## Section: Application Domains

### Robotic

The (parallel) manipulators we study are general parallel robots: the hexapods are complex mechanisms made up of six (often identical) kinematic chains, of a base (fixed rigid body including six joints or articulations) and of a platform (mobile rigid body containing six other joints). The design and the study of parallel robots require the resolution of direct geometrical models (computation of the absolute coordinates of the joints of the platform knowing the position and the geometry of the base, the geometry of the platform as well as the distances between the joints of the kinematic chains at the base and the platform) and inverse geometrical models (distances between the joints of the kinematic chains at the base and the platform knowing the absolute positions of the base and the platform).

Since the inverse geometrical models can be easily solved, we focus on the resolution of the direct geometrical models. The study of the direct geometrical model is a recurrent activity for several members of the project. One can say that the progress carried out in this field illustrates perfectly the evolution of the methods for the resolution of algebraic systems. The interest carried on this subject is old. The first work in which the members of the project took part in primarily concerned the study of the number of (complex) solutions of the problem [61] , [60] . The results were often illustrated by Gröbner bases done with Gb software.

One of the remarkable points of this study is certainly the classification suggested in [52] .