Showing papers on "Configuration space published in 2018"

Relaxation and domain formation in incommensurate two-dimensional heterostructures

Abstract: We introduce configuration space as a natural representation for calculating the mechanical relaxation patterns of incommensurate two-dimensional (2D) bilayers. The approach can be applied to a wide variety of 2D materials through the use of a continuum model in combination with a generalized stacking fault energy for interlayer interactions. We present computational results for small-angle twisted bilayer graphene and molybdenum disulfide (${\mathrm{MoS}}_{2}$), a representative material of the transition-metal dichalcogenide family of 2D semiconductors. We calculate accurate relaxations for ${\mathrm{MoS}}_{2}$ even at small twist-angle values, enabled by the fact that our approach does not rely on empirical atomistic potentials for interlayer coupling. The results demonstrate the efficiency of the configuration space method by computing relaxations with minimal computational cost. We also outline a general explanation of domain formation in 2D bilayers with nearly aligned lattices, taking advantage of the relationship between real space and configuration space. The configuration space approach also enables calculation of relaxations in incommensurate multilayer systems.

Unwinding the Amplituhedron in Binary

Abstract: We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized “winding number”. Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of “sign flips” of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural “dual” of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar $$ \mathcal{N} $$ = 4 SYM as a differential form on the space of physical kinematical data.

Riemannian Motion Policies.

Abstract: We introduce the Riemannian Motion Policy (RMP), a new mathematical object for modular motion generation. An RMP is a second-order dynamical system (acceleration field or motion policy) coupled with a corresponding Riemannian metric. The motion policy maps positions and velocities to accelerations, while the metric captures the directions in the space important to the policy. We show that RMPs provide a straightforward and convenient method for combining multiple motion policies and transforming such policies from one space (such as the task space) to another (such as the configuration space) in geometrically consistent ways. The operators we derive for these combinations and transformations are provably optimal, have linearity properties making them agnostic to the order of application, and are strongly analogous to the covariant transformations of natural gradients popular in the machine learning literature. The RMP framework enables the fusion of motion policies from different motion generation paradigms, such as dynamical systems, dynamic movement primitives (DMPs), optimal control, operational space control, nonlinear reactive controllers, motion optimization, and model predictive control (MPC), thus unifying these disparate techniques from the literature. RMPs are easy to implement and manipulate, facilitate controller design, simplify handling of joint limits, and clarify a number of open questions regarding the proper fusion of motion generation methods (such as incorporating local reactive policies into long-horizon optimizers). We demonstrate the effectiveness of RMPs on both simulation and real robots, including their ability to naturally and efficiently solve complicated collision avoidance problems previously handled by more complex planners.

Topological kinematics of origami metamaterials

Abstract: A variety of electronic phases in solid-state systems can be understood by abstracting away microscopic details and refocusing on how Fermi surface topology interacts with band structure to define available electron states1. In fact, topological concepts are broadly applicable to non-electronic materials and can be used to understand a variety of seemingly unrelated phenomena2–6. Here, we apply topological principles to origami-inspired mechanical metamaterials7–12, and demonstrate how to guide bulk kinematics by tailoring the crease configuration-space topology. Specifically, we show that by simply changing the crease angles, we modify the configuration-space topology, and drive origami structures to dramatically change their kinematics from being smoothly and continuously deformable to mechanically bistable and rigid. In addition, we examine how a topologically disjointed configuration space can be used to constrain the locally accessible deformations of a single folded sheet. While analyses of origami structures are typically dependent on the energetics of constitutive relations11–14, the topological abstractions introduced here are a separate and independent consideration that we use to analyse, understand and design these metamaterials. Origami-inspired metamaterial design gives rise to structures with kinematic properties dictated by the topology of their configuration space. The approach allows for well-defined metamaterial properties even in the presence of unpredictable forces.

Statistical Damage Mechanics of Polymer Networks

Abstract: The macroscopic mechanical response of polymers can be traced down to the microscale physics of the network by using a statistical approach for the description of the configuration state of the polymer chains. In this paper we present a micromechanical model to capture the macroscopic behavior of polymers by tracking the evolution of a distribution function describing chain configurations, more specifically the statistics of the end-to-end distance on the network chains. Damage, manifested in the softening and hysteresis under cyclic loading, is accounted for through the scission of chains, whose occurrence is evaluated on the basis of the probability of failure, also settled in the configuration space. The proposed micromechanical model can easily accommodate also the mechanics of dynamic network with reversible cross-links, thereby providing a general and physics-based approach to the study of polymers and polymer-like materials.

