  A linkage mechanism consists of rigid bodies assembled by joints which can be
used to translate and transfer motion from one form in one place to another. In
this paper, we are particularly interested in a family of spacial linkage
mechanisms which consist of $n$-copies of a rigid body joined together by
hinges to form a ring. Each hinge joint has its own axis of revolution and
rigid bodies joined to it can be freely rotated around the axis. The family
includes the famous threefold symmetric Bricard6R linkage also known as the
Kaleidocycle, which exhibits a characteristic "turning over" motion. We can
model such a linkage as a discrete closed curve in $\mathbb{R}^3$ with a
constant torsion up to sign. Then, its motion is described as the deformation
of the curve preserving torsion and arc length. We describe certain motions of
this object that are governed by the semi-discrete mKdV equations, where
infinitesimally the motion of each vertex is confined in the osculating plane.
