Showing papers on "Configuration space published in 2020"

Asymptotically Optimal Planning under Piecewise-Analytic Constraints

Abstract: We present the first asymptotically optimal algorithm for motion planning problems with piecewise-analytic differential constraints, like manipulation or rearrangement planning. This class of problems is characterized by the presence of differential constraints that are local in nature: a robot can only move an object once the object has been grasped. These constraints are not analytic and thus cannot be addressed by standard differentially constrained planning algorithms. We demonstrate that, given the ability to sample from the locally reachable subset of the configuration space with positive probability, we can construct random geometric graphs that contain optimal plans with probability one in the limit of infinite samples. This approach does not require a hand-coded symbolic abstraction. We demonstrate our approach in simulation on a simple manipulation planning problem, and show it generates lower-cost plans than a sequential task and motion planner.

Finding Faster Configurations Using FLASH

Abstract: Finding good configurations of a software system is often challenging since the number of configuration options can be large. Software engineers often make poor choices about configuration or, even worse, they usually use a sub-optimal configuration in production, which leads to inadequate performance. To assist engineers in finding the better configuration, this article introduces Flash , a sequential model-based method that sequentially explores the configuration space by reflecting on the configurations evaluated so far to determine the next best configuration to explore. Flash scales up to software systems that defeat the prior state-of-the-art model-based methods in this area. Flash runs much faster than existing methods and can solve both single-objective and multi-objective optimization problems. The central insight of this article is to use the prior knowledge of the configuration space (gained from prior runs) to choose the next promising configuration. This strategy reduces the effort (i.e., number of measurements) required to find the better configuration. We evaluate Flash using 30 scenarios based on 7 software systems to demonstrate that Flash saves effort in 100 and 80 percent of cases in single-objective and multi-objective problems respectively by up to several orders of magnitude compared to state-of-the-art techniques.

Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization. (arXiv:math-ph/0408005v1 CROSS LISTED)

Abstract: A nonholonomic system consists of a configuration space Q, a Lagrangian L, and an nonintegrable constraint distribution H, with dynamics governed by Lagrange-d'Alembert's principle. We present two studies both using adapted moving frames. In the first study we apply Cartan's method of equivalence to investigate the geometry underlying a nonholonomic system. As an example we compute the differential invariants for a nonholonomic system on a four-dimensional configuration manifold endowed with a rank two (Engel) distribution. In the second part we study G-Chaplygin systems. These are systems where the constraint distribution is given by a connection on a principal fiber bundle with total space Q and base space S=Q/G, and with a G-equivariant Lagrangian. These systems compress to an almost Hamiltonian system on $T^{*}S$. Under an $s \in S$ dependent time reparameterization a number of compressed systems become Hamiltonian. A necessary condition for Hamiltonization is the existence of an invariant measure on the original system. Assuming an invariant measure we describe the obstruction to Hamiltonization. Chaplygin's "rubber" sphere, a ball with unequal inertia coefficients rolling without slipping or spinning (about the vertical axis) on a plane is Hamiltonizable when compressed to $T^{*}SO(3)$. Finally we discuss reduction of internal symmetries. Chaplygin's "marble" (where spinning is allowed) is not Hamiltonizable when compressed to $T^{*}SO(3)$; we conjecture that it is also not Hamiltonizable when reduced to $T^{*}S^{2}$.

dRRT * : Scalable and informed asymptotically-optimal multi-robot motion planning

Abstract: Many exciting robotic applications require multiple robots with many degrees of freedom, such as manipulators, to coordinate their motion in a shared workspace. Discovering high-quality paths in such scenarios can be achieved, in principle, by exploring the composite space of all robots. Sampling-based planners do so by building a roadmap or a tree data structure in the corresponding configuration space and can achieve asymptotic optimality. The hardness of motion planning, however, renders the explicit construction of such structures in the composite space of multiple robots impractical. This work proposes a scalable solution for such coupled multi-robot problems, which provides desirable path-quality guarantees and is also computationally efficient. In particular, the proposed $$\mathtt{dRRT^*}$$ is an informed, asymptotically-optimal extension of a prior sampling-based multi-robot motion planner, $$\mathtt{dRRT}$$. The prior approach introduced the idea of building roadmaps for each robot and implicitly searching the tensor product of these structures in the composite space. This work identifies the conditions for convergence to optimal paths in multi-robot problems, which the prior method was not achieving. Building on this analysis, $$\mathtt{dRRT}$$ is first properly adapted so as to achieve the theoretical guarantees and then further extended so as to make use of effective heuristics when searching the composite space of all robots. The case where the various robots share some degrees of freedom is also studied. Evaluation in simulation indicates that the new algorithm, $$\mathtt{dRRT^*}$$ converges to high-quality paths quickly and scales to a higher number of robots where various alternatives fail. This work also demonstrates the planner’s capability to solve problems involving multiple real-world robotic arms.

