  In this paper, a hybrid model for single-crystal Shape Memory Alloy (SMA)
wire actuators is presented. The result is based on a mathematical
reformulation of the M\"uller-Achenbach-Seelecke (MAS) model, which provides an
accurate and interconnection-oriented description of the SMA hysteretic
response. The strong nonlinearity and high numerical stiffness of the MAS
model, however, hinder its practical use for simulation and control of complex
SMA-driven systems. The main idea behind the hybrid reformulation is based on
dividing the mechanical hysteresis of the SMA into five operating modes, each
one representing a different physical state of the material. By properly
deriving the switching conditions among those modes in a physically-consistent
way, the MAS model is effectively reformulated within a hybrid dynamical
setting. The main advantage of the hybrid reformulation is the possibility of
describing the material dynamics with a simplified set of state equations while
maintaining all benefits of the physics-based description offered by the MAS
model After describing the novel approach, simulation studies are conducted on
a flexible robotic module actuated by protagonist-antagonist SMA wires. Through
comparative numerical analysis, it is shown how the hybrid model provides the
same accuracy as the MAS model while saving up to 80% of the simulation time.
Moreover, the new modeling framework opens up the possibility of addressing SMA
control from a hybrid systems perspective.
