Nonlinear Dynamics

, Volume 93, Issue 2, pp 689–703 | Cite as

The closed-form motion equation of redundant actuation parallel robot with joint friction: an application of the Udwadia–Kalaba approach

  • Jizhuang Hui
  • Muxuan Pan
  • Ruiying Zhao
  • Li Luo
  • Linlin Wu
Original Paper


The closed-form motion equation of the redundant actuation parallel robot with joint friction is established by an extended application of the Udwadia–Kalaba modeling. Based on the cascading nature of the Udwadia–Kalaba equation, the motion equation of the parallel robot is obtained in a hierarchical manner. The redundant actuation parallel robot is segmented into several leg subsystems, which are connected by the kinematic constraints. The constraint forces between the subsystems can be calculated by the Udwadia–Kalaba equation. In virtue of the derived constraint forces, the explicit joints friction models, described by Coulomb friction and Stribeck friction, are formulated separately. There are no auxiliary variables (such as Lagrange multipliers or pseudo-generalized speeds) needed in the method. The established dynamic modeling technique evades the curse of dimensionality when dealing with Moore–Penrose inverse. A 2-DOF redundant actuation parallel robot is chosen to demonstrate the method.


Parallel robot Motion equation Udwadia–Kalaba equation Joint friction 



Ruiying Zhao is supported by National Natural Science Foundation of China (Grant No. 51605038) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2017JQ5034). Muxuan Pan is supported by Fundamental Research Funds for Central Universities (No. NJ20160020).


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Engineering Laboratory for Highway Maintenance EquipmentChang’an UniversityXi’anPeople’s Republic of China
  2. 2.College of Energy and Power EngineeringNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China

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