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Dynamic behaviour of pneumatic linear actuators
Mechatronics 45 (2017) 37–48
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Dynamic behaviour of pneumatic linear actuatorsR E. Palomares∗, A.J. Nieto, A.L. Morales, J.M. Chicharro, P. Pintado Department of Mechanical Engineering, University of Castilla – La Mancha Avda. Camilo José Cela s/n, 13071, Ciudad Real (Spain)
a r t i c l e
i n f o
Article history: Received 13 July 2016 Revised 10 May 2017 Accepted 11 May 2017
Keywords: Pneumatic linear actuator Nonlinear dynamics Hysteresis behaviour
a b s t r a c t The use of pneumatic linear actuators is generalised in engineering applications because of their many advantages, but modelling the force they supply may become more of a challenge due to their nonlinear behaviour and the hysteresis their energy losses cause. The authors propose a straightforward model to accurately predict force–displacement behaviour using as a basis experimental observations for several pressures and harmonic displacements of the rod. The model proposed includes two dissipative terms: one due to Coulomb friction and another due to structural damping. The force is proportional to relative pressure when acting as an actuator but nonlinear (modelled as a polytropic transformation) when acting as a pneumatic spring (with a closed pressurised chamber). The model accurately reproduces experimental results (Normalised Root Mean Square Errors lower than 2.5%) and may be used in control systems as well as in adaptive stiffness systems. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Fields such as smart structures, robotics, or mechatronics are nowadays focusing on obtaining increasingly accurate designs for their actuators in order to fulfil requirements of high precision, light weight, etc. Smart actuators such as piezoelectric materials, magnetostrictive materials, shape memory alloys or magnetorheological fluids are being developed in order to address these challenges [1], though conventional actuators can also be useful if suitably characterised. That is the case of pneumatic actuators which, despite having been invented in the mid–twentieth century, have awoken a renewed interest in mechatronic applications in the last two decades. Pneumatic Linear Actuators (PLA) are involved in a variety of industrial, automobile, aerospace, and marine applications. A typical PLA consists of a stationary cylindrical tube, a reciprocating piston and a rod. Airtightness is achieved by means of dynamic seals. These are usually made of thermoplastic materials and are able to undergo large deformations under a wide range of pressures, exerting contact pressure on the mating surfaces and avoiding the leakage of fluid. However, this component causes significant frictional forces and hence complicates the achievement of an accurate model for the PLA. Stribeck curves indicate that, for classical seals, friction at the seal–metal interface is directly proportional to speed and inversely proportional to load [2]. In addition, the R ∗
This paper was recommended for publication by associate editor Wei-Hsin Liao. Corresponding author. E-mail address: [email protected] (E. Palomares).
http://dx.doi.org/10.1016/j.mechatronics.2017.05.007 0957-4158/© 2017 Elsevier Ltd. All rights reserved.
sealing performance in terms of friction is dependent on the type of fluid, the geometry of the sealing interface, and the seal material. The case of rectangular seals under elastohydrodynamic lubrication conditions with uniform contact pressure distribution along the seal width has been studied by Nikas [3], and other studies have considered viscous heat generation for rotary seals [4]. Nevertheless, research is still weak on frictional or hysteretic aspects of pneumatic linear actuators. Van Damme et al. [5] proposed a Preisach model in order to better determine the hysteretic behaviour, but their results were only accurate for a narrow range of displacements. Other authors used a Maxwell–slip model to reproduce the force–displacement hysteresis behaviour [6], or even the well–known generalised Bouc–Wen model [7], which shows lower RMS tracking errors when compared to the quasi–static Maxwell–slip model and the Prandtl–Ishlinskii model. An empirical approach, which turned out to be more accurate than most analytical alternatives, was carried out by Pujana-Arrese et al. [8] by means of fitting a fourth degree polynomial in which the coefficients varied linearly with pressure. The importance of these models in pneumatic systems for different applications has been stressed by many authors. Barth et al. [9] present a PLA tracking sinusoidal inputs using a sliding control. Messina et al. [10] present mathematical dynamic models of positioner PLAs controlled by on–off solenoid valves. The nonlinear dynamics of this problem is due to its transient behaviour, but they achieved a mean positioning error of about 2 mm. Bone et al. [11] control the position of a novel hybrid pneumatic–electric actuator with inexpensive on–off solenoid valves. The addition of the DC motor connected in parallel is required to improve performance
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E. Palomares et al. / Mechatronics 45 (2017) 37–48
modelling PLAs, but this strategy commonly leads to failure of the dynamic behaviour predictions [15]. Pneumatic linear actuators require a good hysteresis model which accurately simulates its force–displacement dynamic behaviour. In order to fill this gap, albeit partially, a nonlinear model for a pneumatic linear actuator is proposed in this study when used as a force actuator or as a pneumatic spring. It is simple, straightforward, easy to fit and useful for control purposes. 2. Characterisation of a pneumatic linear actuator A double acting pneumatic linear actuator can behave either as pneumatic spring (by closing the pressurised chamber) or as an actuator (by supplying the required pressure). In the former case, the nonlinear behaviour of the pressure within the closed chamber and the dissipative forces need to be studied; in the latter case, pressurising and depressurising times may be measured and used in control systems. Fig. 1 shows the double acting pneumatic linear actuator chosen to conduct the experimental studies. It is a commercial double–effect actuator with only one rod, model PRA/182050/M/40, manufactured by Norgren following the standard ISO 15552. The piston is 50 mm in diameter and the rod is 20 mm in diameter with a 40 mm stroke. Fig. 1 also shows how the actuator is attached to the hydraulic test machine. Both cylinder ends are attached to the hydraulic grips by means of spherical joints in order to avoid misalignment effects. The testing machine is equipped with a 10 kN full scale load cell and a LVDT with a calibrated displacement range of ±84 mm. Compressed air to feed the actuator is supplied by an air compressor equipped with a precision pressure–control valve. A valve is placed close to the chamber inlet port (the front chamber in the case of Fig. 1) so that the mass of air could remain constant during the test if required. 2.1. Characterisation as a pneumatic spring
Fig. 1. Pneumatic linear actuator characterised for the case study.
and errors. Other authors present linearised PLA models where the parameters are estimated for different amplitudes of the input signal [12]. Nevertheless, there are still aspects that require further studies, especially those related to the transient phenomena due to the discontinuous nature of the on–off input and its influence on the dynamic behaviour of the PLA. In fact, many authors assume negligible dry friction in their control laws, which leads to unavoidable errors [13,14]. Others linearise Coulomb friction when
a)
The dynamic force exerted by the PLA when the rod is subjected to different harmonic displacements d(t) and different initial pressures are set in either the front or the rear chamber, is characterised in this subsection. The pressurised chamber remains closed during the tests, so that the pressure (and hence the force) varies during the displacement cycle. The tests cover a combination of three displacement amplitudes D (3, 5, and 7 mm), five frequencies f (0.5, 1.0, 1.5, 2.0 and 4.0 Hz), and four initial gauge pressures P0 (1, 2, 3, and 4 bar) for each chamber (front and rear) of the PLA
b) 4
F(t)
F(t)
d(t)
Force, F(t)
Force, F(t)
d(t)
1
3
1
3
2
d-
0 Displacement, d(t)
2
d+
4
d-
0 Displacement, d(t)
Fig. 2. Nominal shape of curves of a PLA: a) Rear chamber pressurised b) Front chamber pressurised.
d+
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b)
1200 5· 10
5
1200 5· 10
Pa abs.
4· 10 Pa abs. 5 3· 10 Pa abs. 5 2· 10 Pa abs.
1000
1000
Pa abs.
800 Force [N]
Force [N]
800
5
4· 105 Pa abs. 5 3· 10 Pa abs. 5 2· 10 Pa abs.
5
600
600
400
400
200
200
0 -6
39
-4
-2
0
2
4
0 -6
6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 3. Force–displacement curves of a PLA varying initial value of pressure P0 , keeping constant both amplitude of displacement (5 mm) and frequency (1 Hz). a) Rear chamber; b) Front chamber.
a)
b)
650
650 4.0 Hz 2.0 Hz 1.0 Hz 0.5 Hz
550
450
Force [N]
Force [N]
550
350
250
150 -6
4.0 Hz 2.0 Hz 1.0 Hz 0.5 Hz
450
350
250
-4
-2
0
2
4
6
Displacement [mm]
150 -6
-4
-2
0
2
4
6
Displacement [mm]
Fig. 4. Force–displacement curves of a PLA varying the excitation frequency of harmonic displacement, keeping constant both amplitude of displacement (5 mm) and the initial value of pressure (3 · 105 Pa abs.). a) Rear chamber; b) Front chamber.
