Showing papers on "Configuration space published in 1999"

Randomized kinodynamic planning

Abstract: The paper presents a state-space perspective on the kinodynamic planning problem, and introduces a randomized path planning technique that computes collision-free kinodynamic trajectories for high degree-of-freedom problems. By using a state space formulation, the kinodynamic planning problem is treated as a 2n-dimensional nonholonomic planning problem, derived from an n-dimensional configuration space. The state space serves the same role as the configuration space for basic path planning. The bases for the approach is the construction of a tree that attempts to rapidly and uniformly explore the state space, offering benefits that are similar to those obtained by successful randomized planning methods, but applies to a much broader class of problems. Some preliminary results are discussed for an implementation that determines the kinodynamic trajectories for hovercrafts and satellites in cluttered environments resulting in state spaces of up to twelve dimensions.

Path planning in expansive configuration spaces

Abstract: We introduce the notion of expansiveness to characterize a family of robot configuration spaces whose connectivity can be effectively captured by a roadmap of randomly-sampled milestones. The analysis of expansive configuration spaces has inspired us to develop a new randomized planning algorithm. This new algorithm tries to sample only the portion of the configuration space that is relevant to the current query, avoiding the cost of precomputing a roadmap for the entire configuration space. Thus, it is well-suited for problems where only a single query is submitted for a given environment. The algorithm has been implemented and successfully applied to complex assembly maintainability problems from the automotive industry.

MAPRM: a probabilistic roadmap planner with sampling on the medial axis of the free space

Abstract: Probabilistic roadmap planning methods have been shown to perform well in a number of practical situations, but their performance degrades when paths are required to pass through narrow passages in the free space. We propose a new method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. We give algorithms that perform this retraction while avoiding explicit computation of the medial axis, and we show that sampling and retracting in this manner increases the number of nodes found in small volume corridors in a way that is independent of the volume of the corridor and depends only on the characteristics of the obstacles bounding it. Theoretical and experimental results are given to show that this improves performance on problems requiring traversal of narrow passages.

Singularity Analysis of Closed Kinematic Chains

Abstract: This paper presents a coordinate-invariant differential geometric analysis of kinematic singularities for closed kinematic chains containing both active and passive joints. Using the geometric framework developed in Park and Kim (1996) for closed chain manipulability analysis, we classify closed chain singularities into three basic types: (i) those corresponding to singular points of the joint configuration space (configuration space singularities), (ii) those induced by the choice of actuated joints (actuator singularities), and (iii) those configurations in which the end-effector loses one or more degrees of freedom of available motion (end-effector singularities). The proposed geometric classification provides a high-level taxonomy for mechanism singularities that is independent of the choice of local coordinates used to describe the kinematics, and includes mechanisms that have more actuators than kinematic degrees of freedom.

Potential energy landscape of a model glass former: thermodynamics, anharmonicities, and finite size effects.

Abstract: It is possible to formulate the thermodynamics of a glass forming system in terms of the properties of inherent structures, which correspond to the minima of the potential energy and build up the potential energy landscape in the high-dimensional configuration space. In this work we quantitatively apply this general approach to a simulated model glass-forming system. We systematically vary the system size between $N=20$ and $N=160.$ This analysis enables us to determine for which temperature range the properties of the glass former are governed by the regions of the configuration space, close to the inherent structures. Furthermore, we obtain detailed information about the nature of anharmonic contributions. Moreover, we can explain the presence of finite size effects in terms of specific properties of the energy landscape. Finally, determination of the total number of inherent structures for very small systems enables us to estimate the Kauzmann temperature.

On the classical statistical mechanics of non-Hamiltonian systems

Abstract: A consistent classical statistical mechanical theory of non-Hamiltonian dynamical systems is presented. It is shown that compressible phase space flows generate coordinate transformations with a nonunit Jacobian, leading to a metric on the phase space manifold which is nontrivial. Thus, the phase space of a non-Hamiltonian system should be regarded as a general curved Riemannian manifold. An invariant measure on the phase space manifold is then derived. It is further shown that a proper generalization of the Liouville equation must incorporate the metric determinant, and a geometric derivation of such a continuity equation is presented. The manifestations of the nontrivial nature of the phase space geometry on thermodynamic quantities is explored.

A consistent third-order propagator method for electronic excitation

Abstract: A propagator method referred to as third-order algebraic–diagrammatic construction [ADC(3)] for the direct computation of electronic excitation energies and transition moments is presented. This approach is based on a specific reformulation of the diagrammatic perturbation expansion for the polarization propagator, and extends the existing second-order [ADC(2)] scheme to the next level of perturbation theory. The computational scheme combines diagonalization of a Hermitian secular matrix and perturbation theory for the matrix elements. The characteristic properties of the method are compact configuration spaces, regular perturbation expansions, and size-consistent results. The configuration space is spanned by singly and doubly excited states, while the perturbation expansions in the secular matrix extend through third order in the p-h block, second order in the p-h/2p-2h coupling block, and first order in the 2p-2h block. While the simpler ADC(2) method, representing a counterpart to the MP2 (second-orde...

