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Modeling and Control of an Electrorheological Actuator
Copyright © IFAC Mechatronic Systems, Sydney, Australia, 2004
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MODELING AND CONTROL OF AN ELECTRORHEOLOGICAL ACTUATOR
Wolfgang Kemmetmiiller and Andreas Kugi
Chair of System Theory and Automatic Control. Saarland University. PO Box 15 11 50, 66041 Saarbriicken. Germany. E-Mail: 1I.olfgang.kemmetmlleller@lst:lIni-saarland.de. andreas. [email protected]
Abstract: This paper deals with the modeling and nonlinear control of an ER (electrorheological) actuator consisting of a double-rod cylinder and four ER valves in a full-bridge configuration . Basically, we have to face two difficulties within the controller design : First of all , the ER effect is inherently nonlinear and secondly, the ER full-bridge provides more control inputs than necessary for solving the primary control task. We will show that these additional degrees-of-freedom can be exploited to circumvent undesirable operation and to optimize the overall closed-loop performance. Furthermore, the nonlinearities of the mathematical model will be systematically included in the controller design . Measurement results performed on an experimental test-stand will demonstrate the feasibility of the proposed strategy. Copyright © 2004 [FAC Keywords: ER fluid, ER valve, hydraulic actuator, nonlinear control
et aI., 2000) and the references cited therein . This
I. INTRODUCTION An electrorheological fluid (ER fluid) is in general a suspension (solid particles in a fluid phase) that undergoes dramatic changes in the apparent viscosity and other material properties when subjected to sufficiently strong electric fields. The ER effect is known to be essentially reversible and it exhibits a very rapid response time upon application or removal of electric fields in the order of a few milliseconds, see, e.g., (Whittle et aI. , 1996). Although the exact mechanisms of ER fluids on particle scale are not completely understood, at least two facts are generally accepted (Eckart, 2000): (I) The particles in the fluid form chains when exposed to an electric field, and (2) these chains are responsible for the increase in the apparent viscosity. The possibility to change the rheological properties of a moving ER fluid very quickly by means of electric fields offers the potential to design completely new active and semi-active actuators. Thus in the last years ER fluids have attracted a wide range of industrial applications, as for instance dampers, shock absobers, clutches or servo drives, see, e.g., (Fees, 2001), (Gavin, 2001 ), (Hoppe
inner electrodes of ER valve cylinder block (outer electrode)
sealings
piston
Fig. I . ER actuator (FLUDICON, 200 I). paper is concerned with the modeling and control of a prototype ER piston actuator manufactured by FL UDICON GMBH (FLUDICON, 2001). Fig. I depicts a schematic view of the ER actuator. Essentially, the ER
265
actuator is composed of a double-rod cylinder located in the cross path of a full-bridge consisting offour ER valves, see Fig. 2 for a schematic diagram. Roughly speaking, an ER valve comprises two concentric cylindrical electrodes forming an annular channel where the ER fluid is passing through. Applying a voltage to the inner cyl indrical electrode, with the outer electrode being earthed, an electric field perpendicular to the direction of the flow is generated. If a constant flow of ER fluid is maintained along the ER valve, then an increase in the voltage (in the electric field) causes an increase in the apparent viscosity and hence also in the pressure drop across the valve. Thus, the resistance of the ER fluid to flow through the valve can be controlled by the voltage. This way of using the ER effect is also known as the flow- or valve-operation mode. Due to the rapid response times of the ER effect this type of actuator promises to outperform conventional hydraulic servo-drives with electro-magnetic mechanisms in some fields of operation. in particular in those with high demands on the dynamic performance. Compared to conventional hydraulic piston actuators with a three-land four-way servo valve, where in general the servo current acts as the control input, the ER actuator of Fig. I has four independent control inputs, i.e. the voltages applied to the four ER valves. At first sight it seems to be useful to control the ER valves in the full-bridge in such a way that the behavior of a conventional servo-valve is emulated. However, it turns out that this procedure is only successful up to a cel1ain point. Because of the intrinsic nonlinear nature of the ER effect, the full capability of the ER actuator can only be exploited by taking into account the essential nonlinearities in the controller design. Furthermore, the degrees-of-freedom in form of the four independent control inputs must be utilized in order to optimize the performance of the closed-loop system. As already mentioned in (Spencer, 1996) and (Butz and von Stryk, 2002) the development of suitable (non linear and optimal) control strategies for ER devices is one of the scientific main challenges to press ahead the practical utilization of ER fluids.
• the full-bridge comprising four ER valves with the volume flows qi, i E {a, b, c, d}, • the double-rod cylinder, • the pressure supply consisting of a gear pump with a constant volume flow qo, an adjustable pressure relief valve and a hydraulic accumulator with the pressure Pace and the volume flow qacc, and • the high-voltage amplifiers with the voltages Ui , i E {a, b, c, d}, as the control inputs.
4"
Fig. 2. Schematic diagram of the experimental set-up . FurthemlOre, the experimental set-up is equipped with three pressure transducers for the supply pressure Ps and the chamber pressures Pl and P2; a position sensor for the piston position x; a force sensor for the load force 7/ acting on the piston; and a temperature sensor for measuring the fluid temperature
e,
3. MATHEMATICAL MODEL
3.1 ER Valve Many works dealt with the phenomenological modeling of an ER fluid . The first model for the description of the macroscopic behavior of an ER fluid was proposed in (Klass and Martink, 1967). It is the linear Bingham model with only two parameters, the field dependent yield stress and the viscosity. Even though the model is restricted to cases where the flow is orthogonal to the electric field, it is still used to describe the behavior of an ER fl uid in many cases. A phenomenological modeling in the context of continuum mechanics started with the paper (Rajagopal and Wineman, 1992) where a general model for the Cauchy stress tensor including the electric field was proposed. Based on these results a very systematic mathematical formulation of the governing equations of electrorheology considering a three-dimensional nonlinear constitutive equation for the Cauchy stress tensor can be found e,g, in (Rilzicka, 2000) and (Eckart, 2000). However, for a specific choice of the material parameters these models include the classical Bingham model of (Klass and Martink, 1967). Apart from this continuum mechanics approach several parametric models which rely on experimental results are proposed in the literature, see, e,g., (Butz and von Stryk, 2002) .
