Local Coordinate Frame Research Articles

Noise and control decoupling of Advanced LIGO suspensions

Ground-based interferometric gravitational wave observatories such as Advanced LIGO must isolate their optics from ground vibrations with suspension systems to meet their stringent noise requirements. These suspensions typically have very high quality-factor resonances that require active damping. The sensor noise associated with this damping is a potential significant contributor to the sensitivity of these interferometers. This paper introduces a novel scheme for suspension damping that isolates much of this noise and permits greater amounts of damping. It also decouples the damping feedback design from the interferometer control. The scheme works by invoking a change from a local coordinate frame associated with each suspension, to a coordinate frame aligned with the interferometric readout. In this way, degrees of freedom invisible to the readout can employ effective, but noisy damping. The degree of freedom measured by the readout is then damped using low noise interferometer signals, eliminating the need to use the usual noisy sensors. Simulated and experimental results validate the concepts presented in this paper.

Linear Approach for Initial Recovery of the Exterior Orientation Parameters of Randomly Captured Images by Low-Cost Mobile Mapping Systems

Abstract. In this paper, we present a novel linear approach for the initial recovery of the exterior orientation parameters (EOPs) of images. Similar to the conventional Structure from Motion (SfM) algorithm, the proposed approach is based on a two-step strategy. In the first step, the relative orientation of all possible image stereo-pairs is estimated. In the second step, a local coordinate frame is established, and an incremental image augmentation process is implemented to reference all the remaining images into a local coordinate frame. Since our approach is based on a linear solution for both the relative orientation estimation as well as the initial recovery of the image EOPs, it does not require any initial approximation for the optimization process. Another advantage of our approach is that it does not require any prior knowledge regarding the sequence of the image collection procedure, therefore, it can handle a set of randomly collected images in the absence of GNSS/INS information. In order to illustrate the feasibility of our approach, several experimental tests are conducted on real datasets captured in either a block or linear trajectory configuration. The results demonstrate that the initial image EOPs obtained are accurate and can serve as a good initialization for an additional bundle adjustment process.

Electrostatic Forces: Formulas for the First Derivatives of a Polarizable, Anisotropic Electrostatic Potential Energy Function Based on Machine Learning

Explicit formulas are derived analytically for the first derivatives of a (i) polarizable, (ii) high-rank multipolar electrostatic potential energy function for (iii) flexible molecules. The potential energy function uses a machine learning method called Kriging to predict the local-frame multipole moments of atoms defined via the Quantum Chemical Topology (QCT) approach. These atomic multipole moments then interact via an interaction tensor based on spherical harmonics. Atom-centered local coordinate frames are used, constructed from the internal geometry of the molecular system. The forces involve derivatives of both this geometric dependence and of the trained kriging models. In the near future, these analytical forces will enable molecular dynamics and geometry optimization calculations as part of the QCT force field.

Ray tracing in a finite-element domain using nodal basis functions.

A method is presented for tracing rays through a medium discretized as finite-element volumes. The ray-trajectory equations are cast into the local element coordinate frame, and the full finite-element interpolation is used to determine instantaneous index gradient for the ray-path integral equation. The finite-element methodology is also used to interpolate local surface deformations and the surface normal vector for computing the refraction angle when launching rays into the volume, and again when rays exit the medium. The procedure is applied to a finite-element model of an optic with a severe refractive-index gradient, and the results are compared to the closed-form gradient ray-path integral approach.

Task Polar Coordinate Frame-Based Contouring Control of Biaxial Systems

Contouring control is crucial in high-speed and high-precision manufacturing. In this paper, a novel task polar coordinate frame (TPCF), moving along the desired contour, is proposed to naturally calculate and control the estimated contouring error by the circular approximation, a second-order approximation. The dynamics in the world Cartesian coordinate frame is transformed into radial and angular dynamics in the local polar coordinate frame. By the feedback linearization technique and an input feedforward compensation, the closed-loop dynamics are decoupled in terms of the estimated contouring error and the angular error, respectively. Proportional-plus-derivative controllers can be assigned to stabilize the individual axis dynamics in the TPCF. By tuning the control parameters, different strengthening on estimated contouring error and angular error can be imposed explicitly and directly. Various experiments on an XY-stage biaxial system with typical contours, a circle and a figure-“8,” were conducted. Comparative studies are carried out for the TPCF- and traditional Frenet frame-based controls. The contouring errors were drastically reduced by the proposed approach, particularly in high-speed and large-curvature contouring cases.

