General Configuration Space Research Articles

Supersolvable posets and fiber-type abelian arrangements

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell–Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. This is consistent with Terao’s fibration theorem connecting bundles of hyperplane arrangements to Stanley’s lattice supersolvability. We obtain a combinatorially determined class of K(π,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K(\\pi ,1)$$\\end{document} toric and elliptic arrangements. Under a stronger combinatorial condition, we prove a factorization of the Poincaré polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk–Randell’s formula relating the Poincaré polynomial to the lower central series of the fundamental group.

An elementary proof of the homotopy invariance of stabilized configuration spaces

In this paper we give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space F ( M , k ) F(M,k) for a topological manifold M M . Our technique is to compute the Spanier-Whitehead dual of Σ + ∞ F ( M , k ) \Sigma ^\infty _+ F(M,k) and use the results of Spivak and Wall on normal spherical fibrations to deduce that the Spanier-Whitehead dual is a proper homotopy invariant. This stable invariance was recently proved by Knudsen using factorization homology. Aside from being elementary, our proof has the advantage that it readily extends to “generalized configuration spaces” which have recently undergone study.

NFOMP: Neural Field for Optimal Motion Planner of Differential Drive Robots With Nonholonomic Constraints

Optimal motion planning is one of the most critical problems in mobile robotics. On the one hand, classical sampling-based methods propose asymptotically optimal solutions to this problem. However, these planners cannot achieve smooth and short trajectories in reasonable calculation time. On the other hand, optimization-based methods are able to generate smooth and plain trajectories in a variety of scenarios, including a dense human crowd. However, modern optimization-based methods use the precomputed signed distance function for collision loss estimation, and it limits the application of these methods for general configuration spaces, including a differential drive non-circular robot with non-holonomic constraints. Moreover, optimization-based methods lack the ability to handle U-shaped or thin obstacles accurately. We propose to improve the optimization methods in two aspects. Firstly, we developed an obstacle neural field model to estimate collision loss; training this model together with trajectory optimization allows improving collision loss continuously, while achieving more feasible and smoother trajectories. Secondly, we forced the trajectory to consider non-holonomic constraints by adding Lagrange multipliers to the trajectory loss function. We applied our method for solving the optimal motion planning problem for differential drive robots with non-holonomic constraints, benchmarked our solution, and proved that the novel planner generates smooth, short, and plain trajectories perfectly suitable for a robot to follow, and outperforms the state-of-the-art approaches by 25% on normalized curvature and by 75% on the number of cusps in the MovingAI environment.

Homological stability and densities of generalized configuration spaces

We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.

On generalized configuration space and its homotopy groups

Let [Formula: see text] be a subset of vector space or projective space. The authors define generalized configuration space of [Formula: see text] which is formed by [Formula: see text]-tuples of elements of [Formula: see text], where any [Formula: see text] elements of each [Formula: see text]-tuple are linearly independent. The generalized configuration space gives a generalization of Fadell’s classical configuration space, and Stiefel manifold. Denote generalized configuration space of [Formula: see text] by [Formula: see text]. For studying topological property of the generalized configuration spaces, the authors calculate homotopy groups for some special cases. This paper gives the fundamental groups of generalized configuration spaces of [Formula: see text] for some special cases, and the connections between the homotopy groups of generalized configuration spaces of [Formula: see text] and the homotopy groups of Stiefel manifolds. It is also proved that the higher homotopy groups of generalized configuration spaces [Formula: see text] and [Formula: see text] are isomorphic.

Cohomology of generalized configuration spaces

Let $X$ be a topological space. We consider certain generalized configuration spaces of points on $X$, obtained from the cartesian product $X^{n}$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on $X$. Suppose that $X$ is a ‘nice’ topological space, $R$ is any commutative ring, $H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$ is the zero map, and that $H_{c}^{\bullet }(X,R)$ is a projective $R$-module. We prove that the compact support cohomology of any generalized configuration space of points on $X$ depends only on the graded $R$-module $H_{c}^{\bullet }(X,R)$. This generalizes a theorem of Arabia.

