Tensegrity Flight Simulator

Transcription

1 JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vo. 3, No. 6, November December 000 Tensegrity Fight Simutor Corne Sutn nd Mrtin Coress Purdue University, West Lfyette, Indin nd Robert E. Sketon University of Ciforni, Sn Diego, Ciforni In thispperwe propose new motionsimutorbsed on tendon-controedtensegrity structure. The simutor is equipped with noniner controer tht chieves robust trcking of desired motions. The controer prmeters cn be tuned to gurntee trcking to within prespeci ed toernce nd with prescribed rte of exponenti convergence. The design is veri ed through numeric simutions for speci c ongitudinmotions of symmetric ircrft. Nomencture b = side ength of the bse nd cbin tringes C d = erodynmic coef cient of drg C = erodynmic coef cient of ift C M = erodynmic coef cient of the pitching moment c = men erodynmic chord of ircrft wing D = drg D r = rest-ength of digon tendons D 0 = ength of digon tendons in prestressbe con gurtion g = grvittion cceertion constnt of the Erth h g = overp I = identity mtrix J = inerti mtrix of the simutor s second stge J = ircrft moment of inerti round the pitch xis K = Hessin of the potenti energy k = tendon stiffness k i = stiffness of the ith ctive tendon k n = stiffness of the nth tendon L = ift = ength of brs i = ength of the ith ctive tendon n = ength of the nth tendon r i = rest-ength of the ith ctive tendon rn = rest-ength of the nth tendon M(q) = tensegrity simutor inerti mtrix M t = mss of the second stge M y = erodynmic pitching moment bout the ircrft mss center m = ircrft mss n u = number of ctive tendons P = pretension coef cient q = generized coordintes vector for simutor q = generized coordintes vector for ircrft q g = simutor equiibrium con gurtion q 0 = simutor prestressbe con gurtion q = trcking error r = trcking toernce S = ifting surfce of the wing S r = rest-ength of sdde tendons S 0 = ength of sdde tendons in prestressbe con gurtion Received 10 My 1999; revision received Februry 000; ccepted for pubiction 7 Februry 000. Copyright c 000 by the Americn Institute of Aeronutics nd Astronutics, Inc. A rights reserved. Reserch Assistnt, Schoo of Aeronutics nd Astronutics; currenty t Moecur Geodesics Inc., Boston, MA Member AIAA. Professor, Schoo of Aeronutics nd Astronutics. Professor, Deprtment of Engineering Mechnics. Feow AIAA T D = tension in digon tendons in prestressbe con gurtion T S = tension in sdde tendons in prestressbe con gurtion T n = tension in the nth tendon T r = ircrft thrust T = vector of tensions in ctive tendons T p = vector of tensions in pssive tendons t = time V = potenti estic energy v = ircrft speed X, Y, Z = Crtesin coordintes of the mss center of the second stge X = ircrft horizont rnge Z = ircrft titude = ircrft nge of ttck = rte of exponenti convergence e = eevtor de ection nge f = p de ection nge = ircrft pitch nge = ir density = time constnt,, = Euer nges of the second-stge reference frme y = ircrft pitch rte Introduction The credit for mking simuted ight reity goes to Edwin Link for providing vition with the rst xed-bsed ight triner. Link s triner of 193 ws, in the inventor s words, prt pino, prt pipe orgn, nd itte bit of irpne, 1 nd it pyed mjor prt in vition trining for the rst 0 yers of commerci nd miitry ight. Since then the technoogy of ight simution hs kept pce with the technoogic dvnces in the erospce industry. The six-degree-of-freedom (DOF) Stewrt ptform is the most popur motion bse for ight/motion simutors. A very dvnced six-dof Stewrt ptform ight simutor is the one operted by NASA Ames t Moffett Fied, Ciforni. This simutor, which becme opertion in 1993, is n exct repic of the cockpit of United Airines Boeing nd it hs unique reserch cpbiities. A detied description of this fciity cn be found in Suivn nd Soukup 3 nd Bke. 4 The Stewrt ptform is compex nd expensive mechnism becuse the contro of its motion presents gret technoogic dif - cuties becuse of the teescopicctutors. It is we-known fct in motion simutors circes tht the opertion of teescopic ctutor presents dif cuties, especiy t rge mpitudes nd rge cceertions. This is becuse of the rge nd rpid excursions of the ctutors, generting rge ods on the structure of the simutor. These ods, of both n inerti nd dissiptive nture, resut in hevy structure nd rge ssocited cooing system. The ife of

2 1056 SULTAN, CORLESS, AND SKELTON teescopicctutoris so imited becuse of the signi cnt wering tht occurs during its opertion. To reduce the compexity of the motion bse, there hs been renewed interest in reduced-dof simutors. However, retivey itte work hs been devoted to the design of reduced-dof motion bses or to the evution of the quity of motion senstions tht cn be produced by these devices. 5 Pouiot et. 6 nyzed the simution reism tht cn be chieved using motion bses with ony three DOF; the dvntge of the reduced compexity of the design is importnt for rge trnsport ircrft for which it produces good quity motion simution. The wek point of the proposed three-dof simutors is their inbiity to simute high frequency cceertions (see Pouiot et. 6 ). Aso, becuse the motion of the reduced DOF simutors is controed by teescopic ctutors, the probems ssocited with these ctutors re not eiminted. Another importnt probem for motion simutors is tht of contro system design. Mny contro synthesistechniqueshve been ppied to simutor motion contro system design, incuding cssic, dptive, 7,8 nd optim contro. 