Showing papers on "Configuration space published in 1989"

Planning smooth paths for mobile robots

Abstract: The authors consider the problem of planning paths for a robot which has a minimum turning radius. This is a first step towards accurately modeling a robot with the kinematics of a car. The technique used is to define a set of canonical trajectories which satisfy the nonholonomic constraints imposed. A configuration space can be constructed for these trajectories in which there is a simple characterization of the boundaries of the obstacles generated by the workspace obstacles. The authors describe a graph search algorithm which divides the configuration space into sample trajectories. The characterization of the boundaries makes it possible to calculate an approximate path in time O(n/sup 3// delta log n+Alog (n/ delta )), where n is the number of obstacle vertices in the environment, A is the number of free trajectories, and delta describes the robustness of the generated path and the closeness of the approximation. The authors also describe a plane sweep for computing the configuration space obstacle for a trajectory segment. They use this to generate robust paths using a quadtree based algorithm in time O(n/sup 4/log n+(n/ delta /sup 2/)). >

On the inverse kinematics of redundant manipulators: characterization of the self-motion manifolds

Abstract: The author takes a global rather than instantaneous look at the inverse kinematics of redundant manipulators. This approach is based on a manifold mapping reformulation of manipulator kinematics. While the kinematic problem has an infinite number of solutions for redundant manipulators, the infinity of solutions can be grouped into a finite and bounded set of disjoint continuous manifolds. Each of these manifolds, termed self-motion manifolds, physically corresponds to a distinct self-motion of the manipulator, and the number, geometry, and characterizations of the self-motion manifolds are investigated. >

On nonholonomic mobile robots and optimal maneuvering

Abstract: The authors consider the robot path planning problem in the presence of nonintegrable kinematic constraints, known as nonholonomic constraints. Such constraints are generally caused by one or several rolling contacts between rigid bodies and express that the relative velocity of two points in contact is zero. They make the dimension of the space of achievable velocities smaller than the dimension of the robot's configuration space. Using standard results in differential geometry (Frobenius integrability theorem) and nonlinear control theory, the authors first give a formal characterization of holonomy (and nonholonomy) for robot systems subject to linear differential constraints and state some related results about their controllability. They then apply these results to 'car-like' and 'trailer-like' robots. Finally, they present an implemented planner, which generates collision-free paths with a minimal number of maneuvers for car-like and trailer-like robots among obstacles. >

KAM theory in configuration space.

Abstract: A new approach to the Kolmogorov-Arnold-Moser theory concerning the existence of invariant tori having prescribed frequencies is presented. It is based on the Lagrangian formalism in configuration space instead of the Hamiltonian formalism in phase space used in earlier approaches. In particular, the construction of the invariant tori avoids the composition of infinitely many coordinate transformations. The regularity results obtained are applied to invariant curves of monotone twist maps. The Lagrangian approach has been prompted by a recent study of minimal foliations for variational problems on a torus by J. Moser.

Numerical determination of the electronic structure of atoms, diatomic and polyatomic molecules

Abstract: Main Lectures.- Basic Mathematical Properties of Electronic Wave Functions in Configuration Space.- Basic Mathematical Properties of Electronic Wave Functions in Momentum Space.- The Analytical Structure of Atomic and Molecular Wavefunctions and its Impact on the Rate of Convergence of Variational Calculations.- Stochastic Methods in Quantum Mechanic.- Computational Strategies and New Applications in Green's Function Monte Carlo.- Numerical Determination of non-Relativistic and Relativistic Pair Correlation.- Fully Numerical Calculations for Diatomic Systems.- Very Accurate Calculations for Diatomic, Neutral and Anionic Systems with Numerical Orbitals.- The Development of an Efficient Numerical Orbital Algorithm for Polyatomic Systems: A Review of the Various Options.- Electronic Structure Theory in Momentum Space.- Short Contributions.- Should Numerical Orbital Calculations be done with Basis Sets?.- Quantum Chemistry by Random Walk: High Accuracy for Large Molecules.- Prolate Spheroidal Wavefunctions.- Error Estimate in Variational Calculations of Eigenvalues and Eigenvectors.- A Momentum Space Approach to Improve ab initio Hartree-Fock Results Based on the LCAO-GTF Approximation.- Isotope Shift MCHF Calculations in Strontium.- Finite Element Method for the Accurate Solution of Diatomic Molecules.- Momentum Space Coordinate Transformations and their use in Numerical Orbital Calculations.- Moller-Plesset Calculations with Explicitly Correlated Wave Functions.- On the Coulomb Sturmian Basis.- An Analytical L2 Method for Continuum and Autoionizing States.- Application of the Two-dimensional Fully-Numerical RHF Method to Open-Shell Hydrides.- Interpolation of Numerical Orbitals in Momentum Space.- On the Accuracy of the Algebraic Approximation in Relativistic Electronic Structure Calculations.- Numerical MCSCF in One and Two Dimensions.- Numerical Methods for Calculating Multicenter Integrals for Arbitrary Orbitals.- Nonlinear Sequence Transformations for the Efficient Evaluation of Auxiliary Functions for GTO Molecular Integrals.- Concluding Remarks.- List of Participants.

