  It has become obvious that certain singular phenomena cannot be explained by
a mere investigation of the configuration space, defined as the solution set of
the loop closure equations. For example, it was observed that a particular 6R
linkage, constructed by a combination of two Goldberg 5R linkages, exhibits
kinematic singularities at a smooth point in its configuration space. Such
problems are addressed in this paper. To this end, an algebraic framework is
used in which the constraints are formulated as polynomial equations using
Study parameters. The algebraic object of study is the ideal generated by the
constraint equations (the constraint ideal).
  Using basic tools from commutative algebra and algebraic geometry (primary
decomposition, Hilbert's Nullstellensatz), the special phenomenon is related to
the fact that the constraint ideal is not a radical ideal. With a primary
decomposition of the constraint ideal, the associated prime ideal of one
primary ideal contains strictly into the associated prime ideal of another
primary ideal which also gives the smooth configuration curve. This analysis is
extended to shaky and kinematotropic linkages, for which examples are
presented.
