficiency of existing motion-planning algorithms for other more complex robot systems.

On motion planning with six degrees of freedom: Solving the intersection problems in configuration space

Abstract: The Movers' problem is to find a continuous, collision-free path for a moving object through an environment containing obstacles. The classical formulation of the three-dimensional Movers' problem is as fellows: given an arbitrary rigid polyhedral moving object P with three translational and three rotational degrees of freedom, find a contineous, collision-free path taking P from some initial configuration to a desired goal configuration. The six degree or freedom Movers' problem may be transformed into a point, navigation problem in a six-dimensional configuration space (called C-Space). The C-Space obstacles, which characterize the physically unachievable configurations, are directly represented by six-dimensional manifolds whose boundaries are five dimensional C-surfaces. By characterizing these surfaces and their intersections, collision-free paths may be found by the closure of three operators which (i) slide along 5-dimensional level C-surfaces parallel to C-Space obstacles; (ii) slide along 1- to 4-dimensional intersections of level C-surfaces; and (iii) jump between 6-dimensional obstacles. We show how to construct and represent C-surfaces and their intersection manifolds. We also demonstrate how to intersect trajectories with the boundaries of C

A Generalized Solution to the Inverse Kinematics of Robotic Manipulators

Abstract: In the context of kinematic control of a robotic manipulator if a certain set of task space coordinates (end effector position and orientation) are commanded then the corresponding configuration space coordinates (joint variables) must be provided. The joint variables are obtained by solving the “inverse kinematics problem.” Typically a solution to the problem can be obtained in closed-form; however, such a solution is inherently manipulator-dependent. The paper presents an approach for providing a generalized inverse kinematics solution which is manipulator-independent. The solution is based on an iterative procedure. An algorithm was developed and demonstrated using the PUMA-560 and Stanford manipulator models.

Quantum scattering theory for two- and three-body systems with potentials of short and long range

Abstract: We give a full proof of asymptotic completeness for Schrodinger operators of two- and three-particle quantum systems The interaction is given by pair potentials which may be of short and of long range, including Coulomb forces We apply geometrical time-dependent methods where propagation of scattering states in phase space and in configuration space is essential The main new results are the inclusion of long-range potentials of the two-body estimates in Section VI and for three-particle systems But also where we recover known results some of our methods are new Where possible we have chosen the methods which admit generalization to higher particle numbers

Phase space considerations for interacting particles obeying the Fokker–Planck equation

Abstract: The N‐particle Fokker–Planck equation has been cast into a form which is invariant to coordinate transformations. For a pair of particles interacting through a radial potential V(R), the equation of motion may be projected onto a one‐dimensional problem in R when the conjugate momenta of the center of mass and internal coordinates have symmetric distributions, as in Boltzmann distributions. However, the R2 dependence of the density of configuration space demands that V(R) be supplemented with an effective repulsive potential −2kT ln R, a result which has consequences for Langevin simulations of chemical processes. For N interacting particles a subset of coordinates can often be identified as having such symmetric distributions, and if these are not of interest, they can be eliminated with the result that the potential containing the remaining coordinates is supplemented by −kT ln(‖gkl‖)1/2, where ‖gkl‖ is the determinant of that portion of the covariant metric tensor pertaining to unwanted coordinates k a...

Electrostatic energy minimisation by simulated annealing

Abstract: The authors have determined the minimum energy configuration of N point charges confined to the interior of a circle. The minimisation problem in the multi-dimensional configuration space is solved by a technique based on the simulated-annealing method. They observe striking effects, which could lead to a better understanding of phenomena such as crystallisation, symmetry breaking, commensurate-incommensurate transition, etc. Moreover, an experimental verification of their results appears to be possible.

Quantum field theory and a new universal high-energy scale

Abstract: It is demonstrated that in the framework of quantum field theory with non-Euclidean momentum space there exist several different equally acceptable expressions for the Dirac wave operator, which depend parametrically on the fundamental massM. As a result there appears the necessity to consider fermion fields of different types, includingexotic fields which increase in the flat limitM→∞ as √M. The description of all fermionic fields is made along the lines accepted in the previous article of this series. The main feature of the developed approach —the locality of the theory in configuration space of five dimensions— is conserved.

