Dead Center Identification of Single-DOF Multi-Loop Planar Manipulator and Linkage Based on Graph Theory and Transmission Angle

The dead center position (or singular position) is an important kinematic characteristic in mechanical design. However, its identification is a challenging task and becomes even more complex in multi-loop planar linkages (or manipulators). According to graph theory and transmission angle, this paper proposes a method to identify the dead center positions of single-degree-of-freedom (DOF) planar linkages. The planar linkages can be denoted by a set of independent loops based on topological structural analysis of the kinematic chains and a structural decomposition. According to the relationship between the mobility factor \text{w}_{\mathrm {i}} and loop factor \text{f}_{\mathrm {i}} in the same independent loop, the dead center positions can be located when the new planar linkages containing the corresponding instantaneous virtual loop are formed by adjusting \text{f}_{\mathrm {i}} to equal to \text{w}_{\mathrm {i}} or ( \text{w}_{\mathrm {i}} -1), in which the corresponding transmission angle is equal to 0° and 180°. The Jacobian method for identifying the dead center position is then used to verify the proposed method. Furthermore, the dead center positions of the single-DOF 10-bar and 12-bar planar linkages are solved. The proposed method is systematic and programmable and can be applied to any single-DOF multi-loop linkage regardless of the number of independent loops or types of joints.