Tap cJiC Till Jioc vA Dieu lihien hoc, T.28, S.l (2012), 31-40
GIAI BAI TOAN DONG HOC NGl/OC ROBOT DU" DAN DONG BANG PHiraNG PHAP CHIEU TOA DO VA CHIEU VAN TOG
NGUYEN QUANG HOANG, THAI PHU-ONG THAO Trucrng -Dai hgc Bach khoa Hd Npi; hoangn(]@mail.hut.edu.vn
Tdm tSt. Mot phuang phap gidi bai toan dong hoc ngirac Robot du dan dong duoc trinh bay trong bai bao nay. Trudc het bai toan nguac dong hoc tay may du dan dgng duoc trinh bay 6 hai cap do:
muc lien he van toe va muc lien he gia toe Nghiem ciia eac bai toan nay ducre dua ra nho ma tran tua nghich ddo ciia ma tran Jacobi, trong do c6 chii y den khong gian bii de tranh diroc cac gioi han khap. Gia tri bien khdp tim duoc nlio cac phep tinh tich phan dugfc hieu chinh bang phuang phap chieu. Nha phuang phap cliieu nay, cac bien khop va van toe khop duac hieu chinh ddm bdo chiing nam tren cac da tap lien ket ve vi tri va van toe. Qua do do chinh xac ciia ngliiem can tim duoc cdi thien dang ke. Mot so mo phdng so duac trinh bay de minh hoa cho thuat toan.
Abstract. A method for solving inverse kinematics of redundant robot is proposed in tills paper.
Firstly, the inverse idnematics of redundant robotic manipulators is represented in two levels: velocity level and acceleration level Solutions of these problems are given by pseudo inverse of the Jacobian matrix under consideration of null space in order to avoid joint limitations. Secondly, the values of joint variables are found by the integration which is revised by the coordinate and velocity projection methods. By using this method, the joint variables and joint velocities are adjusted so that they are forced onto manifolds, which are defined by position constraints and velocity constraints. Therefore, the accuracy of solutions is significantly improved. The numerical simulations are illustrated for the oflfectiveness of the proposed method.
1. M6 DAU
Doi voi robot dang chuoi ngirai ta co the phan thanh hai loai: robot chuan va robot du dan dong. Robot chuan la robot co so bac t u do bang so bac t u do cua khau tac dong cuoi hay ban kep. IVai lai, robot du dan dpng co so bac tir do Ion han so bac t u do ciia khau tac dong cuoi. Robot du dan dong co nhieu uu diem so vai robot chuan vl chiing cho phep co the toi iru qui dao chuyen dong, tranh duac vat c ^ , tranh ducrc cac diem ky di, va tranh dugc cac giai han khop [1, 9, 10, 11]. Doi vai robot du dan dpng, do so an nhieu han so phucmg trinh, nen co nhieu phirang an g i ^ quyet bai toan dong hpc ngugc. Phuang an dira tren ma tran Jacobi ciia phuang trinh lien ket hay dugc sii dung nhat, do tinh chat dan gidn ciia phu-ang phap. Voi phu-ang phap nay ta chi can g i ^ he phuang trinh dai so tuyen tinh cd so an nhieu han sd phuang trinh. Cac tpa dp kh6p sau do nhan dugc bang each tich phan cac van toe bien khdiJ theo thai gian, voi dieu kien dau tuang thich. Tuy nhien, cac tpa dp nhan dugc qua phep tfch phan co the lam cho ban kep khong con bam theo qui dao mong
.'i2 NGUYEN QUANG H O A N C , T I I A I PHUaNG 'I'llAo
muon do (•(') i ;i( siii sn licii luy trout; (]iia triiili tfiili (o/m lich pli/in, Mpt sd phirang an khac plnie liien lin/np, nay da dinrc chii y ngliieii ciirti. Do la jjliirang phAp phAn hoi dong hpc [10], ])limrii^ phiij) hieu cllillli gia lirgilg sai sd vre\ir to;i do suy tout; [rj.fj,?].