The paper is organized as follows : In Section 2 a short description of the experimental set-up unde(consideration is given . Section 3 is devoted to the derivation of a mathematical model of the ER actuator. Based on this mathematical model, in Section 4, a nonlinear control concept is proposed which fully exploits the degrees-of-freedom provided by the ER actuator. Some measurement results are presented in Section 5. Finally, Section 6 contains some conclusions and gives a short outlook.
2. EXPERIMENTAL SET-UP Fig. 2 presents a schematic diagram of the experimental set-up. The overall system can be subdivided into the following subsystems:
266
For the subsequent considerations let us consider the 3-dimensional Euclidean space with Cartesian coordinates Xj, j = 1,2,3, and the canonical orthogonal basis {el,e2,e:J}, which meets (ei,ej) = Oij for the standard inner product (-, -). In order to keep the formulas short and readable, we will use Einstein's convention for sums by summing over double repeated indices from 1 to 3. Furthermore, we will use the notation 8 j = {j~j' j = 1,2,3, and 8 0 = In this 3
8 OXj
all a22 a12
---- ~~.~- .-
"",
..
04
'2
= -4> + 4'1', 2
04
a33 2
= -4>, 0 6
= -4> + 02E2 + 4'1' + 21'
1 ( 03 + 0 5 E 22) 1', =2
a13
2
2
E 2,
(4)
= a23 = 0 ,
p80 U l = 8 l a l l + O;W12 () = 8[a12 + ihan o = 83a33.
(5)
Combining (4) with the last two equations in (5), we can derive an explicit expression for the spherical stress in the form
4> =
-_._- ._.
-PXl
04 2 06 2 2 + 0 2 E22 + -1' + -1' E2 4 2
(6)
where P = (Pin - Pout) / L denotes the pressure gradient, with Pin as the inlet and Pout as the outlet pressure of the ER valve. With this the first equation in (5) reads as
P,..., - - Il-
x,
(3)
with the shear rate l' = 82Ul. Since the external body forces are assumed to be zero and the electric field E is independent of Xj, j = 1,2,3, the balance of momentum yields p (80Ui + UjajUi) = ajaij or
aj'
As already mentioned before the ER valve is an annular channel being composed of two cylindrical electrodes with an inner radius Ht, an outer radius Ro and length L. Since the channel gap H = Ro - Ht (in our case 0.6 mm) is small compared to the mean radius !4n = (Ht + Ro) /2 (in our case 4.8 mm), the ER valve can be modeled as a pair of parallel plates with length L and width W = 2!4n1r. Fig. 3 depicts a schematic representation of the flow through the ER valve. Taking the temperature of the fluid 6, L
E=~(t)e2'
Under these restrictions the components of the (symmetric) Cauchy stress tensor (I) simplifY to
It.
sense the expression 8j aj stands for Lj=l
and
U=Ul(t,X2)el
l
(7) By choosing the material parameters 0.3 = 2TJ, with TJ as the constant dynamic viscosity at zero field, and 05 = 2To (E2) / (Ei 11'1), with the field dependent yield stress TO (E2 ), we obtain the "Binghamlike" constitutive equation according to (Rajagopal and Wineman, 1992)
fig. 3. Longitudinal section of the ER valve. the density p, the components of the electric field Ej and the components of the stretching tensor Dij = (8i Uj + 8 j U i), i, j = 1,2,3, where U denotes the velocity field of the fluid, as independent variables, we may write the constitutive equation ofthe isotropic ER fluid by means of the Cauchy stress tensor a in the form, see (Rajagopal and Wineman, 1992), (RilZicka, 2000), (Eckart, 2000)
1
aij
= TO (~) sign (')') + TJ1' lad < TO (E2), the case
a12
l' ~o().
(8)
For when l' = 0, the material behaves like an elastic solid, i.e. the shear stress is given by a12 = GC12, where G is the shear modulus and C12 denotes the strain. If we assume pointwise interaction of the dipoles of the ER fluid then the yield stress T O (E2 ) is proportional to In the case the dipole interaction reaches some kind of saturation at a certain field strength, TU (E2) only increases linearly with E 2 . Measurements have shown that a constitutive relation of the form
= 010ij + 02EiEj + 03Dij + 04DikDkj+
El.
05 (EiDjkEk + DikEkEj ) + 06 (EiDjkDklEI + DikDklEIEj) .
(I) Thereby, the material parameters Om, m = 1, ... ,6, are scalar functions of the invariants 1= {6,p,EjEj,Djj,DjkDkj,DjkDkIDlj, EjDjkEk, EjDjkDklEt} .
for
(9)
(2)
with suitable constant parameters a2 and a3 can be used to approximate the yield stress in the interesting area of operation.
Suppose that the temperature 6 is constant and the ER fluid in the valve is incompressible. Then, the continuity equation imposes a constraint on the velocity field in the form 8 j uj = D jj = O. From this we may deduce that 01 = -4> (Xl, X2, X3), where 4> is the indeterminate part of the stress tensor a also known as the spherical stress (Rajagopal and Wineman, 1992), (Ruzicka, 2000). The geometry of the ER valve as presented in Fig. 3 and the fact that the flow through the annular channel of the ER valve remains laminar give rise to the assumption that the velocity field U and the electric field E can be expressed in the form
In the literature it is documented that the dynamics of the ER effect, i.e. the formation of the chains in the fluid upon a step change in the voltage applied to the electrodes. are in the range of a few milliseconds. Furthermore, it can be shown that in the worst case, when the dynamic viscosity TJ takes its smallest value at zero field , the influence of the fluid inertia on the speed of response is in the same dimension of \-2 milliseconds, see, e.g., (Whittle et aI., 1996). This is
267
why we will henceforth neglect the dynamics of the ER valve for the purpose of a controller design. Next we will calculate the analytical solution of the steadystate velocity field within the gap of the ER valve, see Fig. 3. For an isothermal laminar flow of an incompressible Bingham plastic between a parallel plates gap, plug flow occurs in the center of the channel, i.e. O'2UI = 0 for H,,/ $ X2 $ H - H,,/, with H,,/ depending on the electric field E2 and the pressure gradient P . As we will see in a moment, outside the plug zone the velocity field is parabolic satisfYing the no-slip boundary conditions on the walls of the electrodes, i.e. Uj (0) = 0 and UI (H) = O. Thus, the operation of the ER valve is based on changing the height of the plug zone by the electric field or the voltage applied to the electrodes, respectively. An increase in the electric field beyond a certain value causes the plug zone to reach the value H,,/ = 1J- and then, no fluid can pass through the ER valve anymore. [n this case the valve is closed.
