An Emulator Based Prediction of Dynamic Stiffness for Redundant Parallel Kinematic Mechanisms

Abstract

The accuracy of a parallel kinematic mechanism (PKM) is directly related to its dynamic stiffness, which in turn is configuration dependent. For PKMs with kinematic redundancy, configurations with higher stiffness can be chosen during motiontrajectory planning for optimal performance. Herein, dynamic stiffness refers to the deformation of the mechanism structure, subject to dynamic loads of changing frequency. The stiffnessoptimization problem has two computational constraints: (i) calculation of the dynamic stiffness of any considered PKM configuration, at a given taskspace location, and (ii) searching for the PKM configuration with the highest stiffness at this location. Due to the lack of available analytical models, herein, the former subproblem is addressed via a novel effective emulator to provide a computationally efficient approximation of the highdimensional dynamicstiffness function suitable for optimization. The proposed method for emulator development identifies the mechanism's structural modes in order to breakdown the highdimensional stiffness function into multiple functions of lower dimension. Despite their computational efficiency, however, emulators approximating highdimensional functions are often difficult to develop and implement due to the large amount of data required to train the emulator. Reducing the dimensionality of the approximation function would, thus, result in a smaller training data set. In turn, the smaller training data set can be obtained accurately via finiteelement analysis (FEA). Moving leastsquares (MLS) approximation is proposed herein to compute the lowdimensional functions for stiffness approximation. Via extensive simulations, some of which are described herein, it is demonstrated that the proposed emulator can predict the dynamic stiffness of a PKM at any given configuration with high accuracy and low computational expense, making it quite suitable for most highprecision applications. For example, our results show that the proposed methodology can choose configurations along given trajectories within a few percentage points of the optimal ones.