Spatial Stiffness of the End Effector of a Machine Tool with Combined Kinematics

The article presents theoretical formulations for determining the stiffness matrix of the end effector of a machine tool with combined kinematics based on a 2-dof parallel manipulator with variable-length links. The stiffness matrix is determined using the simple spring decomposition method, wherein the links of the parallel structure mechanism are modeled as simple linear and torsional springs (or their combinations). The end effector of the machine tool with a 2-dof parallel manipulator, connected to the machine base by several active or passive links, is represented as a rigid body linked to the base through multiple simple springs with known linear or torsional stiffness. The stiffness matrix of the spatial parallel manipulator with the end effector connected to the base by multiple links subjected only to compression and tension or transmitting rotational torque, can be determined as a combination of the stiffness matrices of the links, treated as simple springs. The geometric parameters required for calculating the stiffness matrices of individual links — namely, the link axis vectors and the coordinates of their endpoints — are obtained using the inverse kinematics of the manipulator. As a result, analytical expressions are derived for determining the spatial stiffness matrix as a function of the mechanism’s geometric parameters, the position coordinates of the end effector, and the translational and torsional stiffness of the mechanism’s links. The dependence of translational stiffness along the coordinate axes and torsional stiffness about the coordinate axes on the position of the end effector, as well as the influence of the stiffness of variable-length links on the coordinate stiffness of the end effector, are established. These findings provide a tool for optimizing the geometric and structural parameters of machine tool based on stiffness criteria.