MODELING AND IDENTIFICATION OF A PNEUMATIC MUSCLE ACTUATOR SYSTEM CONTROLLED BY AN ON/OFF SOLENOID VALVE

Transcription

1 Proceedings of 7 th International Fluid Power Conference March -4, 1, Aachen, Gerany MODELING AND IDENTIFICATION OF A PNEUMATIC MUSCLE ACTUATOR SYSTEM CONTROLLED BY AN ON/OFF SOLENOID VALVE V. Jouila Deartent of Mechanics and Design Taere University of Technology 3311 Taere, Finland ville.jouila@tut.fi S. A. Gadsden Deartent of Mechanical Engineering McMaster University Hailton, Ontario, Canada L8S 4L7 gadsdesa@caster.ca A. Ellan Det. of Mechanics and Design Taere University of Technology 3311 Taere, Finland aso.ellan@tut.fi ABSTRACT Pneuatic actuators offer desirable roerties for any alications, such as coactness, low costs, high ower-to-weight ratios, reliability, and silicity. However, due to any nonlinearities (air coressibility, friction, air flow through valve), accurate osition and force control of neuatic actuators is extreely difficult and exensive to achieve. There is a growing interest in PWM-controlled neuatic systes using low cost on/off solenoid valves instead of servo valves in order to develo less exensive neuatic servo systes. In addition, a new tye of neuatic McKibben uscle actuator ossesses significant advantages lie a very high force/weight and force/volue erforance, quic resonse, and wide oerational ranges in a variety of environents. In this aer, a high seed on/off valve is alied to control a neuatic McKibben uscle actuator syste. However, the colex nonlinear dynaics of the actuator in addition to those already entioned ae the odeling and accurate control of the neuatic syste a difficult challenge. As a result, the designed odel is nonlinear and ay still contain unnown araeters that require identification in order to obtain reasonable dynaical atching with the real syste. Furtherore, the discontinuous switching nature of the on/off valve causes transients in the syste, aing the analytic odeling of the syste even ore colex. The objective of this research is to develo an analytical odel of the syste which includes the nonlinearities of the syste, and the transforation of the discontinuities into a continuous for. The use of analytical odels enables the ileentation of conventional analytical control aroaches, such as sliding ode control, and rovides a tool for the analysis of stability and robustness. In this aer, the odeling rocess is alied to a one degree of freedo neuatic syste for which the analytical nonlinear syste odel is develoed by a cobination of hysical and eirical ethods. An extensive set of exeriental tests are erfored to characterize the dynaics of the overall syste. A non-analytic and analytic odel of the syste are develoed and validated by a coarison of the siulated results with the exeriental ileentation of the syste. KEYWORDS McKibben uscle actuator, on/off solenoid valve, odeling, PWM control INTRODUCTION Pneuatic actuators are coonly avoided for advanced alications due to robles with control caused by the coressibility of air and other nonlinear effects. Pneuatic control systes are ainly used in sile industrial alications with liited requireents for accurate control of otion and force. However, high ower-to-weight ratios, coactness, ease of aintenance, and the safety of neuatic actuators, offer desirable features for any industrial designs. The neuatic McKibben uscle actuator is a new tye of actuator that offers a high force-to-weight ratio and is able to oerate in a wide range of environents. The coressibility of air, the nonlinear air flow characteristics through the valves, 1

2 friction and the nonlinear characteristics of the McKibben uscle actuator result in a colex and difficult syste to odel and control. In the theoretical analysis of neuatic systes, a cobination of therodynaics, fluid dynaics and the dynaics of the otion is required. The atheatical analysis requires the consideration of the ass flow rates through the valve, the deterination of the ressure, volue and teerature of the air in the actuator, and the deterination of the dynaics of the load. Furtherore, identification techniques, although usually based on linear ethods, can be used for finding the atheatical odel of the neuatic syste. An accurate odel of the actuator is an iortant condition for both control design and for otiizing its oeration. In recent years, a considerable aount of research has been erfored to develo inexensive servo-neuatic systes using on/off solenoid valves with a ulse-width odulation (PWM) technique. In a PWM-controlled syste, the ower is delivered to the actuator in discrete acets of fluid ass, as the valve is either