Design Optimization of Stewart Platform Configuration for Active Vibration Isolation

Transcription

1 Indan Journal of Scence and echnology, Vol 8(2, DOI:.1748/jst/1/v82/896, September 1 ISSN (Prnt : ISSN (Onlne : Desgn Optmzaton of Stewart Platform Confguraton for Actve Vbraton Isolaton Rahmath Ulla Bag* and S. Pugazhenth School of Mechancal Engneerng, SASRA Unversty hanjavur - 611, aml Nadu, Inda; rub786@gmal.com, pugazhenth@mech.sastra.edu Abstract Optmzaton of desgn s of Stewart Platform (SP for Actve Vbraton Isolaton (AVI usng Genetc Algorthm (GA traned neural network was carred out and effect of desgn consdered for optmzaton are studed. Modellng of Stewart platform for sx Degree Of Freedom (DOF AVI was carred out usng MALAB/SIMULINK. he need for the study of sx DOF vbraton control s felt serously n modern day applcatons such as hgh resoluton measurement, mcro manufacturng, etc. Stewart platform has several features whch make them partcularly attractve for sx DOF AVI. he man feature of Stewart platform for AVI s that t can be drectly placed n the path of solaton. he SIMULINK model developed s optmzed usng GA traned neural network and the effect of desgn s consdered for optmzaton are studed. he effects of desgn are analyzed for solaton of foundaton from machnery vbraton and senstve equpment from vbratng floor. he optmal s of SP offer a better solaton performance whle the control s are unaltered. Keywords: Actve Vbraton Isolaton, Genetc Algorthm, Neural Network, Optmzaton, Stewart Platform 1. Introducton It s an mperatve requrement of any engneerng desgn to manage vbraton wthn acceptable levels, as vbratons degrade the performance of equpment and machnery. Vbraton suppresson at the desgn phase s lmted because of functonalty of the equpment or vbraton may be transmtted from the surroundng to the equpment 1. Applcatons of vbraton management could be seen n our day to day lfe, from cars to buldngs and from spacecrafts to submarnes. Improper vbraton management would lead to performance degradaton and poor qualty. o downscale the catastrophc effect of vbraton, vbraton solaton s conceved. In vbraton solaton the equpment or the machne of concern s solated from the harmful vbraton thus achevng good vbraton control. A classc approach to the vbraton solaton problem s by passve solaton.e., by provdng rubber pads, mechancal sprngs and dampers to suppress the ampltude of vbraton. Although passve solaton offers a smple and relable soluton of protectng precson equpment from a vbraton envronment, t has ts own performance lmtatons. Passve solaton lacks n versatlty & effectveness and also results n large sze & weght when used for lowfrequency 2. hs necesstates Actve Vbraton Isolaton (AVI system. In AVI the equpment of concern s actvely solated by actuatng n opposte drecton to that of vbraton. hus AVI system wll have consderably stronger suppresson of vbratons as compared to ordnary passve solaton. Actve solaton of equpment or the surroundng from harmful vbraton enhances the functonalty of the system. Vbraton s generally conceved as a multple Degree Of Freedom (DOF phenomenon as the dsturbng force may emerge from dfferent drectons 4. Efforts are on, n utlzng the concept the concept of Stewart platform (a sx DOF manpulator for mult DOF AVI -11. Researchers are on the lookout for utlzng the potental advantages of Stewart Platform (SP for AVI. he man advantage of *Author for correspondence

