Kinematic Bifurcation

In the theory of mechanisms, many linkages are known whose instantaneous degree of freedom in certain positions of the linkage is greater than their degree of freedom (in our terms: the infinitesimal degree of freedom of a finite mechanism is greater than its finite degree of freedom). The sign of this is that the Jacobian matrix of the constraint functions has a rank deficiency. For a four-bar linkage with equal opposite bars, Litvin (1980) called the attention to the fact that, if one of the bars is fixed in a horizontal position, then the two bars joining to it incline to the horizontal at an angle. Then the relationship between these two angles determines two curves. One is associated to the parallelogram, the other to the anti-parallelogram shape of the linkage. The two curves have a point in common if the four joints lie on a straight line. In this position, the Jacobian matrix has a rank deficiency. Litvin (1980), however, did not attribute any additional significance to this point.