coletely on or off. If the switching frequency of the valve is significantly higher than the syste dynaics, the syste will act as a low-ass filter resonding siilarly as to a continuous ass flow inut. However, due to the discontinuous switching, the develoent of an analytical dynaic odel of the syste is rather difficult, and often revents the direct use of analytical control designs. Although revious wor has shown the otential of PWMcontrolled neuatics, they have suffered due to the lac of an analytical aroach for analyzing the syste [1 4]. However, soe effort has been ade in the area of analytical odeling of such systes [5 7]. A state sace averaging aroach was resented in [8] for odeling a ulse-width odulation (PWM) based on neuatic systes. It rovided the analytic ethod necessary to reove the discontinuities associated with switching and resulted in a odel suitable to standard nonlinear control design techniques. In [9] a ethodology for deriving a nonlinear dynaic odel for a neuatic servo syste was resented. The odel includes cylinder dynaics, ayload otion, friction, and valve characteristics. Exeriental results deonstrating the ability of the odel to redict the easured osition and cylinder chaber ressure were included. In one article, the nonlinearities of the syste were handled by roosing a switching controller based on the reduced order nonlinear odel of the syste [1]. Another notable aer introduced an exerientally develoed discrete-tie odel of a PWMcontrolled neuatic servo syste, for which a controller was develoed based on discrete-tie control ethods [11]. Another strategy used a linear state-sace averaged odel and a linear robust controller based on a loo shaing aroach [1]. This aroach was later followed by a nonlinear averaged odel and a sliding ode controller design [13, 14]. A linearization aroach was subsequently used in an attet to reove the need for colicated nonlinear controllers [15]. The objective of this research is to develo an analytical odel of the neuatic syste which includes the nonlinearities of the syste, and the transforation of the discontinuities into a continuous for. The use of analytical odels enables the ileentation of conventional analytical control aroaches, such as sliding ode control, and rovides a tool for the analysis of stability and robustness. In this aer, the odeling rocess is alied to a one degree of freedo neuatic syste where a McKibben uscle actuator echanis is controlled by a single highseed solenoid valve. A non-analytical and analytical ass flow rate odel through the valve are develoed and cobined with the nonlinear odel of the actuator and the echanis. An extensive set of exeriental tests are erfored to characterize the dynaics of the overall syste. The odels are validated by a coarison of the siulated results with the exeriental ileentation of the syste. Finally, a traditional PI-controller is tuned with the hel of a atheatical odel, and is ileented in the real syste. SYSTEM MODELING Syste Setu and Structure The syste hardware is illustrated scheatically in Fig. 1. The Festo fluidic uscle (MAS1-3 ) is hanging vertically, actuating (lifting) the attached ayload. The suly ressure (.65 MPa abs.) for the syste is rovided by the roortional ressure regulator (Festo VPPM-6L-L1-G18-L6H-V1N). A single 3/ high seed on/off solenoid valve (Festo MHE-1/8-MS1H-3/G-M7) is controlled to actuate the uscle actuator and the ayload. The solenoid is driven by ulsed valve control signal generated in DSace and Matlab environent. An electronic alifier is used to rovide sufficient ower to actuate the valve. A ressure sensor (Festo SDE1- D6-G-H18-C-PU-M8) rovides a feedbac signal for the controller. The dislaceent of the actuator and ayload is easured by an electrical otentioeter. Flow restrictors shown between the actuator and valve are otional and can be used to reduce excessive ressure chattering, if necessary. The tubing between the valve and the actuator is et short as ossible, and thus can be neglected in the overall odel. Fig. 1. Syste setu McKibben Muscle Actuator odel The McKibben uscle is an actuator that consists of a rubber tube with a non-extensible fiber surrounding [16]. This hysical configuration causes the uscle to have variable-stiffness sring-lie characteristics, nonlinear assive elasticity, hysical flexibility, and very lightweight coared to other tyes of actuators [17]. The only coercially available uscle actuator (MAS) by Festo differs slightly fro the general McKibben tye uscle. The fiber of the fluidic uscle is nit into the tube, offering easy assebly and iroved hysteretic behavior and linearity coared to conventional designs [18].

