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Optimal design of kinematic performance for a novel 2R1T parallel mechanism with pantograph units

  • Yeping Lv
  • Yong XuEmail author
  • Jiali Chen
Technical Paper
  • 129 Downloads

Abstract

A new type of 2R1T 3-PzPxS parallel mechanism is presented in this paper to satisfy the needs of configuration innovation for multi-axis machining platform. The limb configuration with a planar pantograph unit and the 3-PzPxS mechanism configuration are proposed and optimized. The analytical solutions of the forward and inverse position for the mechanism are derived, which are convenient to subsequent motion planning and control. After obtaining the workspace of the mechanism by the boundary search algorithm, the influence of key scale parameters on the rotation capacity of the mechanism in the whole workspace is investigated, and the reasonable ranges of the scale parameters required for large rotation capacity of the mechanism are confirmed. Furthermore, based on the Jacobian matrix condition number, the global performance indices of the angular velocity and the angular acceleration of 3-PzPxS parallel mechanism are defined, and ultimately the optimal ranges of the scale parameters are determined according to these two performance indices. The above results lay a theoretical foundation for the further development of high-efficiency hybrid NC machining center.

Keywords

Pantograph unit 2T1R motion Angular velocity performance index Angular acceleration performance index Scale optimization 

List of symbols

R

Rotation joint or rotational motion

T

Translational motion

P

Prismatic joint

S

Spherical joint

Rm

Rotating around the m-axis, m: x, y, z

Tm

Translating along the m-axis, m: x, y, z

Oa-xyz

Moving coordinate system of the moving platform

Ob-XYZ

Fixed coordinate system of the base

Mi-xiyizi

The coordinate system in the ith limb

Ai

The centriod of prismatic joint in the ith limb

Bi, Ci, Di, Ei

The centriod of revolute joint in the ith limb

Si

The centriod of spherical joint in the ith limb

Mi

The cross-point of the ith and the base

Rc

ΔM1M2M3 circumcircle radius

r

ΔS1S2S3 circumcircle radius

zi

The input displacement of the active prismatic joint in the ith limb

α

The angle between the AiCi linkage and the guide path of the active prismatic joint

βi

The relative angle between the linkage BiDi and BiCi

l1

The lengths of linkage AiBi and EiSi in the ith limb

l2

The lengths of linkage BiCi and DiEi in the ith limb

l3

The lengths of linkage BiDi and CiEi in the ith limb

θz, θy, θx

The Euler angles of the moving platform

A

Inverse Jacobian matrix

B

Positive Jacobian matrix

\(\dot{\rho }\)

Input velocity vector of the driving joint

\({\dot{\mathbf{p}}}\)

Output velocity vector of the moving platform

G

Jacobian matrix

K

The ratio of r to Rc

\(\eta_{J}\)

Global performance index of the mechanism

W

Reachable workspace of the mechanism

\(\eta_{{G_{\omega } }}\)

Global angular velocity performance index

\(\eta_{{G_{v} }}\)

The global linear velocity performance index

Gω

Angular velocity Jacobian matrix

Gv

Linear velocity Jacobian matrix

Hω

Angular acceleration Hessian matrix

Hv

Linear acceleration Hessian matrix

K*

The condition number of G*

\(\eta_{{G_{\omega } + H_{\omega } }}\)

Global angular acceleration performance index

\(\eta_{{G_{v} + H_{v} }}\)

Global linear acceleration performance index

Notes

Acknowledgements

Overseas Visiting Study Program for Young and Middle-aged Teachers in Universities in Shanghai (2018), Shanghai Special Foundation Project for Industrial Internet Innovation and Development (201822930), Shanghai Graduate Foundation Project for Academic Innovation (E3-0903-18-01020).

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019
corrected publication 2019

Authors and Affiliations

  1. 1.School of Mechanical and Automotive EngineeringShanghai University of Engineering ScienceShanghaiChina

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