(a total of 120 tests). Tests start with the actuator initially placed at mid–stroke. Fig. 2 shows the qualitative behaviour of the force– displacement curves of the actuator obtained when either the rear (a) or the front (b) chamber is pressurised. The location of the rod for four specific positions is also shown schematically: the rod at mid–stroke (positions labelled 1 and 3 in Fig. 2, i.e., d (t ) = 0), its upper position (position labelled 2, maximum positive d(t)) and its lower position (position labelled 4, maximum negative d(t)). The pressurised chamber is shaded: the higher the pressure, the darker the chamber. The direction in which the hysteresis loop is traversed is also shown. The upper branch corresponds to compression whereas the lower branch corresponds to expansion. Fig. 3 shows the influence of the initial pressure in the force– displacement diagrams for a constant displacement of amplitude D = 5 mm and a constant frequency f = 1 Hz. It can be seen
that the force–displacement diagrams show a significantly nonlinear behaviour along with hysteresis caused by energy dissipation processes. This dissipation is probably due to some sort of friction which must be pressure–dependent. The nonlinear force behaviour is caused by the variation of pressure in the chamber, and this can be modelled as a polytropic transformation. Fig. 4 shows the effect of frequency on the force–displacement behaviour for the case in which the initial absolute pressure is 3 bar and the displacement amplitude is 5 mm. Negligible differences are observed between the curves plotted in Fig. 4, for frequencies from 0.5 to 4 Hz. Therefore, it might be concluded that energy dissipation is not caused by a viscous damping term. Fig. 5 shows the influence of the displacement amplitude D on the force–displacement diagrams for a specific absolute pressure (3 bar) and frequency (1 Hz). It can be concluded that hysteresis
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E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b)
700
700 7 mm 5 mm 3 mm
600
600
500 Force [N]
Force [N]
500
400
400
300
300
200
200
100 -8
7 mm 5 mm 3 mm
-6
-4
-2
0
2
4
6
100 -8
8
-6
Displacement [mm]
-4
-2
0
2
4
6
8
Displacement [mm]
a)
b)
7
7
6
6
5
5
Pressure [bar abs.]
Pressure [bar abs.]
Fig. 5. Force–displacement curves of a PLA varying displacement amplitude D, keeping constant both initial value of pressure (3 · 105 Pa abs.) and frequency (1 Hz). a) Rear chamber; b) Front chamber.
4 3 2
3 2 1
1 0
4
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
Time [s]
Fig. 6. Rear chamber pressurising (a) and depressurising (b) times for three different target and initial pressures.
FPLA (t)
Fnl
2.2. Characterisation as a pneumatic actuator
d(t)
Fs
Fc
Fig. 7. Model scheme.
grows when the displacement amplitude D increases, which could be attributed to some type of structural damping phenomena.
In a pneumatic actuator the force is proportional to the supplied relative pressure. In many cases, the settling time between two different levels of pressure is critical since it may significantly influence the behaviour of a control system. Fig. 6 shows the instantaneous pressure response in the rear chamber when different step commands are sent to the proportional pressure control valve. Overshooting occurs for high target pressures, but was not observed in cases of pressure reduction.
3. Model This section describes the proposed PLA model using as a basis the previous experimental observations. First, dissipative sources will be described since they are common to the spring and actu-
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b) 350
350 Numerical Experimental
300
Numerical Experimental
300
150 Force [N]
150 Force [N]
41
200
200 t=0
t=0
150
150
100
100
50 -6
-4
-2
0
2
4
50 -6
6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 8. Model validation: force–displacement curves for an initial pressure of 2 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
a)
b)
1100
1100 Numerical Experimental
1000 900
900 Force [N]
Force [N]
Numerical Experimental
1000
800 700
t=0
800 700 t=0
600
600
500
500
400 -6
-4
-2
0
2
4
6
400 -6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 9. Model validation: force–displacement curves for an initial pressure of 5 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
ator behaviour. Then, particular models of the dynamic force for these two cases will be described.
3.1. Dissipative forces In accordance with the results depicted in Fig. 3, hysteresis increases, although slightly, with the instantaneous pressure. This behaviour may be due to friction between the inner parts of the actuator and its seals, so the higher the pressure of the chamber, the higher the friction between the seals and the cylinder. This effect is slightly more pronounced when the front chamber is pressurised due to the added effect of friction in the piston seal and friction in the rod seal. This force can be modelled with the help of a Coulomb friction force Fc written as follows:
Fc (t ) = sign d˙ (t )
(P (t ) − Pa )
(1)
where P(t) is the instantaneous absolute pressure in the pressurised chamber, Pa is the ambient pressure, d(t) is the rod displacement and parameter is adjusted for each chamber. As shown in Fig. 5, the area inside a force–displacement curve seems to be directly proportional to the displacement amplitude D. This fact motivates the inclusion of a structural damping term in which the dissipative force Fs can be written as:
Fs (t ) =
δ
2π f
d˙ (t )
(2)
where parameter δ is also numerically determined for each chamber and f is the excitation frequency. Division by f makes Fs (t) independent of frequency for harmonic displacements. 3.2. Pneumatic spring model Since the air mass remains constant (in a closed pressurised chamber) while a displacement is imposed, the main contributor
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E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b) 500
500 Numerical Experimental
450
450
400
350
Force [N]
Force [N]
400
t=0
350 t=0
300
300
150
250
200 -4
Numerical Experimental
-3
-2
-1
0
1
2
3
200 -4
4
-3
-2
-1
Displacement [mm]
0
1
2
3
4
Displacement [mm]
Fig. 10. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 3 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
a)
b) 700
700 Numerical Experimental
600
600
500 Force [N]
500 Force [N]
Numerical Experimental
400
400 t=0
t=0
300
300
200
200
100 -8
-6
-4
-2
0
2
4
6
8
100 -8
-6
-4
-2
Displacement [mm]
0
2
4
6
8
Displacement [mm]
Fig. 11. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 7 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
to the force in the model is the variation of the pressure within the chamber. This variation may be assumed to correspond to a polytropic transformation, i.e.,
V n 0 P (t ) = P0 V (t )
(3)
where n is the polytropic index, V(t) and P(t) are the instantaneous volume and absolute pressure, respectively, and V0 and P0 the initial volume and absolute pressure, respectively. Note that some of these variables or parameters may be “chamber–dependent”, as it were. On the one hand, assuming that the piston of the PLA is located at mid–stroke, the initial volume of each chamber is given by the product of half the stroke (C) and the effective area of the pressurised chamber, whereas the instantaneous volume will also be dependent on the rod displacement d(t). On the other hand, the polytropic index is a parameter which will be determined by curve fitting and may result in different values for each chamber. Then, since the initial pressure P0 is known,
the instantaneous air pressure in the rear or front chamber (depending on which one was pressurised) can be written as follows:
n
P (t ) = P0
Ae
C 2
V0 + d (t )
= P0
n
P (t ) = P0
Ae
C 2
V0 − d (t )
C 2
Fp (t ) = [P (t ) − Pa ]Ae = P0
C 2 C 2
n
− Pa Ae
+ d (t )
when the rear chamber is pressurised, and
Fp (t ) = [P (t ) − Pa ]Ae = P0
C 2 C 2
(5)
− d (t )
and the force exerted by the PLA is
n
C 2 C 2
(4)
+ d (t )
= P0
n
C 2
− d (t )
n
(6)
− Pa Ae
(7)
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b) 600
600 Numerical Experimental
Numerical Experimental
500
500
400
400
Force [N]
Force [N]
43
t=0
300
t=0
300
200
200
100 -6
-4
-2
0
2
4
100 -6
6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 12. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 0.5 Hz. a) Rear chamber; b) Front chamber.
a)
b) 600
600
Numerical Experimental
500
500
400
400
Force [N]
Force [N]
Numerical Experimental
t=0
300
t=0
300
200
200
100 -6
-4
-2
0
2
4
6
Displacement [mm]
100 -6
-4
-2
0
2
4
6
Displacement [mm]
Fig. 13. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 2 Hz. a) Rear chamber; b) Front chamber.
