actuators Article
Simple Controller Design Based on Internal Model Control for Twisted and Coiled Polymer Actuator Motoya Suzuki
ID
and Norihiro Kamamichi *
Department of Robotics and Mechatronics, Tokyo Denki University, Tokyo 120-8551, Japan;
[email protected] * Correspondence:
[email protected]; Tel.: +81-3-5284-5602 Received: 25 May 2018; Accepted: 21 June 2018; Published: 25 June 2018
Abstract: A twisted and coiled polymer actuator (TCPA) is a novel soft actuator. TCPA is fabricated by twisting nylon fibers. The TCPA extends and contracts by heating and cooling. By applying conductive nylon fibers to the actuator, the electroactive TCPA can be driven by the Joule heating of the applied voltage. It has noteworthy properties such as a high power/mass ratio, large deformation, and low hysteresis. By applying conductive nylon fibers to the actuator, it can be driven by the electrical input. From these properties, many soft robots using the electroactive TCPA have been demonstrated, such as robotic hands, locomotion robots, robot skins, biomimetic robots, and so on. In this paper, to realize a simple controller design, an internal model control based on the identified model is applied. The applied controller can be designed easily without experience in parameter-tuning based on controls theory. The validity of the applied method is investigated through experiments. Keywords: soft actuator; electroactive polymer (EAP); control system design; twisted and coiled polymer actuator (TCPA); modeling; internal model control
1. Introduction To realize the safe and smooth motion of robotics and nursing care application, a light and soft actuator is desired [1,2], which is often called a soft actuator. In particular, an electroactive polymer actuator made from polymeric materials and conductive materials is promising because of high power density and the possibility of miniaturization. Various types of the electroactive polymer actuators have been reported such as an ionic polymer metal composites actuator [3,4], a dielectric elastomer actuator [5], a conductive polymer actuator [6], a bucky-gel actuator [7,8], and so on. In 2014, Haines et al. reported a soft actuator made from nylon fibers [9]. This actuator can be easily fabricated by twisting and coiling nylon fibers. From the fabrication method and structure of the actuator, this actuator is called as the twisted and coiled polymer actuator (TCPA) [10], the super-coiled polymer (SCP) actuator [11], or the twisted and coiled actuator (TCA) [12]. The actuation mechanism is based on a thermomechanical property that twisted and coiled nylon fibers contract and extend by heating and cooling. To make the actuation and control system simple, an electroactive TCPA is also reported by coating the TCPA with conductive materials or twining heating wires with the normal TCPA; thus, electric actuation by Joule heating can be realized easily [13]. Driving the actuator electrically, a driving system can be implemented simply. In addition, the actuator can be easily controlled by simple voltage or a current control method, similar to the control methods of DC motors. TCPA has various notable properties. It is easily fabricated by commercially available nylon fibers such as nylon threads or fishing lines. Therefore, it is relatively low cost to fabricate the actuator. Compared with shape memory actuators, the TCPA has a low hysteresis property [9]. In addition, the TCPA realizes larger deformation over 20% and a high power/mass ratio. From these Actuators 2018, 7, 33; doi:10.3390/act7030033
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properties, many applications using the electroactive TCPA have been demonstrated, such as robotic human hands [11,12,14–16], power assist devices [17], wearable devices [18], deformation locomotion robots [19], flying robot [20], robot skin for human robots [21], and biomimetic robots [22]. Many types of control system design methods for the electroactive TCPA had been proposed. In many previous papers on the control system design, proportional-integral-differential (PID) control is applied for control of the displacement or output force. By tuning PID gains, the desired control performance is realized. Yip and Niemeyer proposed a linear time invariant (LTI) model of the actuator based on the physical phenomena and designed a proportional-differential (PD) controller [23]. Arakawa et al. modeled the thermomechanical and thermoelectric property as a linear time invariant black box model, and designed a PID controller based on the identified model [24,25]. Sutton et al. designed a simple force control system based on the LTI first order model [17]. Luong et al. applied an adaptive PID sliding mode control to achieve high tracking and robust performance [26]. We identified the model of the actuator by a grey-box modeling, and applied the PID control to the antagonistic-type TCPA [27], and applied the PI control with an anti-windup compensator to the single TCPA [28]. These research studies indicate that the PID control can realize high tracking and robust performance if the controller gains are tuned adequately. However, tuning of the appropriate PID gains is difficult. Although various auto-tuning methods were reported, such as the Ziegler–Nichols method, the Chien-Hrones-Reswick (CHR) method, and optimization based on an evaluation function [29], the PID gains to realize the desired response need to be determined by