Showing papers on "Configuration space published in 2019"

Exploring implicit spaces for constrained sampling-based planning:

Abstract: We present a review and reformulation of manifold constrained sampling-based motion planning within a unifying framework, IMACS (implicit manifold configuration space). IMACS enables a broad class ...

Distance-based sampling of software configuration spaces

Abstract: Configurable software systems provide a multitude of configuration options to adjust and optimize their functional and non-functional properties. For instance, to find the fastest configuration for a given setting, a brute-force strategy measures the performance of all configurations, which is typically intractable. Addressing this challenge, state-of-the-art strategies rely on machine learning, analyzing only a few configurations (i.e., a sample set) to predict the performance of other configurations. However, to obtain accurate performance predictions, a representative sample set of configurations is required. Addressing this task, different sampling strategies have been proposed, which come with different advantages (e.g., covering the configuration space systematically) and disadvantages (e.g., the need to enumerate all configurations). In our experiments, we found that most sampling strategies do not achieve a good coverage of the configuration space with respect to covering relevant performance values. That is, they miss important configurations with distinct performance behavior. Based on this observation, we devise a new sampling strategy, called distance-based sampling, that is based on a distance metric and a probability distribution to spread the configurations of the sample set according to a given probability distribution across the configuration space. This way, we cover different kinds of interactions among configuration options in the sample set. To demonstrate the merits of distance-based sampling, we compare it to state-of-the-art sampling strategies, such as t-wise sampling, on $10$ real-world configurable software systems. Our results show that distance-based sampling leads to more accurate performance models for medium to large sample sets.

Integrable sigma models and 2-loop RG flow

Abstract: Following arXiv:1907.04737 , we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d σ- models. We focus on the “λ-model,” an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an “interpolating model” for non-abelian duality. The parameters are the WZ level k and the coupling λ, and the fields are g, valued in a group G, and a 2d vector A± in the corresponding algebra. We formulate the λ-model as a σ-model on an extended G × G × G configuration space (g, h, $$ \overline{h} $$ ), defining h and $$ \overline{h} $$ by A+ = h∂+h−1, A_ = $$ \overline{h} $$ ∂− $$ \overline{h} $$ −1. Our central observation is that the model on this extended configuration space is renormalizable without any deformation, with only λ running. This is in contrast to the standard σ-model found by integrating out A±, whose 2-loop renormalizability is only obtained after the addition of specific finite local counterterms, resulting in a quantum deformation of the target space geometry. We compute the 2-loop β-function of the λ-model for general group and symmetric spaces, and illustrate our results on the examples of SU(2)/U(1) and SU(2). Similar conclusions apply in the non-abelian dual limit implying that non-abelian duality commutes with the RG flow. We also find the 2-loop β-function of a “squashed” principal chiral model.

Neural Path Planning: Fixed Time, Near-Optimal Path Generation via Oracle Imitation

Abstract: Fast and efficient path generation is critical for robots operating in complex environments. This motion planning problem is often performed in a robot’s actuation or configuration space, where popular pathfinding methods such as A*, RRT*, get exponentially more computationally expensive to execute as the dimensionality increases or the spaces become more cluttered and complex. On the other hand, if one were to save the entire set of paths connecting all pair of locations in the configuration space a priori, one would run out of memory very quickly. In this work, we introduce a novel way of producing fast and optimal motion plans for static environments by using a stepping neural network approach, called OracleNet. OracleNet uses Recurrent Neural Networks to determine end-to-end trajectories in an iterative manner that implicitly generates optimal motion plans with minimal loss in performance in a compact form. The algorithm is straightforward in implementation while consistently generating near-optimal paths in a single, iterative, end-to-end roll-out. In practice, OracleNet generally has fixed-time execution regardless of the configuration space complexity while outperforming popular pathfinding algorithms in complex environments and higher dimensions1.

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 \drrtstar\ is an informed, asymptotically-optimal extension of a prior sampling-based multi-robot motion planner, \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, \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, \drrtstar\, 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.

