王亞午葉雯珺張一龍賴旭芝吳 敏蘇春翌

(1.中國地質(zhì)大學(武漢)自動化學院,湖北武漢 430074;2.復雜系統(tǒng)先進控制與智能自動化湖北省重點實驗室,湖北武漢 430074;3.康考迪亞大學工程與計算機科學學院,蒙特利爾魁北克H3G 1M8;4.東北電力大學自動化工程學院,吉林吉林 132012)

1 Introduction

Due to the capability of large deformation and shape changes,the dielectric elastomer actuator(DEA)shows promising applications in the field of soft robotics[1].The DEA consists of a soft membrane sandwiched between two compliant electrodes on two surfaces[2].When a voltage is applied to the electrodes,the dielectric elastomer membrane reduces in thickness and expands in area[3].Because the DEA has the advantages of high-strain response,high energy density and fast response time[4],the soft robots driven by the DEA are widely used in various applications,such as artificial muscles[5],bionic robots[6],microelectromechanical systems(MEMS)[7]and so on.

The mathematical modeling of the DEA is the basis for understanding its characteristics,and is also the premise to design the model based controller.In general,the mathematical modeling methods of the DEA are divided into two categories:the physical-based modeling approach and the phenomenological modeling approach.The physical-based modelling approach is mainly based on the physical principles.The mathematical model is obtained by analyzing the energy conversion mechanism during the deformation process of the DEA.This modeling technique has the advantages of explicitness and preciseness.Based on the continuum mechanics theory and the thermodynamic theory,[8]explains the physical-based modeling approach of the DEA in detail.On the other hand,unlike the physical-based approach,the phenomenological modeling approach is mainly based on the experimental phenomenons.By analyzing the experimental data,one can use the combination of physical components (such as,resistor,capacitor,spring and dashpot)to represent the model of the DEA.This technique has the advantages of simplicity and efficiency.[9]employs capacitors and resistors to describe the electrical model of the DEA,and employs springs and dashpots to describe the mechanical model of the DEA.These two models can be connected through the Maxwell force,together to provide a mathematical model of the DEA.

Currently,the complete understanding of the nonlinear dynamic behaviors of the DEA with a general model is still an open challenge.Researches on the DEA mainly focus on materials and physics,while few efforts are made from the control point of view.However,[10]proposes a feedforward control strategy for the DEA.[11]designs the PID controller for the DEA.In general,the model of the DEA usually contains unmeasurable parameters(or parameter perturbations)and external disturbances,so it is meaningful to invistigate the adaptive robust control of the DEA.Meantime,in actual control,some states of the DEA are difficult to obtain,so an effective state observer is required for the actual control of the DEA.In this respect,it is an urgent demand to develop an implementable controller for the DEA,which can take the dynamic of the DEA into the consideration,tolerate the parameter uncertainties,and also work with limited measurable states.

In this paper,we propose an adaptive robust control strategy for the DEA to realize its trajectory tracking control objective.According to the principle of virtual work,the elastic energy of the DEA is described by the Gent model,and then the dynamic model of the DEA is developed.Since the model parameters of the DEA are difficult to obtain,we use two approximators based on the radial basis function neural networks(RBFNNs)to estimate the unknown items of the model.Meantime,since the rate of the stretch of the DEA is difficult to measure,we design a state observer to estimate the system states only according to the measured value of the stretch.According to the approximation results and the observed states,we design the sliding mode controller(SMC)to realize the trajectory tracking control of the DEA.Finally,the simulation results demonstrate the effectiveness of the proposed control strategy.

2 Dynamic model

In this section,the dynamic model of the DEA is developed based on the principle of virtual work.

Figure 1 shows the actuation mechanism of the DEA[12],where Fig.1(a)shows the un-deformed state of the DEA and Fig.1(b)shows the deformed state of the DEA.In the Fig.1,L1,L2andL3are dimensions corresponding to the un-deformed state;l1,l2andl3are dimensions corresponding to the deformed state;P1andP2are tensile forces;Φis the voltage;is the charge.

Fig.1 Model of DEA

We defineλ1=l1/L1,λ2=l2/L2andλ3=l3/L3.Since the DEA is incompressible,

The relationship between the chargeand the voltageΦis[8]

whereTis the environment temperature;ε(λ1,λ2,T)is the permittivity of the DEA,which is a nonlinear function ofλ1,λ2andT.

