Optimal control of dielectric elastomer actuated multibody dynamical systems

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.