cl dtP2
(0)
In the isentropic case the pressure Pacc in the hydraulic accumulator obeys the differential equation d dt Pacc =
(X2)
P = -:;;X2
(
H-y -
W (TO (~)
( 12)
4. CONTROL CONCEPT
12p21]
(13) (Eh) according
The primary objective of the controller design is to track a desired trajectory Xd for the piston position x . From the mathematical model (13), (14) it can be directly seen that the ER actuator in the full-bridge configuration makes available four control inputs in form of the four voltages Ui , i E {a, b, c, d} , applied to the ER valves. At first sight it seems that the ER actuator provides more degrees-of-freedom for the controller design than necessary, in particular if we think of conventional hydraulic servo-drives, where we only have one control input. Thus, we will subsequently present a control concept I which exploits these additional degrees-of-freedom. Obviously, the equilibrium point of the overall system (13) - (16)
In contrast to the ER valve the compressibility of the fluid will be taken into account in the cylinder chambers by the bulk modulus (3 = p~, with the density p. In the absence of leakage flows the mathematical model of the double-rod cylinder takes the form
=
v
m-v
=
A (PI-P2)-TI
cl dt P1
=
VI
cl
clt
((3 - /'i,Pacc)
where Vs comprises the volume of the connection block and the volume of the pipes from the gear pump to the accumulator. The high-voltage amplifiers being used are considerably fast (response time approximately 100 p,s) and thus can be modeled in form of a static gain and a rate limiter counting for the maximum possible current.
3.2 Duuble-Rod Cylinder
cl
K
+ PH) (-2To (E2 ) + PH)2
with the field dependent yield stress TO to (9) and Eh = UIH.
- x c1t
.1
as qacc = CdAo/f, lips - Pacc lsign (ps - Pacc ), with the supply pressure Ps, the discharge coefficient Cd and the orifice area Aa. [n normal operation the adjustable pressure relief valve in Fig. 2 is closed and can therefore be neglected for the controller design. The continuity equation for the supply pressure reads as cl (3 cltPS = Vs (qO - qacc - qa - qc), (16)
Integration over the gap yields the stationary volume flow q through the ER valve
q=
+ (pacc<)
( 15) with the isentropic coefficient /'i" the effective accumulator volume Vacc and a constant c depending on the pre-charge and initial conditions of the gas in the accumulator. The volume flow through the orifice reads
(10)
x.)) . 2'
(3PaccqaccK KVaccPacc
= 0, (02utl (H-y) = 0, given by Ul
qd,
3.3 Pressure Supply and High- Voltage Amplifiers
Since the velocity profile is symmetric with respect to the center of the channel we just consider the flow zone between the inner electrode and the plug 0 $ X2 < H-y, the region where'Y > O. Combining (7) and (8) the steady-state velocity profile is the solution of the ODE (11) P + 1]O~Ul = 0, Uj
= qc -
(14)
In the steady state, i.e. OOUI = 0, we get from together with the symmetric boundary conditions 0"12 (H,,/) = TO and 0" 12 (H - H-y) = -TO the following expression for H,,/
with
q2
where x and v denote the piston position and velocity, PI and P2 are the chamber pressures, T[ denotes the external load force containing the friction force and qi, i E {a, b, C, d}, are the volume flows through the ER valves according to (13). Furthermore, m denotes the piston mass and all masses rigidly connected to the piston, A is the effective piston area and VI and V2 stand for the chamber volumes for x = O.
(7)
H _ H _ TO (E2) -y 2 P'
(3
= \I.i _ Ax (Av + q2),
(3
+ Ax (-Av + qtl ,
ql
= qa - qb
1 patent pending
268
is not unique. Considering the symmetry of the fullbridge, we determine the equilibrium point for Tt = 0 in the form x = 0, v = 0, PI = P2 = P,f, Pace = Ps and iji = ~, i E {a,b,c,d}. From this we may calculate the voltages Ui, i E {a,b,c,d}, from (13) with q = ~ and P = ~. However, since the value of Ps is still undetermined by this choice, we will fix PS within the admissible range above the pre-charge pressure of the accumulator.
clearly proves that the the overall closed-loop system of the cascaded controller is exponentially stable. Up to now we have designed a controller based on the control inputs ql and q2 in (14). In the next step, we have to divide ql and q2 into the flows through the ER valves, qa, qb and qc, qd, respectively. These remaining degrees-of-freedom will be utilized to take into account the following considerations: (A) In the range of vanishing volume flows the ER valve shows an undesirable hysteretic behavior. Therefore, the control strategy should circumvent a complete closing of the valves . (8) The volume flows ql and q2 needed for tracking fast trajectories are much higher than the constant volume flow qo provided by the gear pump. Thus it is clear that these volume flows can only be delivered by the hydraulic accumulator. Furthermore, in order to ensure the operation of the ER actuator the mean value of the supply pressure, denoted by Ps, has to be kept within certain limits 2 , i.e. PS,min < Ps < PS,max ' Therefore, we must include a supply pressure controller in the control concept which does not control the pressure peaks caused by the fast position trajectories but forces the mean value Ps to stay near a prescribed value PS,d. The requirements (A) and (8) give rise to the following partitioning of the volume flows qo + qs qa = sg( qd + - 2 -
The control concept for tracking a desired piston position Xd is based on a cascaded structure, with an inner control loop for the chamber pressures and an outer control loop for the piston position. Let us consider ql and q2 in (14) as the control inputs. Then the feedback law ql
Vi + Ax = Av + . f3 (-c5p (PI -
q2
= -Av +
V2 -Ax
f3
PI,d)
.