the trials and errors of preliminary experiments. From these problems, a more simple control method for stabilizing the control system of the TCPA and realizing the desired response is required. In this paper, in order to solve these problems, a control system design method known as an internal model control is applied. This controller structure is simple, and can be designed easily without experience in parameter tuning based on controls theory. The internal model control consists of the feedforward controller and feedback controller based on the model. The control system can realize a high tracking performance, and the desired response of the TCPA is obtained due to a feedforward control signal and a feedback of estimated disturbances. In this paper, the control system performance is verified through experiments. The rest of the paper is organized as follows: a control system design method of the TCPA is shown in Section 2. Experiment verification of the proposed control system design is shown in Section 3. The conclusion of this paper is shown in Section 4. 2. Method In this section, we explain control system design methods for the TCPA. To design the controller based on the model, a gray-box modeling is applied. In addition, the internal model control method is introduced. 2.1. Modeling For control system design, a physical model of the actuator is derived, although the detailed physical model [30–33] and the detailed model based on experiment data [34,35] have been reported. However, these detailed models are unsuitable for control system design due to the complexity of the model structure and a number of parameters. In the case of using these detailed models, the LTI model for the control system design is computed by linearization methods. To derive the simple model for control system design, a grey-box modeling based on the LTI physical model is applied with reference to methods of Yip and Niemeyer [23]. In this paper, the electroactive TCPA is activated by Joule heating and controlled by the applied voltage. The response of the TCPA has a relationship to temperature variation. In the case of Joule heating, the temperature has a relationship to the applied electric power. Therefore, the model of TCPA is assumed to consist of nonlinear statics of Joule’s law and LTI dynamics from applied power to displacement. Here, the electric power is assumed to be proportional to the square of the applied voltage in the unchanged condition of electric resistance, which is the same as the reference of
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al. [23] and Arakawa et al. [24]. Here, as with reference of Yip and Niemeyer [23], the model of the TCPA is assumed to be expressed as a series connection of the thermomechanical model and the Yip et al. [23] and Arakawa et al. [24]. Here, as with reference of Yip and Niemeyer [23], the model of thermoelectric model: the TCPA is assumed to be expressed as a series connection of the thermomechanical model and the thermoelectric model: d 2 y (t ) dy (t ) (1) md2 y(t)2 + ddy(t) + ky (t ) = cΔ T (t ) , dt + ky(t) = c∆T (t), dt + d m (1) 2 dt dt 2 d∆T (t) ) v2 ((tt)) −−αS ((tt))+ CvC dΔT (t= c ∆T S T = α Δ + v dt c dt RR
(2) (2)
where y(t) is a displacement, ∆T(t) is a temperature variation from an ambient temperature, v(t) is where y(t) is a displacement, ΔT(t) is a temperature variation from an ambientcoefficient, temperature, is an an input voltage, m, d, and k are mechanical parameters, i.e., mass, damping andv(t) spring input voltage, d, and k are mechanical i.e., mass, and of spring constant, Cv is m, a heat capacity, α is a heat parameters, transfer coefficient, Sc damping is an areacoefficient, of the surface the constant, C v is a heat capacity, α is a heat transfer coefficient, Sc is an area of the surface of the TCPA, TCPA, c is a proportional coefficient from a temperature variation to a generated force, and R is an c is a proportional electric resistance. coefficient from a temperature variation to a generated force, and R is an electric resistance. By combining Equations (1) and (2), the dynamical model of the TCPA from the input voltage v(t) combining y(t) Equations and (2),asthe dynamical ordinary model of differential the TCPA from the input voltage to theBy displacement can be (1) expressed a third-order equation: v(t) to the displacement y(t) can be expressed as a third-order ordinary differential equation: d3 y ( t )
d2 y ( t )
dy(t)
d 3 y (t3) + a2 d 2 y2 (t )+ a1 dy (t+) a0 y(t) = b0 v2 (t)2 dt + a 2 dt 2 + a1 dt + a 0 y (t ) = b 0 v (t ) dt dt 3 dt
(3) (3)
where a2 = (dCv + mαSc )/mCv , a1 = (kCv + dαSc )/mCv a0 = kαSc /mCv, and b0 = c/mRCv . The model where a2 = (dC v + mαSc)/mCv, a1 = (kCv + dαSc)/mCv a0 = kαSc/mCv, and b0 = c/mRCv. The model parameters parameters of Equation (3) are identified from the input–output data of the experiment results. of Equation (3) identified from the input–output data of experiment results. For controlare system design, Equation (3) is described as the a system in Figure 1. The model of the For control system design, Equation (3) is described as a system in Figure 1. The model of the TCPA consists of two sub-systems: nonlinear statics N and LTI dynamics GD , which are connected in TCPA N consists of twoas sub-systems: nonlinear statics N and LTI dynamics GD, which are connected in series. is expressed a square function: series. N is expressed as a square function: N (·) = v2 2.
N (⋅) = v .
