Document [original]

ORIGINAL RESEARCH

published: 13 December 2019

doi: 10.3389/frobt.2019.00133

Frontiers in Robotics and AI | www.frontiersin.org 1December 2019 | Volume 6 | Article 133

Edited by:

Herbert Shea,

École Polytechnique Fédérale de

Lausanne, Switzerland

Reviewed by:

Senentxu Lanceros-Mendez,

University of Minho, Portugal

Janno Torop,

University of Tartu, Estonia

*Correspondence:

Jürgen Maas

[email protected]

Specialty section:

This article was submitted to

Soft Robotics,

a section of the journal

Frontiers in Robotics and AI

Received: 06 September 2019

Accepted: 18 November 2019

Published: 13 December 2019

Citation:

Hoffstadt T and Maas J (2019)

Self-Sensing Control for Soft-Material

Actuators Based on Dielectric

Elastomers. Front. Robot. AI 6:133.

doi: 10.3389/frobt.2019.00133

Self-Sensing Control for

Soft-Material Actuators Based on

Dielectric Elastomers

Thorben Hoffstadt and Jürgen Maas*

Mechatronic System Laboratory, Institute of Machine Design and Systems Technology, Technische Universität Berlin, Berlin,

Germany

Due to their energy density and softness that are comparable to human muscles dielectric

elastomer (DE) transducers are well-suited for soft-robotic applications. This kind of

transducer combines actuator and sensor functionality within one transducer so that

no external senors to measure the deformation or to detect collisions are required.

Within this contribution we present a novel self-sensing control for a DE stack-transducer

that allows to control several different quantities of the DE transducer with the same

controller. This flexibility is advantageous e.g., for the development of human machine

interfaces with soft-bodied robots. After introducing the DE stack-transducer that is

driven by a bidirectional flyback converter, the development of the self-sensing state and

disturbance estimator based on an extended Kalman-filter is explained. Compared to

known estimators designed for DE transducers supplied by bulky high-voltage amplifiers

this one does not require any superimposed excitation to enable the sensor capability

so that it also can be used with economic and competitive power electronics like the

flyback converter. Due to the behavior of this converter a sliding mode energy controller is

designed afterwards. By introducing different feed-forward controls the voltage, force or

deformation can be controlled. The validation proofs that both the developed self-sensing

estimator as well as the self-sensing control yield comparable results as previously

published sensor-based approaches.

Keywords: dielectric elastomers, self-sensing, control, soft material actuator, extended Kalman filter, stack-

actuator, flyback-converter

1. INTRODUCTION

Entirely soft-bodied robots exploit the full potential of robotic systems in terms of safe human-

machine-interactions and, thus, are in the scope of research. However, novel mechanical designs in

conjunction with smart and soft materials as well as innovative approaches for modeling and the

development of control strategies to handle such a highly sophisticated robot species are necessary

(Navarro et al., 2013; Robla-Gomez et al., 2017). Due to their behavior that resembles human

muscles, dielectric elastomers (DEs) are a promising approach that could pave the way for soft-

bodied robots. As a DE transducer consists of a very thin, elastomeric dielectric film covered with

compliant electrodes, its behavior can be described by a shape varying capacitor.

By applying a voltage vpto the electrodes of the DE transducer with permittivity ε0·εrand

thickness dthe resulting electrostatic pressure

Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

σel =ε0·εr·vp

d2(1)

compresses the elastomer. This pressure is used to operate a

DE transducer in actuator mode. However, if the change of

the transducer’s capacitance is detected that is caused by its

deformation, a simultaneous operation as sensor is enabled. If

the deformation dependency of the capacitance is known, the

mechanical transducer state can be determined. By exploiting this

self-sensing capability, soft and smart transducers can be realized

that do not require additional external sensors and, thus, can be

comparably easy integrated into various applications not limited

to soft robotics.

For various types of DE transducers different approaches to

control their displacement or force (Maas et al., 2011; Sarban and

Jones, 2012; Rizzello et al., 2015; Wilson et al., 2016; Hoffstadt

and Maas, 2017, 2018b) or to use them for active vibration

attenuation (Dubois et al., 2008; Kaal and Herold, 2011; Sarban,

2011) have been presented previously. Within these approaches

the control variables are directly measured with external sensors,

so that the DE transducer is only operated as actuator. Due to the

additional sensor, these controls are referred to as sensor-based

control schemes.

Within this paper, the focus is given on the development of a

model-based self-sensing control for DE transducers that allows

to control the voltage, force and deformation of the transducer

without measuring any mechanical quantities. Figure 1 gives an

overview of the overall developed control circuit.

As shown in the center and on the right hand side of

Figure 1 the terminal voltage vDE and current iDE have to be

measured to enable the combined actuator-sensor-operation.

In order to determine the mechanical state based on these

measurement quantities adequate self-sensing algorithms are

required. Anderson et al. (2012) summarizes different approaches

for this purpose. The goal of most self-sensing algorithms is to

identify the capacitance of the DE transducer in a first step and

afterwards estimate the deformation and force based on a model

or experimentally obtained information about the deformation

dependency of the capacitance. For almost all approaches the

driving voltage vDE is superimposed with a harmonic excitation

that is used for the sensor functionality.

Chuc et al. (2008) and Jung et al. (2008) published

first frequency domain based approaches by experimentally

identifying changes of the electrical impedance of a DE

transducer under deformation when it is excited by a harmonic

voltage vDE. Beside the capacitance Cpthey also considered losses

in the polymer and the electrode by adding the resistances Rsand

Rp, respectively, see Figure 1.

In Hoffstadt et al. (2014) another model-based identification

algorithm in the frequency domain is presented that estimates

the electrical parameters of a DE transducer by evaluating the

amplitudes of and the phase shift between the superimposed

terminal voltage and current. Furthermore, it was shown that

the behavior of a DE transducer can be sufficiently modeled by

neglecting the parallel resistance Rprepresenting losses in the

dielectric, if the DE transducer is excited with a comparable

high frequency.

The extended Kalman-filter introduced in Hoffstadt and Maas

(2018a) estimates the strain of a DE transducer without any

superimposed excitation so that it can be used independent of the

utilized power electronics. Other approaches in the time domain

estimate the charge qpof the capacitance Cp. Under further

consideration of the measured voltage vDE the capacitance Cp≈

qp/vDE can be determined (Matysek et al., 2011; Gisby et al.,

2013).

Rizzello et al. (2017) developed a self-sensing algorithm based

on the recursive least squares (RLS) method. For this purpose,

he takes into account the equivalent circuit diagram with three

parameters (see Figure 1). In a first step, the parameters of

a discrete transfer function describing the behavior of the

considered circuit are estimated. As these parameters depend on

the electrical parameters, they can be calculated afterwards. For

the identification a harmonic excitation signal is superimposed.