+ PI ,d)
(-c5 p (P2 - P2,d) + P2,d)
(17) yields an exponentially stable linear error dynamics for the tracking behavior of the piston pressures for every c5 p > 0, i.e. d
= -c5 pep;,
ep;
= Pj
- pj,d ,
(18)
where the desired trajectories pj,d are assumed to be sufficiently smooth. For the outer position control loop the desired piston force T d = A (PI ,d - P2,d) serves as a new control input. Since we want to include an integral action in the position tracking controller we choose the control law for T d in the form Td
= m (Xd - c5 x,2 ex -
c5 x ,le x
-
ox,o
qb
Ps =2 +
Td
2A
and
P2,d
=
ps
J
Td
+ -qo2+- qS (22)
+ qo ~ qs qo + qs
qd = sg (-q2)
exdt) ,
2 - 2A '
(-qd
qc = sg (q2 )
+ -2-'
with the function sg (q) = q for q 2: 0 and sg (q) = 0 otherwise and a volume flow qS which will serve as the control input for the supply pressure controller. From the definition of ql and q2, cf. (14), it can be seen that qS has no influence on the position controller (17) - (20). Due to space limitations we will not elaborate the details of the supply pressure controller, we rather try to point out the underlying idea. However, a rigorous proof of the stability is based on the averaging theory in combination with an energy-based formulation of the accumulator and will be published elsewhere. Substituting (22) into (16), we get
(19) with the sufficiently smooth reference trajectory Xd and the position error e", = x - Xd. Furthermore, the controller parameters 15",,; > 0, j = 0, 1,2, are chosen to render the position error dynamics exponentially stable. Note that (19) does not uniquely determine PI ,d and P2,d. Therefore, in view of the symmetric structure of the double-rod cylinder in the ER fullbridge configuration we set P I ,d
= si!;
(20)
d dtPS
At this point it is worth mentioning that the cascaded
controller as being proposed is fully compatible with the system flatness, see, e.g., (Fliess et al, 1995) for more details. Thus, if we take the piston position x as the so-called flat output, the controller (17), (19), (20) entails a linear error dynamics for the piston position given by
=
f3
-Vs (qacc + qS
+ Si!; (ql) + sg (q2»
.
(23) Let us assume that the position reference trajectory Xd is T-periodic, i.e. Xd (t ) = Xd (t + T). Then under certain non-restrictive assumptions it can be shown that the right-hand side of (23) is almost T-periodic, with a slowly varying strictly decreasing mean value. The supply pressure controller is chosen in the form
e~4 )
+ (c5p + 15",,2) 'e x + (opc5 x ,2 + c5 x ,d ex + (21) (c5 p c5 x ,1 + c5 x ,o ) ex + c5pc5 x ,oe x = O.
qs = -os (PS,d - ps) ,
Furthermore, the characteristic polynomial of(21) can be factorized in the form (A3 + c5;,2A2 + c5 x ,IA + 8",,0) (A + c5 p ) . This result combined with (18) and (19)
Os
> 0,
(24)
2 Roughly spenking. the lower limit is determined by the precharge pressure of the nccumulator and the upper Iimit is given by the maximum possibl e pressure ditrerence of the ER valve.
269
where Ps is the moving average value of the supply pressure Ps given by
ps =
the mathematical models of the various components of the ER actuator were derived, where special emphasis was put on a concise modeling of the ER valve. Based on this model a nonlinear control concept was designed which optimally utilizes the potentiality of the ER actuator. The control concept was implemented on a test-stand and the first measurement results prove its usefulness and efficiency. However, due to certain restrictions in the construction of the present test-stand we were not able to exploit the full dynamics of the ER effect. Nonetheless, we can at least reach dynamic responses in the range of comparable conventional hydraulic systems. Currently, we are setting up a new test-stand and with this we are confident to further increase the dynamics of the closed-loop ER actuator.
rt+Ts
.Jt
Ps (T) dT
with
Ts » T .
(25)
The voltages Ui , i E {a, b, c, d}, can be directly calculated by an analytic inversion of (13) with qi from (22). 5. MEASUREMENTS The control concept was implemented on the experimental set-up as described in Section 2 by means of the real-time hardware DS I 103 from the company oSPACE in combination with MATLAB/SIMULlNK and was tested for different reference trajectories Xd with a sampling time of 200 p,s. The measured responses to a faster rectangular-like reference trajectory with 5 mm amplitude and a slow sinusoidal reference trajectory with an amplitude of 20 mm are shown in Fig.4 and Fig. 5, respectively.
7. ACKNOWLEDGEMENT The authors would like to thank FLUOI CO N GMBH in particular Dr. R. Adenstedt, Dr. H. Rosenfeldt and Dipl.-Ing. M. Stork for the fruitful cooperation. 8. REFERENCES
~
Butz, T. and O. von Stryk (2002). Modelling and simulation of electro- and magnetorheological fluid dampers. ZAMM 82, 3-20. Eckart, w. (2000). Phenomenological modelling of electrorheological fluids with an extended Casson-model. Journal o/Continuum Mechanics and Thermodynamics 12, 341-362. Fees, G. (200 I). Statische und dynamische Eigenschaften eines hochdynamischen ER-Servoantriebes. Olhydraulik und Pnellmatik 45, 45-48. Fliess, M., J. Levine, P. Martin and P. Rouchon (1995). Flatness and defect of non-linear systems: introductory theory and examples. IEEE Trans. Automatic Control 61 , 1327-1361 . FLUDlCON (2001). Rheact. Technical Product Description 1,1 - 13. Gavin, H.P. (2001). Annular Poiseuille flow of ER and MR materials. J ~r Rheology 45(4),983- 994. Hoppe, R.H.W., G. Mazurkevitch, U. Rettig and O. von Stryk (2000) . Modeling, simulation and control of e1ectrorheological fluid devices. In : Lectures on Applied Mathematics. pp. 251-276. Springer. Berlin. Klass, D.L. and T. W. Martink (1967). Electroviscous fluids I: Rheological properties . Journal of Applied Physics 38( 1), 67-74. Rajagopal, K.R. and A.S. Wineman (1992). Flow of eIectrorheological materials. Acta Mechanica 91,57-75 . Ruzicka, M. (2000). Electrorheological Fluids: Modeling and Mathematical Theory. Springer. Berlin. Spencer, B.F. (1996) . Recent trends in vibration control in the U.s.A .. In: Proc. 3rd Int. Con! Motion Vibration Control. Vo\. 2. pp. K I- K6 . Japan . Whittle, M., R.J . Atkin and W.A. Bullough (1996). Dynamics of an electrorheological valve. Int. Journal ~f Modern Physics B 10,2933-2950.
3
oS c
2
l
2
005
0. 1
0.15
0.2
0.25
03
0.35
0.4
0.45
0.5
tins
Fig. 4. Measured response x for a rectangular-like reference trajectory Xd . 20
E E .S c
.g 'in
o
0.
·5
·'0
-15 :
·20
.---0.2
0 .4
0.6
0.8
.•... - ..
