Showing papers on "Configuration space published in 1984"

Obstacle avoidance using an octree in the configuration space of a manipulator

Abstract: An automatic system for planning safe trajectories for a computer controlled manipulator among obstacles is a key component of robot assembly operations. This paper describes an algorithm transforming cartesian obstacles into obstacles in the space of the first three joints of a manipulator with six revolute joints (e.g. a ACMA-CRIBIER V80), and giving a hierarchical description of the free space by mean of an octree. Such a description is very useful in testing for collision between the arm of the manipulator end obstacles since it is represented by a point in this space.

Motion Planning with Six Degrees of Freedom

Abstract: : The motion planning problem is of central importance to the fields of robotics, spatial planning, and automated design. In robotics we are interested in the automatic synthesis of robot motions, given high-level specifications of tasks and geometric models of the robot and obstacles. The Mover's problem is to find a continuous, collision-free path for a moving object through an environment containing obstacles. This thesis describes the first known implementation of a complete algorithm (at a given resolution) for the full six degree of freedom Movers' problem. The algorithm transforms the six degree of freedom planning problem 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. Implementing the point navigation operators requires solving fundamental representational and algorithmic questions: we will derive new structural properties of the C-space constraints and show how to construct and represent C-surfaces and their intersection manifolds. Originator-Supplied keywords include: Motion planning, Configuration space, Generalized Voroni diagram, Piano mover's problem, Computational geometry, Path planning, Robotics, Spatial reasoning, Geometric modelling, Obstacle avoidance, Geometric planning, Collision avoidance.

The Everett Interpretation

Abstract: We begin with an essentially technical notion that will be used repeatedly in what follows. Indeed, it will serve as our conduit between physical observations and the mathematical formalism of quantum mechanics. Fix a quantum system. To make the discussion concrete, we suppose it to be nonrelativistic, and we describe it in the Schr6dinger representation. Thus, we have a configuration manifold C for the system, complex-valued wave functions on C (representing states of the system), a Hamiltonian operator H on such wave functions, and the Schr6dinger equation (giving the dynamics of the system). We say that a region R of configuration space is precluded if the wave function 1, as a consequence of its dynamical evolution from an initial state, becomes at some time "small" in the region R. This is intended, not as a precise definition, but rather as a summary of the meaning we have in mind, a meaning to be clarified below. We begin with an example.3 Let the system consist of, say, 101 one-dimensional particles, i.e., let the configuration manifold C be [IO'1 (coordinates x, ...., x,oo and y). Then a typical Schrodinger wave function is b (xi, . . ., x,oo, y). Let the Hamiltonian be given by

Comparing the efficiency of Metropolis Monte Carlo and molecular-dynamics methods for configuration space sampling

Abstract: Computer methods for sampling statistical ensembles generate chains of configurations in which subsequent members differ only slightly. Statistical errors are determined by the number of independent configurations contained in the sample. A quantitative treatment of the persistence of correlation shows how the first two moments of the autocorrelation function of a variablef along the chain are connected with the expected variance of its mean. The variance of the potential energy in the canonical ensemble is shown be to larger than that in the microcanonical one by a factor which is the ratio of the system heat capacity to that of an ideal gas. A comparison has been made of the efficiency of Metropolis Monte Carlo (MC), the usual microcanonical molecular dynamic (MDM) and a modification of molecular dynamics for canonical ensemble sampling (MDC). The analysis is focused on three aspects: the mean square displacement of a representative point in configuration space, the persistence of correlation in the potential energyV and also in a function of interest in free-energy-difference calculations. In MDM simulations of crystalline solids, it was found thatV behaves as an «oscillatory» variable and that the variance of its mean is reduced by antithetic variations of its values.