3 During ressurization of the uscle with air, the uscle widens in diaeter and shortens in length. The axiu force is obtained at the beginning of the contraction and decreases with increasing contraction [19]. The actuator is unidirectional and its axiu contraction without load is tyically % to 5%. The noinal forceto-contraction at different ressure levels is nonlinear, and adds to the difficulty of effectively odeling the uscle actuator. As with all actuation systes, effective alication of neuatic uscle actuators relies on being able to accurately odel and redict the forces that will be generated under any oerating conditions. In general, the roerties of the uscle actuator deend on the geoetric araeters shown in Fig.. Fig.. Geoetric odel of McKibben actuator [17] Fro the geoetry of the uscle, the overall length of the actuator and the diaeter are given by the following two equations: L b s cos (1) bs sin D n () s 3 a tan 1 b sin L L L Where r and θ are the iniu radius and braid angle, resectively. L is the axiu (initial) length, and ε is the contraction ratio. Equations (5) and (6) give a basis for redicting the generated uscle force. However, they fail to coletely odel the behaviour of braided uscle actuators due to the assution of lossless oeration. Subsequently, various hyotheses have been develoed to account for the effects of tubing elasticity, internal frictions, braid thicness, stretching of the fibres, end ca diaeter (i.e. non-cylindrical uscle shae) and aterial odeling in order to rovide ore accurate odels [16, 17, 1 5]. Desite the iroveents, errors between the redicted and easured force still exists. Esecially in the case of Festo fluidic uscles, the odels have been too inaccurate leading to a use of various corrective factors and exonential curve fitting ethods [1, 5]. It has also been observed that there exists hysteresis in the uscles during oeration caused ainly by friction resent in the syste. To account for the friction in the atheatical odels, a constant force offset can be subtracted fro or added into the calculated static force deending on whether the uscle is contracting or exanding [17]. In [1] a araeter was introduced, which tunes the sloe of the considered static odel and atches the odeled data with the exeriental values. Where b s is the length of one braid strand, considered to be inextensible, and n s is the nuber of ties a strand encircles the uscle s circuference fro end-ca to end-ca. Assuing an ideal cylindrical shae, the enclosed volue is defined as follows: 3 bs V 4n s cos sin (3) Fro the rincile of virtual wor and the conservation of energy, and assuing quasi-static and lossless conditions, the force required to defor the ebrane can be exressed by: dv F (4) dl Substitution of equation (3) into (4) leads to the force generation equation, first roosed in [19]: D F (3cos 1) 4 (5) Where F is the contractile uscle force, D is the diaeter of the actuator at the braid angle of 9 (theoretical axiu), and is the uscle ressure. An iroved force equation taes the for [, 1]: r [ a(1 ) ] F (6) b Fig. 3. Measured uscle force including hysteresis behavior In Figure 3, the easured uscle force as a function of the dislaceent and ressure is shown, which also includes the hysteretic behavior of the Festo fluidic uscle. At the oerating oint x=, the actuator is at rest and the ositive dislaceent x refers to the aount of dislaceent/shortening of the actuator. The negative dislaceent indicates the stretching of the actuator fro its noinal length. It should be noted that the actuator can rovide even higher initial forces when re-stretched. The force roduced by the actuator decreases non-linearly as the contraction/dislaceent increases. When the actuator reaches its iniu length/axiu dislaceent the actuator does not rovide any force. 3

4 The uscle actuator introduces a variable sring and a daer in arallel. The variable sring coonent is described by the static uscle force equation including hysteresis henoena. The daer coonent describes the dynaics of the actuator including the viscous friction effect. For odeling the static force characteristics, averaged force/dislaceent curves at each ressure level are deterined by calculating the average of the hysteresis loos shown in Figure 4. In order to cature the nonlinear force characteristics, an alternative fitting aroach fro the aforentioned ethods is used. The axiu available force as a function of dislaceent is introduced by fitting a 4th-order olynoial function for the curve at the axiu ressure.6 MPa. As a result, the nonlinear curve shae of the force/dislaceent characteristics is catured. When the uscle dislaceent is held constant, the actuator force deends alost linearly on the ressure. However, the sloe of the force er unit ressure changes as a function of the dislaceent and thus the uscle force can be described by: F uscle F ax ( x) ( ax )( 1x dx ) FC sign( ) dt (8) F F static ax F ax ( x) a ( x) ( 1 ax a x a x )( 3 1x ) 3 4 a x a x 4 (7) Where ax is the axiu available uscle ressure and is the current uscle ressure. Coefficients [N], 1 [N/], and [Pa] are found by using least-squares ethods. Fig. 5. Coarison between the odeled and actual force including hysteresis In Figure 5, the uscle force odel (equation (8)) is coared with the easured data. As a conclusion, the odel is able to rovide a reasonably good rediction for the uscle force, including the effect of hysteresis. Fig. 4. Static uscle force odel Figure 4 illustrates the redicted force lotted against the easured force data at different ressure levels (.1 to.6 MPa). The odel is able to redict the force reasonably well for alost every ressure. Soe deterioration exists between the odel and actual data at lower ressure levels (less than. MPa). For odeling the hysteresis, a siilar aroach as in [17] is used. The shae and the width of the hysteresis loos are alost the sae for each ressure level as shown in Figure 3. An average value for the width of the hysteresis loo resulted in 3 N. Thus a friction force offset F C (Coulob friction) of 16 N can be added into the static force odel described by the equation (7), as follows: The identification of the viscous friction of the actuator is often a challenging tas. The friction ay be tie-variant and deendent on the teerature and the ressure, as well as on the velocity. However, in our case, the effects of teerature and ressure are neglected. The goal here is to deterine the viscous friction as a function of velocity with a reasonable aount of accuracy. In the analysis, a low-friction neuatic cylinder is used in the easureent setu. First, the friction characteristics of the cylinder is deterined, by driving it at different constant velocities and calculating the friction force with the hel of the easured ressure in chabers of the iston side and iston rod side. When the cylinder iston is oving at a constant velocity, the differential force between the chabers can be assued to be equal with the friction force. The friction characteristics of the cobination of the uscle actuator and the cylinder can be easured directly by the load cell attached to the other end of the uscle actuator. The free end is connected to the cylinder iston. In the easureents, first the cylinder chaber is ressurized. Then at the given tie ste the chaber ort is oened to abient ressure and the uscle actuator is ressurized to create the otion. During this oeration the dislaceent, force and cylinder ressure are easured which are then used to calculate the total friction force as a function of the velocity. A rough estiate of the viscous friction of the uscle actuator can be obtained by subtracting the cylinder friction fro the total friction. Figure 6 shows the estiated friction of the cylinder, uscle actuator, and the cobination of the two. Furtherore, it should be noted that the viscous friction is assued to be syetrical for ositive and 4

5 negative velocities. A descrition of the total uscle force, which includes the behavior of friction, ay be defined as follows: F uscle dx dx Fstatic FC sign( ) C f (9) dt dt A C C q flow rate coefficient u C C q T u 1 1, R 1 R 1 / u u 1, u u cr cr (1) Fig. 6. Estiated friction characteristics of the uscle actuator Non-analytic valve odel The PWM neuatic valve, that controls the airflow into and out of the actuator, is a fundaental coonent of the syste. The valve considered in this study is a 3/-high seed solenoid valve (Festo MHE-1/8-MS1H-3/G-M7) with a switching tie of aroxiately s. The woring rincile of the solenoid valve is described as follows: the tensile force of the reloaded sring, and the force exerted by the air, tend to close the valve. Conversely, the agnetic force created by the current assing through the coils ushes the core and then the oet, which oens the valve. The area of the air assage is a function of the osition of the oet, and it is also deendent on the diaeter of the restricted assage, as well as the geoetric for of the oet. However, the internal structure of the solenoid valve is not usually nown by the user and is not easily easured. Thus, it is not very efficient to establish a odel that taes into account the electrical, agnetic, and echanical subsystes of the coonent. In general, the therodynaics equations considered for valve odeling are those shown in literature for a gas through a nozzle, while assuing an adiabatic rocess, absence of losses, and convergent nozzle [6]. Thus, the ass flow rate assing through the valve can be exressed as follows: Where T u is the ustrea air teerature and u is the ustrea ressure, =1.4 is the secific heat ratio of air, is the strea ressure, and R=87 J/g/K is the air constant. C q and C are, resectively, the flow rate coefficient and the flow rate araeter of the solenoid valve. The critical ressure cr divides the flow into sonic ( / u < cr ) and subsonic ( / u > cr ) flow regies. The flow equation wors well usually for a short convergent nozzle where the friction and coressibility effects are negligible. However, due to the flow rate losses in coercial valves, the critical ressure does not have a refixed exression (usually.58) but it changes deending on the articular tye of valve considered. As