A polarisation based approach to model strain dependent electrostatic pressure of dielectric elastomer actuators - Schlögl, Tristan; Leyendecker, Sigrid (Proc. Appl. Math. Mech. (PAMM), 2017)
A wide‐spread lumped parameter model describing the electrostatic pressure is presented by Pelrine et. al. in 1998 [1]. In Pelrine's model, the electrostatic pressure is affected by the relative permittivity of the material, also known as dielectric constant. However, many researchers found that the dielectric constant of DEAs is not constant at all, but decreasing with increasing pre‐stretch of the material. In this work, an alternative modelling approach is presented, explaining the stretch dependent actuation pressure. It is shown that this new model fits experimental data found in literature quite well. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)<br

A polarisation based approach to model the strain dependent permittivity of dielectric elastomers - Schlögl, Tristan; Leyendecker, Sigrid (Sensors and Actuators A-Physical, 2017)
A wide-spread lumped parameter model describing the electrostatic pressure present in dielectric elastomer actuators is presented by Pelrine et al. in 1998. In Pelrine’s model, the electrostatic pressure is affected by the relative permittivity of the material, also known as dielectric constant. However, many researchers found that the dielectric constant of dielectric elastomers is not constant at all, but decreasing with increasing pre-stretch of the material. This holds especially for acrylic materials such as VHB 4910 from 3M. From a physical point of view, polarisation within the dielectric material is responsible for the material’s permittivity and in general, polarisation is deformation dependent. In this work, an alternative modelling approach is presented, explaining the stretch dependent electrostatic pressure. It is shown that Pelrine implicitly assumes that the polarisation of the material is linear in the imposed electric field strength. If this assumption is modified to allow for a more general polarisation field that is based on invariants of the electromechanically coupled problem, a new polarisation based lumped parameter model is obtained. It is shown that this new model fits experimental data found in literature quite wel
Keywords: dielectric elastomer; electrostatic pressure; permittivity; polarization; hyperelasticity; electromechanical coupling

Comparison of non-locking incompressible multi-field finite element models for dielectric actuators - Schlögl, Tristan; Leyendecker, Sigrid (EuroEAP, 2016)
Four Neo-Hookean material models are investigated, each model is tested with three numerical examples that are typical for dielectric actuator simulation. Model I is based on a classical displacement formulation, extended by electromechanical coupling terms. Model II is formulated in analogy to a nearly incompressible three-field formulation for pure mechanical problems. Model III extends model II by another field accounting for incompressibility. Finally, model IV is an attempt to decrease the amount of additional fields necessary to obtain incompressible behaviour. The first numerical example is a DEA cube where all sides except the top are fixed and an applied voltage exerts pressure to part of the top surface. The second example is a switch in form of a small stacked dielectric actuator that is connected to a rigid body with a revolute joint. The last example is a revolute joint controlled by two stacked actuators in agonist/antagonist configuration. The results show that material model I is not suited to approximate the incompressible material behaviour of dielectric elastomers. Large Poisson ratios inevitably lead to volume locking, even though the model is quite easy to implement and computationally very quick. Model II covers nearly incompressible behaviour very well in all applications, but the computational cost rises by a factor of about 2.6 compared to model I. Model III performs quite well, especially in combination with the structure preserving time integration scheme, allowing for exact incompressibility without any significant increase in computational cost. Model IV remarkably not decreases the computational cost compared to model III and hence does not offer any notable advantage.