2 Desgn Optmzaton of Stewart Platform Confguraton for Actve Vbraton Isolaton usng the Stewart platform for AVI s that t can be drectly kept n the path of dsturbance. Fgure 1 and 2 llustrates the use of Stewart platform for AVI. Present work s amed at studyng the effect of desgn s used n optmzng the confguraton of Stewart platform based AVI, for better solaton. In order to study the effect of s, modelng of SP s carred out usng MALAB/SIMULINK as detaled n secton II and optmzaton of the model developed n MALAB/SIMULINK s carred usng Genetc Algorthm (GA traned Artfcal Neural Network (ANN as detaled n secton. Secton 4 descrbes the effect of desgn s. Fgure. A general Stewart platform. 2. Modellng of Stewart Platform hs secton presents the modelng of Stewart platform for AVI usng MALAB/SIMULINK. A Stewart platform bascally comprse of two platforms connected by means of sx actve legs. he moton of sx actve legs determnes the poston and orentaton of the movng platform wth respect to the fxed base. he sketch of a general Stewart platform s depcted n Fgure. Modellng was carred out consderng the confguraton detals of the prototype workng model of a Stewart platform developed at SASRA Unversty. Fgure 4 shows the physcal model based on whch MALAB model has been created to explore the benefts of Stewart platform Fgure 1. Stewart platform solatng foundaton from machnery vbraton. Fgure 4. A PC controlled workng model of a SP. for AVI. In order to use Stewart platform for AVI to counter any dsturbances, nverse knematcs of Stewart platform has to be performed. From Fgure the transformaton of the movng platform wth respect to the fxed base can be descrbed by the poston vector p of the centrod, P and the rotaton matrx, A R B, of the movng platform. A R B for a RPY (Roll, Ptch, and Yaw wrst s gven by, cos q2.cosq sn q1.sn q2.cos q cos q1.snq cos q1.sn q2.cosq + sn q.snq 1 R = cos q2.snq sn q1.sn q2.sn q + cos q1.cosq cos q1.sn q2.snq sn q1.cos q snq2 sn q1.cos q2 cos q1.cos q 2 Where q, q and q are the roll, ptch and yaw motons respectvely. A B 1 2 (1 Let a = [a x a y a z ] and B b = [b u b v b w ] be the poston vectors of pont A and B n the coordnate frames A and B, respectvely. he vector loop equaton for the th lmb of the manpulator can be wrtten as, Fgure 2. floor. AVI of Senstve equpment from vbratng AB = p+ R. b a (2 A he length of the th lmb s obtaned by takng the dot product of the vector AB wth tself and, B d 2 A B A B = p+ RB. b a p+ RB. b a, for = 1,2,...,6 ( B 2 Vol 8 (2 September 1 Indan Journal of Scence and echnology

3 Rahmath Ulla Bag and S. Pugazhenth Where d denotes the length of the th lmb. Expandng equaton ( yelds, 2 B B A B A B d = p p+ b b + a a + 2p RB b 2p a 2 RB b a (4 For the nverse knematcs problem, the poston vector p and rotaton matrx A R B of frame B wth respect to A are gven and the lmb lengths d, = 1, 2 6, are to be found. he square root the equaton (4 gves, B B A B A B d =± p p+ b b + a a + 2p RB b 2p a 2 RB b a ( Equaton ( s the nverse knematc equaton, where d s the requred dsplacement n each leg correspondng to the dsturbance. he nverse knematc equaton s modelled n MALAB/SIMULINK. he poston and orentaton changes of moble platform are taken as the nput to the model and the requred changes n length of each leg are computed. he overall model for solatng the foundaton from machnery vbraton s modelled n MALAB/SIMULINK and s depcted n Fgure. he excter acts as the source of dsturbance gvng nput to the nverse knematc equaton model, whch calculates the desred changes n leg lengths to actvely solate the base platform from the vbraton gven by the excter on the movng platform. In the model for the second case where senstve equpment needs to be solated from ground vbratons, the excter at the base acts as the source of dsturbance gvng nput to the nverse knematc equaton model, whch calculates the desred changes n leg lengths to actvely solate the movng platform as depcted n Fgure 6.. Optmzaton of Desgn Parameters for Performance Enhancement of AVI he objectve s to fnd the optmal desgn s whch govern the performance of Stewart platform for AVI. he physcal s of Stewart