when the front chamber is pressurised. Note that this term of the force does not produce energy dissipation and, therefore, does not replicate the hysteresis observed in experiments. As a result, the total force FPLA (t) exerted by the PLA is made up of a main force term modelled as a polytropic transformation within the chambers of the actuator Fp , and the two dissipative forces previously described: one due to Coulomb friction Fc and another due to structural damping Fs (see Fig. 7). The expression of the resultant force is:
FPLA (t ) = Fp (t ) ± Fc (t ) ± Fs (t )
(8)
which can be written in detail as follows:
FPLA (t ) =
P0
n
C 2 C 2
± d (t )
± sign d˙ (t )
− Pa Ae
(P (t ) − Pa ) ±
δ
2π f
d˙ (t )
(9)
where the plus sign applies if the front chamber is pressurised and the minus sign if the rear chamber is pressurised. The parameters that must be adjusted in the proposed model are the polytropic index n, the effective area Ae , the parameter which defines the Coulomb friction and the parameter of structural damping δ . All of them may be “chamber–dependent”. A nonlinear least square fitting procedure is required to obtain accurate values of these parameters, but a suitable seed may be easily provided. The effective area can be simply obtained by dividing the applied force in several static tests by the gauge pressure in each case. The polytropic index n can be initially estimated by fitting the force–displacement curve with the sole contribution of Fp . Since the dissipated energy for structural damping is U = π δ D2 , the structural parameter δ can be estimated by measuring the hysteresis area in a test with low initial pressure in order to minimise the effect of the Coulomb friction. Finally, parameter is the most difficult to estimate since it depends on both the friction coeffi-
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E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
c)
7
7 Experimental Numerical
6
5
Pressure [bar abs.]
Pressure [bar abs.]
6
4 3 2 1 0
Experimental Numerical
5 4 3 2 1
0
0.1
0.2
0.3
0.4
0
0.5
0
0.1
0.2
Time [s]
b)
0.5
7 Experimental Numerical
6
Experimental Numerical
6
5
Pressure [bar abs.]
Pressure [bar abs.]
0.4
d)
7
4 3 2
5 4 3 2 1
1 0
0.3 Time [s]
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
Time [s]
Fig. 14. Model validation for rear chamber pressurising and depressurising times for different target and initial pressures.
x(t) M
Table 1 Fitted model parameters for rear and front chambers. Chamber
K
C
n Ae [m2 ] (10−3 ) [m2 ] (10−5 ) δ [N/m] (103 )
PLA
y(t)
Front
Rear
1.123 1.541 2.143 3.509
1.063 1.795 2.136 2.963
Fig. 15. Single Degree of Freedom System with variable stiffnes system using a PLA.
3.3. Actuator model The force Fl applied by the PLA is proportional to the instantaneous relative pressure. That is: cient and the contact area of the deformed seal, but can be estimated by trial and error after deducting the effect of the previously estimated structural damping in tests with low displacement amplitudes. Table 1 gathers the values of these parameters for each chamber after the nonlinear least square fitting process.
Fl (t ) = (P (t ) − Pa )Ae
(10)
The dynamics shown in Fig. 6 when considering changes in the reference pressure Pref suggests a second–order behaviour of the
E. Palomares et al. / Mechatronics 45 (2017) 37–48 Table 2 Fitted model parameters for chamber charge and discharge.
ωn [rad/s] ξ
type:
Table 3 Normalised root mean square error (NRMSE) for different experimental tests.
rear
Charge
Discharge
P0 [bar abs.]
D [mm]
f [Hz]
Chamber
NRMSE [%]
52 1.00
52 0.69
2 3 3 3 3 5 2 3 3 3 3 5
5 3 5 5 7 5 5 3 5 5 7 5
1 1 0.5 2 1 1 1 1 0.5 2 1 1
Front Front Front Front Front Front Rear Rear Rear Rear Rear Rear
1.70 2.39 2.35 1.82 1.79 1.16 1.81 1.99 1.77 1.98 1.39 1.34
P¨ (t ) = −2ξ ωn P˙ (t ) − P˙ref − ωn2 (P (t ) − Pref )
(11)
where ωn and ξ are parameters which define overshooting and settling time. The total force FPLA (t) exerted by the PLA when the pressurised chamber is open must include the effect of the dissipative forces:
FPLA (t ) = Fl (t ) ± Fc (t ) ± Fs (t )
(12)
where the plus and minus signs correspond, again, to pressurised front or rear chamber. Thus, the instantaneous force FPLA (t) can be obtained by solving the following system of equations:
FPLA (t ) = (P (t ) − Pa )Ae ± sign d˙ (t )
δ ˙ d (t ) (P (t ) − Pa ) ± 2π f (13)
45
P¨ (t ) = −2ξ ωn P˙ (t ) − P˙ref − ωn2 (P (t ) − Pref )
the pressure to ambient pressure or to a pressurised state, respectively. Qualitative comparisons show a good agreement between experiments and simulations. Differences can be made out at the scale of the plots, but absolute differences are in the order of milliseconds. 5. Case studies 5.1. Use of a PLA as a variable stiffness pneumatic spring
(14)
In this case, the parameters that must be adjusted in the proposed model are the effective area Ae and the parameters ωn and ξ for the second–order system. The estimation of the effective area was discussed in the previous section and is presented in Table 1. The two latter parameters depend on both the sign and the value of the pressure variation. The fitted values are shown in Table 2. 4. Model validation 4.1. Pneumatic spring validation The validation of the model, via comparing the obtained results from the simulated model to the experimental tests, has been carried out for a wide range of pressures (from 2 to 5 bar abs.) and displacement amplitudes (from 3 to 7 mm). Since force– displacement curves were proven to be non–dependent on the excitation frequency (see Fig. 4), all the comparisons are made with f = 1 Hz. Figs. 8–13 show experimental and simulated results. Subplots (a) and (b) show the comparison in the force–displacement diagram for the rear and front chamber, respectively. Qualitative and quantitative comparisons show that simulations are in good agreement with experimental results. Although it is not visually noticeable, instantaneous peak errors are higher when the rod approaches the dead ends. The slope of the force–displacement curve tends to infinity at these singular points and, therefore, the vertical distance between two, otherwise close curves, increases. Nevertheless, the instantaneous error is always lower than 7% at any given location and, in all cases, the Normalised Root Mean Square Error (NRMSE) with respect to the peak–to–peak force amplitude is lower than 2.5% (Table 3). 4.2. Actuator validation Fig. 14 shows comparisons between experimental tests and model simulations for several cases of step commands. Subplots (a) and (b) show comparisons when increasing the pressure from ambient pressure or from an initial pressurised state, respectively; whereas subplots (c) and (d) show comparisons when decreasing
Fig. 15 shows a single degree of freedom system subjected to excitation at the base. This example is useful to understand how a PLA used as a pneumatic spring can modify the transmissibility function (ratio of the amplitude of the response to that of the base motion) of the system and analyse the influence of neglecting dissipative forces. The single degree of freedom system studied has a sprung mass M = 150 kg, a linear spring with stiffness K = 50 kN/m and a viscous damper characterised by a damping constant C = 10 0 0 Ns/m. A PLA (model PRA/182050/M/40) is attached in parallel to this suspension. The resulting suspension may be used without pressure, pressurising the front chamber or pressurising the rear chamber. By changing the pressure in the pressurised chamber one can modify the resonance frequency of the system, which can be useful in adaptive controls systems. Fig. 16 shows the transmissibility modulus of the described system for different cases. Considering the friction forces modelled in this paper, if the rear or the front chamber is pressurised with 5 bar abs. the resonance frequency increases up to 45%. Lower or higher frequencies may be achieved by setting the adequate pressure. Damping is lower in both cases but especially when pressuring the front chamber. Models neglecting friction forces show transmissibility functions peaks up to 20% higher, which would lead to large errors in dynamics behaviour predictions. 5.2. Control of a PLA Fig. 17 shows the control set–up where the PLA is attached to a hydraulic actuator which provides a displacement x(t) while the pressure in the rear chamber is controlled by a proportional pressure control valve. The objective is to control the pressure (using a standard PID controller) in order to maintain the force as close as possible to a variable target. Fig. 18 shows the PID control strategy. Fig. 19 shows comparison between experimental and simulated force records. The PLA is subjected to a 5 mm amplitude, 0.25 Hz frequency, displacement excitation. The force should remain stable at 200 N for 13 s, jump to 800 N and remain at that level for 20 s before dropping to 200 N. The figure shows the experimental force measurements with and without the proposed con-
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E. Palomares et al. / Mechatronics 45 (2017) 37–48
4.5 PLA (P=Pa ) PLA (P=Pa ) with friction forces PLA rear chamber (P=5 bar abs.) PLA rear chamber (P=5 bar abs.) with friction forces PLA front chamber (P=5 bar abs.) PLA front chamber (P=5 bar abs.) with friction forces
4.0
Transmissibility
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0
1
2
3
4
5
6
Frequency [Hz] Fig. 16. Transmissibility modulus of the Single Degree of Freedom System with variable stiffnes for different pressures, chambers and models.