Dynamic simulation of dielectric elastomer actuated multibody systems - Schlögl, Tristan; Leyendecker, Sigrid (Proceedings of the ASME 2016 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, 2016)
A three-dimensional electro-mechanically coupled finite ele- ment model for dielectric elastomers is used to actuate multibody systems. This setting allows exploring the complex behavior of humanoid robots that are driven by artificial muscles instead of electrical drives. The coupling between the finite element muscle model and the rigid bodies is formulated at configuration level, where Lagrange multipliers account for constraint forces, lead- ing to differential algebraic equations of index-3. A well-chosen set of redundant configuration variables for the multibody system avoids any rotational degrees of freedom and leads to linear cou- pling constraints. As a result, the coupling between the artificial muscles and the multibody system can be formulated in a very modular way that allows for easy future extension. The applied structure preserving time integration scheme provides excellent long time energy behavior. In addition, the index-3 system is solved directly with numerical accuracy, avoiding index reduc- tion approximation

Electrostatic-elastodynamic finite element modelling of stacked dielectric actuators - Schlögl, Tristan; Leyendecker, Sigrid (EuroEAP, 2014)
Generally, the three-dimensional behaviour of dielectric elastomer actuators (DEA) is covered by the Maxwell equations and the balance of momentum as shown by Dorfmann [1]. Building on a variational finite element formulation of the static coupled problem introduced by Vu [5], in this work inertia terms are added in order to obtain a description of the deformation process depending on time. This allows for a structure preserving time integration of fully three-dimensional DEAs [2]. The obtained scheme is used to simulate stacked actuators at finite strains. A reduction method is introduced that allows for the simulation of thin single layers at low computational cost.

Electrostatic-viscoelastic finite element model of dielectric actuators - Schlögl, Tristan; Leyendecker, Sigrid (Computer Methods in Applied Mechanics and Engineering, 2015)
The aim of this work is to set up a numerical framework to characterise the deformation process and effective forces when voltage is applied to dielectric elastomer actuators. Based on an existing model for non-linear electro-elasticity that covers the static case only, inertia terms are included in order to obtain a description of the deformation process depending on time. A potential energy function that is composed of Neo-Hooke material behaviour, electric field energy and coupling terms covers the material properties. Combined with the kinetic energy, a Lagrange function forms the basis in a variational setting of the model. Viscoelastic effects are included using non-conservative forces and account for time dependent strains. The action is approximated using quadrature rules and discretising with finite elements in space. A discrete version of Hamilton’s principle leads to a structure preserving integration scheme for DEAs. The integration scheme is implemented as C++ code and applied to various example
Keywords: Dielectric elastomer; Electroactive polymer; Finite element model; Viscoelastic; Electrostatic; Hyperelastic

Modelling and simulation of dielectric elastomer actuated multibody systems - Schlögl, Tristan; Leyendecker, Sigrid (EuroEAP, 2015)
An important field for dielectric elastomer (DE) actuators is the use as artificial muscles in humanoid robots. DE actuated systems allow for flexible kinematics and safe and natural motions. In contrast to pneumatic and hydraulic systems, artificial muscles facilitate the design of autarkic systems. Besides the automated manufacturing and lightweight power electronics, the simulation of DEs is investigated within the collaborative research project bionicum. A three dimensional finite element model covers the time dependent behaviour of dielectric actuators. A hyperelastic Neo-Hookean material model is fully coupled with electrostatics. Viscoelastic stresses account for damping effects. A structure preserving time integration scheme provides accurate simulation results for energetic investigations. The actuated multibody system is modelled using rigid bodies. A redundant set of state variables avoids rotational degrees of freedom and provides a robust and generic formulation. The finite element model is coupled with the multibody system by constraints formulated at position level. Lagrange multipliers take coupling forces into account. The structure preserving integration scheme allows for the numerically exact fulfilment of constraints without index reduction. The coupled model serves as a basis for future work regarding optimal control of DE actuated robot