Reinforced dynamics for enhanced sampling in large atomic and molecular systems

Abstract: A new approach for efficiently exploring the configuration space and computing the free energy of large atomic and molecular systems is proposed, motivated by an analogy with reinforcement learning. There are two major components in this new approach. Like metadynamics, it allows for an efficient exploration of the configuration space by adding an adaptively computed biasing potential to the original dynamics. Like deep reinforcement learning, this biasing potential is trained on the fly using deep neural networks, with data collected judiciously from the exploration and an uncertainty indicator from the neural network model playing the role of the reward function. Parameterization using neural networks makes it feasible to handle cases with a large set of collective variables. This has the potential advantage that selecting precisely the right set of collective variables has now become less critical for capturing the structural transformations of the system. The method is illustrated by studying the full-atom explicit solvent models of alanine dipeptide and tripeptide, as well as the system of a polyalanine-10 molecule with 20 collective variables.

Generalized Voronoi Diagrams for a Ladder: II. Efficient Construction of the Diagram

Abstract: We present a collection of algorithms, all running in timeO(n2 logn α (n)o(α(n)3)) for some fixed integers(where α(n) is the inverse Ackermann's function), for constructing a skeleton representation of a suitably generalized “Voronoi diagram” for a ladder moving in a two-dimensional space bounded by polygonal barriers consisting ofn line segments. This diagram, which is a two-dimensional subcomplex of the dimensional configuration space of the ladder, is introduced and analyzed in a companion paper by the present authors. The construction of the diagram described in this paper yields a motion-planning algorithm for the ladder which runs within the same time bound given above.

CT-MQC – a coupled-trajectory mixed quantum/classical method including nonadiabatic quantum coherence effects

Abstract: Upon photoexcitation by a short light pulse, molecules can reach regions of the configuration space characterized by strong nonadiabaticity, where the motion of the nuclei is strongly coupled to the motion of the electrons. The subtle interplay between the nuclear and electronic degrees of freedom in such situations is rather challenging to capture by state-of-the-art nonadiabatic dynamics approaches, limiting therefore their predictive power. The Exact Factorization of the molecular wavefunction, though, offers new perspectives in the solution of this longstanding issue. Here, we investigate the performance of a mixed quantum/classical (MQC) limit of this theory, named Coupled Trajectory-MQC, which was shown to reproduce the excited-state dynamics of small systems accurately. The method is applied to the study of the photoinduced ring opening of oxirane and the results are compared with two other nonadiabatic approaches based on different Ansatze for the molecular wavefunction, namely Ehrenfest dynamics and Ab Initio Multiple Spawning (AIMS). All simulations were performed using linear-response time-dependent density functional theory. We show that the CT-MQC method can capture the (de)coherence effects resulting from the dynamics through conical intersections, in good agreement with the results obtained with AIMS and in contrast with ensemble Ehrenfest dynamics.

Confined dynamical systems with Carroll and Galilei symmetries

Abstract: We introduce a general method to construct, directly in configuration space, classes of dynamical systems invariant under generalizations of the Carroll and of the Galilei groups. The method does not make use of any nonrelativistic limiting procedure, although the starting point is a Lagrangian Poincar\'e invariant in the full space. It consists in considering a spacetime in $D+1$ dimensions and partitioning it in two parts, the first Minkowskian and the second Euclidean. The action consists of two terms that are separately invariant under the Minkowskian and Euclidean partitioning. One of those contains a system of lagrangian multipliers that confine the system to a subspace. The other term defines the dynamics of the system. The total lagrangian is invariant under the Carroll or the Galilei groups with zero central charge.

