Analyzing Higher-Order Curvature of Four-Bar Linkages with Derivatives of Screws

Curvature theory, a fundamental subject in kinematics, is typically addressed through the concepts of instantaneous invariants and canonical coordinates, which are pivotal for the generation of mechanical paths. This paper delves into this subject with a higher-order analysis of screws, employing both canonical and natural coordinates. Through this exploration, a new Euler–Savary equation is derived, one that does not rely on canonical coordinates. Additionally, the paper provides a comprehensive classification of the degenerate conditions of the cubic of stationary curves of four-bar linkages at rotational positions. A thorough examination of four-bar linkages in translational positions—the couple links translate instantaneously—is also presented, with analyses extending up to the sixth order. The findings reveal that the Burmester’s points at translational positions can be extended to Burmester’s points with excess one, provided that all pivot points are symmetrically distributed about the pole norm with the two cranks in corotating senses. However, the extension to Burmester’s points with excess two is not possible. Similarly, the Ball’s point with excess one does not progress to Ball’s point with excess two. The paper also highlights that the traditional method, which is based on canonical coordinates, is ineffective when the four-bar linkage forms a parallelogram. Fortunately, this scenario can be effectively analyzed using the screw-based approach. Ultimately, the results presented can also serve as analytical solutions for 3-RR platforms with higher-order shakiness.