Showing papers on "Configuration space published in 2003"

Topological Complexity of Motion Planning

Abstract: . In this paper we study a notion of topological complexity TC(X) for the motion planning problem. TC(X) is a number which measures discontinuity of the process of motion planning in the configuration space X . More precisely, TC(X) is the minimal number k such that there are k different "motion planning rules," each defined on an open subset of X× X , so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik—Schnirelman theory) to study the topological complexity TC(X) . We give an upper bound for TC(X) (in terms of the dimension of the configuration space X ) and also a lower bound (in terms of the structure of the cohomology algebra of X ). We explicitly compute the topological complexity of motion planning for a number of configuration spaces: spheres, two-dimensional surfaces, products of spheres. In particular, we completely calculate the topological complexity of the problem of motion planning for a robot arm in the absence of obstacles.

Random point fields associated with certain Fredholm determinants II: Fermion shifts and their ergodic and Gibbs properties

Abstract: We construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants. Under a mild condition they turn out unique Gibbs measures. Also some ergodic properties, including the entropy positivity, are discussed in the lattice case.

Endogenous configuration space approach to mobile manipulators: A derivation and performance assessment of Jacobian inverse kinematics algorithms

Abstract: By a mobile manipulator we mean a robotic system composed of a non-holonomic mobile platform and a holonomic manipulator fixed to the platform. A taskspace of the mobile manipulator includes positions and orientations of its end effector relative to an inertial coordinate frame. The kinematics of a mobile manipulator are represented by a driftless control system with outputs. Admissible control functions of the platform along with joint positions of the manipulator constitute the endogenous configuration space. Endogenous configurations have a meaning of controls. A map from the endogenous configuration space into the taskspace is referred to as the instantaneous kinematics of the mobile manipulator. Within this framework, the inverse kinematic problem for a mobile manipulator amounts to defining an endogenous configuration that drives the end effector to a desirable position and orientation in the taskspace. Exploiting the analogy between stationary and mobile manipulators we present in the paper a colle...

Quantification of order in the Lennard-Jones system

Abstract: We conduct a numerical investigation of structural order in the shifted-force Lennard-Jones system by calculating metrics of translational and bond-orientational order along various paths in the phase diagram covering equilibrium solid, liquid, and vapor states. A series of nonequilibrium configurations generated through isochoric quenches, isothermal compressions, and energy minimizations are also considered. Simulation results are analyzed using an ordering map representation [Torquato et al., Phys. Rev. Lett. 84, 2064 (2000); Truskett et al., Phys. Rev. E 62, 993 (2000)] that assigns both equilibrium and nonequilibrium states coordinates in an order metric plane. Our results show that bond-orientational order and translational order are not independent for simple spherically symmetric systems at equilibrium. We also demonstrate quantitatively that the Lennard-Jones and hard sphere systems sample the same configuration space at supercritical densities. Finally, we relate the structural order found in fast-quenched and minimum-energy configurations (inherent structures).

Singularities of parallel manipulators: a geometric treatment

Abstract: A parallel manipulator is naturally associated with a set of constraint functions defined by its closure constraints. The differential forms arising from these constraint functions completely characterize the geometric properties of the manipulator. In this paper, using the language of differential forms, we provide a thorough geometric study on the various types of singularities of a parallel manipulator, their relations with the kinematic parameters and the configuration spaces of the manipulator, and the role redundant actuation plays in reshaping the singularities and improving the performance of the manipulator. First, we analyze configuration space singularities by constructing a Morse function on some appropriately defined spaces. By varying key parameters of the manipulator, we obtain homotopic classes of the configuration spaces. This allows us to gain insight on configuration space singularities and understand how to choose design parameters for the manipulator. Second, we define parametrization singularities which include actuator and end-effector singularities (or other equivalent definitions) as their special cases. This definition naturally contains the closure constraints in addition to the coordinates of the actuators and the end-effector and can be used to search a complete set of actuator or end-effector singularities including some singularities that may be missed by the usual kinematics methods. We give an intrinsic classification of parametrization singularities and define their topological orders. While a nondegenerate singularity poses no problems in general, a degenerate singularity can sometimes be a source of danger and should be avoided if possible.

