Showing papers on "Configuration space published in 1994"

A motion planner for nonholonomic mobile robots

Abstract: This paper considers the problem of motion planning for a car-like robot (i.e., a mobile robot with a nonholonomic constraint whose turning radius is lower-bounded). We present a fast and exact planner for our mobile robot model, based upon recursive subdivision of a collision-free path generated by a lower-level geometric planner that ignores the motion constraints. The resultant trajectory is optimized to give a path that is of near-minimal length in its homotopy class. Our claims of high speed are supported by experimental results for implementations that assume a robot moving amid polygonal obstacles. The completeness and the complexity of the algorithm are proven using an appropriate metric in the configuration space R/sup 2//spl times/S/sup 1/ of the robot. This metric is defined by using the length of the shortest paths in the absence of obstacles as the distance between two configurations. We prove that the new induced topology and the classical one are the same. Although we concentrate upon the car-like robot, the generalization of these techniques leads to new theoretical issues involving sub-Riemannian geometry and to practical results for nonholonomic motion planning. >

Instantaneous normal mode analysis of liquid water

Abstract: We present an instantaneous‐normal‐mode analysis of liquid water at room temperature based on a computer simulated set of liquid configurations and we compare the results to analogous inherent‐structure calculations. The separate translational and rotational contributions to each instantaneous normal mode are first obtained by computing the appropriate projectors from the eigenvectors. The extent of localization of the different kinds of modes is then quantified with the aid of the inverse participation ratio—roughly the reciprocal of the number of degrees of freedom involved in each mode. The instantaneous normal modes also carry along with them an implicit picture of how the topography of the potential surface changes as one moves from point to point in the very‐high dimensional configuration space of a liquid. To help us understand this topography, we use the instantaneous normal modes to compute the predicted heights and locations of the nearest extrema of the potential. The net result is that in liquid water, at least, it is the low frequency modes that seem to reflect the largest‐scale structural transitions. The detailed dynamics of such transitions are probably outside of the instantaneous‐normal‐mode formalism, but we do find that short‐time dynamical quantities, such as the angular velocity autocorrelation functions, are described extraordinarily well by the instantaneous modes.

Shortest path synthesis for Dubins non-holonomic robot

Abstract: We calculate the partition of the configuration space R/sup 2//spl times/S/sup 1/ of a car-like robot, only moving forwards, with respect to the type of the length optimal paths. This kind of robot is subject to kinematic constraints on its path curvature and its orientation. Starting from the results on shortest paths, we give new optimality conditions on these paths, and compute the partition for any horizontal plane of the configuration space. >

Dynamics of abelian Higgs vortices in the near Bogomolny regime

Abstract: The aim of this paper is to give an analytical discussion of the dynamics of the Abelian Higgs multi-vortices whose existence was proved by Taubes ([JT82]). For a particular value of a parameter of the theory, λ, called the Higgs self-coupling constant, there is no force between two vortices and there exist static configurations corresponding to vortices centred at any set of points in the plane. This is known as the Bogomolny regime. We will develop some formal asymptotic expansions to describe the dynamics of these multi-vortices for λ close, but not equal to, this critical value. We shall then prove the validity of these asymptotic expansions. These expansions allow us to give a finite dimensional Hamiltonian system which describes the vortex dynamics. The configuration space of this system is the “moduli space”—the space of solutions of the static equations modulo gauge equivalence. The kinetic energy term in the Hamiltonian is obtained from the natural metric on the moduli space given by theL2 inner product of the tangent vectors. The potential energy gives the intervortex potential which is non-zero when λ is not given by its critical value. Thus the reduced equations for the evolution of the vortex parameters take the form of geodesics, with force terms to express the departure from the Bogomolny regime. The geodesics are geodesics on the moduli space with respect to the metric defined by theL2 inner product of the tangent vectors, in accordance with Manton's suggestion ([Man82]). This allows an understanding of the two main phenomenological issues—first of all there is the right angle scattering phenomenon, according to which two vortices passing through one another scatter through ninety degrees. Secondly there is the conjecture from numerical calculations that vortices repel for λ greater than the critical value, and attract for λ less than this value. The results of this paper allow a rigorous understanding of the right angle scattering phenomenon ([Sam92, Hit88]) and reduce the question of attraction or repulsion in the near Bogomolny regime to an understanding of the potential energy term in the Hamiltonian ([JR79]).

Kinetic theory of runaway air-breakdown

Abstract: The kinetic theory for an air breakdown mechanism advanced in a previous paper [Phys. Lett. A 165, 463 (1992)] is developed. The relevant form of the Boltzmann equation is derived and the particle orbits in both velocity space and configuration space are computed. A numerical solution of the Boltzmann equation, assuming a spatially uniform electric field, is obtained and the temporal evolution of the electron velocity distribution function is described. The results of our analysis are used to estimate the magnitude of potential x-ray emissions from discharges in thunderstorms.

