Accurate Motion Control of a Pneumatic Linear Peristaltic Actuator

Article Accurate Motion Control of a Pneumatic Linear Peristaltic Actuator

João Falcão Carneiro 1,*, João Bravo Pinto 2 and Fernando Gomes de Almeida 1

1 LAETA-INEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal; [email protected] 2 LAETA-INEGI, Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal; [email protected] * Correspondence: [email protected]

Received: 26 June 2020; Accepted: 27 July 2020; Published: 30 July 2020

Abstract: Pneumatic linear peristaltic actuators can offer some potential advantages when compared with conventional ones. The low cost, virtually unlimited stroke and easy implementation of curved motion profiles are among those benefits. On the downside, these actuators suffer high mechanical stress that can lead to short service life and increased leakage among chambers during the actuator lifetime. One way to cope with this problem is to impose the force—instead of the displacement—between rollers, as this has been shown to improve the endurance of the hose while reducing leakage during the actuator lifetime. This paper presents closed control loop results using such a setup. Previous studies with linear peristaltic actuators have revealed that, although it is possible to reach zero steady state error to constant references with closed loop control, the dynamic response obtained is very slow. This paper is mainly focused on this topic, namely on the development of several control laws to improve the dynamic performance of the system while avoiding limit cycles. The new developed control law leads to an average time of 1.67 s to reach a 0.1 mm error band in an experiment consisting of a series of 16 steps ranging from 0.02 to 0.32 m in amplitude.

Keywords: linear peristaltic pneumatic actuators; accurate motion control; PID control

1. Introduction Pneumatic systems have always been attractive for industrial environments due to their low cost, high power to weight ratio, high velocity, no overheat, explosion or contamination risk and non-generation of magnetic fields. Despite this, pneumatic solutions are often set aside due to some inherent disadvantageous characteristics. One of these characteristics relates to the difficulty of obtaining fine motion control with conventional pneumatic actuators. This is caused by several coexisting causes, like the nonlinear behavior of servo-valves, the low stiffness exhibited by pneumatic systems and the nonlinear and uncertain behavior of friction. As such, the use of more conventional control techniques typically leads to low positioning performance. Several control strategies have been tested over the years, from classic PID control [1] to fuzzy logic [2]. A common approach to the control of such systems is state feedback, which has been used since the 1980s [3,4] up until more recently [5]. In this last study, positioning errors around 10 µm, the resolution of the used encoder, could be achieved even in the presence of high mass load variations. Despite this, in addition to the positioning error varying with the piston’s position, the technique used in [5] seems to be heavily dependent on experimental non-procedural techniques. For applications like welding, robotic manipulation or fluid injection, fine motion control is mandatory. Pneumatic systems are traditionally very hard to model and control due to their inherent

