### Algebraic Geometry and Robotics - Recent Results and Applications

Manfred L. Husty

University Innsbruck

Austria

February 7, 2019 at 3:00 PM

McConnell Engineering Room 437

Various mathematical formulations are used to describe the kinematics of mechanisms and
robots. The mathematical modelling is the basis for kinematic analysis and synthesis, e.g.
the computation of motion capabilities, singularities, workspaces, operation modes, velocities
and accelerations on one hand and to obtain design parameters on the other. Vector/matrix
formulation with trigonometric functions is the most favoured approach in the engineering
research community to solve these tasks. A less well known, but nevertheless very successful
approach relies on an algebraic formulation via the point model of Euclidean displacements in
the kinematic image space. In this approach mechanism constraints are described with algebraic
(polynomial) equations and these sets of equations pertaining to some given mechanism or robot,
are solved with the powerful tools of algebraic and numerical algebraic geometry.
Within the talk some new results concerning the kinematics of serial manipulators and lower
dimensional parallel manipulators, such as direct kinematics, workspace, operation and assembly
modes will be discussed. In detail the following topics will be addressed:

- methods to establish the sets of equations – the canonical equations, image space transformations,

- inverse kinematic mapping and motion interpolation,

- Jacobian, singularities and operation modes,

- some examples.

Manfred Husty received a Mag. rer nat.(master degree) 1979 and in 1983 a Dr. techn. both from TU-Graz; he joined the Institut of Mathematics and Applied Geometry of Montan University Leoben, Austria and got the Habilitation in geometry in 1988 from Montan University Leoben. 1993/94 he obtained an Erwin Schrödinger scholarship and went to McGill University, Montreal. Since 1994 he is associate member of the Centre of Intelligent Machines (CIM) of McGill University. He became Full Professor of geometry in 2000 at University Innsbruck. From 2004-2009 he was Dean of Engineering, University Innsbruck. In 2013 he got an honorary doctorate from Technical University Cluj Napoca, Romania.

He teaches several courses in geometry, CAD, and kinematics at both the undergraduate and graduate level. His research activities are supported by grants from several agencies. His main research interests are: robot kinematics, computational kinematics, non-Euclidean differential geometry, theory of surfaces, Lie groups and kinematics. He is member of Austrian mathematical society, Chairmen of IFToMM Austria and past chairman of IFToMM TC Computational Kinematics, editor in chief of IBDG, member of scientific committees of several international conferences (ARK, Computational Kinematics, EUCOMES, MUSME, CCToMM, etc.).