1
~--
1.2
1.4
'.6
1.8
tin s
Fig. 5. Measured response x for a sinusoidal reference trajectory Xd. 6. CONCLUSION AND OUTLOOK
In this paper, we presented the modeling and nonlinear control of an electrorheological actuator. In a first step,
270
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IFAC PUBLICATIONS
www.elsevier.comllocatelifac
MODELING AND CONTROL OF AN ELECTRORHEOLOGICAL ACTUATOR
Wolfgang Kemmetmiiller and Andreas Kugi
Chair of System Theory and Automatic Control. Saarland University. PO Box 15 11 50, 66041 Saarbriicken. Germany. E-Mail: 1I.olfgang.kemmetmlleller@lst:lIni-saarland.de. andreas. [email protected]
Abstract: This paper deals with the modeling and nonlinear control of an ER (electrorheological) actuator consisting of a double-rod cylinder and four ER valves in a full-bridge configuration . Basically, we have to face two difficulties within the controller design : First of all , the ER effect is inherently nonlinear and secondly, the ER full-bridge provides more control inputs than necessary for solving the primary control task. We will show that these additional degrees-of-freedom can be exploited to circumvent undesirable operation and to optimize the overall closed-loop performance. Furthermore, the nonlinearities of the mathematical model will be systematically included in the controller design . Measurement results performed on an experimental test-stand will demonstrate the feasibility of the proposed strategy. Copyright © 2004 [FAC Keywords: ER fluid, ER valve, hydraulic actuator, nonlinear control
et aI., 2000) and the references cited therein . This
I. INTRODUCTION An electrorheological fluid (ER fluid) is in general a suspension (solid particles in a fluid phase) that undergoes dramatic changes in the apparent viscosity and other material properties when subjected to sufficiently strong electric fields. The ER effect is known to be essentially reversible and it exhibits a very rapid response time upon application or removal of electric fields in the order of a few milliseconds, see, e.g., (Whittle et aI. , 1996). Although the exact mechanisms of ER fluids on particle scale are not completely understood, at least two facts are generally accepted (Eckart, 2000): (I) The particles in the fluid form chains when exposed to an electric field, and (2) these chains are responsible for the increase in the apparent viscosity. The possibility to change the rheological properties of a moving ER fluid very quickly by means of electric fields offers the potential to design completely new active and semi-active actuators. Thus in the last years ER fluids have attracted a wide range of industrial applications, as for instance dampers, shock absobers, clutches or servo drives, see, e.g., (Fees, 2001), (Gavin, 2001 ), (Hoppe
inner electrodes of ER valve cylinder block (outer electrode)
sealings
piston
Fig. I . ER actuator (FLUDICON, 200 I). paper is concerned with the modeling and control of a prototype ER piston actuator manufactured by FL UDICON GMBH (FLUDICON, 2001). Fig. I depicts a schematic view of the ER actuator. Essentially, the ER
265
actuator is composed of a double-rod cylinder located in the cross path of a full-bridge consisting offour ER valves, see Fig. 2 for a schematic diagram. Roughly speaking, an ER valve comprises two concentric cylindrical electrodes forming an annular channel where the ER fluid is passing through. Applying a voltage to the inner cyl indrical electrode, with the outer electrode being earthed, an electric field perpendicular to the direction of the flow is generated. If a constant flow of ER fluid is maintained along the ER valve, then an increase in the voltage (in the electric field) causes an increase in the apparent viscosity and hence also in the pressure drop across the valve. Thus, the resistance of the ER fluid to flow through the valve can be controlled by the voltage. This way of using the ER effect is also known as the flow- or valve-operation mode. Due to the rapid response times of the ER effect this type of actuator promises to outperform conventional hydraulic servo-drives with electro-magnetic mechanisms in some fields of operation. in particular in those with high demands on the dynamic performance. Compared to conventional hydraulic piston actuators with a three-land four-way servo valve, where in general the servo current acts as the control input, the ER actuator of Fig. I has four independent control inputs, i.e. the voltages applied to the four ER valves. At first sight it seems to be useful to control the ER valves in the full-bridge in such a way that the behavior of a conventional servo-valve is emulated. However, it turns out that this procedure is only successful up to a cel1ain point. Because of the intrinsic nonlinear nature of the ER effect, the full capability of the ER actuator can only be exploited by taking into account the essential nonlinearities in the controller design. Furthermore, the degrees-of-freedom in form of the four independent control inputs must be utilized in order to optimize the performance of the closed-loop system. As already mentioned in (Spencer, 1996) and (Butz and von Stryk, 2002) the development of suitable (non linear and optimal) control strategies for ER devices is one of the scientific main challenges to press ahead the practical utilization of ER fluids.
• the full-bridge comprising four ER valves with the volume flows qi, i E {a, b, c, d}, • the double-rod cylinder, • the pressure supply consisting of a gear pump with a constant volume flow qo, an adjustable pressure relief valve and a hydraulic accumulator with the pressure Pace and the volume flow qacc, and • the high-voltage amplifiers with the voltages Ui , i E {a, b, c, d}, as the control inputs.
4"
Fig. 2. Schematic diagram of the experimental set-up . FurthemlOre, the experimental set-up is equipped with three pressure transducers for the supply pressure Ps and the chamber pressures Pl and P2; a position sensor for the piston position x; a force sensor for the load force 7/ acting on the piston; and a temperature sensor for measuring the fluid temperature
e,
3. MATHEMATICAL MODEL
3.1 ER Valve Many works dealt with the phenomenological modeling of an ER fluid . The first model for the description of the macroscopic behavior of an ER fluid was proposed in (Klass and Martink, 1967). It is the linear Bingham model with only two parameters, the field dependent yield stress and the viscosity. Even though the model is restricted to cases where the flow is orthogonal to the electric field, it is still used to describe the behavior of an ER fl uid in many cases. A phenomenological modeling in the context of continuum mechanics started with the paper (Rajagopal and Wineman, 1992) where a general model for the Cauchy stress tensor including the electric field was proposed. Based on these results a very systematic mathematical formulation of the governing equations of electrorheology considering a three-dimensional nonlinear constitutive equation for the Cauchy stress tensor can be found e,g, in (Rilzicka, 2000) and (Eckart, 2000). However, for a specific choice of the material parameters these models include the classical Bingham model of (Klass and Martink, 1967). Apart from this continuum mechanics approach several parametric models which rely on experimental results are proposed in the literature, see, e,g., (Butz and von Stryk, 2002) .
The paper is organized as follows : In Section 2 a short description of the experimental set-up unde(consideration is given . Section 3 is devoted to the derivation of a mathematical model of the ER actuator. Based on this mathematical model, in Section 4, a nonlinear control concept is proposed which fully exploits the degrees-of-freedom provided by the ER actuator. Some measurement results are presented in Section 5. Finally, Section 6 contains some conclusions and gives a short outlook.
