Multi-furcation in a derivative queer-square mechanism

Abstract This paper investigates the phenomena of multi-furcation through the derivative mechanism from the “queer square” paper fold by allowing a full rotation of the joints and reveals the four categories and fourteen states of multi-furcation based on two phenomena that result in change of mobility and motion type. Specifically, phenomenon one occurs when the mechanism moves past the constraint singularity and subsequently changes the mobility though directions of the joint axes remain the same. Phenomenon two occurs by passing the constraint singularity when the direction of joint axes changes, leading to different mobility. Both phenomena are demonstrated by the derivative queer-square mechanism, on account of its first six states belonging to phenomenon one and its last eight states belonging to phenomenon two. More precisely, for the derivative queer-square mechanism, the change of angle relations for the links, which may form one or two parallel combinations, realises phenomena one and two. Changing from phenomenon one to two is achieved by changing the corresponding angles which makes one of the two parallel combinations to change to antiparallel combination. In each phenomenon, two categories are presented as two different motions, specifically, single translation and double translations belong to phenomenon one and one screw motion about x -axis and one screw motion about y -axis belong to phenomenon two. Different angle ranges make each category split into two states and one state changes to the other state by passing though the singular posture. The analysis of the state configurations is achieved by the platform motion–screw system of the derivative queer-square mechanism via screw theory and the platform motion–screw system presents its ability in mobility change.