Chassis USB NI CompactDAQ 9174 Analog input NI 9215
Load cell
Analog output NI 9263
Front chamber
PLA Rear chamber
P(t) Hydraulic actuator
x(t)
2
Proportional pressure control valve P
Load unit MTS 810 Hydraulic power unit
1
PC + Virtual Instrument LabVIEW
=
3
Air compressor
Fig. 17. Scheme of the control set–up.
Fref (t)
Error (t)
PID Controller
P(t)
PLA Model
Factual (t)
Factual (t)
Fig. 18. PID control scheme.
trol strategy, as well as the simulated controlled force record. Predicted force behaviour with the control strategy agrees with the experimental controlled force measurements. Both which replicate the target force more accurately than the uncontrolled case. 6. Conclusions Pneumatic linear actuators are used in many engineering applications and they show significant force–displacement hystere-
sis which may hinder model accuracy. Nevertheless, pneumatic actuator modelling has not received sufficient attention: authors either consider too simple models or carry out experimental controls which do not take into account the actuator model. This paper proposes accurate models for two different modes: one acting as a pneumatic spring, where the main force is modelled as a polytropic process within the pressurised chamber, and another acting as a variable force actuator. In both cases, a force due to Coulomb friction and a force due to structural damping
E. Palomares et al. / Mechatronics 45 (2017) 37–48
47
1000 PID Simulated PID Controlled Uncontrolled
900 800
Force [N]
700 600 500 400 300 200 100 0 0
5
10
15
20
25
30
35
40
45
Time [s] Fig. 19. Comparison between the experimental force measurements (with and without the proposed control strategy) and the simulated controlled force record. The PLA is subjected to a 5 mm amplitude, 0.25 Hz frequency, displacement excitation while the force should remain stable at 200 N for 13 s, jump to 800 N and remain at that level for 20 s before dropping to 200 N.
must be considered. After suitably fitting the required parameters, not only does the model reproduce the force–displacement behaviour in many situations to a satisfactory degree (NRMSE lower than 2.5%), but it is also simple, straightforward, and permits its use in the design of control strategies. Two case studies have been presented to show the viability of using the models in control schemes.
Acknowledgements The authors are grateful for the support received from National Project TRA–2014–53552–R, “Mejora del confort del transporte ferroviario de alta velocidad mediante suspensiones neumáticas adaptativas y amortiguadores magnetoreológicos” financed by MINECO (España) and from Regional Project PEII–2014–034–P, “Análisis y diseño de elementos neumáticos activos para el control de vibraciones” financed by “Junta de Comunidades de Castilla – La Mancha” (España).
[7] Aschemann H, Schindele D. Comparison of model–based approaches to the compensation of hysteresis in the force characteristic of pneumatic muscles. IEEE Trans Ind Electron 2014;61:3620–9. [8] Pujana-Arrese A, Mendizabal A, Arenas J. Modelling in modelica and position control of a 1–dof set–up powered by pneumatic muscles. Mechatronics 2010;20:535–52. [9] Barth EJ, Zhang J, Goldfarb M. Sliding mode approach to PWM–controlled pneumatic systems. In: Proceedings of the American control conference, Anchorage; 2002. p. 2362–7. [10] Messina A, Giannoccaro NI, Gentile A. Experimenting and modelling the dynamics of pneumatic actuators controlled by the pulse width modulation (PWM) technique. Mechatronics 2005;15:859–81. [11] Bone GM, Xue M, Flett J. Position control of hybrid pneumatic–electric actuators using discrete–valued model-predictive control. Mechatronics 2015;25:1–10. [12] Ljung L. System identification – theory for the user. Englewood Cliffs, NJ: Prentice Hall; 1987. [13] Bouri M, Thomasset D. Sliding control of an electro–pneumatic actuator using an integral switching surface. IEEE Trans Control Syst Technol 2001;9:368–75. [14] Parnichkun M, Ngaecharoenkul C. Kinematics control of a pneumatic system by hybrid fuzzy PID. Mechatronics 20 01;11:10 01–23. [15] Dupont PE, Dunlap EP. Friction modeling and PD compensation at very low velocities. J Dyn Syst Measur Control 1995;117:8–14. E. Palomares graduated as a mechanical engineer from the University of Castilla-La Mancha (Spain) in 2014 and he is currently working on his Ph.D. He also studied a Master in Engineering of Competition in Monlau Repsol Technical School in 2015. His current research interests include pneumatic suspensions for road vehicles, magnetorhelogical dampers and negative stiffness systems.
References [1] Datta R, Jain A, Bhattacharya B. A piezoelectric model based multi–objective optimization of robot gripper design. Struct Multidiscip Optim 2016;53(3):453–70. [2] Bhaumik S, Kumaraswamy A, Guruprasad S, Bhandari P. Investigation of friction in rectangular nitrile–butadiene rubber (NBR) hydraulic rod seals for defence applications. J Mech Sci Technol 2015;29(11):4793–9. [3] Nikas GK. Elasto–hydrodynamics and mechanics of rectangular elastomeric seals for reciprocating piston rods. ASME J Tribol 2003:60–9. [4] Harp SR, Salant RF. Analysis of mechanical seal behaviour during transient operation. ASME J Tribol 1998;120:191–7. [5] Damme MV, Beyl P, Vanderborght B, Ham RV, Vanderniepen I, Versluys R. Modeling hysteresis in pleated pneumatic artificial muscles. In: Proceedings IEEE international conference robotics and mechatronics; 2008. p. 471–6. [6] Vo-Minh T, Tjahjowidodo T, Ramon H, Brussel HV. A new approach to modeling hysteresis in a pneumatic artificial muscles using the maxwell–slip model. IEEE/ASME Trans Mechatron 2011;16:177–86.
A.J. Nieto graduated as a mechanical engineer from the University of Castilla-La Mancha (Spain) in 2003 and obtained his Ph.D. from the same university in 2008. He has been assistant professor with the University of CastillaLa Mancha (2005–2010), and he is currently an associate professor of Mechanical Engineering at the same university. His research interests are now focused on pneumatic suspensions for railway vehicles. He has published over thirty conference papers, thirteen articles in peer reviewed journals, and is author of one patent in Spain.
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E. Palomares et al. / Mechatronics 45 (2017) 37–48 A.L. Morales graduated as a mechanical engineer from the University of Castilla-La Mancha (Spain) in 2005 and obtained his Ph.D. from the same university in 2009. He has been assistant professor with the University of Castilla-La Mancha (2009–2012), and he is currently an associate professor of Mechanical Engineering at the same university. His research interests are focused on active control suspensions for road and railway vehicles. He has published eleven articles in peer reviewed journals, over thirty conference papers and is author of one patent in Spain.
J.M. Chicharro is currently associate professor of mechanical engineering the Universidad de Castilla-La Mancha (Ciudad Real, Spain). He was born in Madrid, Spain, in 1970. He received his Engineer degree in 1994 and the Ph.D. degree from Universidad Politcnica de Madrid (Spain) in 20 0 0. His research interests include Metrology, Mechatronics, Vibrations and Mechanical Design.
P. Pintado graduated as a mechanical engineer from the University of Seville (Spain) in 1986 and obtained his Ph.D. from the same university in 1989. He was a Fulbright Fellow in the Virginia Polytechnic Institute during two academic years (1989–1991). He has been associate professor with the University of Seville (1991–1998), and he is currently a full professor of mechanical engineering at the University of Castilla-La Mancha (Spain). His research interests are now focused on pneumatic suspensions for railway vehicles. He has published over a hundred articles (forty one of them in peer reviewed journals) and three books, is author of four patents in Spain, and one commercial computer application. He has worked as consultant for private corporations.