Path homotopy invariants and their application to optimal trajectory planning

We consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in robotics applications, path-planning with reasoning about homotopy classes of trajectories has typically focused on subsets of the Euclidean plane in the robotics literature. The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult. In this paper we propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups. We present explicit results that apply to knot and link complements in 3-space, discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension, and also present results in the coordination space of robots navigating on an Euclidean plane.

The Concept of “Attractive Region in Environment” and its Application in High-Precision Tasks With Low-Precision Systems

High-precision manipulation is very important for a variety of tasks in manufacturing. In general, high-precision manipulation is usually achieved by the guidance of high-precision sensors; however, sensorless methods, for some special cases, are more reliable and effective. There are many important works in this aspect [1] – [3] . The concept of “attractive region in environment” (ARIE), which was proposed in our previous works [4] – [6] , provides a prospective approach for low-precision systems to achieve high-precision manipulation. It has been used in various industrial applications, such as robotic assembly [7] , grasping [8] , and localization [9] . ARIE is a region from any point of which the uncertainty of the system can be eliminated by a state-independent input. The utilization of attractive region can be easily understood by the case of a bean in a bowl. For a bean in a bowl, no matter where the initial position of the bean is, it will move to and finally stay at the bottom of the bowl under gravity. Thus, the uncertainty of the bean is eliminated by gravity without any sensor feedback. Inspired by the above case, the general concept of ARIE is proposed and is applied to industry. In this paper, we review the application of ARIE and give its formal definition. Furthermore, we establish conditions for the existence of ARIE in the generalized configuration space, which is complex and nonideal. The relationship between the high-dimensional attractive region and the low-dimensional one is also discussed, which is helpful to make feasible and reliable manipulation strategies in the low-dimensional configuration space. The main contribution of this paper is the generalization of the concept of ARIE, which provides a theoretical foundation for applications including the high-precision manipulation with low-precision system. The experimental simulations on different kinds of complex manipulations show the effectiveness of the proposed method for industrial applications.

Topological trajectory classification with filtrations of simplicial complexes and persistent homology

In this work, we present a sampling-based approach to trajectory classification which enables automated high-level reasoning about topological classes of trajectories. Our approach is applicable to general configuration spaces and relies only on the availability of collision free samples. Unlike previous sampling-based approaches in robotics which use graphs to capture information about the path-connectedness of a configuration space, we construct a multiscale approximation of neighborhoods of the collision free configurations based on filtrations of simplicial complexes. Our approach thereby extracts additional homological information which is essential for a topological trajectory classification. We propose a multiscale classification algorithm for trajectories in configuration spaces of arbitrary dimension and for sets of trajectories starting and ending in two fixed points. Using a cone construction, we then generalize this approach to classify sets of trajectories even when trajectory start and end points are allowed to vary in path-connected subsets. We furthermore show how an augmented filtration of simplicial complexes based on an arbitrary function on the configuration space, such as a costmap, can be defined to incorporate additional constraints. We present an evaluation of our approach in 2-, 3-, 4- and 6-dimensional configuration spaces in simulation and in real-world experiments using a Baxter robot and motion capture data.

The Onsager Equation for Corpora

We consider extensions of excluded volume interactions for complex corpora that generalize simple rod-like particles. The Onsager equation can be defined for quite general configuration spaces, and the dimension reduction of the phase space in the limit of highly intense interaction can be shown. The formalism describes both freely articulated and interacting N-rods and the example of interacting 2rods is given in detail.

2T Physics, Scale Invariance and Topological Vector Fields

We construct, in classical two-time physics, the necessary structure for the most general configuration space formulation of quantum mechanics containing gravity in d+2 dimensions. This structure is composed of a symmetric Riemannian metric tensor and of a vector field that defines a section of a flat U(1) bundle over space-time. This construction is possible because of the existence of a finite local scale invariance of the Hamiltonian and because two-time physics contains, at the classical level, a local generalization of the discrete duality symmetry between position and momentum that underlies the structure of quantum mechanics.

Self-similar random processes and infinite-dimensional configuration spaces

We discuss various infinite-dimensional configuration spaces that carry measures quasi-invariant under compactly supported diffeomorphisms of a manifold M corresponding to a physical space. Such measures allow the construction of unitary representations of the diffeomorphism group, which are important to nonrelativistic quantum statistical physics and to the quantum theory of extended objects in M = ℝd. Special attention is given to measurable structure and topology underlying measures on generalized configuration spaces obtained from self-similar random processes (both for d = 1 and d > 1), which describe infinite point configurations having accumulation points.