9,10 Reid nd Nhon 11 nd Nhon nd Reid 1 impemented cssic, dptive,nd optim contro gorithms on the University of Toronto Institute for Aerospce Studies six-dof ight simutor to obtin piot evutionsof the corresponding motion quity. They indicted tht cssic nd dptive gorithms re genery preferred by piots. As resut, combintion of the best fetures of cssic nd dptive schemes into hybrid scheme ws ter performed, incuding the incorportion of noniner dptive ters. 13 Addition dif cuties in the contro of motion simutors resut from uncertinties in the simutor dynmics. These uncertinties re due, for exmpe, to chnges in the simutor inerti properties nd uncertintiesin the drive system dynmics. Hence, robust contro design strtegy shoud be ppied. Idn nd Shr 14 proposed robust controer for six-dof ight simutor. The controer design is bsed on inerized mode of the simutor dynmics together with n uncertinty mode tht describes the vrition of the simutor weight. The design is performed using -synthesis techniques nd the controer is designed for ter motion simution, for which the inerized mode of DC-8 ircrft is used. Numeric simutions show tht good quity ter motion simution is chieved. In this pper we propose six-dof motion simutor tht eimintes the probems creted by teescopic ctutors. The structure of the simutor is tensegrity structure tht hs no br to br connections nd no rigid bodies siding with respect to ech other. The motion of the simutor is controed by tendons, eding to the eimintion of teescopic ctutors. A tensegrity structure is speci type of spce structure composed of set of tendons tht re prestressed ginst system of rigid bodies,usuybrs. 15 Tensegritystructuresoffer exceentopportunities for physicy integrted structure nd controer design becuse the estic components provide exceent opportunities for the sensing nd ctuting functions. Athough the origins of tensegrity structurescn be pinpointedto 197 (see Sneson 16 ), the min investigtionshve been crried out during the st 40 yers. Tensegrity structures were ooked on from n engineering perspective for the rst time by Fuer. 17 Geometric investigtionsfoowed, most of them being reported in Fuer 17 nd Pugh. 18 Approches using mechnics hve been deveoped recenty nd reserch into tensegritystructureshs become systemtic nd imed t estbishingthe theoreticfrmework for the nysis nd design of these structures. Peegrino nd Cdine, 19 Motro et., 0 nd Hnor 1 hve mde importnt contributions towrd further knowedge of the sttics of these structures. Liner dynmic nysis resuts hve been pubishedby Motro et. 0 nd Furuy. Noniner dynmics nd contro design studies hve been reported by Sketon nd Sutn 3 nd Sutn nd Sketon. 4 Appictions of tensegrity structures re now being proposed, rnging from tensegrity domes (see Hnor 5 nd Wng nd Liu 6 ) to tensegrity sensors (Sutn nd Sketon 7 ) nd spce teescopes (Sutn et. 8 ). In this pper, we design noniner robust trcking controerfor simutor motion contro. The controer, which hs been derived in Zenieh nd Coress, 9 ssures exponenti convergence of the trcking error to b of prespeci ed rdius, with prespeci ed rte of convergence. This controer hs two importnt dvntges over other trcking controers:it is continuouscontroer, nd it is simpe to impement becuse it does not invove the computtionof the regressor mtrix. 30 It is known fct tht exct trcking usuy resuts in discontinuous controers tht re undesirbe for sever resons (see Coress 31 ). The pper is orgnized s foows. First description of the proposed tensegritysimutor is given, foowed by the derivtion of its equtions of motion. The sttics of the simutor is nyzed nd the equiibrium con gurtions re mthemticy chrcterized.next, the design of the noniner robust trcking controer is presented. The performnce of the simutor equipped with this controer is then evutedby simutingspeci c ongitudinmotions of symmetric ircrft. Tensegrity Simutor A perspective view of two-stge tensegrity simutor is given in Fig. 1. The rst stge consists of bse with three brs (A i1 B i1, i = 1,, 3) rigidy ttched (cmped) to it. The second stge consistsof cbinwith threebrs(a i B i, i = 1,, 3) cmped to it. Tweve tendons connect the end points of the brs. The six tendons chrcterized by B i 1 A j re cssi ed s sdde tendons. The remining six, A i 1 A j nd B i 1 B j, re ced digon tendons. For mthemtic modeing we ssume tht the tendons re mssess nd iner estic, wheres the bse, cbin, nd brs re rigid. For simpicity, the bse nd cbin tringes A 11 A 1 A 31 nd B 1 B B 3, respectivey, re chosen to be congruent equiter tringes. We so ssume tht brs re identic nd tht they re connected to the bse nd cbin so tht the nges mde by their xes of symmetry with the perpendicur to tringe A 11 A 1 A 31 or B 1 B B 3 re equ; we c this nge d nd we restrict it to stisfy 0 < d < 90 deg. For d = 0 the brs re orthogonto the tringes nd for d = 90 deg their xes of symmetry beong to the tringes. The nges mde by the projections of the brs A i 1 B i 1 (i = 1,, 3) onto the A 11 A 1 A 31 pne with the vector A 11 A 31 re, + 40, + 10, respectivey (see Fig. for the de nitions of nd d ). Simiry the nges mde by the projections of the brs A i B i (i = 1,, 3) onto the B 1 B B 3 pne with B 1 B 3 re, + 40, + 10, respectivey. As n inerti reference frme, we choose n orthonormdextr set of vectorse 1, e, e 3 with origint the geometriccenterof tringe A 11 A 1 A 31. The vector e 3 is orthogon to this tringe, pointing upwrd, wheres e 1 is pre to A 11 A 31. We introduce reference frme s 1, s, s 3, ced the second-stge reference frme nd xed in this stge. Its origin is t the mss center of the second stge, which is ssumed to coincide with the geometric center of tringe Fig. 1 Two-stge tensegrity simutor.