Although several self-sensing approaches have been developed

only a few closed-loop self-sensing controller designs have been

published, so far. Gisby et al. (2011) controls the deformation

of a single-layer circular DE transducer by using the already

mentioned self-sensing approach (Gisby et al., 2013). Here, the

terminal voltage is PWM generated. While the deformation of

the DE transducer mainly depends on the mean of this voltage,

the included higher harmonics are used to enable the sensor

functionality. The manually adjusted proportional gain controller

yields comparable low dynamics and accuracy. Therefore, Rosset

et al. (2013) extends this controller to a PI-controller, using

the same self-sensing approach (Gisby et al., 2013). Here, the

parameters of the controller are optimized for one particular

operating point of the nonlinear control plant. The derived

controller is used to control an optical grid.

Rizzello et al. (2016) systematically combines his RLS-

based self-sensing approach (Rizzello et al., 2017) with his

robust position controller (Rizzello et al., 2015) to control

the deformation of a DE membrane actuator. For the

combined actuator-sensor-operation the required driving voltage

determined by the controller is superimposed with a harmonic

excitation with a high frequency of 1 kHz and an amplitude of

75 V. Compared to the sensor-based control (Rizzello et al., 2015)

almost no drawbacks in terms of the accuracy are observed, while

the bandwidth of the closed-loop self-sensing control is reduced

due to the dynamics of the parameter identification.

Within the referenced publications costly and bulky high-

voltage amplifiers were used to feed the DE transducer. However,

due to the capacitive behavior of DE transducers voltage-fed

current sources are well suited instead of high-voltage amplifiers

(Eitzen et al., 2011a). Here, compact and efficient driving

electronics can be realized when using switched-mode operated

topologies like the bidirectional flyback converter. This converter

allows not only to supply the DE transducer with a certain voltage

but also to recover the energy stored in the DE transducer when

discharging it.

Under consideration of the properties of the bidirectional

flyback converter and the DE transducer, we previously published

sensor-based position and force controls in Hoffstadt and Maas

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 1 | Fundamental structure of the proposed closed-loop position control for DE stack-actuators fed by a bidirectional flyback-converter.

(2017, 2018b) that use the directly measured deformation as

feedback-signal, cf. Figure 1. Within this publication we extend

them to a self-sensing controller that is able to universally

control the voltage, force or deformation of the DE transducer

by just measuring the terminal voltage vDE and current iDE. For

this purpose, in the following section 2 the considered control

plant comprising a DE stack-transducer (Maas et al., 2015)

fed by a bidirectional flyback converter (Eitzen et al., 2011b;

Hoffstadt and Maas, 2016) is introduced and modeled. The

design of the novel self-sensing state and disturbance estimator

is presented in section 3. Due to the non-linear behavior of the

control plant an extended Kalman-filter (EKF) is used for this

purpose (Welch and Bishop, 2001). The developed estimator

does not require any superimposed excitation. The subsequently

presented controller design (Hoffstadt and Maas, 2017, 2018b)

is based on the sliding mode control approach (DeCarlo et al.,

1988) as this is well suited for the considered control plant and

its characteristic behavior. The self-sensing estimator and control

are experimentally validated in section 5. Finally, section 6

summarizes the developed approaches and the result.

2. MODEL OF THE DE TRANSDUCER

SYSTEM

Figure 2A shows a schematic representation of the considered

DE stack-transducer with Nlayers. This multilayer design is used

to scale the deformation 1zin z-direction, as one single layer

has an initial thickness of only d0=50 µm. Details about the

design and the manufacturing were published by Maas et al.

(2015). The static strain-force-behavior is shown in Figure 2B.

The transducer generates higher tensile forces Fact at smaller

strains εz=1z/z0, with z0=N·d0. By increasing the initial

electric field strength E0=vDE/d0the electrostatic pressure

according to Equation (1) increases so that higher forces and

strains are obtained. The blocking-force Fact(εz=0) and the no-

load strain εz(Fact =0) represent two characteristic points of the

strain-force behavior.

An analytical model for this transducer is published in

Hoffstadt and Maas (2015). In Figure 2 the modeled results of

the static strain-force behavior are compared with measurement

results and a finite element analysis (FEA) published by Kuhring

et al. (2015). The analytical model is based on the structure shown

on the right of Figure 1. The actuator tension σact is given by the

force equilibrium:

σact =β·σel −σelast −ηE· ˙εz−E1·εE1, with β=Ae

A. (2)

Here, σelast is the elastic material tension that is calculated using

the Neo-Hookean approach with the Young’s modulus Yto

consider the hyperelastic, non-linear material behavior:

σelast =Y

3·1

1−εz

−(1−εz)2. (3)

Beside this reversible elastic behavior, viscoelastic properties are

taken into account with the viscosity ηEand the Maxwell element

with stiffness E1and viscosity η1. Furthermore, with the area ratio

βit is considered that the electrostatic pressure σel acts only on

the area Aecovered with electrode, while all other tensions are

assumed to homogeneously act on the whole transducer area Ain

z-direction. Instead of applying Equation (1) for the electrostatic

pressure σel, here it is determined depending on the energy Uc,diel

in the electric field of the capacitance Cp:

β·σel =2

V·Uc,diel. (4)

The bidirectional flyback converter control proposed in Hoffstadt

and Maas (2016) enables three discrete input states in terms of

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 2 | Schematic design (A) and static strain-force characteristics (B) of the considered silicone-based DE stack-actuators.

the feeding power ¯

p. Beside an off-state, the DE can be charged

and discharged with almost constant power depending on the

characteristic energy increment 1Umax transfered during every

switching period TSof the converter:

p=









+¯

pmax = +1Umax

TS, charging

0, off-state

−¯

pmax = −1Umax

TS, discharging

with 1Umax =1

2·Lm·Im,max. (5)

The energy increment 1Umax depends on the magnetizing

inductance Lmof the converter and the magnetizing current

Im,max adjusted by its inner control. Under further consideration

of losses pRe dissipated in the electrode material the power ¯

p′=

p−pRe feeds the capacitance of the DE transducer. With this,

the electromechanically coupled behavior of a DE transducer

can be modeled based on a power balance yielding the state

space representation

˙x=





˙εz

mB·z2

·σact−σload

1−εz

˙εz−E1

η1·εE1

−2·Uc,diel ·˙εz

1−εz+1

τp







+









·¯

p′,

with x=





εz

˙εz

εE1

Uc,diel





.