Actuators 2020, 9, 63; doi:10.3390/act9030063 www.mdpi.com/journal/actuators Actuators 2020, 9, 63 2 of 13 nonlinearities, rendering them unsuitable for such applications. To reach low positioning and tracking errors, the control laws are typically very model-dependent, turning the implementation of the controller into a very time-consuming task. These model-dependent controllers tend to be excessively tailored to a particular setup and may result in unsmooth closed loop motion [6–8]. Studies [6,7] address the motion control of a conventional actuator, with payload changes from 2.69 to 13.1 kg without controller retuning. In [6], the tracking errors to a sinusoidal reference position input with variable frequency (up to π rad/s) were only of a few mm. In [7], the steady state errors for step position reference inputs within a 280 mm stroke were as low as 5 µm, the encoder’s resolution. Nevertheless, the controller used is of a high degree of complexity with two control loops: a sliding mode controller is used for the motion control whilst, for the pneumatic force control, a nonlinear state feedback controller is used. Models of friction force and servo-valves are used to cope with the systems nonlinearities. In study [8], for a sinusoidal position reference of 0.5 and 1 Hz, tracking accuracy of about 1 mm is achieved whilst for a 60 mm step input with a 200 ms rise time, positioning accuracy was lower than 50 µm. However, to achieve such results, a highly complex controller is used: a sliding mode controller for force control and a model reference adaptive controller for motion control. An adaptive friction model is used to estimate friction forces. As such, conventional actuators require complex, model-dependent laws that typically must include highly active control laws to surpass the underlying uncertainty [7,9,10]. Furthermore, the noise inherent to velocity and acceleration estimation might also affect the motion performance [11]. This means that accurate motion control can be achieved but often at the cost of motion smoothness. One way to circumvent this problem is the use of hybrid actuators. This approach tries to capture the best features of electrical and pneumatic actuators: fine motion control is ensured by the more linearly behaved electrical actuators while the high force, speed, high compliance and lightweight are ensured by the pneumatic ones. For instance, in [12] the design of a hybrid actuator to be used in a badminton playing robot is presented. The authors of [12] claim that the actuator is substantially lighter than previous existing ones, due not only to the use of lightweight materials (for instance, the inner cylinder is made of glass fiber-reinforced plastics) but also to the use of a wire instead of a rod to transmit the motion from the piston to the outside. The actuator is designed in an integrated structure, making it compact enough to be incorporated in the links of a seven degrees of freedom robot arm. This arm is capable of moving the badminton racket at speeds as high as 19 m/s. Although undesirable for control purposes, the inherent high compliance of pneumatic systems is a benefit in applications that require human–robot collaboration. Having this in mind, a hybrid pneumatic–electric actuator was developed in [13]. In this case, the rotary actuator is not designed in an integrated structure. Instead, it is composed by linear actuators coupled to rack gears which in turn mesh with a pinion gear linked to the output shaft. An electrical motor is also connected to the output shaft. According to the authors of [13], this design significantly produces higher torque than previous designs, while maintaining the low mechanical impedance required for collaborative robotic applications. Another interesting aspect of the study presented in [13] is the fact that it uses on–off valves, thereby reducing the costs when compared to the use of servo-valves as in [12]. Both studies [12,13] report that the use of hybrid control strategies greatly reduces the motion control error when compared to the use of pneumatic control alone. This is because if, on the one hand, high compliance is required for human interaction, it is also undesirable from a control perspective as it reduces the available bandwidth. Consequently, the use of hybrid actuators and control strategies allows a successful combination of the higher bandwidth of electrical actuators with the high compliance of pneumatic actuators. In [12], PID-based controllers were used, whilst in [13] a more complex strategy, based on model based controllers, was followed. Another claimed downside of pneumatic systems, when compared to servo electromechanical alternatives, is their low energetic efficiency [14–16]. For instance, in [16] an assessment of the energetic efficiency of a typical pneumatic solution is made. This study reveals that, even considering an ideal Actuators 2020, 9, 63 3 of 13 compressor and neglecting the motor’s efficiency and friction effects on the actuator, the pneumatic solution efficiency was below 28%. Despite this, there is still some disagreement in the literature regarding which is the best solution (pneumatic or electric) for a given application, since for instance, in [17], it is possible to find applications where pneumatic solutions are more favorable than their electrical counterparts, if the right criteria for dimensioning of the servo-pneumatic device are considered. One interesting study claiming significant energy improvements in a pneumatic actuator is the one presented in [18]. The main novel idea behind that study is the use of a deformable piston structure covered by a flexible membrane—the so-called tension piston. This configuration allows pneumatic energy to be converted into mechanical energy using two different mechanisms: the traditional one, where the pressure acting on the piston rigid areas is converted into a force, and a pneumatic-induced tension on the piston flexible membrane. This configuration presents several potential advantages over conventional techniques. For instance, given the use of a flexible membrane for sealing, it has no leakages and exhibits low friction forces, making it potentially less complex for servo control. In [18], it is shown that the tension piston can develop higher forces when compared to a conventional piston at the same pressure, leading to substantially higher power and energy efficiency. It should be noted, however, that the conventional actuator used for comparison in [18] presents conventional seals, therefore leading to high friction forces that apparently were not accounted for, in at least some of the comparisons made in [18]. Although this aspect is not analyzed in [18], this configuration potentially leads to lower endurance as it involves a flexible membrane. Finally, the piston architecture presented in [18] requires a higher actuator length for the same stroke, when compared to conventional actuators. The use of flexible membranes can also be found in pneumatic artificial muscles (PAM) [19–21] but typically this solution only allows short stroke actuators and presents a stroke-dependent output force. To deal with this problem, in [21] a speed-increasing gear is coupled to the PAM in order to increase the range of motion of the PAM while also flattening the output force characteristics. A different approach was followed by the authors of this study in [22,23] to devise novel pneumatic actuators. In those studies, directly related to the one presented in this paper, a pneumatic linear peristaltic actuator (PLPA), similar to the one used in this work, was proposed. A PLPA comprises a hose compressed by two rollers that act as a piston and presents several advantages over conventional actuators: it is a potentially low-weight and -cost actuator, it has a virtually unlimited stroke and it can easily perform curved motion profiles. Another important advantage of a PLPA, as shown in [22,23] is that that the friction characteristic of such an actuator is not as nonlinear as their conventional counterparts, making the task of devising control laws significantly less complex. In fact, the simple use of a PID type controller led to zero steady state error in [23]. It should be noticed that the use of similar controllers with conventional low-friction actuators caused limit cycles [23]. One of the downsides of a PLPA lies in the fact that it suffers high mechanical stress given the need to compress the hose between the rollers to achieve good sealing performance. In order to experimentally determine the main causes of failure of such actuators, in [24] an experimental characterization of some mechanical properties of three different types of hoses was performed. Those hoses are then used in experimental endurance tests to determine the failure causes and analyze how different parameters may influence the longevity of the actuator. It was found that the distance covered by the PLPA is two orders of magnitude lower than the typical one of conventional pneumatic cylinders. Consequently, further investigation on different types of hoses materials and shapes is required. Another interesting finding of [24] is that the thickness of the hose walls might be significantly reduced during operation. Since in [22–24] the distance between rollers is fixed, the inter-chamber leakage increases due to this effect. The present paper builds upon previous studies [22–24]. Its main contributions are as follows. First, in this study a PLPA prototype where the force between rollers is imposed is used, so that the thickness reduction previously described becomes compensated. Second, although the controllers Actuators 2020, 9, x FOR PEER REVIEW 4 of 13 substantial decrease in the settling time whilst avoiding limit cycles. Specifically, this paper presents a controller with an integral gain modulation strategy that has been experimentally shown to prevent limit cycles in positioning tasks while presenting a very fast response. ActuatorsThis2020 paper, 9, 63 is organized as follows: Section 2 describes the experimental setup, Section 43 of its 13 model and Section 4 the new controllers developed as well as the results obtained in closed-loop positiondeveloped control. in [23 ]Finally, led to steady Section state 5 presents zero error, the the main overall conclusions behavior obtained. of the system was slow. To surpass that problem, this paper presents the development of new controllers that lead to a substantial decrease 2.in Experimental the settling time Setup whilst avoiding limit cycles. Specifically, this paper presents a controller with an integralThe same gain working modulation principle strategy of the that actuators has been studied experimentally in [22−24] shownis used: to compressed prevent limit air cyclesis fed to in onepositioning end of tasksa hose, while which presenting is divided a very into fast two response. chambers by two rollers pressed against it. The compressedThis paper air iscreates organized a pressure as follows: difference Section between2 describes chambers the experimental resulting in setup, a force Section (𝐹) 3that its modelacts on theand rollers. Section 4 the new controllers developed as well as the results obtained in closed-loop position control.The Finally,experimental Section setup5 presents used thein th mainis work conclusions is depicted obtained. in Figure 1. It comprises a PLPA mounted on a carriage, which runs over a monorail guidance system with an integrated position encoder 2. Experimental Setup (Bosch Rexroth IMS series). This encoder has a 5 μm resolution, ±0.75 μm interpolation accuracy and ±0.25The μm samerepeatability. working In principle addition, of the the setup actuators includ studiedes two inFESTO [22–24 servo-valves] is used: compressed (SV) MPYE-5-1/8-HF- air is fed to 010-Bone end The of adescribed hose, which valves is divided and sensors into two are chambers connect byed two to rollersa data pressed acquisition against and it. Thecontrol compressed system comprisingair creates a a pressure PC with di datafference acquisition between boards chambers and all resulting the necessary in a force signal (Fp) conditioning that acts on the hardware. rollers. The two experimental aluminum setup rollers used on inthe this PLPA work have is depicted a urethane in Figurecoating1 .to It reduce comprises the stress a PLPA in mountedthe hose. Theon a bottom carriage, roller which is fixed runs overto the a monorailcarriage while guidance the top system one’s with height an integratedis adjustable position with the encoder help of (Bosch two screwsRexroth on IMS each series). side. ThisBy using encoder (or hasnot) a a 5 µsetm of resolution, four spring0.75 washersµm interpolation on each of accuracythese screws, and force0.25 µ(orm ± ± distance)repeatability. between In addition, the rollers the setupcan be includes imposed. two The FESTO hose servo-valveschosen for this (SV) study MPYE-5-1 was FLEXIGOM/8-HF-010-B AIR, The manufactureddescribed valves by andProductos sensors Mesa, are connected the most to durable a data acquisitionhose tested and in [24]. control This system hose, comprisingdesigned for a PC30 barwith working data acquisition pressure, boards is comprised and all of the a necessaryflexible nitrile signal rubber conditioning sleeve, reinforced hardware. with a high-tenacity polyester fabric and an abrasion-resistant external coating.

