Mo hinh hoa va dieu khien robot di dpng non-hoionomic c6 triidt ngang
Nguyin Van Tinh^*, Pham Thu-gng Cat', Phgm Minh T\ian^
'Vien Cong ngh^ Thong lin, Vi$n Han lam KH&CN Viet Nam
^VienC6ngngh4 Vutru. Vien Hdn lam KH&CN Vi4tNam
^O
Ngay nhan bai 5.10.2015, ngay chuySn phan bien 12 10.2015, ng^y nhgn phan bien 18 11,2015, ngay chip nh^n dang 25.11.2015
Trong bai bao nay, nhdm nghien ciru da xay dung mot c^ch co h§ thong phuong phap md hinh hoa he ddng luc hoc ciia mot robot di ddng banh xe non-holonomic co trugt ngang. Sau do, nhom tac gia thiet ke mot lu^t dieu khien bang phtnmg phap tuyen tinh hoa phan hoi vao ra dl di^u khiln robot di dong nay bam theo m6t quy dao cho trufrc ma co thi bii duffc trupt ngang. Cac ket qua mo phong duffc thuc hiSn b5ng Matlab-Simulink da chung minh tinh dung d^n cua luat dilu khien.
Tii- khoa: robot di dpng non-holonomic, trw0 ngang, tuyen tinh hda phdn hdi Chi so phdn loai 2.2
MODELLING AND CONTROLLING A NON-HOLONOMIC WHEELED MOBILE ROBOT WITH LATERAL SUP
Summary
This paper presents the systematic development to model the dynamics of a non-holonomic wheeled mobile ! robot with lateral slip, followed by the design of a control law using the \ input-output feedback linearization method to drive the mobile robot to track a given trajectory while lateral [ slipping exists. Matiab-Simulink simulation results showed the correctness and performances of the j control law.
Keywords: input-output feedback linearization, lateral slip, non- holonomic wheeled mobile robot Classification number 2.2
Dat van dl
Robot di dpng banh xe da dugc nghien ciru va mig dung trong nhieu nganh c6ng nghiep va la mot lmh vuc thu hiit sir quan tam cua nhieu nha khoa hoc tren the gioi. Ly do robot di dpng dupe ung dung rpng rai tren the giai la do no co the chuyen dpng thong minh ma khong co tac dpng ciia con nguoi, pham vi hoat dpng khong bi gioi han. Dac biet, no c6 the thay the con nguoi trong cac nhiem vu nguy hiem nhu: tim kiem vat lieu no, van chuyen hang hoa trong moi tnrang dpc hai, giam sat an ninh... Nhieu eong trinh nghi8n ciiu ve robot di dpng tap trung vao viec giai quyet bai toan dieu khien chuyen dpng. Nhieu eong trinh nghien curu da thiet ke cac bp dieu khien tuang ting ma chung da tich hpp mo hinh dpng hpe CO rang buoc non-holonomic voi mo hinh dpng luc hpe cua robot di dpng [1-4], O do, cac tac gia da gia su dieu kien rang buoc non-holonomic luon duac dam bao (cac banh xe chi co chuyen dong lan ma khong truot). Tuy nhien, trong thuc te, khong phai luc nao dieu kien rang huge non-holonomic cung lu6n dupe thoa man.
Rang buoc non-holonomic phuc thupc vao rdt nhieu yeu t6 nhu dp cang cua lop, dp tran cua mat san, dp phang cua dia hinh... Khi do, neu muon giai quyet bai toan dieu khien chuyen dpng thi dpng hoc, dpng lire hpe trugt phai duac tinh den khi thiet ke bp dieu khien cho robot di dpng. Trong [5], cac tac gia da phat trien mot mo hinh dpng hoc suy rpng ma a do da chiia dung cac loai trupt khac nhau nhir trupt dpc, tnrpi ngang, trugt quay. Trong [6], dieu khien luc ngang da dugc de xuat bang cac bg dieu khien lire va vi tri, trong do cac yeu to trupt da dupe tinh den. Trong [7], cac tac gia gioi thieu mpt bp dieu khien ben vihig de bam theo quy dao bang each tich hgp dgng hoc trugt vao dpng hpe robot di dpng banh xe dual dang cac ham \k tinh on dinh dugc kiem chung bang toan tii Lie.