(6)

Beside the strain εzand the energy Uc,diel the state vector x

includes the velocity ˙εzas well as the strain εE1of the stiffness E1

of the Maxwell element. Depending on the supplied input power

p′and an external load σload the inner states of the DE transducer

with volume Vand accelerated mass mBcan be calculated with

Equation (6).

For the subsequently developed self-sensing state estimator

models describing the strain dependency of the electrical

transducer parameters are required, too. The series resistance Rs

mainly comprises losses in the contacting of the DE transducer

and electrodes that are applied on the initial area Ae,0 of every

layer. It was shown Hoffstadt et al. (2016) that this resistance is

almost constant in the relevant range of deformation. In contrast,

the capacitance Cpfor the Nlayers connected in parallel is

given by:

Cp=N·ε0·εr·Ae,0

·1

(1−εz)κ=Cp,0 ·1

(1−εz)κ(7)

The change of the initial capacitance Cp,0 also depends on the

factor κ. In case of an absolutely homogeneous deformation

without constraints, κ=2 would apply. However, due to a

passive area around Aethat is required for insulation purposes, as

well as due to stiff mechanical interfaces applied on the top and/or

bottom of the transducer, here the factor is slightly decreased

to κ=1.85.

In analogy, the strain dependency of the parallel resistance Rp

reads as follows:

Rp=1

N·ρp·d0

Ae,0

·(1−εz)κ=Rp,0 ·(1−εz)κ. (8)

This resistance represents losses in the dielectric with the specific

resistance p.

Although Cpand Rpvary with the strain εzthe resulting time

constant τpis independent of the strain:

τp=Rp·Cp=ε0·εr·ρp=const. (9)

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

3. EKF-BASED SELF-SENSING

ALGORITHM

In Hoffstadt and Maas (2018a) we already published a self-

sensing estimator based on a discrete, extended Kalman-

filter that estimates the strain of the DE transducer without

superimposed excitation. However, for the closed-loop operation

beside the inner states of the transducer also the disturbance σload

has to be estimated. For example, this load tension might result

from a collision of an external device or human with a soft-

bodied robot equipped with DE transducers. Therefore, a new

and extended approach based on Equation (6) is applied here. As

mentioned above, the goal is to determine the electromechanical

state of the DE transducer based on the measured terminal

voltage vDE and the current iDE. However, in Equation (6)

the energy Uc,diel represents the electrical state. Therefore,

a modification of the model is required to design the self-

sensing estimator.

For this purpose, the change of the charge qpon the

capacitance Cpis taken into account. It can be calculated under

consideration of the current iDE and the leakage current vp/Rp=

qp/τp, see Figure 1:

˙qp=iDE −vp

=iDE −1

τp

·qp, with qp=Cp·vp. (10)

As the charge depends on the measured current and the invariant

time constant τpfrom Equation (9), it is used as input variable

uqv in the following. Furthermore, if instead of the energy Uc,diel

the charge qpis considered, the electrostatic pressure can be

expressed by:

β·σel =2

V·Uc,diel =q2

V·Cp(εz), with Uc,diel =1

2·q2

Cp(εz).

(11)

Additionally, the voltage vpacross Cpdepends on the terminal

voltage vDE reduced by the voltage drop Rs·iDE across the series

resistance Rsthat is assumed to be constant here:

vp=qp

Cp(εz)=vDE −Rs·iDE. (12)

This voltage will be used as output variable yqv afterwards.

Beside the mechanical states included in xand Equation (6)

the external load σload has to be estimated as disturbance, too. As

it represents an unknown disturbance it is assumed (according

to Isermann and Munchhof, 2011) that it is constant during one

sample time Tof the discrete EKF implemented on a DSP. By

applying ˙σload =0 in combination with Equations (10)–(12) a

fourth order system can be established for the estimation:

˙xqv =fqv xqv,uqv=





˙εz

γ1·σact(qp)−σload

1−εz

˙εz−E1

η1·εE1







with xqv =





εz

εE1

σload





,γ1=V

mB·z2

and

yqv =gqv xqv,uqv=(1−εz)κ

Cp,0

·qp=vp=vDE,m −Rs·iDE,m.

(13)

According to Adamy (2014) the observability of the nonlinear

system (13) is given if the determinant of the observability

matrix QB,qv is not zero. This matrix can be calculated under

consideration of the Lie derivatives Li

fqv gqv, with i=0, ..., 3:

QB,qv xqv,uqv=







∂L0

fqv gqv(xqv,uqv)

∂xqv

∂L1

fqv gqv(xqv,uqv)

∂xqv

∂L2

fqv gqv(xqv,uqv)

∂xqv

∂L3

fqv gqv(xqv,uqv)

∂xqv













∂gqv(xqv,uqv)

∂xqv

∂L1

fqv gqv(xqv,uqv)

∂xqv

∂L2

fqv gqv(xqv,uqv)

∂xqv

∂L3

fqv gqv(xqv,uqv)

∂xqv







(14)

Beside material parameters that are different from zero, the

determinant of QB,qv depends on the charge qpand the strain εz:

det QB,qv xqv,uqv=q4

p·γ2

1·κ4·E2

η1·C4

p(εz)·(1−εz)66= 0,

for εz<1 and qp6= 0. (15)

The strain εzis always smaller than one, and thus does not

influence the observability. However, the uncharged state with

qp=0 is not observable. In contrast for example to piezoelectric

materials, this is due to the fact that the DE materials do

not contain inherent dipoles causing a charge separation under

deformation. Instead, a DE transducer has to be electrically

pre-charged so that a current flow or change of voltage can

be detected when it is deformed. Furthermore, the restricted

observability for qp=0 is not only a drawback of the proposed

approach. All referenced self-sensing methods have the same

issue, but the usually superimposed voltage excitations ensure

that this operating point does not occur. As this superimposed

excitation is not required for the EKF-based estimator a certain

amount of charge qp,min is always applied, here.

This results in the structure of the EKF-based self-sensing state

and disturbance estimator shown in Figure 3. As the EKF will

be implemented on a DSP its discrete implementation according

to Welch and Bishop (2001) is applied. Using the external

estimation of the charge by filtering the measured current

iDE,m has the advantage, that the state vector xqv only includes

mechanical states that have to be estimated with the EKF.

Furthermore, the parameterization effort increases significantly

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 3 | Structure of the proposed self-sensing state and disturbance estimator based on an extended Kalman-filter.

with increasing system order so that it is meaningful to use a

system with order n=4 instead of n=5.