PLPA Hose

Carriage

Encoder measuring sensor Encoder magnetic strip

Figure 1. ExperimentalExperimental setup.

3. SystemThe two Model aluminum rollers on the PLPA have a urethane coating to reduce the stress in the hose. The bottomThe model roller of the is ﬁxed system to thehas carriagebeen presented while thein detail top one’s in [25] height and can is adjustable be summarized with theas follows help of two screws on each side. By using (or not) a set of four spring washers on each of these screws, force (or distance) between the rollers can be𝑀𝑥 imposed. = 𝑃 −𝑃 The hose𝐴 −𝐹 chosen for this study was FLEXIGOM AIR,(1) wheremanufactured 𝑀 represents by Productos the moving Mesa, mass, the most 𝑥, 𝑥 durable , 𝑥 represent hose tested its position, in [24]. Thisvelocity hose, and designed acceleration, for 30 respectively,bar working pressure,𝑃, represent is comprised the pressure of a ﬂexible acting nitrileinside rubberthe PLPA sleeve, chambers reinforced A and with B, respectively, a high-tenacity 𝐴 representspolyester fabric the hose and cross-section an abrasion-resistant area, 𝐹 external is the coating.total friction force acting on the x direction on the system (see Figure 2), which can be detailed as 3. System Model 𝐹 =𝐹| +𝐹| +𝐹 𝛽 (2) The model of the system has been presented in detail in [25] and can be summarized as follows

.. x Mx = (PA PB)A F f r (1) − − . .. where M represents the moving mass, x, x, x represent its position, velocity and acceleration, respectively, PA,B represent the pressure acting inside the PLPA chambers A and B, respectively, A represents the Actuators 2020, 9, 63 5 of 13

x hose cross-section area, F f r is the total friction force acting on the x direction on the system (see ActuatorsFigure 22020), which, 9, x FOR can PEER bedetailed REVIEW as 5 of 13 where 𝐹| is the friction due to the baxll bearings of the rollers,y 𝐹| is the friction between the F f r = F f r r + F f r g + F f r β (2) | | monorail guide and the carriage, 𝐹 is the sliding friction in the transversal direction and 𝛽 is a factorwhere relatingF f r r is thethe frictionhose flattening due to the ve balllocity bearings in the transversal of the rollers, directionF f r g is the(𝑦) frictionto the moving between mass the monorailvelocity | y | inguide the horizontal and the carriage, directionF f r(𝑥).is the𝐹 sliding was experimentally friction in the transversaldetermined direction in [25]. and β is a factor relating . the hoseAssuming ﬂattening that velocity the pressure in the inside transversal each directionof the PLPA (y) tochambers the moving is higher mass velocitythan 0.2 in bar, the the horizontal hose’s . x cross-sectiondirection (x). canF f r bewas considered experimentally constant determined [22] and, as in such, [25]. the roller motion is the unique responsible for the volume change inside each chamber.

