Moton Control of Pneumatc Muscle Actuator usng Fast Swtchng Valve Shenglong Xe, Jangpng Me, Hatao Lu, Panfeng Wang (Key Laboratory of Mechansm Theory and Equpment Desgn, Mnstry of Educaton, Tanjn Unversty, Tanjn 300072) Abstract: Consderng the nonlnear and tme-varyng phenomena exstng n pneumatc muscle actuators (PMAs), ths paper deals wth the modelng of trackng control of PMA usng fast swtchng valves. A close-loop control scheme combned wth feed-forward and feedback controllers s proposed to acheve hgh accuracy trajectory trackng control. Frst, the statc model of the PMA s establshed usng the data obtaned from sometrc experment, and the dynamc model s developed based on the polytropc equaton. Then, the hysteress model and ts nverse model s establshed, n whch the ar mass flow rate through the fast swtchng valve s evaluated usng the Sanvlle equaton. The PWM sgnal used to control the fast swtchng valves s generated referrng to the pulse sgnal modulaton method. Sequentally, the trajectory trackng control models of the PMA are derved by means of close-loop control scheme, whch are mplemented n the envronment of MATLAB/Smulnk. The results ndcate that the control model can acheve satsfactory performance and accuracy, whch valdates the feasblty of the proposed model and control scheme, provdng an effectve approach for hgh accuracy trajectory trackng control of PMA. Keywords: fast swtchng valve; pneumatc muscle actuator; hysteress dentfcaton; trackng control 1 Introducton Pneumatc muscle actuators (PMA) possess several advantages over servo-motors, such as smple structure, compactness, hgh power-to-weght rato, etc. [1], whch have been wdely used n medcal rehabltaton robots, bonc robots, and orthotcs. However, the hysteress [2], threshold pressures [3], creep [4], complance [5], and low bandwdth [6] make the desgn of the correspondng controller complcated. Especally, snce the hysteress may cause energy loss and reduce the contractng force, there wll be errors n trackng control. Therefore, t has to establsh a precse hysteress model to mprove the dynamc behavor of a PMA [7]. Typcally, there are two types of electro-pneumatc valves to control the flud flow of a PMA,.e. the contnuously actng servo/proportonal valves and the on off swtchng valves [8]. Although the servo/proportonal valve has hgh control accuracy, t s expensve and tends to be bulky compared to the on off swtchng (or fast swtchng) valve. Therefore, the trajectory control methods of PMA utlzng fast swtchng valves have been extensvely studed n recent years. Kmura [10] appled feedback lnearzaton control method to deal wth the nonlnearty caused by hysteress of the PMA, whch was verfed by the experment of a sngle pneumatc muscle system. Repperger [11] developed a three-element phenomenologcal model for the PMA, and ntroduced a varable structure controller based on feedback lnearzaton control method to predct ts response. However, the feedback lnearzaton method requres an accurate model and all states, f there were uncertan parameters or unmodeled dynamc characterstcs, the robustness of the control system could not be ensured. Based on the three-element phenomenologcal model, Amato [12] used the robust control strategy to study the Correspondng author. E-mal: luht@tju.edu.cn Supported by Natonal Natural Scence Foundaton of Chna (Grant No. 51405331)
trajectory trackng control of a robotc arm actuated by PMAs. Zhu [13] presented an adaptve robust posture controller to compensate the parametrc uncertantes and uncertan nonlneartes of a parallel manpulator actuated by PMAs. Although these proposed controllers can effectvely handle the nonlneartes caused by the frcton forces of PMAs, they are unable to compensate the rapd change uncertan quanttes. Shen [14] developed a sldng model controller, of whch the effectveness s demonstrated expermentally by step response and snusodal trackng at dfferent frequences. However, due to the nerta, delay, and measurement errors of the system, ths controller wll generate hgh frequency vbraton, and the gas consumpton s large. Therefore, there s stll a need to further nvestgate the precson trackng control of PMA usng fast swtchng valves. In consderaton of ths ssue, ths paper deals wth the moton control of PMA usng fast swtchng valve. The rest of ths paper s organzed as follows. In secton 2 the expermental system s brefly ntroduced. In secton 3, the statc model, dynamc model, and hysteress model of the PMA are systematcally derved. Then, a close-loop control scheme s proposed to acheve hgh accuracy trajectory trackng control of the PMA, and the smulaton s carred out n the envronment of MATLAB/Smulnk n secton 4 before conclusons are drawn n secton 5. 