# Dielectric Elastomer Actuator for Soft Robotics Applications and Challenges
## 3 Modeling of DEA
Nevertheless, the modeling of the spring roll actuator shows a similar trend to that of the circular actuator. The spring roll actuator is also being modeled through the analytical model using the neo-Hookean and the rheological model, or through the numerical model. In recent years, attention to DEA's dynamic actuation has grown. Several experiments have observed and reported the time-dependent behavior of DEAs due to their viscoelastic effect [47, 48, 49, 50, 51, 52, 53]. Cyclic loading simulations and experiments reveal a distinct temporal evolution behavior, which can be characterized by transient and stable regions. As mentioned previously, to model a DEA's viscoelastic behavior, methods such as using the Prony series or the three-parameter solid model were attempted. Figure 4: DE actuator configurations: (**a**) Structure of a circular actuator; (**b**) Electrically activated circular actuator; (**c**) Internal structure of a spring roll actuator; (**d**) Electrically activated spring roll actuator with a hanging mass. These methods may be adequate in modeling static behavior. However, they fail to accurately model and follow a DEA's cyclic loading behavior. To solve this issue, several modeling methods were proposed. The rheological model shown in Figure 2c has one elastic spring module in parallel with one viscoelastic module, composed of a spring and a dashpot. In 2017, [PERSON] et al. introduced an extended rheological model. This model consists of one elastic module in parallel with multiple viscoelastic modules. Forming the general model with the Gent model to describe the strain stiffening effect, it was experimentally validated that increasing the number of viscoelastic modules in the rheological model increases the model's prediction accuracy when describing a DEA's response [47]. Another method of describing a DEA's time-dependent behavior was proposed by [PERSON] et al. In this paper, to describe both the viscoelastic creep and relaxation, a Kelvin-Voigt-Maxwell (KVM) model was proposed. From the paper, it was experimentally shown that the KVM model is more accurate in capturing a DE's initial rapid creeping behavior upon step voltage input [48]. However, long-term relaxation behavior prediction performs similarly to the three-parameter model. In 2018, the viscoelastic modeling approach using phenomenological models was introduced by [PERSON] and [PERSON]. The paper proposed a modified rate-dependent Prandtl-Ishlinkii model (MRPIM) that can describe the asymmetric viscoelastic hysteresis effect with a maximum error of 9.03% [49]. Previously described modeling methods, based on the rheology of the elastomer, require a thorough understanding of the actuator. In contrast, phenomenological approaches are advantageous in that only experimental data are required to characterize a DEA's complex viscoelastic behavior. As a DE's nonlinearity and time-dependent behavior are being analyzed and understood, methods of controlling such nonlinear actuators are being studied as well. In 2019, a high-precision control method using inverse viscoelastic hysteresis compensation was proposed [50; 51]. Authors used a feedforward inverse hysteresis controller with a phenomenological hysteresis model to compensate for the DEA's hysteresis, and combined it with a classical proportional-integral feedback controller to compensate for the creeping behavior. Experiments were performed with conical out-of-plane DEAs, and results show a high-precision control with a maximum tracking error of 3.95% [51]. Along with a closed-loop control method, a comparatively simple open-loop control of DEAs was proposed by [PERSON] and his colleagues. In this approach, the open-loop control law is devised using quasi-linear viscoelasticity. For the material model, a series of Kelvin-Voigt elements are used. Experimental results show that, even without an additional self-sensing mechanism, the actuator follows the target strain with a small tracking error [52].