platform consdered for optmsng the confguraton of Stewart platform for AVI are: Base trangle dstance (B t, Base jont dstance (B j, Moble platform trangle dstance (P t, Moble platform jont dstance (P j and heght of the Stewart platform (h at the home confguraton when all the legs are of dentcal length, as shown n Fgure 7. In order to arrve at an optmal desgn, the physcal desgn s of the Stewart platform are vared keepng the control aspect of the model constant. A smple PID controller wth constant controller gan s consdered. he goal of actve vbraton solaton s to have low ampltude vbraton at resonance ( r and provde good attenuaton of db at hgh frequency. Fgure 6. Model of the 6 DOF AVI system to solate senstve equpment. h Fgure. Model of the 6 DOF AVI system to solate the foundaton. Fgure 7. Desgn s consdered for optmzaton of SP for AVI. Vol 8 (2 September 1 Indan Journal of Scence and echnology

4 Desgn Optmzaton of Stewart Platform Confguraton for Actve Vbraton Isolaton A new term transmssblty ndex for AVI s coned to address both the aspects of vbraton solaton smultaneously and s gven by: ransmssblty at resonancendb ransmssblty ndex for AVI, a = ransmssblty at hgh frequency ndb Hgher the value of α, better wll be qualty of the solaton (assumng that the hgh frequency transmssblty wll always be on the negatve sde and transmssblty at resonance s on the postve sde. For practcal purposes and calculaton, the hgh frequency s consdered to be Hz for the present model. Intally, he SIMULINK model s run wth desgn s generated by GA and correspondng transmssblty ndex s noted. When GA s used by keepng the SIMULINK model of SP n the search loop, t takes longer computatonal tme to fnd the transmssblty ndex for each confguraton of the platform. So n order to quckly as well as more exhaustvely search for the optmal set of s of SP for AVI, Artfcal Neural Network (ANN based desgn tool s used. Fve hundred dfferent sets of desgn s and correspondng transmssblty ndex those obtaned through GA optmzaton are used to tran the neural network. o fnd the optmal desgn s of SP for AVI usng GA traned ANN, randomly generated set of desgn s (B t, B j, P t, P j, h wthn the specfed range as specfed n able I are gven as nput to the traned network and the correspondng output of transmssblty ndex s obtaned. he maxmum value of transmssblty ndex obtaned durng random search represents the optmal confguraton of SP for the gven nput desgn s. he optmal s thus found usng GA traned ANN are tabulated n able II for solatng foundaton from machnery vbraton and for solatng senstve equpment from ground vbraton. ransmssblty characterstc of the optmal confguraton for all frequences are acqured and the transmssblty curve s plotted. For solatng foundaton from machnery vbraton the transmssblty wthout AVI control s hgh at resonance (6 db and hgh frequency attenuaton s only -14 db at Hz whch shows a compromse between transmssblty at resonance and hgh frequency, a typcal characterstc of passve solaton as shown n Fgure. 8. Wth AVI control there s a smaller peak of 7 db at resonance and hgh frequency attenuaton s -47 db at Hz. It s evdent able 1. Sl. No Range of desgn s and constrants Parameter Range of value vared n the optmzaton process (cm 1 Heght (h able 2. Sl. No Base trangle dstance (B t Dstance between the jonts of base (B j Movng platform trangle dstance (P t Dstance between the jonts of movng platform (P j Assocated constrants Physcal constrant mposed by the stroke of the actuator Jont angle lmtaton - Mnmum dstance between two sphercal jonts and jont angle lmtaton -2 Jont angle lmtaton -1 Optmal s of SP for AVI Parameter Mnmum dstance between two unversal jonts and Jont angle lmtaton Optmal Optmal s of s of SP for solatng SP for solatng foundaton (cm equpment (cm 1 Heght (h