A general framework for sampling on the medial axis of the free space

Abstract: We propose a general framework for sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. Generalizing our previous work, this framework provides a template encompassing all possible retraction approaches. It also removes the requirement of exactly computing distance metrics thereby enabling application to more realistic high dimensional problems. In particular, our framework supports methods that retract a given configuration exactly or approximately onto the medial axis. As in our previous work, exact methods provide fast and accurate retraction in low (2 or 3) dimensional space. We also propose new approximate methods that can be applied to high dimensional problems, such as many DOF articulated robots. Theoretical and experimental results show improved performance on problems requiring traversal of narrow passages. We also study tradeoffs between accuracy and efficiency for different levels of approximation, and how the level of approximation effects the quality of the resulting roadmap.

Manifold theoretic compactifications of configuration spaces

Abstract: We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.

A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics

Abstract: A Lie-group formulation for the kinematics and dynamics ofholonomic constrained mechanical systems (CMS) is presented. The kinematics ofrigid multibody systems (MBS) is described in terms of the screw system of theMBS. Using Lie-algebraic properties of screw algebra, isomorphicto se(3), allows a purely algebraic derivation of the Lagrangian motion equations. As such the Lie-group SE(3) ⊗... ⊗ SE(3) (n copies) is theambient space of a MBS consisting of n rigid bodies. Any parameterizationof the ambient space corresponds to a chart on the MBS configuration space ℝn. The key to combine differential geometric and Lie-algebraic approaches is the existence of kinematic basic functions whichare push forward maps from the tangent bundle Tℝn to the Lie-algebra of the ambient space.

Phase Transitions and Topology Changes in Configuration Space

Abstract: The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly “strong” topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: (i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; (ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.

Self-guided enhanced sampling methods for thermodynamic averages

Abstract: In the self-guided molecular dynamics (SGMD) simulation method, a continuously updated average force is used to bias the motions of the system. The method appears to sample the configuration space of a number of complex systems more efficiently than ordinary molecular dynamics, and it was argued that it yields canonical averages of observable quantities with only negligible errors. We analyze the dynamic mapping associated with the SGMD algorithm and find that the dynamics lacks reversibility because the effective potential that governs the motion is a functional of the trajectory rather than a function of the coordinates (i.e., the dynamics is not uniquely specified by the initial conditions but depends on past history as well). This irreversibility is shown to result in substantial errors in canonical averages for model systems. Motivated by this analysis, we introduce an alternative self-guided scheme (the momentum-enhanced hybrid Monte Carlo method) that does converge to the canonical distribution in principle. The method differs from the original SGMD algorithm in that momenta, rather than forces, are averaged to bias the initial choice of momenta at each step in a hybrid Monte Carlo procedure. The relation of the method to other enhanced sampling algorithms is discussed.

Safe path planning in an uncertain-configuration space

Abstract: The objective of this paper is to bring an effective response to the safe path planning problem which should be solved in an uncertain-configuration space. First, a path planning method dealing with localization uncertainties is proposed, where the uncertainties in both position and orientation of a non-holonomic mobile robot are considered. The safety of this method is due to the mixing of the planning phase and the navigation phase using the same process of localization (the Kalman filter). Next, while previous works planned safe paths in the configuration space, we show that it is necessary to plan safe paths in an uncertain-configuration space. Then, we introduce the novel concept of "towers of uncertainties" and show the effectiveness of this concept with some examples.

A bounded uncertainty approach to multi-robot localization

Abstract: We offer a new approach to the multi-robot localization problem. Using an unknown-but-bounded model for sensor error, we are able to define convex polytopes in the configuration space of the robot team that represent the set of configurations consistent with all sensor measurements. Estimates for the uncertainty in various parameters of the team's configuration such as the absolute position of a single robot, or the relative positions of two or more nodes can be obtained by projecting this polytope onto appropriately chosen subspaces of the configuration space. We propose a novel approach to approximating these projections using linear programming techniques. The approach can handle both bearing and range measurements with a computational complexity scaling polynomially in the number of roots. Finally, the workload is readily distributed - requiring only the communication of sensor measurements between robots. We provide simulation results for this approach implemented on a multi-robot team.