Computing metric and topological properties of configuration-space obstacles

Abstract: The author describes an algorithm that takes two polygons as input, and computes a representation of their corresponding configuration-space obstacle, including contact information. The algorithm's output includes a full metric and topological description of the obstacle surface, as well as the set of polygon features that are in contact for each point of the surface. The representation is exact, up to the limits of floating-point arithmetic. The algorithm has been implemented and test-run on over 40 input pairs; run times varied between 12 and 135 s. >

Path-integral formulation for stochastic processes driven by colored noise

Abstract: A detailed discussion of the path-integral formalism for stochastic processes described by a stochastic differential equation driven by a nonwhite noise is given. The path-integral representation in the configuration space of the transition probability for a process driven by Ornstein-Uhlenbeck noise is derived. We show how to treat in this approach any kind of initial conditions, including the question of the coupling with the noise at initial time. Known approximations are reobtained in this context. Markovian approximations based on the Lagrangian are also discussed. The stationary distribution of the process in the weak-noise limit is obtained from the Lagrangian without relying on the use of Fokker-Planck or Markovian approximations.

Path planning using a Jacobian-based freespace generation algorithm

Abstract: A simple and efficient algorithm is developed to generate a 2"-tree (generalized quad-tree) representation of the free configuration space for a manipulator moving in a workspace with obstacles. The algorithm is based on the existence of uniform bounds on the Jacobians relating the differential motions of points on the manipulator to differential joint motions. This representation can be searched to generate a collision-free path and optimized with dynamic constraints to produce an executable trajectory.

A new algorithm for detecting the collision of moving objects

Abstract: Iterative algorithms for detecting the collision of convex objects whose motion is characterized by a path in configuration space are described. They use as an essential substep the computation of the distance between the two objects. When the objects are polytopes in either two-dimensional or three-dimensional space, an algorithm is given which terminates in a finite number of iterations. It either determines that no collision occurs or locates the first collision point on the path. For practical problems it appears that the computational time is short and grows only linearly in the total number of vertices of the two polytopes. Numerical examples are presented. >

Robot motion planning with nonholonomic constraints

Abstract: A study is made of a fundamental problem in dextrous manipulation by a robot hand: the motion of two rigid bodies rolling relative to one another. A systematic procedure for deriving the configuration space of contact and the differential equation for rolling is presented. This approach is applicable to objects of arbitrary shapes and contact constraints. An algorithm that determines the existence of an admissible path between two contact configurations is given. First, the distribution generated by the two constrained vector fields is computed. One then checks to see if the distribution is nonsingular. If so, an admissible path exists between any two contact configurations. It is also shown that the path-finding problem is equivalent to a nonlinear control problem. Thus, existing work in nonlinear control theory can be used. A geometric algorithm that finds a path when one object is flat is given. >

Atom-diatom reactive scattering. I. Quantum theory

Abstract: We describe a time‐independent quantum theory for atom–diatom reactive scattering using a procedure based upon a division of configuration space into three distinct chemical channels. Within each chemical channel, coordinate systems defined as transformations of Jacobi coordinates are described which allow the use of R‐matrix propagation in solving for the unbound motion along a scattering coordinate. The remaining five degrees of freedom are treated variationally, making use of the efficient discrete variable representation to describe the vibration–rotation interaction. The scattering information from the three chemical channels is matched on the common boundary, and the full S matrix is obtained at fixed total energy, angular momentum, and parity. In the second paper of this series, accurate results for the reactions of H+H2 and its isotopomers will be presented.

Optimal Path Planning By Cost Wave Propagation In Metric Configuration Space

Abstract: In this paper we give a solution to the problem of finding an optimal path for an arbitrary robot among static obstacles, from its present state to the closest of a set of goal states. The path is optimal in the sense of minimizing some criterion (minimum motion, distance, effort, etc.). Our solution is the well-known heuristic search method `A* with h = 0' (no heuristic), applied to a graph representing configuration space, with a metric incorporating the optimization criterion. It is applicable to any robot, unrestricted in the number of degrees of freedom, and capable of handling a broad class of optimization criteria. The method works fairly fast for a few degrees of freedom. Examples in two dimensions are given.

