Showing papers on "Configuration space published in 2021"

Positive Configuration Space

Abstract: We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.

Non-perturbative geometries for planar $\mathcal{N}=4$ SYM amplitudes

Abstract: There is a remarkable well-known connection between the G(4, n) cluster algebra and n-particle amplitudes in $$ \mathcal{N} $$ = 4 SYM theory. For n ≥ 8 two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters from among the infinitely many cluster variables, and to find an explanation for certain algebraic functions, such as the square roots of four-mass-box type, that are expected to appear in symbols but are not cluster variables. In this letter we use the notion of “stringy canonical forms” to construct polytopal realizations of certain compactifications of (the positive part of) the configuration space Confn(ℙk−1) ≅ G(k, n)/T that are manifestly finite for all k and n. Some facets of these polytopes are naturally associated to cluster variables, while others are naturally associated to algebraic functions constructed from Lusztig’s canonical basis. For (k, n) = (4, 8) the latter include precisely the expected square roots, revealing them to be related to certain “overpositive” functions of the kinematical invariants.

Many-Body Topological and Skin States without Open Boundaries

Abstract: Robust boundary states have been the focus of much recent research, both as topologically protected states and as non-Hermitian skin states. In this paper, we show that many-body effects can also induce analogs of these robust states in place of actual physical boundaries. Particle statistics or suitably engineered interactions, i.e., in ultracold atomic lattices can restrict the accessible many-body Hilbert space, and introduce effective boundaries in a spatially periodic higher-dimensional configuration space. We demonstrate the emergence of topological chiral modes in a two-fermion hopping model without open boundaries, with fermion pairs confined and asymmetrically propagated by suitably chosen fluxes. Heterogeneous nonreciprocal hoppings across different particle species can also result in robust particle clumping in a translation invariant setting, reminiscent of skin mode accumulation at an open boundary. But unlike fixed open boundaries, effective boundaries correspond to the locations of impenetrable particles and are mobile, giving rise to fundamentally different many-body versus single-body spectra and corresponding dynamics. Since nonreciprocal accumulation is agnostic to the dimensionality of restricted Hilbert spaces, our many-body skin states generalize directly in the thermodynamic limit. The many-body topological states, however, are nontrivially dimension-dependent, and their detailed exploration will stimulate further studies in higher dimensional topological invariants.

Kinematic modeling and control of variable curvature soft continuum robots

Abstract: The compliant structure and influence of external forces usually result in complex deformation of soft continuum robots, which makes the accurate modeling and control of the robot challenging In this work, we present a new variable curvature kinematic modeling approach for soft continuum robots by taking the external forces into consideration, achieving both accurate motion simulation and feedforward control of the robot To this end, the variable curvature configuration is firstly parameterized based on the absolute nodal coordinate formulation (ANCF) Then, a kinematic model is developed to describe the mappings between the defined configuration space and the actuation space with payloads With this model, we achieve accurate and fast motion simulation for the soft continuum robot with different payloads and input pressures within 1 millisecond, which is verified by a set of experiments Finally, an inverse-model-based feedforward controller is developed for a two-section soft continuum robot The experimental results of tracking complex trajectories verify the effectiveness of our model and control strategies The average position error of the end-effector is 289% of the robot length This work can also be served as a tool to design and analyze soft continuum robots with desired workspace

Chaos and Ergodicity across the Energy Spectrum of Interacting Bosons

Abstract: We identify the chaotic phase of the Bose-Hubbard Hamiltonian by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions. The eigenvectors are shown to become ergodic in the thermodynamic limit, in the configuration space Fock basis, in which random matrix theory offers a remarkable description of their typical structure. The distributions of the generalized fractal dimensions, however, are ever more distinguishable from random matrix theory as the Hilbert space dimension grows.

Physically inspired deep learning of molecular excitations and photoemission spectra

Abstract: Modern functional materials consist of large molecular building blocks with significant chemical complexity which limits spectroscopic property prediction with accurate first-principles methods. Consequently, a targeted design of materials with tailored optoelectronic properties by high-throughput screening is bound to fail without efficient methods to predict molecular excited-state properties across chemical space. In this work, we present a deep neural network that predicts charged quasiparticle excitations for large and complex organic molecules with a rich elemental diversity and a size well out of reach of accurate many body perturbation theory calculations. The model exploits the fundamental underlying physics of molecular resonances as eigenvalues of a latent Hamiltonian matrix and is thus able to accurately describe multiple resonances simultaneously. The performance of this model is demonstrated for a range of organic molecules across chemical composition space and configuration space. We further showcase the model capabilities by predicting photoemission spectra at the level of the GW approximation for previously unseen conjugated molecules.

