Coordinate-free Representation Research Articles

Inverse scattering problem for the tensor fields in a coordinate-free representation

The inverse scattering problem is to reconstruct or recover some physical and/or geometric properties of an object from the measured scattered field in the process of illumination by an incident wave. The probing radiation can be an electromagnetic wave (e.g., microwave, optical wave, and X-ray), an acoustic wave, or some other waves. The inverse scattering problem is important when detailed information about the structure and composition of an object is required, but it cannot be obtained directly from the measurements. The paper is devoted to the electromagnetic inverse scattering problem for a dielectric anisotropic and magnetically isotropic media. The properties of an anisotropic medium with respect to electromagnetic waves are defined by the tensors, which give the relation between the inductions and the fields. The tensor Fourier diffraction theorem derived in the paper can be considered as a useful tool for studying tensor fields in inverse problems of electromagnetic scattering. The Fourier diffraction theorem is obtained in a coordinate-free representation by the use of tensor (dyadic) analysis techniques. The method of inversion is based on the first Born approximation.

Inverse scattering problem for the tensor fields in a coordinate-free representation

The paper is devoted to the inverse scattering problem in order to study the complex media whose electrical and magnetic properties are the tensor functions. The tensor Fourier diffraction theorem, derived in the paper, gives the relations connecting the scattered field and the inherent properties of the scattering medium. The method is based on the first Born approximation. The tensor (dyadic) analysis for derivation of the tensor Fourier diffraction theorem in the coordinate-free form is used.

Parameterizations of immiscible two-phase flow in porous media

A fundamental variable characterizing immiscible two-phase flow in porous media is the wetting saturation, which is the ratio between the pore volume filled with wetting fluid and the total pore volume. More generally, this variable comes from a specific choice of coordinates on some underlying space, the domain of variables that can be used to express the volumetric flow rate. The underlying mathematical structure allows for the introduction of other variables containing the same information, but which are more convenient from a theoretical point of view. We introduce along these lines polar coordinates on this underlying space, where the angle plays a role similar to the wetting saturation. We derive relations between these new variables based on the Euler homogeneity theorem. We formulate these relations in a coordinate-free fashion using differential forms. Finally, we discuss and interpret the co-moving velocity in terms of this coordinate-free representation.

Geometric Integral Attitude Control on SO(3)

This article proposes a novel integral geometric control attitude tracking scheme, utilizing a coordinate-free representation of attitude on the Lie group of rigid body rotations, SO(3). This scheme exhibits almost global asymptotic stability in tracking a reference attitude profile. The stability and robustness properties of this integral tracking control scheme are shown using Lyapunov stability analysis. A numerical simulation study, utilizing a Lie Group Variational Integrator (LGVI), verifies the stability of this tracking control scheme, as well as its robustness to a disturbance torque. In addition, a numerical comparison study shows the effectiveness of the proposed geometric integral term, when compared to other state-of-the-art attitude controllers. In addition, software-in-the-loop (SITL) simulations show the advantages of utilizing the proposed attitude controller in PX4 autopilot compared to using PX4’s original attitude controller.

Coordinate-Free Jacobian Motion Planning: A 3-D Space Robot

We address the motion planning problem for a robotic system whose configuration manifold contains a group of rotations. Our approach is applied to a free-floating space robot composed of a three-dimensional base (a spacecraft) and an anthropomorphic onboard manipulator. The robot is actuated by the torques exerted at the joints of the onboard manipulator. A coordinate-free representation of rotations is utilized. The Lagrangian formalism is employed in order to derive a dynamics model of the robot that takes the form of a control system defined on the group of rotations and the joint space of the onboard manipulator. Using the conservation of angular momentum of the robot, a Jacobian motion planning algorithm is designed relying on the Endogenous Configuration Space Approach. The performance of the algorithm is verified by computer simulations.

Coordinate-Invariant Modeling and Control of a Three-DOF Robot Manipulator

Abstract This article presents coordinate-free modeling and control law synthesis for a 3-DOF, revolute-joint robotic manipulator. Most conventional robot analysis and design methods still operate in a parametric framework, which makes all results in a sense local. In contrast to this, we introduce a coordinate-free representation of the robot’s kinematics on S1 × S1 × S1, which allows to express the continuous-time dynamics in a global, structured form. In parallel to the continuous case, a discrete-time model based on the discrete mechanics approach and the approximation of the Lagrangian using the implicit midpoint rule is introduced in the form of a variational integrator. On the basis of the derived models, we cover geometric potential shaping and PD control in continuous and discrete time, where remarkably, the discrete control laws share the structure of their continuous-time counterparts. Numerical experiments illustrate the benefits of the discrete-time geometric approach for simulation and control.

Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover’s distance computation, for efficient and robust computation.

Princeton_TIGRESS 2.0: High refinement consistency and net gains through support vector machines and molecular dynamics in double-blind predictions during the CASP11 experiment.

Protein structure refinement is the challenging problem of operating on any protein structure prediction to improve its accuracy with respect to the native structure in a blind fashion. Although many approaches have been developed and tested during the last four CASP experiments, a majority of the methods continue to degrade models rather than improve them. Princeton_TIGRESS (Khoury et al., Proteins 2014;82:794-814) was developed previously and utilizes separate sampling and selection stages involving Monte Carlo and molecular dynamics simulations and classification using an SVM predictor. The initial implementation was shown to consistently refine protein structures 76% of the time in our own internal benchmarking on CASP 7-10 targets. In this work, we improved the sampling and selection stages and tested the method in blind predictions during CASP11. We added a decomposition of physics-based and hybrid energy functions, as well as a coordinate-free representation of the protein structure through distance-binning Cα-Cα distances to capture fine-grained movements. We performed parameter estimation to optimize the adjustable SVM parameters to maximize precision while balancing sensitivity and specificity across all cross-validated data sets, finding enrichment in our ability to select models from the populations of similar decoys generated for targets in CASPs 7-10. The MD stage was enhanced such that larger structures could be further refined. Among refinement methods that are currently implemented as web-servers, Princeton_TIGRESS 2.0 demonstrated the most consistent and most substantial net refinement in blind predictions during CASP11. The enhanced refinement protocol Princeton_TIGRESS 2.0 is freely available as a web server at http://atlas.engr.tamu.edu/refinement/. Proteins 2017; 85:1078-1098. © 2017 Wiley Periodicals, Inc.

Remarks on isotropic extension of anisotropic constitutive functions via structural tensors

In their original formulation of the method of isotropic extension via structural tensors, which is meant for applications to the derivation of coordinate-free representation formulas for anisotropic constitutive functions, both Boehler and Liu start with the assumption that the invariance group of structural tensors is the symmetry group that defines the anisotropy of the constitutive function in question. As a result, the method (with structural tensors of order not higher than two) is applicable only when the anisotropy is characterized by a cylindrical group or belongs to the triclinic, monoclinic, or rhombic crystal classes. In this note we present a reformulation of the method in which the aforementioned assumption of Boehler and of Liu is relaxed, and we show by examples in finite elasticity and anisotropic linear elasticity that the method of isotropic extension via structural tensors could be applicable beyond the original limitations.

Chemical Space Visualization: Transforming Multidimensional Chemical Spaces into Similarity-Based Molecular Networks

The concept of chemical space is of fundamental relevance for medicinal chemistry and chemical informatics. Multidimensional chemical space representations are coordinate-based. Chemical space networks (CSNs) have been introduced as a coordinate-free representation. A computational approach is presented for the transformation of multidimensional chemical space into CSNs. The design of transformation CSNs (TRANS-CSNs) is based upon a similarity function that directly reflects distance relationships in original multidimensional space. TRANS-CSNs provide an immediate visualization of coordinate-based chemical space and do not require the use of dimensionality reduction techniques. At low network density, TRANS-CSNs are readily interpretable and make it possible to evaluate structure-activity relationship information originating from multidimensional chemical space.

Design of chemical space networks on the basis of Tversky similarity.

Chemical space networks (CSNs) have been introduced as a coordinate-free representation of chemical space. In CSNs, nodes represent compounds and edges pairwise similarity relationships. These network representations are mostly used to navigate sections of biologically relevant chemical space. Different types of CSNs have been designed on the basis of alternative similarity measures including continuous numerical similarity values or substructure-based similarity criteria. CSNs can be characterized and compared on the basis of statistical concepts from network science. Herein, a new CSN design is introduced that is based upon asymmetric similarity assessment using the Tversky coefficient and termed TV-CSN. Compared to other CSNs, TV-CSNs have unique features. While CSNs typically contain separate compound communities and exhibit small world character, many TV-CSNs are also scale-free in nature and contain hubs, i.e., extensively connected central compounds. Compared to other CSNs, these hubs are a characteristic of TV-CSN topology. Hub-containing compound communities are of particular interest for the exploration of structure-activity relationships.