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Dynamic behaviour of pneumatic linear actuatorsR E. Palomares∗, A.J. Nieto, A.L. Morales, J.M. Chicharro, P. Pintado Department of Mechanical Engineering, University of Castilla – La Mancha Avda. Camilo José Cela s/n, 13071, Ciudad Real (Spain)
a r t i c l e
i n f o
Article history: Received 13 July 2016 Revised 10 May 2017 Accepted 11 May 2017
Keywords: Pneumatic linear actuator Nonlinear dynamics Hysteresis behaviour
a b s t r a c t The use of pneumatic linear actuators is generalised in engineering applications because of their many advantages, but modelling the force they supply may become more of a challenge due to their nonlinear behaviour and the hysteresis their energy losses cause. The authors propose a straightforward model to accurately predict force–displacement behaviour using as a basis experimental observations for several pressures and harmonic displacements of the rod. The model proposed includes two dissipative terms: one due to Coulomb friction and another due to structural damping. The force is proportional to relative pressure when acting as an actuator but nonlinear (modelled as a polytropic transformation) when acting as a pneumatic spring (with a closed pressurised chamber). The model accurately reproduces experimental results (Normalised Root Mean Square Errors lower than 2.5%) and may be used in control systems as well as in adaptive stiffness systems. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Fields such as smart structures, robotics, or mechatronics are nowadays focusing on obtaining increasingly accurate designs for their actuators in order to fulfil requirements of high precision, light weight, etc. Smart actuators such as piezoelectric materials, magnetostrictive materials, shape memory alloys or magnetorheological fluids are being developed in order to address these challenges [1], though conventional actuators can also be useful if suitably characterised. That is the case of pneumatic actuators which, despite having been invented in the mid–twentieth century, have awoken a renewed interest in mechatronic applications in the last two decades. Pneumatic Linear Actuators (PLA) are involved in a variety of industrial, automobile, aerospace, and marine applications. A typical PLA consists of a stationary cylindrical tube, a reciprocating piston and a rod. Airtightness is achieved by means of dynamic seals. These are usually made of thermoplastic materials and are able to undergo large deformations under a wide range of pressures, exerting contact pressure on the mating surfaces and avoiding the leakage of fluid. However, this component causes significant frictional forces and hence complicates the achievement of an accurate model for the PLA. Stribeck curves indicate that, for classical seals, friction at the seal–metal interface is directly proportional to speed and inversely proportional to load [2]. In addition, the R ∗
This paper was recommended for publication by associate editor Wei-Hsin Liao. Corresponding author. E-mail address: [email protected] (E. Palomares).
http://dx.doi.org/10.1016/j.mechatronics.2017.05.007 0957-4158/© 2017 Elsevier Ltd. All rights reserved.
sealing performance in terms of friction is dependent on the type of fluid, the geometry of the sealing interface, and the seal material. The case of rectangular seals under elastohydrodynamic lubrication conditions with uniform contact pressure distribution along the seal width has been studied by Nikas [3], and other studies have considered viscous heat generation for rotary seals [4]. Nevertheless, research is still weak on frictional or hysteretic aspects of pneumatic linear actuators. Van Damme et al. [5] proposed a Preisach model in order to better determine the hysteretic behaviour, but their results were only accurate for a narrow range of displacements. Other authors used a Maxwell–slip model to reproduce the force–displacement hysteresis behaviour [6], or even the well–known generalised Bouc–Wen model [7], which shows lower RMS tracking errors when compared to the quasi–static Maxwell–slip model and the Prandtl–Ishlinskii model. An empirical approach, which turned out to be more accurate than most analytical alternatives, was carried out by Pujana-Arrese et al. [8] by means of fitting a fourth degree polynomial in which the coefficients varied linearly with pressure. The importance of these models in pneumatic systems for different applications has been stressed by many authors. Barth et al. [9] present a PLA tracking sinusoidal inputs using a sliding control. Messina et al. [10] present mathematical dynamic models of positioner PLAs controlled by on–off solenoid valves. The nonlinear dynamics of this problem is due to its transient behaviour, but they achieved a mean positioning error of about 2 mm. Bone et al. [11] control the position of a novel hybrid pneumatic–electric actuator with inexpensive on–off solenoid valves. The addition of the DC motor connected in parallel is required to improve performance
38
E. Palomares et al. / Mechatronics 45 (2017) 37–48
modelling PLAs, but this strategy commonly leads to failure of the dynamic behaviour predictions [15]. Pneumatic linear actuators require a good hysteresis model which accurately simulates its force–displacement dynamic behaviour. In order to fill this gap, albeit partially, a nonlinear model for a pneumatic linear actuator is proposed in this study when used as a force actuator or as a pneumatic spring. It is simple, straightforward, easy to fit and useful for control purposes. 2. Characterisation of a pneumatic linear actuator A double acting pneumatic linear actuator can behave either as pneumatic spring (by closing the pressurised chamber) or as an actuator (by supplying the required pressure). In the former case, the nonlinear behaviour of the pressure within the closed chamber and the dissipative forces need to be studied; in the latter case, pressurising and depressurising times may be measured and used in control systems. Fig. 1 shows the double acting pneumatic linear actuator chosen to conduct the experimental studies. It is a commercial double–effect actuator with only one rod, model PRA/182050/M/40, manufactured by Norgren following the standard ISO 15552. The piston is 50 mm in diameter and the rod is 20 mm in diameter with a 40 mm stroke. Fig. 1 also shows how the actuator is attached to the hydraulic test machine. Both cylinder ends are attached to the hydraulic grips by means of spherical joints in order to avoid misalignment effects. The testing machine is equipped with a 10 kN full scale load cell and a LVDT with a calibrated displacement range of ±84 mm. Compressed air to feed the actuator is supplied by an air compressor equipped with a precision pressure–control valve. A valve is placed close to the chamber inlet port (the front chamber in the case of Fig. 1) so that the mass of air could remain constant during the test if required. 2.1. Characterisation as a pneumatic spring
Fig. 1. Pneumatic linear actuator characterised for the case study.
and errors. Other authors present linearised PLA models where the parameters are estimated for different amplitudes of the input signal [12]. Nevertheless, there are still aspects that require further studies, especially those related to the transient phenomena due to the discontinuous nature of the on–off input and its influence on the dynamic behaviour of the PLA. In fact, many authors assume negligible dry friction in their control laws, which leads to unavoidable errors [13,14]. Others linearise Coulomb friction when
a)
The dynamic force exerted by the PLA when the rod is subjected to different harmonic displacements d(t) and different initial pressures are set in either the front or the rear chamber, is characterised in this subsection. The pressurised chamber remains closed during the tests, so that the pressure (and hence the force) varies during the displacement cycle. The tests cover a combination of three displacement amplitudes D (3, 5, and 7 mm), five frequencies f (0.5, 1.0, 1.5, 2.0 and 4.0 Hz), and four initial gauge pressures P0 (1, 2, 3, and 4 bar) for each chamber (front and rear) of the PLA
b) 4
F(t)
F(t)
d(t)
Force, F(t)
Force, F(t)
d(t)
1
3
1
3
2
d-
0 Displacement, d(t)
2
d+
4
d-
0 Displacement, d(t)
Fig. 2. Nominal shape of curves of a PLA: a) Rear chamber pressurised b) Front chamber pressurised.
d+
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b)
1200 5· 10
5
1200 5· 10
Pa abs.
4· 10 Pa abs. 5 3· 10 Pa abs. 5 2· 10 Pa abs.
1000
1000
Pa abs.
800 Force [N]
Force [N]
800
5
4· 105 Pa abs. 5 3· 10 Pa abs. 5 2· 10 Pa abs.
5
600
600
400
400
200
200
0 -6
39
-4
-2
0
2
4
0 -6
6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 3. Force–displacement curves of a PLA varying initial value of pressure P0 , keeping constant both amplitude of displacement (5 mm) and frequency (1 Hz). a) Rear chamber; b) Front chamber.
a)
b)
650
650 4.0 Hz 2.0 Hz 1.0 Hz 0.5 Hz
550
450
Force [N]
Force [N]
550
350
250
150 -6
4.0 Hz 2.0 Hz 1.0 Hz 0.5 Hz
450
350
250
-4
-2
0
2
4
6
Displacement [mm]
150 -6
-4
-2
0
2
4
6
Displacement [mm]
Fig. 4. Force–displacement curves of a PLA varying the excitation frequency of harmonic displacement, keeping constant both amplitude of displacement (5 mm) and the initial value of pressure (3 · 105 Pa abs.). a) Rear chamber; b) Front chamber.
