Nghien ciiu
N A N G C A O D Q C H I N H X A C D I N H V I G P S D Q N G B A N G L O C K A L I M A N
Dinh Xuan Vinh, Cao Minh Thuy Trudng Dai hgc Tii nguySn va Mdi trudng Ha Ndi Tom t i t
Npi dung thdo lugn vi nhiing phuong trinh viphan trong bo lgc Kalmanphu hpp vdi qud trinh dmh vi GPS dgng. Qud trinh dinh vi GPS dgng duac mo td nhu nhiing chuyen dong ngdu nhien theo thdi gian. Cdc tri do GPS duac bieu diin trong mgt phuang trinh viphan kem theo nhieu thuc. Mgt mo hinh bieu diin thoi gian ciia chuyen dong duac xdy dung kem theo nhiiu trdng. Sic kit hop cdc phuang trinh vi phdn ciia the gidi thuc vd mo hinh duac xdy dung dua trin cdc phuang trinh bieu diin suphii hap voi chuyen dgng cua anten mdy ihu GPS dong. Khdo sdt ndy co thi gop phdn ndng cao do chinh xdc thdnh lap bdn do ty li lan, gidm chi phi nhdn luc, tdng tiin do thi cong vd bo sung img difng cho bo loc Kalman.
Tir khoa: Lpc Kalman; GPS dpng.
Abstract
Enhance the accuracy of kinematic GPS positioning with Kalman filtering This paper discusses the differential equations in the Kalman filter that are suitable for kinematic GPS positioning. Kinematic GPS positioning is described as random motion over time. GPS measurements are expressed in a differential equation with real noise. A time model of motion is constructed with white noise. The combination of real world and model is based on equations, that are representing the motion of the kinematic GPS receiver antenna. This paper can contribute to improving the accuracy of large scale mapping, reducing employee costs, increasing schedules and adding Kalman filter applications.
Keywords: Kalman filter; Kinematic GPS.
A. Gidi thieu lpc Kalman lam mlm va lpc theo quy trinh ngiu Phep Ipc tuyln tinh cac gia tri do "^^^n Markov. Kalman (I960) dl xuit dupc ciia mdt tap hgp cac biln ngiu phuang phap lpc tuyen tinh mdi [1], nhiSn d l udc lugng, hay ndi chinh xic giai quylt dugc bai toan vi phin bac hai ban l i d l du bao cic gia tri khdng do I^Y^^ tinh. T r ^ g thai tuc thdi ciia md dugc cua mdt tip hpp khac, da dugc cac hinh tuyln tinh dgng vdi su tham gia nha khoa hpc thi gidi quan tam tir rit ciia nhieu tring dugc udc lugng bing sdm. Phuang phap diu tien dinh hinh su dung cac tri do d trang thii tuang udc lugng tdi uu tu dii lieu cd nhiSu quan tuyen tinh xen lan nhieu trang.
la phuang phip Binh phuang nhd nhit Phuang hinh he thdng lgc Kalman ciiaGauss(1777-1855).Tinhchitchic rdi rac l i mdt udc lugng trang thai chan cua tri do cd chiia sai s l (nhiSu) x&R" theo mdt quy hinh bi chi phdi bdi dupc xac nhan bdi Galileo (1564 - phuang trinh vi phan haySn tinh ngiu 1642). Diu the ki 20, Kolmogorov nhien sau-
(1903 1987) v i Wiener (1894
1964) da sing tao ra Iy thuylt dir bao, "^^ -Fx,_,+Gu,_,+w,_, (1)
20 . . Tgp chi Khoa hgc Tdi nguyin vd Mdi tru&ng - Sd 19 - ndm 2018
Nghien cUu Vdi tri do z^R"' tuin theo phucmg
trinh sau
'^k=Hx,+v, (2) trong do: :x:^ la vector chi trang thai he thdng; ma tran F kich thudc {n x n) trong phucmg trinh vi phan l i ma trin he sd cua i n tai trang thai trudc dd (k-1) so vdi trang thai hien thdi k.