For the implementation of the Kalman-filter algorithm

according to Figure 3 the system (13) has to be linearized in the

predicted state ˆxqv,p,k:

Aqv,k=∂fqv xqv,uqv

∂xqv ˆxqv,p,k,uqv,k

=





0 1 0 0

aqv,21 −γ1·ηE

1−ˆεz,p

−γ1·E1

1−ˆεz,p

−γ1

1−ˆεz,p

0 1 −E1

η10

0 0 0 0







, (16)

with aqv,21 =γ1

1− ˆεz,p

·"ˆσact,p uqv,k− ˆσload,p

1− ˆεz,p

−dˆσelast,p

dˆεz,p

−dˆσel,p

dˆεz,p #,

and dˆσel,p

dˆεz,p

(11)

(7) −κ·q2

p·1− ˆεz,pκ−1

V·Cp,0

Based on this the discrete transition matrix 8qv can be

approximated by (Ifeachor and Jervis, 2002):

8qv ≈I+Aqv,k·T, (17)

where Irepresents the unity matrix of order n=4. The output

vector cT

qv,kis calculated by the jacobian of the output function

gqv in Equation (13) with respect to the state vector xqv:

qv,k=∂gqv xqv,uqv

∂xqv ˆxqv,rmp,k,uqv,k

=h−κ·(1−ˆεz,p)κ−1

Cp,0 ·uqv,k0 0 0 i.

(18)

With these information the predicted state ˆxqv,p,kand the related

covariance matrix Pp,kcan be determined in the prediction step

(denoted by the index p) of the algorithm shown in Figure 3.

In the following correction the Kalman matrix Kand covariance

matrix Pkare calculated to update the estimated state vector ˆxqv,k.

The covariance matrices of the measurement and system noise

Rvv and Qww, respectively, will be parameterized in the validation

section 5. With the information, included in the state vector xqv,k

and Equation (4) to calculate the energy Uc,diel based on the

charge qp, all state variables in xfrom Equation (6) as well as the

load σload can be determined.

4. SELF-SENSING SLIDING MODE

CONTROL

The considered control plant modeled with Equation (6) has

a strongly non-linear behavior. Furthermore, the bidirectional

flyback converter allows to supply discrete feeding powers ¯

pso

that it can be described by the three-point switch in Equation (5).

Due to these properties the design of a variable structure control

is well suited. In Hoffstadt and Maas (2017) a position controller

based on the model (6) was introduced that uses the sliding

mode control (SMC), for this purpose. Additionally, a SMC force

controller was published in Hoffstadt and Maas (2018b). In the

following it will be shown that this controller cannot be used to

solely control the force Fact of the DE transducer but also the

strain εzand the voltage vpby applying different feed-forward

structures to one and the same controller. This flexibility makes

the approach advantageous for sophisticated applications like

in soft robotics. The detailed design of the controller shown in

Figure 4 can be found in Hoffstadt and Maas (2018b) and will be

summarized in the following.

In case of the SMC a static setpoint state vector x∗has to

be defined including setpoints for every state variable. Under

consideration of the static force equilibrium

lim

t→∞ σact (t)=β·σel −σelast

(4)

V·Uc,diel −σelast

=σload (19)

resulting from Equation (2) setpoints for the energy U∗

c,diel can be

derived. On the one hand, the energy can be calculated depending

Frontiers in Robotics and AI | www.frontiersin.org 6December 2019 | Volume 6 | Article 133

Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 4 | Detailed structure of the controller (blue box in Figure 1) including a feed-forward structure to either control the voltage, strain or force and a three-point

controller with hysteresis and adaption of the inner flyback converter control.

on a setpoint strain ε∗

U∗

c,diel ε∗

z=V

2·σelast ε∗

z+σload. (20)

To achieve this strain the electrostatic pressure caused by the

energy according to Equation (4) has to compensate the elastic

material tension σelast(ε∗

z) given by Equation (3) as well as the

influence of the disturbance σload. On the other hand, if the DE

transducer should generate a certain force F∗

act =A·σact the

corresponding energy U∗

c,diel is given by:

U∗

c,diel F∗

act=V

2·F∗

act

A(εz)+σelast (εz),

with σload =F∗

act

A(εz)and A(εz)=A0

1−εz

(21)

In this case, the influence of the elastic deformation has to be

compensated, i.e., the energy has to be increased with increasing

strain εz(see Figure 2B).

Beside these approaches, the energy U∗

c,diel can be also

determined depending on a setpoint voltage v∗

pacross the

capacitance Cp(εz) in Equation (7):

U∗

c,diel v∗

p=1

2·Cp(εz)·v∗

2. (22)

With Equations (20)–(22) three approaches exist to define a

setpoint value for the energy U∗

c,diel. For the system (6) also a

setpoint for the strain ε∗

zis required, while the other two state

variables are zero during steady state ˙εz=εE1=0, respectively.

However, especially if the force or voltage should be controlled

by applying Equations (21) or (22), they should be independent

of the strain, i.e., that no setpoint ε∗

zcan be defined in this case.

To overcome this issue, the control design is based on a reduced

system (23) with ˙εz,εE1and Uc,diel as state variables while the

strain εztogether with the load tension σload is considered to be a

disturbance, here:

˙xU=



¨εz

˙εE1

Uc,diel 

=





mB·z2

·σact−σload

1−εz

˙εz−E1

η1·εE1

−2·Uc,diel ·˙εz

1−εz+1

τp





|{z }

fU(xU,z)

+



1



|{z}

·¯

p′,

with z=εz

σload .

(23)

In this case, the setpoint state vector reads as:

˙x∗

U=0 0 U∗

c,diel T. (24)

4.1. Design of the Sliding Mode

The control operation with a SMC is characterized by two

phases. During the sliding mode the system is led toward its

setpoint x∗on the switching function S(1x)=S(x−x∗)=0.

Within the reaching phase it is ensured, first, that this switching

function is reached from any arbitrary initial state. According to

DeCarlo et al. (1988) one comparable simple approach for the

design of the switching function is obtained if the system is in

standard canonical form (denoted by the index R). To determine

a corresponding transformation matrix T, the system (23) has

to be linearized yielding the system matrix AUfor the estimated

state ˆxU:

AU=∂fU(xU,z)

∂xUxU=ˆxU

=





−γ1·ηE

1−ˆεz

−γ1·E1

1−ˆεz

γ1·2/V

1−ˆεz

1−E1

η10

−2·ˆ

Uc,diel

1−ˆεz0−2·ˆ

˙εz

1−ˆεz−2

τp





.

(25)

As the system behaves linear concerning the input u, the

constant input vector bUis already given in Equation (23). With

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

these information the following transformation matrix Tcan be

derived as proposed by Kalman (1960):

T= −V

2·1− ˆεz

γ1

·





0−1 0

−1E1

η10

t31 t32 −2

V·γ1

1−ˆεz





(26)

with t31 =E1

η1

+γ1·ηE

1− ˆεz

and t32 =E1

η1

·γ1·η1

1− ˆεz

−E1

η1.