On a Representation of Friction in Configuration Space

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

Control problems in super-articulated mechanical systems

Abstract: Control of super-articulated mechanical systems (SAMS: controlled mechanical system in which the dimension of the configuration space exceeds the dimension of the control input space) is studied. As a starting point, a graphical characterization of general SAMS is developed in which a so-called control flow diagram (CFD) is constructed to represent the interaction forces among the degrees of freedom of the system; three types of structure, namely, the chain structure, the tree structure and isolated points, are identified; degree of complexity of the system is defined based on the structure of the system. A systematic backstepping type controller design is proposed for systems with a chain structure to achieve global asymptotic stability and trajectory tracking. Such a controller design itself illustrates the sufficient conditions for the existence of smooth feedback control such that the systems with a chain structure can be globally asymptotically stabilized. The connection of the authors' control algorithm with existing results is discussed. >

Motion planning with many degrees of freedom-random reflections at C-space obstacles

Abstract: This paper describes a motion planner that automatically avoids collisions with obstacles. A transformation of obstacles into configuration space (C-space) is not necessary. Given a polyhedral description of robot, load and environment the algorithm first computes (off-line) a graph which roughly represents the skeleton of the freespace. Due to the way of its computation this graph does not preserve connectivity, although the freespace is a connected set, which is assumed throughout this paper. Reflecting randomly at C-space obstacles a connection between subgraphs is generated. In a second step (online) a connection between a start and a goal configuration to the graph is searched for. The impact of this algorithm concerning movable obstacles is discussed. This method is general and four examples are presented to demonstrate its feasibility by applying it to different kinematic structures with 3, 6 and 12 degrees of freedom. >

The design of shape interactions using motion constraints

Abstract: In this paper we examine an approach to the visualization, analysis and design of functional shape interactions represented as motion constraints in configuration space. A graphical constraint representation, together with a set of computational tools we have developed, permits a designer to visualize and directly manipulate motion constraints in order to achieve desired functional properties. The resulting system has been implemented and applied to the analysis and design of a set of orienting, fixturing and assembly devices for the automated assembly of planar parts. >

A Surface Integral Approach to the Motion Planning of Nonholonomic Systems

Abstract: Nonholonomic mechanical systems are governed by constraints of motion that are nonintegrable differential expressions. Unlike holonomic constraints, these constraints do not reduce the number of dimensions of the configuration space of a system. Therefore a nonholonomic system can access a configuration space of dimension higher than the number of the degrees of freedom of the system. In this paper, we develop an algorithm for planning admissible trajectories for nonholonomic systems that will take the system from onepoint in its configuration space to another. In our algorithm the independent variables are first converged to their desired values. Subsequently, closed trajectories of the independent variables are used to converge the dependent variables

A Geometric Model of Arbitrary Spin Massive Particle

Abstract: A new model of relativistic massive particle with arbitrary spin (($m,s$)-particle) is suggested. Configuration space of the model is a product of Minkowski space and two-dimensional sphere, ${\cal M}^6 = {\Bbb R}^{3,1} \times S^2$. The system describes Zitterbewegung at the classical level. Together with explicitly realized Poincar\'e symmetry, the action functional turns out to be invariant under two types of gauge transformations having their origin in the presence of two Abelian first-class constraints in the Hamilton formalism. These constraints correspond to strong conservation for the phase-space counterparts of the Casimir operators of the Poincar\'e group. Canonical quantization of the model leads to equations on the wave functions which prove to be equivalent to the relativistic wave equations for the massive spin-$s$ field.

Method and apparatus for identifying or controlling travel to a rendezvous

Abstract: Paths are planned for one or more actors, through at least two dimensions of time or space, to identify rendezvous locations meeting a global criterion. At least two scenarios are defined, including at least one for each actor, and a configuration space is created for each of these scenarios. A scenario includes identification of the actor, a source direction for planning, a set of states identifying the source or obstacle locations, and the respective cost metrics for each possible transition between a configuration state in the corresponding configuration space, and its neighbors. Cost waves are propagated in each configuration space, to generate a cost-to-source for each state. A Boolean evaluation is then made of configuration states, according to a global criterion, to identify all possible rendezvous states. Finally, the actors are controlled to travel to that rendezvous chosen according to optimization criteria, or the candidate rendezvous states are displayed for further evaluation or use.

An Algebraic Approach to Discrete Mechanics

Abstract: Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular automata in which time proceeds in integer steps and the configuration space is discrete. In particular, we derive an analog of the Euler-Lagrange equation from a variational principle, and prove an analog of Noether's theorem. We also construct a symplectic structure on the analog of the phase space, and prove that it is preserved by time evolution.

A Complete Perturbative Expansion for Constrained Quantum Dynamics

Abstract: A complete perturbative expansion for the Hamiltonian describing the motion of a quantomechanical system constrained to move on an arbitrary submanifold of its configuration space $R^n$ is obtained.

Geometrical derivation of Lagrange’s equations for a system of particles

Abstract: A concise but general derivation of Lagrange’s equations is given for a system of finitely many particles subject to holonomic and nonholonomic constraints. Based directly on Newton’s second law, it takes advantage of an inertia‐based metric to obtain a geometrically transparent statement of Lagrange’s equations in configuration space. Illustrative examples are included.

