High-dimensional Configuration Space Research Articles

The Theory of (Exclusively) Local Beables

It is shown how, starting with the de Broglie–Bohm pilot-wave theory, one can construct a new theory of the sort envisioned by several of QM’s founders: a Theory of Exclusively Local Beables (TELB). In particular, the usual quantum mechanical wave function (a function on a high-dimensional configuration space) is not among the beables posited by the new theory. Instead, each particle has an associated “pilot-wave” field (living in physical space). A number of additional fields (also fields on physical space) maintain what is described, in ordinary quantum theory, as “entanglement.” The theory allows some interesting new perspective on the kind of causation involved in pilot-wave theories in general. And it provides also a concrete example of an empirically viable quantum theory in whose formulation the wave function (on configuration space) does not appear—i.e., it is a theory according to which nothing corresponding to the configuration space wave function need actually exist. That is the theory’s raison d’etre and perhaps its only virtue. Its vices include the fact that it only reproduces the empirical predictions of the ordinary pilot-wave theory (equivalent, of course, to the predictions of ordinary quantum theory) for spinless non-relativistic particles, and only then for wave functions that are everywhere analytic. The goal is thus not to recommend the TELB proposed here as a replacement for ordinary pilot-wave theory (or ordinary quantum theory), but is rather to illustrate (with a crude first stab) that it might be possible to construct a plausible, empirically viable TELB, and to recommend this as an interesting and perhaps-fruitful program for future research.

Geometrical phase effect in Jahn–Teller systems: Twofold electronic degeneracies and beyond

The question of how many states are subject to geometrical phase in a N -fold electronic degeneracy is examined for Jahn–Teller systems. It is shown that it can be diagnosed from the sign change of the adiabatic electronic vector states when a rotation operator is applied to them in functional space defined in N -dimensional spherical polar coordinates, rather than by adiabatic transportation along a closed path in the higher-dimensional configuration space. It is also demonstrated that the Longuet-Higgins theorem for twofold conical intersections (namely that the electronic wave function changes sign on completing a closed path around a doubly-degenerate intersection) can be used to rationalize the results, irrespectively of N .

Triple RRTs: An Effective Method for Path Planning in Narrow Passages

Although Rapidly-exploring Random Trees (RRTs) have been successfully applied in path planning of robots with many degrees of freedom under non-holonomic and differential constraints, rapidly identifying and passing through narrow passages in a robot's configuration space remains a challenge for RRTs-based planners. This paper presents a novel two-stage approach to address the problem of multi-d.o.f. robot path planning in high-dimensional configuration space with narrow corridors. The first stage introduces an efficient sampling algorithm called Bridge Test to find a global roadmap that identifies the critical region. The second stage presents two varieties of RRTs, called Triple-RRTs, to search for a local connection under the guidance of the global landmark. The two-stage strategy keeps a fine balance between global heuristics and local connection, resulting in high performance over the previous RRTs-based path planning methods. We have implemented the Triple-RRTs planners for both rigid and articulated robots in two- and three-dimensional environments. Experimental results demonstrate the effectiveness of the proposed method.

Navigating molecular worms inside chemical labyrinths

Predicting whether a molecule can traverse chemical labyrinths of channels, tunnels, and buried cavities usually requires performing computationally intensive molecular dynamics simulations. Often one wants to screen molecules to identify ones that can pass through a given chemical labyrinth or screen chemical labyrinths to identify those that allow a given molecule to pass. Because it is impractical to test each molecule/labyrinth pair using computationally expensive methods, faster, approximate methods are used to prune possibilities, "triaging" the ability of a proposed molecule to pass through the given chemical labyrinth. Most pruning methods estimate chemical accessibility solely on geometry, treating atoms or groups of atoms as hard spheres with appropriate radii. Here, we explore geometric configurations for a moving "molecular worm," which replaces spherical probes and is assembled from solid blocks connected by flexible links. The key is to extend the fast marching method, which is an ordered upwind one-pass Dijkstra-like method to compute optimal paths by efficiently solving an associated Eikonal equation for the cost function. First, we build a suitable cost function associated with each possible configuration, and second, we construct an algorithm that works in ensuing high-dimensional configuration space: at least seven dimensions are required to account for translational, rotational, and internal degrees of freedom. We demonstrate the algorithm to study shortest paths, compute accessible volume, and derive information on topology of the accessible part of a chemical labyrinth. As a model example, we consider an alkane molecule in a porous material, which is relevant to designing catalysts for oil processing.

