CGA-Based approach to direct kinematics of parallel mechanisms with the 3-RS structure

Ying Zhang*, Xianwen Kong, Shimin Wei, Duanling Li, Qizheng Liao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)
97 Downloads (Pure)


This paper presents a unified geometric modeling and solution procedure for direct kinematic analysis of a class of parallel mechanisms based on conformal geometric algebra (CGA). After locking the actuated joints, such parallel mechanisms will be turned into a 3-RS structure, which is composed of two triangular platforms connected by three RS serial chains in parallel. Using the proposed approach, the univariate polynomial equation for the direct kinematic analysis of these parallel mechanisms can be derived in three steps. Firstly, the positions of two of the three spherical joints on the moving platform are formulated by the intersection, dissection and dual of the basic geometric entities under the frame of CGA. Secondly, a coordinate-invariant equation expressed in terms of geometric entities is derived via CGA operation. Thirdly, a univariate polynomial equation is obtained directly from the aforementioned coordinate-invariant equation by using tangent-half-angle substitution. Several case studies are then presented to verify the solution procedure. The novelties of this approach lie in that: (1) The formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA; (2) No algebraic elimination procedure is required to derive the univariate polynomial equation; and (3) The proposed approach is applicable to the direct kinematics of this family of parallel mechanisms with any link parameters.

Original languageEnglish
Pages (from-to)162-178
Number of pages17
JournalMechanism and Machine Theory
Early online date8 Mar 2018
Publication statusPublished - Jun 2018


  • Conformal geometric algebra
  • Coordinate-invariant
  • Direct kinematics
  • Elimination-free
  • Parallel mechanisms

ASJC Scopus subject areas

  • Bioengineering
  • Mechanics of Materials
  • Mechanical Engineering
  • Computer Science Applications


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