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Research Papers: Mechanisms and Robotics

# Analysis of Stiffness and Elastic Deformation for Some 3–5-DOF PKMs With SP R or RP S-type legs

Yi Lu and Bo Hu
[+] Author and Article Information
Yi Lu

Robotics Research Center, College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004 P.R. Chinaluyi@ysu.edu.cn

Bo Hu

Robotics Research Center, College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004 P.R. China

J. Mech. Des 130(10), 102307 (Sep 11, 2008) (8 pages) doi:10.1115/1.2918918 History: Received August 05, 2007; Revised December 12, 2007; Published September 11, 2008

## Abstract

This paper proposes an equivalent mechanism approach for establishing the stiffness matrices of some 3–5DOF (degree of freedom) parallel kinematic machines (PKMs) with $SP̱R$- or $RP̱S$-type legs and solving their elastic deformations. First, the geometric constraints of constrained wrench of these PKMs are analyzed, and the poses of the active/constrained forces are determined. Second, based on the principle of virtue work and the determined active/constrained forces, the formulas are derived for solving the $6×6$ Jacobian matrices of these PKMs and the stiffness matrices of $SP̱R$ or $RP̱S$-type active legs. Third, based on the elastic deformations of the $SP̱R$ or $RP̱S$ active legs, the equivalent 6-DOF rigid PKMs of these elastic PKMs are constructed, and their $6×6$ Jacobian matrices are derived. Finally, the formulas are derived for solving the total stiffness matrices and the elastic deformation of the 3DOF $3SP̱R$, 3DOF $3RP̱S$, 4DOF $2SP̱S+2SP̱R$, and 5DOF $4SP̱S+SP̱R$ PKMs.

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## Figures

Figure 1

The 3SP̱R PKM and its force situation

Figure 2

The elastic deformations of the SP̱R-type leg ri under Fai (a) and under Fci (b)

Figure 3

An elastic 3SP̱R PKM and its equivalent rigid PKM

Figure 4

The 2SP̱S+2SP̱R PKM and its equivalent rigid 6-DOF 2SP̱S+2SPP̱R PKM

Figure 5

The 4SP̱S+SP̱R PKM and its equivalent rigid 6-DOF 4SP̱S+SPP̱R PKM

Figure 6

The analytic solved results of the 3SP̱R PKM

Figure 7

The analytic solved results of the total stiffness matrix for the 3-SP̱R PKM

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