For the considered single input single output (SISO) system a

linear switching function is defined:

S(1xR)=cT·1xR,U=c1c2c3·T·ˆxU−x∗

U,

with 1xR,U=T·ˆxU−x∗

U. (27)

During the sliding mode S(1xR)=0 as well as ˙

S(1xR)=

0 applies. This behavior is obtained by the equivalent input

(DeCarlo et al., 1988)

ueq = −cT·bR,U−1·cT·AR,U·1xR,U,

with AR,U=T·AU·T−1and bR,U=T·bU. (28)

With this input the dynamics during the sliding mode only

depend on the coefficients ci, with i=1, 2, 3, of the switching

function in Equation (27):

1˙xR,U=AR,U·1xR,U+bR,U·ueq

=I−b·hcT·bi−1·cT·AR,U·1xR,U

=



0 0 0

0 0 1

0−c1

c3−c2



·1xR=00T

0˜

A1·1xR,U.

(29)

An other characteristic property of the SMC approach is that

during the sliding mode the system order nis reduced by the

number of inputs p(here p=1). Thus, the dynamics during

the sliding mode can be defined by a pole placement under

consideration of ˜

A1. For a second order element with damping

coefficient Dand cut-off frequency ωgthis results in:

det s·I−˜

A1,U=s2+c2

·s+c1

=s2+2·D·ωg·s+ω2

with c3=1, ⇒c1=ω2

g,c2=2·D·ωg.

(30)

4.2. Reachability

To reach this sliding mode a proper controller function

u1xR,Uhas to be determined and parametrized under

consideration of the properties of the feeding power electronics.

One approach to prove the reachability is based on an

investigation of the Laypunov function V1xR,U=1/2·

S21xR,U. To ensure stable steady-state behavior the time

derivate of the Lyapunov function has to be negative:

V1xR,U=S1xR,U·˙

S1xR,U!

<0. (31)

The derivative of the switching function is given by:

S1xR,U=cT·T·AU·1xU+bU·u1xR,U

=ζ1·1˙εz+ζ2·1εE1+ζ3·1Uc,diel +u1xR,U, with

(32)

ζ1=V

2·γ1ω2

g−2·D·ωg·γ1·ηE+E1

η1

+γ2

1·η2

E+γ1·E1·ηE

η1

−1+E2

η2

1,

(33a)

ζ2=V·E1

2·γ1·η1

·ω2

g+2·D·ωg·γ1·η1−E1

η1

−2·γ1·E1−γ2

1·ηE·η1+E2

η2

1and

(33b)

ζ3=2·D·ωg−2

τp

−E1

η1

−γ1·ηE, for ˆxU=0.(33c)

The coefficients ζ1,ζ2and ζ3depend on material parameters as

well as the damping ratio Dand cut-off frequency ωg. These two

controller parameters are chosen in such a way that the influence

of the state variables 1˙εzand 1εE1on Equation (32) vanishes. By

solving ζ1=0 and ζ2=0 the following parameters result:

ωg,0 =rγ1·E1·η1+ηE

η1

and (34a)

D0=1

2·s(E1+γ1·η1·ηE)2

γ1·E1·η1·(η1+ηE). (34b)

According to Equation (5) the input power ¯

psupplied by the

bidirectional flyback converter can be described by a three-point

controller. However, for the design of the SMC the off-state with

p=0 can be neglected in a first step. Under consideration

p= ±¯

pmax a two-point controller is defined:

u1xR,U=sgn S1xR,U·. (35)

The parameter = ±¯

pmax will be chosen so that the reachability

is ensured.

By inserting Equations (34), (34b), and (35) into Equation (32)

the time derivative of the switching function simplifies to:

S1xR,U= − 2

τp·1Uc,diel +u1xR,U

τp·U∗

c,diel +sgn S1xR,ε·. (36)

The control parameter is determined by applying a case-by-case

analysis to satisfy Equation (31):

I.: S1xR,U>0, ⇒u1xR,U= +,

⇒˙

S1xR,U=2

τp

·U∗

c,diel +!

<0,

II.: S1xR,U<0, ⇒u1xR,U= −,

⇒˙

S1xR,U=2

τp

·U∗

c,diel −!

>0.

(37)

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

As the energy U∗

c,diel will always be equal to or larger than zero,

both inequalities are solved by choosing:

= −¯

p′

max ⇒¯

p′

max >2

τp

·U∗

c,diel. (38)

Especially during steady state the introduced two-

point controller will permanently switch between the

positive and negative input power ±¯

pmax. To avoid this

chattering, the controller is extended to a three-point

controller with hysteresis, as already shown in Figure 4:

p∗=





+¯

pmax,for S1xR,U≤ −δSor −δS<S1xR,U<0∧˙

S1xR,U>0

0, else

−¯

pmax for S1xR,U≥δSor 0 <S1xR,U< δS∧˙

S1xR,U<0

. (39)

On the one hand, the off-state of the flyback converter

is now taken into account, while on the other hand the

hysteresis with threshold δSwill significantly reduce the

switching frequency in closed-loop operation. In Figure 4

an output limitation is also depicted that switches off the

control, when the energy ˆ

Uc,diel exceeds a maximum value

Uc,diel,max. Furthermore, to improve the steady state behavior

the inner control of the flyback converter is adapted. Depending

on the absolute value of the switching function |S(1xR,U)|,

the maximum magnetizing current I∗

m,max and thus the

feeding power ¯

paccording to Equation (5) is varied. This

ensures, that for large control deviations corresponding to

large values of |S(1xR,U)|the maximum feeding power is

supplied for achieving the maximum dynamics. In contrast,

for small control deviations the power is reduced for a

higher accuracy by also adapting the hysteresis threshold δS.

Further details can be found in Hoffstadt and Maas (2017,

2018b).

5. EXPERIMENTAL VALIDATION

5.1. Test Setup for the Experimental

Validation

Figure 5 schematically depicts the test setup used for the

experimental validation of the self-sensing estimator and the self-

sensing control. It consists of a bidirectional flyback converter

that supplies the DE transducer with voltages up to 2.5 kV

(Hoffstadt and Maas, 2016). The voltage vDE,m is measured with

the voltage probe TT-SI 9010 from Testec, while the current iDE,m

is determined by the voltage drop across the shunt resistance

Ris=1 k. Details about the utilized DE transducers can be

found in Maas et al. (2015). If no-load scenarios are investigated

in the following, the displacement of the DE transducer is directly

TABLE 1 | Parameters of the utilized silicone based DE stack-transducer and the

self-sensing controller.