(4) (4)
2 The (s) from from the thesquared squaredinput inputvoltage voltagevv2(t) (t) to to the the The transfer transferfunction functionof ofthe thelinear lineardynamics dynamicsGGDD(s) displacement displacementy(t) y(t)isisexpressed expressedas asaathird-order third-ordertransfer transferfunction; function;
bb00 . s) = 3 GD (Gs)D (= . 2 3 2 s a s a s a + + + s + a22s + a11 s +0a0
(5) (5)
Incase caseof ofoverheating, overheating,these thesemodel modelparameters parametersmight mightbe bevaried varied in in response response to to the the input input voltage, voltage, In sincethe thematerials materials property of TCPA the TCPA depends the temperature. Furthermore, case of since property of the depends on the on temperature. Furthermore, in case ofin excessive excessive contraction, these parameters might be varied due to the coil–coil contact. These factors contraction, these parameters might be varied due to the coil–coil contact. These factors should be should be considered for the wide range activation [31,33]; however, our modeling method considered for the wide range activation [31,33]; however, our modeling method is effective for normalis effective normal use ofsince the feedback since the simple controller design and adequate use of the for feedback control, the simplecontrol, controller design and adequate control performance can control performance can be realized, as described later. be realized, as described later.
Figure Figure1.1. The The block blockdiagram diagramof ofthe themodel modelof ofthe theTCPA. TCPA.
2.2. 2.2.Controller ControllerDesign Design In Inthis thissection, section,aacontroller controllerdesign designmethod methodisisexplained. explained.The Theblock blockdiagram diagramof ofthe thecontrol controlsystem system −−1 1, isisshown shownin inFigure Figure2.2.The Thecontrol controlsystem systemconsists consistsofofan aninverse inversecompensator compensatorofofstatic staticnonlinearity nonlinearityNN , aafeedforward feedforwardcontroller controllerGGCC,, and and aa nominal nominal model modelGGDD. First, the inverse compensator N−1 for canceling the nonlinearity of Joule’s law is implemented by the square root function:
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First, the inverse compensator N −1 for canceling the nonlinearity of Joule’s law is implemented by the square root function: q v(t) =
u(t)
(6)
where u(t) is the control signal calculated by the controller, and v(t) is the input voltage. By inserting the series compensation of Equation (6) after the controller GC , the control system can be designed by linear control theory. To avoid breaking the actuator by overheating, the maximum input voltage is limited to within riskless driving voltage. The maximum input voltage is set so that the actuator isn’t overheated nor does it reach the displacement limitations of the coil–coil contact. In addition, the minimum input voltage also is limited, since the negative control signal is unrealized in normal voltage driving. Therefore, the saturated control signal u(t) is given as: umax u(t) = uc (t) u min
(umax < uc (t)) (umin ≤ uc (t) ≤ umax ) (uc (t) < umin )
(7)
where umax and umin are the limits of the square value of input voltage and uc is the control signal of the controller GC . Note that the saturation umax is determined by the preliminary experiment, and umin is set to zero. The internal model control based on the LTI dynamical model is designed to be the same with the reference of Morari and Zafirou [36]. The arbitrary reference model can be applied under the conditions of a minimum phase system. In this paper, for simplification of the controller design, the reference model F(s) is given as a conventional low-pass filter: F (s) =
1
(λs + 1)3
(8)
where λ is a time constant. Here, the time constant λ is adjusted so that the controller can realize the desired response from the reference signal to the output. Note that the reference model is expressed as the third-order transfer function so that the controller GD (s) becomes proper. Next, to cancel the linear dynamics GD (s), the controller GC (s) from the reference signal to the control signal is given as the inverse model of linear dynamics: −1 GC (s) = F (s) GD ( s ).
(9)
Note that GD (s) is assumed to have stable poles and zeros. If the modeling error and any disturbance do not exist, the response from the reference signal to the output becomes the desired response based on the reference F(s), since the controller GC (s) balances the poles and zeros of the linear dynamics. However, the output is not able to track under the conditions with the unconsidered factors such as nonlinear property, disturbances, and nominal error in actual environments. Therefore, a mechanism for considering these factors is needed. The feedback mechanism compensating the influence of the modeling error and some disturbance is designed. Here, the deviation eM between the actuator output and the estimated output of the model is expressed as: e M ( s ) = y ( s ) − GD ( s ) u ( s ). (10) To compensate eM , the control signal uc is given as: uC (s) = GC (s){r (s) − e M (s)}
(11)
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where r is a reference signal. By applying the above control method, the response from the reference signal to the output is equal to the desired response of the reference model F(s), since the eM is eliminated due to the feedback compensation. The2018, internal model Actuators 7, x FOR PEERcontrol REVIEWhas the following advantages: 5 of 13 1. 1. 2. 2. 3.3.
It is extremely simple to design the control system, since the control system can be implemented It is extremely simple to design the control system, since the control system can be implemented by only adjusting the reference model if the model of the TCPA can be derived. by only adjusting the reference model if the model of the TCPA can be derived. This control method can realize a good control performance that is close to the desired This control method can realize a good control performance that is close to the desired feedforward control where the modeling error and some disturbance don’t exist. feedforward control where the modeling error and some disturbance don’t exist. The The internal internal stability stability of of the the control control system system is is guaranteed, guaranteed, since since the the LTI LTI physical physical model model of ofthe the TCPA TCPAdoesn’t doesn’thave havethe theunstable unstablezeros zerosand andpoles. poles.
