On the Hierarchy of screw systems

Donelan, Peter; Gibson, C. G.

doi:10.1007/bf01082452

Cited by 18 publications

(17 citation statements)

References 7 publications

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“…This was first done in a heuristic way by Hunt and then a formal proof was given by Gibson and Hunt [5]. A cleaner version of the classification and proof was then given by Donelan and Gibson [6].…”

Section: B Screw Systemsmentioning

confidence: 99%

A Class of Explicitly Solvable Vehicle Motion Problems

Selig

2015

IEEE Trans. Robot.

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Abstract-A small but interesting result of Brockett is extended to the Euclidean group SE(3) and is illustrated by several examples. The result concerns the explicit solution of an optimal control problem on Lie groups, where the control belongs to a Lie triple system in the Lie algebra. The extension allows for an objective function based on an indefinite quadratic form.Applying the result requires explicit knowledge of the Lie triple systems of the Lie algebra se(3). Hence, a complete classification of the Lie triple systems of this Lie algebra is derived.Examples are considered for optimal trajectories in 3 cases. The first case concerns cars moving in the plane. The second looks at motions that rigidly follow the Bishop frame to a space curve. The final example does not have a particular name as it does not seem to have been studied before.The appendix gives a brief introduction to Screw theory. This is essentially the study of the Lie algebra se(3).

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Section: B Screw Systemsmentioning

confidence: 99%

A Class of Explicitly Solvable Vehicle Motion Problems

Selig

2015

IEEE Trans. Robot.

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“…In [24], it is shown that all the classes described above are submanifolds in the relevant Grassmannian and their adjacency diagrams are established. In particular:…”

Section: Screw Systemsmentioning

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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“…This situation was rectified in [27], and a classification was described intrinsically by the present authors in [14] following the original philosophy of Klein. In [15] we showed that the listing gave rise to natural Whitney stratifications, and gave complete descriptions of the specialisations. These results are outlined below.…”

Section: Screw Systemsmentioning

confidence: 99%

“…Thus a 2-system will intersect the complex in a pencil of points, a 3-system in a pencil of conics, and so on. The details of the classification appear in [14,15]. The most interesting case is the (self-dual) 3-systems.…”

Section: Screw Systemsmentioning

confidence: 99%

“…Given that the classes are unions of orbits of a Lie group acting on the Grassmannian, all the necessary information is obtained by constructing minimal transversals (unfoldings) to orbits at normal forms. Finding transversals is a lengthy but essentially mechanical process whose details can be found in [15]. By inspection one finds that each stratum intersects a transversal in an affine subspace, so is smooth.…”

Section: Regularity and Genericitymentioning

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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On the Hierarchy of screw systems

Cited by 18 publications

References 7 publications

A Class of Explicitly Solvable Vehicle Motion Problems

A Class of Explicitly Solvable Vehicle Motion Problems

Singularities of Robot Manipulators

Singular Phenomena in Kinematics

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