When there are minor changes of dimensions of the DEA(δλ1andδλ2),the works done by the tensile forces areP1L1δλ1andP2L2δλ2,and the work done by the voltage isΦδ.Thus,the variation of the charge is

The inertia forces in each material element along thex-direction andy-direction areand,respectively[13].ρis the density of the DEA.The damping forces in each material element along thex-direction andy-direction arecxandcyrespectively.cis the damping coefficient of the DEA[14].Thus,the works done by the inertial force and the damping force are

According to the thermodynamics theory and the principle of virtual work,the variation of the free energy (W)of the DEA is equal to the works done by the voltage,the tensile forces,the inertia forces,and the damping forces.So,

Moreover,Wis consisted of the elastic energy(Wela)and the electric energy(Wele)[13],that is

whereμ(T)is the shear modulus of the DEA,which depends on the environment temperatureT;Jmis the deformation limit.

Remark 1According to the superelastic material theory,the Gent model is chosen to describe the elastic energy in this paper[15].However,one can choose the other model,such as,Neo-Hookean model,Mooney-Rivlin model,Ogden model,Arruda-Boyce model,and so on.

Submitting(3)into(5),we obtain

By solving (7),the partial differentials of the free energy are given by

Submitting (6)into (8),the dynamic model of the DE is derived as

We assume thatL1=L2=LandP1=P2=P.Meantime,we also assume that the DEA is isotropic.So,λ1=λ2=λ,and(9)is reduced to

where

anddtis the external disturbance,|dt|≤dd.

3 Approximators and observer design

When(11)is obtained,we can design the controller that is based on the model parameters and all system states to realize the control of the DEA.However,in actual control,the model parameters of the DEA(i.e.ρ,μ(T),c,ε(λ,T),andJm)are unknown.Althoughλcan be measured by using the laser displacement sensor,is not measurable.As a result,f(x),g(x)andare unknown.

In order to meet the demand of the actual control,the approximators should be constructed to estimatef(x)andg(x),and the state observer should be designed to acquire.Considering that the RBFNN can approximate any nonlinear function with any precision[16],and it has the advantages of fast convergence speed and avoiding the local minimum problem,we prefer to choose the RBFNN to construct the approximator.Since bothf(x)andg(x)are unknown,a simple way is to design two RBFNN-based approximators to approximatef(x)andg(x),respectively.Moreover,when the number of the hidden layer nodes is sufficient,the approximation error of the three-layer RBFNN (inputlayer,hidden-layer and output layer)for any nonlinear function is small enough[17].So,we use the RBFNN with such simple topologies for convenience.

It is noted that there exist other effective parameter estimation methods(such as[18-20]and so on),which can be employed to construct the approximator.However,the purpose of this paper is to present a feasible tracking control strategy for the DEA utilized in soft robots.So,next we will only show the construct procedure of the RBFNN-based approximator in detail.Interested readers may solve such open problem by employing other parameter estimation methods.

(11)can be rewritten as

whereA=[0 1;0 0],b=[0 1]TandC=[1 0]T;yis the output of the system.

The state observer is designed to be

Moreover,it is known that

whereW1andW2are the ideal weight vectors of two RBFNNs,respectively;σ1andσ2are the ideal output vectors of Gauss functions of two RBFNNs,respectively;are approximation errors of two RBFNNs,respectively;W1,W2,are bounded,that is

Using(15)minus(16)yields

According to(22),

whereH(s)=CT(sI?A)?1b;sdenotes the differential operatorL?1(s)is a transfer function with stable poles andL(s)is chosen to ensure thatH(s)L(s)is strictly positive realness(SPR);

δ1andδ2are bounded,that is

and‖Θ1‖F(xiàn)is Frobenius norm ofΘ1.

LettingH(s)L(s)=then(24)can be rewritten as

SinceH(s)L(s)is SPR,there existsP=PT>0,which makes

whereQ=QT>0.

The Lyapunov function is constructed as

The derivative ofVis

To ensure the convergences of the state observer and the approximators,by employing the trial and error method,we design the update laws to be

whereκ1>0,κ2>0.

Submitting(31)into(30)yields

The robustifying item in(16)is designed to be

whereD1≥β1σM,D2≥β2σMud,σM=σmax[L?1(s)],σmax[Θ3]is the maximum singular value ofΘ3;|u|≤ud;sgn(·)is the symbol function.

