The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion

1987 ◽  
Vol 109 (1) ◽  
pp. 95-100 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Motion ◽  
Distribution Parameter ◽  
Spatial Motion ◽  
Euler Parameters ◽  
Kinematic Theory ◽  
Unit Hypersphere ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Curve Form

The dual Euler parameters of a rigid spatial motion are used to define a curve on a dual unit hypersphere. The dual velocity, curvature, and torsion of this curve form a set of instantaneous parameters related to the instantaneous invariants of the motion. In this paper these new parameters are used to reformulate the kinematic theory of line trajectories. The distribution parameter, Disteli formulas, and the inflection congruence are examined in detail.

The Image of the Coupler Motion of a Special Spherical Four Bar Linkage

1987 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rational Functions ◽  
Global Properties ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Four Bar Linkage ◽  
Spherical Motion

Abstract This paper uses a kinematic mapping of spherical motion to derive an image curve which represents the coupler motion of a doubly folding spherical four bar linkage. The image curve of this linkage, the so called “kite” linkage, can be parameterized by rational functions. This parameterization is presented as well as formulas which allow the computation of its curvature and torsion at any point. These formulas provide a link between the global properties of the coupler motion as represented by the image curve itself and its instantaneous properties given by the curvature and torsion functions.

Planar and Spatial Rigid Motion as Special Cases of Spherical and 3-Spherical Motion

1983 ◽  
Vol 105 (3) ◽  
pp. 569-575 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Motion ◽  
Arc Length ◽  
Spatial Motion ◽  
Special Cases ◽  
Curvature Theory ◽  
Rigid Motions ◽  
Spherical Motion ◽  
Smooth Transformation

This paper examines spherical and 3-spherical rigid motions with instantaneous invariants approaching zero. It is shown that these motions may be identified with planar and spatial motion, respectively. The instantaneous invariants are ratios of arc-length along the surface of the sphere to its radius, thus the process of shrinking their value may be viewed as expanding the sphere while bounding the instantaneous displacements on the sphere. This allows a smooth transformation of the results of the curvature theory of spherical and 3-spherical motion into their planar and spatial counterparts.

The Image Curve of the Coupler of a Special Spherical Four Bar Linkage

1988 ◽  
Vol 110 (3) ◽  
pp. 276-280
Author(s):  
J. M. McCarthy
Keyword(s):  
Rational Functions ◽  
Global Properties ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Four Bar Linkage ◽  
Spherical Motion

This paper uses a kinematic mapping of spherical motion to derive an image curve which represents the coupler motion of a doubly folding spherical four bar linkage. The image curve of this linkage, the so called “kite” linkage, can be parameterized by rational functions. This parameterization is presented as well as formulas which allow the computation of its curvature and torsion at any point. These formulas provide a link between the global properties of the coupler motion as represented by the image curve itself and its instantaneous properties given by the curvature and torsion functions.

A Euclidean Invariants Based Study of Circular Surfaces With Fixed Radius

2007 ◽  
Author(s):  
Lei Cui ◽  
Delun Wang
Keyword(s):  
Complete System ◽  
Distribution Parameter ◽  
Moving Frame ◽  
Necessary Condition ◽  
Spherical Indicatrix ◽  
Circular Surface ◽  
Fixed Radius ◽  
Curvature And Torsion ◽  
The Relationship ◽  
Sufficient And Necessary Condition

In this paper a complete system of Euclidean invariants is presented to study circular surfaces with fixed radius. The study of circular surfaces is simplified to the study of two curves: the spherical indicatrix of the unit normals of circle planes and the spine curve. After the geometric meanings of these Euclidean invariants are explained, the distribution parameter of a circular surface is defined. If the value of the distribution parameter of a circular surface is 0, the circular surface is a sphere. Then the relationship between the moving frame {E1, E2, E3} and the Frenet frame {t, n, b} of the spine curve is investigated, and the expressions of the curvature and torsion of the spine curve are obtained based on these Euclidean invariants. The fundamental theorem of circular surfaces is first proved. Next the first and second fundamental forms of circular surfaces are computed. The last part of this paper is devoted to constraint circular surfaces. The sufficient and necessary condition for a general circular surface to be one that can be generated by a series-connected C’R, HR, RR, or PR mechanism is proved.