Interactive selection of optimal fenestration materials for schematic architectural daylighting design

Complex fenestration systems, such as prismatic and laser cut panels, are emerging as attractive options in architectural design thanks to their high potential to assist in energy and comfort issues. These systems can be used to redirect intense illumination from the sun but have complex transmissive properties that in turn depend on continuously changing illumination conditions due to the dynamic nature of natural light. The resulting non-intuitive interactions with the built environment make it necessary to develop tools that adequately represent these systems' behavior to the architect. The method presented in this paper enables simulation of the direct and indirect illumination from the sun and sky throughout each day for different months Preprint submitted to Automation in Construction 17 December 2007 of the year. The user can interactively explore the high-dimensional configuration space to select optimal materials early in the design process, using both qualitative and quantitative design criteria to make adjustments.

Creating High-quality Paths for Motion Planning

Many algorithms have been proposed that create a path for a robot in an environment with obstacles. Most methods are aimed at finding a solution. However, for many applications, the path must be of a good quality as well. That is, a path should be short and should keep some amount of minimum clearance to the obstacles. Traveling along such a path reduces the chances of collisions due to the difficulty of measuring and controlling the precise position of the robot. This paper reports a new technique, called Partial shortcut, which decreases the path length. While current methods have difficulties in removing all redundant motions, the technique efficiently removes these motions by interpolating one degree of freedom at a time. Two algorithms are also studied that increase the clearance along paths. The first one is fast but can only deal with rigid, translating bodies. The second algorithm is slower but can handle a broader range of robots, including three-dimensional free-flying and articulated robots, which may reside in arbitrary high-dimensional configuration spaces. A big advantage of these algorithms is that clearance along paths can now be increased efficiently without using complex data structures and algorithms. Finally, we combine the two criteria and show that high-quality paths can be obtained for a broad range of robots.

Articulated motion reconstruction from feature points

A fundamental task of reconstructing non-rigid articulated motion from sequences of unstructured feature points is to solve the problem of feature correspondence and motion estimation. This problem is challenging in high-dimensional configuration spaces. In this paper, we propose a general model-based dynamic point matching algorithm to reconstruct freeform non-rigid articulated movements from data presented solely by sparse feature points. The algorithm integrates key-frame-based self-initialising hierarchial segmental matching with inter-frame tracking to achieve computation effectiveness and robustness in the presence of data noise. A dynamic scheme of motion verification, dynamic key-frame-shift identification and backward parent-segment correction, incorporating temporal coherency embedded in inter-frames, is employed to enhance the segment-based spatial matching. Such a spatial–temporal approach ultimately reduces the ambiguity of identification inherent in a single frame. Performance evaluation is provided by a series of empirical analyses using synthetic data. Testing on motion capture data for a common articulated motion, namely human motion, gave feature-point identification and matching without the need for manual intervention, in buffered real-time. These results demonstrate the proposed algorithm to be a candidate for feature-based real-time reconstruction tasks involving self-resuming tracking for articulated motion.

Thermally activated transport of a dislocation loop within an elastic model

We report thermally activated transport of a dislocation loop in terms of a line tension model, where the dislocation line is assumed to be a flexible string. According to conventional rate theory, the features of thermal activation are determined by the saddle-point geometry in high dimensional configuration space. If the circumference of a dislocation loop L is longer than a critical length L c, the selected saddle-point configuration is the well known double-kink type solution. On the other hand, the manner of the thermal activation of a dislocation loop shorter than L c is rather point-defect-like. In the present work, we pay attention to the temperature dependence of transition rate which is represented such as ν 0 ∗ exp ( - E / k B T ) . The pre-exponential factor depends on temperature like ν 0 ∗ ∼ T - 1 / 2 for sufficiently long dislocation loops on the basis of the analysis.