Tiong bfli brui ii;\v, cluing toi stV dt.ing phuang pli;i|) cliicu toa dp sny rpng va chieu van Inc siiv rpuK de hieu clihili itghiriii ciia bai toan dpng hoe dim tren ma Iran Jacobi. Y tuci^ng cir brill ciia plnrtnig phap la chien ket riii^i Ion do siiy rpng va v;ni toe Hiiy rpng nhan dirge sail khi (ich ])liaii treu eiie da lap xae dinli b(Vi eac phirang trinh lien kel tao IxVi chuyen dpng niniit; muon li;iv lien ket cliinnig trinh va dao liaiii eiia no.
Bai biio dugc Itiiili bay nhu san: vice lliiet la]) va phucrng phfip giiii bai toan dugc trinh bay Irong muc 2 v?i 3. Muc I trinli b;'iv pliirtrrig |)lia[) chicii Iga dp vh chieu van Inc de tang dp chinh xac nghiem eiia bai ln;iii C!;ic mo [»!inii); sd bjiiig phan mem da nang Matlab ddi vai mot la\' may pliaiif; 5 bae tir dn dugc trinh bav trong muc 5. ('niii ning \k mpt so ket luan dugc diru ra.
2. D A T B A I T O A N V A P H U O N G P H A P G l A l
.\ei ta\ iii.i\ n bac tir do, gpi q e K" la vecttr ehira ciic tpa dp khop. Ban kep ciia robot van hanh trong khnii^; gian thao tdc hay khong gian lam vice goi x G R'" la vecta chiia vi tri va huang cila bim kep R'" (m < (J). Bai toan dpng hot thuan inbnt dirge gi.ii (iiivet b^ng cac phirang phap hiuh li(x\ qui hie Drnavil-Hartenberg hoac Craig [2, 3. 4. 9, 10]. Ket qu^
ciia bai toan dpng lip<- thuan cho ta lieu hr sau
f{x.q) =i). o-./G W'.qeR". (1) Fay niav la dir dan dniig khi m < n, sd bac I ir do t iia tay may Icm han so bac tu do cua
ban kep.
Dao ham phuang trinh (1) theo thai gian ta nhau dirge phmrng trinh lien he van tdc
J^x + J„q = 0. (2) vai cac nia tran Jacobi nhu sau
JAx.q) = <)f/i)x. J.,{x.q) Of'Oq
Tiep tuc dao ham phuang trinli (2) elm ta phirang trinh lien he (V cap do gia tdc
J^x + J.,q -I- J^x + J„q ^ 0. (3) Vm trirang ligp robot |>liaiif;. m = :i. x = [x.g.<p\. (]uan he (1,2,3) cd the dugc viet cr
dang tuang minh nhu sau
x=f{g), (1')
X - J{q)q, (2')
GlAl BAI T O A N D O N G HQC NGUOC ROBOT DU DAN DONG 33
X = J{q)q + j{q)q. CA')
Hinh 1. Tav may ciu inie e,i>
Ngoai ra. nhu ta biet phuang trinh vi phan chuyen dgng cii-i tay may dang chuoi \\^ (d dang [2, 4, 9, 10]
u = M{q)q + C[q. q)q + g{q). (4a) trong do u la vecta chua cac Itrc/momen dan dpng, M{q) la ma tran khdi lugng, C{q.q)q la
vecta chua cac luc coriolis va luc ly tam, va g{q) la vecta chua luc suy rong do trong truang.
Bai toan dat ra a day la: Cho biet chuyen dpng cua ban kep tuc la biet cac ham x{t), x{t), x(t) ta can tim chuyen dpng ciia cac tpa dp khap q{t), va veeto momen/luc dan dpng «
2.2. Phu'cmg an gidi q u y e t
Gi^ sii" rang ma tran Jacobi J^ ca m x n co hang day du. rank{Jq) = m. Neu biet x va q tir phuang trinh (2, 2') hoac t u (3, 3') ta se gidi dugc cac van toe khap q hoac gia toe khap '4-
Neu su dung phuang trinh (2) ta giai dugc q. thuc hien tich phan va dao ham ta nhan dugc q vk'q.
Neu su dung phuang trinh (3) ta gidi dugc q, thuc hien tich phan ta nhan dugc q va q.
Thay cac gia tn tim dugc vao phuang trinh vi phan chuyen dpng (4a) ta nhan dugc luc/momen dan dpng ciia cac dpng ca dan.
