Path Planning in Kinematic Image Space Without the Study Condition

Pfurner, Martin; Schröcker, Hans-Peter; Husty, Manfred L.

doi:10.1007/978-3-319-56802-7_30

Cited by 12 publications

(28 citation statements)

References 6 publications

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“…Below we make this statement precise. We also verify that the map defined in [5] respects these linear subspaces; a point in P 7 , not on the Study quadric but lying in a linear subspace which defines a subgroup or symmetric subspace, will be mapped to a point in the intersection of the subspace and Q S . So the interpolation method of Pfurner, Schröcker and Husty (PSH method) is ideally suited as an interpolation method on the symmetric subspaces.…”

Section: Introductionmentioning

confidence: 64%

“…The extended map will be denoted Cay −1 6 . In [5] Pfurner et al introduced a simple method for interpolating rigid-body motions. The algorithm consists of writing the control points of the motion as dual quaternions and then performing the interpolation in the ambient P 7 .…”

Section: Cayley Mapsmentioning

confidence: 99%

“…Finally, the motion is found by projecting the curve into the group. The map given in [5] takes an arbitrary point of P 7 to Q S in P 7 , so let us write the coordinates in the first P 7 as a andc, so the map can be written as,…”

Section: Cayley Mapsmentioning

confidence: 99%

“…As a first example, consider linearly interpolating between the identity in the group and a 2θ screw motion with pitch p, note that linear interpolation was also considered in [5]. We can choose coordinates so that the screw axis is the x-axis and then the group element will be,…”

Section: Interpolationmentioning

confidence: 99%

“…Recently [5], a new simple method was presented for interpolating motions based on the embedding of the group of rigid-body displacements SE(3) in the seven dimensional projective space P 7 as a non-singular quadric, known as the Study quadric Q S . This method is reminiscent of a method proposed by Belta and Kumar where interpolation was carried out on matrices and then the matrices mapped back to the group of rigid-body displacements, see [9] and references therein.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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: Introductionmentioning

confidence: 64%

Section: Cayley Mapsmentioning

confidence: 99%

Section: Cayley Mapsmentioning

confidence: 99%

Section: Interpolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Selig

Carricato

2017

Computational Kinematics

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Space Kinematics and Projective Differential Geometry over the Ring of Dual Numbers

Schröcker

Pfurner

Siegele

2020

Advances in Intelligent Systems and Computing

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We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers. Some seemingly weird phenomena in this space have lucid kinematic interpretations. An example is the existence of nonstraight curves with a continuum of osculating tangents which correspond to motions in a cylinder group with osculating vertical Darboux motions. We also look at the set of osculating conics of a curve in projective space, suggest geometrically meaningful examples and briefly discuss and illustrate their corresponding motions.

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Kinematic Interpretation of the Study Quadric’s Ambient Space

Nawratil

2018

Advances in Robot Kinematics 2018

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It is well known that real points of the Study quadric (sliced along a 3-dimensional generator space) correspond to displacements of the Euclidean 3-space. But we still lack of a kinematic meaning for the points of the ambient 7-dimensional projective space P 7 . This paper gives one possible interpretation in terms of displacements of the Euclidean 4-space. From this point of view we also discuss the extended inverse kinematic map, motions corresponding to straight lines in P 7 and linear complexes of SE(3)-displacements. Moreover we present an application of this interpretation in the context of interactive motion design.

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Path Planning in Kinematic Image Space Without the Study Condition

Cited by 12 publications

References 6 publications

Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Motion Interpolation in Lie Subgroups and Symmetric Subspaces

Space Kinematics and Projective Differential Geometry over the Ring of Dual Numbers

Kinematic Interpretation of the Study Quadric’s Ambient Space

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