The kinematic roadmap: a motion planning based global approach for inverse kinematics of redundant robots

Abstract: This paper proposes a novel and global approach to solving the point-to-point inverse kinematics problem for highly redundant manipulators. Given an initial configuration of the robot, the problem is to find a reachable configuration that corresponds to a desired position and orientation of the end-effector. Central to our approach is the novel notion of kinematic roadmap for a manipulator. The kinematic roadmap captures the connectivity of the connected component of the free configuration space of the manipulator in a finite graph like structure. The point-to-point inverse kinematics problem is then solved using this roadmap. We provide completeness results for our algorithm. Our implementation of SEARCH is an efficient closed form solution, albeit local, to inverse kinematics that exploits the serial kinematic structure of serial manipulator arms. Initial experiments with a 7-DOF manipulator have been extremely successful.

Motion planning for a rigid body using random networks on the medial axis of the free space

Abstract: Several motion planning methods using networks of randomly generated nodes in the free space have been shown to perform well in a number of cases, however their performance degrades when paths are required to pass through narrow passages in the free space. In previous work we proposed MAPRM, a method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space without having to first compute the medial axis; this was shown to increase sampling in narrow passages. In this paper we give details of the MAPRM algorithm for the case of a rigid body moving in three dimensions, and show that the retraction may be carried out without explicitly computing the C-obstacles or the medial axis. We give theoretical and experimental results to show this improves performance on problems involving narrow corridors and compare the performance to uniform random sampling from the free space.

Resonant Orbits in Triaxial Galaxies

Abstract: Box orbits in triaxial potentials are generically thin, that is, they lie close in phase space to a resonant orbit satisfying a relation of the form lω1 + mω2 + nω3 = 0 between the three fundamental frequencies. Resonant orbits are confined for all time to a membrane in configuration space; they play roughly the same role in structuring the phase space of three-dimensional systems that periodic orbits play in two dimensions. Stable resonant orbits avoid the center of the potential; orbits that are thick enough to pass near the destabilizing center are typically stochastic. Resonances in triaxial potentials are most important at energies far outside the region of gravitational influence of a central black hole. Near the black hole, the motion is essentially regular, although resonant orbits exist in this region as well, including at least one family whose elongation is parallel to the long axes of the triaxial figure.

Reduction in principal fiber bundles: covariant Euler-Poincaré equations.

Abstract: In Lagrangian mechanics the simplest example of reduction is the Euler-Poincare reduction In this case, the configuration space is a Lie group G and the Lagrangian function L:TG→R is invariant under the natural action of the group on TG by right translations Then L induces a function l:(TG)/G≅g→R, g being the Lie algebra of G, and the Euler-Lagrange equations defined by L transform into a new group of equations for l called the Euler-Poincare equations For example, this is the case for the dynamics of the rigid body In the present paper, the authors extend the idea of reduction to Lagrangian field theory In this framework, the analogous configuration is a principal bundle π:P→M with structure group G (a matrix group) and a Lagrangian L:J1P→R invariant under the natural action of G on the 1-jet bundle defined by (j1xs)⋅g=j1x(Rg∘s), where Rg denotes the right translation by g on P Let l:(J1P)/G→R be the induced mapping It is proved that the Euler-Lagrange equations define a group of equations for critical sections, which generalize the Euler-Poincare equations of mechanics As is well known, the quotient manifold (J1P)/G can be identified with the bundle of connections of π:P→M This fact gives a geometrical meaning to the previous reduction In particular, the critical sections of the Euler-Poincare equations are sections of this bundle, and therefore they can be understood as principal connections of π:P→M The authors exploit this idea in order to explain the compatibility conditions needed for reconstruction The Euler-Poincare equations do not suffice to reconstruct the Euler-Lagrange equations Some extra conditions must be imposed, namely, the vanishing of the curvature of the critical sections This fact is characteristic of field theory and does not appear in classical mechanics Finally, the authors study the Euler-Poincare equations in two examples from the variational approach to harmonic mappings