Grid-based calculation for perturbation theory of large-scale structure

Abstract: Perturbation theory calculation of large-scale structures has been used to interpret the observed nonlinear statistics of large-scale structures at the quasilinear regime. In particular, the so-called standard perturbation theory (SPT) provides a basis for the analytical computation of the higher-order quantities of large-scale structures. Here, we present a novel grid-based algorithm for the SPT calculation, hence named GridSPT, to generate the higher-order density and velocity fields from a given linear power spectrum. Taking advantage of the fast Fourier transform, the GridSPT quickly generates the nonlinear density fields at each order, from which we calculate the statistical quantities such as nonlinear power spectrum and bispectrum. Comparing the density fields (to fifth order) from GridSPT with those from the full $N$-body simulations in the configuration space, we find that GridSPT accurately reproduces the $N$-body result on large scales. The agreement worsens with smaller smoothing radius, particularly for the underdense regions where we find that the second-order Lagrangian perturbation theory algorithm performs well.

Deep Generative Markov State Models

Abstract: We propose a deep generative Markov State Model (DeepGenMSM) learning framework for inference of metastable dynamical systems and prediction of trajectories. After unsupervised training on time series data, the model contains (i) a probabilistic encoder that maps from high-dimensional configuration space to a small-sized vector indicating the membership to metastable (long-lived) states, (ii) a Markov chain that governs the transitions between metastable states and facilitates analysis of the long-time dynamics, and (iii) a generative part that samples the conditional distribution of configurations in the next time step. The model can be operated in a recursive fashion to generate trajectories to predict the system evolution from a defined starting state and propose new configurations. The DeepGenMSM is demonstrated to provide accurate estimates of the long-time kinetics and generate valid distributions for molecular dynamics (MD) benchmark systems. Remarkably, we show that DeepGenMSMs are able to make long time-steps in molecular configuration space and generate physically realistic structures in regions that were not seen in training data.

Configuration Space of Geometric Objects

Abstract: The concept of configuration space of geometric objects is introduced. Its generalized variables are metric parameters of the spatial form and parameters of the location of objects. The properties of configuration spaces of complex geometric objects are considered. The structures of configuration spaces for various classes of geometric object placement problems, including packing and covering problems, are analyzed. The concept of Φ-function of geometric objects with variable metrical parameters is generalized.

Doubling, T-Duality and Generalized Geometry: a simple model

Abstract: A simple mechanical system, the three-dimensional isotropic rigid rotator, is here investigated as a 0+1 field theory, aiming at further investigating the relation between Generalized/Double Geometry on the one hand and Doubled World-Sheet Formalism/Double Field Theory, on the other hand. The model is defined over the group manifold of SU(2) and a dual model is introduced having the Poisson-Lie dual of SU(2) as configuration space. A generalized action with configuration space SL(2, C), i.e. the Drinfel’d double of the group SU(2), is then defined: it reduces to the original action of the rotator or to its dual, once constraints are implemented. The new action contains twice as many variables as the original. Moreover its geometric structures can be understood in terms of Generalized Geometry.

From Koopman–von Neumann theory to quantum theory

Abstract: Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function $$S^{(K)}(q,p,t)$$ whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrodinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing $$\frac{\partial }{\partial p}$$ by 0 and p by $$\frac{\hbar }{\imath } \frac{\partial }{\partial q}$$ , thus providing an explanation for the common quantization rules.

Fuzzy Greedy RRT Path Planning Algorithm in a Complex Configuration Space

Abstract: A randomized sampling-based path planning algorithm for holonomic mobile robots in complex configuration spaces is proposed in this article. A complex configuration space for path planning algorithms may cause different environmental constraints including the convex/concave obstacles, narrow passages, maze-like spaces and cluttered obstacles. The number of vertices and edges of a search tree for path planning in these configuration spaces would increase through the conventional randomized sampling-based algorithm leading to exacerbation of computational complexity and required runtime. The proposed path planning algorithm is named fuzzy greedy rapidly-exploring random tree (FG-RRT). The FG-RRT is equipped with a fuzzy inference system (FIS) consisting of two inputs, one output and nine rules. The first input is a Euclidean function applied in evaluating the quantity of selected parent vertex. The second input is a metaheuristic function applied in evaluating the quality of selected parent vertex. The output indicates the competency of the selected parent vertex for generating a random offspring vertex. This algorithm controls the tree edges growth direction and density in different places of the configuration space concurrently. The proposed method is implemented on a Single Board Computer (SBC) through the xPC Target to evaluate this algorithm. For this purpose four test-cases are designed with different complexity. The results of the Processor-in-the-Loop (PIL) tests indicate that FG-RRT algorithm reduces the required runtime and computational complexity in comparison with the conventional and greedy RRT through fewer number of vertices in planning an initial path in significant manner.