a result, we have found that the given flow rate equation does not corresond well to the ass flow rate characteristics of the solenoid valve under study. Instead, a theoretical odel introduced in [7, 7] is used. Here, the flow rate is considered constant in a sonic flow zone, while it decreases with a quadratic behavior aroxiated by a quarter of ellise in the subsonic flow zone. C v A C v A u RT u RT u u b u 1 1 bv v u, u b v b v (11) C v is the valve discharge flow coefficient, and b v is the critical ressure. In addition, the flow aths of the valve ust be considered searately. In other words, the odel should account for two ossible flows. When the valve is oen the flow ath is through the orifice 1 - > (inflation), and when the valve is closed the flow ath is through the orifice -> 3 (deflation). While inflating, the ustrea ressure is a constant suly ressure, and the strea ressure is the ressure inside the actuator. When exhausting, the ustrea ressure is the actuator ressure, and the strea ressure is the abient ressure. In order to identify the neuatic behavior of the valve a set of exerients according to the rocedure introduced by ISO6358 were carried out. In easureents, three ustrea ressure levels were used and the relevant exeriental oints were fitted by tuning two araeters (C v, b v ) through equation (11). The geoetric and tuned flow rate araeters are reorted in Table 1. Figure 7 shows a good 5

6 overla between the siulated and exeriental curves. It should also be noticed that the valve s flow ways are not syetrical. Paraeter Descrition Value R Air constant 87 [J/(g*K)] T u Ustrea 93 [K] teerature u Ustrea ressure.5,.6,.7 [MPa] (abs) A Valve diaeter 3.14e-6 [ ] C v (1 -> ) Flow coefficient.36 C v ( -> 3) Flow coefficient.39 b v (1 -> ) Critical ressure.8 b v ( -> 3) Critical ressure.49 Table 1. Paraeters used for valve identification ratio, it is necessary to deterine the average ass flow rate as a function of actuator ressure and duty ratio control signal. In its lace, a rocedure siilar to the one introduced in [8] is followed, where an equivalent ass flow rate odel was deterined for a roortional servo valve. The ass flow rate has nonlinear characteristics and is a function of ressure inside the volue, and the control signal u (duty ratio [-1]). Thus, one obtains a traditional reresentation for the ressure change as follows: RT V eq ( u, ) (1) V V In equation (1), the second ter can be couted once the volue V is nown, and the ressure is given. In the first ter, the nonlinear valve function is difficult to easure. Alternatively, the nonlinear valve characteristic can be aroxiated exerientally by charging and discharging a constant volue chaber. This causes the second ter in equation (1) to disaear, allowing the ass flow rate to be calculated fro the rate of change of ressure. A set of inut signals with different duty cycles were alied to the valve and the ressure resonse in the constant volue chaber was easured. Due to the PWM switching, the ressure signal contains a significant aount of vibrations. Thus, the ressure resonse requires digital filtering in order to obtain an averaged resonse. The average ressure signal ay then be differentiated in order to obtain the ressure change at different ties. By distributing the couted sloes of the ressure curve at the corresonding araeter airs (u and ), a araetric reresentation of the surface of the ressure change can be obtained. Using this surface, the ass flow rate can be estiated using equation (1). Figure 8 illustrates the estiated/easured ass flow rate lotted as a function of inut signal (duty ratio) and the relative actuator ressure. Note that a negative ass flow rate indicates a discharging flow. Fig. 7. Flow rate fits for assages 1-> and ->3 Analytic valve odel It is quite obvious that the ass flow rate odel of the 3/ high seed on/off valve is an essential art of the syste odel. The solenoid valve is controlled with the duty ratio of the PWModulated signal. The tie eriod of the PWM-signal is deterined as T PWM and is the inverse of the switching frequency T PWM =1/f PWM. The switching tie for oening and closing the valve is aroxiately s, which naturally reduces the axiu available duty cycle range. The switching frequency and the duty cycle deterine how long the valve is oen and closed during tie eriod T PWM. Valve delays and the discontinuous high frequency switching increase the colexity of the valve odel, and are difficult to handle in the view oint of controller design. Thus the non-analytic odel introduced (equation (11)) is not feasible and an alternative valve odel is needed. In a PWM-controlled syste, the ower is delivered to the actuator in discrete acets of fluid ass, as the valve is either coletely oen (on) or closed (off). If the switching frequency of the valve is significantly higher than the syste dynaics, the syste resonds siilarly as in the case of continuous ass flow. As the control signal for the valve is actually the duty Fig. 8. Estiated ass flow rate for on/off valve In order to odel the ass flow rate, a nd order bi-olynoial function was used, as follows: 6