2. EXPERIMENTAL SET-UP Fig. 2 presents a schematic diagram of the experimental set-up. The overall system can be subdivided into the following subsystems:
266
For the subsequent considerations let us consider the 3-dimensional Euclidean space with Cartesian coordinates Xj, j = 1,2,3, and the canonical orthogonal basis {el,e2,e:J}, which meets (ei,ej) = Oij for the standard inner product (-, -). In order to keep the formulas short and readable, we will use Einstein's convention for sums by summing over double repeated indices from 1 to 3. Furthermore, we will use the notation 8 j = {j~j' j = 1,2,3, and 8 0 = In this 3
8 OXj
all a22 a12
---- ~~.~- .-
"",
..
04
'2
= -4> + 4'1', 2
04
a33 2
= -4>, 0 6
= -4> + 02E2 + 4'1' + 21'
1 ( 03 + 0 5 E 22) 1', =2
a13
2
2
E 2,
(4)
= a23 = 0 ,
p80 U l = 8 l a l l + O;W12 () = 8[a12 + ihan o = 83a33.
(5)
Combining (4) with the last two equations in (5), we can derive an explicit expression for the spherical stress in the form
4> =
-_._- ._.
-PXl
04 2 06 2 2 + 0 2 E22 + -1' + -1' E2 4 2
(6)
where P = (Pin - Pout) / L denotes the pressure gradient, with Pin as the inlet and Pout as the outlet pressure of the ER valve. With this the first equation in (5) reads as
P,..., - - Il-
x,
(3)
with the shear rate l' = 82Ul. Since the external body forces are assumed to be zero and the electric field E is independent of Xj, j = 1,2,3, the balance of momentum yields p (80Ui + UjajUi) = ajaij or
aj'
As already mentioned before the ER valve is an annular channel being composed of two cylindrical electrodes with an inner radius Ht, an outer radius Ro and length L. Since the channel gap H = Ro - Ht (in our case 0.6 mm) is small compared to the mean radius !4n = (Ht + Ro) /2 (in our case 4.8 mm), the ER valve can be modeled as a pair of parallel plates with length L and width W = 2!4n1r. Fig. 3 depicts a schematic representation of the flow through the ER valve. Taking the temperature of the fluid 6, L
E=~(t)e2'
Under these restrictions the components of the (symmetric) Cauchy stress tensor (I) simplifY to
It.
sense the expression 8j aj stands for Lj=l
and
U=Ul(t,X2)el
l
(7) By choosing the material parameters 0.3 = 2TJ, with TJ as the constant dynamic viscosity at zero field, and 05 = 2To (E2) / (Ei 11'1), with the field dependent yield stress TO (E2 ), we obtain the "Binghamlike" constitutive equation according to (Rajagopal and Wineman, 1992)
fig. 3. Longitudinal section of the ER valve. the density p, the components of the electric field Ej and the components of the stretching tensor Dij = (8i Uj + 8 j U i), i, j = 1,2,3, where U denotes the velocity field of the fluid, as independent variables, we may write the constitutive equation ofthe isotropic ER fluid by means of the Cauchy stress tensor a in the form, see (Rajagopal and Wineman, 1992), (RilZicka, 2000), (Eckart, 2000)
1
aij
= TO (~) sign (')') + TJ1' lad < TO (E2), the case
a12
l' ~o().
(8)
For when l' = 0, the material behaves like an elastic solid, i.e. the shear stress is given by a12 = GC12, where G is the shear modulus and C12 denotes the strain. If we assume pointwise interaction of the dipoles of the ER fluid then the yield stress T O (E2 ) is proportional to In the case the dipole interaction reaches some kind of saturation at a certain field strength, TU (E2) only increases linearly with E 2 . Measurements have shown that a constitutive relation of the form
= 010ij + 02EiEj + 03Dij + 04DikDkj+
El.
05 (EiDjkEk + DikEkEj ) + 06 (EiDjkDklEI + DikDklEIEj) .
(I) Thereby, the material parameters Om, m = 1, ... ,6, are scalar functions of the invariants 1= {6,p,EjEj,Djj,DjkDkj,DjkDkIDlj, EjDjkEk, EjDjkDklEt} .
for
(9)
(2)
with suitable constant parameters a2 and a3 can be used to approximate the yield stress in the interesting area of operation.
Suppose that the temperature 6 is constant and the ER fluid in the valve is incompressible. Then, the continuity equation imposes a constraint on the velocity field in the form 8 j uj = D jj = O. From this we may deduce that 01 = -4> (Xl, X2, X3), where 4> is the indeterminate part of the stress tensor a also known as the spherical stress (Rajagopal and Wineman, 1992), (Ruzicka, 2000). The geometry of the ER valve as presented in Fig. 3 and the fact that the flow through the annular channel of the ER valve remains laminar give rise to the assumption that the velocity field U and the electric field E can be expressed in the form
In the literature it is documented that the dynamics of the ER effect, i.e. the formation of the chains in the fluid upon a step change in the voltage applied to the electrodes. are in the range of a few milliseconds. Furthermore, it can be shown that in the worst case, when the dynamic viscosity TJ takes its smallest value at zero field , the influence of the fluid inertia on the speed of response is in the same dimension of \-2 milliseconds, see, e.g., (Whittle et aI., 1996). This is
267
why we will henceforth neglect the dynamics of the ER valve for the purpose of a controller design. Next we will calculate the analytical solution of the steadystate velocity field within the gap of the ER valve, see Fig. 3. For an isothermal laminar flow of an incompressible Bingham plastic between a parallel plates gap, plug flow occurs in the center of the channel, i.e. O'2UI = 0 for H,,/ $ X2 $ H - H,,/, with H,,/ depending on the electric field E2 and the pressure gradient P . As we will see in a moment, outside the plug zone the velocity field is parabolic satisfYing the no-slip boundary conditions on the walls of the electrodes, i.e. Uj (0) = 0 and UI (H) = O. Thus, the operation of the ER valve is based on changing the height of the plug zone by the electric field or the voltage applied to the electrodes, respectively. An increase in the electric field beyond a certain value causes the plug zone to reach the value H,,/ = 1J- and then, no fluid can pass through the ER valve anymore. [n this case the valve is closed.