Almost global finite-time stabilization of rigid body attitude dynamics using rotation matrices

Summary This work considers continuous finite-time stabilization of rigid body attitude dynamics using a coordinate-free representation of attitude on the Lie group of rigid body rotations in three dimensions, SO(3). Using a Holder continuous Morse–Lyapunov function, a finite-time feedback stabilization scheme for rigid body attitude motion to a desired attitude with continuous state feedback is obtained. Attitude feedback control with finite-time convergence has been considered in the past using the unit quaternion representation. However, it is known that the unit quaternion representation of attitude is ambiguous, with two antipodal unit quaternions representing a single rigid body attitude. Continuous feedback control using unit quaternions may therefore lead to the unstable unwinding phenomenon if this ambiguity is not resolved in the control design, and this has adverse effects on actuators, settling time, and control effort expended. The feedback control law designed here leads to almost global finite-time stabilization of the attitude motion of a rigid body with Holder continuous feedback to the desired attitude. As a result, this control scheme avoids chattering in the presence of measurement noise, does not excite unmodeled high-frequency structural dynamics, and can be implemented with actuators that can only provide continuous control inputs. Numerical simulation results for a spacecraft in low Earth orbit, obtained using a Lie group variational integrator, confirm the theoretically obtained stability and robustness properties of this attitude feedback stabilization scheme. Copyright © 2015 John Wiley & Sons, Ltd.

Performance-Oriented Design of Inverse Kinematics Algorithms: Extended Jacobian Approximation of the Jacobian Pseudo-Inverse

For redundant robotic manipulators, we study the design problem of Jacobian inverse kinematics algorithms of desired performance. A specific instance of the problem is addressed, namely the optimal approximation of the Jacobian pseudo-inverse algorithm by the extended Jacobian algorithm. The approximation error functional is derived for the coordinate-free representation of the manipulator’s kinematics. A variational formulation of the problem is employed, and the approximation error is minimized by means of the Ritz method. The optimal extended Jacobian algorithm is designed for the 7 degrees of freedom (dof) POLYCRANK manipulator. It is concluded that the coordinate-free kinematics representation results in more accurate approximation than the coordinate expression of the kinematics.

Berezin integration on non-compact supermanifolds

We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free representation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes’s theorem is derived—here, the boundary integral contains transversal derivatives of arbitrarily high order.

Covariant formulation of the governing equations of continuum mechanics in an Eulerian description

We present the balance relations for a continuum in the Eulerian formulation in a pure covariant fashion. Based on the analysis of nonrelativistic particle mechanics, we adapt the covariant description to the case of a continuum. The use of the covariant Nijenhuis differential as well as the splitting of the vertical configuration bundle are the key objects that allow a coordinate-free representation. We state the balance equations such that they are valid, also when time variant transformations are applied, which leads to a nontrivial space-time connection and a metric which explicitly depends on the time.

Radiation and low-frequency scattering of EM waves in a general anisotropic homogeneous medium

This paper focuses on a boundless homogeneous medium with general anisotropy of electromagnetic properties. The explicit exact form of the four spectral Green's dyads G/spl circ//sub /spl nu//spl xi//(k) is obtained and a coordinate-free representation of the four spatial Green's functions G/spl circ//sub /spl nu//spl xi//(x-x') in terms of one scalar potential W(x-x') is developed. On this basis, asymptotic expressions for the radiation field due to arbitrary sources are derived that show that the associated modes and the eigenmodes determine radiation along a singular optic axis and in all other directions, respectively. Using an integral equation approach and the theory of Newtonian potential, the problem of low-frequency scattering by a small anisotropic ellipsoidal body immersed into an anisotropic medium is solved analytically.