On applications of quantum computing to plasma simulations

Abstract: Quantum computing is gaining increased attention as a potential way to speed up simulations of physical systems, and it is also of interest to apply it to simulations of classical plasmas. However, quantum information science is traditionally aimed at modeling linear Hamiltonian systems of a particular form that is found in quantum mechanics, so extending the existing results to plasma applications remains a challenge. Here, we report a preliminary exploration of the long-term opportunities and likely obstacles in this area. First, we show that many plasma-wave problems are naturally representable in a quantumlike form and thus are naturally fit for quantum computers. Second, we consider more general plasma problems that include non-Hermitian dynamics (instabilities, irreversible dissipation) and nonlinearities. We show that by extending the configuration space, such systems can also be represented in a quantumlike form and thus can be simulated with quantum computers too, albeit that requires more computational resources compared to the first case. Third, we outline potential applications of hybrid quantum–classical computers, which include analysis of global eigenmodes and also an alternative approach to nonlinear simulations.

Hamiltonians Without Ultraviolet Divergence for Quantum Field Theories

Abstract: We propose a way of defining Hamiltonians for quantum field theories without any renormalization procedure. The resulting Hamiltonians, called IBC Hamiltonians, are mathematically well defined (and in particular, ultraviolet finite) without an ultraviolet cutoff such as smearing out the particles over a nonzero radius; rather, the particles are assigned radius zero. These Hamiltonians agree with those obtained through renormalization whenever both are known to exist. We describe explicit examples of IBC Hamiltonians. Their definition, which is best expressed in the particle–position representation of the wave function, involves a kind of boundary condition on the wave function, which we call an interior–boundary condition (IBC). The relevant configuration space is one of a variable number of particles, and the relevant boundary consists of the configurations with two or more particles at the same location. The IBC relates the value (or derivative) of the wave function at a boundary point to the value of the wave function at an interior point (here, in a sector of configuration space corresponding to a lesser number of particles).

Lieb-Thirring inequalities for wave functions vanishing on the diagonal set

Abstract: We propose a general strategy to derive Lieb-Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb-Thirring inequality to wave functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.

Learning Proofs of Motion Planning Infeasibility

Abstract: We present a learning-based approach to prove infeasibility of kinematic motion planning problems. Sampling-based motion planners are effective in high-dimensional spaces but are only probabilistically complete. Consequently, these planners cannot provide a definite answer if no plan exists, which is important for high-level scenarios, such as task-motion planning. We propose a combination of bidirectional sampling-based planning (such as RRT-connect) and machine learning to construct an infeasibility proof alongside the two search trees. An infeasibility proof is a closed manifold in the obstacle region of the configuration space that separates the start and goal into disconnected components of the free configuration space. We train the manifold using common machine learning techniques and then triangulate the manifold into a polytope to prove containment in the obstacle region. Under assumptions about learning hyper-parameters and robustness of configuration space optimization, the output is either an infeasibility proof or a motion plan. We demonstrate proof construction for 3-DOF and 4-DOF manipulators and show improvement over previous algorithms.

Large deviations for the skew-detailed-balance lifted-Markov processes to sample the equilibrium distribution of the Curie–Weiss model

Abstract: Among the Markov chains breaking detailed-balance that have been proposed in the field of Monte-Carlo sampling in order to accelerate the convergence towards the steady state with respect to the detailed-balance dynamics, the idea of 'Lifting' consists in duplicating the configuration space into two copies $\sigma=\pm$ and in imposing directed flows in each copy in order to explore the configuration space more efficiently. The skew-detailed-balance Lifted-Markov-chain introduced by K. S. Turitsyn, M. Chertkov and M. Vucelja [Physica D Nonlinear Phenomena 240 , 410 (2011)] is revisited for the Curie-Weiss mean-field ferromagnetic model, where the dynamics for the magnetization is closed. The large deviations at various levels for empirical time-averaged observables are analyzed and compared with their detailed-balance counterparts, both for the discrete extensive magnetization $M$ and for the continuous intensive magnetization $m=\frac{M}{N}$ for large system-size $N$.

Dominant Reaction Pathways by Quantum Computing.

Abstract: Characterizing thermally activated transitions in high-dimensional rugged energy surfaces is a very challenging task for classical computers. Here, we develop a quantum annealing scheme to solve this problem. First, the task of finding the most probable transition paths in configuration space is reduced to a shortest-path problem defined on a suitable weighted graph. Next, this optimization problem is mapped into finding the ground state of a generalized Ising model. A finite-size scaling analysis suggests this task may be solvable efficiently by a quantum annealing machine. Our approach leverages on the quantized nature of qubits to describe transitions between different system's configurations. Since it does not involve any lattice space discretization, it paves the way towards future biophysical applications of quantum computing based on realistic all-atom models.

