Showing papers on "Configuration space published in 2007"
TL;DR: In this article, the authors present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetric ideally hinged parts: two wheels, a frame and a front assembly.
Abstract: We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetric ideally hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal knife-edge rolling point contact with the ground level. The mass distribution and geometry are otherwise arbitrary. This conservative non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrized by rates of frame lean, steer angle and rear wheel rotation. We construct the terms in the governing equations methodically for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equations by hand in two ways and also checked them against two nonlinear dynamics simulations. In the century-old literature, several sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternative formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as a check for general purpose dynamic programs. For the benchmark bicycles, we accurately calculate the eigenvalues (the roots of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-old result that this conservative system can have asymptotic stability.
301 citations
TL;DR: In this paper, it was shown that the possible Bohmian paths are naively observable from a large enough ensemble, assuming that one obtains the dynamics of BM, and that one requires that a particular path be determined experimentally.
Abstract: Bohmian mechanics (BM) is a popular interpretation of quantum mechanics (QM) in which particles have real positions. The velocity of a point x in configuration space is defined as the standard probability current j(x) divided by the probability density P(x). However, this 'standard' j is in fact only one of infinitely many that transform correctly and satisfy . In this paper, I show that a particular j is singled out if one requires that j be determined experimentally as a weak value, using a technique that would make sense to a physicist with no knowledge of QM. This 'naively observable' j seems the most natural way to define j operationally. Moreover, I show that this operationally defined j equals the standard j, so, assuming , one obtains the dynamics of BM. It follows that the possible Bohmian paths are naively observable from a large enough ensemble. Furthermore, this justification for the Bohmian law of motion singles out x as the hidden variable, because (for example) the analogously defined momentum current is in general incompatible with the evolution of the momentum distribution. Finally I discuss how, in this setting, the usual quantum probabilities can be motivated from a Bayesian standpoint, via the principle of indifference.
180 citations
01 Nov 2007
TL;DR: A sampling-based path planning algorithm capable of efficiently generating solutions for high-dimensional manipulation problems involving challenging inverse kinematics and complex obstacles, known as Jacobian transpose-directed rapidly exploring random trees (JT-RRTs).
Abstract: We present a sampling-based path planning algorithm capable of efficiently generating solutions for high-dimensional manipulation problems involving challenging inverse kinematics and complex obstacles. Our algorithm extends the rapidly-exploring random tree (RRT) algorithm to cope with goals that are specified in a subspace of the manipulator configuration space through which the search tree is being grown. Underspecified goals occur naturally in arm planning, where the final end effector position is crucial but the configuration of the rest of the arm is not. To achieve this, the algorithm bootstraps an optimal local controller based on the transpose of the Jacobian to a global RRT search. The resulting approach, known as Jacobian transpose-directed rapidly exploring random trees (JT-RRTs), is able to combine the configuration space exploration of RRTs with a workspace goal bias to produce direct paths through complex environments extremely efficiently, without the need for any inverse kinematics. We compare our algorithm to a recently-developed competing approach and provide results from both simulation and a 7 degree-of-freedom robotic arm.
123 citations
TL;DR: In this paper, the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity are derived using a variational approach where variations are defined on the Lie group of rigid body configurations.
120 citations
TL;DR: In this article, the authors propose to project the dynamics onto a submanifold of the configuration space, which allows to eliminate the non-linear holonomic constraints and to preserve the Lagrangian structure.
Abstract: In flexible multibody dynamics, advanced modelling methods lead to high-order non-linear differential-algebraic equations (DAEs). The development of model reduction techniques is motivated by control design problems, for which compact ordinary differential equations (ODEs) in closed-form are desirable. In a linear framework, reduction techniques classically rely on a projection of the dynamics onto a linear subspace. In flexible multibody dynamics, we propose to project the dynamics onto a submanifold of the configuration space, which allows to eliminate the non-linear holonomic constraints and to preserve the Lagrangian structure. The construction of this submanifold follows from the definition of a global modal parameterization (GMP): the motion of the assembled mechanism is described in terms of rigid and flexible modes, which are configuration-dependent. The numerical reduction procedure is presented, and an approximation strategy is also implemented in order to build a closed-form expression of the reduced model in the configuration space. Numerical and experimental results illustrate the relevance of this approach.
