Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics

Wu, Yuanqing; Löwe, Harald; Carricato, Marco; Li, Zigang

doi:10.1109/tro.2016.2522442

Cited by 39 publications

(82 citation statements)

References 44 publications

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“…The Lie triple systems of se(3) were classified in [4,7,12], details of symmetric subspaces of SE(3) can also be found in [14]. It was observed in [4], that most of the symmetric subspaces of SE(3) are linear spaces or the intersection of the Study quadric Q s with a linear subspace.…”

Section: Subgroups and Symmetric Subspacesmentioning

confidence: 99%

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Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Selig

Carricato

2017

Computational Kinematics

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We show that a map defined by Pfurner, Schröcker and Husty, mapping points in 7dimensional projective space to the Study quadric, is equivalent to the composition of an extended inverse Cayley map with the direct Cayley map, where the Cayley map in question is associated to the adjoint representation of the group SE(3). We also verify that subgroups and symmetric subspaces of SE(3) lie on linear spaces in dual quaternion representation of the group. These two ideas are combined with the observation that the Pfurner-Schröcker-Husty map preserves these linear subspaces. This means that the interpolation method proposed by Pfurner et al can be restricted to subgroups and symmetric subspaces of SE(3).

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Section: Subgroups and Symmetric Subspacesmentioning

confidence: 99%

“…The theorem can be proved by straightforward inspection of all possible cases. All possibilities were found in [12,4] and [7]. To find points in the symmetric subspaces we need to be able to exponentiate elements of the Lie triple system.…”

Section: Subgroups and Symmetric Subspacesmentioning

confidence: 99%

Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Selig

Carricato

2017

Computational Kinematics

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“…Motivated by this observation, Stramigioli [150] uses this as the defining property of lower pairs. Recently, Selig [145] and Wu et al [158] seized on a concept by Brockett [28] called Lie triple systems. The latter refers to systems of vector fields that are closed under the triple Lie bracket (whereas a Lie algebra is closed under the Lie bracket).…”

Section: Relative Motions As Screw Motionsmentioning

confidence: 99%

“…The latter refers to systems of vector fields that are closed under the triple Lie bracket (whereas a Lie algebra is closed under the Lie bracket). Wu [158] pointed out that this concept allows for studying the kinematic of non-lower pair joints, such as constant velocity joints.…”

Section: Relative Motions As Screw Motionsmentioning

confidence: 99%

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Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

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Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

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Unified Pose Parametrization for 1T2R Parallel Manipulators

Carricato

2018

Mechanism Design for Robotics

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Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics

Abstract: If it is the author's pre-published version, changes introduced as a result of publishing processes such as copy-editing and formatting may not be reflected in this document. For a definitive version of this work, please refer to the published version.

Cited by 39 publications

References 44 publications

Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Geometric methods and formulations in computational multibody system dynamics

Unified Pose Parametrization for 1T2R Parallel Manipulators

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