3:00pm - 5:00pmAlgebraic geometry for kinematics, mechanism science, and rigidity
Chair(s): Matteo Gallet (SISSA, Trieste, Italy), Josef Schicho (JKU University Linz, Austria), Hans-Peter Schröcker (University of Innsbruck, Austria)
Mathematicians became interested in problems concerning mobility and rigidity of mechanisms as soon as study of the subject began. Algebraists and geometers among them, notably Clifford and Study, developed tools still used today to investigate pertinent questions in the field. Recent renewed interest in techniques of algebraic geometry applied to kinematics and rigidity led to a modern classification of mechanisms, discovery of new families, development of algorithms for path planning and overall better understanding of rigid structures and configurations. A wide variety of techniques has been used in this regard and it is reasonable to expect that further influence of algebraic geometry upon kinematics and rigidity will produce deeper understanding leading to useful advancement of technology. We will focus on topics in algebraic geometry motivated by kinematics and rigidity or algebraic geometry methodology with potential application in kinematics and rigidity.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
On four-bar linkages, elliptic functions, and flexible polyhedra
Ivan Izmestiev
Université de Fribourg, Switzerland
Darboux discovered that the (complexified) configuration space of a four-bar linkage is an elliptic curve. We present an explicit parametrization of the configuration space (in terms of the angles between the bars) by Jacobi elliptic functions and some geometric applications of this parametrization: an interpretation of Bottema's zigzag theorem, a derivation of the Dixon angle condition in the Burmester linkage, and examples of flexible quad-surfaces.
Singularity distance computation for parallel manipulators of Stewart Gough Type
Georg Nawratil
Technische Universität Wien, Austria
The number of applications of parallel robots, ranging from medical surgery to astronomy, has increased enormously during the last decades due to their advantages of high speed, stiffness, accuracy, load/ weight ratio, etc. One of the drawbacks of these parallel robots are their singular configurations, where the manipulator has at least one uncontrollable instantaneous degree of freedom. Furthermore, the actuator forces can become very large, which may result in a breakdown of the mechanism. Therefore singularities have to be avoided. As a consequence the kinematic/robotic community is highly interested in evaluating the singularity closeness, but geometric a meaningful distance measure between a given manipulator configuration and the next singular configuration is still missing.
We close this gap for parallel manipulators of Stewart Gough type by introducing such measures. Moreover the favored metric has a clear physical meaning, which is very important for the acceptance of this index by mechanical/constructional engineers.
Every proposed singularity distance results from the solution of an algebraic system of equations, whose computational aspects are discussed on the basis of examples.
Analysis of kinematic singularities through roadmap computations
Mohab Safey El Din1, Eric Schost2
1Sorbonne Universités, Universite Pierre et Marie Curie, France, 2University of Waterloo, Canada
The analysis of kinematic singularities can be modeled as the problem of counting the connected components of a semi-algebraic set or finding a path joining two points in this set whenever such a path exists. These algorithmic problems are known to be difficult and usually tackled through the computation of a roadmap. This is an algebraic curve which will capture the connectivity of the semi-algebraic set under study. In this talk, I will review some recent progress on the state-of-the algorithms for computing roadmaps and report on their implementations which were used to analyze the kinematic singularities of some robots.
Computing cognates of mechanisms
Samantha Sherman1, Jonathan Hauenstein2, Charles Wampler3
1University of Notre Dame, USA, 2University of Notre Dame, 3General Motors
A coupler cognate of a planar linkage is a different mechanism that has the same coupler curve. Roberts showed that there are 3 four-bar mechanisms that generate the same coupler curve. Dijksman provided a list of cognates for six-bar mechanisms but without proof the list was complete. This talk will describe a geometric approach to easily understand cognates which yields a simple method to generate cognates. We combine this with numerical algebraic geometry to give a method to produce a complete list of all coupler cognates. Examples on six - bar mechanisms will be shown to demonstrate the method. This is joint work with Jon Hauenstein and Charles Wampler.