Figure 2. The block diagram of the control system for the twisted and coiled polymer actuator (TCPA). Figure 2. The block diagram of the control system for the twisted and coiled polymer actuator (TCPA).
3. Experiment 3. Experiment The validity of the control system is investigated through experiments. In this section, we The validity of the control system is investigated through experiments. In this section, we explain explain the experimental environment, the system identification results, and the experiment results. the experimental environment, the system identification results, and the experiment results. 3.1.Experimental ExperimentalEnvironment Environment 3.1. Theelectroactive electroactiveTCPA TCPAused usedininthe theexperiments experimentswas wasfabricated fabricatedby bythe thesilver-plated silver-platednylon nylonthread thread The (AGposs100/2, 100/2,Mitsufuji, Mitsufuji,Kyoto, Kyoto,Japan). Japan). The fabrication method is the same as previous our previous (AGposs The fabrication method is the same as our paperpaper [28]. [28]. The nylon thread in an uncoiled state is 300 mm in length and 0.1 mm in diameter. nylon The nylon thread in an uncoiled state is 300 mm in length and 0.1 mm in diameter. The nylonThe thread in thread in a coiled state is 85 mm in length and 0.4 mm in diameter after overtwisting. The a coiled state is 85 mm in length and 0.4 mm in diameter after overtwisting. The configuration of the configuration of the is 100 mm in in diameter. length andThe 0.3electric mm inresistance diameter. isThe electric resistance actuator is 100 mm inactuator length and 0.3 mm 74.2 Ω in an unloadedis 74.2 Ω in an unloaded condition. The behavior of electric activation of the electroactive TCPA condition. The behavior of electric activation of the electroactive TCPA is shown in Figure 3. The lengthis shown in Figure 3. The length ofathe actuator mm. When 20-gofweight is attached to the end of the actuator is 100 mm. When 20-g weightisis100 attached to theaend the actuator, the maximum of the actuator, the maximum strain of the actuator is about 10% by 10 V. strain of the actuator is about 10% by 10 V. schematicdiagram diagramof ofour ourexperimental experimentalenvironment environmentisisshown shownin inFigure Figure4. 4. In In the the environment, environment, AAschematic real-timemeasurement measurement and control system is implemented by using a digital signal processor aareal-time and control system is implemented by using a digital signal processor system system (MTT, Tokyo, Japan, SBOX2). A control program is built in MATLAB/Simulink and loaded to (MTT, Tokyo, Japan, SBOX2). A control program is built in MATLAB/Simulink and loaded to the the digital signal processor system. The control input signal is computed in a digital signal processor digital signal processor system. The control input signal is computed in a digital signal processor and and applied the power amplifier. The power amplifier outputs the driving voltage according to applied to thetopower amplifier. The power amplifier outputs the driving voltage according to the the control signal. The output displacement of the actuator is measured by a laser displacement meter control signal. The output displacement of the actuator is measured by a laser displacement meter (KEYENCE,Osaka, Osaka,Japan, Japan,IA-100), IA-100),and andthe the displacement displacement data data is is input input to to the the digital digital signal signal processor. processor. (KEYENCE, The displacement is defined as the shrinkage ratio [%] of the initial length of the actuator. The sample The displacement is defined as the shrinkage ratio [%] of the initial length of the actuator. The sample time of the controller is set as 1 ms. The actuator is placed in a chamber (As-one). In the experimental time of the controller is set as 1 ms. The actuator is placed in a chamber (As-one). In the experimental environment, the the temperature temperature in and of of 20 20 g isg attached to the ◦ C, environment, in the the chamber chamberisisset setasas2323°C, anda weight a weight is attached to actuator. the actuator.
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Figure 3. The TCPA used in the experiments. Figure Figure 3. 3. The The TCPA TCPAused usedin inthe theexperiments. experiments.
