The Kinematics of a Bipod R2RR Coupling between Two Non-Coplanar Shafts

Abstract

The paper presents a new solution for motion transmission between two shafts with non-intersecting axes. The structural considerations fundament the existence in the structure of the mechanism of three revolute pairs and a bipod contact. Compared to classical solutions, where linkages with cylindrical pairs are used, our solution proposes a kinematical chain also containing higher pairs. Due to the presence of a higher pair, the transmission is much simpler, the number of elements decreases, and as a consequence, the kinematical study is straightforward. Regardless, the classical analysis of linkages cannot be applied because of the presence of the higher pair. For the proposed spatial coupling, the transmission ratio is expressed as a function of constructive parameters. The positional analysis of the mechanism cannot be performed using the Hartenberg–Denavit method due to the presence of a bipod contact, and instead, the geometrical conditions of existence for the bipod contact are applied. The Hartenberg–Denavit method requires the replacement of the bipodic coupling with a kinematic linkage with cylindrical (revolute and prismatic) pairs, resulting in complicated analytical calculus. To avoid this aspect, the geometrical conditions required by the bipod coupling were expressed in vector form, and thus, the calculus is significantly reduced. The kinematical solution for the proposed transmission can be obtained in two ways: first, by considering the equivalent transmission containing only cylindrical pairs and applying the classical analysis methods; second, by directly expressing the condition of definition for the higher pairs (bipodic pair) in vector form. The last method arrives at a simpler solution for which analytical relations for the positional parameters are obtained, with one exception where numerical calculus is needed (but the precision of this parameter is controlled). The analytical kinematics results show two possibilities of building the actual mechanism with the same constructive parameters. The rotation motions from the revolute pairs, internal and driven, and the motions from the bipod joint were obtained through numerical methods since the equations are very intricate and cannot be solved analytically. The excellent agreement validates the theoretical solutions obtained and the possibility of applying such mechanisms in technical applications. The constructive solution exemplified here is simple and robust.