YηEE1η1V mBN·d0=z0τp

1.08 MPa 490 Pa·s 155 kPa 1.7 kPa·s 1.4 cm30.5 g 9.6 mm 24 s

Cp,0 RsRvv ωgD Imin Imax δS

6 nF 135 k4 V22.430 rad/s 3 4 A 8 A 4 ·1Umax

FIGURE 5 | Test setup for the experimental validation comprising a bidirectional flyback converter, a voltage and current measurement, a DE stack-transducer and a

test rig with linear drive, while the data logging and the different controls are implemented on the real-time system.

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 6 | Comparison of the EKF-based estimation results of the proposed self-sensing filter with a sensor-based filter for the no-load scenario.

measured with the laser sensor OptoNCDT ILD 2300 from

Micro-Epsilon. To apply loads to the DE transducer the test rig

on the right hand side of Figure 5 will be used. It consists of

a force measurement with the force sensor 9217A from Kistler

and a voice coil linear drive VM8054-630 from Geeplus. The

DE transducer can be attached between the force measurement

and the linear drive. Via the voice coil load profiles with high

dynamics can be applied to the DE transducer, while the resulting

actuator force Fact,m is measured. Here, the same laser sensor as

for the no-load scenarios measures the displacement of the rigidly

coupled voice coil and DE transducer.

The proposed self-sensing algorithm and the energy control

are implemented on the DSP of a real-time system from dSPACE

operating with a sample rate of fDSP =20 kHz. The system

contains also a fast FPGA board. On this board the control of the

flyback converter and the signal conditioning for the measured

voltage and current vDE,m and iDE,m are performed.

5.2. Validation of the EKF-Based

Self-Sensing Algorithm

Before the closed-loop self-sensing operation is investigated,

the estimation results obtained with the suggested self-sensing

approach are compared to results estimated with the sensor-

based observer introduced in Hoffstadt and Maas (2017, 2018b).

The parameters of the silicone based DE stack-transducer with

N=192 layers are listed in Table 1. This table also includes

parameters for the controller used in the following section.

Figure 6 compares the estimation results of the proposed self-

sensing approach with the sensor-based estimator. The voltage

controlled bidirectional flyback converter supplies the DE stack-

transducer stepwise with voltages of vDE =1.5, 2.5, and 2 kV,

respectively. The charge qpdetermined by filtering the measured

current iDE,m according to Equation (10) is used as input for

the self-sensing filter, while the sensor-based estimator uses the

energy Uc,diel as input.

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 7 | Comparison of the EKF-based estimation results of the proposed self-sensing filter with a sensor-based filter when a load force is applied to the DE

transducer.

The measurement noise Rvv =4 V2required for the

implementation of the EKF can be determined experimentally.

For this, the output function gqv in Equation (13) and the

properties of the voltage probe and current measurement via the

shunt have to be taken into account. One of the main issues

when designing an EKF is to find an appropriate choice of

Qww. Here, the numerical optimization approach presented by

Powell (2002) is used to minimize the error between simulated

and estimated state variables by varying the entries of the

symmetric matrix Qww. For the system introduced in section 3

this optimization yields:

Qww,qv

=ζqv ·





4, 8 ·10−8−1, 7 ·10−9−2, 2 ·10−8−9, 3 ·10−4

−1, 7 ·10−94, 9 ·10−51, 5 ·10−44, 4 ·10−1

−2, 2 ·10−81, 5 ·10−41, 6 ·10−9−1, 7 ·10−5

−9, 3 ·10−44, 4 ·10−1−1, 7 ·10−57, 6 ·104





.

(40)

The entries represent in a certain way the uncertainty of the

model (13) to describe the dynamics of the state variables. While

all entries of the matrix are comparable small, the one in the

fourth row and column is very large. This is due to the unknown

dynamics of the load tension that is considered with ˙σload =

0 in Equation (13). As the dynamics of the state estimation

can be adjusted by the absolute values of the entries in Qww

the scaling factor ζqv is introduced. It gives the opportunity to

adjust a compromise between sufficient dynamics, reliable state

estimation and noise suppression.

In Figure 6 the no-load scenario with Fload =A·σload =

0 is considered. As can be seen in the comparison of the

measured and estimated strains εzin the top right plot, almost

no deviations between the approaches in terms of dynamics

and accuracy occur. Due to parameter deviations the sensor-

based filter estimates small load forces especially during transient

operation. For the self-sensing filter with ζqv =10−3a

comparable small factor is applied here. With this negligible

deviations in the estimated load force occur without affecting the

estimation results of the state variables shown on the right.

Figure 7 compares the estimation results obtained when a

load force of Fload =2 N is stepwise applied to the DE stack-

transducer with the force controlled voice coil actuator. When

the tensile load is applied the strain of the DE transducer reduces

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 8 | Comparison of the sensor-based and self-sensing sliding mode energy control. In both cases two-point controllers (2PC) with I∗

m,max =8 A and

I∗

m,max =4 A as well as the three-point controller (3PC) with hysteresis and adaption of the inner flyback converter control are considered.

from εz≈1.9% to εz≈1.1%. In voltage controlled operation

this causes a reduction of charge and energy as can be seen in

the top left plot. The saw tooth profile in the charge qpand

energy Uc,diel is caused by the voltage control of the flyback

converter that is based on a hysteresis controller. The sensor-

based filter estimates the strain as well as the load force with

errors less than |errε| ≤ 1% and |errF| ≤ 4%, respectively. For

the self-sensing filter two parameterization with ζqv =10−3

and ζqv =1 are investigated. While with ζqv =10−3the

strain and force are estimated with errors below |errε| ≤ 1%

before the load is applied and after it is released, the dynamics

of the estimation is not sufficient to consider the influence of

the load correctly. In contrast, with ζqv =1 the influence of the

load is accurately estimated. However, with this setting the noise

suppression especially for charge states below qp≤5µAs is not

sufficient. Therefore, for the following investigations of the self-

sensing control the scaling factor is switched from ζqv =10−3to

ζqv =1 if the charge exceed qp≤5µAs. This ensures an accurate

estimation of the inner transducer states at low charge states as

well as an accurate detection of a load force and its influence on

the states.

5.3. Validation of the Self-Sensing Control

The parameters of the sliding mode energy controller designed

in section 4 are listed in Table 1. The damping coefficient D=3

and cut-off frequency ωg=2.430 rad/s were determined with

Equation (34). The hysteresis threshold δS=4·1Umax for the

three-point controller in Equation (39) is set to a multiple of the

energy increment 1Umax transfered during one switching period

TSof the flyback converter. Figure 8 compares the closed-loop

operation of the sensor-based controller published in Hoffstadt

and Maas (2018b) and the proposed self-sensing controller.