2 System descrpton Fg. 1 shows the schematc dagram of the pneumatc muscle actuator. The components used here are gven n Table 1.The ar compressor connects the fast swtchng valve wth throttle valve and reservor. The fast swtchng valve 1 s nlet valve, called nlet valve for short; the fast swtchng valve2 s exhaust valve, called exhaust valve for short. Wthout the exctaton of the solenod, the valve wll keep closed due to return sprng. The workng process s as follows. Intally, the PMA connects wth the external envronment, and the nternal pressure of PMA s equal to atmospherc pressure. When the nlet valve s open, the compressed ar flows nto the PMA whle the exhaust valve s closed, resultng n the contracton of the PMA. When the PMA reaches the desred poston, the nlet and exhaust valves are closed at the same tme, then the PMA keeps the compressed ar nsde and mantans the current state. When nlet valve s closed and exhaust valve s open, the compressed ar s dscharged through the exhaust valve. Durng ths process, fast swtchng valves are controlled by PWM sgnals. In the begnnng, the nlet valve s fully opened to let the PMA quckly reach the desred poston; when t s approachng to the desred poston, the duty cycle of PWM s reduced proportonally to the dsplacement devaton. When the dsplacement devaton s smaller than a gven value, the duty cycle of PWM s set zero to avod the PMA oscllatng around the desred poston. Fg. 1. Schematc dagram of the pneumatc muscle actuator
Table 1. Components used n the system Component Type Company PMA DMSP-20-500N-RM-CM Festo Dsplacement sensor TEX 0150 415 002 205 Novotechnk Pressure sensor SDE1-D10-G2-WQ4-L-PU-M8-G5 Festo Fast swtchng valve MHE2-MS1H-3/2G-QS-4-K Festo Data acquston card PCI-6254-779071-01 Natonal Instruments 3 Modelng 3.1 Statc Model of PMA The statc model of a PMA gves a nonlnear relatonshp between the contractng force, the appled pressure, and the PMA length. There are several methods for developng statc models of a PMA, such as geometry analyss, the prncple of energy conservaton, and emprcal method. These models always rely on smplfyng assumptons, for example, neglectng of the rugby-ball shape forms at ether end of the PMA or the thckness of the PMA bladder, etc. These assumptons results n the nconsstency of predcted result and practcal measurement, and make the precse poston control of pneumatc muscle hard to be acheved. Therefore, n ths paper, the statc model s establshed by means of sometrc experment. Fg. 2 shows the measured data of contractng force F const aganst nternal pressure P at dfferent constraned lengths x. It can be seen that for a gven length the varaton of P vs. F const s nearly lnear, but the slopes are vared for x. It ndcates that the contractng force can be formulated as a lnear functon of the nternal pressure and length [15] Fconst a( x) P b( x) (1) where a(x) and b(x) are the slope and the ntercept, respectvely. a(x) and b(x) can be evaluated as follows Fg. 2. Contractng force va the nternal pressure at dfferent constraned lengths
a( x) b( x) 2 0 3 j 0 a x b x j j (2) where a and b j are the coeffcents of the polynomal functon. These coeffcents can be obtaned by usng the least squares curve fttng tools n MATLAB, and are gven as follows 2 a( x) 0.000953 x 0.7979 x 370.7 b x x x x 3 2 ( ) 0.0004561 0.5668 234.6 32560 (3) Substtutng Eq. (3) nto Eq. (1), one can obtan the statc model of PMA. Based on ths model, the varaton of the length of the PMA vs. ts nternal pressure can be obtaned once gven the load and nternal pressure usng the block-dagram shown n Fg. 3. Fg. 3. Block-dagram of PMA s statc characterstc model 3.2 Dynamc Model of PMA Snce the PMA s constructed by an elastc nylon materal rubber covered by a mesh of nextensble threads, t can be assumed that durng chargng and dschargng process are n sothermal and adabatc states. The relatonshp of the mass of ar, muscle volume, and nternal pressure obeys the polytropc gas law [16] k V P const m (4) where P and V are the nner pressure and volume of PMA; m s the ar mass nsde the PMA; k s the polytropc exponent. Takng the total dfferental of Eq. (4) leads to m PV kpv kpv (5) m Then, substtutng the deal gas law nto Eq. (4) results n PV mrt (6) k P mrt PV (7) V
Fg. 4. Block-dagram of PMA s dynamc characterstc model From Eq. (7), the correspondng Smulnk block-dagram can be developed as shown n Fg. 4, where the nput s the ar mass flow rate and the output s the nternal pressure of PMA. Assumng that the volume of PMA s constant durng chargng and dschargng, then pressure of PMA can be obtaned by takng ntegraton of Eq. (7). PV 0 and the nternal 3.3 Hysteress Model of PMA In ths secton, the Prandtl-Ishlnsk (P-I) model [17] wll be used to derve the hysteress of the PMA. It has two advantages: frst, t s smpler compared wth other models because t only consst of lnear play operators; second, the nverse PI model can be obtaned analytcally, whch s easer for realzaton of hysteress compensaton. The elementary operator of the PI model s lnear play operator, whch can be mathematcally llustrated by Fg. 5. Fg. 5. Lnear-play operator Its th lnear play operator can be expressed as whle the ntal condton s The output of PI model s n y ( k) max x( k) r,mn x( k) r, y ( k 1) (8) 0 y (0) max x(0) r,mn x(0) r, y (9) T ω Hr y 0 (10) y( k) y ( k) max x( k) r,mn x( k) r, y ( k 1) = [ x( k), ] 1 1 n where H r denotes the lnear play operator; ω = [ω 1,, ω n] T s the weghtng vector; r = [r 1,, r n] T s the threshold vector; x and y are the nput and output of the operator, respectvely; y 0 s the ntal state; k s the samplng number of the operator; and n s the number of the lnear play operator.
