Higher Order Local Analysis of Singularities in Parallel Mechanisms

A purely algebraic approach to higher order analysis of (singular) configurations of rigid mulitbody systems with kinematic loops (CMS) is presented. It takes advantage of the special structure of the matrix Lie algebra se(3). The generating algebra of the special Euclidean group SE(3). The CMS configuration space is considered as analytic variety V. In regular configurations V has manifold structure but this is lost in singular points. In such points the concept of a tangent vector space does not makes sense but the tangent space(a cone) to V can still be defined. It is shown that this tangent cone can be determind algebraically. Moreover the tangent space to the configuration space is a surface of maximum degree 4, a vector space for regular points. It is the structure of the tangent cone to V that gives the complete geometric picture of the configuration space around a (singular) point. The approach is valid for general multiloop CMS and examplified in detail on the four- and fife-bar mechanism. Results are also shown shown for a Hexapod strucure.