UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

September 30, 2008

939 Evans Hall

3:45PM: Exceptional Steward-Gough platforms of geometric genus 7

Frank-Olaf Schreyer

We call a mechanical gadget exceptional, if its degree of freedom exceeds the dimension of a general mechanism of the same underlying topological type. In this paper we establish the existence of exceptional Steward-Gough platforms by a combination of algebraic geometric and number theoretic methods. A Steward-Gough platform consists of two rigid bodies, one of which is grounded, linked to each other by six legs in ball joins. Usually for given leg lengths and rigid bodies, there is only a finite number of ways to assemble the parts. In application in robotics, the ungrounded body is moved around by changing the leg lengths. Since we have six legs and the group of movements SE(3) is six dimensional, the ungrounded body moves freely a region corresponding to an open part of SE(3). In the talk we are explore in a different situation: the configuration of leg lengths and bodies such that the mechanism moves without changing the leg lengths. We call such mechanism exceptional. Apart from rather trivial examples, one family of exceptional mechanism are well-known and thoroughly studied, the Griffith-Duffy platforms [Griffith, Duffy], [Karger-Husty]. The family of Griffith-Duffy platforms form a unirational variety. In this talk we ask whether there are families of exceptional Stewart-Gough platforms of different kind. In our main result, we establish the existence of a family of mechanisms whose coppler curves in SE(3) are smooth algebraic curves of degree 12 and genus 7.

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