0.25 0.2 Contracton Rato 0.15 0.1 To determne the parameters of PI model, the threshold vector r s frstly determned by the followng equaton 0.05 Expermental Result Identfcaton Result 0 0 1 2 3 4 5 6 Pressure(bar) Fg. 6. Comparson of the expermental result and the PI model r max{ x( t)} 1,, n n 1 (11) Then, usng the least square method (LSM) the weghtng vector ω can then be determned. Fg. 6 shows the comparson of the pressure/length hysteress characterstcs obtaned from the experment and the PI model usng the dentfed parameters. The result clearly shows that the PI model can effectvely characterze the hysteress loop of the PMA. From Eq. (8), the nverse model of PI model can be formulated as x( k)=max{ y( k) r,mn{ y( k) r, x( k 1)}} (12) n x( k) max{ y( k) r,mn{ y( k) r, x ( k 1)}} (13) The nverse hysteress model can then be formulated as follows. 1 x k H y k ω H y k x (14) 1 T ( ) ( ( )) r[ ( ), 0] where H r denotes the nverse PI operator; ω =[ω 1,, ω n] T s the weghtng vector; r =[r 1,, r n] T s the threshold vector; x 0 s the ntal state. The parameters of the nverse hysteress model can be derved as 1 1 = 1 = 2,, n 1 1 j 1 j j2 j2 r r r 1,, n j j j1 1 n j j j j1 j x (0) y (0) y (0) 1,, n (15) Obvously, the weghtng vector and threshold vector of PI model are used to obtan parameters of the nverse PI hysteress model that wll be used for feed-forward compensaton control.
3.4 Fast Swtchng Valve Model The process of ar flowng through the valve port s very complex, whch s often modelled as Sanvlle flow equatons [18]. It has been shown that the nfluence of the change of PWM sgnal frequency to gas flow rate s neglgble [19], when the frequency ranges are between 100Hz to 180Hz. Thus, the mass flow rate can be expressed as a functon of the duty cycle and the effectve orfce area. 2 k 1 p k k u 2k pd pd pd da m T R( k 1) p u u p u pu m 1 p 2 k 1 u 2k dam T u k 1 R( k 1) p p d u 0.528 0.528 (16) where m s the ar mass flow rate of fast swtchng valve; p u s the upstream pressure; p d s the downstream pressure; T u s the upstream temperature; k s the rato of specfc heat; d s duty cycle of PWM sgnals; A m s the effectve orfce area of fast swtchng valve. In reference [20], the effectve orfce area A m of the MHE2-MS1H-3/2G-M7-K fast swtch valve s 1.8194 10-6 m 2. Fg. 7. Smulnk model of fast swtchng valve Here, the Functon model n MATLAB/Smulnk s used to buld the smulaton model of Eq. (16) for the ar mass flow rate of fast swtchng valve. The correspondng Smulnk model s gven n Fg. 7, where u s the duty cycle of PWM sgnals, t s the temperature of ar, and p 1 s the nternal pressure of PMA. The smulaton of the chargng and dschargng of PMA s carred out as follows: durng the chargng process, u>0 and p up s ar source pressure, whle p down s the nternal pressure, p down= p 1; durng the dschargng process, u<0 and p up s the nternal pressure, p up = p 1, and p down s the atmospherc pressure. 3.5 Generaton of PWM There are varous methods to generate PWM sgnal n MATLAB/Smulnk. Here, the pulse sgnal modulaton method ntroduced n reference [21] s used, whch has the advantage of lnear relatonshp. Fg. 8. Block-dagram of generaton of PWM sgnal
The steps of generaton are gven as follows: frst, generatng a reference pulse sgnal sequence wth 50% duty cycle, then gettng the standard trangular wave sgnal after bas and ntegral, and fnally obtanng the requred PWM sgnal after bas of the former standard trangular wave sgnal. Fg. 6 shows the block-dagram of ths method. The pulse generator module s used for obtanng a 50% duty cycle square wave wth magntude of 2. After bas and ntegraton, a trangular wave modulaton sgnal can be acheved for generatng modulated PWM wave. The resdual between the nput sgnal and the trangular wave modulaton sgnal s nputted nto the Relay module. If the nput s larger than 0, the output of Relay module s 1 (smulatng the nflaton process); otherwse, the output s -1 (smulatng the deflaton process). Fnally, the nput sgnal s modulated n the PWM wave to control the fast swtchng valves. 