Systematization of open, serial kinematic chains with three degrees of freedom

• 216 orthogonal kinematic chains with 3DOF, containing revolute and/or prismatic pairs. • Classification of chains into groups according to the direction of kinematic pair axes. • Designation of chains by joint type, axis direction, and group membership. • 40 base chains transformed into identical chains using quaternions. • Reduction of 40 base chains to 36 by changing the rotation angle. The paper presents a systematization of kinematic chains with three degrees of freedom, consisting of revolute and/or prismatic pairs with perpendicular and/or parallel axes. Kinematic chains are classified into groups based on the orientation of the axes. Chains within each group are organized in tables. To simplify notation and facilitate referencing of individual chains, multiple chains, or entire groups, a designation method based on group membership is introduced. It is demonstrated that the base chains selected in each group can be transformed into chains of subsequent subgroups by a single rotation, forming a subgroup of identical chains that differ only in spatial orientation. Quaternions are used to represent these rotations. Out of a total of 216 chains, 40 subgroups of identical chains are selected. Additionally, altering the rotation angle of the second kinematic pair further reduces the number of identical chain subgroups to 36. The proposed systematization of kinematic chains lays the groundwork for future research on workspace determination, which could serve as a basis for designing robotic arms tailored to specific applications.