An Emulator-Based Prediction of Dynamic Stiffness for Redundant Parallel Kinematic Mechanisms
Journal article, 2015

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 motion-trajectory planning for optimal performance. Herein, dynamic stiffness refers to the deformation of the mechanism structure, subject to dynamic loads of changing frequency. The stiffness-optimization problem has two computational constraints: (i) calculation of the dynamic stiffness of any considered PKM configuration, at a given task-space 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 high-dimensional dynamic-stiffness function suitable for optimization. The proposed method for emulator development identifies the mechanism's structural modes in order to breakdown the high-dimensional stiffness function into multiple functions of lower dimension. Despite their computational efficiency, however, emulators approximating high-dimensional 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 finite-element analysis (FEA). Moving least-squares (MLS) approximation is proposed herein to compute the low-dimensional 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 high-precision 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.

parallel kinematic mechanisms

dynamic stiffness


machine learning


Mario Luces

University of Toronto

Pinar Boyraz Baykas

Olycksanalys och prevention

Masih Mahmoodi

University of Toronto

Farhad Keramati

University of Toronto

James K. Mills

University of Toronto

Beno Benhabib

University of Toronto

Journal of Mechanisms and Robotics

1942-4302 (ISSN) 1942-4310 (eISSN)

Vol. 2 8

Areas of Advance

Information and Communication Technology


Subject Categories

Applied Mechanics

Computational Mathematics

Vehicle Engineering



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