non-Cartesian Coordinate System Research Articles

Jerk limited continuous tool path generation for flexible systems in non-cartesian coordinate systems

Inspired by computer numerical control (CNC) technology and drastic changes in the world of electronics and microcontrollers, the idea of using non-cartesian coordinate system for CNC machines became a topic of interest. Non-cartesian coordinate systems provide freedom of movement in systems with large working spaces where high dimensional accuracy is not a prime concern. Applications of such systems can be found in murals, where a painting is applied to a large wall of any arbitrary build material. In the present paper, a new design was introduced to automate the process of drawing murals on large surfaces. The proposed mechanism utilized chain and sprocket to overcome the size limitations of traditionally available x-y tables. In addition to that, the developed mechanism does not utilize the common robust structures used in the conventional CNC systems and the tool motion is created using flexible arms that are not heavily constrained. The effect of systems unwanted vibration has been eliminated using continuous trajectory and kinematic profiles. Parametric curve interpolation algorithms for trajectory planning were implemented to increase the flexibility of drawing complex curves. Parametric curves such as B-spline and non-uniform rational B-spline (NURBS) were employed to generate a jerk limited trajectory for motion control and extraction of kinematic profiles. Such trajectory constrains the jerk of the system and guarantees a smooth motion free of feed-rate fluctuations. Compared to other methods available for extracting the kinematic profiles, jerk limited method decreases the travel time and trajectory error. Ultimately, motion commands were produced by using kinematic profiles and the second order Taylor interpolator. STM32746ZG card was used as the main processor and two stepper motors controlled by a Bit Pattern interpolation algorithm were utilized to drive the proposed mechanism. The input pulses of stepper motors were stored as binary bits and transmitted to drivers in real-time. The feedback was also evaluated by two rotary encoders, placed at the end of the motors’ shafts. Encoder data were acquired and transmitted using a TCP/IP protocol to guarantee zero data loss during the transmission.

Robotic Arm Position Computing Method in the 2D and 3D Spaces

This paper presents a method on how to compute the position of a robotic arm in the 2D and 3D spaces. This method is slightly different from the well-known methods, such as forward or inverse kinematics. The method presented in this paper is an optical method, which uses two video cameras in stereo vision configuration to locate and compute the next move of a robotic arm in space. The method recognizes the coordinates of the markers placed at the joints of the robotic arm using the two video cameras. The coordinate points of these markers are connected with straight lines. Around certain points, circles are drawn. From the tangent to the circles, a non-Cartesian (orthogonal) coordinate system is drawn, which is enough to compute the target position of the robotic arm. All of these drawings are overlaid on the live video feed. This paper also presents another method for calculating the stereo distance using the triangulation method. An alternative method is also presented when a non-Cartesian (orthogonal) 3D coordinate system is created, which is used to compute the target position of the robotic arm in the 3D space. Because the system is in a loop, it can make micro-adjustments of the robotic arm, in order to be exactly in the desired position. In this way, there is no need to make calibrations for the robotic arm. In an industrial system, there is no need to stop the production line, which can be a really big cost saver.

Generation of Rossby waves off the Cape Verde Peninsula: the role of the coastline

Abstract. In December 2002 and January 2003 satellite observations of chlorophyll showed a strong coastal signal along the west African coast between 10 and 22∘ N. In addition, a wavelike pattern with a wavelength of about 750 km was observed from 20 December 2002 and was detectable for 1 month in the open sea, south-west of the Cape Verde Peninsula. Such a pattern suggests the existence of a locally generated Rossby wave which slowly propagated westward during this period. This hypothesis was confirmed by analysing sea surface height provided by satellite altimeter during this period. To decipher the mechanisms at play, a numerical study based on a reduced-gravity shallow-water model has first been conducted. A wind burst, broadly extending over the region where the offshore oceanic signal is observed, is applied for 5 d. A Kelvin wave quickly develops along the northern edge of the cape, then propagates and leaves the area in a few days. Simultaneously, a Rossby wave whose characteristics seem similar to the observed pattern forms and slowly propagates westward. The existence of the peninsula limits the extent of the wave to the north. The spatial extent of the wind burst determines the extent of the response and correspondingly the timescale of the phenomenon (about 100 d in the present case). When the wind burst has a large zonal and small meridional extent, the behaviour of a wave to the north of the peninsula differs from that to the south. These results are corroborated and completed by an analytical study of a linear reduced-gravity model using a non-Cartesian coordinate system. This system is introduced to evaluate the potential impact of the coastline shape. The analytical computations confirm that a period of around 100 d can be associated with the observed wave considering the value of the wavelength; they also show that the role of the coastline remains moderate at such timescales. By contrast, when the period becomes shorter (smaller than 20–30 d), the behaviour of the waves is modified because of the shape of the coast. South of the peninsula, a narrow band of sea isolated from the rest of the ocean by two critical lines appears. Its meridional extent is about 100 km and Rossby waves could propagate there towards the coast.