Pb 𝒙 𝑭𝒇𝒓 𝒙, 𝒙,𝒙

FigureFigure 2. 2. MechanicalMechanical model model nomenclature. nomenclature.

TakingAssuming this thatinto theaccount pressure and inside considering each of that the PLPAtemperature chambers fluctuations is higher thanover 0.2the bar, equilibrium the hose’s temperaturecross-section are can negligible, be considered the following constant [equation22] and, asca such,n be written the roller to motiondescribe is the the pressure unique responsible dynamics offor chambers the volume A and change B [26] inside each chamber. Taking this into account and considering that temperature fluctuations over the equilibrium 𝑑𝑃, 𝑃, 𝑑𝑉, 𝑅 temperature are negligible, the=− following equation+𝛾 can𝑇 be written𝑚 −𝑚 to describe the pressure dynamics(3) of 𝑑𝑡 𝑉 𝑑𝑡 𝑉 , , chambers A and B [26] , , where 𝑉, represents each chamber’s volume, 𝛾 is the ratio of specific heats for air, 𝑇 the dP P dV R . . ambient temperature and 𝑚A ,B , 𝑚 A,B theA,B mass flow of air entering and leaving each chamber, ,= , + γ Tamb mA,Bin mA,Bout (3) dt −VA,B dt VA,B − respectively. In order to determine the mass flow of air 𝑚 ,, a balance should be performed between (i) the flow retrieved or delivered by the servo valves ([27−29]) and (ii) the leakage flow 𝑚 . In the where VA,B represents each chamber’s volume, γ is the ratio of specific heats for air, Tamb the ambient present work however,. the. leakage between chambers is negligible, it was experimentally shown in temperature and mA,Bin , mA,Bout the mass flow of air entering and leaving each chamber, respectively. [25]. Under these assumptions, the model for. each working port of the servo-valve can be obtained In order to determine the mass flow of air mA,B, a balance should be performed between (i) the flow [25] . retrieved or delivered by the servo valves ([27–29]) and (ii) the leakage flow ml. In the present work however, the leakage between chambers. is negligible, it was experimentally shown in [25]. Under these 𝑚 , =𝜌 𝐶,𝑢𝑃𝑌 −𝐶,𝑢𝑃𝑌 (4) assumptions, the model for each working port of the servo-valve can be obtained [25]

Equations (1) through (4) rdefine a nonlinear model of the system. This model can be . 293.15 P Patm approximated by the followingmA,B = ρ 0third-orderC linear1,3(u) PmodelsY1 C2,4(u)PsY1 (4) Tamb Ps − P 1 𝑘 𝑀 Equations𝑥⃛ (1)= through𝐴𝜓 (4) −𝜓 deﬁne − a nonlinear𝑥 − model+𝑘𝑥 of the system. This model can be approximated 𝑀 𝜏 𝜏 by the following third-order linear model (5) 𝐴𝛾𝑅𝑇 𝐴𝛾𝑅𝑇 1 + 𝐺 + 𝐺 𝑢 −𝐹 _ − 𝐹_ ... 𝑉h 𝑉 . .. 𝜏 x = 1 A(ψ ψ ) ka x M + k x M A B τm τm a − − − . where 𝜓, are equilibrium constants,AγRTA0 𝑘 isA γtheRTB 0sliding friction coefficient,1 i 𝜏 is the harmonic(5) + G + GuB u F F VA0 uA VB0 f r_st τm f r_st mean of the pressure time constants of the0 PLPA, 𝑅 is0 the− constant− of air as a perfect gas, 𝑇, represent each chamber’s temperature, 𝐺, are the flow gain values for each of the servo-valve where ψA,B are equilibrium constants, ka is the sliding friction coeﬃcient, τm is the harmonic mean working ports and 𝐹_ is the nonlinear static friction force. It should be noted that in equilibrium of the pressure time constants of the PLPA, R is the constant of air as a perfect gas, TA,B represent conditions 𝐹_ =𝐹_ =0. Equation (5) can be further simplified as 𝑋𝑠 𝑘𝜔 = (6) 𝑈𝑠 𝑠𝑠 +2𝜁𝜔𝑠+𝜔

