Mechanism singularities and shakiness from an algebraic viewpoint

Kinematic singularities are configurations where the configuration space (c-space) V is not smooth or configurations where the rank of the constraint Jacobian drops while the c-space is locally smooth. Both can be analyzed by means of a higher-order analysis, for which the kinematic tangent cone provides a conceptual framework. It is desirable to complement such an analysis with more insight into the origin of specific singular phenomena. Such additional insight is obtained by investigating the kinematics from an algebraic viewpoint. Points of interest are then the singularities of a variety, referred to as algebraic singularities. Algebraic singularities are reflected as kinematic singularities or as shakiness. It is shown that these algebraic singularities can be attributed to properties of the ideal generated by the loop constraints - the constraint ideal. Particular situations arise depending on whether this constraint ideal is radical. If it is not radical, a singular branch is locally defined by a primary ideal. If it is radical, it must be distinguished whether real and complex dimensions of intersecting branches are different. Real intersections indicate kinematic singularities; in all other cases, the mechanism may be in a singularity or may be shaky. Various examples are presented.