  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.
