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 chapter, we are particularly interested in a family of spatial 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 Bricard 6R 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 ℝ3 of 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 and sine-Gordon equations, where infinitesimally the motion of each vertex is confined in the osculating plane.