First of all, no feed-forward control approaches as suggested in

Equations (20)–(22) are considered. Instead, three setpoint steps

for the energy U∗

c,diel are applied that correspond to voltages of

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 9 | Bandwidth of the sensor-based and self-sensing sliding mode energy control for the three investigated controller settings.

FIGURE 10 | Comparison of a simple hysteresis voltage control with the sensor-based and self sensing energy three-point controller using a voltage

feed-forward control.

vDE =1.5 kV, 2.5 kV and 2 kV for the silicone based DE-

stack-transducer, respectively. For both, the sensor-based and

the self-sensing control two-point controllers (2PC) according

to Equation (35) with I∗

m,max =8 A and I∗

m,max =4 A are

investigated as well as the three-point controller (3PC) with

hysteresis and adaption of the inner flyback converter control

from Equation (39) and Figure 4. The DE stack-transducer is

attached between the force measurement and the blocked voice

coil so that it cannot deform (εz=0) to avoid disturbances.

Via the setpoint I∗

m,max for the current control of the flyback

converter its feeding power is adjusted according to Equation (5).

Due to the reduced power it takes a longer time to adjust the

setpoint energies with the two-point controller with I∗

m,max =4 A

compared to the one with I∗

m,max =8 A. In contrast, the reduced

feeding power results in a higher accuracy during steady state.

The standard deviation for the time interval between 50 and

60 ms increases from 0.03 mJ (2PC, I∗

m,max =4 A) to 0.05 mJ

(2PC, I∗

m,max =8 A) for the sensor-based control and from 0.1 to

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 11 | Comparison of the sensor-based and self-sensing energy control with position feed-forward control and an explicit sliding mode position control.

0.3 mJ for the self-sensing control, respectively. The adaptive

three-point controller with hysteresis combines the advantageous

of the two mentioned two-point controllers by automatically

choosing the maximum current I∗

m,max =8 A right after setpoint

steps and reducing this current to I∗

m,max =4 A at steady state.

This fundamental behavior applies for both the sensor-based and

self-sensing control. Although the dynamics of both approaches

are comparable, a small oscillation around the setpoint can be

observed in case of the self-sensing control that results in the

higher standard deviation.

Furthermore, it can be seen that the two-point controllers

permanently switch between the maximum charging and

discharging power ¯

p= ±¯

pmax during steady state. By extending

the controller to a three-point controller with hysteresis, the

switching frequency can be significantly reduced by more than

80% in case of the sensor-based control and 30% in case of the

self-sensing control.

Figure 9 depicts the comparison of the bandwidth of the

introduced controller settings. For this purpose, the small

signal behavior is considered. A harmonic setpoint U∗

c,diel with

increasing frequency, an offset of ¯

Uc,diel =12 mJ and an

amplitude of Uc,diel,amp =2 mJ is applied. The sensor-based two-

point controller with I∗

m,max =8 A and the three-point controller

have a high -3 dB cut-off frequency of about 400 Hz. This is

also obtained with the self-sensing control. However, disruptive

amplitude peaks of about 5 dB result in the already observed

oscillation. By reducing the feeding power ¯

pwith I∗

m,max =4 A,

the cut-off frequency is reduced to 200 Hz, while the amplitude

peaks are suppressed.

5.4. Energy Control With Voltage

Feed-Forward Control

By applying Equation (22) for the feed-forward control depicted

in Figure 4 the voltage vpacross the capacitance Cpcan be

controlled. In Figure 10 the results of the sensor-based and

self-sensing energy three-point controller are compared to the

behavior obtained with the hysteresis voltage control for the

bidirectional flyback converter suggested in Hoffstadt and Maas

(2016).

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 12 | Disturbance reaction of the sensor-based and self-sensing energy control with position feed-forward control and an explicit sliding mode position control.

For the hysteresis voltage control a threshold of 1vDE =30 V

was chosen. If the control deviation |v∗

DE −vDE,m|exceeds this

threshold the control activates the flyback converter to charge or

discharge the DE transducer. Afterwards the converter is turned

into idle state again. The three-point controller suggested here

behaves more or less the same. The only difference is that with the

controller settings from Table 1 a threshold of vp≈16 V results.

This smaller threshold increases on the one hand the steady state

accuracy. However, the switching frequency is on the other hand

a bit higher compared to the simple hysteresis voltage control.

Concerning the sensor-based and self-sensing energy control a

comparable behavior as shown and explained in Figure 8 can be

observed here, too.

5.5. Energy Control With Position

Feed-Forward Control

If the setpoint energy U∗

c,diel of the control structure in Figure 4

is determined with Equation (20) the proposed energy control

can be used to adjust a certain strain ε∗

z, although the strain

is not part of the state vector xU. To compensate the influence

of a disturbance, the estimated load ˆσload is considered in

Equation (20). In contrast, an explicit position control based

on the model (6) was derived in Hoffstadt and Maas (2017).

Figure 11 shows the comparison of the explicit (Position-3PC)

and energy-based position control (Energy-3PC) for the no-load

case of the DE stack-transducer. Both approaches are realized

as sensor-based and self-sensing control with the adaptive three-

point controller from Equation (39).

The explicit and energy-based position control show

comparable dynamics and accuracy for both the sensor-

based and the self-sensing control. The different setpoints

are adjusted within a few milliseconds. By increasing the

strain setpoint ε∗

zthe energy U∗

c,diel also increases according

to Equation (20). Instead of the energy, the voltage vDE is

depicted in Figure 11 as it is measured directly and can be

interpreted more intuitively. With Equation (22) a relationship

between the voltage vp≈vDE is given. As the no-load

case is considered here, for a constant setpoint of the strain

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 13 | Comparison of the sensor-based and self-sensing sliding mode energy control with force feed-forward control. In both cases two-point controllers (2PC)

with I∗

m,max =8 A and I∗

m,max =4 A as well as the three-point controller (3PC) with hysteresis and adaption of the inner flyback converter control are considered.

ε∗

za constant setpoint for the energy U∗

c,diel or the voltage

results, respectively.