7 ( u, P ) eq 8 1 u P u P P P u up up u (13) For calculating the ressure inside the uscle, it is assued that the air is ideal gas and the change of air is adiabatic, such that the ressure change follows: Where 1-9 are the coefficients found using the least-squares ethod. RT V ( x) eq dv ( x) ( u, ) x V ( x) dx (15) Where (1.4 for adiabatic rocess), R, T, V and denotes the secific heat ratio, gas constant, air teerature, volue of the uscle, and uscle ressure, resectively. The second ain exression of equation (15) considers the ower balance of the ressurized flow rate. The first ter (following the subtraction sign) reresents the ressure change due to the ass flow in or out of the uscle chaber. The second ter reresents the ressure change due to the change of the uscle chaber volue. The recirocal volue taes into account the coressibility of the air. Overall syste odel The otion equation of the uscle driving a constant ayload attached in a vertical direction is defined (using Newton s Second Law) as follows: Fig. 9. Fitted odel for ass flow rate The outut obtained fro this function is lotted in Fig. 9. It can be observed that the odel aroxiates the averaged ass flow rate behavior of the valve quite well. The axiu fitting error is g/s, or 4.13 % of the range. The RMSE is 5.51 g/s or 1.16 %. Pressure dynaics Knowledge of the actual ressure inside the uscle actuator is essential for understanding the dynaic behavior. The ressure deends on the ass of the air and the volue of the uscle. The diaeter and length of the uscle were easured, and the volue of the uscle was calculated assuing a cylindrical shae. The volue shows a nearly linear behavior, deendent on dislaceent: V ( 1x x) v v, (14) dx dx M x Fstatic Fc sign( ) C f Mg (16) dt dt Where F static is the static uscle force given in equation (7), M is the total ass of the syste and ayload, and g is the gravitational constant. F c is the Coulob friction, and C f is an exerientally aroxiated daing factor of the uscle actuator. In conventional analytical control aroaches, such as sliding ode control, a state-sace descrition of the syste is referred. Thus, suose that the state vector for the syste is defined as follows: x [ ] T states P x x x (17) Fro the nonlinear odels described in the revious sections (articularly equations (7), (13), (15) and (16)), we have the following discrete-tie equations which can be used in the control and estiation rocesses [31]: T S x 1, 1 [ RT eq ( u, x1, ) x1, 1x3, ] x1, 1x, x, 1 TS x3, 1 x, x 3, 1 (18) (19) T S 1x, [ Fax ( x, ) ( Pax x1, ) M () F sign( x ) C x Mg] x c 3, x 4, 1 f 3, S 3, x3, 1 x3, (1) T Fig. 1. Muscle volue in correlation with dislaceent It should be noted that the analytic ass flow rate odel (equation (13)) enables the use of this ind of syste resentation. 7

8 SIMULATION AND EXPERIMENTAL RESULTS This section describes the results of validating the develoed valve odels and overall syste odel with easureents. In addition, the final syste odel is used for tuning the araeters of a traditional PI-controller, which is used to control the actuator to follow a desired dislaceent trajectory. Valve odel validation In the revious section, two solenoid valve odels were develoed: non-analytical and analytical. The non-analytical valve odel is based on equation (11) for which the tunable araeters were found by fitting the equation to atch with exeriental data. In the odel, the flow aths of the valve are described searately, as the air flow is through assage 1-> when the valve is on and through ->3 when the valve is off. During the on state, the valve inflates the actuator volue, and the suly ressure is the ustrea ressure while the actuator ressure is the strea ressure. During the off stage, they are actuator ressure and abient ressure, resectively. The non-analytical odel also includes the valve delay (~ s) in the switching henoena when the valve changes its state. The easureents are carried out with a nown constant volue. During the inflation rocess, the suly ressure is 5.5 Ma (relative) and the initial ressure in the volue is zero. The volue is ressurized oerating the solenoid valve with different duty ratios (5, 5 and 75 %) of 5 Hz PWM-signal. The chosen frequency is fast enough for the syste dynaics and also rovides a reasonable resolution for controlling the available duty ratio values. It should be noted, that the valve delay decreases the axiu available duty ratio range into 1-9% using this frequency. During the deflation rocess, the volue is first ressurized to a axiu value (5.5 MPa rel.), and then the valve is oerated siilarly with different duty ratios as in the inflation rocess. Fig. 11. The validation of valve odels when inflating a constant volue with different PWM-duty ratios Fig. 1. The validation of valve odels when deflating a constant volue with different PWM-duty ratios Figures 11 and 1 illustrate the siulated and easured ressure resonses for the inflation and deflation rocesses. The nonanalytical valve odel gives a good overall estiation for inflating ressure curves. However, the