Ma tran G la ma Iran he sd diu vio diSu chinh tuy y cua iryu&R' liSn he vdi trang thai cua i n x, trong trie dia thi nd bieu thi cic nguyen nhan giy nSn biln ddi he thdng, anh hudng tdi quy trinh ngau nhien cua he thing. Ma trsin H kich thudc (m x n) trong phuang trinh tri do la ma tran he sd cua tri do z^, w^^
l i nhiSu trang he thdng va nd dugc bieu diln nhu mdt vector; v^ l i nhiSu trang trj do dupc bieu dien dudi dang vector. Chi sd k chi thdi diSm cua he thdng v i k-l l i thdi diem trudc dd.
Phuang trinh (1) phu hgp vdi md hinh van dgng (la md hinh cd ngoai luc tac ddng g i y bien ddi van tde va gia tdc) va khdng the tim thay trong md hinh ddng (li md hinh khdng cd ngoai luc tac dpng) thanh phin Gw^^ vi khong cd nguySn nhan gay bien dang nao dugc tinh din trong md hinh. Cung khdng thS tun thay trong md hinh tTnh thinh phin Fx^j vi vit thS phin iing ngay tire thi vdi nhiing thay ddi diu vio. Trong md hinh ddng nhit khdng cd nguySn nhin giy bien dang, nen ma trin he thdng dugc xac dinh la ma tran don vj.
Vector trang thai tu nhien cda x^ le dT nhien la biln khdng do dugc, cdn z^ l i gii tri do dugc. Biln ngau nhiSn w^^ va v^ bilu dien nhieu he thdng va nhieu tri do, chiing dugc gia thiet la ddc lap vdi nhau, la nhieu trang v i tuin theo phin phdi chuan, nghTa la
P(w)-N(0,Q) (3) p(v)~N(0,Q) ^ (4) Ta CO ma trin nhiSu he thong Q lien quan tdi vector nhilu hS thdng theo:
Q=E[ww'] (5) Ma tran nhilu tri do R cd liSn he vdi vector nhieu tri do v theo:
R=E[vv^J (6) NSu chiing ta mang nhung hi do vdi chu ky T^ dS dua vao phep lpc, thi viec diu tiSn l i ta phii tim dugc ma trin CO sd <^. Ma tran ca sd ciia he thdi gian bat bien cd the tim dugc tir ma tran he thdng ddng [3] nhu sau:
^(t)=L:'[(sl-F)-'] (7) Ci day, I l i ma tran dan vi, I^' la biSn ddi Laplace nghich dio, F l i ma tran he thing dpng.
C6 the chiing minh dugc cic phuong trinh Riccati bieu diSn Hiep phuang sai tien nghiem, Hiep phuang sai hiu nghiSm v i gii tri Hieu ich ciia cac budc lpc Kalman. Phuang trinh Riccati nhu sau:
M^=0^P^_^0l+Q^. (8) K^=M^H''(HM^H'^+RJ-' (9)
P,=(I-K,H)M, (10) 6 diy, P^ la ma trin hiep phuang sai md ta sai sd trong udc lugng trang thai sau khi c|.p nhat; M^ la ma tran hiep phuang sai md t i sai sd trong udc lupng trang thai trudc khi cap nhat. Ma trin nhilu rdi rac g^ cd the tim dugc tir ma tran nhieu lien tuc Q va ma tran ca sd theo
Qk^\^^0(t)Q0^(t)dt (10) De bat dau phuang trinh Riccati, ta cin ma tran hiep phuang sai ban dau P . Tap chi Khoa hgc Tdi nguyen vd Moi tru&ng - Sd 19 - ndm 2018
Nghien cuv
B. Thuc nghiem quan t r i e GPS ddng
Muc tieu i p dung phuang phip xir Iy sau trong cdng tac thu tin hieu GPS ddng vdi loai m i y thu GPS thdng dung, re tien, nhung cd thS cho ta chit lupng vi tri diem dat dp chinh xac ca xen ti met, phu hgp ySu cau x i y dung ludi khdng chS do ve ty le Idn, hoac do ve chi tiSt thanh lap ban dd ty IS 1:500. Chiing tdi tiln hanh mdt each cin than cdng tic thu tin hieu GPS tai khu virc trudng Dai hgc Tii nguySn v i Mdi trudng Ha Ndi, ngiy 07 thing 9 nam 2017. Thilt bi thu tin hiSu GPS gdm 3 may thu loai tin hieu mdt t i n s l X20 ciia hang Huace - Trung Qudc, s l hieu cac may la: 100957, 100961 va 101533. Loai m i y X20 tuong ddi cii, chi thu dupc duy nhat tin hieu GPS khoing cich g i i v i GPS pha sdng tii, khdng thu dugc tin hieu Glonass va Beidou. Dat gdc chan trdi 15", tin sd liy m i u 5 giay.