Figure 4. Experimental environment. Figure4. 4. Experimental Experimental environment. environment. Figure
3.2. Model Identification 3.2. Model Model Identification Identification 3.2. In this section, we explain the identification method of the dynamical model. The model In this this section, section, we we explain explain the the identification identification method methodofofthe thedynamical dynamicalmodel. model. The The model model In parameters were derived using the input–output data of the experiments. In the experiment, the parameters were derived using the input–output data of the experiments. In the experiment, the parameters were derived using the input–output data of the experiments. In the experiment, input voltage–output displacement of pulse response with multilevel amplitudes was measured. For input voltage–output displacement of pulse response withwith multilevel amplitudes waswas measured. For the input voltage–output displacement of pulse response multilevel amplitudes measured. parameter identification, a discrete-time state space model of the actuator is identified by using the parameter identification, a discrete-time state space model of the actuator is identified by using the For parameter identification, a discrete-time state space model of the actuator is identified by using N4SID function of MATLAB. The parameter of the transfer function GD(s) is derived by using the N4SID function of of MATLAB. The parameter using the the the N4SID function MATLAB. The parameterofofthe thetransfer transferfunction function G GDD(s) (s)isis derived derived by by using discrete-continuous conversion method with zero-order hold. discrete-continuousconversion conversionmethod methodwith withzero-order zero-orderhold. hold. discrete-continuous By using a system identification toolbox of MATLAB, the model parameters are identified as a2 By using using aasystem system identification identificationtoolbox toolboxof ofMATLAB, MATLAB,the themodel modelparameters parametersare areidentified identifiedas asaa22 By = 371.1, a1 = 1003, a0 = 44.09, and b0 = 42.09. The identification results are shown in Figure 5. In this 371.1, aa11 == 1003, 1003, aa00 = = 44.09, and bb0 == 42.09. are shown shown in in Figure Figure 5. 5. In In this this == 371.1, 44.09, and 42.09. The The identification identification results results are figure, the experimental data and0simulation result of the identified model are plotted. To evaluate figure, the experimental data and simulation result of the identified model are plotted. To evaluate figure, the experimental data and simulation result of the identified model are plotted. To evaluate the precision of the identified physical model, a fitting ratio [%], which is an index of precision of the the precision precisionof ofthe theidentified identifiedphysical physicalmodel, model,aafitting fittingratio ratio[%], [%],which whichisisan anindex indexof of precision precisionof ofthe the the identified model, is calculated as: identifiedmodel, model,isiscalculated calculatedas: as: identified
T (((yyy (((iii)))−−− yyy(((ii)i))))) 2 ∑ sim i =0 q (yy((ii))−− yy ))2 T ( ( y (i ) − y ) ∑
q
Fit = 1001 −
1− = 100 1− Fit Fit = 100
Tsim Tsim sim i =0 sim sim i =0 Tsim Tsim i=0 sim i =00 i=
2 2
2 a2 aa
(12) (12) (12)
where ysim(i) and y(i) are the output data of simulated model, the experimental output data is at time whereyysim(i) (i) and and y(i) y(i) are are the the output data data of of simulated simulated model, the the experimental output output data is is at at time time where i, Tsim issimthe simulation time,output and ya is the average of model, the output experimental data. The fitting ratiodata calculated by i, T sim is the simulation time, and ya is the average of the output data. The fitting ratio calculated by i,Equation Tsim is the(12) simulation ya is theresults average of the output data. The fitting ratio calculated by is 89.25%.time, The and simulation agree well with the experimental result. Equation(12) (12)isis89.25%. 89.25%. The The simulation simulationresults resultsagree agreewell wellwith withthe theexperimental experimentalresult. result. Equation
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(a)
(b) Figure Identification result: displacement; input voltage. Figure 5. 5. Identification result: (a)(a) displacement; (b)(b) input voltage.
3.3. Experimental Result 3.3. Experimental Result In this experiment, the validity of the internal model control is investigated, as the controller can In this experiment, the validity of the internal model control is investigated, as the controller can balance the actuator properties adequately, and realizes the desired response. balance the actuator properties adequately, and realizes the desired response. 3.3.1. Experimental Results with Sinusoidal Reference Signals 3.3.1. Experimental Results with Sinusoidal Reference Signals The reference signals sinusoidal waves with a frequency and 0.025 Hz, and The reference signals areare setset asas sinusoidal waves with a frequency of of 0.10.1 HzHz and 0.025 Hz, and aa peak value from 2 mm 6 mm. The reference signal the composite wave with the sinusoidal peak value from 2 mm to to 6 mm. The reference signal is is setset asas the composite wave with the sinusoidal waves of 0.1 Hz and 0.025 Hz. The time constant λ is set to 0.5 s, as the control signal is not saturated. waves of 0.1 Hz and 0.025 Hz. The time constant λ is set to 0.5 s, as the control signal is not saturated. To verify the validity of the proposed control system, the experiment result of the internal model To verify the validity of the proposed control system, the experiment result of the internal model control is compared to the conventional on the model. In the is control compared to the conventional feedforwardfeedforward control basedcontrol on the based LTI model. In LTI the conventional conventional feedforward control, the control signals u c is given as uc = Gc(s)r(s) [24,27]. feedforward control, the control signals uc is given as uc = Gc(s)r(s) [24,27]. Experiment results feedforward control shown Figure this figure, solid line Experiment results of of thethe feedforward control areare shown in in Figure 6. 