Singularities of general one-dimensional motions of the plane and space

Gibson, C. G.; Hobbs, C. A.

doi:10.1017/s030821050003273x

Cited by 12 publications

(20 citation statements)

References 10 publications

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“…In a sequence of papers [25,[27][28][29][30][31]33,41,43], Gibson and co-workers filled out details of this programme for planar and spatial motions up to 3-dof, in the process generating new lists of (multi-)singularities of maps between spaces of dimension up to 3. It was shown in [25] that the stable singularity type of the kinematic mapping germ itself imposes restrictions on the singularity type of its trajectory germs.…”

Section: Trajectory Singularities and A Transversality Theoremmentioning

confidence: 99%

Singularities of Robot Manipulators

Donelan

2007

Singularity Theory

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Engineers have for some time known that singularities play a significant role in the design and control of robot manipulators. Singularities of the kinematic mapping, which determines the position of the end-effector in terms of the manipulator's joint variables, may impede control algorithms, lead to large joint velocities, forces and torques and reduce instantaneous mobility. However they can also enable fine control, and the singularities exhibited by trajectories of the points in the end-effector can be used to mechanical advantage.A number of attempts have been made to understand kinematic singularities and, more specifically, singularities of robot manipulators, using aspects of the singularity theory of smooth maps. In this survey, we describe the mathematical framework for manipulator kinematics and some of the key results concerning singularities. A transversality theorem of Gibson and Hobbs asserts that, generically, kinematic mappings give rise to trajectories that display only singularity types up to a given codimension. However this result does not take into account the specific geometry of manipulator motions or, a fortiori , to a given class of manipulator. An alternative approach, using screw systems, provides more detailed information but also shows that practical manipulators may exhibit high codimension singularities in a stable way. This exemplifies the difficulties of tailoring singularity theory's emphasis on the generic with the specialized designs that play a key role in engineering.

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Section: Trajectory Singularities and A Transversality Theoremmentioning

confidence: 99%

Singularities of Robot Manipulators

Donelan

2007

Singularity Theory

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“…Descriptions of generic n-dof planar motions, including complete listings of the (mono and multi) germs which can appear on their trajectories, fall into three distinct cases, n = 1 treated in [23], n = 2 treated in [24] and n ≥ 3 treated in [17]. Planar motions are maps into the 3-dimensional group SE(2).…”

Section: Trajectories Of Planar Motionsmentioning

confidence: 99%

“…As to multigerms, the case when both branches are immersive appeared in [57]. [23] provides a convenient reference for pictures of the versal unfoldings. In particular, the ramphoid cusp illustrates the danger (mentioned in Section 3.4) of trying to obtain local pictures of bifurcation sets by finding defining equations.…”

Section: Theorem 51 ([23])mentioning

confidence: 99%

“…The case n = 1 was covered in [23] and gives rise to the remarkably short list in Table 7: the two monogerms represent the beginning of the list of A-simple monogerms of space curves obtained in [25]. Bifurcation takes place on a nodal surface on which lies a cuspidal curve, and exceptional points distinct from the curve corresponding to tacnodes and triple points: however as we have seen in Section 4 the cuspidal curve is forced by the geometry of the motion to take a special form, namely to be a union of lines.…”

Section: One Dof Spatial Motionsmentioning

confidence: 99%

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Singular Phenomena in Kinematics

Donelan¹,

Gibson²

1999

Singularity Theory

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Les principes généraux de la theorie des systèmes articulés n'ont pas encoreétééclairais et on est reduità un ensemble de recherches isolées et de résultats curieux sans liens apparents entre eux. C'est une raison de plus pour engager les géomètresàéclairer cette question encore obscure: les progrès de la science se font parfois par les côtés les plus inattendus. Gabriel Koenigs, Leçons de Cinématique.

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“…We know from [21] that this stratum is diffeomorphic to half a line. To find its parametrization, suppose that ℎ( , ) = ⟨ ( ), ⟩ be the height function along the unit vector .…”

Section: The Ramphoid Cuspmentioning

confidence: 99%

Flat and Round Singularity theory

Salarinoghabi¹

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Singularities of general one-dimensional motions of the plane and space

Cited by 12 publications

References 10 publications

Singularities of Robot Manipulators

Singularities of Robot Manipulators

Singular Phenomena in Kinematics

Flat and Round Singularity theory

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