Actuators 2020, 9, 63 6 of 13

each chamber’s temperature, GuA,uB are the ﬂow gain values for each of the servo-valve working ports and F f r_st is the nonlinear static friction force. It should be noted that in equilibrium conditions Actuators 2020. , 9, x FOR PEER REVIEW 6 of 13 F f r_st = F f r_st = 0. Equation (5) can be further simpliﬁed as

where 2 X(s) klinωn = (6) U(s) 2 2 s s + 2ζωns + ωn

1 𝑘 where 𝜔 = 𝐴𝜓 −𝜓 + (7) 𝑀" 𝜏# 2 1 ka ωn = A(ψB ψA) + (7) M −1 𝑘 τm 2𝜁𝜔 = + (8) 𝜏 𝑀 1 ka 𝐴2𝛾𝑅𝑇ζωn = + 𝐴𝛾𝑅𝑇 (8) τm M 𝑘𝜔 = 𝐺 + 𝐺 (9) 𝑀𝑉 𝑀𝑉 2 AγRTA0 AγRTB0 klinωn = GuA + GuB (9) The parameters of the model describedMVA0 in (6) 0wereMV experimentallyB0 0 identified in [25] and are presentedThe parameters in Table 1. of the model described in (6) were experimentally identiﬁed in [25] and are presented in Table1. Table 1. Identified system parameters.

SystemTable Parameter 1. Identiﬁed system Value parameters. Unit

System𝜔 Parameter [rad/s] Value 16.7 [rad/s]Unit

ωn [rad𝜁/ s] 16.70.51 [rad-/s] ζ 0.51 - 𝑘 [m/(Vs)] 0.82 [m/(Vs)] klin [m/(Vs)] 0.82 [m/(Vs)]

4.4. Position Closed Loop Control ToTo testtest thethe closedclosed looploop controlcontrol ofof thethe developeddeveloped actuator,actuator, anan increasingincreasing amplitudeamplitude squaresquare wavewave referencereference signalsignal waswas used.used. Each step input had a 20 s durationduration andand betweenbetween stepssteps thethe amplitudeamplitude increaseincrease waswas ofof 2020 mm,mm, upup toto aa maximummaximum ofof 320320 mm,mm, inin aa totaltotal ofof 1616 steps.steps. TheThe errorerror atat thethe endend ofof each step (𝑒), overshoot (𝑀), settling time to 2% of the step amplitude (𝑡) and settling time to each step (eend), overshoot (Mp), settling time to 2% of the step amplitude (tss) and settling time to ±0.10.1 mm of the set point value (t𝑡.), were registered.registered. At the end of each test,test, thethe maximummaximum absoluteabsolute ± 0.1 error ( 𝑒_ ) and overshoot (𝑀_ ) for all steps were registered, as well as the average settling error ( eend_max ) and overshoot ( Mp_max ) for all steps were registered, as well as the average settling times (𝑡 and 𝑡 ) and respective standard deviation (𝑡 e 𝑡 ). times (tss__m and t0.1_._m) and respective standard deviation (tss__σ e t0.1_._σ). SeveralSeveral controllerscontrollers were were tested. tested. Given Given its simplicity its simplicity and its wideand usageits wide in industrial usage environments,in industrial aenvironments, controller from a controller the PID family, from th ane PID I-PD family, [30] controller an I-PD in[30] the controller present case,in the was present the ﬁrst case, one was to the be tested.first one The to I-PD be controllertested. The block I-PD diagram controller is shown block in diagram Figure3. Theis shown controller in Figure parameters 3. The were controller initially determinedparameters usingwere initially Ziegler–Nichols determined tuning using rules Ziegler–Nichols and then experimentally tuning rules ﬁne-tuned. and then Figureexperimentally4 shows thefine-tuned. systems outputFigure position4 shows and the error, systems and Tableoutput2 presents position the and performance error, and metrics Table obtained. 2 presents the performance metrics obtained.

Figure 3. Block diagram of the I-PD controller.

Figure 4 shows that the response speed of the system is slow, being unable to reach the ±0.1 mm range in some of the 20 s steps. In order to accelerate the system, a state feedback controller with integral action (SFCI) was tested. The controlled system block diagram is presented in Figure 5. The SFCIs parameters were chosen to impose a 4th order Bessel filter dynamic with a characteristic frequency of 𝜔 =20 𝑟𝑎𝑑/𝑠. The systems output position is displayed in Figure 6.

ActuatorsActuators 20202020, ,99, ,x 63 FOR PEER REVIEW 77 of of 13 13

(a)

(b)

eend

Mp_max

(c)

FigureFigure 4. 4. I-PDI-PD positioning positioning test test results: results: (a (a) )position position vs. vs. reference; reference; (b (b) )error error evolution evolution in in time; time; ( (cc)) detail detail ofof (b), (b), showing showing the the step step where where MMp_maxp_max occurred.occurred.

TableTable 2. 2. I-PDI-PD controller controller performance performance metrics. metrics.

|eend_max| [µm] |Mp_max| [%] tss_m [s] tss_σ [s] t0.1_m [s] t0.1_σ [s] 𝒆𝒆𝒏𝒅_𝒎𝒂𝒙 [μm] 𝑴𝒑_𝒎𝒂𝒙 [%] 𝒕𝒔𝒔_𝒎 [s] 𝒕𝒔𝒔_𝝈 [s] 𝒕𝟎.𝟏_𝒎 [s] 𝒕𝟎.𝟏_𝝈 [s] 210210 0.9 0.9 4.22 4.22 0.18 0.18 16.66 16.66 * * 2.59 2.59 * * ** In In 55 ofof the the 16 16 steps, steps, the errorthe error was bigger was thanbigger 0.1 mm.than These0.1 mm. steps These were not steps considered were not for theconsidered calculation for of t0.1_them and t0.1_σ. calculation of 𝑡._ and 𝑡._.

