On the Configurations of Closed Kinematic Chains in three-dimensional Space
This work addresses motion planning and singularity analysis for closed kinematic chains, but it appears incremental as it builds on existing parameterization methods.
The paper tackles the problem of describing the configuration space of closed kinematic chains in 3D Euclidean space, and it finds that in special cases, the space can be parameterized by a simple cube of dimension n-3 using diagonal lengths.
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.