Multiconfiguration Pair-Density Functional Theory for Iron Porphyrin with CAS, RAS, and DMRG Active Spaces

Abstract: Porphyrins are present in many metalloproteins, and they are also important components of a variety of nonbiological functional materials. Furthermore, they are representative of the kind of large, strongly correlated system that is especially difficult for accurate calculations. For example, predicting the order of their spin states has been challenging. Here we study the energetic order of four states (one singlet, two triplets, and one quintet) of iron porphyrin, FeP, by the multiconfiguration pair-density functional theory (MC-PDFT). Five active space prescriptions, namely, CAS(8, 6), CAS(8, 11), CAS(16, 15), RAS(34,2,2;13,6,16), and DMRG(34, 35), are used to obtain the kinetic energy, density, and on-top density. Although the prediction of which spin state of FeP is the ground state depends on the selection of the active space when one uses multireference second-order perturbation theory and such calculations lead incorrectly to a quintet ground state with the largest studied active space, all five active spaces correctly lead to a triplet ground state when one uses MC-PDFT. We conclude that the (34,35) active space is large enough to give a qualitatively correct description of the orbital space and configuration space such that one obtains the correct spin state prediction when the external correlation energy is added accurately in a post-SCF step. We also conclude that MC-PDFT can provide an efficient and accurate approach to treat the electron correlation in large, strongly correlated systems with the complexity of iron porphyrin.

Dual approach for effective potentials that accurately model structure and energetics

Abstract: Because they eliminate unnecessary degrees of freedom, coarse-grained (CG) models enable studies of phenomena that are intractable with more detailed models. For the same reason, the effective potentials that govern CG degrees of freedom incorporate entropic contributions from the eliminated degrees of freedom. Consequently, these effective potentials demonstrate limited transferability and provide a poor estimate of atomic energetics. Here, we propose a simple dual-potential approach that combines "structure-based" and "energy-based" variational principles to determine effective potentials that model free energies and potential energies, respectively, as a function of the CG configuration. We demonstrate this approach for 1-site CG models of water and methanol. We accurately sample configuration space by performing simulations with the structure-based potential. We accurately estimate average atomic energies by postprocessing the sampled configurations with the energy-based potential. Finally, the difference between the two potentials predicts a qualitatively accurate estimate for the temperature dependence of the structure-based potential.

Physically consistent numerical solver for time-dependent Fokker-Planck equations.

Abstract: We present a simple thermodynamically consistent method for solving time-dependent Fokker-Planck equations (FPE) for overdamped stochastic processes, also known as Smoluchowski equations. It yields both transition and steady-state behavior and allows for computations of moment-generating and large-deviation functions of observables defined along stochastic trajectories, such as the fluctuating particle current, heat, and work. The key strategy is to approximate the FPE by a master equation with transition rates in configuration space that obey a local detailed balance condition for arbitrary discretization. Its time-dependent solution is obtained by a direct computation of the time-ordered exponential, representing the propagator of the FPE, by summing over all possible paths in the discretized space. The method thus not only preserves positivity and normalization of the solutions but also yields a physically reasonable total entropy production, regardless of the discretization. To demonstrate the validity of the method and to exemplify its potential for applications, we compare it against Brownian-dynamics simulations of a heat engine based on an active Brownian particle trapped in a time-dependent quartic potential.

Neural Path Planning: Fixed Time, Near-Optimal Path Generation via Oracle Imitation

Abstract: Fast and efficient path generation is critical for robots operating in complex environments. This motion planning problem is often performed in a robot's actuation or configuration space, where popular pathfinding methods such as A*, RRT*, get exponentially more computationally expensive to execute as the dimensionality increases or the spaces become more cluttered and complex. On the other hand, if one were to save the entire set of paths connecting all pair of locations in the configuration space a priori, one would run out of memory very quickly. In this work, we introduce a novel way of producing fast and optimal motion plans for static environments by using a stepping neural network approach, called OracleNet. OracleNet uses Recurrent Neural Networks to determine end-to-end trajectories in an iterative manner that implicitly generates optimal motion plans with minimal loss in performance in a compact form. The algorithm is straightforward in implementation while consistently generating near-optimal paths in a single, iterative, end-to-end roll-out. In practice, OracleNet generally has fixed-time execution regardless of the configuration space complexity while outperforming popular pathfinding algorithms in complex environments and higher dimensions

Generation of Synchronized Configuration Space Trajectories of Multi-Robot Systems

Abstract: We pose the problem of path-constrained trajectory generation for the synchronous motion of multi-robot systems as a non-linear optimization problem. Our method determines appropriate parametric representation for the configuration variables, generates an approximate solution as a starting point for the optimization method, and uses successive refinement techniques to solve the problem in a computationally efficient manner. We have demonstrated the effectiveness of the proposed method on challenging simulation and physical experiments with high degrees of freedom robotic systems.

