On the Configurations of Closed Kinematic Chains in Three-dimensional Space

Gerhard Zangerl, Alexander Steinicke

Research output: Contribution to journalArticleResearchpeer-review


A kinematic chain in three-dimensional Euclidean space consists of n links that are connected by spherical joints. Such a chain is said to be within a closed configuration when its link lengths form a closed polygonal chain in three dimensions. We investigate the space of configurations, described in terms of joint angles of its spherical joints, that satisfy the the loop closure constraint, meaning that the kinematic chain is closed. In special cases, we can find a new set of parameters that describe the diagonal lengths (the distance of the joints from the origin) of the configuration space by a simple domain, namely a cube of dimension n−3. We expect that the new findings can be applied to various problems such as motion planning for closed kinematic chains or singularity analysis of their configuration spaces. To demonstrate the practical feasibility of the new method, we present numerical examples.
Original languageEnglish
Pages (from-to)96-115
Number of pages20
JournalInternational Electronic Journal of Geometry
Issue number1
Publication statusPublished - 30 Apr 2022


  • Configuration space
  • Closed kinematic chain
  • closed continuous random walk
  • path planning

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