Showing papers on "Configuration space published in 1990"

Robot Motion Planning

Abstract: 1 Introduction and Overview.- 2 Configuration Space of a Rigid Object.- 3 Obstacles in Configuration Space.- 4 Roadmap Methods.- 5 Exact Cell Decomposition.- 6 Approximate Cell Decomposition.- 7 Potential Field Methods.- 8 Multiple Moving Objects.- 9 Kinematic Constraints.- 10 Dealing with Uncertainty.- 11 Movable Objects.- Prospects.- Appendix A Basic Mathematics.- Appendix B Computational Complexity.- Appendix C Graph Searching.- Appendix D Sweep-Line Algorithm.- References.

Local symmetries and constraints

Abstract: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space‐time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the ‘‘field‐independent’’ local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are ‘‘field dependent.’’ As by‐products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.

Path planning using Laplace's equation

Abstract: A method for planning smooth robot paths is presented. The method relies on the use of Laplace's equation to constrain the generation of a potential function over regions of the configuration space of an effector. Once the function is computed, paths may be found very quickly. These functions do not exhibit the local minima which plague the potential field method. Unlike decompositional and algebraic techniques. Laplace's equation is very well suited to computation on massively parallel architectures. >

Motion of two rigid bodies with rolling constraint

Abstract: The motion of two rigid bodies under rolling constraint is considered. In particular, the following two problems are addressed: (1) given the geometry of the rigid bodies, determine the existence of an admissible path between two contact configurations; and (2) assuming that an admissible path exists, find such a path. First, the configuration space of contact is defined, and the differential equations governing the rolling constraint are derived. Then, a generalized version of Frobenius's theorem, known as Chow's theorem, for determining the existence of motion is applied. Finally, an algorithm is proposed that generates a desired path with one of the objects being flat. Potential applications of this study include adjusting grasp configurations of a multifingered robot hand without slipping, contour following without dissipation or wear by the end-effector of a manipulator, and wheeled mobile robotics. >

A Monte-Carlo algorithm for path planning with many degrees of freedom

Abstract: A stochastic technique is described for planning collision-free paths of robots with many degrees of freedom (DOFs). The algorithm incrementally builds a graph connecting the local minima of a potential function defined in the robot's configuration space and concurrently searches the graph until a goal configuration is attained. A local minimum is connected to another one by executing a random motion that escapes the well of the first minimum, succeeded by a gradient motion that follows the negated gradient of the potential function. All the motions are executed in a grid shown through the robot's configuration space. The random motions are implemented as random walks which are known to converge toward Brownian motions when the steps of the walks tend toward zero. The local minima graph is searched using a depth-first strategy with random backtracking. In the technique, the planner does not explicitly represent the local-minima graph. The path-planning algorithm has been fully implemented and has run successfully on a variety of problems involving robots with many degrees of freedom. >

Towards an Optimal Mutation Probability for Genetic Algorithms

Abstract: In this paper the optimal parameter setting of Genetic Algorithms (GAs) is investigated Particular attention has been paid to the dependence of the mutation probability P M upon two parameters, the dimension of the configuration space l and the population size M Assuming strict conditions on both the problem to be optimized and the GA, P M converges to 0 as the population size M or the dimension of the configuration space l converges to infinity For direct application a heuristic comprising these results is presented The parameter settings obtained by applying this heuristic are in accordance with those which have been obtained earlier by experiment

New approach to find exact solutions for cosmological models with a scalar field.

Abstract: We study the cosmological models whose ordinary differential equations can be given a Lagrangian description on a two-dimensional "configuration space." By requiring the existence of a N\"other symmetry for such a Lagrangian, we are able to show that the potential must have an expotential form. With the help of the constants of motion we can get from that Lagrangian, we integrate the models and analyze the behavior for all the possible varying free parameters and plot some of the solutions. We also compare our results with others available in the literature.

Orbifolds as configuration spaces of systems with gauge symmetries

Abstract: In systems like Yang-Mills or gravity theory, which have a symmetry of gauge type, neither phase space nor configuration space is a manifold but rather an orbifold with singular points corresponding to classical states of non-generically higher symmetry. The consequences of these symmetries for quantum theory are investigated. First, a certain orbifold configuration space is identified. Then, the Schrodinger equation on this orbifold is considered. As a typical case, the Schrodinger equation on (double) cones over Riemannian manifolds is discussed in detail as a problem of selfadjoint extensions. A marked tendency towards concentration of the wave function around the singular points in configuration space is observed, which generically even reflects itself in the existence of additional bound states and can be interpreted as a quantum mechanism of symmetry enhancement.