97 citations
TL;DR: In this paper, the authors present an analysis of the Feynman diagrams directly in configuration space, which leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space.
95 citations
TL;DR: In this article, the authors derived the leading four-nucleon force in chiral effective field theory using the method of unitary transformation, which can be used in many-and many-nuclidean studies.
Abstract: We discuss in detail the derivation of the leading four-nucleon force in chiral effective field theory using the method of unitary transformation. The resulting four-nucleon force is given in both momentum and configuration space. It does not contain any unknown parameters and can be used in few- and many-nucleon studies.
87 citations
TL;DR: This paper addresses the path planning problem with general end-effector constraints (PPGEC) for robot manipulators by proposing the Adapted-RGD method, which is adapted from an existing randomized gradient descent method for closed-chain robots.
85 citations
TL;DR: This work proves classification and realization theorems for state complexes, using CAT(0) geometry as the primary tool, for reconfiguration problems in robotics, biology, computer science, combinatorics, and group theory.
77 citations
TL;DR: The notion of PD is extended to take into account both translational and rotational motion to separate the intersecting objects, namely generalizedPD, which is used for performing C-obstacle query and checking path non-existence.
Abstract: Penetration depth (PD) is a distance measure that is used to describe the extent of overlap between two intersecting objects. Most of the prior work in PD computation has been restricted to translationalPD, which is defined as the minimal translational motion that one of the overlapping objects must undergo in order to make the two objects disjoint. In this paper, we extend the notion of PD to take into account both translational and rotational motion to separate the intersecting objects, namely generalizedPD. When an object undergoes a rigid transformation, some point on the object traces the longest trajectory. The generalized PD between two overlapping objects is defined as the minimum of the longest trajectories of one object, under all possible rigid transformations to separate the overlapping objects. We present three new results to compute generalized PD between polyhedral models. First, we show that for two overlapping convex polytopes, the generalized PD is the same as the translational PD. Second, when the complement of one of the objects is convex, we pose the generalized PD computation as a variant of the convex containment problem and compute an upper bound using optimization techniques. Finally, when both of the objects are non-convex, we treat them as a combination of the above two cases and present an algorithm that computes a lower and an upper bound on the generalized PD. We highlight the performance of our algorithms on different models that undergo rigid motion in the 6-dimensional configuration space. Moreover, we utilize our algorithm for complete motion planning of rigid robots undergoing translational and rotational motion in a plane or in 3D space. In particular, we use generalized PD computation for performing C-obstacle query and checking path non-existence.
75 citations
10 Apr 2007
TL;DR: A utility-guided algorithm is presented for the online adaptation of the random tree expansion strategy that guides expansion towards regions of maximum utility based on local characteristics of state space.
Abstract: Randomly expanding trees are very effective in exploring high-dimensional spaces. Consequently, they are a powerful algorithmic approach to sampling-based single-query motion planning. As the dimensionality of the configuration space increases, however, the performance of tree-based planners that use uniform expansion degrades. To address this challenge, we present a utility-guided algorithm for the online adaptation of the random tree expansion strategy. This algorithm guides expansion towards regions of maximum utility based on local characteristics of state space. To guide exploration, the algorithm adjusts the parameters that control random tree expansion in response to state space information obtained during the planning process. We present experimental results to demonstrate that the resulting single-query planner is computationally more efficient and more robust than previous planners in challenging artificial and real-world environments.
01 Apr 2007
TL;DR: In this paper, the authors showed that the Chaplygin system can be reduced to an almost Hamiltonian system in T*S petertodd 2 with a non-closed 2-form ωNH.
Abstract: “Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G
2). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S
2 and base S
2, that can be reduced to an almost Hamiltonian system in T*S
2 with a non-closed 2-form ωNH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I
j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ωNH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different I
j are known.