(a total of 120 tests). Tests start with the actuator initially placed at mid–stroke. Fig. 2 shows the qualitative behaviour of the force– displacement curves of the actuator obtained when either the rear (a) or the front (b) chamber is pressurised. The location of the rod for four specific positions is also shown schematically: the rod at mid–stroke (positions labelled 1 and 3 in Fig. 2, i.e., d (t ) = 0), its upper position (position labelled 2, maximum positive d(t)) and its lower position (position labelled 4, maximum negative d(t)). The pressurised chamber is shaded: the higher the pressure, the darker the chamber. The direction in which the hysteresis loop is traversed is also shown. The upper branch corresponds to compression whereas the lower branch corresponds to expansion. Fig. 3 shows the influence of the initial pressure in the force– displacement diagrams for a constant displacement of amplitude D = 5 mm and a constant frequency f = 1 Hz. It can be seen
that the force–displacement diagrams show a significantly nonlinear behaviour along with hysteresis caused by energy dissipation processes. This dissipation is probably due to some sort of friction which must be pressure–dependent. The nonlinear force behaviour is caused by the variation of pressure in the chamber, and this can be modelled as a polytropic transformation. Fig. 4 shows the effect of frequency on the force–displacement behaviour for the case in which the initial absolute pressure is 3 bar and the displacement amplitude is 5 mm. Negligible differences are observed between the curves plotted in Fig. 4, for frequencies from 0.5 to 4 Hz. Therefore, it might be concluded that energy dissipation is not caused by a viscous damping term. Fig. 5 shows the influence of the displacement amplitude D on the force–displacement diagrams for a specific absolute pressure (3 bar) and frequency (1 Hz). It can be concluded that hysteresis
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E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b)
700
700 7 mm 5 mm 3 mm
600
600
500 Force [N]
Force [N]
500
400
400
300
300
200
200
100 -8
7 mm 5 mm 3 mm
-6
-4
-2
0
2
4
6
100 -8
8
-6
Displacement [mm]
-4
-2
0
2
4
6
8
Displacement [mm]
a)
b)
7
7
6
6
5
5
Pressure [bar abs.]
Pressure [bar abs.]
Fig. 5. Force–displacement curves of a PLA varying displacement amplitude D, keeping constant both initial value of pressure (3 · 105 Pa abs.) and frequency (1 Hz). a) Rear chamber; b) Front chamber.
4 3 2
3 2 1
1 0
4
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
Time [s]
Fig. 6. Rear chamber pressurising (a) and depressurising (b) times for three different target and initial pressures.
FPLA (t)
Fnl
2.2. Characterisation as a pneumatic actuator
d(t)
Fs
Fc
Fig. 7. Model scheme.
grows when the displacement amplitude D increases, which could be attributed to some type of structural damping phenomena.
In a pneumatic actuator the force is proportional to the supplied relative pressure. In many cases, the settling time between two different levels of pressure is critical since it may significantly influence the behaviour of a control system. Fig. 6 shows the instantaneous pressure response in the rear chamber when different step commands are sent to the proportional pressure control valve. Overshooting occurs for high target pressures, but was not observed in cases of pressure reduction.
3. Model This section describes the proposed PLA model using as a basis the previous experimental observations. First, dissipative sources will be described since they are common to the spring and actu-
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b) 350
350 Numerical Experimental
300
Numerical Experimental
300
150 Force [N]
150 Force [N]
41
200
200 t=0
t=0
150
150
100
100
50 -6
-4
-2
0
2
4
50 -6
6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 8. Model validation: force–displacement curves for an initial pressure of 2 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
a)
b)
1100
1100 Numerical Experimental
1000 900
900 Force [N]
Force [N]
Numerical Experimental
1000
800 700
t=0
800 700 t=0
600
600
500
500
400 -6
-4
-2
0
2
4
6
400 -6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 9. Model validation: force–displacement curves for an initial pressure of 5 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
ator behaviour. Then, particular models of the dynamic force for these two cases will be described.
3.1. Dissipative forces In accordance with the results depicted in Fig. 3, hysteresis increases, although slightly, with the instantaneous pressure. This behaviour may be due to friction between the inner parts of the actuator and its seals, so the higher the pressure of the chamber, the higher the friction between the seals and the cylinder. This effect is slightly more pronounced when the front chamber is pressurised due to the added effect of friction in the piston seal and friction in the rod seal. This force can be modelled with the help of a Coulomb friction force Fc written as follows:
Fc (t ) = sign d˙ (t )
(P (t ) − Pa )
(1)
where P(t) is the instantaneous absolute pressure in the pressurised chamber, Pa is the ambient pressure, d(t) is the rod displacement and parameter is adjusted for each chamber. As shown in Fig. 5, the area inside a force–displacement curve seems to be directly proportional to the displacement amplitude D. This fact motivates the inclusion of a structural damping term in which the dissipative force Fs can be written as:
Fs (t ) =
δ
2π f
d˙ (t )
(2)
where parameter δ is also numerically determined for each chamber and f is the excitation frequency. Division by f makes Fs (t) independent of frequency for harmonic displacements. 3.2. Pneumatic spring model Since the air mass remains constant (in a closed pressurised chamber) while a displacement is imposed, the main contributor
42
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b) 500
500 Numerical Experimental
450
450
400
350
Force [N]
Force [N]
400
t=0
350 t=0
300
300
150
250
200 -4
Numerical Experimental
-3
-2
-1
0
1
2
3
200 -4
4
-3
-2
-1
Displacement [mm]
0
1
2
3
4
Displacement [mm]
Fig. 10. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 3 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
a)
b) 700
700 Numerical Experimental
600
600
500 Force [N]
500 Force [N]
Numerical Experimental
400
400 t=0
t=0
300
300
200
200
100 -8
-6
-4
-2
0
2
4
6
8
100 -8
-6
-4
-2
Displacement [mm]
0
2
4
6
8
Displacement [mm]
Fig. 11. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 7 mm and frequency of 1 Hz. a) Rear chamber; b) Front chamber.
to the force in the model is the variation of the pressure within the chamber. This variation may be assumed to correspond to a polytropic transformation, i.e.,
V n 0 P (t ) = P0 V (t )
(3)
where n is the polytropic index, V(t) and P(t) are the instantaneous volume and absolute pressure, respectively, and V0 and P0 the initial volume and absolute pressure, respectively. Note that some of these variables or parameters may be “chamber–dependent”, as it were. On the one hand, assuming that the piston of the PLA is located at mid–stroke, the initial volume of each chamber is given by the product of half the stroke (C) and the effective area of the pressurised chamber, whereas the instantaneous volume will also be dependent on the rod displacement d(t). On the other hand, the polytropic index is a parameter which will be determined by curve fitting and may result in different values for each chamber. Then, since the initial pressure P0 is known,
the instantaneous air pressure in the rear or front chamber (depending on which one was pressurised) can be written as follows:
n
P (t ) = P0
Ae
C 2
V0 + d (t )
= P0
n
P (t ) = P0
Ae
C 2
V0 − d (t )
C 2
Fp (t ) = [P (t ) − Pa ]Ae = P0
C 2 C 2
n
− Pa Ae
+ d (t )
when the rear chamber is pressurised, and
Fp (t ) = [P (t ) − Pa ]Ae = P0
C 2 C 2
(5)
− d (t )
and the force exerted by the PLA is
n
C 2 C 2
(4)
+ d (t )
= P0
n
C 2
− d (t )
n
(6)
− Pa Ae
(7)
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
b) 600
600 Numerical Experimental
Numerical Experimental
500
500
400
400
Force [N]
Force [N]
43
t=0
300
t=0
300
200
200
100 -6
-4
-2
0
2
4
100 -6
6
-4
Displacement [mm]
-2
0
2
4
6
Displacement [mm]
Fig. 12. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 0.5 Hz. a) Rear chamber; b) Front chamber.
a)
b) 600
600
Numerical Experimental
500
500
400
400
Force [N]
Force [N]
Numerical Experimental
t=0
300
t=0
300
200
200
100 -6
-4
-2
0
2
4
6
Displacement [mm]
100 -6
-4
-2
0
2
4
6
Displacement [mm]
Fig. 13. Model validation: force–displacement curves for an initial pressure of 3 · 105 Pa abs., displacement amplitude of 5 mm and frequency of 2 Hz. a) Rear chamber; b) Front chamber.
