Integrated type and approximate dimensional synthesis of four-bar planar mechanisms for rigid body guidance

A method is presented that integrates type and approximate dimensional synthesis of planar four-bar mechanisms for rigid-body guidance. In this work, the term four-bar mechanism denotes a linkage comprised of any two of RR, PR, RP and PP dyads. As a precursor, attempts are made to linearize this particular synthesis problem such that linear algebra techniques may be applied to obtain a solution. This method uses kinematic mapping to map planar displacements to three dimensional coordinates in a projective image space. Limited success is achieved in this regard. An improved novel approach is then developed by correlating the positions of key points of the mechanism in two different coordinate frames. By doing so, the number of independent variables defining a suitable dyad for the desired rigid-body guidance is reduced from five to two. After applying these geometric constraints, numerical methods are used to size link lengths, locate joint axes, and decide between RR, PR, RP and PP dyads that, when combined, guide a rigid body through the best approximation, in a least squares sense, of n specified positions and orientations, where n ≥ 5. No initial guesses of type or dimension are required. Several examples are presented illustrating the effectiveness and robustness of this new approach.