cl dtP2
(0)
In the isentropic case the pressure Pacc in the hydraulic accumulator obeys the differential equation d dt Pacc =
(X2)
P = -:;;X2
(
H-y -
W (TO (~)
( 12)
4. CONTROL CONCEPT
12p21]
(13) (Eh) according
The primary objective of the controller design is to track a desired trajectory Xd for the piston position x . From the mathematical model (13), (14) it can be directly seen that the ER actuator in the full-bridge configuration makes available four control inputs in form of the four voltages Ui , i E {a, b, c, d} , applied to the ER valves. At first sight it seems that the ER actuator provides more degrees-of-freedom for the controller design than necessary, in particular if we think of conventional hydraulic servo-drives, where we only have one control input. Thus, we will subsequently present a control concept I which exploits these additional degrees-of-freedom. Obviously, the equilibrium point of the overall system (13) - (16)
In contrast to the ER valve the compressibility of the fluid will be taken into account in the cylinder chambers by the bulk modulus (3 = p~, with the density p. In the absence of leakage flows the mathematical model of the double-rod cylinder takes the form
=
v
m-v
=
A (PI-P2)-TI
cl dt P1
=
VI
cl
clt
((3 - /'i,Pacc)
where Vs comprises the volume of the connection block and the volume of the pipes from the gear pump to the accumulator. The high-voltage amplifiers being used are considerably fast (response time approximately 100 p,s) and thus can be modeled in form of a static gain and a rate limiter counting for the maximum possible current.
3.2 Duuble-Rod Cylinder
cl
K
+ PH) (-2To (E2 ) + PH)2
with the field dependent yield stress TO to (9) and Eh = UIH.
- x c1t
.1
as qacc = CdAo/f, lips - Pacc lsign (ps - Pacc ), with the supply pressure Ps, the discharge coefficient Cd and the orifice area Aa. [n normal operation the adjustable pressure relief valve in Fig. 2 is closed and can therefore be neglected for the controller design. The continuity equation for the supply pressure reads as cl (3 cltPS = Vs (qO - qacc - qa - qc), (16)
Integration over the gap yields the stationary volume flow q through the ER valve
q=
+ (pacc<)
( 15) with the isentropic coefficient /'i" the effective accumulator volume Vacc and a constant c depending on the pre-charge and initial conditions of the gas in the accumulator. The volume flow through the orifice reads
(10)
x.)) . 2'
(3PaccqaccK KVaccPacc
= 0, (02utl (H-y) = 0, given by Ul
qd,
3.3 Pressure Supply and High- Voltage Amplifiers
Since the velocity profile is symmetric with respect to the center of the channel we just consider the flow zone between the inner electrode and the plug 0 $ X2 < H-y, the region where'Y > O. Combining (7) and (8) the steady-state velocity profile is the solution of the ODE (11) P + 1]O~Ul = 0, Uj
= qc -
(14)
In the steady state, i.e. OOUI = 0, we get from together with the symmetric boundary conditions 0"12 (H,,/) = TO and 0" 12 (H - H-y) = -TO the following expression for H,,/
with
q2
where x and v denote the piston position and velocity, PI and P2 are the chamber pressures, T[ denotes the external load force containing the friction force and qi, i E {a, b, C, d}, are the volume flows through the ER valves according to (13). Furthermore, m denotes the piston mass and all masses rigidly connected to the piston, A is the effective piston area and VI and V2 stand for the chamber volumes for x = O.
(7)
H _ H _ TO (E2) -y 2 P'
(3
= \I.i _ Ax (Av + q2),
(3
+ Ax (-Av + qtl ,
ql
= qa - qb
1 patent pending
268
is not unique. Considering the symmetry of the fullbridge, we determine the equilibrium point for Tt = 0 in the form x = 0, v = 0, PI = P2 = P,f, Pace = Ps and iji = ~, i E {a,b,c,d}. From this we may calculate the voltages Ui, i E {a,b,c,d}, from (13) with q = ~ and P = ~. However, since the value of Ps is still undetermined by this choice, we will fix PS within the admissible range above the pre-charge pressure of the accumulator.
clearly proves that the the overall closed-loop system of the cascaded controller is exponentially stable. Up to now we have designed a controller based on the control inputs ql and q2 in (14). In the next step, we have to divide ql and q2 into the flows through the ER valves, qa, qb and qc, qd, respectively. These remaining degrees-of-freedom will be utilized to take into account the following considerations: (A) In the range of vanishing volume flows the ER valve shows an undesirable hysteretic behavior. Therefore, the control strategy should circumvent a complete closing of the valves . (8) The volume flows ql and q2 needed for tracking fast trajectories are much higher than the constant volume flow qo provided by the gear pump. Thus it is clear that these volume flows can only be delivered by the hydraulic accumulator. Furthermore, in order to ensure the operation of the ER actuator the mean value of the supply pressure, denoted by Ps, has to be kept within certain limits 2 , i.e. PS,min < Ps < PS,max ' Therefore, we must include a supply pressure controller in the control concept which does not control the pressure peaks caused by the fast position trajectories but forces the mean value Ps to stay near a prescribed value PS,d. The requirements (A) and (8) give rise to the following partitioning of the volume flows qo + qs qa = sg( qd + - 2 -
The control concept for tracking a desired piston position Xd is based on a cascaded structure, with an inner control loop for the chamber pressures and an outer control loop for the piston position. Let us consider ql and q2 in (14) as the control inputs. Then the feedback law ql
Vi + Ax = Av + . f3 (-c5p (PI -
q2
= -Av +
V2 -Ax
f3
PI,d)
.