New perspectives on self-linking

Abstract: We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the space of knots as a subspace of what we call the n-th mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.

Nuclear level statistics: Extending shell model theory to higher temperatures

Abstract: The shell model Monte Carlo (SMMC) approach has been applied to calculate level densities and partition functions to temperatures up to ,1.5‐2 MeV, with the maximal temperature limited by the size of the configuration space. Here we develop an extension of the theory that can be used to higher temperatures, taking into account the large configuration space that is needed. We first examine the configuration space limitation using an independent-particle model that includes both bound states and the continuum. The larger configuration space is then combined with the SMMC under the assumption that the effects on the partition function are factorizable. The method is demonstrated for nuclei in the iron region, extending the calculated partition functions and level densities up to T,4 MeV. We find that the back-shifted Bethe formula has a much larger range of validity than was previously believed. The present theory also shows more clearly the effects of the pairing phase transition on the heat capacity.

Statistical mechanics and phase diagrams of rotating self-gravitating fermions

Abstract: We compute statistical equilibrium states of rotating self-gravitating fermions by maximizing the Fermi-Dirac en- tropy at fixed mass, energy and angular momentum. We describe the phase transition from a gaseous phase to a condensed phase (corresponding to white dwarfs, neutron stars or fermion balls in dark matter models) as we vary energy and temperature. We increase the rotation up to the Keplerian limit and describe the flattening of the configuration until mass shedding occurs. At the maximum rotation, the system develops a cusp at the equator. We draw the equilibrium phase diagram of the rotat- ing self-gravitating Fermi gas and discuss the structure of the caloric curve as a function of degeneracy parameter (or system size) and angular velocity. We argue that systems described by the Fermi-Dirac distribution in phase space do not bifurcate to non-axisymmetric structures when rotation is increased, in continuity with the case of polytropes with index n > 0.808 (the Fermi gas at T = 0 corresponds to n = 3/2). This differs from the study of Votyakov et al. (2002) who consider a Fermi-Dirac distribution in configuration space appropriate to stellar formation and find "double star" structures (their model at T = 0c or- responds to n = 0). We also consider the case of classical objects described by the Boltzmann entropy and discuss the influence of rotation on the onset of gravothermal catastrophe (for globular clusters) and isothermal collapse (for molecular clouds). On general grounds, we complete previous investigations concerning the nature of phase transitions in self-gravitating systems. We emphasize the inequivalence of statistical ensembles regarding the formation of binaries (or low-mass condensates) in the microcanonical ensemble (MCE) and Dirac peaks (or massive condensates) in the canonical ensemble (CE). We also describe an hysteretic cycle between the gaseous phase and the condensed phase that are connected by a "collapse" or an "explosion".

Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering?

Abstract: We examine the notion and properties of the non-Hermitian effective Hamiltonian of an unstable system using as an example potential resonance scattering with a fixed angular momentum. We present a consistent self-adjoint formulation of the problem of scattering on a finite-range potential, which is based on the separation of the configuration space into two segments, internal and external. The scattering amplitude is expressed in terms of the resolvent of a non-Hermitian operator H. The explicit form of this operator depends on both the radius of separation and the boundary conditions at this place, which can be chosen in many different ways. We discuss this freedom and show explicitly that the physical scattering amplitude is, nevertheless, unique, although not all choices are equally adequate from the physical point of view. The energy-dependent operator H should not be confused with the non-Hermitian effective Hamiltonian H(eff) which is usually exploited to describe interference of overlapping resonances. We note that the simple Breit-Wigner approximation is as a rule valid for any individual resonance in the case of few-channel scattering on a flat billiardlike cavity, leaving no room for nontrivial H(eff) to appear. The physics is appreciably richer in the case of an open chain of L connected similar cavities whose spectrum has a band structure. For a fixed band of L overlapping resonances, the smooth energy dependence of H can be ignored so that the constant LxL submatrix H(eff) approximately describes the time evolution of the chain in the energy domain of the band and the complex eigenvalues of H(eff) define the energies and widths of the resonances. We apply the developed formalism to the problem of a chain of L delta barriers, whose solution is also found independently in a closed form. We construct H(eff) for the two commonly considered types of boundary conditions (Neumann and Dirichlet) for the internal motion. Although the final results are in perfect coincidence, somewhat different physical patterns arise of the trend of the system with growing openness. Formation in the outer well of a short-lived doorway state shifted in energy is explicitly demonstrated together with the appearance of L-1 long-lived states trapped in the inner part of the chain.

