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Research Papers

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

[+] Author and Article Information
Mario Luces

Department of Mechanical
and Industrial Engineering,
University of Toronto,
5 King's College Road,
Toronto, ON M5S 3G8, Canada
e-mail: mario.luces@mail.utoronto.ca

Pinar Boyraz

Mechanical Engineering Department,
Istanbul Technical University,
Inonu Cd., No: 65,
Gumussuyu,
Istanbul 34437, Turkey
e-mail: pboyraz@itu.edu.tr

Masih Mahmoodi

Department of Mechanical
and Industrial Engineering,
University of Toronto,
5 King's College Road,
Toronto, ON M5S 3G8, Canada
e-mail: masih.mahmoodi@utoronto.ca

Farhad Keramati

Department of Mechanical
and Industrial Engineering,
University of Toronto,
5 King's College Road,
Toronto, ON M5S 3G8, Canada
e-mail: farhad.keramatimoezabad@mail.utoronto.ca

James K. Mills

Department of Mechanical
and Industrial Engineering,
University of Toronto,
5 King's College Road,
Toronto, ON M5S 3G8, Canada
e-mail: mills@mie.utoronto.ca

Beno Benhabib

Department of Mechanical
and Industrial Engineering,
University of Toronto,
5 King's College Road,
Toronto, ON M5S 3G8, Canada
e-mail: benhabib@mie.utoronto.ca

1Corresponding author.

Manuscript received July 27, 2015; final manuscript received October 14, 2015; published online November 24, 2015. Assoc. Editor: Byung-Ju Yi.

J. Mechanisms Robotics 8(2), 021021 (Nov 24, 2015) (15 pages) Paper No: JMR-15-1208; doi: 10.1115/1.4031858 History: Revised October 14, 2014; Accepted October 20, 2014; Received July 27, 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.

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Figures

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Fig. 1

Proposed prediction methodology

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Fig. 2

(a) A 6DOF 3 × PPRS PKM and (b) example FRF of the amplitude of the displacement

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Fig. 3

Example FRFs of the compliance of a PKM configuration for (a) Cxx, (b) Cyx =Cxy, (c) Czx=Cxz, (d) Cyy, (e) Czy=Cyz, and (f) Czz

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Fig. 4

Architecture of the 3 × PPRS PKM

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Fig. 5

Sample path of PKM tool-platform

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Fig. 6

Training poses and sample path of PKM tool-platform

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Fig. 7

Prediction of displacement of tool-platform data at P02

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Fig. 8

Displacements at configurations near optimum

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Fig. 9

Optimal configuration selection by emulator in comparison to random selection

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Fig. 10

Symmetric PKM configurations: (a) qa, (b) qb, and (c) qc

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Fig. 11

Side-by-side comparison of a PKM configuration to three symmetric configurations

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Fig. 12

Joint-space trajectories: (a) θ1, (b) θ2, (c) θ3, (d) δ1, (e) δ2, and (f) δ3

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