Trong [8], cac tae gia da xay dung mo hinh dgng luc hpe robot di dgng banh xe ma 6 do da chiia dimg dgng luc hgc trugt ngang. Sau
'Tdc gid hin he: Email. nvlinh@ioita
^ N 6 H | V i f t l N a i T ] 4(1) 1.2016
do, mo hinh d6ng luc bpc nay dupe sii dung de thiet ke bg lap quy dao va bg dieu khien de cho phep dSn duong (navigation) co hieu qua robot di dpng trong dieu kien co trupl ngang.
Trong trucmg hgp co trupt ngang hoae trugt dpc thi yeu to ma sat o diem tiep xuc giua banh xe va mat sdn dugc chii y den. He so ma sat phu thugc manh me vko dang dia hinh, do cang ciia lop va van toe robot di dgng. Doi voi dieu khien tnigt dgc, trong [9], he so ma sat duac xem nhu mgt ham cua ty so trugt. Trong [10], cac tac gia da trinh bay mgt xe tu hanh cho nong nghiSp vai gia sii van t6e eua xe nay nho, chi co trugt dgc dupe chii y den, con trugt ngang bj bo qua. Trong [11], da nghien eiiu bang thuc nghiem cac anh huong eua cac tham so nhu ban kinh banh xe, khoang each giua hai banh, tai trgng len hieu qua dieu khien khi t6n tai trugt dpc. Trong [12], cac tac gia da xay dung mpt bai toan path-following khi ton tai ea trugt ngang va trugt dpe. Dua tren mo hinh trupt, bg dieu khien ehuyen dpng dupe tong hgp co tinh d8n trupt dpe. D6i vai trugt ngang, bp dieu khiln dupe dua tren mo hinh ma sat ngang.
Noi dung nghien cuu Md hinh dgng hoc
Xet 1 robot di dpng banh xe c6 trugt ngang nhu hinh 1, voi gia thi6t dp trupt dpc ciia xe c6 the bo qua, mo hinh dpng hpe eiia xe dugc mo ta nhu sau:
r^g = Xj^ cosS + yufSinO + bO (1) r^i_ =x^ cos$ + y/^sm9-b0 (2)
(3) Trong do, r] la do trugt ngang eua robot di dgng.
Banh trai
7 j
Hhih I. mhot dt ddng banh
Md hinh dgng lire hgc cUa robot di dpng Dpng nang ciia than robot di dpng la:
' « M ( 4 + > ^ ) + | / W ^ ' (4)
Trong do, m^ la khoi lupng eiia than robot di dpng, l^ la mo men quan tinh cua thSn nay xung quanh true thang diing di qua diem M.
Dpng nang ciia banh trai va banh phai lan lupt la:
Tong dpng nang cua be la:
^ -^M '^^L '*'^R
2
+'"w^^^^flv(4+^iyUD + 2^MW^ 1 . (5)
(6)
(7)
Trong do, / ^ va /^ lan lugt la mo men quan tinh ciia banh xe xung quanh true quay va true thang diing.
Vi the nang ciia robot di dpng bang 0, nen ham Lagrange eiia no la: L=K.
