Six-dimensional Configuration Space Research Articles

On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The scheme is particularly suitable for applications in manufacturing, such as assembly planning, where typically motion planning needs to be carried out in very tight quarters. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms Probabilistic Roadmaps (PRM), with over 40-fold speedup in a six-dimensional coordination-tight setting. (ii) A probabilistic completeness proof for the case of MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages, which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.

Rapid Mapping and Exploration of Configuration Space

We describe a graphics processing unit (GPU)-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three- and six-dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six-dimensional configuration space that are efficiently computed using the fast Fourier transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such configuration space computations practical, and often interactive.

Free-energy landscape for cage breaking of three hard disks

We investigate cage breaking in dense hard-disk systems using a model of three Brownian disks confined within a circular corral. This system has a six-dimensional configuration space, but can be equivalently thought to explore a symmetric one-dimensional free-energy landscape containing two energy minima separated by an energy barrier. The exact free-energy landscape can be calculated as a function of system size by a direct enumeration of states. Results of simulations show the average time between cage breaking events follows an Arrhenius scaling when the energy barrier is large. We also discuss some of the consequences of using a one-dimensional representation to understand dynamics through a multidimensional space, such as diffusion acquiring spatial dependence and discontinuities in spatial derivatives of free energy.

Conditional symmetries and the quantization of Bianchi Type I vacuum cosmologies with and without cosmological constant

In this paper, the quantization of the most general Bianchi Type I geometry, with and without a cosmological constant, is considered. In the spirit of identifying and subsequently removing as many gauge degrees of freedom as possible, a reduction of the initial six-dimensional configuration space is presented. This reduction is achieved by imposing as additional conditions on the wavefunction, the quantum version of the, linear in momenta, classical integrals of motion (conditional symmetries). The vector fields inferred from these integrals induce, through their integral curves, motions in the configuration space which can be identified to the action of the automorphism group of Type I, i.e. GL(3,ℛ). Thus, a wavefunction depending on one degree of freedom, namely the determinant of the scale factor matrix, is found.A measure for constructing the Hilbert space is proposed. This measure respects the above-mentioned symmetries and is also invariant under the classical property of covariance under arbitrary scalings of the Hamiltonian (quadratic constraint).

Decomposition of the Coulomb interaction into self-consistent fields and local scattering terms

The electronic Hamiltonian is used to derive the equations of motion for the one-band density matrix in the six-dimensional configuration space. Projecting the Hamiltonian onto a local basis makes it possible to decompose the Coulomb interaction into a contact and a monopole - monopole part. An approach is presented which allows us to identify the two parts as scattering terms and self-consistent fields in the limiting case of the Boltzmann transport equation. The approach does not assume spatial homogeneity and includes (i) the self-consistent field and (ii) local electron - electron collision terms and (iii) permits us in the extension to a two-band model to incorporate consistently impact ionization in a form analogous to the electron - electron scattering term.

Configuration space of a body moving on a surface and the kinematics of rolling

The problem of contact between objects and object manipulation is reduced to the simplest yet fundamental case of a body moving on a surface in Cartesian space. Certain smoothness assumptions are made such that the body and the surface have the properties of submanifolds embedded in Cartesian space. The configuration space of the body constrained to move on the surface is constructed as a submanifold of the Cartesian product of the six-dimensional configuration space of the free body and the two-dimensional submanifold of the body itself. This is done considering initially the motion of “general sliding”. Consequently “rolling” motion is considered as an additional kinematic constraint. Certain mathematical properties are proved and algebraic relations derived from the contact conditions are interpreted in the context of the physics of the problem. In this work we developed the foundations for the investigation of the dynamics and control of contacting bodies.

Two-band density matrix theory: the effect of external fields on transition and transport processes

We present a derivation of the two-band density matrix dynamics which considers the effect of external fields on both the intraband drift motion in the six-dimensional configuration space and the interband transitions. This includes in particular the contribution of the light field to the drift term of the interband density matrix which is, for example, responsible for the 'resonant' Stark effect. A relation between the density matrix and electrodynamic quantities is established where our approach directly accounts for the decomposition into interband and intraband polarization. The consideration of a static magnetic field without invoking the 'Peierls substitution' turns out to yield deviations from the usual formulations in magneto-optics and transport theory. In the regime of quantum transport we obtain a general expression for the quantum correction of the energy density which is valid for nonequilibrium conditions and arbitrary band structure and can be used for the derivation of quantum hydrodynamic equations.

On a Representation of Friction in Configuration Space

This article provides a geometric representation of friction for a rigid planar part with two translational and one rotational degrees of freedom. The article constructs a generalized friction cone by imbedding into the part's configuration space the force constraints that define the classic Coulomb friction cone in real space. The resulting representation provides a simple geometric method for determining the possible motions of a part subjected to an applied force and torque. The representation has been used both for simulating part motions and for planning assem bly operations. The approach generalizes to the six-dimensional configuration space of a three-dimensional part.