Transition state in atomic physics

Abstract: The transition state is fundamental to modern theories of reaction dynamics: essentially, the transition state is a structure in phase space that all reactive trajectories must cross. While transition-state theory (TST) has been used mainly in chemical physics, it is possible to apply the theory to considerable advantage in any collision problem that involves some form of reaction. Of special interest are systems in which chaotic scattering or half-scattering occurs such as the ionization of Rydberg atoms in external fields. In this paper the ionization dynamics of a hydrogen atom in crossed electric and magnetic fields are shown to possess a transition state: We compute the periodic orbit dividing surface (PODS) which is found not to be a dividing surface when projected into configuration space. Although the possibility of a PODS occurring in phase space rather than configuration space has been recognized before, to our knowledge this is the first actual example: its origin is traced directly to the presence of velocity-dependent terms in the Hamiltonian. Our findings establish TST as the method of choice for understanding ionization of Rydberg atoms in the presence of velocity-dependent forces. To demonstrate this TST is used to (i) uncover a multiple-scattering mechanism for ionization and (ii) compute ionization rates. In the process we also develop a method of computing surfaces of section that uses periodic orbits to define the surface, and examine the fractal nature of the dynamics.

Investigation of $^{4}He_{3}$ trimer on the base of Faddeev equations in configuration space

Abstract: Precise numerical calculations of bound states of a three-atomic Helium cluster are performed. The modern techniques of solution of Faddeev equations are combined to obtain an efficient numerical scheme. Binding energies and other observables for ground and excited states are calculated. Geometric properties of the clusters are discussed.

Realizing holonomic constraints in classical and quantum mechanics

Abstract: We consider the problem of constraining a particle to a submanifold Sigma of configuration space using a sequence of increasing potentials. We compare the classical and quantum versions of this procedure. This leads to new results in both cases: an unbounded energy theorem in the classical case, and a quantum averaging theorem. Our two step approach, consisting of an expansion in a dilation parameter, followed by averaging in normal directions, emphasizes the role of the normal bundle of Sigma, and shows when the limiting phase space will be larger (or different) than expected.

A generalization of Snaith-type filtration

Abstract: In this paper we describe the Goodwillie tower of the stable homotopy of a space of maps from a finite-dimensional complex to a highly enough connected space. One way to view it is as a partial generalization of some wellknown results on stable splittings of mapping spaces in terms of configuration spaces. 0. INTRODUCTION It has been known for a while (see [1] for a survey article and a list of references) that given a parallelizable, compact m-dimensional manifold M with a nonempty boundary, and given a connected, pointed space Z, there is a configuration space model for the space of unbased maps Map(M, SmZ), which stably splits. More precisely, there is a weak equivalence: (0.1) Q? E' (Map(M, SmZ)) 1 QJJ E' ((C(M, OM; n) AEn ZAn)), n>1 where C(M, OM; n) stands for the space of n-tuples of distinct points in M, where all n-tuples whose intersection with OM is not empty have been identified to a point. There is an analogous splitting for the space of based maps. A closely related result is the stable splitting of spaces of the form QmEmX. This later splitting is sometimes refered to as the Snaith splitting (at least in the case m =oo), and we refer to (0.1) as Snaith-type splitting. It is, therefore, natural to ask if for a based space K, that is not a manifold, but, say, a finite CW-complex, anything can be said about the functor X F-+ QMap*(K, X) (where Map*(K, X) stands for the space of based maps from K to X). One does not expect this functor to split, in general, but it is still reasonable to try to approximate it by more elementary functors in a way that would give the splitting above in the case when K = M and X = SmZ. It turns out that this question (in fact a generalization of it) can be answered positively within the framework of the theory referred to as calculus of functors, which had been developed by T. Goodwillie in [4], [5], [6]. Since [6] has not been published yet, we present a brief outline of the theory of "Taylor towers" in the appendix. In what follows we will freely use notation from there. Consider again the identity (0.1). Observe that the factors Q (C(M, OM; n) AEn ZAn) Received by the editors July 21, 1994 and, in revised form, February 4, 1997. 1991 Mathematics Subject Classification. Primary 55P99. (?)1999 American Mathematical Society 1123 This content downloaded from 157.55.39.235 on Fri, 07 Oct 2016 05:53:45 UTC All use subject to http://about.jstor.org/terms

Topological Origin of the Phase Transition in a Mean-Field Model

Abstract: We argue that the phase transition in the mean-field XY model is related to a particular change in the topology of its configuration space. The nature of this topological change can be discussed on the basis of elementary Morse theory using the potential energy per particle V as a Morse function. The value of V where such a topological change occurs equals the thermodynamic value of V at the phase transition and the number of (Morse) critical points grows very fast with the number of particles N . Furthermore, as in statistical mechanics, the way the thermodynamic limit is taken is crucial in topology. {copyright} {ital 1999} {ital The American Physical Society}

Diffeomorphism groups and current algebras: configuration space analysis in quantum theory

Abstract: The constuction of models of non-relativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space Γ of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on Γ, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on ℝd.

Configuration Spaces of Mechanical Linkages

Abstract: We present an easy to survey constructive method using only basic mathematics which allows us to define a homeomorphism between any compact real algebraic variety and some components of the configuration space of a mechanical linkage. The aim is to imitate addition and multiplication in the framework of weighted graphs in the euclidean plane that permit a ``mechanical description'' of polynomial functions, and thus of varieties.