Gpi vee to tpa dp Lagrange ciia robot di dpng la;
9 = [%'.VM,^.'7,f*ff,f*f,] , phuang trinh rang buoc dugc bieu dien theo dang sau:
A{q)q = 0 (8)
Ket hgp cac phuong trinh (1), (2), (3) va (8), ta xac dinh dupe ma tran A(q) nhu sau;
Ai)'
cos^ sine -6 0 - / - 0 COS0 sine b 0 0 -r -sine cos^ 0 1 0 0
(9)
Phuong trinh Lagrange cua ehuyin dpng ciia robot di dgng la:
Trong do, X=['^,.^,'l3] la mgt vec ta nhan tii Lagrange bieu diln cac luc rang buoc ciia robot di dpng, u la vec ta luc suy rpng tuong ling vai cac tpa dp suy rpng q. BSng each giai phuang trinh Lagrange, phuong trinh dpng lire hgc eiia robot di dpng co dang nhu sau:
nhufti HOC
teNsm^O) 1-2016
vdi-v,=r° « » 0 > »T [0 0 0 0 0 ij ' cac ma tran dSu vao,
(11) J O 0 0 1 0 0] l i
T r o n g d o , m - S, {qf MS, [q) = ^ "
•p - [^j(' ^i. J la vec ta dau vao gom mo men quay banh phai, banh trai; F^^^ la lire day tac dpng vao than robot theo huong ngang nhu hinh 2.
Gpi v = [(*j,,^^] , S,(q) , va ^J^cd 1^ oia tran thoa man phuong trinh sau:
q = S^{q)v + S.^{q)fi (12) DS dang tim dupe ma trin
s,(?)= Mi)-
Dao ham 2 ve phuang trinh (12):
Hcfn nira, ta cung c6:
s,'(,),v, . / Sl[q)N^^(l
Nhan ca 2 ve cua phuong trinh (11) voi sl[c r . [ s , ( , ) ' M S , ( , ) ] v + [s,(9)''MS,(,)]v + [ i , (,)"• Jkffl, (q)\tl*[s, {q)' US, {q)]ii T a d l t h a y ,
S,(,)'MS,(?)=U Thay (17) vao (16) ta duoc:
[i,(,)'*ffr,W]v+[s,(«f»«!(?)]''-'
yo
yu
<%
\x
'^M\
D ^
\py\ \
•'SA! \
^\^<
i^" .
Hinh 2. cdc bien ddu ray(x)
(13)
(14)
(15)
i :
(16)
(17)
(18) (19)
Thiet ke luat dieu hhien
Gpi D{x^y^ la diem muc tieu di chuyen vai van toe dai V^ khong doi theo huong 9^, ta co:
yo-y.^^0, (20) Gpi trang thai ciia robot di dgng la:
S-[%..)'A..^.'7.«',i.f*t'4.^Ly (21) M6 hinh trang thai eua robot di dpng dugc bieu dien
duoi dang sau:
trong do,
' U(-«)J L-™* "s»JL>'i.->'«.
Tinh dao ham bae nhat ciia (23):
(22)
(23)
(24)
tessSiVife
trongdo (H = ^ = -^((*fl-(ii), v = -[^g+^[.) Tiep tuc dao ham 2 ve (24), ta duae:
(25) trong do, f la mgt vec to phu thuge vao quy dao cua muc tieu D. N8u D ehuyin dgng theo duong thang thi f dugc thay boi f^ nhu sau:
e se tien tiem c£in ve 0. Tiic ^-i ->• C;;-; ^ 0.
Tien hanh mo phong bSng Matlab-Simulink. Cac tham so eiia robot di dgng dupe ti6n hanh mo phong dupe chpn nhu sau; m^ = 17 kg; r = 0,095 m;
b = 0,24 m; /^ = 0,023 kgm^; l^= 0,011 kgm^;
l^ = 0,537 kgm^; m^ = 0,5 kg. Khoang each mong muon: C = 0,3 m. Cac tham so dieu khien
/ L =
y2 -CO + >'2 .(y - f'o (U sin (e - ^o)
-_V] Q}-7J- V/^cacos {0-0;^) (26)
thi f dugfc thay boi f^ nhu sau:
>'2.<»+>'j.ii>-i'„(i»-9j,)sm(e-e„) _ -j,<»-i;-F„(<B-9„)cos(fl-e„)
Ta CO the viet lai (25) dual dang:
y =~hv + f
(27)
/ c
trong do h -- 2
N6u_)'|>0, thi h luon kha nghich.