Actuators 2020, 9, 63 8 of 13

Figure4 shows that the response speed of the system is slow, being unable to reach the 0.1 mm ± range in some of the 20 s steps. In order to accelerate the system, a state feedback controller with integral action (SFCI) was tested. The controlled system block diagram is presented in Figure5. The SFCIs parameters were chosen to impose a 4th order Bessel ﬁlter dynamic with a characteristic frequency of ωc = 20 rad/s. The systems output position is displayed in Figure6. Actuators 2020, 9, x FOR PEER REVIEW 8 of 13 Actuators 2020, 9, x FOR PEER REVIEW 8 of 13

Figure 5. Block diagram of the state feedback controller with integral action (SFCI) controller. Figure 5. Block diagram of the state feedback controller with integral action (SFCI) controller. Figure 5. Block diagram of the state feedback controller with integral action (SFCI) controller.

Figure 6. SFCI controller positioning test results: position vs. reference. Figure 6. SFCI controller positioning test results: position vs. reference. From the results presented in Figure 6, it is seen that system response is quicker, but a limit cycle From the the results results presented presented in in Figure Figure 6,6 it, itis isseen seen that that system system response response is quicker, is quicker, but a but limit a limitcycle occursFrom with the the results SFCI. Topresented circumvent in Figure this problem, 6, it is seen a state that systemfeedback response controller is quicker, was designed but a limit where cycle an occurscycle occurs with the with SFCI. the To SFCI. circumvent To circumvent this problem, this problem, a state feedback a state feedback controller controller was designed was designedwhere an integraloccurs with action the isSFCI. added To circumventwhenever thethis systemproblem, is ainside state feedbacka pre-specified controller error was range designed (SFC+I). where This an whereintegral an action integral is actionadded iswhenever added whenever the system the systemis inside is a inside pre-specified a pre-speciﬁed error errorrangerange (SFC+I). (SFC This+I). controllerintegral action behaves is addedas a regular whenever state feedbackthe system controller is inside (SFC) a pre-specified whenever it erroris outside range the (SFC+I). error range This Thiscontroller controller behaves behaves as a regular as a regular state feedback state feedback controller controller (SFC) (SFC)whenever whenever it is outside it is outside the error the range error andcontroller as a PI-DD behaves2 whenever as a2 regular it is inside state feedback it. In this controllerwork, an error (SFC) range whenever of δ = it±2 is mm outside was experimentallythe error range range and as a2 PI-DD whenever it is inside it. In this work, an error range of δ = 2 mm was and as a PI-DD2 whenever it is inside it. In this work, an error range of δ = ±2 mm was experimentally selected,and as a PI-DDsince the whenever SFC can itensure is inside this it. positioning In this work, tolerance. an error Figurerange of 7 δpresents = ±2 mm the was block experimentally± diagram of selected,experimentally since the selected, SFC can since ensure the SFC this can positioning ensure this tolerance. positioning Figure tolerance. 7 presents Figure the7 presents block diagram the block of theselected, SFC+I since controller. the SFC The can controller ensure thisparameters positioning were tolerance. chosen to Figure make the 7 presents system behavethe block as adiagram 3rd order of thediagram SFC+I of controller. the SFC+ TheI controller. controller The parameters controller were parameters chosen to were make chosen the system to make behave the system as a 3rd behave order Besselthe SFC+I filter controller. ( 𝜔 = 40 The 𝑟𝑎𝑑/𝑠 controller) when parameters outside the were error chosen range. to make The theintegral system action behave gain as a (3rd𝑘 ) order was as a 3rd order Bessel ﬁlter (ωc = 40 rad/s) when outside the error range. The integral action gain (k ) Bessel filter ( 𝜔 = 40 𝑟𝑎𝑑/𝑠 ) when outside the error range. The integral action gain ( 𝑘 ) wasi experimentallywas experimentally tuned tuned and andis reset is reset whenever whenever the the error error is isoutside outside the the error error band, band, as as represented represented in experimentally tuned and is reset whenever the error is outside the error band, as represented in Figure 77.. TheThe resultsresults areare shownshown inin FigureFigure8 8 and and the the performance performance metrics metrics are are displayed displayed in in Table Table3. 3. Figure 7. The results are shown in Figure 8 and the performance metrics are displayed in Table 3.

Figure 7. Block diagram of the SFC+I controller. Figure 7. Block diagram of the SFCSFC+I+I controller. controller. Table 3. SFC+I controller performance metrics. Table 3. SFC+I controller performance metrics. 𝒕𝒔𝒔_𝝈 𝒆 𝑴 𝒕 𝒕 𝒕 𝒆𝒏𝒅_𝒎𝒂𝒙 [μm] 𝒑_𝒎𝒂𝒙 [%] 𝒔𝒔_𝒎 [s] 𝒕𝒔𝒔_𝝈 𝟎.𝟏_𝒎 [s] 𝟎.𝟏_𝝈 [s] 𝒆 [μm] 𝑴 [%] 𝒕 [s] [s]𝒔𝒔_𝝈 𝒕 [s] 𝒕 [s] 𝒆𝒆𝒏𝒅_𝒎𝒂𝒙 [μm] 𝑴𝒑_𝒎𝒂𝒙 [%] 𝒕𝒔𝒔_𝒎 [s] 𝒕𝟎.𝟏_𝒎 [s] 𝒕𝟎.𝟏_𝝈 [s] 𝒆𝒏𝒅_𝒎𝒂𝒙 𝒑_𝒎𝒂𝒙 𝒔𝒔_𝒎 [s] 𝟎.𝟏_𝒎 𝟎.𝟏_𝝈 45 5.08 0.53 1.18 6.7 0.75 45 5.08 0.53 1.18 6.7 0.75

