NATURE OF THE JACOBIAN MATRIX OF ONE CHANGER OF TRANSVERSE COORDINATE SYSTEM OF A RIEMANNIAN FOLIATION OF A FLAG OF AN EXTENSION HAVING DENSE LEAVES ON A COMPACT MANIFOLD

Abstract In this paper, the geometric error modeling method of CNC cylindrical grinder based on the differential motion relationship between coordinate systems, the function fitting model method of basic geometric error terms based on cftool toolbox and the error compensation method based on Jacobian matrix are proposed. Firstly, the differential motion theory, which is widely used in the field of robot kinematics error modeling, is used to build the machine tool space machining error model of CNC cylindrical grinder. Different from the multi-body theory, this modeling method can clearly reflect the influence degree of each moving part on the grinding wheel cutter. Secondly, SJ6000 laser interferometer was used to measure and identify the geometric error terms of B2-K3032 CNC precision cylindrical grinder. MATLAB cftool toolbox was used to perform mathematical function fitting on the known error data, and the mathematical relationship between 24 geometric errors and machining instructions was found. Finally, combining with the 24 Sum of Sine function model, the known verticality error and position deviation, the differential motion matrix of each moving part in the tool coordinate system and the corresponding Jacobian matrix, the compensation quantity (dx dz db dc) of the comprehensive geometric error in the tool coordinate system by the CNC precision cylindrical grinder is obtained. In order to verify the feasibility of the above method, RA1000 series roundness meter was used to measure the radial circular runout error before and after the correction. The experimental results show that the precision of each shaft section is increased by 17.54%, 15.22%, 15.71%, 18.4%, 12.87%, respectively, and the average machining accuracy is increased by 15.948%. Therefore, the above methods are effective and reasonable for improving the precision of spindle workpieces, and can also be used for reference in the initial design stage of CNC cylindrical grinder manufacturing enterprises or improving the machining accuracy of existing machine tools.

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Formulation of manipulator Jacobian using the velocity similarity principle

Robotica ◽

10.1017/s0263574700007359 ◽

1990 ◽

Vol 8 (1) ◽

pp. 81-84 ◽

Cited By ~ 4

Author(s):

K. C. Gupta ◽

R. Ma

Keyword(s):

Coordinate System ◽

Jacobian Matrix ◽

Matrix Form ◽

Base System ◽

Regional Structure ◽

Reference Position ◽

Similarity Principle ◽

Spherical Wrist ◽

Zero Reference ◽

Position Representation

SUMMARYVelocity similarity principle V(θ., ṡ, ub, Qb) = DabV (θ., ṡ, ua, Qa)D ab–1 is presented and used to derive several useful forms of the Jacobian matrix for the manipulator from its basic kinematic equations in 4 X 4 matrix form. The zero reference position representation is used and, therefore, the base system is the only coordinate system utilized in the derivations. For manipulators with a spherical wrist, a modified form of the Jacobian is presented in which the Jacobian columns corresponding to the regional structure are completely decoupled from those corresponding to the wrist structure.

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On the sensitivity analysis of the computational kinematics of the robotic manipulators

Robotica ◽

10.1017/s0263574700018658 ◽

1995 ◽

Vol 13 (6) ◽

pp. 575-581

Author(s):

Siamak Vahidi ◽

Kazem Kazerounian

Keyword(s):

Coordinate System ◽

Numerical Experiments ◽

Jacobian Matrix ◽

Robotic Manipulators ◽

Performance Criteria ◽

Euler Angles ◽

Numerical Accuracy ◽

Evaluation Measure ◽

Computational Kinematics ◽

New Formulation

SummaryIn the computational kinematics of robotic manipulators, accuracy and sensitivity of the results are highly dependent on the choice of the coordinate system, the metric system, and the appropriateness of performance evaluation measure. In this paper, these undesired sensitivities are examined and suitable performance criteria are developed to eliminate the coordinate system and metric system dependencies, and adverse numerical effects associated with them. Also a new formulation for the Jacobian matrix, joint velocities, and hand velocities, based on the Euler angles, is developed. This formulation, further improves numerical accuracy of the computations. Numerical experiments are included.

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Riemannian foliations with parallel curvature

Nagoya Mathematical Journal ◽

10.1017/s0027763000020390 ◽

1983 ◽

Vol 90 ◽

pp. 145-153

Author(s):

Robert A. Blumenthal

Keyword(s):

Tangent Bundle ◽

Normal Bundle ◽

Compact Manifold ◽

Riemannian Metric ◽

Riemannian Foliation ◽

Riemannian Foliations ◽

Smooth Compact Manifold

Let M be a smooth compact manifold and let be a smooth codimension q Riemannian foliation of M. Let T(M) be the tangent bundle of M and let E ⊂ T(M) be the subbundle tangent to . We may regard the normal bundle Q = T(M)/E of as a subbundle of T(M) satisfying T(M) = E ⊕ Q. Let g be a smooth Riemannian metric on Q invariant under the natural parallelism along the leaves of .

