Singular aspects of general planar motions with two degrees of freedom

General planar motions with two degrees of freedom differ fundamentally from those with one degree of freedom, in that they may exhibit "cross-caps" singularities. At such singularities the trajectories fail to exhibit four of the generic singularity types normally associated with such motions, namely, lips, beaks, and two of their degenerations. Provided cross-caps are avoided, the union of the bifurcation curves associated with the lips and beaks types are characterized kinematically as envelopes of a two-parameter family of instantaneous singular lines. In the special class of composite planar motions, these lines are described in terms of the classical instantaneous centers of the constituent motions with 1 DOF, and it is shown that not all generic singularity types can appear on the trajectories. Computer-generated pictures illustrate the theory for examples of rational motions.