Landau Levels and Geometric Quantization

Abstract: The geometrical approach to phase-space quantization introduced by Klauder [KQ] is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher dimensional configuration space; quantization corresponds to freezing the particle to its first Landau level. The Geometric Quantization [GQ] scheme appears as the natural technique to define the interaction with the magnetic field for a particle on a general Riemannian manifold. The freedom of redefining the operators' ordering makes it possible to select that particular definition of the Hamiltonian which is adapted to a specific polarization; in this way the first Landau level acquires the expected degeneracy. This unification with GQ makes it clear how algebraic relations between classical observables are or are not preserved under quantization. From this point of view all quantum systems appear as the low energy sector of a generalized theory in which all classical observables have a uniquely assigned quantum counterpart such that Poisson bracket relations are isomorphic to the commutation relations.

The Role of Energy Minimization in Simulation Strategies of Biomolecular Systems

Abstract: It is now possible to calculate the classical energy of a complex system such as a protein as a function of its coordinates. By making many such calculations for various coordinate values, one can explore multidimensional energy surfaces. These energy surfaces are the basis for molecular dynamics and Monte Carlo studies. Another important method for exploring these energy surfaces is to find configurations for which the energy is a minimum. By this, we mean finding a point in configuration space where all of the forces on the atoms are balanced. By simply minimizing the energy of a molecule, we can identify stable conformations. Perhaps more importantly, by adding external to the molecule in the form of restraints and constraints, a wide range of modeling strategies can be developed using minimization techniques as the foundation to answer specific questions. For example, by forcing specific atoms to overlap atoms in a template structure during a molecular geometry minimization, one can answer the question, “how much energy is required for one molecule to adopt the shape of another.” In this chapter, we discuss how minimization techniques are used in a variety of molecular strategies, focusing on the use of constraints and restraints to extend the scope and utility of traditional structure minimization.

Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function

Abstract: A method is presented for determining the stationary phase points for multidimensional path integrals employed in the calculation of finite‐temperature quantum time correlation functions. The method can be used to locate stationary paths at any physical time; in the case that t≫βℏ, the stationary points are the classical paths linking two points in configuration space. Both steepest descent and simulated annealing procedures are utilized to search for extrema in the action functional. Only the first derivatives of the action functional are required. Examples are presented first of the harmonic oscillator for which the analytical solution is known, and then for anharmonic systems, where multiple stationary phase points exist. Suggestions for Monte Carlo sampling strategies utilizing the stationary points are made. The existence of many and closely spaced stationary paths as well as caustics presents no special problems. The method is applicable to a range of problems involving functional integration, where optimal paths linking two end points are desired.

Method and apparatus of free space enumeration for collision avoidance

Abstract: A method and an apparatus of free space enumeration for motion planning, having general applicability while reducing amounts of information and calculations involved, capable of modifying a path, capable of setting up appropriate configuration space quantization, and capable of adopting appropriate strategy for free space enumeration. The method may include the step of selecting cells only between the initial point and the final point, or selecting cells between the initial point and the final point, using plurality of strategies for selecting the cells simultanuously, or dividing the configuration space into multiplicity of cells defined in terms of intervals in the degrees of freedom, or determining a collision-free path in the free space joining the initial point and the final point without the collision, or modifying the collision-free path. The apparatus for performing the method is also disclosed.

Inequivalent quantizations in multiply connected spaces

Abstract: The novel features which show up in the process of quantization of a dynamical system on a multiply (nonsimply)-connected configuration space are analysed in the present paper. After rediscussing the path integral approach to the problem, we show how one can give a fiber bundle classification of the inequivalent quantizations associated with nonsimply connected spaces. We discuss various examples, and the generalization to the case in which, due to internal symmetries, a system can admit of nonscalar quantizations.

Path planning with transition changes

Abstract: A configuration space is used for path planning and for controlling the motion of an object. The configuration space includes states which contain cost to goal and direction arrow values. These values indicate a path or absence of a path from the states to at least one goal state. The configuration space is differentially updated after a cost to goal or direction arrow value is changed.

