Analysis of Stiffness and Elastic Deformation for Some 3–5-DOF PKMs With SPR or RPS-type legs

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.