Nonintrinsicity of References in Rigid-Body Motions

2001 ◽  
Vol 68 (6) ◽  
pp. 929-936 ◽  
Author(s):  
S. Stramigioli
Keyword(s):  
Rigid Body ◽  
Lie Groups ◽  
Relative Motion ◽  
Lie Group ◽  
Rigid Bodies ◽  
Group Approach ◽  
Rigid Body Motions ◽  
Reference Problem

This paper shows that in the use of Lie groups for the study of the relative motion of rigid bodies some assumptions are not explicitly stated. A commutation diagram is shown which points out the “reference problem” and its simplification to the usual Lie group approach under certain conditions which are made explicit.

On the geometry of rigid-body motions: The relation between Lie groups and screws

2002 ◽  
Vol 216 (1) ◽  
pp. 13-23 ◽  
Author(s):  
S Stramigioli ◽  
B Maschke ◽  
C Bidard
Keyword(s):  
Rigid Body ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Approach ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Group Approach ◽  
Rigid Body Motions

This paper gives a synthetic presentation of the geometry of rigid-body motion in a projective geometrical framework. An important issue is the geometric approach to the identification of twists and wrenches in a Lie group approach and their relation to screws. The paper presents a novel formal way to describe the spaces of lines, axials, polars and screws as subsets or subspaces of Lie algebras in order to make clear the relation between screw concepts and Lie group concepts.

Velocity and Acceleration Analysis of Contact Between Three-Dimensional Rigid Bodies

1996 ◽  
Vol 63 (4) ◽  
pp. 974-984 ◽  
Author(s):  
N. Sankar ◽  
V. Kumar ◽  
Xiaoping Yun
Keyword(s):  
Rigid Body ◽  
Relative Motion ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
Local Surface ◽  
Contact Points ◽  
Pure Rolling ◽  
Acceleration Analysis ◽  
Rigid Body Motions ◽  
Two Bodies

During manipulation and locomotion tasks encountered in robotics, it is often necessary to control the relative motion between two contacting rigid bodies. In this paper we obtain the equations relating the motion of the contact points on the pair of contacting bodies to the rigid-body motions of the two bodies. The equations are developed up to the second order. The velocity and acceleration constraints for contact, for rolling, and for pure rolling are derived. These equations depend on the local surface properties of each contacting body. Several examples are presented to illustrate the nature of the equations.

Velocity and Acceleration Analysis of Contact Between Three-Dimensional Rigid Bodies

1994 ◽  
Author(s):  
Nilanjan Sarkar ◽  
Vijay Kumar ◽  
Xiaoping Yun
Keyword(s):  
Rigid Body ◽  
Relative Motion ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
Local Surface ◽  
Contact Points ◽  
Pure Rolling ◽  
Acceleration Analysis ◽  
Rigid Body Motions ◽  
Two Bodies

Abstract During manipulation and locomotion tasks encountered in robotics, it is often necessary to control the relative motion between two contacting rigid bodies. In this paper we obtain the equations relating the motion of the contact points on the pair of contacting bodies to the rigid body motions of the two bodies. The equations are developed up to the second order. The velocity and acceleration constraints for contact, for rolling, and for pure rolling are derived. These equations depend on the local surface properties of each contacting body. Several examples are presented to illustrate the nature of the equations.

scholarly journals Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Z. E. Musielak ◽  
N. Davachi ◽  
M. Rosario-Franco
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Algebraic Structure ◽  
Second Order ◽  
Lagrangian Formalism ◽  
Group Approach ◽  
Second Order Differential Equations ◽  
Gauge Functions ◽  
The Relationship

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

On Higher Order Point-Line and the Associated Rigid Body Motions

2006 ◽  
Vol 129 (2) ◽  
pp. 166-172 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Higher Order ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Dual Quaternion ◽  
Displacement Operator ◽  
Screw Motion ◽  
Rigid Body Motions ◽  
Point Line

This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of ∞2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