A Nonlinear Current Control Method for Resistance Spot Welding

Resistance spot welding (RSW) is one of the most commonly used methods in the manufacturing process for joining sheet metal together. This paper deals with the constant current control for RSW, which is the most common control strategy in actual applications. Since the process of RSW is nonlinear and time-varying, conventional control schemes do not yield satisfactory performance. To cope with this problem, a new control method is proposed in this paper. By solving the governing equation of the equivalent circuit of the RSW electrical system at each control cycle at its local coordinate frame, a nonlinear relationship between input and output variables is obtained. The relationship can be used to estimate an approximate value of the trigger time of the silicon-controlled rectifier for the next control cycle based on the information of the previous control cycle. In order to compensate the estimation error and improve the performance of the closed-loop system, a proportional-derivative (PD) controller is employed. Because the proposed controller regulates trigger time error instead of output current error, the parameters of the PD controller are easy to determine. The effectiveness of the proposed controller was verified through experiments under different conditions. Experimental results showed that the performance of the proposed controller was much better than that of a well-tuned proportional-integral-derivative controller.

Interactive real-time physics: An intuitive approach to form-finding and structural analysis for design and education

Real-time physics simulation has been extensively used in computer games, but its potential has yet to be fully realized in design and education. We present an interactive 3D physics engine with a wide variety of applications.In common with traditional FEM, the use of a local element stiffness matrix is retained. However, unlike typical non-linear FEM routines elements forces, moments and inertia are appropriately lumped at nodes following the dynamic relaxation method. A semi-implicit time integration scheme updates linear and angular momentum, and subsequently the local coordinate frames of the nodes. A co-rotational approach is used to compute the resultant field of displacements in global coordinates including the effect of large deformations. The results obtained compare well against established commercial software.We demonstrate that the method presented allows the making of interactive structural models that can be used in teaching to develop an intuitive understanding of structural behaviour. We also show that the same interactive physics framework allows real-time optimization that can be used for geometric and structural design applications.

A micro cutter auto-alignment system with on-machine positioning error measurement and compensation methods

To perform an accurate micro machining process, accurate alignment of the micro cutter with respect to the local coordinate frame for machining is essential. Due to the fragility of the micro tool, traditional cutter alignment method used in industry is no longer proper for micro machining. Utilizing machine vision technology with 2 CCDs, a micro tool positioning error measurement and auto-compensation method and system were proposed. In the measurement method, Power Law Method was first used to enhance the image taken by a CCD. Canny Edge Method and image projection method were then used to identify the contours of the cutter and the workpiece from the image. Finally, the position errors of the cutter were determined through calculating the number of pixels between the cutter and the workpiece. The proposed error compensation system can automatically generate NC codes which will accurately re-locate the cutter based on the measured position errors. Experiments were conducted on a micro machine tool, and the results have shown that the y-dir. positioning error of the cutter was significantly improved from −0.025 mm to 0.001 mm, and the total machining error was reduced from −0.02 mm to −0.001 mm.

Modified diffusion equation for the wormlike-chain statistics in curvilinear coordinates

One of the essential physical quantities used to study the conformation and structure of polymers is the so-called propagator in polymer theories. On the basis of the wormlike-chain statistical-physics model, we derive the partial diffusion equation that the propagator satisfies, for a curvilinear coordinate system. As it turns out, an additional term exists, that couples the rotating local coordinate frame with an orientation differential operator; this term has not been previously documented. In addition, for a wormlike chain moving on a curved surface, the external-field term needs to be supplemented by a surface curvature energy penalty.

Spin angular momentum tunnelling in coupled anisotropic optical fibres

We have studied the longitudinal evolution of spin angular momentum (SAM) in a system of two coupled anisotropic monomode optical fibres. On the basis of perturbation theory with degeneracy we have obtained the modes of such a system for various cases of mutual orientation of anisotropy axes and relations between anisotropy and exchange constants. We have provided their expressions in both local and global coordinate frames. We have studied the evolution of the SAM of individual fibres if one of the fibres is excited with linearly and circularly polarized light. We have shown that at certain points the specific SAM (SAM reduced to energy) suffers fast changing from +1 to −1, the spatial scale of such variations being small and determined by the ratio of the anisotropy constant to the inter-fibre exchange.

