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.

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.

scholarly journals From deep TLS validation to ensembles of atomic models built from elemental motions

2015 ◽  
Author(s):  
Alexandre Urzhumtsev ◽  
Pavel Afonine ◽  
Andrew H Van Benschoten ◽  
James Fraser ◽  
Paul D Adams
Keyword(s):  
Rigid Body ◽  
Diffraction Data ◽  
Harmonic Vibration ◽  
Mathematical Framework ◽  
Matrix Elements ◽  
Computational Tools ◽  
X Ray ◽  
Structural Ensembles ◽  
The Matrix ◽  
Rigid Body Motions

The widely used Translation Libration Screw (TLS) approximation describes concerted motions of atomic groups in X-ray refinement. TLS refinement often provides a better interpretation of diffraction data and the resulting rigid body motions may subsequently be assigned biochemical significance. In TLS refinement, three matrices (T, L and S) describe harmonic vibration, libration and their correlation. Because these matrices describe specific motions, they impose a number of conditions on their elements. Ignoring these conditions while refining the matrix elements may result in matrices that cannot be interpreted in terms of physically realistic motions. We describe a mathematical framework and the computational tools to analyze refined TLS matrices through their decomposition into descriptors of underlying motions. This allows for straightforward validation and identification of implausible TLS parameters. An algorithm for the generation of structural ensembles that are consistent with given TLS parameters, implemented as a part of the Phenix project, is also described.

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.

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.

Generalized Polar Decompositions in Matrix Exponential

2011 ◽  
Vol 130-134 ◽  
pp. 3023-3026
Author(s):  
Yi Min Tian ◽  
Ao Zhang
Keyword(s):  
Differential Equation ◽  
Lie Group ◽  
Analytic Solution ◽  
Matrix Exponential ◽  
Group Method ◽  
Lie Group Method ◽  
The Matrix ◽  
Numeric Solution ◽  
Basic Ideas ◽  
Solution Manifold

Matrix exponential computstion is a difficulty thing when the order of the matrix get big and big after discretion. When we use Lie group method to get numeric solution of a differential equation, we often face this problem.Li group method is a kind of prosperous method, its basic ideas is to keep the numeric solution in a manifold which is less than the Euclid space while bigger than the analytic solution manifold, so we can get more exact numeric solution than other method. So we discussed the generalized polar decompositions method for matrix exponential.

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 Clifford algebra of points, lines and planes

Robotica ◽  
2000 ◽  
Vol 18 (5) ◽  
pp. 545-556 ◽  
Author(s):  
J. M. Selig
Keyword(s):  
Computer Vision ◽  
Computer Graphics ◽  
Rigid Body ◽  
Clifford Algebra ◽  
Stereo Vision ◽  
Vision System ◽  
Linear Elements ◽  
Rigid Body Motions ◽  
Rigid Motions ◽  
Assembly Problem

The Clifford algebra for the group of rigid body motions is described. Linear elements, that is points, lines and planes are identified as homogeneous elements in the algebra. In each case the action of the group of rigid motions on the linear elements is found. The relationships between these linear elements are found in terms of operations in the algebra. That is, incidence relations, the conditions for a point to lie on a line for example are found. Distance relations, like the distance between a point and a plane are found. Also the meet and join of linear elements, for example, the line determined by two planes and the plane defined by a line and a point, are found. Finally three examples of the use of the algebra are given: a computer graphics problem on the visibility of the apparent crossing of a pair of lines, an assembly problem concerning a double peg-in-hole assembly, and a problem from computer vision on finding epipolar lines in a stereo vision system.

scholarly journals RELATIVE CAMERA POSE ESTIMATION METHOD USING OPTIMIZATION ON THE MANIFOLD

2017 ◽  
Vol XLII-1/W1 ◽  
pp. 41-46
Author(s):  
C. Cheng ◽  
X. Hao ◽  
J. Li
Keyword(s):  
Rigid Body ◽  
State Estimation ◽  
Group Algebra ◽  
Pose Estimation ◽  
Lie Group ◽  
General State ◽  
Estimation Model ◽  
Camera Pose Estimation ◽  
Camera Pose ◽  
Rigid Body Transformation

To solve the problem of relative camera pose estimation, a method using optimization with respect to the manifold is proposed. Firstly from maximum-a-posteriori (MAP) model to nonlinear least squares (NLS) model, the general state estimation model using optimization is derived. Then the camera pose estimation model is applied to the general state estimation model, while the parameterization of rigid body transformation is represented by Lie group/algebra. The jacobian of point-pose model with respect to Lie group/algebra is derived in detail and thus the optimization model of rigid body transformation is established. Experimental results show that compared with the original algorithms, the approaches with optimization can obtain higher accuracy both in rotation and translation, while avoiding the singularity of Euler angle parameterization of rotation. Thus the proposed method can estimate relative camera pose with high accuracy and robustness.

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