Parallel manipulators. Part 2 : Theory. Singular configurations and Grassmann geometry

Parallel manipulators have a specific mechanical architecture where all the links are connected both at the basis and at the gripper of the robot. By changing the lengths of these links we are able to control the positions and orientations of the gripper. In general, for a given set of links lengths there is only one position for the gripper. But in some cases more than one solution may be found for the position of the gripper : this is a singular configuration. To determine these singular configurations the classical method is to find the roots of the determinant of the jacobian matrix. In our case this matrix is complex and it seems to be impossible to find these roots. We propose here a new method based on Grassmann line-geometry. If we consider the set of lines of P3, it constitutes a linear variety of rank 6. We show that a singular configuration is obtained when the variety spanned by the lines associated to the robot links has a rank less than 6. An important feature of the varieties of this geometry is that they can be described by simple geometric rules. Thus to find the singular configurations of parallel manipulators we have to find the configuration where the robot matches these rules. Such an analysis is performed on a special parallel manipulator and we show that we find all the well-known singular configurations but also new ones