A motion planner for car-like robots based on a mixed global/local approach

Abstract: Deals with the problem of motion planning for a car-like robot (i.e. nonholonomic mobile robot whose turning radius is lower bounded). The main contribution is the introduction of a new metric in the configuration space R/sup 2/*S/sup 1/ of such a system. This metric is defined from the length of the shortest paths in the absence of obstacles. The authors study the relations between the new induced topology and the classical one. This study leads to new theoretical issues about sub-Riemannian geometry and to practical results for motion planning. In particular they prove an inclusion relation of neighbourhoods in both topologies, which is the basis of an efficient obstacle avoidance local method. >

Rapid computation of configuration space obstacles

Abstract: Mathematical properties of configuration space are presented, and algorithms invoking those properties for efficient computation of obstacles in configuration space are described. Simple elements in Cartesian space which can be transformed into configuration space rapidly are identified. Transformations of complex workspace shapes into configuration space are described in terms of multiple transformations of such simpler primitives. Computational considerations and examples are presented for the first three degrees of freedom of an industrial robot. >

2D path planning: a configuration space heuristic approach

Abstract: In this paper we describe a heuristic technique for solving the 2D find-path problem for a rigid mobile body amidst a set of fixed obstacles of arbitrary shapes. The proposed approach tackles path planning as an informed search process in dis crete configuration space. Three heuristics are proposed to guide this search process, all of them relying on a global path computed from the R-MAT model of free-space (a retraction of MA T defined specifically for path planning). One of the heuristics guides the evolution of the two Cartesian degrees of freedom of the mobile body along the search, while the remaining two guide the evolution of its rotational degree of freedom. The benefits derived from the use of the proposed heuristics are twofold: on the one hand, a speed-up of the search process as compared to noninformed search algo rithms is attained, and on the other hand, the features of the resulting solution path can be somewhat controlled. Other advantages and shortcomings of the proposed path planning app...

Translationally invariant coupled cluster theory for simple finite systems.

Abstract: The widely used coupled cluster method (CCM) in quantum many-body theory has recently provided very accurate descriptions of a large number of extended systems. Although its earlier applications to closed-shell and neighboring finite nuclei were also very successful, they have been shrouded in algebraic and technical complexity. Furthermore, they are difficult to compare with more traditional calculations of generalized shell-model theory since, at least at the important level of two-body correlations, they have been largely implemented in relative-coordinate space rather than the more usual oscillator configuration space. The CCM is reviewed here in the precise context of applications to simple finite systems. Special attention is paid to formulate it in such a way that comparison may be made with generalized shell-model or configuration-interaction (CI) theories. Particular regard is paid to an exact incorporation of translational invariance, so that any spuriosity associated with the center-of-mass motion is always avoided. An important side benefit is that the number of many-body configurations in the usual oscillator basis is dramatically reduced. We are thereby able to present both CI and CCM calculations on {sup 4}He up to the essentially unprecedented level of 60{h bar}{omega} in oscillator excitation energy, for two popular and quasirealistic choicesmore » of the nucleon-nucleon interaction for which exact Monte Carlo results are available for this nucleus. Although even our simplest approximations attain about 95% of the total binding energy, the convergence in the oscillator configuration space is shown to be both very slow and of a complicated nonuniform nature. Strong implications are drawn for standard implementations of generalized shell-model techniques for heavier nuclei.« less

Generation of configuration space obstacles: moving algebraic surfaces

Abstract: We present an algebraic algorithm to generate the boundary of configuration space obstacles arising from the translatory motion of curved convex objects among curved convex obsta cles. Both the bou...

Correlations in Extended Systems: A Microscopic Multilocal Method for Describing Both Local and Global Properties

Abstract: We review the basic principles of the various coupled cluster (CC) methods based on an exponential form for the many-body wavefunction, and contrast them with the configuration-interaction ( CI) method. Particular emphasis is placed on their applicability to problems in quantum chemistry. We prove that in all cases we can construct an energy functional which variationally determines both the ground-state wavefunction and the dynamic equations of motion for nonstationary states. As a result the equations of motion assume the familiar classical canonical Hamiltonian form in some well-defined (multibody ) configuration space. We also thereby construct the expectation-value functional for an arbitrary operator in such a way that the Feynman-Hellmann theorem is preserved at all natural levels of truncation of the appropriate configuration space. We show in detail that only in the case of the recently introduced extended cc method ( ECCM) is the expectation-value functional expressed fully in terms of linked (multilocal) amplitudes. The ECCM is thereby capable of describing such global phenomena as shape transitions and other stereochemical properties, and the large-scale behavior of the molecular energy surfaces. We illustrate our methodology on the one-body density matrix, which is now much more easily discussed than by conventional methods in quantum chemistry.