3 SULTAN, CORLESS, AND SKELTON 1057 In ddition tendons shoud be in tension. Mthemticy, this is chrcterized by T n (q g ) > 0 for n = 1,..., 1 (8) An equiibriumcon gurtion in which of the tendons re in tension is ced fesibe equiibriumcon gurtion.we re interested in fesibe equiibrium con gurtions for which the cbin nd bse tringes re pre nd their mss centers re verticy igned, i.e., q g is given by q g = [w g Z g ] T (9) Fig. De nition of nd ±. B 1 B B 3. The vector s 3 is orthogon to this tringe, pointing upwrd, wheres s 1 is pre to B 1 B 3. The simutor hs six degrees of freedom. As independentgenerized coordintes we choose w, u, h, the Euer nges for 3-1- sequence to chrcterize the orienttion of the second-stge reference frme retive to the inerti reference frme, nd X, Y, Z, the Crtesin coordintes of the mss center of the second stge retive to the inerti reference frme. Thus, the vector q of generized coordintes is given by q = [w u h X Y Z] T (1) Tensegrity Simutor Dynmics The derivtionof the second-ordernoninerordinrydifferenti equtions tht describe the dynmics of the tensegrity simutor, crried out using the Lgrnge methodoogy, yieds the foowing: M(q) q + c(q, Çq) + A(q)T(q) + Ĝ = 0 () The system inerti mtrix M(q) hs the structure where X = M(q) = X T JX 0 0 M t I cos(u ) sin(h ) cos(h ) 0 sin(u ) 0 1 cos(u ) cos(h ) sin(h ) 0 The inerti mtrix of the second stge J is ccuted with respect to the second-stge reference frme. The components of the vector c(q, Çq) re qudrtic functions of Çq nd re given by c i = 6 6 j = 1 m = i m The vector c(q, Çq) cn be expressed s (3) i Çq j Çq m for i = 1,..., 6 (4) c(q, Çq) = C(q, Çq) Çq (5) The mtrix C(q, Çq) is not unique, however, if it is chosen ccording to C i j = 1 6 m = i m j j i for i = 1,..., 6 nd j = 1,..., 6 (6) then the mtrix ÇM(q) C(q, Çq) is skew-symmetric. The vector A(q)T(q) represents the generized forces becuse of the tendons in tension,where A mn n m (m = 1,..., 6, n = 1,..., 1) nd T n = (k n / rn )( n rn ) (n = 1,..., 1). The expressions for n re given in Appendix A. The vector Ĝ =[ M t g] T represents the generized forces due to the grvittion force ed. Tensegrity Simutor Sttics Using Eq. (), equiibrium con gurtions q g of the tensegrity simutor re given by A(q g )T(q g ) + Ĝ = 0 (7) Çq m At fesibe con gurtion chrcterized by Eq. (9) the digon tendons A i1 A j hve the sme ength, D 1, the digon tendons B i 1 B j hve the sme ength, D, the sdde tendons B i 1 A i hve the sme ength, S 1, nd the sdde tendons B 11 A 3, B 1 A 1, B 31 A hve the sme ength, S. These engths re given by D 1 = (b / 3)[1 + cos(w g 60)] + h g + h g cos(d ) + / p 3 b sin(d )[cos( + 30) + sin(w g )] 1 (10) D = (b / 3)[1 cos(w g)] + h g + h g cos(d ) + / p 3 b sin(d )[cos(w g + 30) cos( 30)] 1 (11) S 1 = (b /3)[1 + cos(w g 60)] + h g + sin (d ) + / p 3 b sin(d )[sin(w g 60) sin( )] 1 (1) S = (b / 3)[1 cos(w g)] + h g + sin (d ) + / p 3 b sin(d )[sin( 60) + cos(w g 30)] 1 (13) respectivey.the overp h g, de ned s the distnce between tringes A 1 A A 3 nd B 11 B 1 B 31 ndretedto Z g by h g = cos(d ) Z g, is positive (h g > 0) if the distnce between A 1 A A 3 nd A 11 A 1 A 31 is smer thn the distnce between B 11 B 1 B 31 nd A 11 A 1 A 31. For simpicity we ssume tht tendons hve the sme stiffness k, the digon tendons hve the sme rest-ength D r, nd the sdde tendons hve the sme rest-ength S r. Hence tensions in tendons of ength D 1, D, S 1, S re equ to T D1 = k( D 1 D r )/ D r, T D = k( D D r )/ D r T S1 = k(s 1 S r )/ S r, T S = k(s S r )/ S r (14) respectivey. The requirement tht the fesibe equiibrium con gurtions re chrcterized by Eq. (9) must hod for Ĝ vrying in certin domin, i.e., the design shoud toerte vritions in the mss of the cbin. If we substitute Eq. (9) into Eqs. (7) nd (8), we obtin the conditions these fesibe equiibrium con gurtions must stisfy A g T g + [0 M t g] T = 0 T D1 > 0, T D > 0, T S1 > 0, T S > 0 (15) Here T g =[T D1 T D T S1 T S ] T nd A g is 4 mtrix given in Appendix B. To sove these conditions for the fesibe equiibrium con gurtions (w g nd Z g ), we hve to choose the rest-engths of the tendons. In the foowing we sh choose the rest-engths of the tendons using certin fesibe equiibrium con gurtions of the tensegrity structure, ced prestressbe con gurtions. Prestressbe Con gurtions Consider the cse when no grvity is present, i.e., Ĝ = 0. The correspondingfesibe equiibriumcon gurtionsre ced prestressbe con gurtions.let q 0 denote prestressbecon gurtion.we impose the condition tht t the equiibrium q 0 the second stge is