odel is unable to coletely estiate the ressure resonse during the deflation rocess. The deviation is caused by the silencer that was attached to the third valve ort. The use of the silencer is necessary because of the very loud noise caused by the high seed valve switching. The silencer clearly changes the characteristics of the flow through -> 3. The effects of the silencer on the ressure resonse are strongest when the ressure inside the volue aroaches the steady-state value. As the ressure dro across the valve decreases the silencer slows the flow. In consequence, the ressure in the siulations dros faster causing a steady-state offset coared to the easureents. The effects of the silencer at higher ustrea ressure levels are alost negligible, and the odel is able to follow the easured ressure curves quite accurately. Clearly, better estiation results would be gained if the valve was oerated without the silencer. The analytical valve odel was described in the revious section. The urose of it is to rovide ass flow estiation through the valve as a function of the duty ratio and the ressure of the controlled volue. The odel assues that the suly ressure at the first valve ort is constant (6.5 MPa abs.) and ort 3 is at the abient ressure. The valve relies heavily on the easureents that are naturally sensitive to the uncertainties. A large nuber of easureents were carried out and the ressure resonses of the constant volue were easured. Fro the easured data, the ressure change was catured and the aroxiation for the average ass flow rate was calculated at different voluetric ressure levels and duty ratios. A nd -order bi-olynoial function was fitted to the obtained data with relatively good accuracy. The coarison of analytical odel to the non-analytical odel and easured ressure resonses are shown in Figures 11 and 1, resectively. Desite soe divergences between the odel and the easureent, the odel is able to estiate the ressure with reasonable accuracy. The advantage of the odel is its suitability to be used for conventional control design, such as sliding ode control. The effectiveness of the odel was roved in [9], where it was used successfully with the sooth variable structure filter (SVSF) to estiate the syste states for the sliding ode controller. In the controller, a odel based feedforward control was used to rovide an equivalent control signal 8

9 (duty ratio) for the valve. Based on the nonlinear state-sace odel (equations (18-1)), the equivalent desired ass flow ay be estiated. The nd -order bi-olynoial function can then be solved to find the desired duty ratio when the desired flow rate and the actuator ressure (easured) are nown. The sliding ode controller and the filter are nown to be very robust to araeter variations and other uncertainties, yielding an error tolerance for the estiated ass flow rate odel. not tae into account the real transition between the inflating and deflating curves. As a result, the accuracy of the odels is not as good coared to the uwards otion. In addition, the un-odeled silencer affects the accuracy of the non-analytical odel. Altogether, both the odels can estiate the real rocess reasonably well, as the axiu dislaceent error is aroxiately 1% and the RMSE values for the dislaceent are around 3-5%. Syste odel validation The overall odel was verified for both the non-analytic and analytic valve odels. In the validation rocess, a set of inut signals (duty ratio) of triangle wavefor was alied to the syste. The frequency of the wavefor was varied and the duty ratio value changed between.5 and.95. In the case of the non-analytical odel, a ulse width odulator is used to convert the duty ratio signal into an aroriate on/off control (f=5 Hz) for the valve. The siulation and easureent results for inut signal frequencies.8 and.5 Hz are shown in Figures 13 and 14. First of all, notice that the easured dislaceent signal contains no excessive chattering. The chosen PWM-frequency is thus high enough coared to the syste dynaics. Although not shown here, the alitude of chattering in the actuator ressure is aroxiately 15 Pa. An exerient with a PWM-frequency of 5 Hz resulted in an unaccetably large aount of chattering into the actuator dislaceent. An even soother trajectory was gained with PWMfrequency of 1 Hz. The drawbac of using that is the oor resolution of the inut signal; even if an.1 s saling tie is used. Fig. 13. Siulated and easured dislaceent resonse for a triangle (f=.8 Hz) inut signal The siulation results show that both odels are able to estiate the dislaceent of the actuator and ayload (9.6 g) with reasonable good accuracy. An extreely good accuracy is obtained during the uward otion. The odels fail to describe the creeing effect near the axiu dislaceent. It is a noinal effect for this tye of actuator caused by aterial deforations. The largest odeling errors occur during the wards otion. This is caused by the silified hysteresis odel where a constant force offset is either reduced fro or added into the static uscle force odel deending on the sign of the actuator