3. T O I t r u C H U A N c t h V E C T O V A N T O C V A G I A T O C SUY R O N G 3.1. T o i iTu c h u a n ciia vecto' v a n t o e suy r o n g
Gidi phuang trinh (2) ket hgp dieu kien chuan ciia vecta van tdc suy rong nh6 nhat. Tiic gidi (2) tim q tij x, vai dieu kien
/ = ]-q^Wq ^ rain, H^ > 0. (4) Ket qud la
q = J + {q)x. (5)
;{ I NGUYEN QUANG I I O A N G , T H A I PHUONG T H A O
vai
./(<,),; W 'j'''{q)\J{q)W-'j'''{q)]-\
dugc ggi la ma trau tua nglncli d;io e(i trpng ao eiia ina tran -lacnbi J(q) [8].
Nen chpn ina lian Irniig sd la ma tran do'n vj, W ^ I, nghiem tinh theo cong thirc (4) se CO cliiian iihn nhat.
Nen chpn ma tnin irp'ig so Ih ma tiiiii khoi hrgng ciia l.ay in;iy, W ^ M{q), thi nghiem tren dugc liin irng v('ri tien chiifin ldi iru dnng nang. dnng nang cue tieu.
Nen chii y don khdng gian bu eiia ma Iran .laenbi, thi nghieni ctia (2) se la
q -- J^, {q)x + (/ J,\ Jjzo. (6)
\(Vi zo G IR" la ve<t(/ tiiy v. \'e( \it nav so tao ra chuyen dgng cho cac khau ma khong dnh hirgng den cliii\en dpng I'lia ban kep. Thong tlnrang vecta nay se dinre chgn de khai thac I hem <ae uu ilieiii ciia tav iiiiiy du dan dnng nhu tranh vat can. tranh diem ky di, tranh va vao cac giai han khap. Thong thuang nguai ta bay tinh ZQ theo cong thiic
ZQ = a—'— 7)
dg
vdi o{q] la cac ham muc tieu phu thugc vao yen cau dat ra. Chanj;, han, de tranh cac diem ky di, tai dd det[J{q)j'^{q)] = 0, ta ehpn (yf>(g) la ham do khd nang thao tae ciia robot:
<i>{g) = ^det{J{q)J''\q)]. (8) Do do, viec cue dai ham nay se giiip robot tranh dugc cac diem ky di trong qua trinh hoat
dong. De tranh va vao cac giai han khap, nguai ta dira vao ham do khodng each tdi giai han khap:
vdi qiMiqtm) la ky hieu cila gidi han ldn nhat (nho nhat) va q, la gia tri giua ciia khodng lam viec ciia khdp; Cj ta cac trong so. Do do, cue tien khodng each nay, ti'nh du dan dpng se dugc khai thac de giir cho cac bien khdp gan gia tri giiia ctia khodng lam viec ciia robot, tranh dugc su va vao cac giai ban khap, De tranh va vao vat can, ta sii dung ham khodng each tdi vat cdn
<f>{q) = inm\\p{q)-o\\, (10) vdi o la vecta vi tri ciia mot diem thich hgp tren chuang ngai vat (vi du tam trong trircrng
hgp mo hinh vat cdn la hinh cau) vkp(q) la vecta vi tri suy rgng cau tnic cda robot. Do do, cue dai khodng each nay se giup robot tranh dugc vat cdn trong qua trinh hoat dpng. Tren thuc te robot khong gian, viec mo td cac vat cdn cung nhu xac dinh gia tri ham nay la kha phiic tap. Trong bai bao nay, viec tranh va vao cac gidi han khdp dugc quan tam khai thac.
Thay (9) vao (7), va viet ggn lai ta nhan dugc
GlAl BAI T O A N DONG HOC NGUgfC ROBOT DU DAN DONG 35
^o = - ^ = - ^ ( , - 5 ) , (11).
vdi Kii = Ci/{q^M ~qimf, i^ 1,2, ...,n.
