Kaleidocycles are continuously rotating n-jointed linkages. We consider a certain class of six-jointed kaleidocycles which have a spring at each joint. For this class of kaleidocycles, stored energy varies throughout the rotation process in a nonconstant, cyclic pattern. The purpose of this paper is to model and provide an analysis of the stored energy of a kaleidocycle throughout its motion. In particular, we will solve analytically for the number of stable equilibrium states for any kaleidocycle in this class.