A General Kinetostatic Model Based Stiffness Estimation for Tripod Parallel Kinematic Machines with Prismatic Actuators

A general stiffness modeling methodology for tripod parallel kinematic machines (PKMs) with prismatic actuators is proposed in this paper. With the technique of substructure synthesis, the whole system of a tripod is divided into a platform, a base and three kinematic limb substructures. Each limb assemblage is modeled as a spatial beam constrained by two sets of six degree-of-freedom (6-DOF) virtual lumped springs with equivalent stiffness at their geometric centers. The equilibrium equation of each individual limb assemblage is derived through finite element formulation, while that of the platform is derived with the Newton’s 2nd law. The governing stiffness matrix is synthesized by introducing the deformation compatibility conditions between the platform and the limbs. By extracting a 6x6 block matrix from the inversion of the governing compliance matrix, a stiffness matrix of the platform is formulated. Taking the Sprint Z3 Head and the A3 Head as examples, the distributions of stiffness values of these two types of PKM modules are predicted and discussed. It is worth mentioning that the proposed methodology of stiffness modeling can further be applied to other types of PKMs for evaluating the global rigidity performance over entire workplace efficiently with minor revisions.