4 1058 SULTAN, CORLESS, AND SKELTON obtinedfrom the rst through cockwiserottionof 60 deg round the xis e 3, i.e., w 0 = 300 deg nd q 0 = [ Z 0 ] (16) With this geometry, sdde nd digon tendons hve the sme engths given by S 0 = h 0 + b /3 + sin (d ) / p 3 b sin(d ) cos( 30) 1 D 0 = + b / 3 + h 0 h 0 cos(d ) / p 3 b sin(d ) sin( ) 1 (17) (18) respectivey.here h 0 = cos(d ) Z 0 is the overpin the prestressbe con gurtion q 0. In this con gurtion, the tensions in the sdde nd digon tendons re equ to T S nd T D, respectivey, hence the rst condition in Eq. (15) reduces to cos( )/ D 0 cos( + 60)/ S 0 [h 0 cos(d )]/ D 0 h 0 / S 0 T D T S = 0 (19) The second of the preceding equtions (h 0 cos(d ))(T D / D 0 ) + h 0 (T S / S 0 ) = 0 nd the condition tht 0 < d < 90 deg show tht in order for the tensions to be positive (T D > 0, T S > 0), we must hve 0 < h 0 < cos(d ) (0) Eqution (19) must hod for nonzero [T D T S ] T, thus the determinnt of the mtrix mutipying[t D T S ] T must be zero; this is equivent to h 0 = cos(d ) cos( + 60) cos( 60) (1) The constrint 0 < h 0 < cos(d ) eds to 30 deg < < 90 deg or 10 deg < < 70 deg. Soving Eq. (19) for the tensions,we obtin tht where T 0S T 0D = 1 p 6 k {[( cos(d ) T S = PT 0S, T D = PT 0D () cos(d ) h 0 h 0 D 0 S 0 1 h 0 )/ h 0 ]( D 0 / S 0 )}k (3) is normized vector such tht the Euciden norm of the vector T of tensions is one for P = 1. Using the expressions for the tensions T S nd T D, T S = (k/ S r )(S 0 S r ), T D = (k/ D r )(D 0 D r ) (4) we get the foowing expressionsfor S r nd D r in terms of h 0 nd P: S r = ks 0 T 0S P + k, D r = k D 0 T 0D P + k (5) An importnt issue in tensegrity structures reserch is the stbiity of the prestressbe con gurtions. 16 It cn be shown tht for (, d ) (30, 90) (0, 90) the prestressbe con gurtions previousy nyzed re stbe. Indeed, consider the potenti energy of the structure in the bsence of the grvittion ed: The second derivtive of the potenti energy with respect to the generized coordintes is given by K i j, i = 1,..., 6, j = 1,..., 6 (7) If we evute K t prestressbe con gurtion chrcterized by q 0 =[ cos(d ) h 0 ] with h 0 given by Eq. (1) nd use the corresponding rest-engths given by Eq. (5) we get where K 0 = dig[k 11 K 11 / K 11 / K 44 K 44 K 44 ] (8) K 11 = b (T S / S 0 + T D / D 0 ) + p 3b sin(d ) [(T S / S 0 ) sin( + 60) + (T D / D 0 ) sin( )] (9) K 44 = 1(T S / S 0 + T D / D 0 ) (30) For (, d ) (30, 90) (0, 90) nd for P > 0 we hve K 11 > 0 nd K 44 > 0, yieding K 0 > 0, which provestht these prestressbecon- gurtions re stbe. For simutor design we choose the vues of nd d such tht suf cient cernce between the brs is gurnteed. For this purpose we proceed s foows. Consider pir (, d ) nd compute the minimum distnce d mn i j between two brs A i j B i j nd A mn B mn. The probemof minimum distncebetween their xes of symmetry eds to constrined qudrtic optimiztion probem: d mn i j = min(w Hw + f T w + c), w = [w 1 w ] w H = k v 1k v 1 v v 1 v k v k f T = [ v 1 v ], c = k k 0 w 1, 1 (31) (3) where = A i j A mn, v 1 = A i j B i j, v = A mn B mn. We denote by d(, d ) the minimum of d mn i j over possibe combintions of brs, i.e., over (i, j ), (m, n) pirs [i = 1,, 3, j = 1,, m = 1,, 3, n = 1,, (i, j) 6= (m, n)]. Foowing this nysis we choose = d = 60 deg tht, for = 5 m, b = 3.33 m, yieds cernce d(, d ) = 1. m. This procedure ssumesthtthe prestressbecon gurtionnd the ctuequiibrium con gurtion (when the grvittion ed is cting) do not differ too much. Fesibe Equiibrium Con gurtions The fesibe equiibriumcon gurtions of interest re chrcterized by Eqs. (15). We ssume tht the rest-engths of the tendons re given by Eq. (5). For M t = 0, G = M t g = 0 nd we know tht Eqs. (15) hve soution; thus we expect them to hve soution for sm M t 6= 0. A continution procedure cn be ppied to sove for these soutions s foows: sighty increse M t nd sove the two noniner equtions A g T g + [0 M t g] T = 0 for w g nd h g (using cssic noniner sovers ike Newton Rphson), then check if T gi > 0 (i = 1,..., 4). If this hppens, then we hve found new fesibe equiibrium con gurtion with q g =[w g Z g ] T where Z g = cos(d ) h g. We continue to increse M t nd repet the procedureunti one of the tendonsbecomessck or when the M t of interest is reched. To iustrte the ppiction of this procedure, grph of the vrition of w g nd Z g with the mss of the second stge (top), M t, is given in Fig. 3 for the foowing chrcteristics: = 5 m, b = 3.33 m, = 60 deg, d = 60 deg P = 000, k = 5000 N, g = 9.81 m s (33) V = 1 n = 1 k n rn n rn (6) At points on these curves the distnces between brs were computed to scertin if the brs intersect. This did not hppen (the minimum distnce between two brs ws 0.7 m).