A New Approach to Quantising Space-Time: I. Quantising on a General Category

Abstract: A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for space-time (or space). The starting point is the observation that entities of this type can be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) can be regarded as the set of objects in a category. In this first of a series of papers, we study this question in general and develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold $Q\simeq G/H$ where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of `arrow fields' on the category. Physically, this means that an arrow between two objects in the category is viewed as some sort of analogue of momentum. After finding the `category quantisation monoid', we show how suitable representations can be constructed using a bundle of Hilbert spaces over the set of objects.

Geometric Algorithms for Kinematic Calibration of Robots Containing Closed Loops

Abstract: We present a coordinate-invariant, differential geometric formulation of the kinematic calibration problem for a general class of mechanisms The mechanisms considered may have multiple closed loops, be redundantly actuated, and include an arbitrary number of passive joints that may or may not be equipped with joint encoders Some form of measurement information on the position and orientation of the tool frame may also be available Our approach rests on viewing the joint configuration space of the mechanism as an embedded submanifold of an ambient manifold, and formulating error measures in terms of the Riemannian metric specified in the ambient manifold Based on this geometric framework, we pose the kinematic calibration problem as one of determining a parametrized multidimensional surface that is a best fit (in the sense of the chosen metric) to a given set of measured points in both the ambient and task space manifolds Several optimization algorithms that address the various possibilities with respect to available measurement data and choice of error measures are given Experimental and simulation results are given for the Eclipse, a six degree-of-freedom redundantly actuated parallel mechanism The geometric framework and algorithms presented in this article have the desirable feature of being invariant with respect to the local coordinate representation of the forward and inverse kinematics and of the loop closure equations, and also provide a high-level framework in which to classify existing approaches to kinematic calibration

A New Approach to Quantising Space-Time: I. Quantising on a General Category

Abstract: A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for space-time (or space). The starting point is the observation that entities of this type can be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) can be regarded as the set of objects Ob(Q) in a category Q. In this first of a series of papers, we study this question in general and develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold Q is isomorphic to G/H where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of 'arrow fields' on Q. Physically, this means that an arrow between two objects in the category is viewed as some sort of analogue of momentum. After finding the 'category quantisation monoid', we show how suitable representations can be constructed using a bundle of Hilbert spaces over Ob(Q).

Sphalerons, spectral flow, and anomalies

Abstract: The topology of configuration space may be responsible in part for the existence of sphalerons. Here, sphalerons are defined to be static but unstable finite-energy solutions of the classical field equations. Another manifestation of the nontrivial topology of configuration space is the phenomenon of spectral flow for the eigenvalues of the Dirac Hamiltonian. The spectral flow, in turn, is related to the possible existence of anomalies. In this review, the interconnection of these topics is illustrated for three particular sphalerons of SU(2) Yang–Mills–Higgs theory.

Uniqueness of Gibbs measures relative to Brownian motion

Abstract: We consider the set of Gibbs measures relative to Brownian motion for given potential V . V is assumed to be Kato-decomposable but general otherwise. A Gibbs measure for such a potential is in many cases given by a reversible Ito diffusion μ . We show that if V is growing at infinity faster than quadratically and in a sufficiently regular way, then μ is the only Gibbs measure that exists. For general V we specify a subset of the configuration space Ω such that μ is the only Gibbs measure for V supported on this subset. We illustrate our results by some examples.