when the front chamber is pressurised. Note that this term of the force does not produce energy dissipation and, therefore, does not replicate the hysteresis observed in experiments. As a result, the total force FPLA (t) exerted by the PLA is made up of a main force term modelled as a polytropic transformation within the chambers of the actuator Fp , and the two dissipative forces previously described: one due to Coulomb friction Fc and another due to structural damping Fs (see Fig. 7). The expression of the resultant force is:
FPLA (t ) = Fp (t ) ± Fc (t ) ± Fs (t )
(8)
which can be written in detail as follows:
FPLA (t ) =
P0
n
C 2 C 2
± d (t )
± sign d˙ (t )
− Pa Ae
(P (t ) − Pa ) ±
δ
2π f
d˙ (t )
(9)
where the plus sign applies if the front chamber is pressurised and the minus sign if the rear chamber is pressurised. The parameters that must be adjusted in the proposed model are the polytropic index n, the effective area Ae , the parameter which defines the Coulomb friction and the parameter of structural damping δ . All of them may be “chamber–dependent”. A nonlinear least square fitting procedure is required to obtain accurate values of these parameters, but a suitable seed may be easily provided. The effective area can be simply obtained by dividing the applied force in several static tests by the gauge pressure in each case. The polytropic index n can be initially estimated by fitting the force–displacement curve with the sole contribution of Fp . Since the dissipated energy for structural damping is U = π δ D2 , the structural parameter δ can be estimated by measuring the hysteresis area in a test with low initial pressure in order to minimise the effect of the Coulomb friction. Finally, parameter is the most difficult to estimate since it depends on both the friction coeffi-
44
E. Palomares et al. / Mechatronics 45 (2017) 37–48
a)
c)
7
7 Experimental Numerical
6
5
Pressure [bar abs.]
Pressure [bar abs.]
6
4 3 2 1 0
Experimental Numerical
5 4 3 2 1
0
0.1
0.2
0.3
0.4
0
0.5
0
0.1
0.2
Time [s]
b)
0.5
7 Experimental Numerical
6
Experimental Numerical
6
5
Pressure [bar abs.]
Pressure [bar abs.]
0.4
d)
7
4 3 2
5 4 3 2 1
1 0
0.3 Time [s]
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
Time [s]
Fig. 14. Model validation for rear chamber pressurising and depressurising times for different target and initial pressures.
x(t) M
Table 1 Fitted model parameters for rear and front chambers. Chamber
K
C
n Ae [m2 ] (10−3 ) [m2 ] (10−5 ) δ [N/m] (103 )
PLA
y(t)
Front
Rear
1.123 1.541 2.143 3.509
1.063 1.795 2.136 2.963
Fig. 15. Single Degree of Freedom System with variable stiffnes system using a PLA.
3.3. Actuator model The force Fl applied by the PLA is proportional to the instantaneous relative pressure. That is: cient and the contact area of the deformed seal, but can be estimated by trial and error after deducting the effect of the previously estimated structural damping in tests with low displacement amplitudes. Table 1 gathers the values of these parameters for each chamber after the nonlinear least square fitting process.
Fl (t ) = (P (t ) − Pa )Ae
(10)
The dynamics shown in Fig. 6 when considering changes in the reference pressure Pref suggests a second–order behaviour of the
E. Palomares et al. / Mechatronics 45 (2017) 37–48 Table 2 Fitted model parameters for chamber charge and discharge.
ωn [rad/s] ξ
type:
Table 3 Normalised root mean square error (NRMSE) for different experimental tests.
rear
Charge
Discharge
P0 [bar abs.]
D [mm]
f [Hz]
Chamber
NRMSE [%]
52 1.00
52 0.69
2 3 3 3 3 5 2 3 3 3 3 5
5 3 5 5 7 5 5 3 5 5 7 5
1 1 0.5 2 1 1 1 1 0.5 2 1 1
Front Front Front Front Front Front Rear Rear Rear Rear Rear Rear
1.70 2.39 2.35 1.82 1.79 1.16 1.81 1.99 1.77 1.98 1.39 1.34
P¨ (t ) = −2ξ ωn P˙ (t ) − P˙ref − ωn2 (P (t ) − Pref )
(11)
where ωn and ξ are parameters which define overshooting and settling time. The total force FPLA (t) exerted by the PLA when the pressurised chamber is open must include the effect of the dissipative forces:
FPLA (t ) = Fl (t ) ± Fc (t ) ± Fs (t )
(12)
where the plus and minus signs correspond, again, to pressurised front or rear chamber. Thus, the instantaneous force FPLA (t) can be obtained by solving the following system of equations:
FPLA (t ) = (P (t ) − Pa )Ae ± sign d˙ (t )
δ ˙ d (t ) (P (t ) − Pa ) ± 2π f (13)
45
P¨ (t ) = −2ξ ωn P˙ (t ) − P˙ref − ωn2 (P (t ) − Pref )
the pressure to ambient pressure or to a pressurised state, respectively. Qualitative comparisons show a good agreement between experiments and simulations. Differences can be made out at the scale of the plots, but absolute differences are in the order of milliseconds. 5. Case studies 5.1. Use of a PLA as a variable stiffness pneumatic spring
(14)
In this case, the parameters that must be adjusted in the proposed model are the effective area Ae and the parameters ωn and ξ for the second–order system. The estimation of the effective area was discussed in the previous section and is presented in Table 1. The two latter parameters depend on both the sign and the value of the pressure variation. The fitted values are shown in Table 2. 4. Model validation 4.1. Pneumatic spring validation The validation of the model, via comparing the obtained results from the simulated model to the experimental tests, has been carried out for a wide range of pressures (from 2 to 5 bar abs.) and displacement amplitudes (from 3 to 7 mm). Since force– displacement curves were proven to be non–dependent on the excitation frequency (see Fig. 4), all the comparisons are made with f = 1 Hz. Figs. 8–13 show experimental and simulated results. Subplots (a) and (b) show the comparison in the force–displacement diagram for the rear and front chamber, respectively. Qualitative and quantitative comparisons show that simulations are in good agreement with experimental results. Although it is not visually noticeable, instantaneous peak errors are higher when the rod approaches the dead ends. The slope of the force–displacement curve tends to infinity at these singular points and, therefore, the vertical distance between two, otherwise close curves, increases. Nevertheless, the instantaneous error is always lower than 7% at any given location and, in all cases, the Normalised Root Mean Square Error (NRMSE) with respect to the peak–to–peak force amplitude is lower than 2.5% (Table 3). 4.2. Actuator validation Fig. 14 shows comparisons between experimental tests and model simulations for several cases of step commands. Subplots (a) and (b) show comparisons when increasing the pressure from ambient pressure or from an initial pressurised state, respectively; whereas subplots (c) and (d) show comparisons when decreasing
Fig. 15 shows a single degree of freedom system subjected to excitation at the base. This example is useful to understand how a PLA used as a pneumatic spring can modify the transmissibility function (ratio of the amplitude of the response to that of the base motion) of the system and analyse the influence of neglecting dissipative forces. The single degree of freedom system studied has a sprung mass M = 150 kg, a linear spring with stiffness K = 50 kN/m and a viscous damper characterised by a damping constant C = 10 0 0 Ns/m. A PLA (model PRA/182050/M/40) is attached in parallel to this suspension. The resulting suspension may be used without pressure, pressurising the front chamber or pressurising the rear chamber. By changing the pressure in the pressurised chamber one can modify the resonance frequency of the system, which can be useful in adaptive controls systems. Fig. 16 shows the transmissibility modulus of the described system for different cases. Considering the friction forces modelled in this paper, if the rear or the front chamber is pressurised with 5 bar abs. the resonance frequency increases up to 45%. Lower or higher frequencies may be achieved by setting the adequate pressure. Damping is lower in both cases but especially when pressuring the front chamber. Models neglecting friction forces show transmissibility functions peaks up to 20% higher, which would lead to large errors in dynamics behaviour predictions. 5.2. Control of a PLA Fig. 17 shows the control set–up where the PLA is attached to a hydraulic actuator which provides a displacement x(t) while the pressure in the rear chamber is controlled by a proportional pressure control valve. The objective is to control the pressure (using a standard PID controller) in order to maintain the force as close as possible to a variable target. Fig. 18 shows the PID control strategy. Fig. 19 shows comparison between experimental and simulated force records. The PLA is subjected to a 5 mm amplitude, 0.25 Hz frequency, displacement excitation. The force should remain stable at 200 N for 13 s, jump to 800 N and remain at that level for 20 s before dropping to 200 N. The figure shows the experimental force measurements with and without the proposed con-
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E. Palomares et al. / Mechatronics 45 (2017) 37–48
4.5 PLA (P=Pa ) PLA (P=Pa ) with friction forces PLA rear chamber (P=5 bar abs.) PLA rear chamber (P=5 bar abs.) with friction forces PLA front chamber (P=5 bar abs.) PLA front chamber (P=5 bar abs.) with friction forces
4.0
Transmissibility
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0
1
2
3
4
5
6
Frequency [Hz] Fig. 16. Transmissibility modulus of the Single Degree of Freedom System with variable stiffnes for different pressures, chambers and models.