A multiresolution work space, multiresolution configuration space approach to solve the path planning problem

Abstract: The algorithm SCOUT is proposed to solve the path-planning problem for robots moving between stationary obstacles. The configuration space of a robot is represented as a binary tree of blocks, each of which can be efficiently tested for collisions using a hierarchy of bubbles in work space. The configuration space is explored independently from start and goal until a contiguous path of free blocks connecting the two is found. The algorithm finds a first path quickly, then improves on the length of the path by testing more parts of the configuration space, and finally converges to the optimal path for any imposed metric. The tradeoff between the length of the path and time/memory usage is explicit and can be used to the advantage of integrated systems. The method is applicable to mobile robots as well as to manipulators. Results are shown for a 2 d.o.f. planar arm and a 5 d.o.f industrial manipulator. >

Reduction, quantization, and nonunimodular groups

Abstract: It is shown that even in relatively nice cases the naive approach to the quantization of constraints is not correct in general [i.e., the procedure that if f=0 is a classical constraint and τ(f) is the associated quantum operator, then the quantum constraint is τ(f)=0]. An explicit procedure for the quantization of constraints in the case of a configuration space with a symmetry group is provided and proven, where the reduced configuration space is the orbit space. It is not thought that the group acts freely, merely that all isotropy subgroups are conjugated to each other.

An algebraic formulation of configuration-space obstacles for spatial robots

Abstract: An algebraic formulation of the boundaries of configuration-space obstacles for wrist-partitioned spatial robots is presented. When the end-effector of the robot moves in contact with an obstacle, it is constrained not only by the robot reachability constraints but also by the link-obstacle contact constraints. The reachability constraint is modeled by a chain of two spherical joints, and the contact constraint is a combination of spherical joint and planar joint. Each of these constraints defines a manifold in the 6D space of rigid displacements. Parameterized and algebraic expressions defining these manifolds are obtained using dual quaternions. The obstacle boundary is obtained from the intersection of the manifolds associated with two types of constraints. An example is provided to show how this formulation leads to equations for the boundary of a joint obstacle for a PUMA robot. >

On the structure of quantum phase space

Abstract: The space of labels characterizing the elements of Schwinger’s basis for unitary quantum operators is endowed with a structure of symplectic type. This structure is embodied in a certain algebraic cocycle, whose main features are inherited by the symplectic form of classical phase space. In consequence, the label space may be taken as the quantum phase space: It plays, in the quantum case, the same role played by phase space in classical mechanics, some differences coming inevitably from its nonlinear character.

R-Matrix-Floquet Theory of Multiphoton Processes

Abstract: A unified theory is developed to analyse both the multiphoton ionization of atoms and laser-assisted electron-atom collisions. It combines the R-matrix method with the Floquet theory, is nonperturbative and is applicable to any atom. It takes advantage of the natural R-matrix division of configuration space into two regions, which allows us to choose the most appropriate form of the interaction Hamiltonian in each region and also to use the Hermitian Floquet theory in the internal region. Our theory is illustrated by the calculation of multiphoton ionization rates for a one-dimensional model.

A method for fast computation of collision-free robot movements in configuration-space

Abstract: Some algorithms for a fast and efficient computation of collision-free robot gross-motions are presented. A technique based on a lookup-table for mapping obstacles into configuration-space, is shown. Some algorithms for the generation of the distance-field in configuration space, which were derived from well-known methods but adapted for c-space, are described and compared with respect to their effectiveness. >

Nonlinear control of multibody systems in shape space

Abstract: Nonlinear control of planar multibody systems motivated by the classical cat-fall problem and the more practical problem of reorientation of multibody satellites in space are studied. A multibody system model reduced by translational and rotational symmetries was assumed in a Hamiltonian setting. A further reduction by the first integral (the system angular momentum) results in a configuration space of relative joint angles. This is equivalent to reducing the system to the symplectic leaf of the previously assumed model. The system after reduction is still Hamiltonian, and a canonical representation can be obtained. Angular-momentum-preserving controls generated by joint motors were introduced. The application of this linearizing input results in a reduced-dimension model and was found to capture the dynamics of the system in the shape space. The state space was extended to track the change in phase shift of the absolute angles. An important reachability result is proved. An optimal control problem was formulated to accomplish reorientation. >

The configurational quantum cat map

Abstract: A charged particle moving in a bounded region of the plane (with periodic boundary conditions) is subject to external periodic electromagnetic fields. Classically, they effect a hyperbolic mapping of the particle configuration space to itself which leads to highly chaotic motion. It is shown that the quantum-mechanical time-evolution operator has anabsolutely continuous spectrum of quasienergies, indicating a strong irregularity in the motion of the quantum system. The quantum time evolution turns out to have nonvanishing algorithmic complexity.