According to(23)and(33),we obtain

Using the property of the trace of the matrix,one has

Moreover,the following inequation always holds.

whereλmin(Q)is smallest eigenvalues ofQ.

Submitting(34)?(36)into(32)yields

Submitting(38)and(39)into(37)yields

whereα1=W1M+c1/κ1andα2=W2M+c2/κ2.

Moreover,we know that

Submitting(41)into(40)leads to

Whenλmin(Q)>0,the requirement for＜0 is

Lemma 1[21]Givenx ∈Rnand a nonlinear functionh(x,t):Rn →R×Rn,the differential equation

has a differential solutionx(t)ifh(x,t)is continuous inx(t)andt.The solutionx(t)is said to be uniformly ultimately bounded(UUB)if there exists a compact setU ∈Rnsuch that,for allx(t0)=x0∈U,there exist aδ >0 and a numberT(δ,x0)such thatx(t)<δfor allt≥t0+T.

Further,the solution of(48)is

where

the state transition matrixΦ(t,τ)is bounded bym0e?a(t?τ),wherem0=eA(t?τ)anda=KCTare positive constants.

Similarly,

Lemma 2[22]Consider the linear time-invariant system in state-space representation

withx(t)∈Rn,u(t)∈Rm,B ∈Rn×m.Then,every solutionx(t)of(56)satisfies

wherek1decays exponentially to zero,k2is a positive constant that depends on the eigenvalues ofA.

4 Controller design

In this section,we design the SMC based on the above approximators and the state observer to realize the trajectory tracking control of the DEA.In fact,there are other control methods that can be regarded as the candidates to achieve this control objective (such as,PID control method,fuzzy control method,backstepping control method and so on).However,the sliding mode control method itself has strong robust performance,and can effectively overcome the influence of the parameter uncertainty and the external disturbance on the system control[23].Since the external disturbance is considered in the model of the DEA,we may as well design the SMC directly.

The sliding mode surface is chosen to be

The Lyapunov function is constructed as

According to(16),the derivative ofV1is

The controller is designed to be

Submitting(62)into(61)yields

Fig.2 Structural diagram of closed-loop control system

5 Simulations

In order to verify the effectiveness of the proposed control strategy,a simulation is carried out by using MATLAB tool.The model parameters of the DEA are shown in Table 1[24].

Table 1 Model parameters of DEA

According to experimental results in[25],

whereε0=8.85×10?12(F/m)is the permittivity of vacuum;

The initial states of the DEA are chosen to be

So,the tensile forces are

The target trajectory is

The external disturbance is

The parameters of two RBFNNs are:

the initial values of the weights are=0.1(i=1,2 andj=1,2,···,7).The parameters of the observer arek1=4×105,k2=8×107.So,H(s)=1/(s2+4s+2200).Since

may less than 0,H(s)is not SPR.We chooseL?1(s)=1/(s+3×105)to ensure thatH(s)L(s)is SPR.The parameters of the controller(62)are

In(31),

In(33),D1=D2=1.2.

The simulation results are shown in Figs.3-6.According to Figs.3(a)and 3(b),λandtrackλdandd,respectively.The tracking error curves are shown in Fig.4.Meanwhile,according to Figs.3(c)and 3(d),the maximum observation error is about 1.76%.So,the designed approximators,state observer and SMC are effective.Moreover,according to Figs.5(a)and 5(b),do not converge tof(x)andg(x),respectively.It is because the input signal does not satisfy persistent excitation condition[26].

Fig.3 Tracking results and observed results

Fig.4 Tracking error

Fig.5 Approximation results

Fig.6 Control input

6 Conclusions

This paper proposes an adaptive robust control strategy for the DEA.Based on the RBFNNs,two approximators are used to estimate the unknown items of the DEA’s model.Meantime,the state observer is designed to estimate the system states.Based on the approximation results and the observed states,the SMC is designed to realize the tracking control of the DEA.

The proposed control strategy only requires the stretch of the DEA,but does not require the the model parameters and the rate of the stretch.Thus,the proposed control strategy has adaptivity and robustness.

Moreover,the dynamic model(9)does not consider the creep and the hysteresis characteristics of the DEA.In the future,we will develop a more complex and more realistic mathematical model of the DEA to take the creep and the hysteresis into consideration.Furthermore,the control strategy proposed in this paper should be appropriately extended to meet the new model.