5 c SULTAN, CORLESS, AND SKELTON 1059 The requirementtht the trcking error q = q q d convergesto zero usuyedsto discontinuouscontroertht is undesirbefor sever resons (see Coress 31 ). However, if we ony require trcking to within some prespeci ed toernce r we cn design continuous controer (see Zenieh nd Coress 9 ). Speci cy, given r, > 0, we cn design controer so tht the cosed-oopsystem is robust r c trcker; nmey, there existsscrsc 1 nd c such tht, for ny desired (twice continuousy differentibe) trjectory q d ( ) nd ny uncertinty D every soution stis es k q(t)k (c 1 k q(t 0 )k + c k Ç q(t 0 )k ) exp( c (t t 0 )) + r for t t 0 (39) Fig. 3 Vrition of à g nd Z g with mss of second stge. which mens tht the trcking error exponentiy converges with rte c to the b of rdius r de ned by k qk r. To ppy the resutsof Zenieh nd Coress 9 for controerdesign, it is ssumed tht there exists positive constnts b 0, b 1, b, b 3, such tht for q, Çq, nd, 0 < b 0 I M(q, ) b 1 I, k C(q, Çq, )k b k Çqk Robust Trcking Controer The contro inputs for the simutor re the rest-engthsof some of the tendons. For further nysis, we seprtethe tendons into ctive tendons of controedrest-engthsnd pssive tendons of xed restengths. Using Eqs. () nd (5), the behvior of the simutor is described by M(q) q + C(q, Çq) Çq + A p (q)t p (q) + A (q)t + Ĝ = 0 (34) where the components of T p nd T re the tensions in the pssive nd ctive tendons, respectivey. For robust contro design we consider tht the mss of the second stgemy be uncertin.the uncertintyis representedby theumped uncertin term D, where D is known nonempty set. Thus the equtions of motion trnsform to M(q, ) q + C(q, Çq, ) Çq + A p (q)t p (q) + A (q)t + Ĝ( ) = 0 (35) For further nysis we write the equtions of motion s where M(q, ) q + C(q, Çq, ) Çq + u 0 + A p (q)t p (q) + Ĝ( ) = u (36) u = A (q)t + u 0, u 0 = A (q g )T (q g ) (37) Here q g is fesibe equiibrium con gurtion. Under the ssumption of invertibiity of A (q), Eq. (37) cn be soved for the ctive rest-engths r 1,..., r nu ; speci cy, r i = k i i k + T i i for i = 1,..., n u (38) where T, the vector of ctive tensions, is given by T = A (q) 1 (u 0 u). The issue of ctive tendon seection is ddressed next. Becuse the number of generized coordintes is six, the system wi be controed with six tendons. The six re chosen by nyzing the properties of the mtrix A(q g ). Speci cy, we consider the set of mtrices creted by combining ny six coumns of A(q g ); et these mtrices be denoted by A i (i = 1,..., 94). We compute the corresponding set of minimum singur vues r i = r min( A i ) (i = 1,..., 94). The A i, which yieds the mximum of {r i, i = 1,..., 94}, providesthe set of ctive tendons. Appying this procedure with the dt in Eqs. (33) nd with q g corresponding to the nomin mss M t = 140 kg of the simutor,we obtintht the set of sdde tendons is the best choice for the set of ctive tendons. Let q d ( ) : IR! IR 6 be desired motion of the simutor. Idey, we wish to design controersuch tht every motion q( ) : IR! IR 6 of the cosed-oopsystem convergesto the desired motion exponentiy. We so wnt to specify the rte of convergence c priori. k u 0 + A p (q)t p (q) + Ĝ( )k b 3 (40) If we express C(q, Çq, ) s C(q, Çq, ) = dig( Çq T ) C(q, ) we get k C(q, Çq, )k k C(q, )k k Çqk. Thus b cn be chosen to n upper bound on the mximum singur vue of C(q, ). Becuse nytic determintion of the bounds b 0, b 1, b, b 3 is not usuy possibe, they re numericy determined. One pproch is to discretize the probem by gridding D nd the domin in the spce of generized coordintes in which it is resonbe to ssume tht the system trjectories ie. Let (q i, i) (i = 1,..., N ) denote N grid points. We evute the minimum nd mximum eigenvuesof M(q i, i ) (ced k min i nd k mx i, respectivey), the mximum singur vue of C(q i, i ) (ced r mx i ), nd the quntity F i =k u 0 + A p (q i )T p (q i ) + Ĝ( i )k. Any positive number smer thn the minimum of {k min i, i = 1,..., N} cn be chosen s b 0, wheres ny number greter thn the mximum of {k mx i, i = 1,..., N} cn be chosen s b 1. Simiry ny number greter thn the mximum of {r mx i, i = 1,..., N} cn be chosen s b nd number greter thn the mximum of {F i, i = 1,..., N } cn be chosen s b 3. A ner grid cn be used to test if the chosen vues re stisfctory. In Zenieh nd Coress 9 the foowing noniner,continuouscontroer hs been proposed for robust trcking contro of gener css of mechnic systems: u = Q (k q k + ²) 1 q Here Q, K, q = b 1k Ǻk + b k ºk + b 3 º = Çq d K q, = Ç q + K q (41), nd ² stisfy Q > c b 1 I, 0 < ² (c r) k min( Q)(b 0/ b 1), k > c I (4) where k min( Q) denotes the minimum eigenvue of the positivede nite symmetric mtrix Q. The resuting cosed-oop system hs been shown to be robust r c trcker with c 1 = 1 + k mx(k )c, c = b 1/ b 0(c 1 c ) c 1 = k min( Q)/b 1 (43) where k mx(k ) is the mximum eigenvue of the positive-de nite symmetric mtrix K (see Zenieh nd Coress 9 ). Simuting Longitudin Motions of Symmetric Aircrft In ight simution, the desired motion q d ( ) to be trcked by the simutor is generted by the motion of n ircrft. We consider here the ongitudinmotion of symmetric ircrft. A ongitudin motion of symmetric ircrft cn occur when the resutnt force beongs to the ongitudin pne of symmetry of the ircrft nd