Efficient Planar Caging Test Using Space Mapping

Abstract: This paper presents an efficient algorithm to test whether a planar object can be caged by a formation of point agents (point fingertips or point mobile robots). The algorithm is based on a space mapping between the 2-D work space ( $\mathcal {W}$ space) and the 3-D configuration space ( $\mathcal {C}$ space) of the given agent formation. When performing caging test on a planar object, the algorithm looks up the space mapping to recover the $\mathcal {C}$ space of the given agent formation, labels the recovered $\mathcal {C}$ space, and counts the number of labeled surfaces to judge the success of caging. The algorithm is able to work with various planar shapes, including objects with convex boundaries, concave boundaries, or holes. It can also respond quickly to varying agent formations and different object shapes. Experiments and analysis on different objects and fingertip formations demonstrate the completeness, robustness, and efficiency of our proposal. Note to Practitioners —This paper proposes an algorithm to solve a geometric problem—find whether a given formation of planar points can constrain (or cage) a planar shape. Users can use the proposed algorithm to actuate a formation of robotic fingertips to perform caging-based grasping tasks or use the proposed algorithm to actuate a formation of mobile robots to perform cooperative transportation tasks. The algorithm inherits the merits of caging and helps users to avoid explicit force analysis. It offers robustness to avoid uncertainty in the tasks. The code of our work is in the supplementary material.

Application of path-integral quantization to indistinguishable particle systems topologically confined by a magnetic field

Abstract: We demonstrate an original development of path-integral quantization in the case of a multiply connected configuration space of indistinguishable charged particles on a 2D manifold and exposed to a strong perpendicular magnetic field. The system occurs to be exceptionally homotopy-rich and the structure of the homotopy essentially depends on the magnetic field strength resulting in multiloop trajectories at specific conditions. We have proved, by a generalization of the Bohr-Sommerfeld quantization rule, that the size of a magnetic field flux quantum grows for multiloop orbits like $(2k+1)\frac{h}{c}$ with the number of loops $k$. Utilizing this property for electrons on the 2D substrate jellium, we have derived upon the path integration a complete FQHE hierarchy in excellent consistence with experiments. The path integral has been next developed to a sum over configurations, displaying various patterns of trajectory homotopies (topological configurations), which in the nonstationary case of quantum kinetics, reproduces some unclear formerly details in the longitudinal resistivity observed in experiments.

Characterization of the NISTmAb Reference Material using small-angle scattering and molecular simulation

Abstract: Both conformational and colloidal stability of therapeutic proteins must be closely monitored and thoroughly characterized to assess the long-term viability of drug products We characterized the IgG1 NISTmAb reference material in its histidine formulation buffer and report our findings on the higher order structure and interactions of NISTmAb under a range of conditions In this paper we present the analysis of experimental small-angle scattering data with atomistic molecular simulations to characterize the monodisperse dilute solution of NISTmAb In part II we describe the characterization of the NISTmAb at high protein concentration (Castellanos et al 2018) The NISTmAb was found to be a flexible protein with a radius of gyration of 490 ± 12 A in histidine formulation buffer using a variety of neutron and X-ray scattering measurements Scattering data were then modeled using molecular simulation After building and validating a starting NISTmAb structure from the Fc and Fab crystallographic coordinates, molecular dynamics and torsion-angle Monte Carlo simulations were performed to explore the configuration space sampled in the NISTmAb and obtain ensembles of structures with atomistic detail that are consistent with the experimental data Our results indicate that the small-angle scattering profiles of the NISTmAb can be modeled using ensembles of flexible structures that explore a wide configuration space The NISTmAb is flexible in solution with no single preferred orientation of Fc and Fab domains, but with some regions of configuration space that are more consistent with measured scattering profiles Analysis of inter-domain atomistic contacts indicated that all ensembles contained configurations where residues between domains are ≤ 4 A, although few contacts were observed for variable and C H 3 regions

A Deep Reinforcement Learning Approach for Dynamically Stable Inverse Kinematics of Humanoid Robots