Motion Planning for a Convex Polygon in a Polygonal Environment

Abstract: We study the motion-planning problem for a convex m -gon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the three-dimensional space of all free placements of P in Q ) in time that is near-quadratic in mn , which is nearly optimal in the worst case. The algorithm is also conceptually simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also near-quadratic in mn . In addition, we describe an algorithm that preprocesses the computed free configuration space so that reachabilityqueries can be answered in polylogarithmic time.

On the geometry of classifying spaces and horizontal slices

Abstract: We study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let U be a neighborhood of the moduli space. Then we know the universal covering space V of U is a smooth manifold. Suppose D is the classifying space of a polarized Calabi-Yau manifold with the automorphism group G. Then we prove that the map from V to D induced by the period map is a pluriharmonic map. We also give a Kahler metric on V, which is called the Hodge metric. We prove that the Ricci curvature of the Hodge metric is negative away from zero. We also proved the nonexistence of the Kahler metric on the classifying space of a Calabi-Yau threefold which is invariant under a cocompact lattice of G. 1. Introduction. Let (X, ) be a polarized simply connected Calabi-Yau manifold. That is, X is a simply connected compact Kahler manifold of dimension n with zero first Chern class and is aKform of X such that ( ) H 2 (X, Z). In this paper, we study the local properties of the moduli space of the polarized Calabi-Yau manifold (X, ). By definition is the parameter space of the complex structures over X for the fixed polarization ( ); is a quasi-projective variety by a theorem of Viehweg (17). Suppose X is a Calabi-Yau manifold. Let N =d im H 1 (X, TX) =0

Diffeomorphism invariant Quantum Field Theories of Connections in terms of webs

Abstract: In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only a finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise-smooth finite paths and loops. In particular, we (a) characterize the spectrum of the Ashtekar-Isham configuration space, (b) introduce spin-web states, a generalization of the spin-network states, (c) extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism-invariant states and finally (d) extend the 3-geometry operators and the Hamiltonian operator.

Configuration space based recurrence relations for sunset-type diagrams

Abstract: We derive recurrence relations for the calculation of multiloop sunset-type diagrams with large powers of massive propagators. The technique is formulated in configuration space and exploits the explicit form of the massive propagator raised to a given power. We write down and evaluate a convenient set of basis integrals. The method is well suited for a numerical evaluation of this class of diagrams. We give explicit analytical formulae for the basis integrals in the asymptotic regime.

Mode regularization of the configuration space path integral in curved space

Abstract: The path integral representation of the transition amplitude for a particle moving in curved space has presented unexpected challenges since the introduction of path integrals by Feynman fifty years ago In this paper we discuss and review mode regularization of the configuration space path integral, and present a three loop computation of the transition amplitude to test with success the consistency of such a regularization Key features of the method are the use of the Lee-Yang ghost fields, which guarantee a consistent treatment of the non-trivial path integral measure at higher loops, and an effective potential specific to mode regularization which arises at two loops We also perform the computation of the transition amplitude using the regularization of the path integral by time discretization, which also makes use of Lee-Yang ghost fields and needs its own specific effective potential This computation is shown to reproduce the same final result as the one performed in mode regularization

The Hermite correction method for nonadiabatic transitions

Abstract: We have performed molecular dynamics simulations on a system where electronic transitions are allowed anywhere in configuration space among any number of coupled states. A classical path theory based on the Hermite correction to the Gaussian wave packet expansion, proposed by Gert D. Billing [J. Chem. Phys. 107, 4286 (1997)] has been used. The calculations are carried out on the same model used by J. C. Tully [J. Chem. Phys. 93, 1061 (1990)] and the transition probabilities agree well with corresponding exact quantum mechanical results.

Topological aspects of geometrical signatures of phase transitions

Abstract: Certain geometric properties of submanifolds of configuration space are numerically investigated for classical phi(4) models in one and two dimensions. Peculiar behaviors of the computed geometric quantities are found only in the two-dimensional case, when a phase transition is present. The observed phenomenology strongly supports, though in an indirect way, a recently proposed topological conjecture about a topology change of the configuration space submanifolds as counterpart to a phase transition.

Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

Abstract: We give lowed bounds on the number of periodic trajectories in strictly convex smooth billiards in $\R^{m+1}$ for $m\ge 3$. For plane billiards (when m=1) such bounds were obtained by G. Birkhoff in the 1920's. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik - Schirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere $S^m$, i.e., the space of n-tuples of points $(x_1, ..., x_n)$, where $x_i\in S^m$ and $x_i e x_{i+1}$ for i=1,2, ..., n.