Modelling size effects using 3D density-based dislocation dynamics

We use density-based continuity equations to model strain patterns and size effects in confined plastic flow, namely, shearing of thin films and microbending. To this end, we use a representation in terms of coupled equations for the densities of screw and edge components. We show how these equations derive from a more general formulation in a higher-dimensional configuration space, and discuss relations with other density-based approaches proposed in the past. The new element here is the incorporation into previous continuum formulations of geometrical features and interactions of dislocation lines that cannot be neglected or ‘averaged out’ within a three-dimensional setting of plasticity at the micron and nano-scales.

A targeted reweighting method for accelerating the exploration of high-dimensional configuration space

Time scales available to biomolecular simulations are limited by barriers among states in a high-dimensional configuration space. If equilibrium averages are to be computed, methods that accelerate barrier passage can be carried out by non-Boltzmann sampling. Barriers can be reduced by modifying the potential-energy function and running dynamics on the modified surface. The Boltzmann average can be restored by reweighting each point along the trajectory. We introduce a targeted reweighting scheme where some barriers are reduced, while others are not modified. If only equilibrium properties are desired, trajectories in configuration space can be generated by Langevin dynamics. Once past a transient time, these trajectories guarantee equilibrium sampling when reweighted. A relatively high-order stochastic integration method can be used to generate trajectories. The targeted reweighting scheme is illustrated by a series of double-well models with varying degrees of freedom and shown to be a very efficient method to provide the correct equilibrium distributions, in comparison with analytic results. The scheme is applied to a protein model consisting of a chain of connected beads characterized by dihedral angles and the van der Waals interactions among the beads. We investigate the sampling of configuration space for a model of a helix-turn-helix motif. The targeted reweighting is found to be essential to permit the original all-helical conformation to bend and generate turn structures while still maintaining the alpha-helical segments.

Narrow passage sampling for probabilistic roadmap planning

Probabilistic roadmap (PRM) planners have been successful in path planning of robots with many degrees of freedom, but sampling narrow passages in a robot's configuration space remains a challenge for PRM planners. This paper presents a hybrid sampling strategy in the PRM framework for finding paths through narrow passages. A key ingredient of the new strategy is the bridge test, which reduces sample density in many unimportant parts of a configuration space, resulting in increased sample density in narrow passages. The bridge test can be implemented efficiently in high-dimensional configuration spaces using only simple tests of local geometry. The strengths of the bridge test and uniform sampling complement each other naturally. The two sampling strategies are combined to construct the hybrid sampling strategy for our planner. We implemented the planner and tested it on rigid and articulated robots in 2-D and 3-D environments. Experiments show that the hybrid sampling strategy enables relatively small roadmaps to reliably capture the connectivity of configuration spaces with difficult narrow passages.

Articulated Body Motion Capture by Stochastic Search

We develop a modified particle filter which is shown to be effective at searching the high-dimensional configuration spaces (c. 30 + dimensions) encountered in visual tracking of articulated body motion. The algorithm uses a continuation principle, based on annealing, to introduce the influence of narrow peaks in the fitness function, gradually. The new algorithm, termed annealed particle filtering, is shown to be capable of recovering full articulated body motion efficiently. A mechanism for achieving a soft partitioning of the search space is described and implemented, and shown to improve the algorithm's performance. Likewise, the introduction of a crossover operator is shown to improve the effectiveness of the search for kinematic trees (such as a human body). Results are given for a variety of agile motions such as walking, running and jumping.

