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.

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.

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.

scholarly journals A note on para-holomorphic Riemannian–Einstein manifolds

2016 ◽  
Vol 13 (09) ◽  
pp. 1650107
Author(s):  
Cristian Ida ◽  
Alexandru Ionescu ◽  
Adelina Manea
Keyword(s):  
Complex Manifold ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Complex Geometry ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Einstein Metric ◽  
Einstein Manifolds ◽  
Norden Metric ◽  
Norden Metrics

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. First, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using a one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler–Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler–Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits a natural para-Kählerian–Norden Einstein metric.

Coordinate Mappings for Rigid Body Motions

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Equations Of Motion ◽  
Time Integration ◽  
Exponential Mapping ◽  
Group Setting ◽  
Dual Quaternions ◽  
Integration Schemes ◽  
Rigid Body Motions ◽  
Lie Group Integration

Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group SE(3) of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group Sp̂(1), and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the SE(3) representation. This relation gives rise to a novel closed form of the dexp mapping on SE(3). In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

scholarly journals Geometric Interpretation of a Non-Linear Beam Finite Element on The Lie Group SE(3)

2014 ◽  
Vol 61 (2) ◽  
pp. 305-329 ◽  
Author(s):  
Valentin Sonneville ◽  
Alberto Cardona ◽  
Olivier Brüls
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Lie Group ◽  
Geometric Interpretation ◽  
Invariance Property ◽  
Exponential Map ◽  
Element Formulation ◽  
Equilibrium Equations ◽  
Geometrically Exact Beam ◽  
Rigid Body Motions

Abstract Recently, the authors proposed a geometrically exact beam finite element formulation on the Lie group SE(3). Some important numerical and theoretical aspects leading to a computationally efficient strategy were obtained. For instance, the formulation leads to invariant equilibrium equations under rigid body motions and a locking free element. In this paper we discuss some important aspects of this formulation. The invariance property of the equilibrium equations under rigid body motions is discussed and brought out in simple analytical examples. The discretization method based on the exponential map is recalled and a geometric interpretation is given. Special attention is also dedicated to the consistent interpolation of the velocities.

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.

scholarly journals Three-dimensional solvsolitons and the minimality of the corresponding submanifolds

2017 ◽  
Vol 28 (06) ◽  
pp. 1750048 ◽  
Author(s):  
Takahiro Hashinaga ◽  
Hiroshi Tamaru
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Solvable Lie Groups ◽  
Left Invariant Riemannian Metric ◽  
Invariant Riemannian Metrics ◽  
Simply Connected

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.

scholarly journals Vectorial parameterizations of pose

Robotica ◽  
2021 ◽  
pp. 1-19
Author(s):  
Timothy D. Barfoot ◽  
James R. Forbes ◽  
Gabriele M. T. D’Eleuterio
Keyword(s):  
Computer Vision ◽  
Rigid Body ◽  
Pose Estimation ◽  
Lie Group ◽  
Matrix Exponential ◽  
Uncertainty Representation ◽  
The Matrix ◽  
Alignment Problem ◽  
Cayley Transformation ◽  
Rigid Body Motions

Abstract Robotics and computer vision problems commonly require handling rigid-body motions comprising translation and rotation – together referred to as pose. In some situations, a vectorial parameterization of pose can be useful, where elements of a vector space are surjectively mapped to a matrix Lie group. For example, these vectorial representations can be employed for optimization as well as uncertainty representation on groups. The most common mapping is the matrix exponential, which maps elements of a Lie algebra onto the associated Lie group. However, this choice is not unique. It has been previously shown how to characterize all such vectorial parameterizations for SO(3), the group of rotations. Some results are also known for the group of poses, where it is possible to build a family of vectorial mappings that includes the matrix exponential as well as the Cayley transformation. We extend what is known for these pose mappings to the $4 \times 4$ representation common in robotics and also demonstrate three different examples of the proposed pose mappings: (i) pose interpolation, (ii) pose servoing control, and (iii) pose estimation in a pointcloud alignment problem. In the pointcloud alignment problem, our results lead to a new algorithm based on the Cayley transformation, which we call CayPer.

Distance Metrics on the Rigid-Body Motions with Applications to Mechanism Design

1995 ◽  
Vol 117 (1) ◽  
pp. 48-54 ◽  
Author(s):  
F. C. Park
Keyword(s):  
Rigid Body ◽  
Physical Space ◽  
Length Scale ◽  
Distance Metrics ◽  
Distance Metric ◽  
Euclidean Group ◽  
Invariant Distance ◽  
Mathematical Question ◽  
Left And Right ◽  
Rigid Body Motions

In this article we examine the problem of designing a mechanism whose tool frame comes closest to reaching a set of desired goal frames. The basic mathematical question we address is characterizing the set of distance metrics in SE(3), the Euclidean group of rigid-body motions. Using Lie theory, we show that no bi-invariant distance metric (i.e., one that is invariant under both left and right translations) exists in SE(3), and that because physical space does not have a natural length scale, any distance metric in SE(3) will ultimately depend on a choice of length scale. We show how to construct left- and right-invariant distance metrics in SE(3), and suggest a particular left-invariant distance metric parametrized by length scale that is useful for kinematic applications. Ways of including engineering considerations into the choice of length scale are suggested, and applications of this distance metric to the design and positioning of certain planar and spherical mechanisms are given.

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