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

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


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.


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

List of symbols


Rotation joint or rotational motion


Translational motion


Prismatic joint


Spherical joint


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


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


Moving coordinate system of the moving platform


Fixed coordinate system of the base


The coordinate system in the ith limb


The centriod of prismatic joint in the ith limb

Bi, Ci, Di, Ei

The centriod of revolute joint in the ith limb


The centriod of spherical joint in the ith limb


The cross-point of the ith and the base


ΔM1M2M3 circumcircle radius


ΔS1S2S3 circumcircle radius


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


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


The relative angle between the linkage BiDi and BiCi


The lengths of linkage AiBi and EiSi in the ith limb


The lengths of linkage BiCi and DiEi in the ith limb


The lengths of linkage BiDi and CiEi in the ith limb

θz, θy, θx

The Euler angles of the moving platform


Inverse Jacobian matrix


Positive Jacobian matrix

\(\dot{\rho }\)

Input velocity vector of the driving joint


Output velocity vector of the moving platform


Jacobian matrix


The ratio of r to Rc


Global performance index of the mechanism


Reachable workspace of the mechanism

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

Global angular velocity performance index

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

The global linear velocity performance index


Angular velocity Jacobian matrix


Linear velocity Jacobian matrix


Angular acceleration Hessian matrix


Linear acceleration Hessian matrix


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



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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