6 h 1060 SULTAN, CORLESS, AND SKELTON the resutnt torque is orthogon to it. Using singe rigid-body mode to describe the ongitudin motion of symmetric ircrft nd ssuming no wind, the foowing equtions of motion cn be derived through the ppiction of the ws of mechnics: ÇX = v cos(h ), ÇZ = v sin(h ), Ç h = x y m Çv = T r cos D m g sin(h ) m v Ç = T r sin L + m g cos(h ) + m vx y J x Ç y = M y (44) The horizont rnge (X ) nd the titude (Z ) of the ircrft re mesured with respect to dextr set of unit vectors f 1, f, f 3, ttched to the Erth. The motion is ssumed to tke pce in the vertic f 1 f 3 pne with f 3 vertic. In this mode, we negect the gyroscopic coupes (becuse of rotting rigid bodies) s we s propusioncontributionto the pitchingmoment (becuseof offsetof the resutnt thrust from the ircrft center of mss). For simutor design we sh use the vector q of generized coordintes of the ircrft de ned by q = [w u h X Y Z ] T (45) By the choiceof the inertireferencefrme ttchedto the Erth, the ircrftmotion tkes pcein verticpne nd we hvew = u = Y = 0. The erodynmic forces L, D, nd the pitching moment M y re given by L = q Sv C, D = q Sv C d M y = q Sv c C M (46) For sm vritions of the Mch number, the erodynmic coef - cients cn be consideredindependentof this prmeter. For certin ircrft, whose dynmic properties re nyzed in Sutn, 3 these coef cients depend on, d e, nd d f, nd re given in Appendix C. The tmosphere mode used here yieds the foowing dependency of the ir density q on the titude Z : q = 1.65exp[4.56 og(1 Z /44300)] kg m 3 (47) where Z is given in meters. Evution of the Tensegrity Simutor In the foowing we nyze the biity of the tensegritysimutor to trck ongitudinmotions of n ircrft. First we de ne wht we men by trcking nd desired motion in this context. The min tsk of ight simutor is to give the piot the sme senstions one woud hve when ying the re ircrft. In the rst pproximtion we negect the in uence of the vestibur system of the piot. For simpicitywe so ssume tht the piot s hed oction with respect to the mss center of the second stge of the simutor is the sme s its oction woud be with respect to the mss center of the ircrft. Thus, in cse the simutor trcks resonby we the cceertion nd ngur veocity of the ircrft, the piot wi hve the sme senstions when ying the simutor s when ying the re ircrft. Mthemticy, the trcking probem mens tht the simutor nd the ircrft shoud hve the sme cceertions nd ngur veocities. Usuy, they strt from different initi con gurtions nd veocities. The simutor is in equiibrium with zero veocity, which is dynmicy equivent to rectiiner uniform trnstion of the ircrftt n rbitrryveocity.it is suf cient for the simutor to trck the ircrft motion becuse of certin commnd s seen from n inerti reference frme ttched to the ircrft tht is in rectiineruniform motion before the commnd is ppied. Thus the desired motion is given by q d (t) = q (t) Çq 0 t q 0 + q 0 + Çq 0 t (48) where q 0, Çq 0, q 0, Çq 0 re the initi conditions of the simutor nd ircrft(usuy Çq 0 = 0). In this pper we re miny interestedin the cpbiity of the tensegrity simutor equipped with the noniner robust trcking controer to trck the desired motion t the onset of the cceertion, i.e., immeditey fter n erodynmic or throtte commnd is ppied. We now considerthe foowingscenrio.the ircrftis in uniform rectiiner trnstion; correspondingy,the simutor is in equiibrium, chrcterized by q g. Then, commnd tht does not tke the ircrft out of its ongitudin ight is performed. As resut the simutor shoud undergo motion to trck the desired trjectory: where q d = [w g 0 h d X d 0 Z d ] T (49) d = h h 0, X d = X ÇX 0 t X 0 Z d = Z ÇZ 0 t Z 0 + Z g (50) For simpicity we ssume tht the second-stge reference frme is centr princip for the second stge such tht its inerti mtrix J is digon, J = dig[j 1 J J 3 ]. We choose the foowing nomin inerti prmeters for the simutor: M t = 140 kg, J 1 = 300 kg m J = 400 kg m, J 3 = 500 kg m (51) The geometric nd estic prmeters re those given in Eqs. (33). The controer is designed using the foowing prmeters: c = 5, r = 0., b 0 = 90, b 1 = 700, b = 500 b K 3 = 40000, ² = 565, = 6I, Q = 17501I (5) The bounds b i (i = 0, 1,, 3) hve been determinedbsed on grid of the set D where D =[90, 190] kg nd = [w g 30, w g + 30] [ 30, 30] [ 30, 30] [ b/ 3, b/3] [ b/ 3, b/3] [Z g b/ 3, Z g + b/3] Here w g nd Z g re the nomin vues (corresponding to M t = 140 kg) of w nd Z t equiibrium. The increments used in gridding were 5 deg for w, u, nd h, b/6 for X, Y, Z, nd kg for = M t. The mss of the ircrftis m = 1400 kg nd the pitchingmoment of inerti is J = 60,000 kg m. The ifting surfce is S = 11.9 m nd the men erodynmic chord is c = 1. m. Eevtor Commnd Here we evute the biity of the system to simute the motion of the ircrft when subjected to n eevtor step input commnd. Assume tht the ircrft is initiy in eve ight chrcterized by the foowing trim conditions: Z 0 = 3000 m, h 0 = 0, 0 = 0, ÇX 0 = 86.4 m s 1 d f 0 = 0, d e 0 = 1.34 deg, T r0 = 1479 N (53) The corresponding simutor equiibrium is chrcterized by w u h g = 30. deg, Z g = 3.46 m, X = Y = = = 0 (54) The eevtor, which is pproximted s rst-order system with time constnt s = 0. s, is given step commnd t time t 0 = 0.5 s. This resuts in the eevtor de ection, A d e = d e 0 + d e {1 exp[ (t t 0)/ s ]} (55) We present next the time histories of the most signi cnt cceertions (Fig. 4), the contro inputs (Fig. 5), nd the ctutor A forces(fig. 6) for d e = deg. The foowingnottionhs been introduced for the tendons: tendon 1 = A 1 B 1, = A 1 B 11, 3 = A B 1, 4 = A B 31, 5 = A 3 B 11, 6 = A 3 B 31.