6. InIn this figure, thethe solid line is the displacement of the TCPA, and the dotted line is a desired value, that is, the reference signal is the displacement of the TCPA, and the dotted line is a desired value, that is, the reference signal after after passing the reference model F(s). From6a,b, Figure 6a,b, although the displacement varied passing throughthrough the reference model F(s). From Figure although the displacement is variedisalong along the reference signal, large tracking error between the displacement and the value desired value the reference signal, the largethe tracking error between the displacement and the desired arises. arises. From Figure 6c, the large tracking error that arises is the same as with the result of the From Figure 6c, the large tracking error that arises is the same as with the result of the sinusoidal wave sinusoidal wave of 0.1 Hz and 0.025 Hz. This tracking error is considered to arise due to the of 0.1 Hz and 0.025 Hz. This tracking error is considered to arise due to the unconsidered factors such factors such as nonlinear thermalproperty conductivity [31], mechanical property [35], noise, asunconsidered nonlinear thermal conductivity [31], mechanical [35], noise, the parameter variation due to the parameter variation due to the temperature dependence and the coil–coil contact, and so on. the temperature dependence and the coil–coil contact, and so on. Furthermore, these parameters might parameters might beon. varied due towith the the coil–coil contact, on. Therefore, beFurthermore, varied due to these the coil–coil contact, and so Therefore, reference of Yipand andso Niemeyer [23], with the reference of Yip and Niemeyer [23], Arakawa et al. [24], and Suzuki and Kamamichi [27], Arakawa et al. [24], and Suzuki and Kamamichi [27], the feedforward control based on the LTI model the feedforward control based on the LTI model cannot realize the desired response. cannot realize the desired response. The experimental results of the internal model control are shown in Figure 7. In each of the control results, the large error decreased, and the displacement of the actuator can be tracked to the desired value compared to the feedforward control. As observed in the results, the desired control performance can be realized due to the feedback of deviation between the actual output and the estimated output of the nominal model. In the experiment, the displacement control can be realized in the same response performance for the other amplitude and frequency of the target signal if the input voltages are within the upper limitation of the input saturation. Therefore, as shown in the
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results, the internal model control can eliminate the nominal error and disturbance, and also realize the desired control performance. Actuators 2018, 7, x FOR PEER REVIEW 8 of 13
(a)
(b)
(a)
(b)
(c) Figure 6. Experiment result of the feedforward control with the sinusoidal wave (the solid line: displacement, the dotted line is the desired value): (a) the sinusoidal wave of 0.1 Hz; (b) the sinusoidal wave of 0.025 Hz; (c) the composite wave with the sinusoidal wave of 0.1 Hz and 0.025 Hz.
The experimental results of the internal model control are shown in Figure 7. In each of the control results, the large error decreased, and the displacement of the actuator can be tracked to the desired value compared to the feedforward control. As observed in the results, the desired control performance can be realized due to the feedback of deviation between the actual output and the estimated output of the nominal model. In the experiment, the displacement control can be realized (c) in the same response performance for the other amplitude and frequency of the target signal if the Figure6.6.Experiment Experiment result resultofofthe the feedforwardcontrol controlwith withthe thesinusoidal sinusoidalwave wave(the (thesolid solidline: line: Figure input voltages are within the upper feedforward limitation of the input saturation. Therefore, as shown in the displacement, the dotted line is the desired value): (a) the sinusoidal wave of 0.1 Hz; (b) the sinusoidal displacement, the dotted is the can desired value): (a) sinusoidal wave of disturbance, 0.1 Hz; (b) the and sinusoidal results, the internal modelline control eliminate thethe nominal error and also realize waveofof0.025 0.025Hz; Hz;(c) (c)the thecomposite compositewave wavewith withthe thesinusoidal sinusoidal wave wave of of 0.1 0.1 Hz Hz and and 0.025 0.025 Hz. Hz. the wave desired control performance. The experimental results of the internal model control are shown in Figure 7. In each of the control results, the large error decreased, and the displacement of the actuator can be tracked to the desired value compared to the feedforward control. As observed in the results, the desired control performance can be realized due to the feedback of deviation between the actual output and the estimated output of the nominal model. In the experiment, the displacement control can be realized in the same response performance for the other amplitude and frequency of the target signal if the input voltages are within the upper limitation of the input saturation. Therefore, as shown in the results, the internal model control can eliminate the nominal error and disturbance, and also realize the desired control performance.
(a)
(b) Figure 7. Cont.
(a)
(b)
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(c) Figure7.7.Experiment Experimentresult resultofofthe theproposed proposedcontrol control methodwith withthe thesinusoidal sinusoidalwave wave(the (thesolid solidline: line: (c)method Figure the sinusoidal wave of 0.1 Hz; (b) the displacement, the dotted line is the desired value): (a) displacement, dotted line is the desired value): (a) method the sinusoidal of 0.1wave Hz; (b) sinusoidal Figure 7.the Experiment result of the proposed control with thewave sinusoidal (thethe solid line: sinusoidal wave of 0.025 Hz; (c) the composite wave of 0.1 Hz and 0.025 Hz. wave of 0.025 Hz; (c)the thedotted composite of 0.1 Hz and 0.025 Hz.sinusoidal wave of 0.1 Hz; (b) the displacement, line iswave the desired value): (a) the sinusoidal wave of 0.025 Hz; (c) the composite wave of 0.1 Hz and 0.025 Hz.