Reducing Multiple Object Motion Planning To Graph Searching

Abstract: In this paper we study the motion planning problem for multiple objects where an object is a 2-dimensional body whose faces are line segments parallel to the axes of $R^{2}$ and translations are the only motions allowed. Towards this end we analyze the structure of configuration space, the space of points that correspond to positions of the objects. In particular, we consider CONNECTED, the set of all points in configuration space that correspond to configurations of the objects where the objects form one connected component. We show that CONNECTED consists of faces of various dimensions such that if there is a path in CONNECTED between two 0-dimensional faces (vertices) of CONNECTED then there is a path between them along 1-dimensional faces (edges) of CONNECTED. It is known that if there is a motion between the configurations. Thus by the result of this paper the existence of a motion between two vertices of CONNECTED implies a motion corresponding to a path along edges of CONNECTED. Hence we have reduced the motion planning problem from a search of a high dimensional space to a graph searching problem. Searching the graph of vertices and edges of CONNECTED for a path has a prohibitive worse-case complexity because of the large number of vertices and edges. However, if the search generates edges and vertices only as they are needed, a practical and efficient algorithm may be possible using some effective heuristic. From this result it is shown that motion planning for rectangles in a rectangular boundary is in PSPACE. Since it is known that the problem is PSPACE-hard we conclude it is a PSPACE-complete problem.

Monopoles and maps from $S^2$ to $S^2$; the topology of the configuration space

Abstract: The configuration space for the SU(2)-Yang-Mills-Higgs equations on ℝ3 is shown to be homotopic to the space of smooth maps fromS 2 toS 2. This configuration space indexes a family of twisted Dirac operators. The Dirac family is used to prove that the configuration space does not retract onto any subspace on which the SU(2)-Yang-Mills-Higgs functional is bounded.

Symmetric space property and an inverse scattering formulation of the SAS Einstein–Maxwell field equations

Abstract: We formulate stationary axially symmetric (SAS) Einstein–Maxwell fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space. This result enables us to embed the configuration space into an eight‐dimensional flat manifold and formulate SAS Einstein–Maxwell fields as a σ‐model. We then give, in a coordinate free way, a Belinskii–Zakharov type of an inverse scattering transform technique for the field equations supplemented by a reduction scheme similar to that of Zakharov–Mikhailov and Mikhailov–Yarimchuk.

Motion Planning with Six Degrees of Freedom. Revision

Abstract: : The motion planning problem is of central importance to the fields of robotics, spatial planning, and automated design An implemented algorithm is presented for the 'classical' formulation of the three-dimensional Movers' problem: Given an arbitrary rigid polyhedral moving object 'p' with three translational and three rotational degrees of freedom, find a continuous, collision free path taking 'p' from some initial configuration to a desired goal configuration This thesis describes the first known implementation of a complete algorithm (at a given resolution) for the full six degree of freedom Movers' problem The algorithm transforms the six degree of freedom planning problem 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 intersection 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

Integrality of the monopole number in SU(2) Yang-Mills-Higgs theory on ℝ3

Abstract: We prove that in classical SU(2) Yang-Mills-Higgs theories on ℝ3 with a Higgs field in the adjoint representation, an integer-valued monopole number (magnetic charge) is canonically defined for any finite-actionL1,loc2 configuration. In particular the result is true for smooth configurations. The monopole number is shown to decompose the configuration space into path components.

Quantum mechanics as a multidimensional Ermakov theory. I. Time independent systems

Abstract: Presents a multidimensional Ermakov theory applicable to both separable and nonseparable time independent quantum mechanical systems, the separability of the system being determined by the separability of the wavefunction in configuration space. A consequence of the theory is the existence of a new exact invariant for multidimensional quantum systems. In one dimension the author shows that this new invariant reduces to the well known Ermakov-Lewis invariant. Two applications of the theory are given. In the first case a new exact invariant is obtained for planar optical and particle channelling systems. In the second case he obtains the Ermakov invariants and electron ray path equations for the hydrogen atom.

Minimal sensitivity optimisation of perturbative wavefunctions

Abstract: Stevenson's principle (1981) of minimal sensitivity is applied directly to configuration space wavefunctions calculated in first-order perturbation theory. The redundant parameters, which comprise the essential elements of minimal sensitivity calculations, actually become functions of configuration space, giving the method considerable flexibility and potential accuracy. It is shown that the method is exact in an illustrative class of simple cases and, for the more realistic case of the ground state wavefunction of the quartic oscillator, it is considerably more accurate than the closely related perturbative variational approach, particularly in the asymptotic region.

