Analysis of Stiffness and Elastic Deformation for Some 3–5-DOF PKMs With SP R or RP S-type legs

Abstract

This paper proposes an equivalent mechanism approach for establishing the stiffness matrices of some 3–5DOF (degree of freedom) parallel kinematic machines (PKMs) with SP̱R- or RP̱S-type legs and solving their elastic deformations. First, the geometric constraints of constrained wrench of these PKMs are analyzed, and the poses of the active/constrained forces are determined. Second, based on the principle of virtue work and the determined active/constrained forces, the formulas are derived for solving the 6×6 Jacobian matrices of these PKMs and the stiffness matrices of SP̱R or RP̱S-type active legs. Third, based on the elastic deformations of the SP̱R or RP̱S active legs, the equivalent 6-DOF rigid PKMs of these elastic PKMs are constructed, and their 6×6 Jacobian matrices are derived. Finally, the formulas are derived for solving the total stiffness matrices and the elastic deformation of the 3DOF 3SP̱R, 3DOF 3RP̱S, 4DOF 2SP̱S+2SP̱R, and 5DOF 4SP̱S+SP̱R PKMs.