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.

Quantum Assisted Simulator

Abstract: Quantum simulation offers a possibility to explore the exponentially large configuration space of quantum mechanical systems and thus help us study poorly understood topics such as high-temperature superconductivity and drug design. Here, we provide a novel hybrid quantum-classical algorithm for simulating the dynamics of quantum systems. Without loss of generality, the Hamiltonian is assumed to be a linear combination of unitaries and the Ansatz wavefunction is taken as a linear combination of quantum states. The quantum states are fixed, and the combination parameters are variationally adjusted. Unlike existing variational quantum simulation algorithms, our algorithm does not require any classical-quantum feedback loop and by construction bypasses the barren plateau problem. Moreover, our algorithm does not require any complicated measurements, such as the Hadamard test. The entire framework is compatible with existing experimental capabilities and thus can be implemented immediately. We also provide an extension of our algorithm to imaginary time evolution.

Learning Dense Visual Correspondences in Simulation to Smooth and Fold Real Fabrics

Abstract: Robotic fabric manipulation is challenging due to the infinite dimensional configuration space, self-occlusion, and complex dynamics of fabrics. There has been significant prior work on learning policies for specific deformable manipulation tasks, but comparatively less focus on algorithms which can efficiently learn many different tasks. In this paper, we learn visual correspondences for deformable fabrics across different configurations in simulation and show that this representation can be used to design policies for a variety of tasks. Given a single demonstration of a new task from an initial fabric configuration, the learned correspondences can be used to compute geometrically equivalent actions in a new fabric configuration. This makes it possible to robustly imitate a broad set of multi-step fabric smoothing and folding tasks on multiple physical robotic systems. The resulting policies achieve 80.3% average task success rate across 10 fabric manipulation tasks on two different robotic systems, the da Vinci surgical robot and the ABB YuMi. Results also suggest robustness to fabrics of various colors, sizes, and shapes. See this https URL for supplementary material and videos.

Combining phonon accuracy with high transferability in Gaussian approximation potential models.

Abstract: Machine learning driven interatomic potentials, including Gaussian approximation potential (GAP) models, are emerging tools for atomistic simulations. Here, we address the methodological question of how one can fit GAP models that accurately predict vibrational properties in specific regions of configuration space while retaining flexibility and transferability to others. We use an adaptive regularization of the GAP fit that scales with the absolute force magnitude on any given atom, thereby exploring the Bayesian interpretation of GAP regularization as an "expected error" and its impact on the prediction of physical properties for a material of interest. The approach enables excellent predictions of phonon modes (to within 0.1 THz-0.2 THz) for structurally diverse silicon allotropes, and it can be coupled with existing fitting databases for high transferability across different regions of configuration space, which we demonstrate for liquid and amorphous silicon. These findings and workflows are expected to be useful for GAP-driven materials modeling more generally.

Motion Planning for Dual-Arm Manipulation of Elastic Rods

Abstract: We present a novel motion planning strategy for the manipulation of elastic rods with two robotic arms. In previous work, it has been shown that the free configuration space of an elastic rod, i.e., the set of equilibrium shapes of the rod, is a smooth manifold of a finite dimension that can be parameterized by one chart. Thus, a sampling-based planning algorithm is straightforward to implement in the product space of the joint angles and the equilibrium configuration space of the elastic rod. Preliminary results show that planning directly in this product space is feasible. However, solving for the elastic rod's shape requires the numerical solution of differential equations, resulting in an excessive and impractical runtime. Hence, we propose to pre-compute a descriptor of the rod, i.e., a roadmap in the free configuration space of the rod that captures its main-connectivity. By doing so, we can plan the motion of any dual-arm robotic system over this roadmap with dramatically fewer solutions of the differential equations. Experiments using the Open Motion Planning Library (OMPL) show significant runtime reduction by an order of magnitude.

