Algebraic analysis of overconstrained single loop four link mechanisms with revolute and prismatic joints

Although spatial four link single loop mechanisms with revolute and prismatic joints are well investigated in many publications, there are only few of them dealing with the geometric conditions that cause these generally rigid structures to become movable. This article deals with the derivation of the necessary geometrical design conditions for mobility using kinematic mapping. Furthermore, the coupler motions of the mechanisms are computed. For the Bennett mechanism and the overconstrained RPRP chain the ruled surfaces representing the base surfaces for the line symmetric motions as well as the moving and fixed axodes are computed using the derived design parameters only. All surfaces are computed explicitly and shown in examples. A further contribution of this article is that in case of the RPRP mechanism algebraic mobility conditions in the design parameters are found which have never been published before.