7 SULTAN, CORLESS, AND SKELTON 1061 Fp Commnd Here we evute the biity of the system to simute the motion of the ircrft becuse of p step input commnd. Assume tht the ircrft is initiy in eve ight chrcterized by Z 0 = 100 m, h 0 = 0, 0 = 0, ÇX 0 = 74.8 m s 1 d f 0 = 0, d e 0 = 1.34 deg, T r0 = 1479 N (56) nd tht pproching nding the ps re given de ection, d f = d f 0 + d A f {1 exp[ (t t 0)/ s ]} (57) Fig. 4 Eevtor commnd: desired ( ) nd simutor ( ) cceertions. with d A f = 5 deg, t 0 = 0.5 s, nd s = 0.5 s. The equiibrium con gurtion of the simutor is the sme s before. Numeric simutions (Fig. 7) show tht the simutor is very effective in trcking the cceertions. As in the eevtor step commnd cse, numeric simutions indicte tht even better trcking is chieved for the generized coordintes, generized veocities, nd ngur veocity of the ircrft, though none of the tendons becomes sck nd suf cient cernce between brs is gurnteed. The correspondingcontro input time histories,given in Fig. 8, show tht their vritions nd rnges re cceptbe (the sme is true of the contro forces). Throtte Commnd Finy, we evute the biity of the system to simute the motion of the ircrft when subjected to throtte step input commnd. We suppose tht the ircrft is initiy in eve ight chrcterized Fig. 5 Eevtor commnd: contro time histories. Fp commnd: desired ( ) nd simutor ( ) cceer- Fig. 7 tions. Fig. 6 Eevtor commnd: ctutor force time histories. The simutor is very effective in trcking the cceertions even throughout the initi phse. Numeric simutions indicte tht even better trcking is chieved for the generized coordintes nd veocities nd for the ngur veocity of the ircrft. The time histories of the contro inputs nd ctutor forces show tht their vritions nd rnges re cceptbe. The simutions show tht throughout the motion none of the tendons becomes sck, nd there is wys suf cient cernce between brs. Fig. 8 Fp commnd: contro time histories.

8 106 SULTAN, CORLESS, AND SKELTON Fig. 9 Throtte commnd: desired ( ) nd simutor ( ) cceertions. Fig. 11 Robustness evution: desired ( ), nomin ( ), nd perturbed (+, ) simutors. response of the one whose inerti properties re vried with 50% is represented by. We scertin tht, even for these rge perturbtions in the inerti properties, the trcking is very good. The perturbed simutors responses re very cose to the response of the nomin simutor nd to the one of the ircrft. Our simutions con rm tht even better trcking is obtined for the generizedcoordintesnd veocitieswhie none of the tendonsbecome sck nd none of the brs touch ech other. It is so importnt to mention tht, even though for controer design we ssumed vritions ony in the mss of the cbin M t, the design is very robust with respect to vritions in inerti prmeters of the simutor. We so remrk tht the perturbed simutors msses considered in the preceding exmpe re out of the rnge used for controer design (D =[90, 190] kg). However the perturbed simutors equipped with the nomin controer trck the desired motion very we. Fig. 10 Throtte commnd: contro time histories. by the prmeters of the previous section nd is given the throtte commnd, T r = T r0 + T A r {1 exp[ (t t 0)/s ]} (58) with T A r = 500 N, t 0 = 0.5 s, nd s = 0.5 s. The time histories of the signi cnt cceertionsnd the contros re given in Figs. 9 nd 10, respectivey. As in the cse of the erodynmic commnds, trcking is very good, nd none of the tendons becomes sck nd there is suf - cient cernce between brs. Here the contro time histories exhibit smoother vritionsthn in the cse of the erodynmiccommnds. Robustness of the Design In the foowingwe evutethe robustnessof our design.we consider tht the controeris designedfor the nomin simutor, whose inertipropertiesre M t = 140 kg, J 1 = 300kg m, J = 400 kg m, J 3 = 500 kg m, nd tht it is used to contro the motion of the simutor when its inerti properties chnge. For iustrtion, we consider n eevtor step commnd with d A e = 5 deg, t 0 = 0.5 s, s = 0. s ppied to the ircrft, which is initiy in eve ight chrcterizedby the prmetersin the subsection Eevtor Commnd. Figure 11 shows the vertic cceertionof the ircrft nd the corresponding vertic cceertions of three simutors: the nomin one nd two perturbed simutors, whose inerti properties re vried between +50% nd 50%, respectivey, from the nomin ones (this mens tht the vues of M t, J 1, J, J 3 re vried between +50% nd 50%, respectivey, from the nomin ones). The