Localizability and Distributed Localization of Sensor Networks using Relative Position Measurements

The paper introduces the localization problem of sensor networks using relative position measurements. It is assumed that relative positions are measured in local coordinate frames of individual sensors, for which different sensors may have different orientations of their local frames and the orientation errors with respect to the global coordinate frame are not known. A new necessary and sufficient condition is developed for localizability of such sensor networks that are modeled as directed graphs. That is, every sensor node should be 2-reachable from the anchor nodes. Moreover, for a localizable sensor network, a distributed, linear, and iterative scheme based on the graph Laplacian of the sensor network is developed to solve the coordinates of the sensor network in the global coordinate frame.

Fast protein structure alignment using Gaussian overlap scoring of backbone peptide fragment similarity

Aligning and comparing protein structures is important for understanding their evolutionary and functional relationships. With the rapid growth of protein structure databases in recent years, the need to align, superpose and compare protein structures rapidly and accurately has never been greater. Many structural alignment algorithms have been described in the past 20 years. However, achieving an algorithm that is both accurate and fast remains a considerable challenge. We have developed a novel protein structure alignment algorithm called 'Kpax', which exploits the highly predictable covalent geometry of C(α) atoms to define multiple local coordinate frames in which backbone peptide fragments may be oriented and compared using sensitive Gaussian overlap scoring functions. A global alignment and hence a structural superposition may then be found rapidly using dynamic programming with secondary structure-specific gap penalties. When superposing pairs of structures, Kpax tends to give tighter secondary structure overlays than several popular structure alignment algorithms. When searching the CATH database, Kpax is faster and more accurate than the very efficient Yakusa algorithm, and it gives almost the same high level of fold recognition as TM-Align while being more than 100 times faster.

New spatial curved beam and cylindrical shell elements of gradient-deficient Absolute Nodal Coordinate Formulation

Based on previous studies, a new spatial curved slender-beam finite element and a new cylindrical shell finite element are proposed in the frame of gradient-deficient Absolute Nodal Coordinate Formulation (ANCF). The strain energy of the beam element is derived by using the definition of the Green–Lagrange strain tensor in continuum mechanics so that the assumption on small strain can be relaxed. By using the differential geometry and the continuum mechanics, the angle between two base vectors of a defined local coordinate frame of the cylindrical shell element is introduced into the strain energy formulations. Therefore, the new shell element can be used to model parallelogram shells. The analytical formulations of elastic forces and their Jacobian for the above two finite elements of gradient-deficient ANCF are also derived via the skills of tensor analysis. The generalized-alpha method is used to solve the huge set of system equations. Finally, four case studies including both static and dynamic problems are given to validate the proposed beam and cylindrical shell elements of gradient-deficient ANCF.

GIRAF: a method for fast search and flexible alignment of ligand binding interfaces in proteins at atomic resolution.

Comparison and classification of protein structures are fundamental means to understand protein functions. Due to the computational difficulty and the ever-increasing amount of structural data, however, it is in general not feasible to perform exhaustive all-against-all structure comparisons necessary for comprehensive classifications. To efficiently handle such situations, we have previously proposed a method, now called GIRAF. We herein describe further improvements in the GIRAF protein structure search and alignment method. The GIRAF method achieves extremely efficient search of similar structures of ligand binding sites of proteins by exploiting database indexing of structural features of local coordinate frames. In addition, it produces refined atom-wise alignments by iterative applications of the Hungarian method to the bipartite graph defined for a pair of superimposed structures. By combining the refined alignments based on different local coordinate frames, it is made possible to align structures involving domain movements. We provide detailed accounts for the database design, the search and alignment algorithms as well as some benchmark results.

Narrow-angle representations of the phase and group velocities and their applications in anisotropic velocity-model building for microseismic monitoring

It is usually believed that angular aperture of seismic data should be at least 20° to allow estimation of the subsurface anisotropy. Although this is certainly true for reflection data, for which anisotropy parameters are inverted from the stacking velocities or the nonhyperbolic moveout, traveltimes of direct P- and S-waves recorded in typical downhole microseismic geometries make it possible to infer seismic anisotropy in angular apertures as narrow as about 10°. To ensure the uniqueness of such an inversion, it has to be performed in a local coordinate frame tailored to a particular data set. Because any narrow fan of vectors is naturally characterized by its average direction, we choose the axes of the local frame to coincide with the polarization vectors of three plane waves corresponding to such a direction. This choice results in a significant simplification of the conventional equations for the phase and group velocities in anisotropic media and makes it possible to predict which elements of the elastic stiffness tensor are constrained by the available data. We illustrate our approach on traveltime synthetics and then apply it to perforation-shot data recorded in a shale-gas field. Our case study indicates that isotropic velocity models are inadequate and accounting for seismic anisotropy is a prerequisite for building a physically sound model that explains the recorded traveltimes.