Randomized Incremental Convex Hull is Highly Parallel

Abstract: The randomized incremental convex hull algorithm is one of the most practical and important geometric algorithms in the literature. Due to its simplicity, and the fact that many points or facets can be added independently, it is also widely used in parallel convex hull implementations. However, to date there have been no non-trivial theoretical bounds on the parallelism available in these implementations. In this paper, we provide a strong theoretical analysis showing that the standard incremental algorithm is inherently parallel. In particular, we show that for n points in any constant dimension, the algorithm has O(log n) dependence depth with high probability. This leads to a simple work-optimal parallel algorithm with polylogarithmic span with high probability. Our key technical contribution is a new definition and analysis of the configuration dependence graph extending the traditional configuration space, which allows for asynchrony in adding configurations. To capture the "true" dependence between configurations, we define the support set of configuration c to be the set of already added configurations that it depends on. We show that for problems where the size of the support set can be bounded by a constant, the depth of the configuration dependence graph is shallow (O(log n) with high probability for input size n). In addition to convex hull, our approach also extends to several related problems, including half-space intersection and finding the intersection of a set of unit circles. We believe that the configuration dependence graph and its analysis is a general idea that could potentially be applied to more problems.

Light-Matter Hybrid-Orbital-Based First-Principles Methods: The Influence of Polariton Statistics.

Abstract: A detailed understanding of strong matter-photon interactions requires first-principle methods that can solve the fundamental Pauli-Fierz Hamiltonian of nonrelativistic quantum electrodynamics efficiently. A possible way to extend well-established electronic-structure methods to this situation is to embed the Pauli-Fierz Hamiltonian in a higher-dimensional light-matter hybrid auxiliary configuration space. In this work we show the importance of the resulting hybrid Fermi-Bose statistics of the polaritons, which are the new fundamental particles of the "photon-dressed" Pauli-Fierz Hamiltonian for systems in cavities. We show that violations of these statistics can lead to unphysical results. We present an efficient way to ensure the correct statistics by enforcing representability conditions on the dressed one-body reduced density matrix. We further present a general prescription how to extend a given first-principles approach to polaritons and as an example introduce polaritonic Hartree-Fock theory. While being a single-reference method in polariton space, polaritonic Hartree-Fock is a multireference method in the electronic space, i.e., it describes electronic correlations. We also discuss possible applications to polaritonic QEDFT. We apply this theory to a lattice model and find that, the more delocalized the bound-state wave function of the particles is, the stronger it reacts to photons. The main reason is that within a small energy range, many states with different electronic configurations are available as opposed to a strongly bound (and hence energetically separated) ground-state wave function. This indicates that under certain conditions coupling to the quantum vacuum of a cavity can indeed modify ground state properties.

Grover Search as a Naturally Occurring Phenomenon.

Abstract: We provide first evidence that under certain conditions, 1/2-spin fermions may naturally behave like a Grover search, looking for topological defects in a material. The theoretical framework is that of discrete-time quantum walks (QWs), i.e., local unitary matrices that drive the evolution of a single particle on the lattice. Some QWs are well known to recover the (2+1)-dimensional Dirac equation in continuum limit, i.e., the free propagation of the 1/2-spin fermion. We study two such Dirac QWs, one on the square grid and the other on a triangular grid reminiscent of graphenelike materials. The numerical simulations show that the walker localizes around the defects in O(sqrt[N]) steps with probability O(1/logN), in line with previous QW search on the grid. The main advantage brought by those of this Letter is that they could be implemented as "naturally occurring" freely propagating particles over a surface featuring topological defects-without the need for a specific oracle step. From a quantum computing perspective, however, this hints at novel applications of QW search: instead of using them to look for "good" solutions within the configuration space of a problem, we could use them to look for topological properties of the entire configuration space.

Machine learning surrogate models for Landau fluid closure

Abstract: The first result of applying the machine/deep learning technique to the fluid closure problem is presented in this paper. As a start, three different types of neural networks [multilayer perceptron (MLP), convolutional neural network (CNN), and two-layer discrete Fourier transform (DFT) network] were constructed and trained to learn the well-known Hammett–Perkins Landau fluid closure in configuration space. We find that in order to train a well-preformed network, a minimum size of the training data set is needed; MLP also requires a minimum number of neurons in the hidden layers that equals the degrees of freedom in Fourier space, despite the fact that training data are being fed into the configuration space. Out of the three models, DFT performs the best for the clean data, most likely due to the existence of the simple Fourier expression for the Hammett–Perkins closure, but it is the least robust with respect to input noise. Overall, with appropriate tuning and optimization, all three neural networks are able to accurately predict the Hammett–Perkins closure and reproduce the intrinsic nonlocal feature, suggesting a promising path to calculating more sophisticated closures with the machine/deep learning technique.

