On the Configurations of Closed Kinematic Chains in three-dimensional Space Gerhard Zangerl Department of Mathematics, University of Innsbruck Technikestraße 13, 6020 Innsbruck, Austria E-mail:
[email protected] Alexander Steinicke Department of Applied Mathematics and Information Technology, Montanuniversitaet Leoben Peter Tunner-Straße 25/I, 8700 Leoben, Austria E-mail:
[email protected] Abstract 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. 1 Introduction This study is the natural further development of [32] in which closed configurations of a two-dimensional kine- matic chain (KC) in terms of its joint angles were considered. As a generalization, we study the configuration spaces of a three-dimensional closed kinematic chain (CKC) with n links in terms of the joint angles of its spherical joints.