Mathematical Modeling of Passive and Active Tensions in Biological Muscles for Soft Robotic Actuators

Biological muscles generate tension from the combined contribution of the passive elastic recoil and the actively controlled contractile mechanisms. Understanding and replicating these passive and active tensions is necessary and beneficial for designing soft robotic actuators that emulate muscle-like behavior. In the current work, the aim is to develop a mathematical framework for modeling both the passive and active tensions in a biological muscle as functions of muscle length and contraction velocity. We will describe the passive tension by a nonlinear monotonically increasing function of length with threshold behavior in order to capture the experimentally observed stiffening occurring in stretched biological muscles. We will model the active tension using the superposition of Gaussian functions that relate bell-shaped tension-length with a flat plateau over the optimal length of the sarcomere. The parameters of this Gaussian representation of the active tension-length relation are determined from formulating a least-squares optimization problem, such that a Characteristic (indicator) function is approximated globally over the optimal length range of the sarcomere by summation