Identification of bifurcation points occurring on the mechanism paths of deployable structures

This paper is concerned with the identification of bifurcation points occurring on the mechanism paths of a linkage, in which the structural system consists of links and revolute joints. The way that the links are assembled is mathematically described by compatibility conditions. The present paper mainly deals with the kinematic paths of mechanism that goes through special configurations of kinematic bifurcations. In general, such special configurations are called uncertainty configurations or singular configurations. The Jacobian derived from the compatibility conditions has been used as a measure of the distance away from the singular configurations. The Jacobian does not always give sufficient results for the detection of bifurcation points. This suggests that it is necessary to consider higher-order infinitesimals for examining approximate solutions of compatibility conditions. The Hessian matrix derived from the Jacobi matrix is considered to be one of the promising measure and may clarify the nature of the singular configuration as to whether the system is at a bifurcation point or not. In this paper, the theory behind the kinematic bifurcation of mechanisms of a linkage is explained and the numerical analysis of illustrative examples is discussed in detail.