Contracted graphs of kinematic chains with multiple joints

This paper presents an approach for the structural synthesis of kinematic chains with multiple joints based on the concept of contracted graphs. A graph representation of multiple joints is introduced and the modified Euler formula is derived. Then, contracted graphs of kinematic chains with m multiple joints and L_ind independent loops are synthesized from the catalogs of contracted graphs of kinematic chains with simple joints and L_ind + m independent loops. Finally, basic kinematic chains with multiple joints are obtained from their corresponding contracted graphs. As a result, the number of kinematic chains with one and two multiple joints up to ten links are listed.