Abstract: Real time calculation of inverse kinematics (IK) with dynamically stable configuration is of high necessity in humanoid robots as they are highly susceptible to lose balance This paper proposes a methodology to generate joint-space trajectories of stable configurations for solving inverse kinematics using Deep Reinforcement Learning (RL) Our approach is based on the idea of exploring the entire configuration space of the robot and learning the best possible solutions using Deep Deterministic Policy Gradient (DDPG) The proposed strategy was evaluated on the highly articulated upper body of a humanoid model with 27 degree of freedom (DoF) The trained model was able to solve inverse kinematics for the end effectors with 90% accuracy while maintaining the balance in double support phase

Topological origin of phase transitions in the absence of critical points of the energy landscape

Abstract: Different arguments led to surmise that the deep origin of phase transitions has to be identified with suitable topological changes of potential-related submanifolds of configuration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of equipotential energy submanifolds of configuration space. However, it has been recently shown that the 2D lattice φ 4-model provides a counterexample that falsifies the mentioned theorems. On the basis of a numerical investigation, the present work indicates the way to overcome this difficulty: in spite of the absence of critical points of the potential in correspondence of the transition energy, also the phase transition of this model stems from a change of topology of both the energy and potential level sets. But in this case the topology changes are asymptotic (N → ∞). This fact is not obvious since the Z 2 symmetry-breaking transition could be given measure-based explanations in presence of trivial topology. * Electronic address:

Rapidly-Exploring Random Vines (RRV) for Motion Planning in Configuration Spaces with Narrow Passages

Abstract: Classical RRT algorithm is blind to efficiently explore configuration space for expanding the tree through a narrow passage when solving a motion planning (MP) problem. Although there have been several attempts to deal with narrow passages which are based on a wide spectrum of assumptions and configuration setups, we solve this problem in rather general way. We use dominant eigenvectors of the configuration sets formed by properly sampling the space around the nearest node, to efficiently expand the tree around the obstacles and through narrow passages. Unlike classical RRT, our algorithm is aware of having the tree nodes in front of a narrow passage and in a narrow passage, which enables a proper tree expansion in a vine-like manner. A thorough comparison with RRT, RRT-connect, and DDRRT algorithm is provided by solving three different difficult MP problems. The results suggest a significant superiority the proposed Rapidly-exploring Random Vines (RRV) algorithm might have in configuration spaces with narrow passages.

Quotient-Space Motion Planning

Abstract: A motion planning algorithm computes the motion of a robot by computing a path through its configuration space To improve the runtime of motion planning algorithms, we propose to nest robots in each other, creating a nested quotient-space decomposition of the configuration space Based on this decomposition we define a new roadmap-based motion planning algorithm called the Quotient-space roadMap Planner (QMP) The algorithm starts growing a graph on the lowest dimensional quotient space, switches to the next quotient space once a valid path has been found, and keeps updating the graphs on each quotient space simultaneously until a valid path in the configuration space has been found We show that this algorithm is probabilistically complete and outperforms a set of state-of-the-art algorithms implemented in the open motion planning library (OMPL)

Completeness and orthonormality of the Volkov states and the Volkov propagator in configuration space

Abstract: Volkov states and Volkov propagator are the basic analytical tools to investigate QED processes occurring in the presence of an intense plane-wave electromagnetic field In the present paper we provide alternative and relatively simple proofs of the completeness and of the orthonormality at a fixed time of the Volkov states Concerning the completeness, we exploit some known properties of the Green’s function of the Dirac operator in a plane wave, whereas the orthonormality of the Volkov states is proved, relying only on a geometric argument based on the Gauss theorem in four dimensions In relation with the completeness of the Volkov states, we also study some analytical properties of the Green’s function of the Dirac operator in a plane wave, which we explicitly prove to coincide with the Volkov propagator in configuration space In particular, a closed-form expression in terms of modified Bessel functions and Hankel functions is derived by means of the operator technique in a plane wave and different asymptotic forms are determined Finally, the transformation properties of the Volkov propagator under general gauge transformations and a general gauge-invariant expression of the so-called dressed mass in configuration space are presented

Characterization of the NISTmAb Reference Material using small-angle scattering and molecular simulation : Part II: Concentrated protein solutions.

