Be´zier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications

1995 ◽  
Vol 117 (1) ◽  
pp. 36-40 ◽  
Author(s):  
F. C. Park ◽  
B. Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Reference Frames ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Fixed Reference ◽  
Spatial Displacements ◽  
Natural Way

In this article we generalize the concept of Be´zier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Be´zier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Be´zier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Be´zier sense) using this recursive algorithm. We apply this alogorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body. The orientation trajectory of motions generated in this way have the important property of being invariant with respect to choices of inertial and body-fixed reference frames.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

scholarly journals Riemannian Means as Solutions of Variational Problems

2006 ◽  
Vol 9 ◽  
pp. 86-103 ◽  
Author(s):  
Luís Machado ◽  
F. Silva Leite ◽  
Knut Hüper
Keyword(s):  
Riemannian Manifold ◽  
Variational Problem ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Single Point ◽  
Variational Problems ◽  
Compact Lie Groups ◽  
Data Set ◽  
The Given ◽  
Best Fit

We formulate a variational problem on a Riemannian manifoldMwhose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points inM. By a limiting process, these solutions converge to a single point inM. which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.

scholarly journals Pose Changes From a Different Point of View

2018 ◽  
Vol 10 (2) ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Robert Mahony ◽  
Sipu Ruan ◽  
Jochen Trumpf
Keyword(s):  
Rigid Body ◽  
Reference Frame ◽  
Semidirect Product ◽  
Group Structure ◽  
Linear Part ◽  
Point Of View ◽  
Affine Transformations ◽  
Fixed Reference ◽  
New Perspective ◽  
Direct Product Group

For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semidirect product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference frame to another is usually not articulated well in the literature. Here, we show that, remarkably, when changes in pose are viewed from a space-fixed reference frame, the space of pose changes can be endowed with a direct product group structure, which is different from the semidirect product structure of the space of motions. We then show how this new perspective can be applied more naturally to problems such as monitoring the state of aerial vehicles from the ground, or the cameras in a humanoid robot observing pose changes of its hands.

Pose Changes From a Different Point of View

2017 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Robert Mahony ◽  
Sipu Ruan ◽  
Jochen Trumpf
Keyword(s):  
Rigid Body ◽  
Reference Frame ◽  
Direct Product ◽  
Group Structure ◽  
Linear Part ◽  
Point Of View ◽  
Affine Transformations ◽  
Fixed Reference ◽  
New Perspective ◽  
Direct Product Group

For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semi-direct product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference frame to another is usually not articulated well in the literature. Here we show that, remarkably, when changes in pose are viewed from a space-fixed reference frame, the space of pose changes can be endowed with a direct product group structure, which is different from the semi-direct product structure of the space of motions. We then show how this new perspective can be applied more naturally to problems such as monitoring the state of aerial vehicles from the ground, or the cameras in a humanoid robot observing motions of its hands.

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

POISSON–LIE GROUPS AND CLASSICAL W-ALGEBRAS

1992 ◽  
Vol 07 (05) ◽  
pp. 853-876 ◽  
Author(s):  
V. A. FATEEV ◽  
S. L. LUKYANOV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Group ◽  
Lie Groups ◽  
Group Structure ◽  
Two Dimensional ◽  
Poisson Structures ◽  
Additional Symmetries ◽  
Conformal Field ◽  
Poisson Lie Groups

This is the first part of a paper studying the quantum group structure of two-dimensional conformal field theory with additional symmetries. We discuss the properties of the Poisson structures possessing classical W-invariance. The Darboux variables for these Poisson structures are constructed.

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