Motion Cycle and Configuration Torus With Their Relationship to Furcation During Reconfiguration

In a classical mobility-one single loop linkage, the motion begins from an original position determined by the assembled condition and runs in cycles. In normal circumstances, the linkage experiences a full cycle when the input-joint completes a full revolution. However, there are some linkages that accomplish a whole cycle with the input-joint having to go through multiple revolutions. Their motion cycle covers multiple revolutions of the input-joint. This paper investigates this typical phenomenon that the output angle is in a different motion cycle of the input angle that we coin this as the multiple input-joint revolution cycle. The paper then presents the configuration torus for presenting the motion cycle and reveals both bifurcation and double points of the linkage, using these mathematics-termed curve characteristics for the first time in mechanism analysis. The paper examines the motion cycle of the Bennett plano-spherical hybrid linkage that covers an $8π$ range of an input-joint revolution, reveals its four double points in the kinematic curve, and presents two motion branches in the configuration torus where double points give bifurcations of the linkage. The paper further examines the Myard plane-symmetric 5R linkage with its motion cycle covering a $4π$ range of the input-joint revolution. The paper, hence, presents a way of mechanism cycle and reconfiguration analysis based on the configuration torus.