Samuli Piipponen

Computational algebraic geometry and global analysis of regional manipulators

The global analysis of the singularities of regional manipulators is addressed in this paper. The problem is approached from the point of view of computational algebraic geometry. The main novelty is to compute the syzygy module of the differential of the constraint map. Composing this with the differential of the forward kinematic map and studying the associated Fitting ideals allows for a complete stratification of the configuration space according to the corank of singularities. Moreover using this idea we can also compute the boundary of the image of the forward kinematic map. Obviously this gives us also a description of the image itself, i.e. the manipulator workspace. The approach is feasible in practice because generators of syzygy modules can be computed in a similar way as Grobner bases of ideals.

Kinematical Analysis of Overconstrained Mechanism Using Computational Geometric Algebra

In this paper we will show how the methods of computational and geometric algebra can be used to analyze the kinematics of multibody systems. As an example we treat thoroughly the well known Bricard’s mechanism which is a classic example of so called overconstrained mechanism, but the same methods can be applied to wide class of rigid multibody systems. It turns out that the configuration space of Bricard’s system is a smooth closed curve which can be explicitly parametrized. Our computations also yield a new formulation of constraints which is better than the original one from the point of view of numerical simulations.Copyright

Classification of Singularities in Kinematics of Mechanisms

In this short article we will discuss methods of finding and classifying singularities of planar mechanisms. The key point is to observe that the configuration spaces of the mechanisms can be understood as analytic and algebraic varieties. The set of singular points of an algebraic variety is itself an algebraic variety and of lower dimension than the original one. The singular variety can be computed using the Jacobian criterion. Once the singular points are obtained their nature can be investigated by investigating the localization of the constraint ideal at the local ring at this point. This will tell us if the singularity is an intersection of several motion modes or a singularity of a particular motion mode. The nature of the singularity can be then analyzed further by computing the tangent cone at this point.