Branching Avoidance in Kinematic Image Space for Linkage Synthesis

Abstract: We consider a finite position synthesis and branching avoidance of four-bar linkages based on kinematic theory derived from dual quaternions. The branching defect is defined as a situation where all specified task positions are not connected by a continuous motion of the linkage end effector. The workspace of a four-bar linkage is derived to be the intersection of two hyperboloids in the kinematic image space. This is used to develop a fast method to determine branching of a linkage and to find an explicit solution to the end effector trajectory. A new synthesis method is developed where one task position is given a lower priority. Using the new branching analysis method and by synthesizing linkages with various values of one parameter of the linkage, we can determine a range of values for the constrained dimension which will give useful linkages for the four most prioritized task positions. Finally a method to find the closest useful linkage to the last task position is derived.