Modelling, simulation and optimal control of dielectric elastomer actuated systems - Schlögl, Tristan (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2018)
The aim of this work is to present a physically motivated simulation framework to predict and control the time dependent behaviour of multibody systems that are actuated via artificial muscles. The artificial muscles are composed of stacked dielectric elastomers that contract due to electrostatic forces when a voltage is applied. As both electrical and mechanical quantities are involved in this interrelation, a multidisciplinary modelling approach is required. Existing electromechanically coupled models for dielectric elastomers can be categorised in two groups. The first group covers general three-dimensional field theory of electromagnetic forces in deformable continua with arbitrary geometry. The second group contains so called lumped parameter models, where spatially discrete configuration variables condense the complex physical relationships by exploiting symmetries, regularities and predicted behaviour. Both approaches are usually based on different assumptions, material models and modelling procedures, resulting in different simulation results. This work tries to bridge these two groups by deriving an energy consistent lumped parameter model directly from the corresponding three dimensional field theory. As both models consequently share the same modelling approach, they can be easily compared and assessed.   The present work starts by introducing a time dependent, versatile and monolithic three-dimensional finite element formulation for the electromechanically coupled problem that serves as the artificial muscle model. A variational time integration scheme ensures structure preservation as well as a good energy behaviour. An electromechanically coupled, visco-hyperelastic material approach provides flexibility and modularity. The artificial muscle model is coupled with a multibody system that represents the actuated structure. This setting allows for exploring the complex behaviour of humanoid structures that are driven by artificial muscles instead of electrical drives. The multibody system is composed of rigid bodies that are connected via joints and based on a redundant formulation that avoids rotational degrees of freedom and singularities. As a result, a very modular coupling between the multibody system and the finite element muscle is obtained. The set-up of multibody systems and the derivation of relevant equations is supported by the C++ library MulDi that emerged from this work. In order to control artificial muscle actuated systems and possibly avoid oscillations that are inherent with the elastic structure of the actuators, optimal control theory is utilised. To reduce the computational cost that is necessary to solve optimal control problems, an energy consistent lumped parameter model for dielectric elastomers is derived, where numerical examples illustrate potential applications.   The utilised variational time integration scheme turned out to be very suitable for solving electromechanically coupled problems. Apart from the preservation characteristics and the good energy behaviour, the integrator allows to solve algebraic constraints on configuration level exactly at the discrete time nodes. This allows for a neat coupling between the artificial muscles and the actuated structure. Moreover, fully incompressible material behaviour can be obtained, avoiding volumetric locking effects. Numerical examples show that the rather small achievable maximum contraction and forces of stacked actuators compared to real muscles still limit their use in human like structures. Optimal control theory, however, has been proven to provide a suitable tool for avoiding oscillations that are inherent with the elastic nature of the actuators and yields optimised voltage control trajectories.</

On electrostatic-viscoelastic simulation of dielectric actuators - Schlögl, Tristan; Leyendecker, Sigrid (Proc. Appl. Math. Mech. (PAMM), 2015)
Describing and controlling the behaviour of dielectric elastomer actuated systems requires a suitable simulation model for the electro active polymers. This work is based on three dimensional finite elements to cover arbitrary actuator geometries. Maxwell’s equations for electrostatics are coupled with the momentum balance by an electrostatic stress tensor. Time de- pendent quantities allow for a description of the deformation process depending on time. A potential energy function that is composed of Neo-Hooke material behaviour, electric field energy and coupling terms covers the material properties. Vis- coelastic effects are included and account for time dependent strains. A discrete version of Hamilton’s principle leads to a structure preserving integration scheme for dielectric elastomers. The integration scheme is implemented in C++ and applied to examples

On modelling and simulation of dielectric elastomer actuators via electrostatic-elastodynamic coupling - Schlögl, Tristan; Leyendecker, Sigrid; Reitelshöfer, Sebastian; Landgraf, Maximilian; Yoo, In Seong; Franke, Jörg (Proceedings of the IMSD, 2014)
The aim of this work is to set up a numerical framework to characterise the deformation process and effective forces when voltage is applied to dielectric elastomer actuators (DEA). As with capacitors, when an external voltage is applied to the conductive layers, an electric field is established. Resulting Maxwell stresses lead to a contraction of the DEA. These effects are described by the Maxwell equations, the balance of momentum and constitutive material laws. Based on a model for non-linear electro-elasticity that covers the static case only, inertia terms are included in order to obtain a description of the deformation process depending on time. A potential energy function that is composed of Neo-Hooke material behaviour, electric field energy and coupling terms covers the material behaviour. Combined with the kinetic energy, a Lagrange function forms the basis in a variational setting of the model. Approximating the action integral using quadrature rules, discretising with finite elements in space and finite differences in time and applying a discrete version of Hamilton’s principle [9], a structure preserving integration scheme for DEAs is obtained and evaluated
UnivIS-Import:2015-04-16:Pub.2014.tech.FT.ltd.onmode