On the asymptotic behaviour of the elastic 3→3 scattering wave function in the hyperspherical basis

Using the hyperspherical function method it is shown that the terms representing the contribution of single and double scattering processes to the three-body scattering wave function corresponding to elastic 3→3 scattering have the asymptotic behaviour ϱ − 5 2 in the six-dimensional configuration space.

Motion planning with six degrees of freedom by multistrategic bidirectional heuristic free-space enumeration

A general and efficient method is presented that uses a configuration space for planning a collision-free path among known stationary obstacles for an arbitrarily moving object with six degrees of freedom. The basic approach is to restrict the free space concerning path planning and to avoid executing unnecessary collision detections. The six-dimensional configuration space is equally quantized into cells by placing a regular grid, and the cells concerning path planning are enumerated by simultaneously executing multiple search strategies. Search strategies of different characteristics are defined by assigning different values to the coefficients of heuristic functions. The efficiency of each search strategy is evaluated during free-space enumeration, and a more promising one is automatically selected and preferentially executed. The free-space cells are efficiently enumerated for an arbitrary moving object in all kinds of working environments. The implementation of this method on several examples that have different characteristics is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the formulation of boundary conditions for rigid nonspherical Brownian particles near solid walls: Applications to orientation-specific reactions with immobilized enzymes

Transport processes involving the motion of rigid nonspherical Brownian particles within porous media and, more generally, proximate to solid surfaces, arise in a multitude of natural and engineered contexts. Detailed modeling proceeds via solution of the microscale convective-diffusion equation in the six-dimensional configuration space composed of particle positions and orientations, subject to boundary conditions accounting for either impenetrability (inertness) or catalytic activity of the bounding surfaces. Owing to the geometrical complexity associated with even simple particle-wall combinations, the construction of such conditions has long been the source of ambiguities in the literature. We present a rigorous, general analysis of issues arising from particle-wall contact within the context of Brownian transport. Abstract configuration-space flux conditions are shown to possess a physically transparent interpretation in terms of rigid-body kinematics. As a specific application of our general theory, explicit asymptotic results are derived for a model problem pertaining to orientation-specific reactions of macromolecular substrates with immobilized enzymes. In the course of the analysis, a robust numerical technique is developed for the solution of configuration-space diffusion problems, the general utility of which transcends its application to the specific physical system considered here.

Symmetry of the nodal hypersurface for 1s2p 3P helium.

The wave function for the 1s2p ${\mathrm{}}^{3}$P state of the helium atom is found to have an unexpectedly simple nodal hypersurface, symmetric about the z axis when the position of one electron is fixed and independent of the interelectron distance. The six-dimensional configuration space of the electrons is divided by the surface into two geometrically similar regions, but the hypersurface is not a plane of symmetry and points on the hypersurface are not at equivalent sites in the two regions.

A search algorithm for motion planning with six degrees of freedom

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 “Movers'” problem is to find a continuous, collision-free path for a moving object through an environment containing obstacles. We present an implemented algorithm 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 paper describes an 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 intersections, collision-free paths may be found by the closure of three operators which (i) slide along five-dimensional level C-surfaces parallel to C-space obstacles; (ii) slide along one- to four-dimensional intersections of level C-surfaces; and (iii) jump between six-dimensional obstacles. These operators are employed by a best-first search algorithm in C-space. We will discuss theoretical properties of the algorithm, including completeness (at a resolution). This paper describes the heuristic search, with particular emphasis on the heuristic strategies that evaluate local geometric information. At the heart of this paper lie the design and implementation of these strategies for planning paths along level C-surfaces and their intersection manifolds, and for reasoning about motions with three degrees of rotational freedom. The problems of controlling the interaction of these strategies, and of integrating diverse local experts for geometric reasoning provide an interesting application of search to a difficult domain with significant practical implications. The representations and algorithms we develop impact many geometric planning problems, and extend to Cartesian manipulators with six degrees of freedom.

Two Electrons in a Coulomb Potential. Double-Continuum Wave Functions and Threshold Law for Electron-Atom Ionization

The Schr\odinger equation for two electrons in a Coulomb field is studied in the critical region where both electrons have near-zero kinetic energies. The main feature of this problem is that the mutual screening between the two electrons determines and is determined by the partition of the available energy between them. This energy-dependent screening can be taken into account to yield a complex potential in the radial variable $R={({r}_{1}^{2}+{r}_{2}^{2})}^{\frac{1}{2}}$ of the six-dimensional configuration space of the two electrons. Solutions of this equation are obtained and are shown to correspond to the classical orbits given in an early paper by Wannier. A possible way is indicated of using these wave functions to establish the Wannier threshold law which, for ionization of neutral atoms, is $\ensuremath{\sigma}\ensuremath{\propto}{E}^{1.127}$. Finally, the interplay between the total energy and the Coulomb potential is discussed both for this problem and for the case of one electron in the field of a nucleus.