Position analysis of the tripod joint: An alternative approach

The position analysis, which is the most challenging phase of the kinematic analysis of a mechanism, was solved for the tripod joint, in exhaustive form, a few decades ago. The method proposed at the time was supposed to be able to find all possible assembly configurations of the joint. Regrettably, it becomes useless when the angle that parametrizes the position of one of the connected shafts reaches some specific values, which happens several times for each turn of the shaft; moreover, an error in the expression of a coefficient of a pivotal equation hampers adoption of the method. This paper presents a new procedure whose steepest step is finding the roots of a fourth order algebraic equation. It is shown that eight assembly configurations do exist for the adopted kinematic model of tripod joint, in the complex domain. The proposed procedure has a more direct approach than the previous one and is not affected by its singularities. Moreover, because no arbitrarily-selected movable reference frame is introduced, it leads to equations whose coefficients reveal the periodicity of the dependence of some parameters of the joint configuration on the angular position of one of the connected shafts. This means that some aspects of the tripod joint behavior can be hinted even without solving the equations. A numerical example shows application of the new procedure to a case study.