Natural frequency analysis of a planar parallel mechanism with task space redundancy for high-precision milling

The objective of this research is to present dynamic modeling and natural frequency analyses of a 3-PRR planar parallel kinematic mechanism (PKM) for CNC machining applications. The orientation of the moving platform is considered a redundant degree of freedom in task space, and its optimal value is obtained during movement to avoid singular configurations, thereby improving accuracy in milling operations. The inverse dynamic problem is also solved and the actuator forces in the presence of milling forces are obtained in both arbitrary and optimal orientations of the moving platform. Moreover, vibration equations of the PKM are determined by considering the distal links as Euler–Bernoulli beams with the prismatic joints modeled as linear springs. Finally, the natural frequencies in different configurations of the moving platform are calculated. The presented formulation can be utilized for the motion planning and control of PKMs, meeting accuracy requirements in applications such as milling. Accordingly, this study makes three key contributions: (1) It leverages task space redundancy to avoid singularities and reduce actuator loads; (2) it presents an analytical method for evaluating vibrations and natural frequencies in PKMs with flexible joints and links; and (3) it improves milling accuracy by tuning natural frequencies through task space redundancy, thereby minimizing vibrations induced by external loads.