Differential Kinematics of Spherical and Spatial Motions Using Kinematic Mapping

1986 ◽  
Vol 53 (1) ◽  
pp. 15-22 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Ravani
Keyword(s):  
Geodesic Curvature ◽  
Intrinsic Properties ◽  
Mechanism Synthesis ◽  
Basic Framework ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Spherical Mechanism ◽  
Spatial Kinematics

This paper develops the basic framework for studying differential kinematics of spherical and spatial motions using a mapping of spatial kinematics. Relationships are derived relating the intrinsic properties of the image curves corresponding to a mapping of spherical and spatial kinematics and the instantaneous invariants of the corresponding spherical and spatial motions. In addition, in the case of spherical motions, the equations for the inflection cone and cubic cone of stationary geodesic curvature, important in spherical mechanism synthesis, are derived in terms of the curvature and torsion of corresponding image curves. Similar relationships defining the polhodes of spherical motions and their curvature at the reference instant are recast as well. A simple example involving a special spherical four-bar motion is also presented.

Kinematic Theory of Spatial Rack Drives

2016 ◽  
Vol 851 ◽  
pp. 265-272
Author(s):  
Emilia Abadjieva
Keyword(s):  
Vector Analysis ◽  
Spatial Motion ◽  
Geometric Characteristics ◽  
Kinematic Theory ◽  
Motion Transformation ◽  
Kinematic Geometry ◽  
Transformation Type ◽  
Analysis And Synthesis

On the basis of performed vector analysis of spatial motion transformation of type rotation into translation is created a kinematic theory of this transformation type. Using the kinematic theory the basic kinematic-geometric characteristics of spatial rack mechanisms, applicable to their analysis and synthesis are initiated. These characteristics: kinematic cylinder of level, kinematic relative helicoids, and kinematic pitch surfaces are elements of kinematic geometry of the spatial rack drives

The Image Curve of the Planet in a Spherical Epicyclic Gear Train

1988 ◽  
Vol 110 (3) ◽  
pp. 281-286
Author(s):  
Q. Ge ◽  
J. M. McCarthy
Keyword(s):  
Reference Frames ◽  
Dimensional Space ◽  
Orthogonal Transformation ◽  
Gear Train ◽  
Euler Parameters ◽  
Four Dimensions ◽  
Epicyclic Gear ◽  
Fixed Reference ◽  
Curvature And Torsion ◽  
Differential Properties

This paper uses the Euler parameters of the motion of the planet of a spherical epicyclic gear train to obtain a curve on the surface of a hypersphere in four dimensions. This curve, called the image curve, represents the rotational motion of the planet as it rolls without slipping on the fixed gear. Two image curves are obtained for two different choices of moving and fixed reference frames and it is shown that they are related by an orthogonal transformation in four dimensional space. The differential properties of the image curve are computed and it is found that the curvature and torsion are constant. A reference position is chosen and the canonical frame and instantaneous invariants of the motion are determined in terms of the dimensions of the gear train.

Kinematic Synthesis of RRSS Spatial Motion Generators using Euler Parameters and Quaternion Algebra

1993 ◽  
Vol 207 (5) ◽  
pp. 355-359 ◽  
Author(s):  
K-W Lee ◽  
Y-S Yoon
Keyword(s):  
Quaternion Algebra ◽  
The Other ◽  
New Method ◽  
Trigonometric Functions ◽  
Spatial Motion ◽  
Euler Parameters ◽  
Kinematic Synthesis ◽  
Motion Generator

This study proposes a new method using Euler parameters and quaternion algebra for the kinematic synthesis of an RRSS spatial motion generator. The proposed method is compared with the other method using rotation angles for two-position and three-position motion guide problems. It is found that the proposed method provides solutions better in most cases tested here because the synthesis equations in the method include no trigonometric functions. The solutions converge more easily than in the other method as well as avoiding rotational singularity problems encountered otherwise.