Mapping activation of resources among upper division physics students in non-Cartesian coordinate systems: A case study

An in-depth investigation to identify cognitive resources upper division physics students use to reason about unit vectors, position vectors, and velocity vectors in polar coordinates.

Vector Circle Clipping Algorithm Based on Polygon Window of Hexagonal Grid System

This paper proposes an algorithm for cropping vector circles in polygon window under hexagonal grid system for the problem of long time consuming and low efficiency of polygon window clipping algorithm in existing Non-Cartesian coordinate system. First, the algorithm converts Cartesian coordinates into hexagonal grid coordinates, then performs intersection detection by vector operation, and uses geometric relations to judge the virtual real intersection points, thus obtaining the coordinates of the real intersection points, and thus cutting the arc results. Finally, the final cropping result is obtained. The experimental results show that the algorithm is an effective vector circle clipping algorithm based on the hexagonal grid system polygon window, and the algorithm has high efficiency and stability.

Some studies on Lorentz transformation matrix in non-cartesian co-ordinate system

The Lorentz matrices for transformation of co-ordinates in Cartesian system are presented for the cases when the relative velocity between two reference frames is along X , Y and Z axes. The general form of the matrix for transformation of co-ordinates from unprimed to primed frame has been deduced in case of Cartesian co-ordinate system with the help of the above matrices. This matrix has not been transformed to the cases of cylindrical and spherical polar co-ordinates due to the fact that the calculations are cumbersome and lengthy. Hence, considering the relative velocity between two frames along a co-ordinate axis the transformation matrix has been found out for cylindrical and spherical co-ordinates.

Three-dimensional transformation optics for arbitrary coordinate systems: transforming conductive materials and boundaries.

A three-dimensional transformation optics method, leading to homogeneous materials, applicable to any non-Cartesian coordinate systems or waveguides/objects of arbitrary cross-sections is presented. Both the conductive boundary and internal material of the desired device is determined by the proposed formulation. The method is applicable to a wide range of waveguide, radiation, and cloaking problems, and is demonstrated for circular waveguide couplers and an external cloak. An advantage of the present method is that the material properties are simplified by appropriately selecting the conductive boundaries. For instance, a right-angle circular waveguide bend is presented which uses only one homogenous material. Also, transformation of conductive materials and boundaries are studied. The conditions in which the transformed boundaries remain conductive are discussed. In addition, it is demonstrated that negative infinite conductivity can be replaced with positive conductivity, without affecting the field outside the conductive boundary. It is also observed that a negative finite conductivity can be replaced with a positive one, by accepting some small errors. The general mathematical procedure and formulation for calculating the parametric surface equations of the conductive peripheries are presented.

Interpolation of Saddle-Shaped Trajectory in Non-Cartesian Coordinate System

Interpolation of Saddle-Shaped Trajectory in Non-Cartesian Coordinate System

A Novel Trajectory Control Algorithm for Intersection Seam Automated Welding

A saddle-shaped curve is formed by intersection of two tubes. This paper proposes a new saddle curve interpolation algorithm for automated welding system with non-Cartesian coordinate system. The trajectory interpolation of saddle-shaped curve is usually carried out through approximating the curve as segments of straight line/circular arc for intersection seam automated CNC welding. Simulation of the direct interpolation algorithm is carried out to testify its effectiveness and superiority. With this algorithm, it is demonstrated that the chord error is within the allowable limits and the algorithm has high reliability.

Curve Fitting Approach of Saddle-Shaped Curve for Automated Welding/Cutting

Curve fitting, or interpolation algorithm is core technology for Computer Numerical Control (CNC) systems. A saddle-shaped curve is formed by intersection of two tubes. The trajectory interpolation of saddle-shaped curve is usually carried out through approximating the curve as segments of straight line for automated CNC welding/cutting. This paper proposes a new saddle curve curve fitting algorithm for automated welding/cutting system with non-Cartesian coordinate system. Simulation of the direct curve fitting algorithm has demonstrated that the algorithm has high curve interpolation precision and high feedrate stability.

