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research-article

# Analytical Elastostatic Stiffness Modeling of Overconstrained Parallel Manipulators Using Geometric Algebra and Strain Energy

[+] Author and Article Information
Qinchuan Li

Mail box 597, Institute of Mechatronics, Zhejiang Sci-Tech University Xiasha Higher Education Zone Hangzhou, Zhejiang 310018 China lqchuan@zstu.edu.cn

Lingmin Xu

Mechatronic Institute, Zhejiang Sci-Tech University, Hangzhou, Zhejiang Province, 310018, P.R. of China Hangzhou, Zhejiang Province 310018 China xulingmin1993@163.com

Qiaohong Chen

Mailbox 597 Hangzhou, Zhejiang 310018 China chen_lisa@zstu.edu.cn

Xinxue Chai

Institute of Mechatronics, Zhejiang Sci-Tech University Xiasha Higher Education Zone Hangzhou, Zhejiang 310018 China chaixx@zstu.edu.cn

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received September 9, 2018; final manuscript received February 19, 2019; published online xx xx, xxxx. Assoc. Editor: Shaoping Bai.

ASME doi:10.1115/1.4043046 History: Received September 09, 2018; Accepted February 19, 2019

## Abstract

A general method for the analytical elastostatic stiffness modeling of overconstrained parallel manipulators (PMs) using geometric algebra and strain energy is proposed. Firstly, an analytical solution of the constraint and actuation wrenches exerted on the moving platform is obtained using the outer product and dual operation of geometric algebra, which avoids solving complex symbolic linear equations. Secondly, considering the compliances of the limbs, an analytical elastostatic model is established using the strain energy to obtain the stiffness matrices of the limbs. Finally, the deformation compatibility equations are added into equilibrium equations to obtain the overall stiffness matrix of the PM, which has concise expression and clear physical meaning. The proposed method is applied to the Tex3 overconstrained PM and the Tex4 overconstrained PM with redundant actuation to prove its validity. Comparable results between theoretical analysis and finite-element analysis (FEA) show that the former could be used as an effective alternative to the FEA method in the pre-design stage. This new approach is universally applicable to elastostatic stiffness analysis of overconstrained PMs.

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