TL;DR: This paper proposes a realistic model with musculotendon complexes and addresses the question of controlling the gaze in the human oculomotor system as a simple mechanical control system with a holonomic constraint.
Abstract: In this paper, we study the human oculomotor system as a simple mechanical control system. It is a well known physiological fact that all eye movements obey Listing's law, which states that eye orientations form a subset consisting of rotation matrices for which the axes are orthogonal to the normal gaze direction. First, we discuss the geometry of this restricted configuration space (referred to as the Listing space). Then we formulate the system as a simple mechanical control system with a holonomic constraint. We propose a realistic model with musculotendon complexes and address the question of controlling the gaze. As an example, an optimal energy control problem is formulated and numerically solved
TL;DR: A detailed analysis of how nonlocal information is transmitted to quantum trajectories in simple systems (free particle and harmonic oscillator) turns out to be very interesting when compared to analogous systems in classical mechanics.
TL;DR: In this article, it was shown that the topology of certain submanifolds of configuration space must necessarily change at the phase transition point, which is a necessary topological condition for first or second-order phase transitions.
TL;DR: A relatively simple description of dark energy based on the dynamics of a scalar field which is exact in the limit that the equation of state approaches a cosmological constant, assuming some degree of smoothness of the potential.
Abstract: Most parametrizations of the dark energy equation of state do not reflect realistic underlying physical models. Here we develop a relatively simple description of dark energy based on the dynamics of a scalar field which is exact in the limit that the equation of state approaches a cosmological constant, assuming some degree of smoothness of the potential. By introducing just two parameters defined in the configuration space of the field, we are able to reproduce a wide class of quintessence models. We examine the observational constraints on these models as compared to linear evolution models and show how priors in the field space translate into priors on observational parameters.
27 Jun 2007
TL;DR: This work presents an efficient algorithm to compute the generalized penetration depth (PDg) between rigid models using displacement vectors and shows that the optimum answer lies on the boundary of the contact space and poses the computation as a constrained optimization problem.
Abstract: We present an efficient algorithm to compute the generalized penetration depth (PDg) between rigid models. Given two overlapping objects, our algorithm attempts to compute the minimal translational and rotational motion that separates the two objects. We formulate the PDg computation based on modeldependent distance metrics using displacement vectors. As a result, our formulation is independent of the choice of inertial and body-fixed reference frames, as well as specific representation of the configuration space. Furthermore, we show that the optimum answer lies on the boundary of the contact space and pose the computation as a constrained optimization problem. We use global approaches to find an initial guess and present efficient techniques to compute a local approximation of the contact space for iterative refinement. We highlight the performance of our algorithm on many complex models.
TL;DR: The transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies was studied in this paper.
Abstract: We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are (a) a proof of the existence of bounded domains in configuration space which are devoid of nodal points, (b) an analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability, (c) a detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point—X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.
TL;DR: In this article, the relative configuration spaces of shapes are n-spheres and complex projective spaces, from which knowledge I construct natural mechanics on these spaces and also show that these coincide with Barbour's indirectly-constructed relational dynamics by performing a full reduction on the latter.
Abstract: Relational particle dynamics include the dynamics of pure shape and cases in which absolute scale or absolute rotation are additionally meaningful. These are interesting as regards the absolute versus relative motion debate as well as discussion of conceptual issues connected with the problem of time in quantum gravity. In spatial dimension 1 and 2 the relative configuration spaces of shapes are n-spheres and complex projective spaces, from which knowledge I construct natural mechanics on these spaces. I also show that these coincide with Barbour's indirectly-constructed relational dynamics by performing a full reduction on the latter. Then the identification of the configuration spaces as n-spheres and complex projective spaces, for which spaces much mathematics is available, significantly advances the understanding of Barbour's relational theory in spatial dimensions 1 and 2. I also provide the parallel study of a new theory for which positon and scale are purely relative but orientation is absolute. The configuration space for this is an n-sphere regardless of the spatial dimension, which renders this theory a more tractable arena for investigation of implications of scale invariance than Barbour's theory itself.