Shannon Entropy for the Hydrogen Atom Confined by Four Different Potentials

Abstract: Spatial confinements induce localization or delocalization on the electron density in atoms and molecules, and the hydrogen atom is not the exception to these results. In previous works, this system has been confined by an infinite and a finite potential where the wave-function exhibits an exact solution, and, consequently, their Shannon entropies deliver exact results. In this article, the Shannon entropy in configuration space is examined for the hydrogen atom submitted to four different potentials: (a) infinite potential; (b) Coulomb plus harmonic oscillator; (c) constant potential; and (d) dielectric continuum. For all these potentials, the Schrodinger equation admitted an exact analytic solution, and therefore the corresponding electron density has a closed-form. From the study of these confinements, we observed that the Shannon entropy in configuration space is a good indicator of localization and delocalization of the electron density for ground and excited states of the hydrogen atom confined under these circumstances. In particular, the confinement imposed by a parabolic potential induced characteristics that were not presented for other confinements; for example, the kinetic energy exhibited oscillations when the confinement radius is varied and such oscillations coincided with oscillations showed by the Shannon entropy in configuration space. This result indicates that, when the kinetic energy is increased, the Shannon entropy is decreased and vice versa.

Self-assembly of geometric space from random graphs

Abstract: We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein–Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein–Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

Spectrum of Itinerant Fractional Excitations in Quantum Spin Ice.

Abstract: We study the quantum dynamics of fractional excitations in quantum spin ice. We focus on the density of states in the two-monopole sector, ρ(ω), as this can be connected to the wave-vector-integrated dynamical structure factor accessible in neutron scattering experiments. We find that ρ(ω) exhibits a strikingly characteristic singular and asymmetric structure that provides a useful fingerprint for comparison to experiment. ρ(ω) obtained from the exact diagonalization of a finite cluster agrees well with that, from the analytical solution of a hopping problem on a Husimi cactus representing configuration space, but not with the corresponding result on a face-centered cubic lattice, on which the monopoles move in real space. The main difference between the latter two lies in the inclusion of the emergent gauge field degrees of freedom, under which the monopoles are charged. This underlines the importance of treating both sets of degrees of freedom together, and it presents a novel instance of dimensional transmutation.

Non-abelian Quantum Statistics on Graphs

Abstract: We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space X. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of X which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.

Quantum coherence and the localization transition

Abstract: A dynamical signature of localization in quantum systems is the absence of transport which is governed by the amount of coherence that configuration space states possess with respect to the Hamiltonian eigenbasis. To make this observation precise, we study the localization transition via quantum coherence measures arising from the resource theory of coherence. We show that the escape probability, which is known to show distinct behavior in the ergodic and localized phases, arises naturally as the average of a coherence measure. Moreover, using the theory of majorization, we argue that broad families of coherence measures can detect the uniformity of the transition matrix (between the Hamiltonian and configuration bases) and hence act as probes to localization. We provide supporting numerical evidence for Anderson and many-body localization (MBL). For infinitesimal perturbations of the Hamiltonian, the differential coherence defines an associated Riemannian metric. We show that the latter is exactly given by the dynamical conductivity, a quantity of experimental relevance which is known to have a distinctively different behavior in the ergodic and in the many-body localized phases.

Formalizing Spatial Configuration Optimization Problems with the Use of a Special Function Class

Abstract: An approach to the analysis of spatial configuration optimization problems by generating configuration spaces of geometric objects is proposed. Depending on the choice of generalized variables, various classes of spatial configurations are investigated. A class of functions in the configuration space of geometric objects is introduced, which allows us to propose new and develop available approaches for the formalization of optimization problems for spatial configurations. The problem of placing circular objects in a bounded domain by the criterion of minimization of the total area of their pairwise intersections is considered.

On the origin of phase transitions in the absence of symmetry-breaking

Abstract: In this paper we investigate the Hamiltonian dynamics of a lattice gauge model in three spatial dimensions. Our model Hamiltonian is defined on the basis of a continuum version of a duality transformation of a three dimensional Ising model. The system so obtained undergoes a thermodynamic phase transition in the absence of a global symmetry-breaking and thus in the absence of an order parameter. It is found that the first order phase transition undergone by this model fits into a microcanonical version of an Ehrenfest-like classification of phase transitions applied to the configurational entropy. It is discussed why the seemingly divergent behaviour of the third derivative of configurational entropy is the effect of a deeper geometrical transition of the equipotential submanifolds of configuration space, which, in its turn, is likely to be the ”shadow” of an even deeper transition of topological kind.

