Synthesis of a non-circular cable spool to realize a nonlinear rotational spring

In this paper we present a cable mechanism which realizes a nonlinear rotational spring from a linear translational spring. The spring is pulled by a cable wound around a non-circular spool, which is rigidly attached to the joint. The non-circular shape of the spool induces a nonlinear relationship between its angular position and the torque created by the tension of the cable. Depending on the shape of the spool, various torque profiles can be realized. We show that for a given nonlinear torque profile, there is a closed-form solution of the shape of the spool which synthesizes this function. In a first part, we present the geometry of the problem. In a second part, we derive the methodology to calculate the shape of the spool to synthesize a prescribed torque profile. In the last part, we verify the design methodology by experiments with three different spools realizing a constant force spring, an exponential softening spring and a cubic polynomial spring. We discuss the possible sources of errors between the theoretical and experimental results.