Path integrals in parametrized theories: Newtonian systems

Abstract: Constraints in dynamical systems typically arise either from gauge or from parametrization. We study Newtonian systems moving in curved configuration spaces and parametrize them by adjoining the absolute time and energy as conjugate canonical variables to the dynamical variables of the system. The extended canonical data are restricted by the Hamiltonian constraint. The action integral of the parametrized system is given in various extended spaces: Extended configuration space or phase space and with or without the lapse multiplier. The theory is written in a geometric form which is manifestly covariant under point transformations and reparametrizations. The quantum propagator of the system is represented by path integrals over different extended spaces. All path integrals are defined by a manifestly covariant skeletonization procedure. It is emphasized that path integrals for parametrized systems characteristically differ from those for gauge theories. Implications for the general theory of relativity are discussed.

An extension of Szebehely's problem to holonomic systems

Abstract: We study the problem of the determination of the potential function of forces generating a given family of orbits in the n-dimensional configuration space of the representative point of a holonomic system. We obtain first-order partial differential equations to solve the problem and we discuss some particular examples.

Configuration space Faddeev continuum calculations: N-d s -wave scattering lengths with tensor-force interactions

Abstract: Formulation of the $s$-wave, zero-energy Faddeev-type scattering equations for a model which includes a spin-triplet tensor-force nucleon-nucleon interaction and a Coulomb force between two protons is discussed. Numerical solution of the equations for the Reid-soft-core, Argonne ${V}_{14}$, and super-soft-core $C$ potential models is obtained using spline techniques. Kohn variational estimates are also presented, and comparison is made with other previously published results. The inclusion of a tensor force does not alter our previous conclusions about the Coulomb force effects on the p-d scattering length, which were based upon $s$-wave nucleon-nucleon potential model studies.

Numerical solution of Salpeter's equation

Abstract: A method for solving Salpeter's relativistic bound-state equation is presented. The interaction can be given in either momentum space or configuration space and may have various Lorentz-Dirac properties. The operators of the equation are represented as matrices in a basis of nonrelativistic harmonic-oscillator states. The resulting non-Hermitian matrix is diagonalized for various values of the oscillator frequency, a variational parameter. To reduce the size of the matrices a two-particle Foldy-Wouthuysen transformation is applied. As an example, the charmonium and b-quarkonium mass spectra are calculated using a linear confining potential plus one-gluon exchange. The effects of the coupling between the positive- and negative-energy components are examined and found to be important for the light mesons.

Exact evaluation of the propagator for the damped harmonic oscillator

Abstract: Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.

The Configuration Space Approach to Robot Path Planning

Abstract: Graphically simulated configuration maps are used to plan manipulator paths in two-dimensions. Configuration maps represent a transformation of the Cartesian workspace into the manipulator joint coordinates, identifying both the free space and the space occupied by obstacles. Visual examination of the configuration map allows simulation users to plan collision-free paths by maneuvering the manipulator point along a series of path line segments which connect starting and final manipulator configurations. A PUMA 600 revolute manipulator experimentally verifies one such path planned through a congested workspace.

Moments of the reduced local energy

Abstract: Compact measures of the local accuracy of a wavefunction are proposed. They are defined in terms of the reduced local energy. These compact measures are applied to examine the local accuracy in different regions of configuration space for some well known Hartree-Fock wavefunctions from helium through argon.

Projection operator techniques in physics

Abstract: A systematic account of projection operators (projectors) and orthogonalization techniques together with applications to selected areas of physics is presented. This unified approach is shown to have advantages over other approaches in that the mathematical statements are more precise. The mathematical level, however, is aimed at the practicing physicist and lies between rigorous mathematics and current use in physics. Further, the techniques presented have practical applications as is demonstrated by examples in the quantum theory of measurement, in the relationship between second quantization and configuration space techniques, and in an account of generalized Wannier and Bloch functions. Attention is paid to the problem of construction of orthogonal projection operators (orthogonal projectors). The construction of orthogonal projectors even in approximate form would allow the solution of many practical problems ranging from the eigenvalue spectrum problem to the construction of states for many‐body systems. One can almost say that a n y problem in quantum mechanics can be formulated as a problem involving the construction of projectors.