A Nonrelativistic Quantum Field Theory with Point Interactions in Three Dimensions

Abstract: We construct a Hamiltonian for a quantum-mechanical model of nonrelativistic particles in three dimensions interacting via the creation and annihilation of a second type of nonrelativistic particles, which are bosons. The interaction between the two types of particles is a point interaction concentrated on the points in configuration space where the positions of two different particles coincide. We define the operator, and its domain of self-adjointness, in terms of co-dimension-three boundary conditions on the set of collision configurations relating sectors with different numbers of particles.

Rapidly-Exploring Quotient-Space Trees: Motion Planning using Sequential Simplifications

Abstract: Motion planning problems can be simplified by admissible projections of the configuration space to sequences of lower-dimensional quotient-spaces, called sequential simplifications. To exploit sequential simplifications, we present the Quotient-space Rapidly-exploring Random Trees (QRRT) algorithm. QRRT takes as input a start and a goal configuration, and a sequence of quotient-spaces. The algorithm grows trees on the quotient-spaces both sequentially and simultaneously to guarantee a dense coverage. QRRT is shown to be (1) probabilistically complete, and (2) can reduce the runtime by at least one order of magnitude. However, we show in experiments that the runtime varies substantially between different quotient-space sequences. To find out why, we perform an additional experiment, showing that the more narrow an environment, the more a quotient-space sequence can reduce runtime.

Linear motion planning with controlled collisions and pure planar braids

Abstract: We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topological complexity ($TC_s$, $s\geq2$) of the "no-$k$-equal" configuration space Conf$_k(\mathbb{R},n)$. This yields (with $k=3$) the LS-cat and the higher topological complexity of Khovanov's group PP$_n$ of pure planar braids on $n$ strands, which is an $\mathbb{R}$-analogue of Artin's classical pure braid group on $n$ strands. Our methods can be used to describe optimal motion planners for PP$_n$ provided $n$ is small.

T-dualities and Doubled Geometry of the Principal Chiral Model

Abstract: The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is here investigated with the aim of getting a further deepening of its relation with Generalized Geometry and Doubled Geometry. A one-parameter family of equivalent Hamiltonian descriptions is analysed, and cast into the form of Born geometries. Then O(3, 3) duality transformations of the target phase space are performed and we show that the resulting dual models are defined on the group SB(2, ℂ) which is the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel’d double SL(2, ℂ). A parent action with doubled degrees of freedom and configuration space SL(2, ℂ) is then defined that reduces to either one of the dually related models, once suitable constraints are implemented.

STAMPEDE: A Discrete-Optimization Method for Solving Pathwise-Inverse Kinematics

Abstract: We present a discrete-optimization technique for finding feasible robot arm trajectories that pass through provided 6-DOF Cartesian-space end-effector paths with high accuracy, a problem called pathwise-inverse kinematics. The output from our method consists of a path function of joint-angles that best follows the provided end-effector path function, given some definition of “best”. Our method, called Stampede, casts the robot motion translation problem as a discrete-space graph-search problem where the nodes in the graph are individually solved for using non-linear optimization; framing the problem in such a way gives rise to a well-structured graph that affords an effective best path calculation using an efficient dynamic-programming algorithm. We present techniques for sampling configuration space, such as diversity sampling and adaptive sampling, to construct the search-space in the graph. Through an evaluation, we show that our approach performs well in finding smooth, feasible, collision-free robot motions that match the input end-effector trace with very high accuracy, while alternative approaches, such as a state-of-the-art per-frame inverse kinematics solver and a global non-linear trajectory-optimization approach, performed unfavorably.

Quasilocal degrees of freedom in Yang-Mills theory

Abstract: Gauge theories possess non-local features that, in the presence of boundaries, inevitably lead to subtleties. In this article, we continue our study of a unified solution based on a geometric tool operating on field-space: a connection form. We specialize to the $D+1$ formulation of Yang-Mills theories on configuration space, and we precisely characterize the gluing of the Yang-Mills field across regions. In the $D+1$ formalism, the connection-form splits the electric degrees of freedom into their pure-radiative and Coulombic components, rendering the latter as conjugate to the pure-gauge part of the gauge potential. Regarding gluing, we obtain a characterization for topologically simple regions through closed formulas. These formulas exploit the properties of a generalized Dirichlet-to-Neumann operator defined at the gluing surface; through them, we find only the radiative components and the local charges are relevant for gluing. Finally, we study the gluing into topologically non-trivial regions in 1+1 dimensions. We find that in this case, the regional radiative modes do not fully determine the global radiative mode (Aharonov-Bohm phases). For the global mode takes a new contribution from the kernel of the gluing formula, a kernel which is associated to non-trivial cohomological cycles. In no circumstances do we find a need for postulating new local degrees of freedom at boundaries. $\text{The partial results of these notes have been completed and substantially clarified in a more recent, comprehensive article from October 2019. (titled: "The quasilocal degrees of freedom of Yang-Mills theory").}$

