On caustics by reflexion

Bruce, J. W.; Giblin, Peter; Gibson, C. G.

doi:10.1016/0040-9383(82)90004-0

Cited by 24 publications

(30 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is already known that there are subtle links between the geometry of mechanisms and the theory of caustics [10] so it is not unreasonable to expect that there are links between symplectic geometry and the singularities of manipulators.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Singularities of Robot Manipulators

Donelan

2007

Singularity Theory

View full text Add to dashboard Cite

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.

show abstract

Section: Resultsmentioning

confidence: 99%

“…• Further exploration of specific finite-dimensional classes of manipulators with a view to finding transversality theorems in the spirit of, for example, [10].…”

Section: Resultsmentioning

confidence: 99%

Singularities of Robot Manipulators

Donelan

2007

Singularity Theory

View full text Add to dashboard Cite

show abstract

“…Let m be a nonsingular point of such that z = 0. Let us prove that there exists a line m ⊆ 2 containing m, m and J F S m · m where J F S m ∈ Mat 3 is the Jacobian matrix of F S at m. Since rank K < 4, there exists A m b m ∈ 3 × \ 0 0 such that A m m = 0, A m m = 0 and such that…”

Section: Reflected Linesmentioning

confidence: 98%

“…Many mathematicians have studied individually different caustics. In [9,17], when S is not at infinity, Quetelet and Dandelin showed that the caustic is the evolute of the S-centered homothety (with ratio 2) of the pedal curve of from S, i.e., the evolute of the orthotomic of with respect to S. This decomposition has also been used in a modern approach by [2][3][4] in the real case.…”

Section: Introductionmentioning

confidence: 97%

On the Degree of Caustics by Reflection

Josse¹,

Pène²

2014

Communications in Algebra

View full text Add to dashboard Cite

Given a point S ∈ 2 = 2 and an irreducible algebraic curve of 2 (with any type of singularities), we consider the lines m obtained by reflection of the lines S m on (for m ∈ ). The caustic by reflection S is classically defined as the Zariski closure of the envelope of the reflected lines m . We identify this caustic with the Zariski closure of , where is some rational map. We use this approach to give general and explicit formulas for the degree (with multiplicity) of caustics by reflection. Our formulas are expressed in terms of intersection numbers of the initial curve (or of its branches). Our method is based on a fundamental lemma for rational map thanks to the notion of -polar and on the computation of intersection numbers. In particular, we use precise estimates related to the intersection numbers of with its polar at any point and to the intersection numbers of with its Hessian curve. These computations are linked with generalized Plücker formulas for the class and for the number of inflection points of .

show abstract

“…First we prove the following fact. Now suppose that the dimension of M is even, peM and a hyperplane H in the pencil is tangent to M at p. We can measure the contact between M and H at p by considering the height function in some local coordinate system in a direction normal to H on M at p. This contact is defined up to contact equivalence (see [2]) and if it is of type A, this means we only have a Morse local normal form up to sign. However since M is of even dimension the sign of the determinant of the Hessian is well defined and we denote it by e p .…”

mentioning

confidence: 99%

A Zeuthen Segre formula for even dimensional submanifolds of real projective space

Bruce

1983

Glasgow Math. J.

View full text Add to dashboard Cite

In this paper we generalise results of Craveiro de Carvalho ([3]) in two ways. First we prove the following fact. Now suppose that the dimension of M is even, peM and a hyperplane H in the pencil is tangent to M at p. We can measure the contact between M and H at p by considering the height function in some local coordinate system in a direction normal to H on M at p. This contact is defined up to contact equivalence (see [2]) and if it is of type A, this means we only have a Morse local normal form up to sign. However since M is of even dimension the sign of the determinant of the Hessian is well defined and we denote it by e p . (In the case when M is a hypersurface e p is the sign of the Gauss curvature of M at p in any affine chart). peMProof of Proposition 1. We first prove part (a). Since openness follows from general principles it is enough to prove density. If P" denotes the dual space of hyperplanes in P" we claim that it is enough to prove that an (open) dense subset of P" meets any submanifold JVfcP" transversally. https://www.cambridge.org/core/terms. https://doi

show abstract

On caustics by reflexion

Cited by 24 publications

References 4 publications

Singularities of Robot Manipulators

Singularities of Robot Manipulators

On the Degree of Caustics by Reflection

A Zeuthen Segre formula for even dimensional submanifolds of real projective space

Contact Info

Product

Resources

About