  In this work, a simulation model for the optimal control of dielectric
elastomer actuated flexible multibody dynamics systems is presented. The
Dielectric Elastomer Actuator (DEA) behaves like a flexible artificial muscles
in soft robotics. It is modeled as an electromechanically coupled geometrically
exact beam, where the electric charges serve as control variables. The DEA-beam
is integrated as an actuator into multibody systems consisting of rigid and
flexible components. The model also represents contact interaction via
unilateral constraints between the beam actuator and e.g. a rigid body during
the grasping process of a soft robot. Specifically for the DEA, a work
conjugated electric displacement and strain-like electric variables are derived
for the Cosserat beam. With a mathematically concise and physically
representative formulation, a reduced free energy function is developed for the
beam-DEA. In the optimal control problem, an objective function is minimized
while the dynamic balance equations for the multibody system have to be
fulfilled together with the complementarity conditions for the contact and
boundary conditions. The optimal control problem is solved via a direct
transcription method, transforming it into a constrained nonlinear optimization
problem. The beam is firstly semidiscretized with 1D finite elements and then
the multibody dynamics is temporally discretized with a variational integrator
leading to the discrete Euler-Lagrange equations, which are further reduced
with the null space projection. The discrete Euler-Lagrange equations and the
boundary conditions serve as equality constraints, whereas the contact
constraints are treated as inequality constraints in the optimization of the
discretized objective. The effectiveness of the developed model is demonstrated
by three numerical examples, including a cantilever beam, a soft robotic worm
and a soft grasper.