4 Moton control method and smulaton To compensate the nonlnear hysteress of the PMA, the nput feed-forward and output feedback are combned n the moton control method. The hysteress nverse model provdes a control nput, whch represents the functon of a desred trajectory to keep the output followng the desred trajectory. It s effcent for low-frequency systems regardless of the creep and vbratons. The accuracy of the feed-forward control depends on the performance of hysteress model. Therefore, the feedback loop s used to deal wth the trackng error caused by hysteress modelng, and the combned control method provdes a hgh gan feedback and overcome creep and vbratons n the systems. The feedback loop s a PID controller, whch s gven as de() t (17) t u( t) K pe( t) K e( ) d K 0 d dt where e(t) s error sgnal; u(t) s output sgnal; K p s proportonal gan; K s ntegral gan; K d s dervatve gan. The control scheme s llustrated n Fg. 9. Hysteress compensaton based on nverse PI model s bult nto the control system through feed-forward processng. The controller of the nternal pressure and contracton rato uses a proportonal-ntegral (PID) controller. The correspondng Smulnk control scheme s shown n Fg. 10. Fg. 9. Moton control scheme for pneumatc muscle actuator Fg. 11 shows the smulaton results of step and snusod trajectory trackng responses. It can be found that the maxmal error s 1% (see Fg.11(b)). It ndcates that the proposed poston control method can acheve satsfactory accuracy. However, t should be noted that there s a poston bas (about 4 mm) n the max length of PMA n Fg.11(a). It s caused by the errors n the statc model of experment. Actually, the maxmal error n Fg.11(b) s also caused by these modelng errors.
Fg. 10. Smulnk control scheme for pneumatc muscle actuator 500 490 Reference Smulaton Result 500 480 480 460 Length(mm) 470 Length(mm) 440 420 460 400 450 440 0 1 2 3 4 5 6 7 8 9 10 Tmes(s) (a) Step response Fg. 11. Smulaton of the moton control method 380 Reference Smulaton Result 360 0 1 2 3 4 5 6 7 8 9 10 Tmes(s) (b) Snusod trackng response 5 Concluson Ths paper presents an alternatve approach for accurate moton control of PMA drven by fast swtchng valves. The statc model of the PMA s derved usng a polynomal based least square fttng of the data from sometrc experment. Parameters of the PMA hysteress model are dentfed usng LMS by fttng the pressure-dsplacement hysteress loop. Subsequently, a close-loop control scheme s desgned to acheve precse trackng control of PMA, whch combned the nverse hysteress compensaton feed-forward and the feedback PID controllers. The smulaton results ndcate that the proposed models and control scheme are able to acheve satsfactory accuracy. References 1. Tsagaraks N G, Caldwell D G. Development and control of a soft-actuated exoskeleton for use n physotherapy and tranng[j]. Autonomous Robots, 2003, 15(1): 21-33. 2. Vo-mnh T, Tjahjowdodo T, Ramon H, etal. A new approach to modelng hysteress n a pneumatc artfcal muscle usng the Maxwell-slp model[j]. IEEE/ASME Transactons on Mechatroncs, 2011, 16(1): 177-186. 3. Obajulu S C, Roche E T, Pgula F A, etal. Soft pneumatc artfcal muscles wth low threshold pressures for a cardac compresson devce[c]//asme 2013 Internatonal Desgn Engneerng Techncal Conferences and Computers and Informaton n Engneerng Conference. Amercan Socety of Mechancal Engneers, 2013, 1-8. 4. Vo-mnh T, Kamers B, Ramon H, et al. Modelng and control of a pneumatc artfcal muscle manpulator jont Part I: Modelng of a pneumatc artfcal muscle manpulator jont wth accountng for creep effect[j]. Mechatroncs, 2012, 22(7): 923-933.
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