Visualizing invisibility: Metamaterials-based optical devices in natural environments

Photorealistic ray tracing methods have been developed that allow us to see how devices such as imperfect invisible spheres and invisibility cloaks would appear if actually constructed and placed in outdoor environments. The methods developed allow photorealistic depiction of devices with gradient indices of refraction and birefringence or trirefringence in non-Cartesian coordinate systems (and hence accurately handle ray splitting/beam walkoff). The resulting images, which can be rendered in real time to produce animations as will be shown, allow subjective assessment of the performance of optical instruments such as invisibility devices in environments in which they are intended to ultimately be used.

Discretization of Macroscopic Transport Equations on Non-Cartesian Coordinate Systems

We discuss discretization schemes for the Poisson equation, the isothermal drift-diffusion equations, and higher order moment equations derived from the Boltzmann transport equation for general coordinate systems. We briefly summarize the method of dimension reduction when the problem does not depend on one coordinate. Discretization schemes for dimension-reduced coordinate systems are introduced, which provide curvilinear coordinate systems. In addition to the reduction of the dimensionality, another benefit of these curved coordinate systems is that the domain approximation is more accurate, and therefore, the mesh point density can be kept smaller compared to the original problem. We obtain a discretization scheme for the isothermal drift-diffusion equation in closed from. For higher order transport equations, we use the approximation method of optimum artificial diffusivity and generalize it for non-Cartesian coordinate systems. For the special case of cylindrical coordinates, we can show that it is not necessary to introduce special discretization schemes apart from the standard Scharfetter-Gummel scheme.

Complex coordinate stretching as a generalized absorbing boundary condition

By complex coordinate stretching and a change of variables, it is shown simply that PML is reflectionless for all frequencies and all angles. Also, Maxwell's equations for PML media reduce to ordinary Maxwell's equations with complex coordinate systems. Many closed-form solutions for Maxwell's equations map to corresponding closed-form solutions in complex coordinate systems. Numerical simulations with the closed-form solutions show that metallic boxes lined with PML media are highly absorptive. These closed-form solutions lend a better understanding to the absorptive properties of PML media. For instance, they explain why a PML medium is absorptive when a dielectric or metallic interface extends to the edge to a simulation region where PML media reside. More importantly, the complex coordinate stretching method can be generalized to non-Cartesian coordinate systems, providing absorbing boundary conditions in these coordinate systems. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 363–369, 1997.

Is the Graph of y = kx Straight?

So as to create a little suspense, we declare in advance that the answer to the innocent question posed in the title depends on what is meant by x and y. In this article, we shall show how the well-known fact that the graph of y = kx in a Cartesian coordinate system is straight can be used as a point of departure for investigations into loci in non-Cartesian coordinate systems.

Vibrational motions of buckminsterfullerene

A non-Cartesian coordinate system is developed which permits the vibrational motions of Buckminsterfullerene to be expressed in terms of four force constants. A 180 × 180 matrix is then derived which, when diagonalized, yields the complete vibrational spectrum. These results are compared with those obtained previously via a MNDO calculation.

Numerical solution of three-dimensional moving boundary problems: Melting and solidification with blanketing of a third layer

A three-dimensional melting and solidification problem is solved in which a second layer of solid is added at variable rates. The physical properties of the system are position and temperature dependent, and there is heat generation within the system. Each phase has a grid scheme where every grid point travels differently with the effect of phase change or addition. The resulting differential equations are solved through an explicit method where the derivatives are approximated in a non-Cartesian coordinate system. The effect of blanketing by a second solid layer having different thermal conductivities is investigated. Numerical results are generated for a specific melting and solidification problem.

Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time

We point out and discuss an ambiguity which arises in the quantum theory of fields when the background metric is not explicitly Minkowskian-in other words, when an external gravitational field, real or apparent, is present. A general theory of a canonical neutral scalar field in a static universe, including the construction of a Fock space, is presented. It is applied to a portion of two-dimensional flat space-time equipped with a non-Cartesian space-time coordinate system with respect to which the metric is nonetheless static. The resulting particle interpretation of the field is shown to be different from the standard one in special-relativistic free-field theory. The ambiguity frustrates an attempt to define uniquely the energy-momentum tensor by the usual method of normal ordering. We discuss various suggestions for (1) distinguishing a unique correct quantization in a given physical situation, or (2) reinterpreting seemingly inequivalent theories as physically equivalent. In passing it is shown that the vacuum state and the energy density of a free field in a box with periodic boundary conditions differ from those associated with a region of the same size in infinite space; this result should be of interest outside the gravitational context.

Weyl Transform in Nonrelativistic Quantum Dynamics

The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.