Base trangle dstance (B t Dstance between the jonts of base (B j Movng platform trangle dstance (P t Dstance between the jonts of movng platform (P j that optmal desgn confguraton of the SP results n effectve solaton. In the case of solatng senstve equpment from floor vbraton the transmssblty wthout AVI control s hgh at resonance (2 db and hgh frequency attenuaton s only -24 db at Hz. Wth AVI control there s a smaller peak of 6 db at resonance and hgh frequency attenuaton s -4 db at Hz as shown n Fgure 9. Optmal confguraton of SP s effectve n solatng the senstve equpment. 4 Vol 8 (2 September 1 Indan Journal of Scence and echnology

5 Rahmath Ulla Bag and S. Pugazhenth 6 Vertcal ransmssblty (db Wth AVI control Wthout AVI control ransmssblty at resonance (db Effect of B j Base Jont Dstance, Bj (cm ransmssblty at Hz, (db Effect of B j Base Jont Dstance, Bj (cm Frequency (Hz Fgure 8. Vertcal ransmssblty (db ransmssblty wth and wthout AVI. W th AV I control W thout AV I control ransmssblty at Resonance, (db ransmssblty at Resonance, (db Effect of B t Base rangle Dstance, Bt (cm Effect of P j Platform Jont Dstance, Pj (cm ransmssblty at Hz, (db ransmssblty at Hz, (db Effect of B t Base rangle Dstance, Bt (cm Effect of P j 2 4 Platform Jont Dstance, Pj (cm 1 Fgure F requency ( H z ransmssblty wth and wthout AVI. ransmssblty at Resonance (db Effect of P t ransmssblty at Hz, (db Effect of P t 4. Effect of Desgn Parameters he effect of desgn s consdered for optmzng the confguraton of SP for AVI for two cases of solaton are dscussed. 4.1 AVI of Foundaton from Machnery Vbraton he effects of desgn s are studed at two dfferent frequences both at resonance (2.7 Hz and at a hgh frequency of Hz, to analyse the qualty of solaton n terms of ampltude reducton at resonance and hgh frequency attenuaton. One by one, a under consderaton alone s vared keepng the other s constant at ther respectve optmal values. Fgure represents the varaton of transmssblty at resonance and at hgh frequency. he value at whch transmssblty s found mnmum s tabulated n able III. As the parametrc study s carred out at two dfferent frequences, the optmal confguraton s result of trade-off of desgn between these two frequences. ransmssblty at Resonance (db Platform rangle Dstance, Pt (cm Effect of h Heght of the Platform, h (cm At resonance Platform rangle Dstance, Pt (cm Heght of the Platform, h (cm Fgure. ransmssblty varaton at resonance and hgh frequency whle solatng foundaton Heght (h ransmssblty at Hz, (db Effect of h At hgh frequency Attenuaton at resonance s found to be good and n the acceptable range f the heght of the platform s between 42 and 2 cm. he effect of heght at hgh frequency called for a heght between and cm. he optmal heght from the ANN results s 48 cm Base rangle Dstance (B t he value of B t was requred to be greater than 48 cm at resonance and 8 cm at hgh frequency. he effect of B t at hgh frequency has a profound effect on optmal confguraton of SP. hs s seen n the optmal confguraton of SP wth B t value of 8 cm. Vol 8 (2 September 1 Indan Journal of Scence and echnology

6 Desgn Optmzaton of Stewart Platform Confguraton for Actve Vbraton Isolaton able. Sl. No Parameters of SP for solatng foundaton Parameter (cm Optmal range at resonance Optmal range at hgh frequency Optmzed confguraton 1 Heght (h Base trangle dstance (B t Dstance between the jonts of base (B j Movng platform trangle dstance (P t Dstance between the jonts of movng platform (P j > possble as mnmum as possble possble possble 7 < 2 16 possble 4.1. Dstance between the Jonts of Base (B j he base jont dstance needs to be smaller to reduce transmssblty at resonance as well as to have good hgh frequency attenuaton. Whle satsfyng the need for smaller jont dstance, the physcal constrant of mnmum