The basis of wavelets for a finite Heisenberg magnet

Abstract: We propose here a method for an immediate diagonalisation of the XXX Heisenberg magnetic ring for the spin 1 2 with a finite number N of nodes. The key ingredient is the basis of wavelets, i.e. Fourier transforms on orbits of the translation group on the set of all magnetic configurations with a definite number r of spin deviations from ferromagnetic saturation. The method demonstrates the Yang–Baxter structure of the relevant classical configuration space, reduces by N the size of the secular equation, and explains the dynamics of the Heisenberg ring in terms of geometry of orbits. We show in particular, that a κ-tuply rarefied orbit, with κ a divisor of N, stimulates neighbour regular (N-element) orbits to twist in a way to become a κ-tuply cover of the adjacent rarefied orbit.

Hydrogenic orbitals in Momentum space and hyperspherical harmonics: Elliptic Sturmian basis sets

Abstract: Momentum space hydrogenic orbitals can be regarded as orthonormal and complete Sturmian basis sets and explicitly given in terms of (hyper)-spherical harmonics on the 4-D hypersphere S 3 . Among the alternative coordinate systems that allow separation of variables, the usual ones involving parameterizations of the sphere S 3 by circular functions correspond to canonical subgroup reduction chains; we also investigate harmonic "elliptic" sets (as, e.g., obtained by parameterizations in terms of Jacobi elliptic functions). In this article we list the canonical hydrogenic Sturmian sets and the orthogonal transformations connecting them. The latter enjoy useful three-term recurrence relationships that allow their efficient calculations even for large strings. We also consider modifications needed when the conservation of the symmetry of Sturmians with respect to parity. Finally, we discuss some properties of elliptic hydrogenic Sturmians and their relations with canonical Sturmians. Because elliptic Sturmians cannot be expressed in closed form, it is important to find expansions in a suitable basis set and calculate the transformation coefficients. We derive three-term recursion relationships fulfilled by the coefficients of the transformation between elliptic Sturmians and canonical Sturmians. A concluding discussion on the connections between configuration space and momentum space hydrogenic Sturmians completes this article. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 92: 212-228, 2003

Symmetry, reduction and swimming in a perfect fluid

Abstract: This thesis presents a geometric picture of a deformable body in a perfect fluid and a way to approximate its dynamics and the motion, resulting from cyclic shape deformations, of the body and, interestingly, the fluid as well. Emphasis is placed on the group structure of the configuration space of the body fluid system and the resulting symmetry in their equations of motion. Symmetry is also used to reduce a series expansion for the flow of a time dependent vector field in order to obtain a novel expansion for the path-ordered exponential. This can be used to approximate holonomy, or geometric phase, in a principal bundle when its evolution is governed by a connection on the bundle and it is subject to periodic shape inputs. Simple models for swimming in and the stirring of a perfect fluid are proposed and examined.

Kinematic dexterity of mobile manipulators: an endogenous configuration space approach

Abstract: A mobile manipulator is treated as a robotic system composed of a non-holonomic mobile platform and a holonomic manipulator mounted on the platform. The kinematics of the mobile manipulator can be represented as a driftless control system with outputs. By adopting the endogenous configuration space approach we propose two kinematic dexterity measures, called local and global dexterity. The local dexterity, modeled upon the manipulability of stationary manipulators, indicates how infinitesimal motions in the configuration space propagate to the taskspace of the mobile manipulator. The global dexterity corresponds to L2-norm of the local dexterity over a prescribed region of the configuration space. Advantages of the endogenous dexterity measures over traditional performance measures of mobile manipulators known from the literature are described. Both the dexterities are employed for determining optimal configurations and optimal geometries of an exemplary mobile manipulator.

On the Cohomology of Configuration Spaces on Surfaces

Abstract: The integral cohomology rings of the configuration spaces of -tuples of distinct points on arbitrary surfaces (not necessarily orientable, not necessarily compact and possibly with boundary) are studied. It is shown that for punctured surfaces the cohomology rings stabilize as the number of points tends to infinity, similarly to the case of configuration spaces on the plane studied by Arnold, and the Goryunov splitting formula relating the cohomology groups of configuration spaces on the plane and punctured plane to arbitrary punctured surfaces is generalized. Moreover, on the basis of explicit cellular decompositions generalizing the construction of Fuchs and Vainshtein, the first cohomology groups for surfaces of low genus are given.