+ PI ,d)
(-c5 p (P2 - P2,d) + P2,d)
(17) yields an exponentially stable linear error dynamics for the tracking behavior of the piston pressures for every c5 p > 0, i.e. d
= -c5 pep;,
ep;
= Pj
- pj,d ,
(18)
where the desired trajectories pj,d are assumed to be sufficiently smooth. For the outer position control loop the desired piston force T d = A (PI ,d - P2,d) serves as a new control input. Since we want to include an integral action in the position tracking controller we choose the control law for T d in the form Td
= m (Xd - c5 x,2 ex -
c5 x ,le x
-
ox,o
qb
Ps =2 +
Td
2A
and
P2,d
=
ps
J
Td
+ -qo2+- qS (22)
+ qo ~ qs qo + qs
qd = sg (-q2)
exdt) ,
2 - 2A '
(-qd
qc = sg (q2 )
+ -2-'
with the function sg (q) = q for q 2: 0 and sg (q) = 0 otherwise and a volume flow qS which will serve as the control input for the supply pressure controller. From the definition of ql and q2, cf. (14), it can be seen that qS has no influence on the position controller (17) - (20). Due to space limitations we will not elaborate the details of the supply pressure controller, we rather try to point out the underlying idea. However, a rigorous proof of the stability is based on the averaging theory in combination with an energy-based formulation of the accumulator and will be published elsewhere. Substituting (22) into (16), we get
(19) with the sufficiently smooth reference trajectory Xd and the position error e", = x - Xd. Furthermore, the controller parameters 15",,; > 0, j = 0, 1,2, are chosen to render the position error dynamics exponentially stable. Note that (19) does not uniquely determine PI ,d and P2,d. Therefore, in view of the symmetric structure of the double-rod cylinder in the ER fullbridge configuration we set P I ,d
= si!;
(20)
d dtPS
At this point it is worth mentioning that the cascaded
controller as being proposed is fully compatible with the system flatness, see, e.g., (Fliess et al, 1995) for more details. Thus, if we take the piston position x as the so-called flat output, the controller (17), (19), (20) entails a linear error dynamics for the piston position given by
=
f3
-Vs (qacc + qS
+ Si!; (ql) + sg (q2»
.
(23) Let us assume that the position reference trajectory Xd is T-periodic, i.e. Xd (t ) = Xd (t + T). Then under certain non-restrictive assumptions it can be shown that the right-hand side of (23) is almost T-periodic, with a slowly varying strictly decreasing mean value. The supply pressure controller is chosen in the form
e~4 )
+ (c5p + 15",,2) 'e x + (opc5 x ,2 + c5 x ,d ex + (21) (c5 p c5 x ,1 + c5 x ,o ) ex + c5pc5 x ,oe x = O.
qs = -os (PS,d - ps) ,
Furthermore, the characteristic polynomial of(21) can be factorized in the form (A3 + c5;,2A2 + c5 x ,IA + 8",,0) (A + c5 p ) . This result combined with (18) and (19)
Os
> 0,
(24)
2 Roughly spenking. the lower limit is determined by the precharge pressure of the nccumulator and the upper Iimit is given by the maximum possibl e pressure ditrerence of the ER valve.
269
where Ps is the moving average value of the supply pressure Ps given by
ps =
the mathematical models of the various components of the ER actuator were derived, where special emphasis was put on a concise modeling of the ER valve. Based on this model a nonlinear control concept was designed which optimally utilizes the potentiality of the ER actuator. The control concept was implemented on a test-stand and the first measurement results prove its usefulness and efficiency. However, due to certain restrictions in the construction of the present test-stand we were not able to exploit the full dynamics of the ER effect. Nonetheless, we can at least reach dynamic responses in the range of comparable conventional hydraulic systems. Currently, we are setting up a new test-stand and with this we are confident to further increase the dynamics of the closed-loop ER actuator.
rt+Ts
.Jt
Ps (T) dT
with
Ts » T .
(25)
The voltages Ui , i E {a, b, c, d}, can be directly calculated by an analytic inversion of (13) with qi from (22). 5. MEASUREMENTS The control concept was implemented on the experimental set-up as described in Section 2 by means of the real-time hardware DS I 103 from the company oSPACE in combination with MATLAB/SIMULlNK and was tested for different reference trajectories Xd with a sampling time of 200 p,s. The measured responses to a faster rectangular-like reference trajectory with 5 mm amplitude and a slow sinusoidal reference trajectory with an amplitude of 20 mm are shown in Fig.4 and Fig. 5, respectively.
7. ACKNOWLEDGEMENT The authors would like to thank FLUOI CO N GMBH in particular Dr. R. Adenstedt, Dr. H. Rosenfeldt and Dipl.-Ing. M. Stork for the fruitful cooperation. 8. REFERENCES
~
Butz, T. and O. von Stryk (2002). Modelling and simulation of electro- and magnetorheological fluid dampers. ZAMM 82, 3-20. Eckart, w. (2000). Phenomenological modelling of electrorheological fluids with an extended Casson-model. Journal o/Continuum Mechanics and Thermodynamics 12, 341-362. Fees, G. (200 I). Statische und dynamische Eigenschaften eines hochdynamischen ER-Servoantriebes. Olhydraulik und Pnellmatik 45, 45-48. Fliess, M., J. Levine, P. Martin and P. Rouchon (1995). Flatness and defect of non-linear systems: introductory theory and examples. IEEE Trans. Automatic Control 61 , 1327-1361 . FLUDlCON (2001). Rheact. Technical Product Description 1,1 - 13. Gavin, H.P. (2001). Annular Poiseuille flow of ER and MR materials. J ~r Rheology 45(4),983- 994. Hoppe, R.H.W., G. Mazurkevitch, U. Rettig and O. von Stryk (2000) . Modeling, simulation and control of e1ectrorheological fluid devices. In : Lectures on Applied Mathematics. pp. 251-276. Springer. Berlin. Klass, D.L. and T. W. Martink (1967). Electroviscous fluids I: Rheological properties . Journal of Applied Physics 38( 1), 67-74. Rajagopal, K.R. and A.S. Wineman (1992). Flow of eIectrorheological materials. Acta Mechanica 91,57-75 . Ruzicka, M. (2000). Electrorheological Fluids: Modeling and Mathematical Theory. Springer. Berlin. Spencer, B.F. (1996) . Recent trends in vibration control in the U.s.A .. In: Proc. 3rd Int. Con! Motion Vibration Control. Vo\. 2. pp. K I- K6 . Japan . Whittle, M., R.J . Atkin and W.A. Bullough (1996). Dynamics of an electrorheological valve. Int. Journal ~f Modern Physics B 10,2933-2950.
3
oS c
2
l
2
005
0. 1
0.15
0.2
0.25
03
0.35
0.4
0.45
0.5
tins
Fig. 4. Measured response x for a rectangular-like reference trajectory Xd . 20
E E .S c
.g 'in
o
0.
·5
·'0
-15 :
·20
.---0.2
0 .4
0.6
0.8
.•... - ..
1
~--
1.2
1.4
'.6
1.8
tin s
Fig. 5. Measured response x for a sinusoidal reference trajectory Xd. 6. CONCLUSION AND OUTLOOK
In this paper, we presented the modeling and nonlinear control of an electrorheological actuator. In a first step,
270