response of the perturbed simutor whose inerti properties re vried +50% is represented in Fig. 11 by +, wheres the Concusions Motivtedby the desire to eiminte compictedstrut ctutors, tensegrity motion simutor is proposed. The min dvntge tht tensegrity motion simutor hs over Stewrt ptform bsed simutor is tht it does not hve teescopicstruts. Tensegrity simutors do not incude rigid bodies siding with respect to ech other, nor do they hve compicted br to br joints. The ctuting nd the sensing functions cn be crried by the tendons. For the proposed simutor, noniner continuous robust trcking controer is designed. The controer gurntees exponenti convergence of the trcking error with prespeci ed rte of convergence to b of prescribed rdius. The physic contros re the rest-engths of six of the 1 tendons of the tensegrity structure. The proposedtensegritysimutorequippedwith the robusttrcking controer is very effective in trcking the ongitudinsymmetric motions of certin ircrft. In gener, the motions generted by throtte commnds re better trcked thn those due to erodynmic contro surfces commnds. Further numeric simutions indicted tht the design is very robust to vritions in the inerti properties of the cbin. Appendix A: Tendon Lengths The engths of the tendons j ( j = 1,..., 1) re given by 1 = q 5 A b p 3/ 6 cos(q 1 ) cos(q ) b p 3/3 + sin(d ) sin( ) + q 4 A 1 + b p 3/ 6 sin(q 1 ) cos(q ) + sin(d ) cos( ) + q 6 + (b/) cos(q ) sin(q 3 ) b p 3/6 sin(q ) cos(d ) 1

9 SULTAN, CORLESS, AND SKELTON 1063 = q 4 A 1 + b p 3/ 6 sin(q 1 ) cos(q ) + sin(d ) sin( 30) + b/ + [q 5 A + A 3 sin(d ) cos( 30)] + q 6 + (b/ ) cos(q ) sin(q 3 ) b p 3/6 sin(q ) cos(d ) 1 3 = q 4 b p 3/3 sin(q 1 ) cos(q ) sin(d ) cos( 60) + [q 5 A 3 sin(d ) sin( 60)] + q 6 + b p 3/ 3 sin(q ) cos(d ) 1 4 = q 4 b p 3/3 sin(q 1 ) cos(q ) b/ + sin(d ) sin( 30) + q 6 + b p 3/3 sin(q ) cos(d ) + q 5 + b p 3/ 6 cos(q 1 ) cos(q ) + b p 3/ 6 sin(d ) cos( 30) 1 5 = q 4 + A 1 + b p 3/ 6 sin(q 1 ) cos(q ) sin(d ) cos( 60) + b/ + [q 5 + A + A 3 sin(d ) sin( 60)] + q 6 (b/ ) cos(q ) sin(q 3 ) b p 3/6 sin(q ) cos(d ) 1 6 = q 5 + A b p 3/ 6 cos(q 1 ) cos(q ) + sin(d ) sin( ) + b p 3/ 6 + q 4 + A 1 + b p 3/ 6 sin(q 1 ) cos(q ) + sin(d ) cos( ) b/ + q 6 b/ cos(q ) sin(q 3 ) b p 3/6 sin(q ) cos(d ) 1 7 = b/ + q 4 + A 1 + b p 3/ 6 sin(q 1 ) cos(q ) + sin(d ) cos( + 60) + [A 3 + q 5 + A + sin(d ) sin( + 60)] + q 6 (b/ ) cos(q ) sin(q 3 ) b p 3/6 sin(q ) cos(d ) 1 8 = b p 3/3 q 5 + A + b p 3/6 cos(q 1 ) cos(q ) + sin(d ) cos( + 30) + A 1 q 4 b p 3/ 6 sin(q 1 ) cos(q ) sin(d ) sin( + 30) + q 6 + (b/) cos(q ) sin(q 3 ) b p 3/6 sin(q ) cos(d ) 1 9 = b/ q 4 + b p 3/3 sin(q 1 ) cos(q ) + sin(d ) cos( ) + b p 3/ 6 + q 5 + b p 3/3 cos(q 1 ) cos(q ) sin(d ) sin( ) + q 6 + b p 3/3 sin(q ) cos(d ) 1 10 = b/ sin(d ) cos( ) + q 4 A 1 + b p 3/6 sin(q 1 ) cos(q ) + [ A 3 sin(d ) sin( ) + q 5 A ] + q 6 + (b/) cos(q ) sin(q 3 ) b p 3/6 sin(q ) cos(d ) 1 11 = sin(d ) cos( + 60) + q 4 b p 3/3 sin(q 1 ) cos(q ) + [ sin(d ) sin( + 60) q 5 + A 3 ] + q 6 + b p 3/ 3 sin(q ) cos(d ) 1 1 = b/ sin(d ) sin( + 30) q 4 A 1 b p 3/6 sin(q 1 ) cos(q ) + [ A 3 + sin(d ) cos( + 30) q 5 A ] + q 6 + (b/) cos(q ) sin(q 3 ) + b p 3/6 sin(q ) + cos(d ) 1 where the tendons re beed s foows: tendon 1 = A 1 B 1, tendon = A 1 B 11, tendon 3 = A B 1, tendon 4 = A B 31, tendon 5 = A 3 B 11, tendon 6 = A 3 B 31, tendon 7 = A 11 A 3, tendon 8 = A 1 A 1, tendon 9 = A 31 A, tendon 10 = B 11 B 1, tendon 11 = B 1 B, tendon 1 = B 31 B 3, nd A 1, A, A 3 re given by A 1 = (b/ )[cos(q 1 ) cos(q 3 ) sin(q 1 ) sin(q ) sin(q 3 )] A = (b/ )[sin(q 1 ) cos(q 3 ) + cos(q 1 ) sin(q ) sin(q 3 )] A 3 = b p 3/ 6 [1 cos(q 1 ) cos(q )] Appendix B: Mtrix A g The eements of mtrix A g re given by A g11 = (b/ D 1 ) b cos(w g + 30) + p 3 sin(d ) cos(w g ) A g1 = (b/ D ) b sin(w g) + p 3 sin(d ) sin( w g 30) A g13 = (b/ S 1 ) b cos(w g + 30) + p 3 sin(d ) cos(w g 60) A g14 = (b/ S ) b sin(w g) + p 3 sin(d ) cos(w g + 60) A g1 = (3/ D 1 )[ cos(d ) h], A g = (3/ D )[ cos(d ) h] A g3 = (3h/ S 1 ), A g4 = (3h/ S ) Appendix C: Aerodynmic Coef cients The erodynmic coef cients of the ircrft re given s foows ( nges in rdins): C d = C d + C d e d C M + C d f d, C = C + C d e + C d f = C M + C d f M + C d e M + C 0 M C = C d C d e = d e C d f = d f = C C C d e d = d e C d f d = d f C 0 M = 0.07, C M = C d e M = d e C d f = d M f

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