Showing papers on "Coordinate system published in 2019"

Data-driven discovery of coordinates and governing equations

Abstract: The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam's razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom deep autoencoder network to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. This method places the discovery of coordinates and models on an equal footing.

Data-driven discovery of coordinates and governing equations

Abstract: The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam's razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom autoencoder to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional dynamical systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. It is the first method of its kind to place the discovery of coordinates and models on an equal footing.

Intrinsic curvature and topology of shadows in Kerr spacetime

Abstract: From the viewpoint of differential geometry and topology, we investigate the characterization of the shadows in a Kerr spacetime. Two new quantities, the length of the shadow boundary and the local curvature radius are introduced. Each shadow can be uniquely determined by these two quantities. For the black hole case, the result shows that we can constrain the black hole spin and the angular coordinate of the observer only by measuring the maximum and minimum of the curvature radius. While for the naked singularity case, we adopt the length parameter and the maximum of the curvature radius. This technique is completely independent of the coordinate system and the location of the shadow, and is expected to uniquely determine the parameters of the spacetime. Moreover, we propose a topological covariant quantity to measure and distinguish different topological structures of the shadows.

Vehicle Speed Prediction Using a Markov Chain With Speed Constraints

Abstract: Modern vehicles are designed to improve fuel consumption while satisfying emissions regulations. As a result, powertrains are becoming increasingly complex and changing rapidly. Optimal control based on the future vehicle speed is one way to address these changes. In this approach, accurate prediction of velocity is closely related to the performance of optimal control. However, there exists uncertainty and complexity in predicting the driver’s behavior, which can be influenced by the surrounding driving environment. In order to overcome such limitations, we propose a prediction algorithm of vehicle speed based on a stochastic model using a Markov chain with speed constraints. The Markov chain, which forms the basis of the proposed algorithm, generates the velocity trajectory stochastically within speed constraints. The constraints are estimated by an empirical model that takes into account the road geometry and is organized by the intuitive form of matrices. To reduce the complexity of the vehicle position and roadway integration, a curvilinear coordinate system is presented using GPS information and a roadway geometry model. Based on these coordinates, the algorithm predicts the future velocity for each cycle from the current vehicle position up to the ahead distance. The proposed algorithm is evaluated through experiments in an urban area, and the test vehicle collects driving data on the roadway position domain. The algorithm was verified in the following conditions: free of surrounding vehicles and traffic lights, the average velocity of the test vehicle is 47.8 km/h, and the maximum velocity of the test vehicle is 66.6 km/h. The experimental results show that a 3.8041 km/h root-mean-square-error is achieved with a prediction horizon of up to 200 m.

Dimensionality, Coordinate System and Reference Frame for Analysis of In-Situ Space Plasma and Field Data

Abstract: In the analysis of in-situ space plasma and field data, an establishment of the coordinate system and the frame of reference, helps us greatly simplify a given problem and provides the framework that enables a clear understanding of physical processes by ordering the experimental data. For example, one of the most important tasks of space data analysis is to compare the data with simulations and theory, which is facilitated by an appropriate choice of coordinate system and reference frame. While in simulations and theoretical work the establishment of the coordinate system (generally based on the dimensionality or dimension number of the field quantities being studied) and the reference frame (normally moving with the structure of interest) is often straightforward, in space data analysis these are not defined a priori, and need to be deduced from an analysis of the data itself. Although various ways of building a dimensionality-based (D-based) coordinate system (i.e., one that takes account of the dimensionality, e.g., 1-D, 2-D, or 3-D, of the observed system/field), and a reference frame moving along with the structure have been used in space plasma data analysis for several decades, in recent years some noteworthy approaches have been proposed. In this paper, we will review the past and recent approaches in space data analysis for the determination of a structure’s dimensionality and the building of D-based coordinate system and a proper moving frame, from which one can directly compare with simulations and theory. Along with the determination of such coordinate systems and proper frame, the variant axis/normal of 1-D (or planar) structures, and the invariant axis of 2-D structures are determined and the proper frame velocity for moving structures is found. These are found either directly or indirectly through the definition of dimensionality. We therefore emphasize that the determination of dimensionality of a structure is crucial for choosing the most appropriate analysis approach, and failure to do so might lead to misinterpretation of the data. Ways of building various kinds of coordinate systems and reference frames are summarized and compared here, to provide a comprehensive understanding of these analysis tools. In addition, the method of building these systems and frames is shown not only to be useful in space data analysis, but also may have the potential ability for simulation/laboratory data analysis and some practical applications.