jont dstance should be satsfed. hs s the reason for optmzed value of B j beng 7 cm. studed by keepng the other desgn s at optmal values as depcted n Fgure 11. he desgn s at whch the transmssblty s found mnmal s tabulated n able IV Heght (h he transmssblty at resonance was mnmal, f the heght of the platform s between 8 and 6 cm, smlar effect has been seen at resonance. he optmal heght of the platform s found to be at 6 cm Base rangle Dstance (B t he transmssblty at resonance s mnmal for values of base trangle dstance rangng from 2 to cm and for good hgh frequency attenuaton the value of B t should be placed ransmssblty at Resonance, (db ransmssblty at Resonance (db Effect of B j Base Jont Dstance, Bj (cm Effect of B t Base rangle Dstance, Bt (cm ransmssblty at Hz (db ransmssblty at Hz (db Base Jont rangle, Bj (cm Effect of B j Effect of B t Base rangle Jont, Bt (cm Movng Platform rangle Dstance (P t he effect of P t at resonance as well as at hgh frequency s smlar, requrng t to be lower whch resulted n a lower P t value n the optmal confguraton of 16 cm Dstance between the Jonts of Movng Platform (P j As the dstance between the jont ncreases the transmssblty also ncreases at resonance as well as at hgh frequency resultng n Pj value to be cm. 4.2 AVI of Senstve Equpment from Floor Vbratons In ths case also, the ndvdual effects of the desgn s are studed at two dfferent frequences, one at resonance and the other at hgh frequency. Each desgn s vared and ts effect on transmssblty s ransmssblty at Resonance (db ransmssblty at Resonance (db ransmssblty at Resonance (db Effect of P j Platform Jont Dstance, Pj (cm Effect of P t Platform rangle Dstance, Pt (cm Heght of the Platform, h (cm At resonance ransmssblty at Hz (db ransmssblty at Hz (db ransmssblty at Hz (db Platform Jont Dstance, Pj (cm Platform rangle Dstance, Pt (cm Effect of P j Effect of P t Heght of the Platform, h (cm At hgh frequency Fgure 11. ransmssblty varaton at resonance and hgh frequency whle solatng equpment. 6 Vol 8 (2 September 1 Indan Journal of Scence and echnology

7 Rahmath Ulla Bag and S. Pugazhenth able 4. Sl. No Parameters of SP for solatng equpment Parameter (cm Optmal range at resonance Optmal range at hgh frequency Optmzed confguraton 1 Heght (h Base trangle dstance (B t Dstance between the jonts of base (B j Movng platform trangle dstance (P t Dstance between the jonts of movng platform (P j possble possble possble 9 < 6 at 2 cm. herefore, the value of B t n the optmal confguraton of SP stands at 2 cm agreeng wth the results Dstance between the Jonts of Base (B j he base jont dstance needs to be smaller to reduce transmssblty at resonance as well as to have good hgh frequency attenuaton. hs s the reason for optmzed value of B j beng 9 cm Movng Platform rangle Dstance (P t he value of P t for lower transmssblty at resonance s 12 cm. ransmssblty at hgh frequency s mnmal for mnmum platform trangle dstance. As we ncrease the platform trangle dstance, the hgh frequency attenuaton remans unchanged tll 1 cm of P t and hgh frequency attenuaton becomes poor afterwards. Combned effect of P t at resonance and at hgh frequency results n the optmal value beng 12 cm. 4. Effect of Stffness on AVI of SP A plot of stffness aganst heght of the Stewart platform s shown n Fgure 12. Stffness ncreases wth ncrease n heght ntally and then t decreases. he workng heght of the platform should satsfy workspace and sngularty constrants. Generally, a cubc confguraton ensures better AVI compared to non-cubc confguraton for same control strategy because cubc confguraton ensures orthogonalty and symmetry n all the drectons 12. On the other hand, t s requred to have a stff platform whle solatng foundaton from machnery vbraton 1. herefore, the optmzed heght of the Stewart