Elastic Strips: A Framework for Motion Generation in Human Environments

Robotic applications are expanding into dynamic, unstructured, and populated environments. Mechanisms specifically designed to address the challenges arising in these environments, such as humanoid robots, exhibit high kinematic complexity. This creates the need for new algorithmic approaches to motion generation, capable of performing task execution and real-time obstacle avoidance in high-dimensional configuration spaces. The elastic strip framework presented in this paper enables the execution of a previously planned motion in a dynamic environment for robots with many degrees of freedom. To modify a motion in reaction to changes in the environment, real-time obstacle avoidance is combined with desired posture behavior. The modification of a motion can be performed in a task-consistent manner, leaving task execution unaffected by obstacle avoidance and posture behavior. The elastic strip framework also encompasses methods to suspend task behavior when its execution becomes inconsistent with other constraints imposed on the motion. Task execution is resumed automatically, once those constraints have been removed. Experiments demonstrating these capabilities on a nine-degree-of-freedom mobile manipulator and a 34-degree-of-freedom humanoid robot are presented, proving the elastic strip framework to be a powerful and versatile task-oriented approach to real-time motion generation and motion execution for robots with a large number of degrees of freedom in dynamic environments.

Elastic Strips: A Framework for Motion Generation in Human Environments

Robotic applications are expanding into dynamic, unstructured, and populated environments. Mechanisms specifically designed to address the challenges arising in these environments, such as humanoid robots, exhibit high kinematic complexity. This creates the need for new algorithmic approaches to motion generation, capable of performing task execution and real-time obstacle avoidance in high-dimensional configuration spaces. The elastic strip framework presented in this paper enables the execution of a previously planned motion in a dynamic environment for robots with many degrees of freedom. To modify a motion in reaction to changes in the environment, real-time obstacle avoidance is combined with desired posture behavior. The modification of a motion can be performed in a task-consistent manner, leaving task execution unaffected by obstacle avoidance and posture behavior. The elastic strip framework also encompasses methods to suspend task behavior when its execution becomes inconsistent with other constraints imposed on the motion. Task execution is resumed automatically, once those constraints have been removed. Experiments demonstrating these capabilities on a nine-degree-of-freedom mobile manipulator and a 34-degree-of-freedom humanoid robot are presented, proving the elastic strip framework to be a powerful and versatile task-oriented approach to real-time motion generation and motion execution for robots with a large number of degrees of freedom in dynamic environments.

Using motion planning to study protein folding pathways.

We present a framework for studying protein folding pathways and potential landscapes which is based on techniques recently developed in the robotics motion planning community. Our focus in this work is to study the protein folding mechanism assuming we know the native fold. That is, instead of performing fold prediction, we aim to study issues related to the folding process, such as the formation of secondary and tertiary structure, and the dependence of the folding pathway on the initial denatured conformation. Our work uses probabilistic roadmap (PRM) motion planning techniques which have proven successful for problems involving high-dimensional configuration spaces. A strength of these methods is their efficiency in rapidly covering the planning space without becoming trapped in local minima. We have applied our PRM technique to several small proteins (~60 residues) and validated the pathways computed by comparing the secondary structure formation order on our paths to known hydrogen exchange experimental results. An advantage of the PRM framework over other simulation methods is that it enables one to easily and efficiently compute folding pathways from any denatured starting state to the (known) native fold. This aspect makes our approach ideal for studying global properties of the protein's potential landscape, most of which are difficult to simulate and study with other methods. For example, in the proteins we study, the folding pathways starting from different denatured states sometimes share common portions when they are close to the native fold, and moreover, the formation order of the secondary structure appears largely independent of the starting denatured conformation. Another feature of our technique is that the distribution of the sampled conformations is correlated with the formation of secondary structure and, in particular, appears to differentiate situations in which secondary structure clearly forms first and those in which the tertiary structure is obtained more directly. Overall, our results applying PRM techniques are very encouraging and indicate the promise of our approach for studying proteins for which experimental results are not available.

Motion Planning for Cooperative Transportation of a Large Object by Multiple Mobile Robots.

In this paper, we propose a motion planning method for cooperative transportation of a large object by multiple mobile robots in a complicated environment. We must plan the motions of the object and mobile robots to avoid obstacles in a three-dimensional environment. A computation time to solve this problem is exploded, since we need search a solution in a high dimensional configuration space (C-space). Accordingly, we reduce the dimensions of C-space using the feature of the transportation by multiple mobile robots. In our method, all things (the object, the robots and the obstacles) are represented by an octree that is the approximate cell decomposition method for a three-dimensional environment. We can find a solution with A* search algorithm in this smaller dimensional C-space using the potential field. We verify the effectiveness of our proposed motion planning method through simulations.