ActuatorsActuators 20202020, 9, ,x9 FOR, 63 PEER REVIEW 9 of9 13 of 13

(a)

(b)

eend

Mp_max

(c)

FigureFigure 8. SFC+I 8. SFC controller+I controller positioning positioning test testresults: results: (a) position (a) position vs reference; vs reference; (b) error (b) error evolution evolution in time; in time; (c) (detailc) detail of (b), of ( bshowing), showing the the step step where where Mp_maxMp_max occurred.occurred.

Comparing the values presentedTable 3. SFCin Tables+I controller 2 and performance 3, this control metrics. strategy allowed the PLPA response to become faster, taking about an eighth of the time to reach the 2% range. |eend_max| [µm] |Mp_max| [%] tss_m [s] tss_σ [s] t0.1_m [s] t0.1_σ [s] One way to reduce the time required to reach the 0.1 mm range would be to increase the integral 45 5.08 0.53 1.18 6.7 0.75 gain, 𝑘. However, it was found that by doing so, the system started to show limit cycle behavior. In order to make it faster and circumvent limit cycle limitations, the integral action was modulated according to (10):

Actuators 2020, 9, 63 10 of 13

Comparing the values presented in Tables2 and3, this control strategy allowed the PLPA response to become faster, taking about an eighth of the time to reach the 2% range. One way to reduce the time required to reach the 0.1 mm range would be to increase the integral Actuators 2020, 9, x FOR PEER REVIEW 10 of 13 gain, ki. However, it was found that by doing so, the system started to show limit cycle behavior. In order to make it faster and circumvent limit cycle limitations, the integral action was modulated according to (10): 𝑢 =𝑠𝑎𝑡 𝑘 ∙ |𝑒𝑡| ∙𝑒𝑡𝑑𝑡 Z (10) ui = sat∞ ke e(t) e(t)dt (10) kmin · · Gain 𝑘 was experimentally tuned while gain 𝑘 was kept at the same value as 𝑘. This was done Gainbecauseke was when experimentally Equation (10) tuned is saturated, while gain thekmin controlwas kept law at is the the same same value as in as theki. previous This was donecase tested.because The when final Equation controller (10) architecture is saturated, (SFC+I theM control) is shown law in is Figure the same 9. as in the previous case tested. Actuators 2020, 9, x FOR PEER REVIEW 10 of 13 The ﬁnalFigure controller 10 shows architecture the results (SFC the+I MSFC+I) is shownM controller in Figure and9. Table 4 presents the positioning performance metrics. 𝑢 =𝑠𝑎𝑡 𝑘 ∙ |𝑒𝑡| ∙𝑒𝑡𝑑𝑡 (10)

Gain 𝑘 was experimentally tuned while gain 𝑘 was kept at the same value as 𝑘. This was done because when Equation (10) is saturated, the control law is the same as in the previous case tested. The final controller architecture (SFC+IM) is shown in Figure 9. Figure 10 shows the results the SFC+IM controller and Table 4 presents the positioning performance metrics.

Figure 9. BlockBlock diagram diagram of of the the SFC+I SFC+IMM controller.controller.

Figure 10 shows the results the SFC+IM controller and Table4 presents the positioning performance metrics. Figure 9. Block diagram of the SFC+IM controller.

((a))

(b)

Figure 10. Cont. (b)

Actuators 2020, 9, 63 11 of 13 Actuators 2020, 9, x FOR PEER REVIEW 11 of 13

eend

Mp_max

(c)

Figure 10. SFCFigure+ I10.M controllerSFC+IM controller positioning positioning test test results: results: ( (aa) )position position vs. vs.reference; reference; (b) error (b evolution) error evolution in in time; (c) detail of (b), showing the step where Mp_max occurred. time; (c) detail of (b), showing the step where Mp_max occurred. Table 4. SFC+I controller with new integral law performance metrics. Table 4. SFC+I controller with new integral law performance metrics.

𝒆𝒆𝒏𝒅_𝒎𝒂𝒙 [μm] 𝑴𝒑_𝒎𝒂𝒙 [%] 𝒕𝒔𝒔_𝒎 [s] 𝒕𝒔𝒔_𝝈 [s] 𝒕𝟎.𝟏_𝒎[s] 𝒕𝟎.𝟏_𝝈[s] |eend_max| [µm] |Mp_max| [%] tss_m [s] tss_σ [s] t0.1_m[s] t0.1_σ[s] 4545 5.08 5.08 0.53 0.53 1.181.18 6.7 6.7 0.75 0.75