Nonholonomic Path Planning Using Harmonic Functions

Abstract: A method is presented for planning obstacle-avoiding paths for a system which exhibits nonholonomic constraints. The method is based on the use of harmonic functions. Linear constraints on the velocity of a nonholonomic system can be directly expressed as Neumann boundary conditions for a harmonic function. Such boundary conditions are easily represented in a resistive network. The resulting potential represents an integration of nonholonomic constraints over an admissible subset of configuration space. The method is applied to path planning for simple wheeled vehicles.

Method and apparatus for controlling maneuvers of a vehicle

Abstract: Maneuvers of a vehicle, in the presence of obstacles, are planned using a three-dimensional configuration space. Axes of the configuration space correspond to x and y coordinate locations of the rear differential of the vehicle and angle of the vehicle. The configuration space is filled with cost to goal and direction arrows values using an exhaustive search strategy. The direction arrows values point to a least cost path to a goal for the vehicle. The exhaustive search strategy involves searching a bow-tie shaped neighborhood of a goal state, and then a bow-tie shaped neighborhood of each state in the first neighborhood, iterating until all reachable states are searched. A precedence order is established so that states which are blocked are not searched.

A note on the path integral for systems with primary and secondary second class constraints

Abstract: It is shown that the phase space path integral for a system with arbitrary second class constraints (primary, secondary ...) can be rewritten as a configuration space path integral of the exponent of the Lagrangian action with some local measure.

Scars in wavefunctions of the diamagnetic Kepler problem

Abstract: The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.

Describing the macroscopic world: Closing the circle within the dynamical reduction program

Abstract: With reference to recently proposed theoretical models accounting for reduction in terms of a unified dynamics governing all physical processes, we analyze the problem of working out a worldview accommodating our knowledge about natural phenomena. We stress the relevant conceptual differences between the considered models and standard quantum mechanics. In spite of the fact that both theories describe systems within a genuine Hilbert space framework, the peculiar features of the spontaneous reduction models limit drastically the states which are dynamically stable. This fact by itself allows one to work out an interpretation of the formalism which makes it possible to give a satisfactory description of the world in terms of the values taken by an appropriately defined mass density function in ordinary configuration space. A topology based on this function and which is radically different from the one characterizing the Hilbert space is introduced, and in terms of it the idea of similarity of macroscopic situations is precisely defined. Finally, the formalism and the interpretation are shown to yield a natural criterion for establishing the psychophysical parallelism. The conclusion is that, within the considered theories and at the nonrelativistic level, one can satisfy all sensible requirements for a completely satisfactory macro-objective description of reality.

Minimum crossing numbers for 3-braids

Abstract: Given a braid on N strings, find an algorithm which generates an Artin braid word B of minimal length. This is an important unsolved problem-a solution would give us the most economical way of notating and drawing braids. The length of an Artin word equals the number of crossings seen in a braid diagram. Minimum crossing numbers provide a measure of complexity for braids. This paper presents an algorithm for N=3. A three-dimensional configuration space for 3-braids will also be defined and analysed.

Configuration space computation for mechanism design

Abstract: We describe the HIPAIR configuration space computation program for higher pairs and show how it automates reasoning about shape and motion for mechanism design. We describe an interactive parametric design module that combines configuration space computation with differential constraint satisfaction. HIPAIR handles pairs of 2.5D parts with two degrees of freedom, including pairs with intermittent, simultaneous, and degenerate contacts. This class contains 90% of 2.5D pairs and 80% of all higher pairs according to our survey of 2500 mechanisms. We have tested HIPAIR on over 100 pairs, including gears, cams, ratchets, and escapements. It analyzes pairs with thousands of contacts in under ten seconds. The configuration spaces encode the relations among part shapes, part motions, and overall behavior in a concise, complete, and explicit format. They help designers analyze part interactions, implement functions, identify failure modes, and modify designs. >

On the fundamental group of the space of harmonic 2-spheres in the n-sphere

Abstract: Harmonic maps φ : S −→ S are of interest from the point of view of differential geometry (as branched minimal immersions), and from the point of view of gauge theory (as classical solutions of the S non-linear sigma model). From the work of Chern, Calabi, Barbosa and others, there is now a complete description of such harmonic maps in terms of holomorphic data. However, very little is known about the global properties of the space Harm(S, S) of all harmonic maps. Since the analogous object in Yang-Mills theory — the space of solutions to the Yang-Mills equations — has proved to be of great significance, the space of harmonic maps is also of some interest. In this paper we shall give an approach to the study of the topology of Harm(S, S).

On translational motion planning in 3-space

Abstract: Let B be a convex polyhedron translating in 3-space amidst k convex polyhedral obstacles A1,…,Ak with pairwise disjoint interiors. The free configuration space (space of all collision-free placements) of B can be represented as the complement of the union of the Minkowski sums Pi=Ai⊕(-B), for i=1,…,k. We show that the combinatorial complexity of the free configuration space of B is O(nklog2k), where n is the total complexity of the individual Minkowski sums P1,…,Pk. The bound is almost tight in the worst case. We also derive an efficient randomized algorithm that constructs this configuration space in expected time O(nklog3k).