Optimal control of a slot car racer - Penner, Johann; Schlögl, Tristan; Leyendecker, Sigrid (Proc. Appl. Math. Mech (PAMM), 2017)
In this contribution, we present a simulation method for the optimal control of a mechatronic system that is based on discrete variational calculus and apply it to compute the time-minimal path of a slot car racer. Here, the DMOC (Discrete Mechanics and Optimal Control) method is used to generate offline optimal trajectories for the electro-mechanically coupled system, i.e. sequences of discrete configurations and sequences of driving voltages. The time-minimal path is achieved by the choice of different cost functions, the sum of time steps, the negative sum of the quadratic momenta and the negative sum of the quadratic velocities. Simulation results show that the momentum formulation yields the lowest number of iterations.<br

Optimal control of a slot car racer - Penner, Johann; Schlögl, Tristan; Leyendecker, Sigrid (Proceedings of the 7th GACM, 2017)
Within this work, we compute and apply control strategies for the time-minimal path of a slot car racer. Here, the DMOC (Discrete Mechanics and Optimal Control) method is used to generate offline optimal trajectories for the electro-mechanically coupled system, i.e. sequences of discrete configurations and driving voltages. These sequences are embedded to a control architecture with an underling camera tracking system which allows to correct the vehicle towards the desired state via compute

Qualifying dielectric elastomer actuators for usage in complex and compliant robot kinematics - Reitelshöfer, Sebastian; Landgraf, Maximilian; Schlögl, Tristan; Franke, Jörg; Leyendecker, Sigrid (3rd international conference on Electromechanically Active Polymer (EAP) transducers & artificial muscles (EuroEAP), 2013)

Qualifying dielectric elastomer actuators for usage in complex and compliant robot kinematics - Reitelshöfer, Sebastian; Landgraf, Maximilian; Schlögl, Tristan; Franke, Jörg; Leyendecker, Sigrid (EuroEAP, 2013)

Strukturerhaltende Simulation und Optimalsteuerung gekoppelter Systeme - Leyendecker, Sigrid; Schlögl, Tristan; Wenger, Theresa (2017)
In diesem Beitrag wird die Simulation künstlicher Muskeln, die beispielsweise humanoide Roboter aktuieren, vorgestellt. Das Modell besteht aus einer elektromechanisch gekoppelten Materialbeschreibung des künstlichen Muskels, der mit einem starren Mehrkörpersystem verbunden ist. Basierend auf strukturerhaltender Zeitintegration und der Finite Elemente Methode wird sowohl die Vorwärtsdyna- mik als auch ein Optimalsteuerungsproblem energiekonsi- stent numerisch simuliert. Nach einer kurzen Einführung werden die Grundlagen variationeller Integration vorgestellt und das elektrome- chanisch gekoppelte Problem dielektrischer Elastomere formuliert. Ein numerisches Beispiel illustriert die Struk- turerhaltung in der vorwärtsdynamischen Simulation eines DEA-aktuierten Mehrkörpersystems, bevor eine Optimal- steuerung den künstlichen Muskel energieeffizient in ei- nen stationären Zustand bringt.

Variational formulation and finite element modelling of electrostatic-elastodynamic coupling - Schlögl, Tristan; Leyendecker, Sigrid (Proc. Appl. Math. Mech. (PAMM), 2014)