CONJUGATION IN THE DISPLACEMENT GROUP AND MOBILITY IN MECHANISMS

2009 ◽  
Vol 33 (2) ◽  
pp. 163-174 ◽  
Author(s):  
Jacques M. Hervé
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Algebraic Structure ◽  
Frame Of Reference ◽  
Mathematical Tool ◽  
Simple Matter ◽  
Rigid Body Motions ◽  
Displacement Group

The paper deals with the Lie group algebraic structure of the set of Euclidean displacements, which represent rigid-body motions. We begin by looking for a representation of a displacement, which is independent of the choice of a frame of reference. Then, it is a simple matter to prove that displacement subgroups may be invariant by conjugation. This mathematical tool is suitable for solving special problems of mobility in mechanisms.

Simultaneous Coordination of Two Rigid Body Motions Through Infinitesimally Separated Positions

1975 ◽  
Vol 97 (2) ◽  
pp. 527-531
Author(s):  
M. N. Siddhanty ◽  
A. H. Soni
Keyword(s):  
Rigid Body ◽  
Fourth Order ◽  
Rigid Bodies ◽  
Mathematical Approach ◽  
Link Mechanism ◽  
Rigid Body Motions ◽  
Design Solutions ◽  
Mathematical Relationships

A generalized mathematical approach is developed to guide two rigid bodies for simultaneous coordination of their infinitesimally separated positions. Mathematical relationships are developed to incorporate up to fourth-order derivatives while specifying infintesimally separated positions. The approach is demonstrated by considering an eight-link mechanism. It is shown that for a maximum of five precision positions of the two rigid bodies, a maximum of 1024 design solutions are possible.

scholarly journals The theory on homogeneous micro-fluid mixtures

1979 ◽  
Vol 1 (3-4) ◽  
pp. 38-48
Author(s):  
Nguyen Van Diep ◽  
Truong Minh Chanh
Keyword(s):  
Rigid Body ◽  
Relative Motion ◽  
Constitutive Equations ◽  
Fluid Mixtures ◽  
Rigid Body Motions ◽  
Thermodynamic Restrictions

This paper deals with the construction x of a generalized- diffusive theory of simple micro-fluid mixtures composed of a reactive constituents in the relative motion to one another and treated as a single simple micro-fluid. The basic laws of balance are given by postulating the balance law of energy together with invariant requirement under superposed rigid body motions or the mixture as a whole. The linear or nonlinear constitutive equations and full thermodynamic restrictions are given.

A Computationally Efficient Planar Rigid Body Velocity Measure

2009 ◽  
Author(s):  
Luis E. Criales ◽  
Joseph M. Schimmels
Keyword(s):  
Rigid Body ◽  
Rigid Bodies ◽  
Body Motion ◽  
Computationally Efficient ◽  
Straight Line ◽  
Body Velocity ◽  
Velocity Measure ◽  
Rigid Body Motions ◽  
Line Path ◽  
Planar Motions

A planar rigid body velocity measure based on the instantaneous velocity of all particles that constitute a rigid body is developed. This measure compares the motion of each particle to an “ideal”, but usually unobtainable, motion. This ideal motion is one that would carry each particle from its current position to its desired position on a straight-line path. Although the ideal motion is not a valid rigid body motion, this does not preclude its use as a reference standard in evaluating valid rigid body motions. The optimal instantaneous planar motions for general rigid bodies in translation and rotation are characterized. Results for an example planar positioning problem are presented.

Choice of Riemannian Metrics for Rigid Body Kinematics

1996 ◽  
Author(s):  
Miloš Žefran ◽  
Vijay Kumar ◽  
Christopher Croke
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Three Dimensions ◽  
Euclidean Group ◽  
Screw Motion ◽  
Screw Motions ◽  
Rigid Body Motions ◽  
Two Parameter

Abstract The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, SE(3). Chasles’s theorem states that there exists a screw motion between two arbitrary elements of SE(3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian metric with such geodesics exists and we show that the metrics whose geodesics are screw motions form a two-parameter family of semi-Riemannian metrics.

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