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Organizers |
Asymptotics of Null Lie Quadratics in Tension
by
Shreya Bhattarai
University of Western Australia
A null Riemannian cubic in tension is a curve that arises as a solution to a variational problem on a Riemannian manifold. If the manifold is a bi-invariant Lie group, there is an associated curve V(t) in the Lie algebra called a null Lie quadratic in tension which satisfies the equation
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In "Asymptotics of Null Lie Quadratics in E3", Noakes determines accurate asymptotics for the case k = 0. I will talk about extending these results to the case k > 0.
Date received: November 3, 2008
Riemannian cubics and Lie quadratics: an introduction
by
Brian Corr
University of Western Australia
Riemannian cubics solve a variational problem for curves in Riemannian manifolds. When the manifold is a bi-invariant Lie group their study reduces to that of a class of curves in the Lie algebra (Lie quadratics). Some basic properties of Lie quadratics will be reviewed, especially in Euclidean 3-space.
Date received: November 16, 2008
Determining polynomial invariants of SE(3)
by
Deborah Crook
Victoria University of Wellington
This talk looks at the problem of determining polynomial invariants of the special Euclidean group SE(3) in its adjoint action on its Lie algebra, se(3). Both the action on a single element of se(3), and on a pair of elements, are examined.
Date received: November 2, 2008
Serial manipulators, screw systems and singularities
by
Peter Donelan
Victoria University of Wellington
Serial manipulators consist of a finite sequence of rigid links connected by a corresponding sequence of joints, starting from a base and terminating at the manipulator's end-effector. Each joint can be represented by a twist-an element of the Lie algebra of the Euclidean group SE(3)-or, more properly, a screw which is an element of the corresponding projective space. The forward kinematics of the manipulator are then represented by a product of exponentials in the Euclidean group. However, as the manipulator moves the screws and their span, the instantaneous screw system, vary. Selig obtained explicit formulae for the exponential map which are exploited to analyse these kinematics, with a view to understanding how the screw system varies for a given joint sequence and, in particular, how the given joints determine the manipulator's singularities. This has implications for the classification of over-constrained manipulators which exhibit unexpected self-motion.
Date received: November 2, 2008
Exterior - A Maple 10/11/12 library for computations in exterior calculus
by
Mark Hickman
Department of Mathematics & Statistics, University of Canterbury
Exterior is a package for Maple 10/11/12 that implements the exterior calculus. It allows the construction of jet bundles and exterior differential systems. The user interface is designed to mimic (as much as possible) standard mathematical notation both for the user input and the output. The package allows the user to compute, for example, symmetries of partial differential equations and exterior differential systems, characteristic vectors, Maurer-Cartan forms, torsion of lifted coframes and invariants that arise in the Cartan method of equivalence.
This talk will give concrete examples of computations using this package.
Date received: October 22, 2008
Representation of mechanical constraints on the Euclidean motion group
by
Manfred Husty
University Innsbruck, Austria
Using Study's representation as a convenient model we will discuss the representation of different mechanical systems on the motion group. We will show how mechanical constraints that represent serial robots, parallel robots and other mechanical systems map to different sets or varieties on the group. Especially we will discuss how these representations can be used to solve either direct or inverse kinematics of the systems. In the last part we will show how robot singularities fit into this framework.
Date received: November 2, 2008
Lie group approximation and quantum control
by
Wayne Lawton
Department of Mathematics, National University of Singapore
Approximation of trajectories in unitary groups by (trigonometric) polynomials has applications in classical control (polarization mode dispersion compensators, wavelet design, integrable systems), and the approximation methods are based on operator splitting formuli developed for quantum mechanics. This talk discusses potential applications to quantum control including a program to extend the recent solution of the Ten Martini Problem for the Almost Mathieu Operator to its time dependent analogue, the Kicked-Harper Operator, that provides a model for quantum chaos.
Date received: October 31, 2008
Geometry of Riemannian cubics
by
Gerrard Liddell
Maths, University of Otago
This talk will describe some of the geometry of Riemannian cubics for SO(3).
Date received: November 3, 2008
Riemannian cubics and friends
by
Lyle Noakes
University of Western Australia
A Riemannian cubic is a curve in a manifold solving a particular second order variational problem. Riemannian cubics are higher order geodesics, with applications in mechanical engineering, classical mechanics, and approximation theory with non-affine constraints. They reduce to cubic polynomials in the case of Euclidean space.
When the manifold is a bi-invariant Lie group, Riemannian cubics are studied in terms of an associated curve (the Lie quadratic) in the corresponding Lie algebra. The theory of Lie quadratics is quite rich (and still incomplete), even in SO(3) and SL(2, R).
The talk will review some central results about Riemannian cubics and (time permitting) some connections with the theory of Riemannian Bezier curves.
Date received: October 15, 2008
Quadratures for null Lie quadratics in sl(2) and so(3)
by
Michael Pauley
University of Western Australia
A Lie quadratic in a Lie algebra is a solution to a differential equation which arises in trajectory planning problems, and in computer graphics. A subclass called null Lie quadratics have special meaning, with regard to the applications, as well as the methods by which we study them. I will talk about how, in the Lie algebras sl(2) and so(3), it is possible to write quadrature formulae for null Lie quadratics.
Date received: October 29, 2008
The Riemannian Cox-de Boor algorithm
by
Tomasz Popiel
University of Western Australia
Coauthors: Lyle Noakes
The well-known Cox-de Boor algorithm for constructing polynomial curves generalises in a natural way to a Riemannian manifold M: line segments are replaced by minimal geodesics. The resulting curves can be used to solve interpolation problems in M, which arise in applications including robotics and computer animation, in which M is the Lie group SO(3) of rotations of Euclidean 3-space. Although these curves are straightforward to construct, information about their derivatives, which is needed for applications where the degree of smoothness of an interpolant is important, is usually difficult to establish. We present some recent developments.
Date received: October 27, 2008
Rational interpolation of rigid-body motions
by
J.M. Selig
London South Bank University
The group manifold of the group of rigid-body motions can be considered as an open set in a six-dimensional non-singular quadric, known as the Study quadric. Rational motions are rational curves in this quadric. Using a birational map the quadric can be transformed to six-dimensional projective space P6. These birational maps are simply derived from Cayley maps from the Lie algebra of the group to the group itself. In this way interpolation problems in the group can be transformed into interpolation problems in the Lie algebra. If only rotations about a fixed point are considered, this procedure restricts to well known methods in Computer Graphics and Computer Aided Design. Velocities can also be considered in a straightforward way and this leads to rational approximations for motions determined by variational principles, such as motions with stationary acceleration.
Date received: October 28, 2008