Abstract: Both conformational and colloidal stability of therapeutic proteins must be closely monitored and thoroughly characterized to assess the long-term viability of drug products. We characterized the IgG1 NISTmAb reference material in its histidine formulation buffer and report our findings on the higher order structure and interactions of NISTmAb under a range of conditions. In this paper we present the analysis of experimental small-angle scattering data with atomistic molecular simulations to characterize the monodisperse dilute solution of NISTmAb. In part II we describe the characterization of the NISTmAb at high protein concentration (Castellanos et al. 2018). The NISTmAb was found to be a flexible protein with a radius of gyration of 49.0 ± 1.2 A in histidine formulation buffer using a variety of neutron and X-ray scattering measurements. Scattering data were then modeled using molecular simulation. After building and validating a starting NISTmAb structure from the Fc and Fab crystallographic coordinates, molecular dynamics and torsion-angle Monte Carlo simulations were performed to explore the configuration space sampled in the NISTmAb and obtain ensembles of structures with atomistic detail that are consistent with the experimental data. Our results indicate that the small-angle scattering profiles of the NISTmAb can be modeled using ensembles of flexible structures that explore a wide configuration space. The NISTmAb is flexible in solution with no single preferred orientation of Fc and Fab domains, but with some regions of configuration space that are more consistent with measured scattering profiles. Analysis of inter-domain atomistic contacts indicated that all ensembles contained configurations where residues between domains are ≤ 4 A, although few contacts were observed for variable and C H 3 regions. Graphical Abstract Heavy atom self contact maps of the NISTmAb indicate a highly-flexible structure.

Algorithm 990: Efficient Atlasing and Search of Configuration Spaces of Point-Sets Constrained by Distance Intervals

Abstract: For configurations of point-sets that are pairwise constrained by distance intervals, the EASAL software implements a suite of algorithms that characterize the structure and geometric properties of the configuration space. The algorithms generate, describe, and explore these configuration spaces using generic rigidity properties, classical results for stratification of semi-algebraic sets, and new results for efficient sampling by convex parametrization. The article reviews the key theoretical underpinnings, major algorithms, and their implementation. The article outlines the main applications such as the computation of free energy and kinetics of assembly of supramolecular structures or of clusters in colloidal and soft materials. In addition, the article surveys select experimental results and comparisons.

Application of Clustering Algorithms to Partitioning Configuration Space in Fitting Reactive Potential Energy Surfaces

Abstract: A large number of energy points add great difficulty to construct reactive potential energy surfaces (PES). To alleviate this, exemplar-based clustering is applied to partition the configuration space into several smaller parts. The PES of each part can be constructed easily and the global PES is obtained by connecting all of the PESs of small parts. This divide and conquer strategy is first demonstrated in the fitting of PES for OH3 with Gaussian process regression (GPR) and further applied to construct PESs for CH5 and O+CH4 with artificial neural networks (NN). The accuracy of PESs is tested by fitting errors and direct comparisons with previous PESs in dynamically important regions. As for OH3 and CH5, quantum scattering calculations further validate the global accuracy of newly fitted PESs. The results suggest that partitioning the configuration space by clustering provides a simple and useful method for the construction of PESs for systems that require a large number of energy points.

On a Hamilton‐Jacobi‐Bellman approach for coordinated optimal aircraft trajectories planning

Abstract: In the context of future air trfic management (ATM), an increasing importance is given to environmental considerations and especially fuel consumption. It is thus advisable to make an optimal use of external conditions knowledge, like wind or temperature, to reduce the total fuel needed to complete a flight. On the other hand,safety must be guaranteed all over the trajectory: encounters below the regulatory separation minima, termed as conflicts, must never occur. In this paper, we consider the problem of optimally planning conflict free aircraft trajectories, based on a minimal time criterion and taking into account an ambient wind field. Aircraft motions are restricted to the horizontal plane only in order to reduce the dimension of the problem and to avoid costly changes of flight level. Since global optimality for the whole set of aircraft is sought after, the admissible space is modeled after a Cartesian product of the individual, two-dimensional state spaces with forbidden configurations removed. An Hamilton-Jacobi-Bellman approach in the so-called configuration space obtained that way is then applied to get a coordinated, conflict-free optimal planning. One of the main advantages of this method is the insurance of getting a global optimum without having to rely on an complex initialization procedure. However, as the size of the instances increases with the power of the number of aircraft considered, a dedicated numerical algorithm must be designed. In this work, the Ultra-Bee scheme is adapted to configuration spaces and solved in a four dimensional plus time setting.