THE DIRECT KINEMATICS OF PLANAR PARALLEL MANIPULATORS - SPECIAL ARCHITECTURES AND NUMBER OF SOLUTIONS

This paper presents new results on the direct kinematic problem of planar three-degree-of-freedom parallel manipulators. This subject has been addressed in the past. Indeed, the latter problem has been reduced to the solution of a minimal polynomial of degree 6 by several researchers working independently. This paper focuses on the direct kinematic problem associated with particular architectures of planar parallel manipulators. For some special geometries, namely, manipulators for which all revolute joints on the platform and on the base are respectively collinear, it has been conjectured that only four solutions are possible, as opposed to six in the general case. However, this fact has never been shown and the polynomial solution derived for the general case still gives six solutions for the special geometry, two of which are spurious and unfeasible. In this paper, a formal proof of the aforementioned conjecture is derived using Sturm's theorem. Then, alternative derivations of the polynomial solutions are pursued and a robust computational scheme is given for the direct kinematics. The scheme accounts for special cases that would invalidate the previous derivations. Finally, possible simplifications of the general polynomial are discussed and related to particular geometries of the manipulator. It is first shown that it is not possible to find an architecture that would lead to a vanishing coefficient for the term of degree 6 in the polynomial. Then, a special geometry different from the one mentioned above and leading to closed-form solutions is introduced. A simplified planar three-degree-of-freedom parallel manipulator can be of great interest, especially for applications in which the manipulator is working on a vertical plane, i.e., when gravity is in the plane of motion.