5. Results Discussion 5. Results Discussion From the several controllers considered in this work, the SFCI controller was the only one From theleading several to a limit controllers cycle. It consideredis known that in even this without work, theStribeck SFCI friction, controller a limit was cycle the can only occur one leading whenever an integral action is present [31]. Furthermore, as also shown in [31], the limit cycle can to a limit cycle.occur for It isa knowngiven set thatof controller even without parameters Stribeck and not friction, occur for aanother limit cycleset of canparameters. occur wheneverThis an integral actionsituation is present happened [31 in]. the Furthermore, present study, since as also the I-PD shown is a particular in [31], thecase limit of the cycleSFCI controller can occur (with for a given set of controller𝑘 =0), parameters and while the and SFCI notexhibited occur limit for cycles, another the I-PD set ofdid parameters. not. This situation happened in Table 5 sums up the performance of the controllers tested in this paper. From this table it can be the present study, since the I-PD is a particular case of the SFCI controller (with k3 = 0), and while the clearly seen that the I-PD controller achieves much lower overshoot values than the remaining ones. SFCI exhibitedHowever, limit it cycles,is a very theslow I-PD controller, did not.taking more than twice the time of the SFC+I and ten times the Table5duration sums upof the the SFC+I performanceM to reach the of 0.1 the mm controllers error band. tested in this paper. From this table it can be clearly seen thatThe the introduction I-PD controller of the modulate achievesd integral much gain, lower in the overshoot SFC+IM controller, values clearly than improved the remaining the ones. However, itperformance is a very slowof the controller,system behavior taking when more compared than to twice the one the obtained time ofwith the the SFC SFC+I+I controller. and ten times the The time to reachM the 0.1 mm range was considerably diminished from 6.7 to 1.67 s, the maximum duration oferror the was SFC reduced+I to from reach 45 to the 15 μ 0.1m, mmwhile errorthe overshoot band. remained essentially the same. It is therefore believed that the controller proposed in this work (SFC+IM), when compared to PID-based controllers, leadsTable to a 5. betterPositioning compromise performance between overshoot metrics: and comparison settling time. between However, diﬀ thiserent controller controllers. requires more parameters, making the tuning of the controller more difficult. Future work will therefore focus µ Controlleron adaptive |parameter-tuningeend_max| [ m] |methodologies.Mp_max| [%] tss_m [s] tss_σ [s] t0.1_m [s] t0.1_σ [s] I-PD 210 0.9 4.22 0.18 16.66 * 2.59 * SFC+ITable 5. 45 Positioning performance 5.08 metrics: comparison 0.53 between 1.18 different controllers. 6.7 0.75 M SFC+I Controller 15𝒆𝒆𝒏𝒅_𝒎𝒂𝒙 [μm] 5.43𝑴𝒑_𝒎𝒂𝒙 [%] 𝒕 0.25𝒔𝒔_𝒎 [s] 𝒕𝒔𝒔_𝝈 0.04[s] 𝒕𝟎.𝟏_𝒎 [s]1.67 𝒕𝟎.𝟏_𝝈 [s] 1.42 I-PD * Not considering 210 steps in which 0.9 the error 4.22 was bigger 0.18 than 16.660.1 mm. * 2.59 * SFC+I 45 5.08 0.53 1.18 ± 6.7 0.75 M The introductionSFC+IM of the modulated15 integral5.43 gain, in0.25 the SFC0.04+I controller,1.67 clearly1.42 improved the performance of the system* Not behavior considering when steps in compared which the error to was the bigger one obtainedthan ± 0.1 mm. with the SFC+I controller. The time to reach the 0.1 mm rangewas considerably diminished from 6.7 to 1.67 s, the maximum error was reduced from 45 to 15 µm, while the overshoot remained essentially the same. It is therefore M believed that the controller proposed in this work (SFC+I ), when compared to PID-based controllers, leads to a better compromise between overshoot and settling time. However, this controller requires more parameters, making the tuning of the controller more diﬃcult. Future work will therefore focus on adaptive parameter-tuning methodologies.

6. Conclusions This paper has presented the servo control of a linear peristaltic pneumatic actuator where the force between rollers is imposed. The linear and nonlinear models of the system were presented. A new linear-based controller was developed and compared with datum controllers already used in Actuators 2020, 9, 63 12 of 13 the literature with peristaltic linear actuators. The new controller developed in this work leads to a much faster response while avoiding limit cycles. The average time required to reach a 0.1 mm error band was 1.67 s, in an experiment consisting of a series of 16 steps ranging from 0.02 to 0.32 m in amplitude. These results are comparable with the ones achieved using conventional piston actuators, thereby showing that linear peristaltic actuators may be an eﬀective alternative. Future work will focus on the development of diﬀerent hoses geometries and materials that might increase the service life.

Author Contributions: Conceptualization, J.F.C. and F.G.d.A.; methodology, J.F.C. and F.G.d.A.; software, J.B.P., J.F.C.; validation, J.F.C., J.B.P. and F.G.d.A.; investigation, J.F.C., J.B.P. and F.G.d.A.; resources, J.F.C., J.B.P., and F.G.d.A.; writing—original draft preparation, J.F.C.; writing—review and editing, J.F.C., J.B.P. and F.G.d.A.; All authors have read and agreed to the published version of the manuscript. Funding: This work was financially supported through contract LAETA – UIDB/50022/2020 by “Fundação para a Ciência e Tecnologia”, which the authors gratefully acknowledge. Conflicts of Interest: The authors declare no conflicts of interest.

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