Brownian dynamics simulation of interacting particles

Abstract: The dynamics of the ions in an electrical double layer has been studied using stochastic dynamics simulation methods. The theory of a brownian particle in an external force field is discussed and a modified form of the fluctuation-dissipation theorem is derived. With slowly varying external forces the simulations can be restricted to configuration space and the correction term to the fluctuation-dissipation theorem, arising from the external force, can be neglected. It is shown that the diffusion coefficient of a brownian particle can be calculated from the force correlation function. In comparison with calculations from mean-square displacements the former way turns out to be much more efficient. Finally, a relation between the external force () and the random force () is derived such that = -2 .

The KS-transformation in hypercomplex form and the quantization of the negative-energy orbit manifold of the Kepler problem

Abstract: In a previous note we have shown that the KS-transformation, introduced by Kustaanheimo and Stiefel into Celestial Mechanics for the regularization of the Kepler problem, may be formulated in terms of hypercomplex numbers as the product of a quaternion and its anti-involute, thus representing a particular morphism of the real algebra of quaternions-having for image the physical configuration space of the Kepler problem. In the present note we show, first, that this formulation allows a straight derivation of the Hopf fibering of the sphere S3 (characterized by unit quaternions) having the base space given by the sphere S2 (characterized by unit vectors), and secondly that the KS-transformation allows the ‘quantization of the symplectic manifold S2’ in the sense of Souriau, the associated quantum manifold S3 having a contact structure given by the bilinear relation characteristic of the KS-theory. Furthermore, after presenting a natural extension of the hypercomplex KS-transformation to the full phase space of the Kepler problem, we show that this extension allows the quantization of the manifold of Kepler orbits of fixed negative energy (manifold diffeomorphic to the symplectic product S2×S2). The energy levels satisfy a well known quantum integrality condition and the associated quantum manifold is diffeomorphic to the product manifold S3×S3 quotiented by a suitable equivalence relation.

Generalized Klein-Gordon equations in d dimensions from supersymmetry

Abstract: The Wess-Zumino model is extended to higher dimensions, leading to a generalized Klein-Gordon equation whose propagator is computed in configuration space.

Configuration space analysis for fully frustrated vector spins

Abstract: Classical vector spins, on the sites of a three-dimensional simple cubic lattice, interacting via a periodic fully frustrated array of exchange interactions, may exhibit an interesting manifold of ground state configurations. In particular, for Heisenberg spins, the manifold has dimension 5, with two continuous degeneracy parameters, in addition to global rotation angles. A configuration space analysis, including pair and triangle overlap statistics, has been performed for this test model Des spins vectoriels classiques sur les sites d'un reseau cubique simple 3D, avec une distribution periodique completement frustree d'interaction d'echange, peuvent posseder une interessante variete de configuration de base. Pour des spins de Heisenberg, la variete a la dimension 5, avec deux parametres de degenerescence continus, en plus des angles de rotation globale. Analyse d'espace de configurations en incluant l'etude statistique des recouvrements pour les paires et les triangles

Channel coupling in the 1De spectrum of beryllium

Abstract: Channel coupling in the 1De spectrum of Be is examined through a combination of hyperspherical and R-matrix techniques. Plots of the adiabatic functions in configuration space provide an interpretation of the channel coupling whereas the R-matrix procedure provides reliable spectroscopic data.

Phase-space eikonal method for treating wave equations.

Abstract: We present a new method for treating classical (or quantal) wave equations in the short-wavelength (semiclassical) regime, based on a description of the wave in the ray phase space. The coherent-state representation is defined, and the equation which it obeys is given and solved under assumptions similar to those of conventional eikonal theory. As indicated by an example, the result is a smooth distribution on phase space which, when ''projected'' onto configuration space, yields a wave field with no caustic singularities.

An alternative computational strategy for direct space hartree–fock calculations on extended model chains

Abstract: Based on previous analyses of slowly convergent exchange lattice sums entering the configuration space restricted Hartree–Fock–Roothaan scheme for chain systems, an alternative computational strategy is developed. Within the present formalism, the traditionally used finite Fourier transform of k-dependent LCAO density matrices are by-passed and an advantageous computational organization is obtained.