Kinematics of Tripode Transmissions. A New Approach

Abstract

We present the kinematics of a two-tripode-joint transmission consisting of a sliding joint and a fixed joint. Beginning with a simplified joint having the output shaft in a fixed direction, the kinematic equations are derived from the closure relationships in a formal way. Due to the threefold symmetry of the tripode, the intrinsic parameters of the tulip-to-tripode matrix transformation are obtained; in particular, with this new approach, it is shown why the sum of the roller-to-tripode centre distances depend on the radius of the pot and the joint angle only. The triple frequency of the offset and the double frequency of the rollers on the tripode legs with respect to the driving shaft are also demonstrated, while the roller-to-tulip distances keep the same frequency. A new geometric approach based on the properties of the 120° angle is introduced and the above properties are demonstrated with the aid of geometric tools. For the whole transmission with the two real joints, the kinematic equations are solved numerically through a Newton–Raphson procedure and the results are compared to the simplified joint calculations. The phase angle between the two joints is also introduced and its influence on the kinematics is discussed through the roller-to-tulip distances.