Self-Assembly of Geometric Space from Random Graphs

Abstract: We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

The reconstructed power spectrum in the Zeldovich approximation

Abstract: Author(s): Chen, SF; Vlah, Z; White, M | Abstract: Density-reconstruction sharpens the baryon acoustic oscillations signal by undoing some of the smoothing incurred by nonlinear structure formation. In this paper we present an analytical model for reconstruction based on the Zeldovich approximation, which for the first time includes a complete set of counterterms and bias terms up to quadratic order and can fit real and redshift-space data pre- and post-reconstruction data in both Fourier and configuration space over a wide range of scales. We compare our model to n-body data at z = 0 from the DarkSky simulation [1], finding sub-percent agreement in both real space and in the redshift-space power spectrum monopole out to k = 0.4 h Mpc-1, and out to k = 0.2 h Mpc-1 in the quadrupole, with comparable agreement in configuration space. We compare our model with several popular existing alternatives, updating existing theoretical results for exponential damping in wiggle/no-wiggle splits of the BAO signal and discuss the usually-ignored effect of higher bias contributions on the reconstructed signal. In the appendices, we re-derive the former within our formalism, present exploratory results on higher-order corrections due to nonlinearities inherent to reconstruction, and present numerical techniques with which to calculate the redshift-space power spectrum of biased tracers within the Zeldovich approximation.

Tunneling without bounce

Abstract: The false vacua of some potentials do not decay via Euclidean bounces. This typically happens for tunneling actions with a flat direction (in field configuration space) that is lifted by a perturbation into a sloping valley, pushing the bounce off to infinity. Using three different approaches we find a consistent picture for such decays. In the Euclidean approach the bottom of the action valley consists of a family of pseudobounces (field configurations with some key good properties of bounces except extremizing the action). The pseudobounce result is validated by minimizing a Wentzel--Kramers--Brillouin action in Minkowski space along appropriate paths in configuration space. Finally, the most natural approach uses the tunneling action method proposed recently with a simple modification of boundary conditions.

Guided Motion Planning for Snake-like Robots Based on Geometry Mechanics and HJB Equation

Abstract: In this paper, a novel guided motion planning method is proposed for snake-like robots, which decomposes the configuration space into the fiber space and the gait space, with the motion planned in the low-dimensional fiber space, so as to avoid the dimensionality curse problem. More specifically, first, based on geometry mechanics, the kinematic connection is derived, which successfully maps the motion in the fiber space into that in the gait space. According to the kinematic connection, a novel bidirectional motion control method for snake-like robots is proposed to improve the flexibility that can produce not only the forward locomotion but also the backward locomotion additionally. The convergence of the bidirectional motion control method is proved based on averaging theory. Then, the motion planning problem is solved in the low-dimensional fiber space, for which the HJB equation is used to generate the optimal control in the fiber space to avoid the well-known curse of dimensionality. Finally, combining the bidirectional motion control and the motion planned in the fiber space, a guided motion planning method in the configuration space is proposed. Numerical simulations and experiments based on the real snake-like robot platform are performed, whose results demonstrate that the proposed algorithm is valid and robust.

String topology and configuration spaces of two points

Abstract: Given a closed manifold $M$. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincar\'e duality model of the manifold, and involves the configuration space of two points on $M$. We moreover, construct an $IBL_\infty$-structure on (a model of) cyclic chains on the cochain algebra of $M$, such that the natural comparison map to the $S^1$-equivariant loop space homology intertwines the Lie bialgebra structure on homology. The construction of the coproduct/cobracket depends on the perturbative partition function of a Chern-Simons type topological field theory. Furthermore, we give a construction for these string topology operations on the absolute loop space (not relative to constant loops) in case that $M$ carries a non-vanishing vector field and obtain a similar description. Finally, we show that the cobracket is sensitive to the manifold structure of $M$ beyond its homotopy type. More precisely, the action of ${\rm Diff}(M)$ does not (in general) factor through ${\rm aut}(M)$.