Jacobian-based natural frequency analysis of parallel manipulators

The dynamic performance of a parallel manipulator is often limited by its natural frequencies. These frequencies are a result of the controlled actuator stiffness and the mechanical design. Typically, only the lowest natural frequencies are of interest. However, existing methods to analyze natural frequencies of parallel manipulators with flexible links have been developed in a high-dimensional generalized coordinate space. This paper presents a novel natural frequency analysis method for parallel manipulators that focuses on the lowest natural frequencies and which expresses the corresponding eigenmodes in an end-effector Cartesian reference frame. The analysis method combines a flexible-body stiffness analysis method with a rigid-body inertia analysis method. As an example, the two lowest natural frequencies of the Heli4 robot are modeled and measured over a range of controlled actuator stiffness values for a single pose. The results show that structural compliance in the mechanical parts of the Heli4 robot is significant for actuator stiffness values above roughly 1000 Nm/rad. For an actuator stiffness value of 4000 Nm/rad, consideration of the structural compliance reduces the maximum prediction error for the two lowest natural frequencies from 33% to 6%.