Task-Space Decomposed Motion Planning Framework for Multi-Robot Loco-Manipulation

Abstract: This paper introduces a novel task-space decomposed motion planning framework for multi-robot simultaneous locomotion and manipulation. When several manipulators hold an object, closed-chain kinematic constraints are formed, and it will make the motion planning problems challenging by inducing lower-dimensional singularities. Unfortunately, the constrained manifold will be even more complicated when the manipulators are equipped with mobile bases. We address the problem by introducing a dual-resolution motion planning framework which utilizes a convex task region decomposition method, with each resolution tuned to efficient computation for their respective roles. Concretely, this dual-resolution approach enables a global planner to explore the low-dimensional decomposed task-space regions toward the goal, then a local planner computes a path in high-dimensional constrained configuration space. We demonstrate the proposed method in several simulations, where the robot team transports the object toward the goal in the obstacle-rich environments.

Dual Fast Marching Tree Algorithm for Human-Like Motion Planning of Anthropomorphic Arms With Task Constraints

Abstract: This article presents a dual fast marching tree algorithm that consists of constrained fast marching tree planning in a Cartesian space (C_FMT∗) and human-like fast marching tree planning in self-motion manifolds (H_FMT∗) for human-like motion planning for anthropomorphic arms with task constraints. The key idea of the proposal is to exploit dual sampling in a Cartesian space and self-motion manifolds to explore the entire configuration space. The C_FMT∗ reduces the constrained planning problem to the unconstrained instance by sampling in the obstacle-free Cartesian space and satisfying the task constraints; it can solve most of these constrained path planning tasks quickly and obtain lower cost solutions compared to the existing techniques. In addition, a validity checking method of Cartesian sampling points based on self-motion manifolds is introduced to ensure the probabilistic completeness of the new planner. By analyzing musculoskeletal models of the human arm and the muscle strength property, a torque effort criterion was deduced to generate biomimetic motion for anthropomorphic arms. Then, an H_FMT∗ that incorporates the FMT∗ algorithm with the torque effort criterion is also proposed and used to bias the tree growth toward human-like movements in the self-motion manifolds of the obtained path. Finally, the proposed approach has been illustrated with several real examples executed with a humanoid robot. The obtained results show that the paths obtained with the proposed approach are quicker and more human-like.

Using Euler Curves to Model Continuum Robots

Abstract: Due to the continuous and flexible nature of continuum robot backbones and the infinite number of parameters required to represent them in configuration space, modeling them accurately and in real-time is challenging. While the constant curvature assumption provides a simple alternative, it is limited in its capabilities as it cannot account for external tip forces. In cases where the backbone deviates from the constant curvature backbone, Euler curves are an interesting alternative for modeling continuum robots. In this paper, we show that a linear approximation of the backbone curvature is sufficiently accurate for estimating the shape of a robot subject to external tip forces. Next, we propose a numerical static model for tendon-driven continuum robots experiencing in-plane external tip forces. In this model, we use Euler arc splines to circumvent the limitations of standard numerical integration schemes required to calculate these curves. The system reduces to solving two nonlinear equations, allowing fast approximation of the backbone shape. The proposed model is validated experimentally on a robot prototype. Average tip error of 3.07% of the robot length is obtained for an average computation time of 0.51 ms.

And yet it moves: Paradoxically moving linkages in kinematics

Abstract: The configuration space of a mechanical linkage, consisting of rigid bodies moving in space constrained by joints, is defined by algebraic conditions. If these equations do not define a complete intersection, then the dimension of the configuration space is higher than expected. These linkages violate the Chebychev-Grubler-Kutzbach formula for the degree of freedom of mobility. Mathematicians developed a variety of methods to describe and understand this phenomenon. This paper explains some of them.

Ergodic sampling of the topological charge using the density of states

Abstract: In lattice calculations, the approach to the continuum limit is hindered by the severe freezing of the topological charge, which prevents ergodic sampling in configuration space. In order to significantly reduce the autocorrelation time of the topological charge, we develop a density of states approach with a smooth constraint and use it to study SU(3) pure Yang Mills gauge theory near the continuum limit. Our algorithm relies on simulated tempering across a range of couplings, which guarantees the decorrelation of the topological charge and ergodic sampling of topological sectors. Particular emphasis is placed on testing the accuracy, efficiency and scaling properties of the method. In their most conservative interpretation, our results provide firm evidence of a sizeable reduction of the exponent z related to the growth of the autocorrelation time as a function of the inverse lattice spacing.

Generation of synchronized configuration space trajectories with workspace path constraints for an ensemble of robots

Abstract: We present an approach to generate path-constrained synchronous motion for the coupled ensemble of robots. In this article, we refer to serial-link manipulators and mobile bases as robots. We assum...

Engineered swift equilibration for arbitrary geometries.

