Three Points on Three Circles in General Pose

Paul Zsombor-Murray Department of Mechanical Engineering and Center of Intelligent Machines, McGill University, Montreal, Canada

Paul Zsombor-Murray is an associate professor at the Center of Intelligent Machines of the McGill University in Montreal, Canada. As a frequent guest of the seminars at IGMR he gives insight in his research activities, which include e.g. “simplification of mechanism analysis and synthesis with kinematic mapping”, “measurement of quadric surfaces using minimum data points” and “shortest path on the torus”.


Inspired by the Paragrip manipulator at IGMR Paul Zsombor-Murray solved a complicated geometric problem:
A possible first time solution to placing three points in a moving or end effector (EE) frame on three corresponding circles arbitrarily disposed in a reference or fixed frame (FF) is exposed. Circles are modeled as sphere-plane pairs, each plane on a sphere centre. Closure of the EE triangle is done with three spheres each centred on a circle and intersecting another circle on its centre. Quadric constraint equations are reduced, three to linear bivariates, three to linearly bivariate in one of the two points' coordinates they contain. Nine equations become six by eliminating three variables with plane equations. A numerical example admits two real solutions of a resultant univariate of degree 16 like that for special cases already reported. The real solutions are shown. Gröbner bases were used. Solutions were verified using kinematic mapping, an alternate approach.


Mittwoch, 20. Juni 2018, 16.30 Uhr, Reuleaux-Seminarraum B108