In addition, Figure 12 depicts the disturbance reaction of the

different position control approaches. For this purpose, a tensile

load force of F∗

load =0.5 N is applied by the linear drive of the

test rig in Figure 5, while the setpoint strain is constantly set to

ε∗

z=1%. Right after the load is applied, the strain deviates from

its setpoint due to the influence of the disturbance. However, the

load is estimated with the sensor-based as well as the self-sensing

EKF. According to Equation (20) the setpoint energy U∗

c,diel, or

voltage v∗

DE, respectively, is increased to compensate the influence

of the disturbance ˆσload. In Figure 12 this behavior can be seen

in the response of the corresponding voltage vDE in the third

subplot. This compensates the influence of the disturbance within

approx. 15 ms. In case of the energy-based position control a

slightly higher control deviation can be observed after the load

steps. This is mainly due to the fact, that the energy control

only reacts on control deviations of the energy Uc,diel, while the

explicit position control considers the control deviation of the

strain εzdirectly.

5.6. Energy Control With Force

Feed-Forward Control

Beside the two validated approaches, Equation (21) offers

the opportunity to realize a force feed-forward control under

consideration of the current elastic material tension σelast(ˆεz)

based on the proposed energy control as already depicted in

Figure 4. As for the previous two approaches, the controller

settings are the same as listed in Table 1. However, in Figure 13

also the two-point controller with Im,max =8 A and Im,max =4 A

is considered again. The deformation of the DE stack-transducer

is blocked in this case to investigate the control behavior

caused by setpoint steps without any disturbance. In general, a

comparable behavior to the pure energy control in Figure 8 can

Frontiers in Robotics and AI | www.frontiersin.org 16 December 2019 | Volume 6 | Article 133

Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

FIGURE 14 | Disturbance reaction of the sensor-based and self-sensing energy control with force feed-forward control.

be observed here, too. According to Equation (21) a constant

energy setpoint U∗

c,diel is obtained for a certain force F∗

act and

εz=0. In comparison to the two-point controllers the adaptive

three-point controller from Equation (39) ensures the highest

possible dynamics by the maximum current Im,max =8 A

during transient operation as well as good steady state accuracy

with significantly reduced switching frequency by reducing the

current to Im,max =4 A. The self-sensing control adjusts

the different setpoint forces with dynamics that are absolutely

comparable to the sensor-based control.

In Figure 14 also the disturbance reaction of the energy-based

force control is shown. In this case a variable strain εzadjusted

by the position-controlled linear drive acts as a disturbance. The

stepwise change of the strain to ε∗

z=1% causes a reduction of

the force Fact in the first moment. However, by increasing the

setpoint energy U∗

c,diel, or voltage v∗

DE, respectively, according

to Equation (21) the influence is compensated comparable

to the behavior observed for the disturbance reaction of the

energy-based position control in Figure 12. The sensor-based

control offers a marginal better control quality what is caused

by the slightly higher dynamics of the sensor-based state and

disturbance estimation.

The investigation of the pure energy control in Figures 8,9as

well as of the different feed-forward controls in the Figures 10–

14 proved that both high dynamics and good steady state

accuracy are obtained with the proposed self-sensing control

approach. The developed self-sensing EKF estimates not only

the inner states of the DE transducer but also an external load

tension. In case of the position feed-forward control this allows

to compensate the influence of a disturbance load. In contrast,

if the force feed-forward control is applied the elastic material

tension caused by a deformation of the DE transducer is reliably

compensated. Furthermore, extensions to an adaptive three-

point controller enabled a reduction of the switching frequency

of up to 80% to increase the energy efficiency without reducing

the bandwidth of about 400 Hz and the steady state accuracy.

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Hoffstadt and Maas Self-Sensing Control for Dielectric Elastomer Actuators

6. CONCLUSION

DE transducer combine high energy densities and multi-

functional operation modes. Multilayer topologies like the DE

stack-actuator considered here have also high force densities

with considerable absolute deformations so that they are well-

suited to be used as active skins or as end effector in soft-

robotic applications. But, beside the transducer design also

appropriate control and sensing algorithms are required to enable

the combined actuator-sensor-operation in closed loop operation

without external sensors to measure mechanical states. The

design of such a self-sensing state and disturbance estimator as

a universal energy control that uses the information from a novel

self-sensing estimator were addressed within this contribution.

For this purpose, in section 2 the control plant comprising a

DE stack-transducer fed by a bidirectional flyback converter and

its model to describe the electormechanically coupled behavior

was summarized. To characterize the electrical behavior the

model includes the energy Uc,diel as one state variable. Based

on this model subsequently a self-sensing state and disturbance

estimator was developed that estimates the mechanical state

of the transducer as well as an external load force by just

measuring the terminal voltage and current. Due to the non-

linear system behavior an EKF was used for this purpose. It

allows to estimate the transducer state without any superimposed

voltage excitation as used for other self-sensing approaches. The

validation results have shown that almost no confinements in

terms of dynamics and accuracy compared to the sensor-based

estimator are obtained. The sensor-based estimator requires a

measurement of the terminal voltage and the displacement.

The developed energy control uses the information provided

by the self-sensing EKF for closed loop operation. Due to the

behavior of the bidirectional flyback converter, that either charges

or discharges a DE transducer with almost constant power when

enabled, the sliding mode control approach was applied. By

controlling the energy in the capacitance of the DE transducer

it is possible to control the voltage, force or displacement of

the transducer by using different feed-forward control structures.

The setpoint energy required to achieve a certain actuator force

or displacement was obtained under consideration of the static

force equilibrium included in the derived model. Within the

validation it was shown that a precise control of the voltage, force

and displacement with high dynamics and a bandwidth of up

to 400 Hz is achieved with this approach. The step response as

well as the disturbance reaction yield comparable dynamics and

accuracy for both the sensor-based and self-sensing control.

Although here a DE stack-transducer was considered, the

developed self-sensing EKF and control approach can also

be applied to other topologies well-suited for soft robotic

applications like DE-based minimum energy structures or

membrane actuators. The utilized bidirectional flyback converter

represents an efficient and competitive converter topology and

can also be used to supply any kind of DE transducer. In

case of soft-bodied robots equipped with DE transducers and

the mentioned converter the suggested self-sensing control

approach can be used to control the impedance of the robot

by applying the proposed force and displacement feed-forward

controls in combination with a human-machine-interface model.

If under consideration of the utilized test setup a charge

of at least qp=5µAs is applied, the proposed self-

sensing filter can also detect collisions or interactions. This

could be used e.g., in human machine interfaces or active

skins, so that the control can react on these events. While

for these applications the force and displacement control

are most important, the voltage control could be used to

avoid exceeding limitations that would cause a damage of

the transducer.

DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this

manuscript will be made available upon request to JM

([email protected]).

AUTHOR CONTRIBUTIONS

The research results included in this contribution are the

outcome of TH’s Ph.d. thesis that was supervised by JM.

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Conflict of Interest: The authors declare that the research was conducted in the

absence of any commercial or financial relationships that could be construed as a

potential conflict of interest.

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