On the Relation Between Kinematic Mappings of Planar and Spherical Displacements

1986 ◽  
Vol 53 (2) ◽  
pp. 457-459 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Mechanism Design ◽  
Dimensional Space ◽  
Rigid Motion ◽  
Orthogonal Matrix ◽  
Task Planning ◽  
Euler Parameters ◽  
Rigid Displacement ◽  
Higher Dimensional ◽  
Robot Task ◽  
Limiting Case

Kinematic mappings, which transform the parameters specifying a rigid displacement into a point in a higher dimensional space, have been the focus of recent study for the purpose of classifying rigid motion, aiding mechanism design, and automating robot task planning. Presented here is the procedure which demonstrates that the particular mapping introduced by Blaschke and Mu¨iller (1956) to study planar kinematics, and used subsequently by other researchers, is a limiting case of a mapping based on the Euler parameters of an orthogonal matrix used by Ravani (1982) for spherical kinematics.

scholarly journals Moving and fixed axoids of frenet thrihedral of directing curve on example of cylindrical line

2020 ◽  
Vol 11 (3) ◽  
pp. 41-48
Author(s):  
T. A. Kresan ◽  
◽  
S. F. Pylypaka ◽  
V. M. Babka ◽  
Ya. S. Kremets ◽  
...  
Keyword(s):  
Angular Velocity ◽  
Solid Body ◽  
Rotational Motion ◽  
Linear Velocity ◽  
The Body ◽  
Helical Line ◽  
Spatial Motion ◽  
Instantaneous Axis Of Rotation ◽  
Axis Of Rotation ◽  
Curvature And Torsion

If the solid body makes a spatial motion, then at any point in time this motion can be decomposed into rotational at angular velocity and translational at linear velocity. The direction of the axis of rotation and the magnitude of the angular velocity, that is the vector of rotational motion at a given time does not change regardless of the point of the solid body (pole), relative to which the decomposition of velocities. For linear velocity translational motion is the opposite: the magnitude and direction of the vector depend on the choice of the pole. In a solid body, you can find a point, that is, a pole with respect to which both vectors of rotational and translational motions have the same direction. The common line given by these two vectors is called the instantaneous axis of rotation and sliding, or the kinematic screw. It is characterized by the direction and parameter - the ratio of linear and angular velocity. If the linear velocity is zero and the angular velocity is not, then at this point in time the body performs only rotational motion. If it is the other way around, then the body moves in translational manner without rotating motion. The accompanying trihedral moves along the directing curve, it makes a spatial motion, that is, at any given time it is possible to find the position of the axis of the kinematic screw. Its location in the trihedral, as in a solid body, is well defined and depends entirely on the differential characteristics of the curve at the point of location of the trihedral – its curvature and torsion. Since, in the general case, the curvature and torsion change as the trihedral moves along the curve, then the position of the axis of the kinematic screw will also change. Multitude of these positions form a linear surface - an axoid. At the same time distinguish the fixed axoid relative to the fixed coordinate system, and the moving - which is formed in the system of the trihedral and moves with it. The shape of the moving and fixed axoids depends on the curve. The curve itself can be reproduced by rolling a moving axoid over a fixed one, while sliding along a common touch line at a linear velocity, which is also determined by the curvature and torsion of the curve at a particular point. For flat curves, there is no sliding, that is, the movable axoid is rolling over a stationary one without sliding. There is a set of curves for which the angular velocity of the rotation of the trihedral is constant. These include the helical line too. The article deals with axoids of cylindrical lines and some of them are constructed.

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