velocity. Thus, the odel does Fig. 14. Siulated and easured dislaceent resonse for a triangle (f=.5 Hz) inut signal PI-Control A PI-controller was tuned by trial-and-error using the nonanalytical and analytical syste odels. During the oeration it was noted that the syste is very sensitive to oscillations and instability. The instability issue liited the axiu gains for the controller resulting in a lac of accuracy. A reasonable erforance was gained with a derivative gain in the controller during the siulations, but due to the easureent noise it could not be used in the real syste. Figures 15 and 16 illustrate the results when the inut signal is sinusoidal with alitude of.1 and frequencies of.5 and 1 rad/s. The tuned gains for the controller were P=5 and I=15. It can be seen that the syste erforance is quite oor with the PIcontroller. The resonse for the initial ste is slow, and the syste is not able to follow the desired trajectory very well. A steady-state error of aroxiately is resent during the entire cycle. The RMSE is over 1%, and increases as the frequency of the desired trajectory is increased. It becoes aarent that neuatic systes are very difficult to control accurately. The erforance of traditional linear aroaches lie PI and PID controllers is oor with highly nonlinear neuatic systes. Thus, nonlinear and robust aroaches are needed for better erforance. However, the odels develoed in this aer of the neuatic uscle actuator syste give a fir basis for the design of ore advanced control strategies. The nonanalytical odel can be used to siulate the real rocess quite accurately. In addition, the analytical odel can be used in the odel-based controller and state estiator design. 9

10 In this aer, a nonlinear odel for a neuatic uscle actuator syste controlled by an on/off solenoid valve was develoed. The solenoid valve is oerated by a ulse width odulated (PWM) schee which gives an interesting alternative to develo low-cost neuatic servo systes. On the other hand, accurate control of highly nonlinear neuatic systes requires advanced control techniques that often use odel-based aroaches. Thus, the ain focus of this research was to develo a odel for control design aroaches which cature the ajor nonlinearities resent in the syste with reasonable accuracy. In PWM-oerated systes, the high seed switching of the valve results in discontinuities which are often difficult to handle fro the viewoint of control design. As a consequence, two valve odels were develoed in this study. The non-analytical odel is able to describe the real oeration of the syste with relatively good accuracy, and including the nonlinear flow regies and valve switching delays. The odel is suitable for syste analysis and for testing of controllers. The analytical odel of the syste, which includes the nonlinearities of the syste, was develoed to transfor the discontinuities into a continuous for. This was accolished by introducing a continuous flow odel through the valve as a function of duty ratio of PWMsignal and actuator ressure. The use of this analytical odel enables the ileentation of conventional analytical control aroaches, such as sliding ode control, and rovides a tool for the analysis of stability and robustness. NOMENCLATURE Fig. 15. Trajectory (ω=.5 rad/s) followed control with tuned PIcontroller Fig. 16. Trajectory (ω=1 rad/s) followed control with tuned PIcontroller CONCLUSIONS A [ ] effective orifice area of the valve C f [Ns/] viscous friction coefficient C [-] flow rate coefficient of the valve C q [-] flow rate araeter of the valve C v [-] discharge flow coefficient of the valve D, D [] uscle actuator diaeter F [N] force in general F c [N] Coulob friction F ax [N] axiu uscle force uscle F uscle [N] force generated by the uscle L,L [] uscle length, initial length M [g] weight of the ayload (9.6 g) cr [Pa] critical ressure ratio [Pa] strea ressure [Pa] ressure inside the uscle u [Pa] ustrea ressure s [Pa] suly ressure [Pa] atoshere ressure R [J/(gK)] gas constant T [K] air teerature T u [K] ustrea air teerature T PWM [s] tie eriod of PWM-signal T S [s] saling tie V [ 3 ] volue in general V [ 3 ] volue of the uscle a -4 [] uscle force coefficients b s [] length of one braid strand b v [] critical ressure f PWM [Hz] switching frequency of the PWM-signal g [/s ] gravity constant [-] secific air heat ratio [N] coefficient for uscle force eq. 1 [N/] coefficient for uscle force eq. [Pa] coefficient for uscle force eq. [g/s] equivalent ass flow rate eq [g/s] ass flow rate 1-9 [-] coefficients for eq. ass flow rate n s [-] nuber of strand encircles [Pa] ressure cr [Pa] critical ressure ratio [Pa] valve strea ressure [Pa] ressure inside the uscle ax [Pa] axiu uscle ressure [Pa] atoshere ressure u [Pa] valve strea ressure r [] initial uscle radius u [-] control signal (duty ratio) u eq [-] equivalent control signal x [] dislaceent of the uscle θ, θ [º] uscle braid angle, initial braid angle ε [-] uscle contraction ratio v, v 1, [ 3, ] uscle volue coefficient [-] denotes a tie derivative 1

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