Consequently, the effective configuration does not feature the geometric properties specified on the theoretical design. Further computation trials shall be performed on the effective 6-6 and theoretical one to improve response times and result files sizes. The selected manipulator is a generic 6-6 in a realistic configuration, measured on a real parallel robot prototype constructed from a theoretically singularity-free design. Comparing the results, three kinematics models shall be retained. In the fifth section, one FKP typical example shall be solved implementing the ten identified kinematics models. The method implements proven variable elimination and the algorithms compute two important mathematical objects which shall be described: a Gröbner Basis and the Rational Univariate Representation including a univariate equation. The fourth section gives a brief description of the theoretical information about the selected exact algebraic method. They are classified into two families: the displacement based models and position based ones. The third section describes the ten formulations for the forward kinematics problem. The second section details the two models for the inverse kinematics problem, addresses the issue of the kinematics modeling aimed at its adequate algebraic resolution. The first part describes the kinematics fundamentals and definitions upon which the exact models are built.
This document is divided into 3 main topics distributed into five sections. This algebraic method will be fully detailed in this chapter. Three journal articles have been covering this question for the general planar and spatial manipulators. The proposed method uses Gröbner bases and the rational univariate representation,, implementing specific techniques in the specific context of the FKP. Hence, this justified the implementation of an exact method based on proven variable elimination leading to an equivalent system preserving original system properties. The geometric iterative method has shown promises,, but, as for any other iterative methods, it needs a proper initial guess. However, these methods are often plagued by the usual Jacobian inversion problems and thus cannot guarantee to find solutions in all non-singular instances. Intervals analyses were also implemented with the Newton method to certify results. Inasmuch, a sole univariate polynomial cannot be proven equivalent to a complete system of several polynomials.
However, resultant or dialytic elimination can add spurious solutions, and it will be demonstrated how these can be hidden in the polynomial leading coefficients. Using variable elimination, for the 3-RPR, 6 complex solutions were calculated and, for the 6-6, Husty and Wampler applied resultants to solve the FKP with success. Computer algebra was then selected in order to manipulate exact intermediate results and solve the issue of numeric instabilities related to round-off errors so common with purely numerical methods. The continuation method was then applied to find the solutions,, however, it will be explained why they are prone to miss some solutions. Ronga, Lazard and Mourrain have established that the general 6-6 hexapod FKP has 40 complex solutions using respectively Gröbner bases, Chern classes of vector bundles and explicit elimination techniques. To compute all the solutions, polynomial equations were justified. They only converge on one real root and the method can even fail to compute it. The numeric iteration methods such as the very popular Newton one were first implemented. Hunt geometrically demonstrated that the 3-RPR could yield 6 assembly modes. For the 3-RPR, 8 assembly modes were first counted. Solving the FKP of general parallel manipulators was identified as finding the real roots of a system of non-linear equations with a finite number of complex roots.