Kk hpp (22) va (29), ta co:
y = -hm~' (T-bijco) + f Ta chpn luat dieu khien:
^ = bijo} + mh~' [f -y^^^^^j +K^e + Kpe) trong d6,e = y- y^^,,,^ , Kp, K^ la cac ma trSn hgng xac dinh duong. Sa do kh6i ciia he thdng digu khi6n dugc mo ta nhu hinh 3.
Yeu cau cua bai toan di6u khiln la di^m P (hinh 2) phai bam tiem can theo di6m D voi sai lech bam tien ve khong. Do vay, ta chpn Yd^^^ed ^ [C,0]
Thay (31) vao (30), ta dupe:
e + Kae + Kpe = 0 (32) Tu phuang trinh dpng luc hpe sai lech nay, sai lech
[4 O'
^^=-^«=[o 4
Neu diem D(x^y^) chuyen dpng voi van toe dai V^
khong doi theo duong tron co dang;
Khong mat tinh tong quat, ty so trupt dugc gia sii la: sa=^ = 0,2.
V Ket qua mo phong
Ta se tiSn hanh mo phong theo 2 trucmg hpp:
Truemg hgp 1: mue tieu D(Xp, y^) di chuyen theo duong thang c6 he so goc 0^ = 71/6 vai van toe 0,2 m/s.
Thai gian (s) Hinh 4- dothisail^ch e -y-y^^^
3 4 5 6 7
-
a 9 10 Thifi gian (s)
Hinh 5 do lln loi do Inral ngang f} kh robot ham ihc-o du/hig ihdng
\m^^M
Then gian (s) n quay a hai banhxi
• L - ^ , !
Tritcmg hop 2: muc tieu di chuyen theo duong tron CO phuang trinh (27) voi van toe dai V^ = 0,4 m/s.
Thai gian (s) h9: ddlh}sailiche=y-y^^^^j
~6 ».s|
s
i ' I I 1™°'^''°'
Thoi gian (s) Hinh JO- dd ihi tdc dp tmpr ngang 7J khi ro i
TrvcX(m) Hinh 11 • guy dao diem P va diem D
Thm gian (s) :« quay d hai banh xe robot di
Then gian (s) Hinh 13 vec to ddu ra y(x)
Toe do trugt ngang tuong iing vai cac truang hgp 1 va 2 duoc mo ta qua eac hinh ve 5 va 10.
Cae hinh 4, 6, 9, 11 da minh hga tinh on dinh tiem can eua luat dieu khien. Trong cac hinh 8 va 13, gia tri y^(x) > 0 voi ^t> 0 nen ma tran h trong phuong trinh (31) luon kha nghich. Hinh 7 va 12 minh hpa do thi mo men quay luon lien tue va hihi han. Do vay, luat dieu khien la kha thi.
Ket luan
Trong nghien ciiu nay, ehiing toi da xay dung thanh eong mo hinh dpng hgc, dgng lire hpe cua robot di dpng khi co trupt ngang. Trong eac mo hinh dgng hpe, dpng lue hpe deu chiia dung dpng hpe, dpng lire hoc ciia trugt ngang. Sau do, chung toi da de xuat mpt luat dieu khien theo phuong phap tuyen tinh hoa phan h6i vao ra. Tinh on dinh cua luat dieu khien da dupe kiSm chiing bang Matlab-Simulink khi ti^n hanh mo phong cho robot bam theo quy dao thang va quy dao tron.
Trong nrong lai, chiing toi se khao sat va thiSt ki bg di6u khien cho robot di dpng khi vira co trupt dpc, vira CO trupt ngang.
l8?SSJ!SSiViii^_^
Tai lieu tham khao
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