Transition path of a lattice dislocation in a high dimensional configuration space

Motion of a dislocation in a simple model lattice has been studied by applying a transition path technique. The minimum energy path (MEP) in a high dimensional configuration-space from one stable configuration to the adjacent one is calculated as a function of applied stress. From the energy profiles along the MEP, the applied stress dependence of the activation energy (Δ H− τ relation) and the Peierls stress are obtained. The Peierls potential can reasonably be defined, when the position of the dislocation is determined along the MEP, taking account of the atomic displacements and the energy inside the dislocation core of radius 3 b. The conventional scheme for determining the stress dependence of the activation energy, which assumes a Peierls potential with the work done term subtracted, reproduces the Δ H− τ relation obtained directly by the transition path calculation.

Polarization and localization in insulators: Generating function approach

We develop the theory and practical expressions for the full quantum-mechanical distribution of the intrinsic macroscopic polarization of an insulator in terms of the ground state wavefunction. The central quantity is a cumulant generating function which yields, upon successive differentiation, all the cumulants and moments of the probability distribution of the center of mass X/N of the electrons, defined appropriately to remain valid for extended systems obeying twisted boundary conditions. The first moment is the average polarization, where we recover the well-known Berry phase expression. The second cumulant gives the mean-square fluctuation of the polarization, which defines an electronic localization length xi_i along each direction i. It follows from the fluctuation-dissipation theorem that in the thermodynamic limit xi_i diverges for metals and is a finite, measurable quantity for insulators. It is possible to define for insulators maximally-localized ``many-body Wannier functions'', which for large N become localized in disconnected regions of the high-dimensional configuration space, establishing a direct connection with Kohn's theory of the insulating state. Interestingly, the expression for xi_i^2, which involves the second derivative of the wavefunction with respect to the boundary conditions, is directly analogous to Kohn's formula for the ``Drude weight'' as the second derivative of the energy.

Potential energy landscape of a model glass former: thermodynamics, anharmonicities, and finite size effects.

It is possible to formulate the thermodynamics of a glass forming system in terms of the properties of inherent structures, which correspond to the minima of the potential energy and build up the potential energy landscape in the high-dimensional configuration space. In this work we quantitatively apply this general approach to a simulated model glass-forming system. We systematically vary the system size between N=20 and N=160. This analysis enables us to determine for which temperature range the properties of the glass former are governed by the regions of the configuration space, close to the inherent structures. Furthermore, we obtain detailed information about the nature of anharmonic contributions. Moreover, we can explain the presence of finite size effects in terms of specific properties of the energy landscape. Finally, determination of the total number of inherent structures for very small systems enables us to estimate the Kauzmann temperature.

Motion Planning with Reversal Manoeuvres for a Non-Holonomic Constrained Robot

This paper presents a new motion planning approach, including reversal manoeuvres, for car-like robots subject to non-holonomic constraints. The paper presents a complete path planning algorithm and describes the procedure for constructing a collision-free path for a mobile robot constrained by its rectangular shape and kinematics. Emphasis is made on the techniques of reversal transfers, which are used to compensate the limited manoeuvrability of a car-like robot. The approach works entirely in the workspace, as opposed to building a higher dimensional configuration space (C-space). It starts by constructing a visibility graph and finding the shortest path for a point robot and then detects areas where collision may occur by minimum distance calculations between obstacles and between the selected path and obstacles. Robot configurations are placed along the shortest path and lemmas are developed for ascertaining transfers from one robot configuration to another. The transfer techniques include direct, indirect and reversal manoeuvres, and ensure that the path is feasible for the robot to travel with a given steering limit. The process runs in time O(nk + n log n) for k obstacles and n vertices. The algorithm is tested in computer simulations and results are given, demonstrating the versatility of the algorithm.