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Unified Kinematics Analysis and Analytic Singularity-Free Workspace of a Metamorphic Parallel Mechanism With Controllable Rotation Center

Volume 5B: 38th Mechanisms and Robotics Conference ◽

10.1115/detc2014-34164 ◽

2014 ◽

Author(s):

Dongming Gan ◽

Jian S. Dai ◽

Jorge Dias ◽

Lakmal D. Seneviratne

Keyword(s):

Coordinate System ◽

Parallel Mechanism ◽

Jacobian Matrix ◽

Central Line ◽

Kinematics Analysis ◽

Base Plane ◽

Analytic Singularity ◽

Rotation Center ◽

Pure Rotation ◽

Rotation Motion

This paper presents a metamorphic parallel mechanism with controllable rotation center in its pure rotation topology. Based on reconfiguration of a reconfigurable Hooke (rT) joint, the rotational center of the mechanism can be altered along the central line perpendicular to the base plane. A unified Dixon resultant based method is proposed to solve the forward kinematics analytically by covering all configurations with variable rotation centers while the rotation motion is expressed using Cayley formula. Then singularity loci are derived and represented in a new coordinate system with the three Rodrigues-Hamilton parameters assigned in three perpendicular directions. Limb-actuation singularity loci are also obtained from row vectors of the Jacobian matrix. By using Cayley formula, analytical workspace boundaries are expressed by including the mechanism structure parameters and input actuation limits. Finally, singularity-free workspace of configurations with variable rotation centers is demonstrated in the proposed coordinate system.

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Reference Coordinate System Requirements for Geophysics

International Astronomical Union Colloquium ◽

10.1017/s0252921100050946 ◽

1975 ◽

Vol 26 ◽

pp. 87-92

Author(s):

P. L. Bender

Keyword(s):

Coordinate System ◽

Polar Motion ◽

Main Part ◽

Measurement Techniques ◽

Reference Points ◽

Angular Motion ◽

Coordinate Systems ◽

Axis Of Rotation ◽

The Earth ◽

Fixed System

AbstractFive important geodynamical quantities which are closely linked are: 1) motions of points on the Earth’s surface; 2)polar motion; 3) changes in UT1-UTC; 4) nutation; and 5) motion of the geocenter. For each of these we expect to achieve measurements in the near future which have an accuracy of 1 to 3 cm or 0.3 to 1 milliarcsec.From a metrological point of view, one can say simply: “Measure each quantity against whichever coordinate system you can make the most accurate measurements with respect to”. I believe that this statement should serve as a guiding principle for the recommendations of the colloquium. However, it also is important that the coordinate systems help to provide a clear separation between the different phenomena of interest, and correspond closely to the conceptual definitions in terms of which geophysicists think about the phenomena.In any discussion of angular motion in space, both a “body-fixed” system and a “space-fixed” system are used. Some relevant types of coordinate systems, reference directions, or reference points which have been considered are: 1) celestial systems based on optical star catalogs, distant galaxies, radio source catalogs, or the Moon and inner planets; 2) the Earth’s axis of rotation, which defines a line through the Earth as well as a celestial reference direction; 3) the geocenter; and 4) “quasi-Earth-fixed” coordinate systems.When a geophysicists discusses UT1 and polar motion, he usually is thinking of the angular motion of the main part of the mantle with respect to an inertial frame and to the direction of the spin axis. Since the velocities of relative motion in most of the mantle are expectd to be extremely small, even if “substantial” deep convection is occurring, the conceptual “quasi-Earth-fixed” reference frame seems well defined. Methods for realizing a close approximation to this frame fortunately exist. Hopefully, this colloquium will recommend procedures for establishing and maintaining such a system for use in geodynamics. Motion of points on the Earth’s surface and of the geocenter can be measured against such a system with the full accuracy of the new techniques.The situation with respect to celestial reference frames is different. The various measurement techniques give changes in the orientation of the Earth, relative to different systems, so that we would like to know the relative motions of the systems in order to compare the results. However, there does not appear to be a need for defining any new system. Subjective figures of merit for the various system dependon both the accuracy with which measurements can be made against them and the degree to which they can be related to inertial systems.The main coordinate system requirement related to the 5 geodynamic quantities discussed in this talk is thus for the establishment and maintenance of a “quasi-Earth-fixed” coordinate system which closely approximates the motion of the main part of the mantle. Changes in the orientation of this system with respect to the various celestial systems can be determined by both the new and the conventional techniques, provided that some knowledge of changes in the local vertical is available. Changes in the axis of rotation and in the geocenter with respect to this system also can be obtained, as well as measurements of nutation.

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2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

International Astronomical Union Colloquium ◽

10.1017/s0252921100050867 ◽

1975 ◽

Vol 26 ◽

pp. 21-26

Keyword(s):

Coordinate System ◽

Working Group ◽

General Character ◽

Coordinate Systems ◽

Reference Coordinate System ◽

Group 2 ◽

Definition Of ◽

Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

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