Singularities in Motion and Displacement Functions of Spatial Linkages

A new approach for the determination of singularities and displacement functions of spatial linkages is proposed. Singularities in motion occur at positions where an overconstrained group of links becomes movable while the driving link is fixed. Two alternative solutions for the conditions of the mobility of the link group are proposed: (i) The first one is based on the differentiation of matrices which describe the coordinate transformation; (ii) The second one is based on the presentation of the link motion as a screw motion. It is proven that singularities in motion occur in the cases where: (i) Some or all links of the overconstrained group of links are aligned; (ii) The links of the group are not aligned but they form a mobile configuration, (iii) A Certain number of axes of revolute or cylindrical pairs belong to a ruled surface. Cases (i) and (ii) cover the situations when the oscillating driving link reaches its extreme positions. The determination of displacement functions is based on the modeling of the linkage by two open kinematic chains formed: (i) by links of the overconstrained group of links and (ii) the driving link and the frame. The structure of equations for the link displacements is related with the structure of the velocity Jacobian for the overconstrained group and simplifies the computation procedure. The proposed methods are illustrated with examples of RCRCR and 7R linkages.