Geometrical Treatment of Periodic Graphs with Coordinate System Using Axis-fiber and an Application to a Motion Planning

In this paper, we proposed a spherical robot with two motors in the horizontal and vertical directions which derive the robot to do omni-directionally roll. Based on the structure of the robot, we derived the kinematic model using inertial and moving coordinate system. In order to minimize the energy of the system, an optimization problem with two optimization variables which are the parameters to control the angular velocity of the motors is given. After that, a particle swarm optimization (PSO) algorithm is used to solve the optimization problem. The simulation shows that the motion planning with the algorithm has high precision.

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Proximity and Motion Planning on ℓ1-Rigid Planar Periodic Graphs

Transactions on Computational Science XX - Lecture Notes in Computer Science ◽

10.1007/978-3-642-41905-8_6 ◽

2013 ◽

pp. 72-91

Author(s):

Norie Fu ◽

Akihiro Hashikura ◽

Hiroshi Imai

Keyword(s):

Motion Planning ◽

Periodic Graphs

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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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