Lie Group Based Type Synthesis Using Transformation Configuration Space for Reconfigurable Parallel Mechanisms With Bifurcation Between Spherical Motion and Planar Motion

Abstract: This paper investigates novel reconfigurable parallel mechanisms with bifurcation between planar subgroup SE(2) and rotation subgroup SO(3) based on a transformation configuration space. Having recollected necessary theoretical fundamentals with regard to compositional submanifolds and kinematic bonds, the motion representation of the platform of reconfigurable parallel mechanisms are investigated. The transformation configuration space of a reconfigurable parallel mechanism with motion branches is proposed with respect to the fact that the intersection of Lie subgroups or submanifolds is the identity element or a non-identity element. The switch conditions of the transformation configuration space are discussed, leading to establishment of a theoretical foundation for realizing a switch of motion branches. The switch principle of reconfigurable parallel mechanisms is further investigated with respect to the common motion between SE(2) parallel-mechanism motion generators and SO(3) parallel-mechanism motion generators. Under this principle, the subchains with common motion generators are synthesized and divided into two types of generators. The first type of generators generates kinematic chains with a common intersection of three joint axes, and the second type of generators generates a common intersection of two joint axes. Following this, two types of reconfigurable parallel mechanisms with three identical subchains are synthesized, resulting in 11 varieties in which platforms can be switched between SE(2) and SO(3) after passing through the singularity configuration space.

High Dimensional Normality of Noisy Eigenvectors

Abstract: We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing the stochastic eigenstate equation (SEE) which describes the Lie group valued flow of eigenvectors induced by matrix valued Brownian motion. We consider the associated colored eigenvector moment flow defining an SDE on a particle configuration space. This flow extends the eigenvector moment flow first introduced in Bourgade and Yau (2017) to the multicolor setting. However, it is no longer driven by an underlying Markov process on configuration space due to the lack of positivity in the semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged decay and contractive properties. This allows us to establish optimal time of relaxation to equilibrium for the colored eigenvector moment flow and prove joint asymptotic normality for eigenvectors. Applications in random matrix theory include the explicit computations of joint eigenvector distributions for general Wigner type matrices and sparse graph models when corresponding eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector distributions for Levy matrices when the eigenvectors correspond to small energy levels.

Notes on biadjoint amplitudes, Trop G(3, 7) and X(3, 7) scattering equations

Abstract: In these notes we use the recently found relation between facets of tropical Grassmannians and generalizations of Feynman diagrams to compute all “biadjoint amplitudes” for n = 7 and k = 3. We also study scattering equations on X (3, 7), the configuration space of seven points on CP2. We prove that the number of solutions is 1272 in a two-step process. In the first step we obtain 1162 explicit solutions to high precision using near-soft kinematics. In the second step we compute the matrix of 360 ×360 biadjoint amplitudes obtained by using the facets of Trop G(3, 7), subtract the result from using the 1162 solutions and compute the rank of the resulting matrix. The rank turns out to be 110, which proves that the number of solutions in addition to the 1162 explicit ones is exactly 110.

Description of Four- and Five-Nucleon Systems by Solving Faddeev-Yakubovsky Equations in Configuration Space

Abstract: Description of four- and five-nucleon systems by solving Faddeev-Yakubovsky equations in configuration space

Realtime Simulation of Thin-Shell Deformable Materials Using CNN-Based Mesh Embedding

Abstract: We address the problem of accelerating thin-shell deformable object simulations by dimension reduction. We present a new algorithm to embed a high-dimensional configuration space of deformable objects in a low-dimensional feature space, where the configurations of objects and feature points have approximate one-to-one mapping. Our key technique is a graph-based convolutional neural network (CNN) defined on meshes with arbitrary topologies and a new mesh embedding approach based on physics-inspired loss term. We have applied our approach to accelerate high-resolution thin shell simulations corresponding to cloth-like materials, where the configuration space has tens of thousands of degrees of freedom. We show that our physics-inspired embedding approach leads to higher accuracy compared with prior mesh embedding methods. Finally, we show that the temporal evolution of the mesh in the feature space can also be learned using a recurrent neural network (RNN) leading to fully learnable physics simulators. After training our learned simulator runs 500–10000× faster and the accuracy is high enough for robot manipulation tasks.

Cohomology of generalized configuration spaces

Abstract: Let be a topological space. We consider certain generalized configuration spaces of points on , obtained from the cartesian product 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 . Suppose that is a ‘nice’ topological space, is any commutative ring, is the zero map, and that is a projective -module. We prove that the compact support cohomology of any generalized configuration space of points on depends only on the graded -module . This generalizes a theorem of Arabia.

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.

