On solving the direct kinematics problem for parallel robots

Abstract

In this paper, we propose efficient methods for solving the direct kinematics problem (DKP) for parallel manipulators. By solving, we mean computing all the real solutions in a certified way, without any assumption on the manipulator. For example, the precision and the real character of the solutions are guaranteed. The proposed algorithms use as black boxes several recent algorithms from computer algebra such as F4/F5 for computing Gröbner bases, recent methods for isolating the real roots by mean of intervals with rational bounds which are mixed with strategies coming from the interval arithmetics field. The resulting solutions are efficient for such kind of output since the running time never exceeds few seconds ( in fact about 1 second for non extreme but general examples) thanks to the algorithms but also to a well adapted translation of DKP problem into algebraic equations. As an example we give a new computational proof of the existence of a parallel robot with 40 real roots.