15 Nov 2007
TL;DR: In this article, the authors derived the Newtonian dynamics from prior information codified into an appropriate statistical model, and the trajectory follows from a principle of inference, the method of Maximum Entropy.
Abstract: Newtonian dynamics is derived from prior information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the state of a particle is defined by a probability distribution. The corresponding configuration space is a statistical manifold the geometry of which is defined by the information metric. The trajectory follows from a principle of inference, the method of Maximum Entropy No additional “physical” postulates such as an equation of motion, or an action principle, nor the concepts of momentum and of phase space, not even the notion of time, need to be postulated. The resulting entropic dynamics reproduces the Newtonian dynamics of any number of particles interacting among themselves and with external fields. Both the mass of the particles and their interactions are explained as a consequence of the underlying statistical manifold.
TL;DR: The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail as mentioned in this paper, and the aim of this paper is to fill in the gaps as well as summarize the importance of these integrals.
Abstract: It is well-known that certain combinations of configuration space integrals defined by Bott and Taubes [11] produce cohomology classes of spaces of knots. The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail. The aim of this paper is to fill in the gaps as well as summarize the importance of these integrals.
TL;DR: An analytical solution is provided to explain the heavy tail of the degree distribution, neighbor connectivity, and clustering coefficient that allows us to understand the universal CSN topology observed in systems ranging from a simple quadratic well to the native state of the beta3s peptide and a two-dimensional lattice heteropolymer.
Abstract: The configuration space network (CSN) of a dynamical system is an effective approach to represent the ensemble of configurations sampled during a simulation and their dynamic connectivity. To elucidate the connection between the CSN topology and the underlying free-energy landscape governing the system dynamics and thermodynamics, an analytical solution is provided to explain the heavy tail of the degree distribution, neighbor connectivity, and clustering coefficient. This derivation allows us to understand the universal CSN topology observed in systems ranging from a simple quadratic well to the native state of the beta3s peptide and a two-dimensional lattice heteropolymer. Moreover, CSNs are shown to fall in the general class of complex networks described by the fitness model.
TL;DR: In this paper, an index theorem in the TP monopole background, defined in the projected space, and providing meaning of the projection operator is presented. And the index theorem is extended to general configurations which do not satisfy the equation of motion, and the configuration space can be classified into topological sectors.
Abstract: In previous papers, we studied 't Hooft-Polyakov (TP) monopole configurations in U(2) gauge theory on the fuzzy 2-sphere, and showed that they have nonzero topological charges in the formalism based on the Ginsparg-Wilson (GW) relation. In this paper, we will show an index theorem in the TP monopole background, which is defined in the projected space, and provides meaning of the projection operator. We also extend the index theorem to general configurations which do not satisfy the equation of motion, and show that configuration space can be classified into topological sectors. We further calculate the spectrum of the GW Dirac operator in TP monopole backgrounds, and consider the index theorem in these cases.
TL;DR: In this paper, the equilibrium statistical mechanics of canonical, non-canonical and non-Hamiltonian equations of motion are analyzed and a phase space measure is defined by means of the Jacobian of the transformation between different kinds of coordinates since such a determinant is different from zero in the noncanonical case even if the phase space compressibility is null.
Abstract: We analyse the equilibrium statistical mechanics of canonical, non-canonical and non-Hamiltonian equations of motion, throwing light on the peculiar geometric structure of phase space. Some fundamental issues regarding time translation and phase space measure are clarified. In particular, we emphasize that a phase space measure should be defined by means of the Jacobian of the transformation between different kinds of coordinates since such a determinant is different from zero in the non-canonical case even if the phase space compressibility is null. Instead, the Jacobian determinant associated with phase space flows is unity whenever non-canonical coordinates lead to a vanishing compressibility, so its use for defining a measure may not always be correct. To better illustrate this point, we derive a mathematical condition for defining non-Hamiltonian phase space flows with zero compressibility. The Jacobian determinant associated with the time evolution in phase space is very useful for analysing time translation invariance. The proper definition of a phase space measure is particularly important when defining the entropy functional in the canonical, non-canonical, and non-Hamiltonian cases. We show how the use of relative entropies can circumvent some subtle problems that are encountered when dealing with continuous probability distributions and phase space measures. Finally, a maximum (relative) entropy principle is formulated for non-canonical and non-Hamiltonian phase space flows.
