Overconstrained mechanisms based on planar four-bar-mechanisms

We study a particular class of planar four-bar mechanisms FBM(Q) which are based on a given quadrilateral (quad) Q=a0a1a2a3. The self-motion of FBM(Q) consists of two different parts - one is the motion of an anti-parallelogram whilst the other one is a pure translation with circular paths. We will refer to this translatoric part in this paper only and demonstrate that this translatoric self-motion has the following property: At any moment the positions of the corresponding four coupler points form quads homothetic to Q. This property can be used to define spatial one-parametric motions of an extruded version of the four-bar mechanism which again generate quads of coupler points homothetic to Q. As the next step we take an arbitrary “saturated chain” of quads in space (each vertex shares a vertex with another quad of the set) and define the corresponding one-parametric spatial motions. Then all can be parametrized by the same parameter t. We will show that these partial motions can be interlinked by spherical 2R-joints without locking the one-parametric self-motion. This way the construction delivers a series of new (overconstrained) mechanisms which generalize results on so-called “Fulleroid” linkages. An example based on four quads in space (in planes of a tetrahedron) is worked out in detail.