Extrinsic Calibration between Camera and LiDAR Sensors by Matching Multiple 3D Planes.

Abstract: This paper proposes a simple extrinsic calibration method for a multi-sensor system which consists of six image cameras and a 16-channel 3D LiDAR sensor using a planar chessboard. The six cameras are mounted on a specially designed hexagonal plate to capture omnidirectional images and the LiDAR sensor is mounted on the top of the plates to capture 3D points in 360 degrees. Considering each camera-LiDAR combination as an independent multi-sensor unit, the rotation and translation between the two sensor coordinates are calibrated. The 2D chessboard corners in the camera image are reprojected into 3D space to fit to a 3D plane with respect to the camera coordinate system. The corresponding 3D point data that scan the chessboard are used to fit to another 3D plane with respect to the LiDAR coordinate system. The rotation matrix is calculated by aligning normal vectors of the corresponding planes. In addition, an arbitrary point on the 3D camera plane is projected to a 3D point on the LiDAR plane, and the distance between the two points are iteratively minimized to estimate the translation matrix. At least three or more planes are used to find accurate external parameters between the coordinate systems. Finally, the estimated transformation is refined using the distance between all chessboard 3D points and the LiDAR plane. In the experiments, quantitative error analysis is done using a simulation tool and real test sequences are also used for calibration consistency analysis.

An Adaptive Framework to Tune the Coordinate Systems in Nature-Inspired Optimization Algorithms

Abstract: The performance of many nature-inspired optimization algorithms (NIOAs) depends strongly on their implemented coordinate system. However, the commonly used coordinate system is fixed and not well suited for different function landscapes, NIOAs thus might not search efficiently. To overcome this shortcoming, in this paper we propose a framework, named ACoS, to adaptively tune the coordinate systems in NIOAs. In ACoS, an Eigen coordinate system is established by making use of the cumulative population distribution information, which can be obtained based on a covariance matrix adaptation strategy and an additional archiving mechanism. Since the population distribution information can reflect the features of the function landscape to some extent, NIOAs in the Eigen coordinate system have the capability to identify the modality of the function landscape. In addition, the Eigen coordinate system is coupled with the original coordinate system, and they are selected according to a probability vector. The probability vector aims to determine the selection ratio of each coordinate system for each individual, and is adaptively updated based on the collected information from the offspring. ACoS has been applied to two of the most popular paradigms of NIOAs, i.e., particle swarm optimization and differential evolution, for solving 30 test functions with 30D and 50D at the 2014 IEEE Congress on Evolutionary Computation. The experimental studies demonstrate its effectiveness.

Further results on dynamic-algebraic Boolean control networks

Abstract: Restricted coordinate transformation, controllability, observability and topological structures of dynamic-algebraic Boolean control networks are investigated under an assumption. Specifically, given the input-state at some point, assume that the subsequent state is certain or does not exist. First, the system can be expressed in a new form after numbering the elements in admissible set. Then, restricted coordinate transformation is raised, which allows the dimension of new coordinate frame to be different from that of the original one. The system after restricted coordinate transformation is derived in the proposed form. Afterwards, three types of incidence matrices are constructed and the results of controllability, observability and topological structures are obtained. Finally, two practical examples are shown to demonstrate the theory in this paper.

Isostable reduction of oscillators with piecewise smooth dynamics and complex Floquet multipliers.

Abstract: Phase-amplitude reduction is a widely applied technique in the study of limit cycle oscillators with the ability to represent a complicated and high-dimensional dynamical system in a more analytically tractable set of coordinates. Recent work has focused on the use of isostable coordinates, which characterize the transient decay of solutions toward a periodic orbit, and can ultimately be used to increase the accuracy of these reduced models. The breadth of systems to which this phase-amplitude reduction strategy can be applied, however, is still rather limited. In this work, the theory of phase-amplitude reduction using isostable coordinates is further developed to accommodate a broader set of dynamical systems. In the first part, limit cycles of piecewise smooth dynamical systems are considered and strategies are developed to compute the associated reduced equations. In the second part, the notion of isostable coordinates for complex-valued Floquet multipliers is introduced, resulting in one phaselike coordinate and one amplitudelike coordinate for each pair of complex conjugate Floquet multipliers. Examples are given with relevance to piecewise smooth representations of excitable cardiomyocytes and the relationship between the reduced coordinate system and the emergence of cardiac alternans is discussed. Also, phase-amplitude reduction is implemented for a chaotic, externally forced pendulum with complex Floquet multipliers and a resulting control strategy for the stabilization of its periodic solution is investigated.