Decoupling Constraints from Sampling-Based Planners

Abstract: We present a general unifying framework for sampling-based motion planning under kinematic task constraints which enables a broad class of planners to compute plans that satisfy a given constraint function that encodes, e.g., loop closure, balance, and end-effector constraints. The framework decouples a planner’s method for exploration from constraint satisfaction by representing the implicit configuration space defined by a constraint function. We emulate three constraint satisfaction methodologies from the literature, and demonstrate the framework with a range of planners utilizing these constraint methodologies. Our results show that the appropriate choice of constrained satisfaction methodology depends on many factors, e.g., the dimension of the configuration space and implicit constraint manifold, and number of obstacles. Furthermore, we show that novel combinations of planners and constraint satisfaction methodologies can be more effective than previous approaches. The framework is also easily extended for novel planners and constraint spaces.

BBS invariant measures with independent soliton components

Abstract: The Box-Ball System (BBS) is a one-dimensional cellular automaton in the configuration space $\{0,1\}^{\mathbb{Z} }$ introduced by Takahashi and Satsuma [8], who identified conserved quantities called solitons. Ferrari, Nguyen, Rolla and Wang [4] map a configuration to a family of soliton components, indexed by the soliton sizes $k\ge 1$. Building over this decomposition, we give an explicit construction of a large family of invariant measures for the BBS that are also shift invariant, including Ising-like Markov and Bernoulli product measures. The construction is based on the concatenation of iid excursions of the associated walk trajectory. Each excursion has the property that the law of its $k$ component given the larger components is product of a finite number of geometric distributions with a parameter depending on $k$. As a consequence, the law of each component of the resulting ball configuration is product of identically distributed geometric random variables, and the components are independent. This last property implies invariance for BBS, as shown by [4].

Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity

Abstract: We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior of these families as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This setting will be used in a subsequent paper to study the space of spherical conic metrics with large cone angles, where the existence theory is still incomplete. Of independent interest is how setting up the analysis on these compactified configuration spaces provides a good framework for analyzing “confluent families” of regular singular, ie conic, elliptic differential operators.

Robotic Swarm Motion Planning for Load Carrying and Manipulating

Abstract: Certain species of ants can carry out tasks in dense work spaces while maintaining their ability to accurately manipulate heavy loads, and these advantages are of interest to the robotics community. We consider a robotic swarm of $N\ge 6$ agents that assumes the task of moving a load through a cluttered space. This forces the swarm to carefully manipulate the orientation of the load, while transporting it to its destination point. We model this scenario as a 6-PPSS (Prismatic-Prismatic-Spherical-Spherical) redundant mobile platform, having six degrees of freedom. As with insects, the multitude of agents enables sharing the burden of the load in the case that one or more agents are blocked by an obstacle. We model this by a semi-algebraic set of constraints on the distances between the agents and the load. We apply an Extended Kalman Filter routine, in order to estimate their relative locations. We show how the estimation-error is reduced when position-information is shared among the agents. These estimations are then used to calculate the full configuration and investigate the effect of position estimation error on the platform heading error. We show how motion planning can then be calculated in the model’s full configuration space and demonstrate this with a distributed control scheme. To reduce the search time, we introduce a variant of the crawling probabilistic road map motion planning algorithm under a set of kinematic constraints and work-space obstacles. Finally, we exemplify our algorithms on several simulated scenarios.

Topological transitions in the configuration space of non-Euclidean origami

Abstract: Origami structures have been proposed as a means of creating three-dimensional structures from the micro- to the macroscale and as a means of fabricating mechanical metamaterials. The design of such structures requires a deep understanding of the kinematics of origami fold patterns. Here we study the configurations of non-Euclidean origami, folding structures with Gaussian curvature concentrated on the vertices, for arbitrary origami fold patterns. The kinematics of such structures depends crucially on the sign of the Gaussian curvature. As an application of our general results, we show that the configuration space of nonintersecting, oriented vertices with positive Gaussian curvature decomposes into disconnected subspaces; there is no pathway between them without tearing the origami. In contrast, the configuration space of negative Gaussian curvature vertices remains connected. This provides a new, and only partially explored, mechanism by which the mechanics and folding of an origami structure could be controlled.

Distribution of rare saddles in the p -spin energy landscape

Abstract: We compute the statistical distribution of index-1 saddles surrounding a given local minimum of the $p$-spin energy landscape, as a function of their distance to the minimum in configuration space and of the energy of the latter. We identify the saddles also in the region of configuration space in which they are subdominant in number (i.e., rare) with respect to local minima, by computing large deviation probabilities of the extremal eigenvalues of their Hessian. As an independent result, we determine the joint large deviation probability of the smallest eigenvalue and eigenvector of a GOE matrix perturbed with both an additive and multiplicative finite-rank perturbation.