The control point concept for nonlinear trajectory-tracking control of autonomous helicopters with fly-bar

An alternative approach for automatic trajectory-tracking control of small unmanned helicopters with fly-bar is proposed. This approach uses the spatial three-dimensional coordinates of a point on the helicopter's local coordinate frame other than its centre of gravity, called the control point, and the helicopter's yaw angle as four control outputs. The helicopter is assumed to have four independent control inputs. With this choice of control outputs, the helicopter's input–output model becomes a square control system, which opens the possibility of implementation of many robust nonlinear control methods that are suitable for such systems. The helicopter, which has six rigid body degrees of freedom (DOFs), has two underactuated DOFs (UA-DOFs). It is proved that the zero-dynamics of the UA-DOFs are inherently stable, leading to a stable control system. A sliding mode controller is designed for trajectory-tracking of the outputs. It is verified via simulations that the response of the control outputs and UA-DOFs are in fact stable.

Module-Based Static Structural Design of a Modular Reconfigurable Robot

In this paper, the structural design of modular reconfigurable robots (MRRs) is studied. This problem is defined as the determination of proper module sizes according to the robot’s payload and end-effector deflection specifications. Because an MRR has multiple configurations, a simple design process is proposed in order to avoid performing the structural design stage at each configuration. The final structural design is only carried out at a single configuration that can guarantee the robot’s satisfactory performance for all remaining feasible configurations. It is shown that the module structural design stage can be performed at the local coordinate frame of each module. While the module local force requirement can be fully determined, the determination of the module local deformation requirement is redundant. Thus, there can exist multiple design solutions. To overcome this problem, a nonlinear approach using a genetic algorithm is used to search for an optimal solution. Finally, a design simulation is performed on an example MRR, and the results show the effectiveness of the proposed design method.

Modeling of ionic polymer metal composite actuator dynamics using a large deflectionbeam model

This paper introduces a novel technique for modeling the dynamics of ionic polymer metalcomposites (IPMCs). The proposed finite element modeling technique combines a largedeflection beam model and lumped RC model. In this approach, a local coordinateframe attached at the first node of each element which undergoes rigid bodymotion and elastic deformation of the element is described with respect to the localcoordinate frame. The proposed model accounts for the large elastic deflection of eachelement. This modeling strategy has the advantage of accurately describing the largedeformation of the IPMC using a limited number of elements. Experimental results arefound to be in close agreement with those of simulating the proposed model.

Geophysical inversion of on board satellite gradiometer data — A feasibility study in the ALPACA region, Central Europe

The structure of the lithosphere of the ALPACA (Alpine-Pannonian-Carpathian) region is described by a model containing about 200000 rectangular volume elements (prisms) of variable dimensions defined in the mapping system of Hungary. Forward computations show that the contribution of the structural units (topography and upper mantle) of the lithosphere to the disturbing potential T of the Earth may reach several tenths of E unit at 300 km elevation. The contribution from the sediments is less by a factor of ten but even its magnitude exceeds the planned sensitivity of the satellite on board measurements. It is expected that some regional information about the horizontal density variation of the crust can be deduced from the GOCE data, especially for the density contrast between the lower crust and upper mantle. Since the density distribution of either the topographical masses or the sedimentary complex is much better known than the density jump on the Moho, therefore their effect on the second derivatives of T can be removed from the measurements. The residuals can be interpreted by inversion using the closed analytical formulae available for rectangular volume elements. The modelling approach based on the local planar co-ordinate frame was compared to the polyhedron representation of the same crustal model defined in a global rectangular co-ordinate system. No significant effect of the Earths curva-ture could be indicated for the Moho interface.

First-principles design of nanomachines

Learning from nature's amazing molecular machines, globular proteins, we present a framework for the predictive design of nanomachines. We show that the crucial ingredients for a chain molecule to behave as a machine are its inherent anisotropy and the coupling between the local Frenet coordinate reference frames of nearby monomers. We demonstrate that, even in the absence of heterogeneity, protein-like behavior is obtained for a simple chain molecule made up of just 30 hard spheres. This chain spontaneously switches between 2 distinct geometries, a single helix and a dual helix, merely because of thermal fluctuations.