Multiple Solutions of Direct Kinematics of 3-RPR Parallel Manipulators

Thursday, May 25, 2023 16:00to17:00
McConnell Engineering Building Zames Seminar Room, MC 437, 3480 rue University, Montreal, QC, H3A 0E9, CA
CIM Seminar in Robotic Mechanical Systems 
Speaker: Prof. Manfred L. Husty
University of Innsbruck, 
Abstract: A planar parallel 3-RPR parallel manipulator consists of three anchor points (A;B;C) in a base connected via three extensible legs (r1; r2; r3) to a triangular platform (D;E; F). In the direct kinematics (DK) one has to compute the pose of the platform when the design of the manipulator (location of the base points and the shape of the moving platform) and the lengths of the legs are given. It is well known that this task allows six solutions. When some of the solutions of the DK coincide the manipulator becomes singular. In the presentation, it will be shown that multiple solutions of the direct kinematics are a new way of looking into the notions of "shakiness" of mechanisms or "flexibility" of pin jointed frameworks of rigid bars. For the first time general conditions for maximal (= 6) coinciding solutions will be given. It will be discussed that multiple solutions of the DK are far beyond the classical singularity theory of a manipulator. They belong to so called constraint singularities which describe special situations in the configuration space of the manipulator.
The discussion is done within the framework of algebraic geometry and polynomial equations, because multiple solutions are closed sets. We have therefore also to explain the advantages and limitations of this approach.
Prof. Dr. Dr.(hc) Manfred Husty is Emeritus Professor from University of Innsbruck, Austria. His research
milestones in kinematic geometry are a) closed form solution to the direct kinematics of a general six
legged Stewart-Gough parallel manipulator which was until 1994 the most challenging unsolved problem in
robotic kinematics and b) closed form solution of inverse kinematics of a general 6R serial robot using
kinematic mapping in 2005 which greatly simplifies and accelerates computation. This makes it more effective
as a real time procedure than earlier results modelled in Euclidean space.
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