Kinematic and Dynamic Vehicle Model-Assisted Global Positioning Method for Autonomous Vehicles with Low-Cost GPS/Camera/In-Vehicle Sensors.

Abstract: Real-time, precise and low-cost vehicular positioning systems associated with global continuous coordinates are needed for path planning and motion control in autonomous vehicles. However, existing positioning systems do not perform well in urban canyons, tunnels and indoor parking lots. To address this issue, this paper proposes a multi-sensor positioning system that combines a global positioning system (GPS), a camera and in-vehicle sensors assisted by kinematic and dynamic vehicle models. First, the system eliminates image blurring and removes false feature correspondences to ensure the local accuracy and stability of the visual simultaneous localisation and mapping (SLAM) algorithm. Next, the global GPS coordinates are transferred to a local coordinate system that is consistent with the visual SLAM process, and the GPS and visual SLAM tracks are calibrated with the improved weighted iterative closest point and least absolute deviation methods. Finally, an inverse coordinate system conversion is conducted to obtain the position in the global coordinate system. To improve the positioning accuracy, information from the in-vehicle sensors is fused with the interacting multiple-model extended Kalman filter based on kinematic and dynamic vehicle models. The developed algorithm was verified via intensive simulations and evaluated through experiments using KITTI benchmarks (A project of Karlsruhe Institute of Technology and Toyota Technological Institute at Chicago) and data captured using our autonomous vehicle platform. The results show that the proposed positioning system improves the accuracy and reliability of positioning in environments in which the Global Navigation Satellite System is not available. The developed system is suitable for the positioning and navigation of autonomous vehicles.

High-Precision Speed and Position Estimation Based on Hall Vector Frequency Tracking for PMSM With Bipolar Hall-Effect Sensors

Abstract: Because of the low cost and the convenient installation, bipolar Hall-effect sensors are commonly applied to permanent magnet synchronous motors. Generally, speed and position are estimated inaccurately due to the installation error of the sensors. This paper addresses two key challenges on improving the estimation precision: 1) proposing the coordinate transformation to get the Hall rotary vector and 2) designing the Hall vector frequency tracking method to estimate the speed and position. First, the Hall vector which is related to the position is obtained by the coordinate transformation. Second, the synchronous frequency tracking filter (SFTF) is used to extract the fundamental waves which are sine function and cosine function of the position from the two orthogonal components of the Hall vector. The SFTF can eliminate the high-frequency interferences. Third, the arctan function is used to calculate the speed and position with extracted fundamental waves. The proposed method can improve estimation precision and reduce estimated error related to the inaccurate installation of Hall-effect sensors, which is parameter-independent and implemented easily. The effectiveness was verified by the experimental results on a turbo molecular pump.

Deep Learning Models for Global Coordinate Transformations that Linearize PDEs

Abstract: We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a nonlinear PDE into a linear PDE. Our architecture is motivated by the linearizing transformations provided by the Cole-Hopf transform for Burgers equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix $\mathbf{K}$. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix $\mathbf{K}$ in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers equation, as well as the substantially more challenging Kuramoto-Sivashinsky equation, showing that our method provides a robust architecture for discovering interpretable, linearizing transforms for nonlinear PDEs.

Two-Dimensional Fourier-Based Modeling of Electric Machines—An Overview

Abstract: An increasing need for fast and reliable models has led to a continuous development of Fourier-based (FB) analytical modeling. This paper presents an overview of the techniques that are currently available in FB modeling for electric machines. By coupling that overview to the most relevant literature related to the subject, an interesting starting point is provided for anyone who wants to use or improve FB models. The following seven aspects of FB models are discussed in detail: 1) the magnetic potential (scalar or vector potential); 2) the coordinate system and the solution of the partial-differential equations for each magnetic potential and for each coordinate system; 3) the way in which time dependence is accounted for; 4) the implementation of the source terms; 5) the possibilities to account for slotted structures; 6) the modeling of eccentricity; and 7) the post-processing computation of physical quantities, such as flux density, electromotive force, torque, losses, and eddy currents in conductive objects. Furthermore, this paper gives the closed form solution of the Laplace, Poisson, and Helmholtz equations in each coordinate system. In addition, this paper tackles other important features of FB models such as computational time reduction and coupling the machine model to an electric circuit.