TL;DR: In this paper, Dyson's Hierarchical Model (HM) is shown to be equivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter).
Abstract: We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigourous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and functional RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large-N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review is to present a configuration space counterpart, suitable for lattice formulations, of functional RG equations formulated in momentum space (often called exact RG equations and abbreviated ERGE).
TL;DR: In this paper, a large class of dissipative systems can be brought to a canonical form by introducing complex co-ordinates in phase space and a complex-valued hamiltonian.
TL;DR: Newtonian dynamics is derived from prior information codified into an appropriate statistical model as discussed by the authors, where there is an irreducible uncertainty in the location of particles so that the state of a particle is defined by a probability distribution.
Abstract: Newtonian dynamics is derived from prior information codified into an appropriate statistical model The basic assumption is that there is an irreducible uncertainty in the location of particles so that the state of a particle is defined by a probability distribution The corresponding configuration space is a statistical manifold the geometry of which is defined by the information metric The trajectory follows from a principle of inference, the method of Maximum Entropy No additional "physical" postulates such as an equation of motion, or an action principle, nor the concepts of momentum and of phase space, not even the notion of time, need to be postulated The resulting entropic dynamics reproduces the Newtonian dynamics of any number of particles interacting among themselves and with external fields Both the mass of the particles and their interactions are explained as a consequence of the underlying statistical manifold
TL;DR: In this paper, a procedure for conversion of second class constraints based on deformation the structure of local symmetries of the Lagrangian formulation is proposed, which does not require extension or reduction of configuration space of the theory.
TL;DR: A particle diffusing in a two-dimensional (2D) container, shaped as a simplified configuration space of two passing 2D circular particles in a flat channel, is considered, and the mean first passage time through one absorbing boundary is calculated using the one-dimensional Fick-Jacobs equation and its modification.
Abstract: A particle diffusing in a two-dimensional (2D) container, shaped as a simplified configuration space of two passing 2D circular particles in a flat channel, is considered. The mean first passage time through one absorbing boundary is calculated using the one-dimensional Fick-Jacobs equation and its modification; both derived by mapping the 2D diffusion equation onto the longitudinal ("reaction") coordinate. The obtained results are compared with the hopping time, defined as the inverted lowest eigenvalue of the full 2D problem. The comparison shows that the mapped equations give reliable results, in contrast to predictions of the simplest concept of the transition state theory.
TL;DR: In this article, the authors describe an implementation of density functional theory that is formulated fully in configuration space, where all wave functions, densities, and potentials are represented on a grid.
Abstract: We describe an implementation of density functional theory that is formulated fully in configuration space, where all wave functions, densities, and potentials are represented on a grid. Central to the method is a fourth-order factorization of the evolution operator for the Kohn-Sham Hamiltonian. Special attention is paid to nonlocal pseudopotentials of the Kleinman-Bylander type, which are necessary for a quantitatively accurate description of molecules, clusters, and solids. It is shown that the fourth-order factorization improves the computational efficiency of the method by about an order of magnitude compared with second-order schemes. We use the Ono-Hirose filtering method to reduce the resolution of the grid used for representing the wave functions. Some care is needed to maintain the fourth-order convergence using filtered projectors, and the necessary precautions are discussed. We apply the method to an isolated carbon atom as well as the carbon-monoxide molecule, the benzene molecule, and the buckminsterfullerene cluster, obtaining quantitative agreement with previous results. The convergence of the method with respect to time step, grid resolution, and filtering method is discussed in detail.