Abstract: Engineered swift equilibration (ESE) is a class of driving protocols that enforce an equilibrium distribution with respect to external control parameters at the beginning and end of rapid state transformations of open, classical nonequilibrium systems. ESE protocols have previously been derived and experimentally realized for Brownian particles in simple, one-dimensional, time-varying trapping potentials; one recent study considered ESE in two-dimensional Euclidean configuration space. Here we extend the ESE framework to generic, overdamped Brownian systems in arbitrary curved configuration space and illustrate our results with specific examples not amenable to previous techniques. Our approach may be used to impose the necessary dynamics to control the full temporal configurational distribution in a wide variety of experimentally realizable settings.

High-throughput systematic topological generation of low-energy carbon allotropes

Abstract: The search for new materials requires effective methods for scanning the space of atomic configurations, in which the number is infinite. Here we present an extensive application of a topological network model of solid-state transformations, which enables one to reduce this infinite number to a countable number of the regions corresponding to topologically different crystalline phases. We have used this model to successfully generate carbon allotropes starting from a very restricted set of initial structures; the generation procedure has required only three steps to scan the configuration space around the parents. As a result, we have obtained all known carbon structures within the specified set of restrictions and discovered 224 allotropes with lattice energy ranging in 0.16–1.76 eV atom−1 above diamond including a phase, which is denser and probably harder than diamond. We have shown that this phase has a quite different topological structure compared to the hard allotropes from the diamond polytypic series. We have applied the tiling approach to explore the topology of the generated phases in more detail and found that many phases possessing high hardness are built from the tiles confined by six-membered rings. We have computed the mechanical properties for the generated allotropes and found simple dependences between their density, bulk, and shear moduli.

Active Learning Configuration Interaction for Excited-State Calculations of Polycyclic Aromatic Hydrocarbons.

Abstract: We present the active learning configuration interaction (ALCI) method for multiconfigurational calculations based on large active spaces. ALCI leverages the use of an active learning procedure to find important electronic configurations among the full configurational space generated within an active space. We tested it for the calculation of singlet-singlet excited states of acenes and pyrene using different machine learning algorithms. The ALCI method yields excitation energies within 0.2-0.3 eV from those obtained by traditional complete active-space configuration interaction (CASCI) calculations (affordable for active spaces up to 16 electrons in 16 orbitals) by including only a small fraction of the CASCI configuration space in the calculations. For larger active spaces (we tested up to 26 electrons in 26 orbitals), not affordable with traditional CI methods, ALCI captures the trends of experimental excitation energies. Overall, ALCI provides satisfactory approximations to large active-space wave functions with up to 10 orders of magnitude fewer determinants for the systems presented here. These ALCI wave functions are promising and affordable starting points for the subsequent second-order perturbation theory or pair-density functional theory calculations.

Nonparametric Analysis of Nonequilibrium Simulations.

Abstract: We extend the nonparametric framework of reaction coordinate optimization to nonequilibrium ensembles of (short) trajectories. For example, we show how, starting from such an ensemble, one can obtain an equilibrium free-energy profile along the committor, which can be used to determine important properties of the dynamics exactly. A new adaptive sampling approach, the transition-state ensemble enrichment, is suggested, which samples the configuration space by "growing" committor segments toward each other starting from the boundary states. This framework is suggested as a general tool, alternative to the Markov state models, for a rigorous and accurate analysis of simulations of large biomolecular systems, as it has the following attractive properties. It is immune to the curse of dimensionality, does not require system-specific information, can approximate arbitrary reaction coordinates with high accuracy, and has sensitive and rigorous criteria to test optimality and convergence. The approaches are illustrated on a 50-dimensional model system and a realistic protein folding trajectory.

Rationally Inattentive Path-Planning via RRT*

Abstract: We consider a path-planning scenario for a mobile robot traveling in a configuration space with obstacles under the presence of stochastic disturbances. A novel path length metric is proposed on the uncertain configuration space and then integrated with the existing RRT* algorithm. The metric is a weighted sum of two terms which capture both the Euclidean distance traveled by the robot and the perception cost, i.e., the amount of information the robot must perceive about the environment to follow the path safely. The continuity of the path length function with respect to the topology of the total variation metric is shown and the optimality of the Rationally Inattentive RRT* algorithm is discussed. Three numerical studies are presented which display the utility of the new algorithm.

Topological Complexity of Configuration Spaces of Fully Articulated Graphs and Banana Graphs

Abstract: We determine the topological complexity of configuration spaces of graphs that are not necessarily trees, which was a crucial assumption in previous results. We do this for two very different classes of graphs: fully articulated graphs and banana graphs. We also complete the computation in the case of trees to include configuration spaces with any number of points, extending a proof of Farber. At the end we show that an unordered configuration space on a graph does not always have the same topological complexity as the corresponding ordered configuration space (not even when they are both connected). Surprisingly, in our counterexamples the topological complexity of the unordered configuration space is in fact smaller than for the ordered one.