Metrics and connections for rigid-body kinematics

Zefran, M; Kumar, V; Croke, C

doi:10.1177/02783649922066187

Title: Metrics and connections for rigid-body kinematics

Full Record
Other Related Research

Abstract

The set of rigid-body motions forms a Lie group called SE(3), the special Euclidean group in three dimensions. In this paper, the authors investigate possible choices of Riemannian metrics and affine connections on SE(3) for applications to kinematic analysis and robot-trajectory planning. In the first part of the paper, they study metrics whose geodesics are screw motions. They prove that no Riemannian metric can have such geodesics, and they show that the metrics whose geodesics are screw motions from a two-parameter family of semi-Riemannian metrics. In the second part of the paper, they investigate affine connections which through the covariant derivative give the correct expression for the acceleration of a rigid body. They prove that there is a unique symmetric connection with this property. Furthermore, they show that there is a family of Riemannian metrics that are compatible with such a connection. These metrics are products of the bi-invariant metric on the group of rotations and a positive-definite constant metric on the group of translations.

Authors:

Zefran, M ^[1]; Kumar, V; Croke, C ^[2]

California Inst. of Tech., Pasadena, CA (United States)
Univ. of Pennsylvania, Philadelphia, PA (United States)

Publication Date:: Mon Feb 01 00:00:00 EST 1999

OSTI Identifier:: 346804

Resource Type:: Journal Article

Journal Name:: International Journal of Robotics Research

Additional Journal Information:: Journal Volume: 18; Journal Issue: 2; Other Information: PBD: Feb 1999

Country of Publication:: United States

Language:: English

Subject:: 32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATION; RIEMANN SPACE; MANIPULATORS; CONTROL THEORY; TRAJECTORIES; ROBOTS; COMPUTERIZED CONTROL SYSTEMS

Citation Formats


                    Zefran, M, Kumar, V, and Croke, C. Metrics and connections for rigid-body kinematics.  United States: N. p., 1999. 
        Web.  doi:10.1177/02783649922066187.

Copy to clipboard


                    Zefran, M, Kumar, V, & Croke, C. Metrics and connections for rigid-body kinematics.  United States.  https://doi.org/10.1177/02783649922066187

Copy to clipboard


                    Zefran, M, Kumar, V, and Croke, C. 1999.  
        "Metrics and connections for rigid-body kinematics".  United States.  https://doi.org/10.1177/02783649922066187.

Copy to clipboard


                    
@article{osti_346804,

  title        = {Metrics and connections for rigid-body kinematics},

  author       = {Zefran, M and Kumar, V and Croke, C},

  abstractNote = {The set of rigid-body motions forms a Lie group called SE(3), the special Euclidean group in three dimensions. In this paper, the authors investigate possible choices of Riemannian metrics and affine connections on SE(3) for applications to kinematic analysis and robot-trajectory planning. In the first part of the paper, they study metrics whose geodesics are screw motions. They prove that no Riemannian metric can have such geodesics, and they show that the metrics whose geodesics are screw motions from a two-parameter family of semi-Riemannian metrics. In the second part of the paper, they investigate affine connections which through the covariant derivative give the correct expression for the acceleration of a rigid body. They prove that there is a unique symmetric connection with this property. Furthermore, they show that there is a family of Riemannian metrics that are compatible with such a connection. These metrics are products of the bi-invariant metric on the group of rotations and a positive-definite constant metric on the group of translations.},

  doi          = {10.1177/02783649922066187},

  url          = {https://www.osti.gov/biblio/346804},
  journal      = {International Journal of Robotics Research},
number       = 2,

  volume       = 18,

  place        = {United States},

  year         = {Mon Feb 01 00:00:00 EST 1999},

  month        = {Mon Feb 01 00:00:00 EST 1999}

}

Copy to clipboard

Journal Article:

https://doi.org/10.1177/02783649922066187

Other availability

Find in Google Scholar

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Similar records in OSTI.GOV collections:

IMPLICIT AND FORMAL USES OF A FLAT METRIC IN GENERAL RELATIVITY

Thesis/Dissertation Katzin, G

The meaning of functions of the Riemannian coordinates and their relation to measurements are investigated. It is shown how symmetries of a Riemannian space-time describable by groups of motions may be given physical interpretation in terms of fiat space-time concepts. The linear first integrals of the differential equations of motion of a test particle in a curved space-time are interpreted in terms of flat space-time counterparts. Analysis of the symmetries admitted by the Friedmann-Lemaitre space-time for an open cosmological model shows that they have the same structure as the homogeneous Lorentz group of a flat space-time expressed in a properlymore »« less
Motion of small bodies in classical field theory

Journal Article Gralla, Samuel - Physical Review. D, Particles Fields

I show how prior work with R. Wald on geodesic motion in general relativity can be generalized to classical field theories of a metric and other tensor fields on four-dimensional spacetime that (1) are second-order and (2) follow from a diffeomorphism-covariant Lagrangian. The approach is to consider a one-parameter-family of solutions to the field equations satisfying certain assumptions designed to reflect the existence of a body whose size, mass, and various charges are simultaneously scaled to zero. (That such solutions exist places a further restriction on the class of theories to which our results apply.) Assumptions are made only onmore »« less
https://doi.org/10.1103/PHYSREVD.81.084060
Metric-connection theories of gravity

Thesis/Dissertation Yasskin, P

In this thesis I study the metric-connection theories of gravity. I am mostly concerned with theories whose connection cannot be completely specified in terms of the metric. Although I sometimes consider completely general connections, I often restrict my attention to Cartan connections (metric-compatible but nontorsion-free). In the context of metric-Cartan connection theories, I try to answer the question, is the torsion observable. Using a method similar to that used by Papapetrou and Dixon in the context of metric theories, I derive propagation equations for a body's momentum and angular momentum. These show that elementary particle spins feel the torsion butmore »« less
Affine connections in special relativity

Journal Article Epstein, K - Phys. Rev., D; (United States)

Affine geometry, implemented by a ''Lorentz connection'' for accelerated frames of reference in pseudo-Euclidean space-times, a ''Fourier connection'' for abstract Hilbert spaces associated with classical Fourier analysis, and a ''quantum connection'' for quantum-mechanical Hilbert spaces, give snew perspectives on special relativity, where affine connections are usually interpreted phenomenologically, in a manner which obscures theri geometric significance. The connections are determined by ''absolute constants,'' whose covariant derivatives, expressed in terms of the connection coefficients, vanish identically. In the deterministic theory of prequantum physics, the space-time connection is expressed in terms of ''local Lorentz transformations,'' which represent the motion of accelerated framesmore »« less
https://doi.org/10.1103/PhysRevD.18.1837
Kinematic approach in the impact problem of rigid bodies

Conference Sinopoli, A - Applied Mechanics Reviews; (United States)

The aim of this work is to analyze, by means of a kinematic approach, the problem of the impact between rigid bodies, when the surfaces involved in the impulsive phenomenon are of finite extent. The formulation here adopted permits to use the Gauss variational principle of least compulsion and to formulate the dynamic evolution of the system, after an impact, as a minimization problem. In this case, among all the possible subsequent motions, the real one is that which minimizes the kinetic energy connected to the sudden velocities variations. Interesting results are obtained in the case of the impact betweenmore »« less

Similar Records