3.3.2. Experimental Results with Step Reference Signals 3.3.2. Experimental Results with Step Reference Signals 3.3.2. Experimental Resultsthe withcontrol Step Reference Signals of the internal model control, the reference To further investigate performance To further investigate the control performance of the internal model control, the reference signals signals are set as investigate the step wave with performance multilevel of amplitude. The experimental of the To further the control the internal model control, the results reference are set as the step wave with multilevel amplitude. The experimental results of the conventional signals are set as the control step wave with multilevel The experimental results thewave, conventional feedforward are shown in Figureamplitude. 8. In each experimental result of theof step feedforward control are shown in Figure 8. In each experimental result of the step wave, the large conventional feedforward control are shown in Figure 8.and In each result of the stepsteady wave, state the large steady state error between the displacement the experimental desired value arises. This steadythe state error between the between displacement and the desired value arises. This steady statestate error is large steady state error the displacement and the desired value arises. This steady error is considered due to the modeling error and disturbances. This tracking error is considered to considered due to the modeling error and disturbances. This tracking error is considered to arise error is considered due to the modeling error and disturbances. This tracking error is considered todue arise due to the unconsidered factors such as nominal errors and disturbance. Therefore, the to the unconsidered factors such as nominal disturbance. Therefore, the feedforward arise due to the unconsidered factors errors such asand nominal errors and disturbance. Therefore, control the feedforward control based on the LTI model cannot realize the desired response. based feedforward on the LTI model realize the desired response. controlcannot based on the LTI model cannot realize the desired response.
(a)
(a)
(b)
(b) feedforward control of step response: (a) Figure 8. Experiment result of the conventional
Figure 8. Experiment result of the conventional feedforward control of step response: (a) displacement displacement (the solid line) and the reference signal (the dotted line); (b) input voltage. (the solid line) and the reference (theconventional dotted line); (b) input voltage. Figure 8. Experiment result signal of the feedforward control of step response: (a)
displacement (the solid line) andofthethe reference (thecontrol dotted line); (b) input The experimental results internalsignal model are shown in voltage. Figure 9. In each experimental result, the overshoot doesn’tmodel arise, control and the are displacement be tracked to the desired The experimental results of the internal shown in can Figure 9. In each experimental The results of the internal control are shown Figure 9. In each Although quite smallarise, errors between the model displacement the desiredtoin value arise in the result,value. the experimental overshoot doesn’t and the displacement canand be tracked the desired value. experimental result, the overshoot doesn’t arise, and the displacement can be tracked to the desired transient response, the steady state errors are eliminated due to the feedback of the internal model. Although quite small errors between the displacement and the desired value arise in the transient response, The displacement is small able to errors be tracked desired value, sinceand the control signal value is adequately value. Although quite between the displacement the desired ariseisin the the steady state errors are eliminated due to to thethe feedback of the internal model. The displacement able adjusted due to the feedback of the output error. From these results, the nominal error and response, the steady state errors are eliminated to the feedback thetointernal model. totransient be tracked to the desired value, since the control signal isdue adequately adjustedof due the feedback disturbances can eliminated due to the internal model feedback. also shownsignal in these The displacement is be able to be tracked to the desired value, sinceAsthe control isresults, adequately
of the output error. From these results, the nominal error and disturbances can be eliminated due to adjusted due to the feedback of the output error. From these results, the nominal error and disturbances can be eliminated due to the internal model feedback. As also shown in these results,
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the internal model control can balance the actuator properties adequately and realize the desired the internal model feedback. also shownthe in these results, the internal modeland control canthe balance the the internal model controlAs can balance actuator properties adequately realize desired control performance. actuator properties adequately and realize the desired control performance. control performance.
(a)
(b) (b) Figure 9. Experimental result of the internal model control in step response: (a) displacement (the Figure result of of thethe internal model control in step response: (a) displacement (the solid Figure9.9.Experimental Experimental result internal model control in step response: (a) displacement (the solid line) and the reference signal (the dotted line); (b) input voltage. line) and theand reference signal (the dotted line); (b) input solid line) the reference signal (the dotted line); (b)voltage. input voltage.