3.2. T o i m i c h u a n c d a gia t o e suy r o n g
Khi gidi bai toan ngugc dpng hpc d cap dp gia tdc, phirang tnnh (3) hoac (3') dugc sii dung cho bai toan nay, Viet lai (3") ta cd
J{q)q = x-J{q)q. (12) Tuang t u nhu ddi vdi (2'). phirang trinli tien co vo sd nghieni, (T day ta se dira them vao
dieu kien nghiem ciia (12) cd chuan nho nhat. Tiic gidi (12') vdi dieu kien
: ^q'Wq^min. (13) Ket qud cho ta bieu thiic
9 = J + ( « ) ( i - j ( 5 ) 9 ) . (14) Neu chii y den khong gian bii ciia ma tran Jacobi, thi nghiem ciia (12) se la
q = J + ( 9 ) [ x - j(q)g] + [I - J + ( , ) J ( 9 ) | z o . (15)
Trong phan nay, viec chpn vecta ZQ dugc de xuat nhu sau
zo = -Kig-K2{q-q), Ki, K2>0, (16) vdi hy vpng tranh dugc va cham vdi gidi han khdp. Thuc vay, khi thay (16) vao (15) ta nhan
dugc phuang trinh vi phan ddi vdi bien khdp q{t) nhu sau
q = J+(?)|x - J{q)q] - (/ - J+J)Kiq - (/ - J+J)K2(q - ,„) hay
q+[{I- J'^J)Ki + J^J]q + {I- J^J)K2{q - go) = J^{q)x (17) vdi hy vong rang nghiem q{t) se tuan hoan khi it tuan hoan, va nhu vay se tranh dugc su tang hoac gidm lien tuc ctia bien khdp va do do se trdnh dugc va cham vao gidi han khdp.
4. H I E U C n i N H B A N G P H U ' O N G P H A P C H I E U
Do cd sai sd cua phuang phap va sai sd lam tron trong qua trinh tim q{t) tir cac phuang trinh (6) hoac (17) bang cac phuang phap sd, nghiem q{t), q{t) tim dugc cd the khong cdn thda man cac phuang trinh lien ket (1') va (2'). Trong phan nay trinh bay phuang phap chieu de hieu chinh cac gia tri q(t), q(t) tim dugc sao cho chung thoa man cac phuang trinh lien ket (1') va (2')-
3() Nt;UYEN QUANG H O A N G , T H A I PHUONG T H A O
4 . 1 . Hieu chinh t o a d o s u y rong
Kel qmi tieh |)lum (ir g clio ta toa dn suy rdng g', gia in nay cd (he khong con th6a man phuang trinh (!') .hi e.ic sai sd lich phan, sai sd him trdn f{x g) ?^ 0. (J day, la se hieu chinh d(> dat dirge q thda miin ciie phuang trinh (1). Theo phirang i)hap chien. ta se tim diem 9 nam hen da tap .xiie dinh bdi (1) v;i each diem q" vdi kiioang each ngan nhat. Khi do bai loiin (nV Ihaiih lim q tlida imin (I) san chn ham V san dAy dat circ lien
V = \{q -q*)''P(q q'} ^ mm, P > (I (18) Sir dung phuang phap ham plial (hay phircnig phaj) nhan Mr Lagrange tang (irang). ta
X('M ham Clin Ini iru san
L=V+^:^f'''(x.q)A/ix.q). (19) vdi A la ma tnin duang cheo, \Ae dinh dufrng (neu chpn cac phan tir bang nhau thi ta co
the thay hang mpt hang sn). dci la e/ic nhan td pliat (penalty fadoi). Dao ham theo bien g ta nhau dugc
fcC?. 0 = ^ = P(« - 9*) + < ( 9 ) > l / ( x . 9) = 0. (20) Day la he m phuang trinh dai sd phi I uyen ddi vdi q va cd the giai bang phuong phap
lap Newton-Raphson. Khai trien Taylor ham h{g, t) 6 lan can go ta nhan dirge
h{qo + Aq. t) = h{qo) + H{qo)Aq + ... (21)
H{qo) = §^iPi^-9*) + J:''igWix,q)U = P+ ^[Jl'Anx.g)U ^ -P + {Jl'AJ.U- (22) Luu y khi tinh ma tran /f(gi)), la da bo di mpt sd hang kin tinh ^[JlAf{x,g)]qo- Ve mat toan hpc hoan toan cd the tinh dugc vdi cong cu Maple, tuy nhien sd hang nay tuong doi phiic tap non (V day t a bd qua no.
Phuang phap lap dugc sir dung de tim nghiem g nhu sau:
1) fc = 0 cho biet sd budc lap A'^, g'*^* = q'.
2) Tinh h{q^^i).