Explicit expressions for Rortex tensor and velocity gradient tensor decomposition

Abstract: The introduction of Rortex provides a new perspective to investigate the local properties of vortical structures in transitional and turbulent flows, as Rortex offers a new and systematic description of the local fluid rotation, including scalar, vector and tensor forms. Unfortunately, the previous definition of Rortex is not straightforward, which requires the explicit calculation of somewhat cumbersome coordinate rotation. In this letter, a new explicit tensor form of Rortex and the relevant explicit velocity gradient tensor decomposition are presented, based on an explicit formula of the Rortex vector. The explicit tensor form represents the real local rotational part of the velocity gradient tensor in the original coordinate system. The explicit calculation of coordinate rotations can be totally avoided, which indicates an important improvement of Rortex based velocity gradient tensor decomposition.The introduction of Rortex provides a new perspective to investigate the local properties of vortical structures in transitional and turbulent flows, as Rortex offers a new and systematic description of the local fluid rotation, including scalar, vector and tensor forms. Unfortunately, the previous definition of Rortex is not straightforward, which requires the explicit calculation of somewhat cumbersome coordinate rotation. In this letter, a new explicit tensor form of Rortex and the relevant explicit velocity gradient tensor decomposition are presented, based on an explicit formula of the Rortex vector. The explicit tensor form represents the real local rotational part of the velocity gradient tensor in the original coordinate system. The explicit calculation of coordinate rotations can be totally avoided, which indicates an important improvement of Rortex based velocity gradient tensor decomposition.

UprightNet: Geometry-Aware Camera Orientation Estimation From Single Images

Abstract: We introduce UprightNet, a learning-based approach for estimating 2DoF camera orientation from a single RGB image of an indoor scene. Unlike recent methods that leverage deep learning to perform black-box regression from image to orientation parameters, we propose an end-to-end framework that incorporates explicit geometric reasoning. In particular, we design a network that predicts two representations of scene geometry, in both the local camera and global reference coordinate systems, and solves for the camera orientation as the rotation that best aligns these two predictions via a differentiable least squares module. This network can be trained end-to-end, and can be supervised with both ground truth camera poses and intermediate representations of surface geometry. We evaluate UprightNet on the single-image camera orientation task on synthetic and real datasets, and show significant improvements over prior state-of-the-art approaches.

Calibration method for panoramic 3D shape measurement with plane mirrors

Abstract: High-speed panoramic three-dimensional (3D) shape measurement can be achieved by introducing plane mirrors into the traditional fringe projection profilometry (FPP) system because such a system simultaneously captures fringe patterns from three different perspectives (i.e., by a real camera and two virtual cameras in the plane mirrors). However, calibrating such a system is nontrivial due to the complicated setup. This work introduces a flexible new technique to calibrate such a system. We first present the mathematical representation of the plane mirror, and then mathematically prove that it only requires the camera to observe a set of feature point pairs (including real points and virtual points) to generate a solution to the reflection matrix of a plane mirror. By calibrating the virtual and real camera in the same world coordinate system, 3D point cloud data obtained from real and virtual perspectives can be automatically aligned to generate a panoramic 3D model of the object. Finally, we developed a system to verify the performance of the proposed calibration technique for panoramic 3D shape measurement.

Determine turntable coordinate system considering its non-orthogonality.

Abstract: In industrial measurement and laboratory research, many measured objects are placed on the three-axis turntable. In this paper, we propose a fast and practical method to determine the turntable coordinate system with the aid of spatial coordinate of point cloud data. By sphere fitting, plane fitting, and point projection, the scattered point cloud data are combined together to obtain initial direction vectors. Considering the non-orthogonality of turntable, the least two pairs of skew lines are used to compute the approximate turntable center. And the intersection angles and distances between each axis are given to judge the degree of non-orthogonality. Then, based on the approximate sphere center and the initial rotation vectors, the direction vectors are, respectively, optimized in a predefined order. An experimental system is set up to validate the proposed method. Attitude parameters computed by spatial point coordinates before and after turntable calibration are employed to give the quantitative evaluation results. And the total attitude errors in three axes motion forms all reduce, which demonstrates the effectiveness and practicality of the proposed method.