platform found n the current research s between the heght of the stff platform and heght of cubc confguraton as depcted n Fgure 12. he optmal heght of 48 cm offers moderate value of stffness wthn the work volume of the Stewart platform satsfyng the geometrc constrants of both reachable jont angles and attanable leg lengths. It s requred to have a soft platform whle solatng senstve equpment from vbratng floor 1. he optmal heght of the platform for AVI of senstve equpment from vbratng floor has been found to be 6 cm, ths heght offers a low value of stffness. he heght of the platform thus optmzed s n between the heght of the soft platform and the heght of cubc confguraton as shown n Fgure 12.. Concluson A methodology to optmze the desgn confguraton of Stewart platform for the best solaton effectveness s developed. Optmzaton of desgn s of Stewart platform for AVI s carred out for both the cases, AVI of foundaton from machnery vbraton as well as for AVI of senstve equpment from vbratng floor. he effect 4.2. Dstance between the Jonts of Movng Platform (P j he requrement of P j at resonance s to have mnmal dstance between sphercal jonts. For hgher attenuaton at hgh frequency P j should be less than cm. For hgher values of platform jont dstance, transmssblty ncreases and attenuaton suffers. Hence, the optmal value s found to be at 6 cm. Fgure 12. Effect of Stffness. Vol 8 (2 September 1 Indan Journal of Scence and echnology 7

8 Desgn Optmzaton of Stewart Platform Confguraton for Actve Vbraton Isolaton of desgn s consdered for optmzaton are studed. In the process of optmzaton t s found that, by just usng optmal confguraton of Stewart platform for AVI, the platform offers a better performance even though the control strategy used s unaltered. It s hghly heartenng to note that by choosng the optmal desgn s whch needs only mnor changes n desgn, the transmssblty can be controlled. 6. References 1. El Madany MM, El-amm A. On a subclass of nonlnear passve and sem-actve dampng for vbraton solaton. Comput Struct. 199; 6(: zou HS. Desgn of a pezoelectrc excter/actuator for mcro-dsplacement control: theory and experment. PRECIS ENG. 1991; 1(2:4.. Sun LL, Hansen CH, Doolan C. Evaluaton of the performance of a passve actve vbraton solaton. MECH SYS SIGNAL PR. 1; 1: Vose H, urpn MH, Dames PM, Umbanhowar P, Lynch KM. Modelng, desgn and control of 6-DoF flexure-based parallel mechansm for vbratory manpulatons. Mech Mach heory. 1; 64:111.. ao Y, Ja M, Zeng-Guang H, Fengshu J, Mn. Robust backsteppng control of actve vbraton solaton usng a Stewart platform. Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton; Japan; 9. p Yao Z, JIngru Z, Shje X. Parameter desgn of vbraton solaton platform for control moment gyroscopes. Acta Astranautca. 12; 81(2: Daley S, Johnson FA, Pearson JB, Dxon R. Actve vbraton control for marne applcatons. Control Eng Pract. 4; 12(4: Preumont A, Francos A, Bossens F, Abu-Haneh A. Force feedback versus acceleraton feedback n actve vbraton solaton. J Sound Vb. 2; 27(4: Gexue R, Quha L, Nng H, Rendong N, Bo P. On vbraton control wth Stewart parallel mechansm. Mechatroncs. 4; 14(1:1 1.. Hauge GS, Campbell ME. Sensors and control of a space based sx axs vbraton solaton system. J Sound Vb. 4; 269(-: Yun Y, L Y. A general dynamcs and control model of a class of mult-dof manpulators for actve vbraton control. Mech Mach heory.11; 46(: Geng ZJ, Haynes LS. Sx Degree-of-Freedom Actve Vbraton Control usng Stewart platforms. IEEE ransacton on Control Systems echnology. 1994; 2(1:4. 1. Preumont A, Horodncaa M, Romanescua I, de Marneffea B, Avraama M, Deraemaekera A, Bossensb F, Abu Haneh A. A sx-axs sngle-stage actve vbraton solator based on Stewart platform. J Sound Vb. 7; (-: Vol 8 (2 September 1 Indan Journal of Scence and echnology