Microscopic theory for cross-linked macromolecules. I. Broken symmetry, rigidity, and topology

Abstract: This paper is concerned with the onset of rigidity in randomly cross-linked macromolecules. We discuss in detail the possible partitionings of configuration space, which may accompany the spontaneous breaking of translational invariance, treating separately the cases of crystals, amorphous solids, and cross-linked macromolecules. We describe the order parameters for these systems, drawing the distinction between solids with discrete translational symmetry and solids with macroscopic translational invariance, such as randomly cross-linked macromolecular solids. We show that the latter may be described by a sequence of probability distributions for the overlaps of equilibrium states. In a cross-linked system of impenetrable linear chains, the configuration space of the solid state is partitioned into two categories of equilibrium states: those related by translational and rotational symmetry, and those unrelated by these symmetries. The latter are a consequence of the distinct topologies of the network, which are consistent with a given set of cross links. We show how the overlap-probability distributions may be calculated.

Differential budding: method and apparatus for path planning with moving obstacles and goals

Abstract: A method is presented for path planning after changes in task space. In one embodiment, the method is applied to planning a path for a robot arm. The method identifies areas in the configuration space which are affected by the changes in task space. Cost waves can then be repropagated in these affected areas to allow for planning in N dimensions and using space variant metrics. The method is also adapted to use in the presence of phantom obstacles.

Differential A*: an adaptive search method illustrated with robot path planning for moving obstacles and goals, and an uncertain environment

Abstract: Differential A* is presented. It is a method that builds on the A*/configuration-space approach to adapt quickly to changes in the space by determining and updating the localized regions affected by those changes rather than regenerating the entire space. This is particularly effective with moving obstacles or goals and in an uncertain environment because only small parts of the space are affected at a time. This technique can provide significant speed improvements over, with the same desired results, as complete space regeneration. The A* search algorithm and its relationship to the configuration space method of path planning are presented. The connection of A* to wave propagation in configuration space for path planning is described. The differential A* method is outlined, with the focus on path planning. Examples of moving obstacles and goals and planning in an uncertain environment are presented. >

Quantum Chemistry in Fock Space

Abstract: The many electron-problem of atomic or molecular theory can be formulated either in configuration space (where operators act on the coordinates of the particles) or in Fock space (where operators act on the occupation numbers of spin orbitals). The representation of operators in Fock space (often also referred to as occupation number representation or 2nd quantization) is essentially equivalent to the representation in configuration space, it is, however, more general insofar as a Fock space Hamiltonian has eigenstates with arbitrary numbers of electrons, while in configuration space the number of electrons is fixed. So Fock space is, in a sense, a direct sum of Hilbert spaces for various particle numbers. Processes like ionization or electron attachment, in which the number of electrons is changed are hence better described in Fock space than in configuration space (i.e. in n- particle Hilbert space).

Computing the configuration space for a robot on a mesh-of-processors☆

Abstract: In this paper, we present a systolic algorithm for computing the configuration space of an arrangement of arbitrary obstacles in the plane for a rectilinearly convex robot. The obstacles and the robot are assumed to be represented in digitized form by a √n × √n nibary image. The algorithm is designed for a Mesh-of-Processors architecture with n processors (using the canonical representation of an image on a processor array) and has an execution time of O(√n) which is asymptotically optimal.

Reasoning about kinematic topology

Abstract: Reasoning about kinematics is an important aspect of common sense physics. In earlier work, we have developed the place vocabulary theory of qualitative kinematics in mechanisms, a formal theory for representing the kinematic behavior of two-dimensional mechanisms. The computation of a place vocabulary is very complex because it takes into account the details of object shapes. In this paper, we present a rep resentation which is much more abstract than a place vocabulary, the kinematic topology. Kinematic topology does not define qualitative inference rules, but provides a characterization of the topology of legal configurations. For example, the kinematic topology of a pair of gears is one or several doubly connected regions, whose shape in configuration space indicates the relative speeds of the two gears. For many applications, reasoning about kinematics at this level is sufficient. Kinematic topology can be computed in a purely qualitative manner and thus gives an existence proof that a purely qualitative kinematics is possible. Like in other qualitative reasoning applications, the qualitative computation has the effect that the result is almost always ambiguous. On the other hand, a kinematic topology can be given even for mechanisms whose designs are only imprecise sketches, and can be generalized to arbitrary object shapes, several degrees of freedom, and three dimensions. We hope that such generalizations of kinematic topology can provide the basis for efficiently computing place vocabularies, and reasoning about general kinematic interactions.