Flat Connection for Rotating Vacuum Spacetimes in Extended Teleparallel Gravity Theories

Abstract: Teleparallel geometry utilizes Weitzenbock connection which has nontrivial torsion but no curvature and does not directly follow from the metric like Levi–Civita connection. In extended teleparallel theories, for instance in f ( T ) or scalar-torsion gravity, the connection must obey its antisymmetric field equations. Thus far, only a few analytic solutions were known. In this note, we solve the f ( T , ϕ ) gravity antisymmetric vacuum field equations for a generic rotating tetrad ansatz in Weyl canonical coordinates, and find the corresponding spin connection coefficients. By a coordinate transformation, we present the solution also in Boyer–Lindquist coordinates, often used to study rotating solutions in general relativity. The result hints for the existence of another branch of rotating solutions besides the Kerr family in extended teleparallel gravities.

Modified couple stress theory in orthogonal curvilinear coordinates

Abstract: The formulations for the modified couple stress theory (MCST) are consistently derived in general orthogonal curvilinear coordinate systems. In particular, the expressions for the rotation vector, higher-order strain, and stress tensors, i.e., the rotation gradient tensor and the deviatoric part of the symmetric couple stress tensor, and the classical strain and stress tensors are derived for an arbitrary orthogonal curvilinear coordinate system. Additionally, using the theory of surfaces, the formulations for the MCST are derived for general doubly curved coordinates, which are more convenient to use for shells of arbitrary curvature. The expressions for special cases, i.e., cylindrical and spherical shells, are obtained. The MCST expressions derived in this study are comprehensive and generally and can be used for consistent utilisation of the MCST in any orthogonal curvilinear coordinate system.

An interaction-asymptotic region decomposition method for general state-to-state reactive scatterings.

Abstract: A single set of coordinates, which is optimal for both asymptotic product and reactant, is difficult to find in a state-to-state reactive scattering calculation using the quantum wave packet method. An interaction-asymptotic region decomposition (IARD) method was proposed in this work to solve this "coordinate problem." In the method, the interaction region and asymptotic regions are applied with the local optimal coordinate system, i.e., hyperspherical and corresponding Jacobi coordinates. The IARD method is capable of efficiently and accurately accomplishing a calculation with a grid box for the Jacobi coordinate R extending several hundred bohrs for both reactant and product arrangements. We demonstrate the effectiveness of the IARD method with the reaction of H + HD, which is the simplest direct reaction, and F + HD, which is a typical reaction involving resonances with products of extremely slow translational energy and requires extremely long absorbing potential in all channels.

3D mapping of discontinuity traces using fusion of point cloud and image data

Abstract: A new methodology is presented for 3-D automated mapping of joints that are exposed primarily as traces in a rock face as opposed to planar facets. The method identifies 3-D points in a photogrammetry or a LiDAR derived point cloud that corresponds to the traces of the joints as observed in image data. First, the 2-D trace texture is extracted from image data using a hybrid global and local threshold method and integrating a series of image-processing algorithms. Second, data matching links the pixel locations corresponding to the identified traces in an image to the 3-D coordinates in the point cloud. This matching is accomplished by a coordinate transformation between the image coordinates and point cloud coordinates. Finally, a 3-D discontinuity trace map is acquired by analysing the 3-D spatial features of the traces. A case study of a rock slope along a highway is presented using the proposed method. The results demonstrate that the fusion of image data and point cloud data improves the mapping of discontinuities that primarily appear as traces in outcrops versus that achieved by existing methods that rely only on point cloud data.

Numerical integration of the contravariant integral form of the Navier–Stokes equations in time-dependent curvilinear coordinate systems for three-dimensional free surface flows

Abstract: We propose a three-dimensional non-hydrostatic shock-capturing numerical model for the simulation of wave propagation, transformation and breaking, which is based on an original integral formulation of the contravariant Navier–Stokes equations, devoid of Christoffel symbols, in general time-dependent curvilinear coordinates. A coordinate transformation maps the time-varying irregular physical domain that reproduces the complex geometries of coastal regions to a fixed uniform computational one. The advancing of the solution is performed by a second-order accurate strong stability preserving Runge–Kutta fractional-step method in which, at every stage of the method, a predictor velocity field is obtained by the shock-capturing scheme and a corrector velocity field is added to the previous one, to produce a non-hydrostatic divergence-free velocity field and update the water depth. The corrector velocity field is obtained by numerically solving a Poisson equation, expressed in integral contravariant form, by a multigrid technique which uses a four-colour Zebra Gauss–Seidel line-by-line method as smoother. Several test cases are used to verify the dispersion and shock-capturing properties of the proposed model in time-dependent curvilinear grids.