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Mathematical Problems in Engineering
Volume 2015, Article ID 957269, 11 pages
Research Article

A New Decoupling Method for Explicit Stiffness Analysis of Kinematically Redundant Planar Parallel Kinematic Mechanism

1School of Robot and Automation Engineering, Dongyang Mirae University, Seoul 08221, Republic of Korea
2School of Mechanical Engineering, Soongsil University, Seoul 07040, Republic of Korea

Received 29 June 2015; Accepted 19 October 2015

Academic Editor: Zoran Gajic

Copyright © 2015 Hyun-Pyo Shin and Donghun Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Optimization and control of stiffness for parallel kinematic mechanisms (PKM) are critical issues because stiffness is directly related to the precision and response characteristics of the end-effector of PKMs. Unlike nonredundant PKMs, redundant PKMs have additional actuators exceeding their essential degrees-of-freedom (DOF), resulting in an increase in the redundancy of control. The stiffness of redundant PKMs is divided into passive and active stiffness. Active stiffness is changeable even in cases of fixed kinematic parameters and end-effector posture. However, it is not easy and intuitive to control the active stiffness of redundant PKMs for the complexity of Hessian matrix operations. This paper describes a new decoupling method for explicit stiffness analysis of redundant PKM with the well-known two-DOF and one-redundant planar five-bar PKM. Three actuating joints are decoupled to three groups containing two actuating joints. With this mathematical configuration, the stiffness matrix for one-redundant actuation is also divided into three stiffness matrices for nonredundant actuation, and the contribution of each actuator can be intuitively investigated. Stiffness matrices for the original and decoupled cases are compared in detail. In particular, this decoupling method is applicable to redundant PKMs with many passive joints. Finding optimal joints for one- or two-redundant actuation with various candidates is more intuitive with this decoupling method.