3) Kiem tra dieu kien dirng, neu ||/i(9^*'')|| < £ hoac k > N thi dirng, trai lai tiep tuc 4) 4) Tinh ma tran H(g^^^), gidi he h(g<^"') + /f (g<^"')Ag = 0 tim Aq.
5) Tinh gia tri mdi 9''^+^) = ^C^) 4- Aq.
6) Tang k, k = k+\. quay lai 2).
4.2. Hieu chinh v a n t o e s u y r o n g
Sau khi hieu chinh tpa dp suy rpng, t a tien hanh hieu chinh van tdc suy rgng. Viec hieu chinh van toe suy rpng duge sii dung khi ta gidi bai toan dpng hgc ngugc 6 miic gia toe, tiic
GlAl BAI T O A N DQNG HOC NGUgrC ROBOT DU DAN DQNG
la sii dung phuang trinh (3) hoac (3')- Gid siV ket qud tieh i)h;iii lir q cho It g', gia tri nay cd the khong con thda man phucriig (rinh (2) hoac (2") do eiii sai sd lam tron, tiic
van toe suy rpng sal sd tich phiin.
J,x + J,,q' ?iO. (23) 6 day ta .si^ hieu chinh de dat dugc q thda man eac i)hiro'ng trinh (2).
Theo phuang phap chieu, la so tim diein q nam tren da lap xiie dinh b^ri (2) vacach diem q* vdi khodng each ngan nhat. Nhir the, bai toan tnV thanh liiii q llida nuin (2) san cho luiiii V dat cue tieu
y=.,ig -q*)'Q{q~g*)^mm, Q > i) (24)
Ciidi bai toan nay bang phirang phap nhan tu Lagrange cho ta ket qud nhu san q = Q-'j]\j,Q-'j'!;r\j,x - J,q-)+q' = J+J,x + (/ ^ JjjJ,?*. (25) Nhu vay. trong qua trinh hieu chinh (chieu) ta da tim dugc trang thai tua he thoa man cac dieu kien lien ket ciia he.
5. ivro P H 6 N G SO
Hinh 2. Tay may phang 5 bac tu do
Trong phan nay, cac ket qud mo phdng so bang phan mem da nang Matlab duge dua ra.
Ddi tugng khdo sat la mot tay may phang 5 bac tu do vdi cac khdp quay chuyen dong trong mat phang diing. Cac khau cd khdi lugng mj, chieu dai i,, vi tri khdi tam xac dinh bdi a,, va momen quan tinh khdi doi vdi true qua khdi lam ciia khau la Jc, Mb hinh \-a cac thdng sd cua tay may dugc dua ra nhu tren Hinh 2 va Bdng 1.
Bdng 1. Cac thong sd cua tay may kliau ;
/[m]
oH mM
Jo. [kg m l 1 0.55 0,21) 2.00 0,0504
2 0,50 0,20 1.75 0,0304
; j
0,45 0 21]
1.50 0,0253
4 0,40 0,2(1 1,00 0,0133
0,20 11,10 ()„'')(l 0,0010
Trong cvic mo phdng, ta cho ban kep chuyen dpng theo qui dao trdn vdi tain la (xc- yc) = (0 8,0.5)m va ban kinh R = Q, 4m. Van toe doc qui dao la l,Om/s, hudng ciia ban kep dugc
38 NGUYEN QUANG IIOANG, THAI PHUONG THAO
gar khong dui <i= 1, r>7(l,Sia<l Ket (piA < iia bai loan dpng hgc thuan cho ta phuang trinh s X -fig), vdi
fiq) \Y,U^m{Y,<h)-Y.'"'"''^Y.'l'^ T.^'^'''
k=l k=l / . I A I / . I
d day x = \x.y.<i)]'' \h \<''( (o clnia vi Iri (.r, y) va hircVng eiia ban kep 4>, g = [9i 92 9,3 94 q^f la M'chr ehira cae bien khdp.
De tlia\ dugc hieu (]iia eiia pliinnig [)hiip ehi('n toa do. ta lhir( hien mo phdng sd cho cac trirdng hgp khnnn chieu va ei) cliien Ina do <"r e;ie nnie van ld( va gia ide. Ket qud khi khong sir dung phep chieu Ina dn dirgi' (he hien den cdc Hinh 3, 4. Hinh 3 chi ra do thi theo thcri gian (lia cac Iga dg khdp q,{l) va Hinh I la sai so batii ([ui dao.