Advances in the theory of collective motion in nuclei

Abstract: The theory of collective motion in nuclei has as its geometric origin the comparison of certain nuclear phenomena with properties of a liquid drop Bohr and Mottelson (1952,53) [1,2] introduced the idea of a irrotational flow and explained a variety of collective phenomena by the deformations and vibrations of a nuclear fluid The nuclear shell theory succeeded later on in the representation of collective excitation by coherent superpositions of many single-particle excitations Elliott (1958) [3] showed that collective levels in the shell theory can be connected to a group SU(3) Independently of the shell theory, various attempts were made to develop the geometric ideas implicit in the BohrMottelson model Weaver, Biedenharn and Cusson [4,5,6] introduced the group SL(3, ~) of volume-preserving deformations into the collective theory With this group, they connected kinematical transformations of the system of A nucleons, the vortex spin, and a spectrum generating algebra Inclusion of the mass quadrupole tensor leads to a natural extension of this group which was also studied by Rowe, Rosensteel and collaborators [7,8] In the geometric models, it is the final goal to explain collective phenomena from the point of view of many-body dynamics Therefore one has to link the collective coordinates to the singleparticle coordinates This program was already started by Lipkin (1955) [9] and by Villars (1957) [10] Whereas these authors tried to keep the single-particle coordinates, new viewpoints were developed later by Zickendraht [11], by Dzyublik et al [12], and by Buck Biedenharn and Cusson [13] by use of the orthogonal intrinsic group SO(n, ~) acting on the particle indices and commuting with $0(3, ~) Rowe and Rosensteel (1980) [14] were the first to analyze this scheme through an orbit analysis in configuration space Vanagas (1977) [15] pointed out the close relation of the group SO(n, ~) to the symmetric group of orbital permutations and proposed the group SO(n, ~) as a symmetry group of the collective hamiltonian In the following sections, the dynamical implications of this proposal will be analyzed, based on work done with Z Papadopolos, M Saraceno and W Schweizer

Calculation Of LIDAR Beam Spread In Stratified Media

Abstract: The irradiance distribution of light propagating in a multiple scattering medium undergoes spatial spreading. A numerical technique for obtaining two-way (transmitted and received combined) spread functions in configuration space is presented. This formulation is useful for modeling lidar systems in media characterized by a strongly forward scattering function such as the ocean. The theoretical basis is a random walk small angle formalism for Fourier space represen-tations of the spread functions. The assumption of all scatters being independent events allows the expression for the Fourier transform of the photon distribution (i.e. the char-acteristic function) to be written as a product of initial conditions and exponentials of integrals of scattering particle density and the Fourier representation of the local scat-tering function of the medium. The method thus applies to a stratified medium where the direction of variation is along the beam axis. We have developed a numerical method to compute configuration space spread functions utilizing two-dimensional Fast Fourier transforms. The one way results agree well with experimental measurements. When applied to the case of few scatters, our results contrast sharply with Gaussian beam models which only approach validity in a diffusion limit.

Application of the method of collective coordinates the the quantisation of the two‐dimensional higgs model

Abstract: A comprehensive analysis of the application of the method of collective coordinates to the two dimensional Higgs model is given. First the instanton solution is derived, and the geometry of configuration space, and the construction of Schrodinger wave functionals are discussed. It is then explicitly verified that the Goldstone mode is the projection of the vacuum state onto the generator of the broken symmetry. The elimination of this Goldstone mode by means of the unitary gauge condition is demonstrated to the the crucial point in the construction of a consistent perturbation procedure. The parameter of the broken symmetry group is then used as the collective coordinate for field configurations around a minimum of the interaction. Throughout, the discussion is sufficiently detailed in order to facilitate the application of the method to other fields.

Analysis of the convergence of lattice sums arising in the configuration space Hartree-Fock-Roothaan method for model chains

Abstract: Some mathematical implications of the extended nature of model chains are reviewed and put in perspective with computational aspects of electronic structure calculations of polymers.

Random flights of massless particles and precursors

Abstract: The diffusion of photons through a cloud of ionised plasma is considered. Phenomenological hyperbolic equations derived earlier from transient thermodynamics predict diffusion profiles which are typically double-peaked. The propagation of the diffusion front is causal and this gives rise to a precursor peak preceding the main diffusion peak. It is shown that this feature finds an explanation in terms of random walk theory. The configuration space for the displacements is taken to be Minkowski space and the random distributions are concentrated on the future light cone.

Some geometrical consequences of physical symmetries

Abstract: Invariant submanifolds of the linear representation space C4m of the physical symmetry group SU(2,2)×SU(m) and its subgroup P×SU(m) are studied in some detail. It is shown that there exists only one such manifold admitting unique projection onto Minkowski space. The structure of this manifold is investigated by using proper local coordinate systems.