Chassis USB NI CompactDAQ 9174 Analog input NI 9215
Load cell
Analog output NI 9263
Front chamber
PLA Rear chamber
P(t) Hydraulic actuator
x(t)
2
Proportional pressure control valve P
Load unit MTS 810 Hydraulic power unit
1
PC + Virtual Instrument LabVIEW
=
3
Air compressor
Fig. 17. Scheme of the control set–up.
Fref (t)
Error (t)
PID Controller
P(t)
PLA Model
Factual (t)
Factual (t)
Fig. 18. PID control scheme.
trol strategy, as well as the simulated controlled force record. Predicted force behaviour with the control strategy agrees with the experimental controlled force measurements. Both which replicate the target force more accurately than the uncontrolled case. 6. Conclusions Pneumatic linear actuators are used in many engineering applications and they show significant force–displacement hystere-
sis which may hinder model accuracy. Nevertheless, pneumatic actuator modelling has not received sufficient attention: authors either consider too simple models or carry out experimental controls which do not take into account the actuator model. This paper proposes accurate models for two different modes: one acting as a pneumatic spring, where the main force is modelled as a polytropic process within the pressurised chamber, and another acting as a variable force actuator. In both cases, a force due to Coulomb friction and a force due to structural damping
E. Palomares et al. / Mechatronics 45 (2017) 37–48
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1000 PID Simulated PID Controlled Uncontrolled
900 800
Force [N]
700 600 500 400 300 200 100 0 0
5
10
15
20
25
30
35
40
45
Time [s] Fig. 19. Comparison between the experimental force measurements (with and without the proposed control strategy) and the simulated controlled force record. The PLA is subjected to a 5 mm amplitude, 0.25 Hz frequency, displacement excitation while the force should remain stable at 200 N for 13 s, jump to 800 N and remain at that level for 20 s before dropping to 200 N.
must be considered. After suitably fitting the required parameters, not only does the model reproduce the force–displacement behaviour in many situations to a satisfactory degree (NRMSE lower than 2.5%), but it is also simple, straightforward, and permits its use in the design of control strategies. Two case studies have been presented to show the viability of using the models in control schemes.
Acknowledgements The authors are grateful for the support received from National Project TRA–2014–53552–R, “Mejora del confort del transporte ferroviario de alta velocidad mediante suspensiones neumáticas adaptativas y amortiguadores magnetoreológicos” financed by MINECO (España) and from Regional Project PEII–2014–034–P, “Análisis y diseño de elementos neumáticos activos para el control de vibraciones” financed by “Junta de Comunidades de Castilla – La Mancha” (España).
[7] Aschemann H, Schindele D. Comparison of model–based approaches to the compensation of hysteresis in the force characteristic of pneumatic muscles. IEEE Trans Ind Electron 2014;61:3620–9. [8] Pujana-Arrese A, Mendizabal A, Arenas J. Modelling in modelica and position control of a 1–dof set–up powered by pneumatic muscles. Mechatronics 2010;20:535–52. [9] Barth EJ, Zhang J, Goldfarb M. Sliding mode approach to PWM–controlled pneumatic systems. In: Proceedings of the American control conference, Anchorage; 2002. p. 2362–7. [10] Messina A, Giannoccaro NI, Gentile A. Experimenting and modelling the dynamics of pneumatic actuators controlled by the pulse width modulation (PWM) technique. Mechatronics 2005;15:859–81. [11] Bone GM, Xue M, Flett J. Position control of hybrid pneumatic–electric actuators using discrete–valued model-predictive control. Mechatronics 2015;25:1–10. [12] Ljung L. System identification – theory for the user. Englewood Cliffs, NJ: Prentice Hall; 1987. [13] Bouri M, Thomasset D. Sliding control of an electro–pneumatic actuator using an integral switching surface. IEEE Trans Control Syst Technol 2001;9:368–75. [14] Parnichkun M, Ngaecharoenkul C. Kinematics control of a pneumatic system by hybrid fuzzy PID. Mechatronics 20 01;11:10 01–23. [15] Dupont PE, Dunlap EP. Friction modeling and PD compensation at very low velocities. J Dyn Syst Measur Control 1995;117:8–14. E. Palomares graduated as a mechanical engineer from the University of Castilla-La Mancha (Spain) in 2014 and he is currently working on his Ph.D. He also studied a Master in Engineering of Competition in Monlau Repsol Technical School in 2015. His current research interests include pneumatic suspensions for road vehicles, magnetorhelogical dampers and negative stiffness systems.
References [1] Datta R, Jain A, Bhattacharya B. A piezoelectric model based multi–objective optimization of robot gripper design. Struct Multidiscip Optim 2016;53(3):453–70. [2] Bhaumik S, Kumaraswamy A, Guruprasad S, Bhandari P. Investigation of friction in rectangular nitrile–butadiene rubber (NBR) hydraulic rod seals for defence applications. J Mech Sci Technol 2015;29(11):4793–9. [3] Nikas GK. Elasto–hydrodynamics and mechanics of rectangular elastomeric seals for reciprocating piston rods. ASME J Tribol 2003:60–9. [4] Harp SR, Salant RF. Analysis of mechanical seal behaviour during transient operation. ASME J Tribol 1998;120:191–7. [5] Damme MV, Beyl P, Vanderborght B, Ham RV, Vanderniepen I, Versluys R. Modeling hysteresis in pleated pneumatic artificial muscles. In: Proceedings IEEE international conference robotics and mechatronics; 2008. p. 471–6. [6] Vo-Minh T, Tjahjowidodo T, Ramon H, Brussel HV. A new approach to modeling hysteresis in a pneumatic artificial muscles using the maxwell–slip model. IEEE/ASME Trans Mechatron 2011;16:177–86.
A.J. Nieto graduated as a mechanical engineer from the University of Castilla-La Mancha (Spain) in 2003 and obtained his Ph.D. from the same university in 2008. He has been assistant professor with the University of CastillaLa Mancha (2005–2010), and he is currently an associate professor of Mechanical Engineering at the same university. His research interests are now focused on pneumatic suspensions for railway vehicles. He has published over thirty conference papers, thirteen articles in peer reviewed journals, and is author of one patent in Spain.
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E. Palomares et al. / Mechatronics 45 (2017) 37–48 A.L. Morales graduated as a mechanical engineer from the University of Castilla-La Mancha (Spain) in 2005 and obtained his Ph.D. from the same university in 2009. He has been assistant professor with the University of Castilla-La Mancha (2009–2012), and he is currently an associate professor of Mechanical Engineering at the same university. His research interests are focused on active control suspensions for road and railway vehicles. He has published eleven articles in peer reviewed journals, over thirty conference papers and is author of one patent in Spain.
J.M. Chicharro is currently associate professor of mechanical engineering the Universidad de Castilla-La Mancha (Ciudad Real, Spain). He was born in Madrid, Spain, in 1970. He received his Engineer degree in 1994 and the Ph.D. degree from Universidad Politcnica de Madrid (Spain) in 20 0 0. His research interests include Metrology, Mechatronics, Vibrations and Mechanical Design.
P. Pintado graduated as a mechanical engineer from the University of Seville (Spain) in 1986 and obtained his Ph.D. from the same university in 1989. He was a Fulbright Fellow in the Virginia Polytechnic Institute during two academic years (1989–1991). He has been associate professor with the University of Seville (1991–1998), and he is currently a full professor of mechanical engineering at the University of Castilla-La Mancha (Spain). His research interests are now focused on pneumatic suspensions for railway vehicles. He has published over a hundred articles (forty one of them in peer reviewed journals) and three books, is author of four patents in Spain, and one commercial computer application. He has worked as consultant for private corporations.