Nonlinear kinematic tolerance analysis of planar mechanical systems

This paper presents a nonlinear kinematic tolerance analysis algorithm for planar mechanical systems comprised of higher kinematic pairs. The part profiles consist of line and circle segments. Each part translates along a planar axis or rotates around an orthogonal axis. The part shapes and motion axes are parameterized by a vector of tolerance parameters with range limits. A system is analyzed in two steps. The first step constructs generalized configuration spaces, called contact zones, that bound the worst-case kinematic variation of the pairs over the tolerance parameter range. The zones specify the variation of the pairs at every contact configuration and reveal failure modes, such as jamming, due to changes in kinematic function. The second step bounds the worst-case system variation at selected configurations by composing the zones. Case studies show that the algorithm is effective, fast, and more accurate than a prior algorithm that constructs and composes linear approximations of contact zones.

CONTINUITY PROPERTIES OF SCHRÖDINGER SEMIGROUPS WITH MAGNETIC FIELDS

The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-continuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownian-bridge expectation. Altogether, the article is meant to extend some of the results in B. Simon's landmark paper [Bull. Amer. Math. Soc.7 (1982) 447] to non-zero vector potentials and more general configuration spaces.

Integration Over a Generic Algebra

In this paper we consider the problem of quantizing theories defined over configuration spaces described by noncommuting parameters. If one tries to do that by generalizing the path-integral formalism, the first problem one has to deal with is the definition of integral over these generalized configuration spaces. This is the problem we state and solve in the present work, by constructing an explicit algorithm for the integration over a generic algebra. The general conditions a given algebra has to satisfy in order to admit our integration are not yet fully understood, but many examples are discussed in order to illustrate our construction.

A NEW KIND OF SINGULAR STIFF PROBLEMS AND APPLICATION TO THIN ELASTIC SHELLS

We consider the asymptotic behavior of the solution of a class of problems involving a small parameter ε and ε2. This generalizes the “singular stiff” problems arising in classical thin shell theory. The new problems appear in theory of composite shells, when the local structure implies coupling between membrane stresses and flexions. According to specific hypotheses, this kind of problems contains singular perturbations and penalty problems where the limit solution belongs to a subspace G1 of the general configuration space V. In addition to the coercive problem, spectral properties are considered in the small and medium frequency ranges, including spectral families in the case without compactness.

A geometric approach to error detection and recovery for robot motion planning with uncertainty

Robots must plan and execute tasks in the presence of uncertainty. Uncertainty arises from sensing errors, control errors, and uncertainty in the geometric models of the environment and of the robot. The last, which we will call model uncertainty, has received little previous attention. In this paper we present a formal framework for computing motion strategies which are guaranteed to succeed in the presence of all three kinds of uncertainty. We show that it is effectively computable for some simple cases. The motion strategies we consider include sensor-based gross motions, compliant motions, and simple pushing motions. We show that model uncertainty can be represented by position uncertainty in a generalized configuration space. We describe the structure of this space, and how motion strategies may be planned in it. It is not always possible to find plans that are guaranteed to succeed. In the presence of model error, such plans may not even exist. For this reason we investigate error detection and recovery (EDR) strategies. We characterize what such strategies are, and propose a formal framework for constructing them. Our theory represents what is perhaps the first systematic attack on the problem of error detection and recovery based on geometric and physical reasoning.

Configuration space topology and quantum internal symmetries

The inequivalent quantizations of a physical system with a general configuration space Q are studied, with particular emphasis on the case when π 1(Q) is nonabelian. The necessary generaliztions of the state space and the set of observables are given. Possible realizations of this scenario in quantum mechanics and quantum field theory are outlined.

Constraints on electronic energy hypersurfaces of higher multiplicities

Upper and lower bounds are derived for electronic energy hypersurfaces of various multiplicities over a general nuclear configuration space. Whereas, in relations derived earlier, only energy hypersurfaces of the same multiplicity could be used as upper or lower bounds for each other, this restriction is now removed in the more general relations proposed in the present study.