• - " ; ^ / V V V ^ / V W
\;r-
. - • • W V A A A A A y ^
Hinh 3. Do thi cae bien khdp Hinh 4- Do tin sai sd bam quy dao theo thai gian q{t), [rad] e{t), [m, m, rad]
Tir do thi tren ta thay rang sai so bam quy dao van con kha ldn. ca 10~^m, va cd xu huang tang dan ve tri sd.
'•fC/v7t7\7t7\7i'
Hinh 5. Dd thi e;ic bien klidfp HiJih 6. Do thi sai so bam theo thai gian g(f), [rad] qui dao e((), [m, m, rad]
TVong trudng hgp co sii dung phuang phap chieu tga dp, cac ket qud nhan dugc khi gi^
bai toan theo phuang trinh (6). Trong do ta chpn ma tran trpng sd W la ma tran dan vi va ma tran A' trong (11) la
GlAl BAI T O A N D O N G HQC NGUaC ROBOT DU DAN D O N G
if = rfsas(10,10,10,10, 10).
a 5 10 IS
= ^•iyv/\y\y\yv/A]
0 S 10 IS
KywvAvvTvy]
[ T W W V l
/f?n/i 7. Do thi mdmen dan dgng Hinh 8. Do thi cac bien khdp u{t), [Nm] theo thai gian q(t), [rad]
Ket qud dua ra bao gom dd thi theo thdi gian ciia cac tpa dp khdp g,(t), sai sd bam qui dao, do thi cac momen dan dpng Ui(t) tuang iing vdi cac Hinh 5, 6, va 7.
Cac Hinh 8, 9, va 10 la ket qua khi gidi bai toan dua tren phuang trinh lien he gia toe (15), vdi cac tham sd sir dung trong bieu thiic
Kl -cfza5(10,10,10,10,10),
K2 = diag{20,20,20,20,20).
TIT cac ket qud tren ta thay rang, sai sd bam qui dao la rat nhd, cd 10~^^ (Hinh 6 va 9).
Dd thi cac bien khdp thay doi cd tinh chat lap lai tuang ting vdi su lap Iai chuyen dong ciia
1 • ' J
° r " • ' ' ••"• - 0 s
110"^^
10 is
1 ., .., - — -J r " •""
0 s m o ' *
10 15 1
1 . J
• " • • ' . • 1
;°L/v-^/'-^--v-Ar-A/l
Hinh 9. Bo thi sai so bam qui i e{t), [m, m, rad]
Hinh 10. Do thi momen dan dpng u{t), [Nm]
40 N(;ll^•6N QUANG MOANG, 'I'IIAI rnuoNG THAO
G. K K I L U A N
Bai biio l a p I n m K giai (!ii,\el biii loiiii d g n g h g c n g u g c r o b o t dir d;m d g n g d u a t r e n cac p h i r a n g trhili IK'II he ye v a n t o e vti g i a u',e I ' l i i r a n g p h a p <hieu liicii c h i n h d u g c d u a v a o da lam l a n g d o e h i n h \;ic nti.liieiii eiia biii (n;in. Nguai r a , k h o n g g i a n bii ciia m a I r a n Jacobi enii,!; d i r g e kliai I hae d e d a m b a n cliu e a e biril k l i d p mini I r n i m IIUVMI gidi h a n eiia n o v a do d d d a t r d n h d i n r e t a» va t h a m \ a o c a c gidi b a n k h d p . Ti'nh d i i n g d a n v a lin c;iy ciia p h u a n g
|)h!ip dli d i r g e k h a n g d i n h I limit; q " ' ' '••!'• m " I ' h d n g s d d d i v m l a y m a y p h a n g 5 b a c t u do.
Su s a n h kel q u a m n p h o n g eiia p h i n m g p h a p eliieu v d i p h i n n i g ph;ip p h d n h d i d g n g h p c thay rd p h u a n g p h a | ) chieu ed d o clii'nh x a c Ii'ni h a n . I'liinrnt; iiirdnt; p h a t t i i e n t i e p t h e o ciia bii Inan Id gidi <|iive( bai t o a n d u n g line n g u g c \ d i c a r h e n c h u a n t(!i u n k h a c n h u t d i im mdmen d a n d p n g va tni u u emit; ^ual (aia e a c d n n g va d a n .
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