Type synthesis of two-degrees-of-freedom (DOF) planar mechanisms has been carried out using graph theory to determine the possible kinematic structures of variable compression ratio (VCR) engine mechanisms with two to three independent loops, and has resulted in the structures of 87 mechanisms satisfying search specification. By applying evaluation criteria to the enumerated mechanisms, the kinematic structures of three mechanisms are selected as suitable VCR engine mechanisms and verified by analysis results. In this research, VCR engine mechanisms with revolute joints and only one prismatic joint are enumerated, and the basic method to determine the VCR engine mechanisms with higher pairs is discussed. The procedure used in this research can be utilized to determine the kinematic structures of desired mechanisms and the results can be used as an atlas of two-DOF adjustable slider-crank mechanisms.

As the kinematic structure of an articulated manipulator affects the characteristics of its motion, rigidity, vibration, and force transmissibility, finding the most suitable kinematic structure for the desired task is important in the conceptual design phase. This paper proposes a systematic method for generating non-isomorphic graphs of articulated manipulators that consist of a fixed base, an end-effector, and a two-degree-of-freedom (DOF) intermediate kinematic chain connecting the two. Based on the analysis of the structural characteristics of articulated manipulators, the conditions that must be satisfied for manipulators to have a desired DOF is identified. Then, isomorphism-free graph generation methods are proposed based on the concepts of the symmetry of a graph, and the number of graphs generated are determined. As a result, 969 graphs of articulated manipulators that have two-DOF non-fractionated intermediate kinematic chains and 33,438 graphs with two-DOF fractionated intermediate kinematic chains are generated, including practical articulated manipulators widely used in industry.

This paper proposes an efficient exact dimensional synthesis method for finding all the link lengths of the Watt II and Stephenson III six-bar slider-crank function generators, satisfying nine prescribed precision points using the homotopy continuation method. The synthesis equations of each mechanism are initially constructed as a system of 56 quadratic polynomials whose Bézout number, which represents the maximum number of solutions, is 256 ≅ 7.21 × 1016. In order to reduce the size of the system, multi-homogeneous formulation is applied to transform the system into 12 equations in 12 unknowns, and the multi-homogeneous Bézout number of the system is 286,720. The Bertini solver, based on the homotopy continuation method, is used to solve the synthesis equations to obtain the dimensions of the two mechanisms. For the arbitrarily given nine precision points, the proposed method yields 37 and 31 defect-free solutions of Watt II and Stephenson III six-bar slider-crank mechanisms, respectively, and it is confirmed that they pass through the prescribed positions.