Elastic waves in homogeneous and layered transversely isotropic media: Plane waves and Gaussian wave packets. A general approach

Solutions to the equation of motion are derived for transversely isotropic media such as fiber composites, ideally fiber-textured austenitic steels, or extruded metal-matrix composites. The approach is most general in that the orientation of the materials’ axis of rotational symmetry is arbitrary. Thus the results obtained using a coordinate-free representation are particularly convenient in view of layered structures, where for the materials of interest the fiber axis is perpendicular to the surface normal, but variable in orientation. Plane elastic waves are characterized by the corresponding wave vectors, making especially possible a quantitative evaluation of the deviation of wave propagation direction and energy flux, which is characteristic for anisotropic materials. Reflection and refraction of plane waves at an interface between two arbitrarily oriented transversely isotropic media is examined yielding an algorithm that provides the respective reflection and transmission coefficients. The propagation of elastic waves of finite spatial and temporal extent is modeled using the concept of Gaussian wave packets. The relations given in the literature for general anisotropic media are specialized to the homogeneous and layered transversely isotropic cases. Numerical evaluation of the analytical results is included.

A COORDINATE‐FREE APPROACH TO THE ANALYSIS OF GROWTH PATTERNS: MODELS AND THEORETICAL CONSIDERATIONS

Developmental biology holds keys to our understanding of morphological pattern formation whether these patterns are expressed in the fossil record or among extant species. Though much is known about osseous growth at the cellular level (e.g. Hall, 1991), we have minimal understanding of the coordinated processes that combine to produce a complex, three-dimensional form. We have proposed a framework for the coordinate-free representation of form, a statistical method for comparing and modelling growth trajectories for complex morphologies, and a means for the eventual elucidation of the role of growth in the evolution of morphology. Our method uses the coordinate locations of biological landmarks to represent form as a matrix of all possible linear distances between landmarks, the form matrix. When two forms are expressed in this way, comparison of these forms is accomplished by computing the ratios of like linear distances, the form difference matrix. When the forms being compared are from a growth series, the matrix of ratios is called a growth matrix. Patterns of growth for two groups can be compared by computing the growth difference matrix. We applied growth difference matrix analysis to the study of sexual dimorphism of ontogeny in the M. fascicularis craniofacial skeleton. Growth matrices describing growth in male and female M. fascicularis were presented along with the growth difference matrix that describes sexual dimorphism of growth to underscore the detailed information available from this analytical technique. The method is quite general and can be applied to two- or three-dimensional data sets of landmark coordinates (cross-sectional or longitudinal) collected from almost any developing structure. The methods that we propose enable us to go beyond a mathematical summary of the comparison of forms and the comparison of growth patterns. We provide examples of how growth patterns might be used in the study of phylogenetic relationships. Our plans for use of this method in the study of evolutionary change assumes that morphological change in the craniofacial skeleton results from evolutionary change in developmental units (as defined by Atchley & Hall, 1991) that underlie morphological structure. We believe we have the basic tools to ultimately propose informed phylogenies based solely on developmental data. This task requires the identification of 'growth features' and the polarization of these features as primitive or derived. It is also advisable to determine a set of primitive growth features for the groups of interest. This will necessitate the inclusion of outgroups in our growth analysis.(ABSTRACT TRUNCATED AT 400 WORDS)

Fourier transform of a linear distribution with triangular support and its applications in electromagnetics

A three-dimensional Fourier transform (FT) of a linear function with triangular support is derived in its coordinate-free representation. The Fourier transform of this distribution is derived in three steps. First, the 2-D FT of a constant (top hat) function is obtained. Next, the distribution is generalized to a linearly varying function. Finally, the formulation is extended to a coordinate-free representation which is the 3-D FT of the 2-D function defined over a surface. This formulation is applied to the near-field computation, yielding accurate numerical solutions. >

Coordinate-free operators based on one vector. I. Formal considerations

In many systems, the tensors used to describe physical properties must acquire their structure from one vector. Knowledge of that fact alone leads to an interesting line of analysis for such systems. The analysis begins with a discussion of the types of dyadics that can be constructed from one vector. Attention is focused on certain exemplary dyadic operators, which, because of their geometrical properties, would appear particularly basic; the algebra of these dyadics is developed in detail. The algebra is then used in a derivation of the expansion of general functions of these dyadics. Several applications of the general expansion are presented: the inverse of a general dyadic operator (when it exists), a general rotation operator that includes strains and reflections, the determinant of a general dyadic operator, and operators, including some new ones, used in constructing Lorentz boosts. Finally, a coordinate-free representation of the Levi-Civita tensor is provided.