3.3.3. Experiment of the Pole Assignment 3.3.3. 3.3.3.Experiment Experimentofofthe thePole PoleAssignment Assignment We verified that the proposed control method can realize the desired control performance. Here, verified that the proposed control method realize the desired control performance. We verified that of the proposed method cancan realize therise desired control performance. Here, asWe with the method the referencecontrol of Yip and Niemeyer [14], the time is recorded when the pole Here, as with the method of the reference of Yip and Niemeyer [14], the rise time is recorded when the as of with the method of the reference of Yip and Niemeyer [14], the rise time is recorded when the the reference model is varied. In the experiment, the time constant λ is changed from 0.25 s to 1pole s pole of reference the reference model is varied. In the experiment, the time constant λ is changed from 0.25 of per the model is varied. In the experiment, the time constant λ is changed from 0.25 s to 0.25 s. Note that the pole is a reciprocal of the time constant λ. The reference signal is the step1s s toper 1wave s0.25 per with 0.25 s. Note the pole is a reciprocal of are the determined, time constant λ. The signal the s. Note that that theofpole is a These reciprocal of the time constant λ.soThe reference signal is the step amplitude 3 mm. parameters that thereference control input isisnot step wave with amplitude 3 mm. These parameters aredetermined, determined,so so that that the control wave with amplitude of 3dependence mm. These are control input inputisisnot not saturated to evaluate theof ofparameters the time constant. saturated totoevaluate ofofthe time constant. saturated evaluatethe thedependence dependence the time constant. The experimental results are shown in Figure 10. In this figure, the solid line is the measured displacement, and the dottedare lineshown is the desired value. the displacement the The results ininFigure 10. In this figure, the solid the Theexperimental experimental results are shown Figure 10.In Ineach thisresult, figure, the solidline lineisisconverges themeasured measured reference signals. The response isdesired higher,value. as theIn time constant smaller. On the other hand, as displacement, and dotted line isisthe result, displacement converges the displacement, andthe the dotted linespeed thedesired value. Ineach each result,isthe the displacement converges the the time constant is larger, the response speed from the reference signal to the displacement is slower. reference signals. The response speed is higher, as the time constant is smaller. On the other hand, reference signals. The response the time constant is smaller. On the other hand, as Furthermore, the error between thespeed desired value thesignal displacement increases as the timeis asthe the time constant is larger, the response theand reference signal the displacement time constant istracking larger, the response speed fromfrom the reference to theto displacement is slower. constant is larger; that is, the tracking performance is higher as the pole is larger. This is considered slower. Furthermore, the tracking error between thevalue desired and the displacement increases Furthermore, the tracking error between the desired andvalue the displacement increases as the time to arise due to small gain theorem, which is a trade-off between the control performance and stability asconstant the timeisconstant is larger; is, theperformance tracking performance higher poleThis is larger. This is larger; that is, thethat tracking is higher asisthe poleas is the larger. is considered [36]. From Figure 10d, is observed that the time increases due the increase of the time constant. considered arise due toit theorem, small gainwhich theorem, which is between a trade-off between the control performance to arise duetoto small gain is arise trade-off theto control performance and stability As shown in the results, the response from the reference signal to displacement depends on the time [36].stability From Figure 10d, itFigure is observed that the rise time the increase constant. and [36]. From 10d, it is observed thatincreases the rise due timetoincreases dueoftothe thetime increase of constant of the reference model. Therefore, the desired control performance can be realized by Astime shown in the results, the response from the reference to displacement depends on the time the constant. As shown in the results, response signal from the reference signal to displacement adjusting the pole of the reference model. constantonofthe thetime reference model. the desired control performance be realizedcan by depends constant of the Therefore, reference model. Therefore, the desired controlcan performance adjusting the pole of the reference model. be realized by adjusting the pole of the reference model.
(a)
(a) Figure 10. Cont.
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(b)
(c)
(d) Figure Theresponse responseproperty propertydue due the variation the time constant displacement(solid (solid Figure 10.10.The toto the variation ofof the time constant λ:λ: (a)(a) displacement line) and the desired value (dotted line); (b) input voltage; (c) tracking error between the displacement line) and the desired value (dotted line); (b) input voltage; (c) tracking error between the displacement and the desired value; rise time time constant. and the desired value; (d)(d) rise time vsvs time constant.
4. Conclusions 4. Conclusions In this paper, we focused on the simple control method of the electroactive TCPA. The internal In this paper, we focused on the simple control method of the electroactive TCPA. The internal model control based on the model of the actuator was proposed. The validity of the method was model control based on the model of the actuator was proposed. The validity of the method was confirmed through the experiments. From various experiment results, the following was confirmed: confirmed through the experiments. From various experiment results, the following was confirmed: • Compared to the conventional feedforward control, the proposed method can cancel the • Compared to the conventional feedforward control, the proposed method can cancel the actuator actuator properties adequately and realize the desired control performance. properties adequately and realize the desired control performance. • The desired control performance can be realized only by adjusting the reference model. • The desired control performance canbebedesigned realized only adjusting the reference model.model Therefore, Therefore, the control system can easilybyonly by adjusting the reference if the the control be designed easily only by adjusting the reference model if the LTI model LTI modelsystem of the can actuator can be derived. of the actuator can be derived. In the future works, robust control methods for realizing the desired performance will be In the future works, robust control realizing performance willorbe considered, even if the modeling error is methods larger duefor to the aging the and desired the parameter variation, if a considered, even if the modeling error is larger due to the aging and the parameter variation, or if a large disturbance is confused. Furthermore, we will apply this control method to applications such large disturbance is confused. Furthermore, we will apply this control method to applications such as as the human robot hand with this actuator, and verify the validity of control methods. the human robot hand with this actuator, and verify the validity of control methods. Author Contributions: M.S. designed the control system, fabricated the actuator, constructed the experimental Author Contributions: M.S. designed the control system, fabricated thepaper. actuator, constructed the experimental environment, and performed the experiment. Both authors wrote this environment, and performed the experiment. Both authors wrote this paper. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.
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