Explicit guidance along an optimal space curve

Abstract: An explicit guidance theory is developed for maneuvering to a prescribed destination with terminal constraints on velocity vector direction. Motion is constrained to an optimal, three-dimensional space curve by constraint forces perpendicular to velocity (lift). Lift components are derived by twice differentiating functions specifying radial distance and geocentric latitude as functions of longitude. An optimal space curve is determined by solving a two-point boundary-value problem in the calculus of variations. Necessary conditions for an extremum are 1) a set of coupled, fourth-order, Euler-Lagrange differential equations for the space curve functions; 2) a single, first-order differential equation for the adjoint variable; and 3) boundary conditions specified at two ends of the trajectory. Although energy is not conserved because of drag, motion along the space curve is integrable because lift-induced drag is determined by trajectory curvature. Velocity along the space curve may be expressed by a quadrature evaluated by the method of successive approximation to refine the accuracy of the compressibility drag slowdown. Background F UTURE maneuvering vehicles, such as the space plane, will require advanced midcourse guidance algorithms to optimize performance and arrive at a prescribed destination with terminal constraints on flight path. During flight in the atmosphere, vehicle orientation relative to the velocity vector (angle of attack) is controlled to generate the required acceleration perpendicular to velocity (lift). Many explicit guidance algorithms have been developed for lift-controlled entry vehicles. l~21 The guided trajectory problem is not generally integrable, except in certain cases (discussed shortly). Integrable cases for lifting trajectories include constant lift-to-drag L/D ratio, constant-bank angle, and equilibrium glide at constant flight-path angle. Hodograph space solutions express velocity magnitude by a function of turning angle, and the configuration space trajectory is determined t>y a quadrature.4"8 These approximate solutions are useful for preliminary design of midcourse trajectories satisfying mission performance objectives within vehicle aerothermodynamic limitations (trim, loads, and heating). Direct methods have been used extensively to develop explicit, optimal guidance algorithms. For a vehicle with bounded lift control, optimal range extension maneuvers consist of maximum and minimum L/D subarcs connected by intermediate cruise segments.9"11 Numerical solutions may be ill-conditioned because switching points must be determined to satisfy the boundary conditions. Green's Theorem may be applied to determine the sequence of maximum and minimum L/D subarcs (without cruise segments) that optimize performance while satisfying end conditions on altitude and flight path.12-13 Optimal, modulated-lift trajectories admit approximate analytic solutions characterized by a change of independent variable from time to an appropriate trajectory variable. For example, proportional navigation guidance minimizes control effort or time-integrat ed lift acceleration magnitude, and closed-form trajectory solutions are obtained when the new independent variable is line-of-sight angle.14"16 For maximum velocity turns to a specified heading, approximate optimal control histories were derived from an integrable system of equations using flight-path angle17 or range18 as new indepen

Spatial Dynamics of Deformable Multibody Systems with Variable Kinematic Structure

Abstract: : A method of the spatial kinematic and dynamic analysis of deformable multibody systems subject to topology changes and impacts is presented. A pieced interval analysis scheme that accounts for the change in the spatial system topology due to the changes on the connectivity between bodies is developed. Deformable bodies in the system are discretized using the finite element method and accordingly a finite set of deformation modes is employed to characterize the system vibration. Even though there are infinitely many arrangements for deformable body axes, computational difficulties may be encountered due to the use of a limited number of deformation modes. Therefore, the deformable body references have to be carefully selected, and accordingly as the system topology changes, new bases for the configuration space to another, a set of spatial interface conditions or compatibility conditions that are formulated using a set of nonlinear algebraic equations are developed. The solution of these equations uniquely define the spatial configuration of the deformable multibody system after the change in the system kinematic structure. The techniques proposed in this research are applied to several technological system such as robotic manipulators and weapon systems. Keywords: Spatial kinematics; Dynamic analysis; Deformable multibody systems; Interval analysis scheme; Nonlinear algebraic equations.

Global Dynamics in Neural Networks II.

Abstract: The Hedlund-Richardsod Theorem states t ha t a global mapping from configuration space to itself can be realized by a Eu­ clidean cellular automaton if and only if it takes the quiescent configu­ ration to itself , commutes with shifts, and is continuous in the product topology. An analogous theorem characte rizing t he realizability of self­ mappings of finite or infinit e configuration space via neural networks is established. It follows that, under natural hypotheses, a uniform limit of global dynamics is a global dynamics . We also give sufficient conditions for the global dynamics of a neural network to be realized by a cellular automaton.

On topological effects in quantum mechanics: The harmonic oscillator in the pointed plane

Abstract: Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold M as its configuration space is based on a quantization method, which, in general, reflects global properties of M, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on M. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold M, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on M, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self‐adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self‐adjoint extensions of the energy operator. An experimental setup...