Thdi gian do bat dau Iiic 10 gid 20 ph6t, gid H i Ndi. Ket thuc do liic 11 gid 38 phiit. Trong 50 phut dau tien, ba may thu d chS dp tTnh. Phuang phip do tucmg ddi cho phep xac dinh chinh xac vi tri diSm miy thu. Khoing 25 phut cudi, may sd 100961 di dpng theo hai hudng g i n vudng gdc nhau.
Dd la do m i y 100961 dat d nga ba dudng. Qua trinh di ddng may theo hai con phd khoang 18 phiit, sau dd dit trd lai may vao chan ba chac v i n giu nguyen tren moc khoang 7 phut, sau dd kSt thuc ca do. Dac thii pho nhd, be ngang phd khoing 5 met. Mdt con phd cd nhieu cay to hai ve dudng, din tdi tin hieu GPS bi mit trong vai phut. DiSu kien ve tinh va chit lugng 22 —
may thu rat khiSm tdn. Hiu het thdi gian do chi thu duge tin hieu 5 ve tinh.
Day gan nhu la gidi ban cudi ciia chit lugng ca do. Doi vdi ca do tTnh thi khdng van de gi, nhung vdi ca do dpng thi chat lugng tin hiSu rit tdi.
Khoang cich giiia cic diem trac dia k h i gin nhau, tu 229 met dSn 280 met va phan bd nhu hinh 1. Ciu tao mdc trac dia nhu hinh 2. Hinh 3 l i may thu 100961 va sinh viSn do dac. Hinh 4 la tap tri do va hudng di chuySn ciia m i y 100961.
Xu ly dii lieu do bing phan mem Compas kem theo may. Sd lieu do tinh theo phuang phip tuong ddi dat ket qua tdt. Sai sd vi tri diem thu dugc cd dp chinh xic ±1 mm. Ly do l i khoang cich giua cac diem kha gan nhau (chua dSn 300 met). Ve tri do ddng, lua chpn 249 tri do dgng cd thdi gian tu 4:15:00 G P S T d l n 4 : 3 8 : l 5 G P S T . Khoing thdi gian tir 4:28:20 GPST dSn 4:30:55 GPST khdng thu dugc tin hieu. Ly do, may ddng di chuyin trSn con phd nhd, cd nhilu ciy xanh ven dudng, m i y X20 chi nhin dugc tin hieu GPS ciia 5 ve tinh, cic ve tinh Glonass va Beidou khdng c i u tnic trong m i y thu X20.
Sau khi di chuyin theo hai con phd va quay trd lai mdc trac dia cii, may 100961 dat trd lai trSn chan may v i n giira nguyen tren mdc trac dia va thu rin hieu GPS vdi thdi gian khoing 7 phiit. Sau dd tit m i y va k i t thiic thuc nghiSm.
Tgp chi Khoa hgc Tdi nguyen vd Mdi truang - So 19-nam 2018
Nghien cdy
1
,-»V V
"4^
* '
Hinh 3: Mdy Ihu 100961
Bang 1. Mft totfn Iri Ito GPS Mng (WGSS4)
Hinh 4: Tgp trf do va hir&ng di chuyen cua 100961
Kinematic 1 2 3 4 5 6 7 8 9
GPST 4:15'00AM 4,15.05 AM 4.15:10AM 4:15:15 AM 4:15:20 AM 4:15:25 AM 4:15:30 AM 4:15:35 AM 4:15:40 AM
x-ecef(m) -1617957 304 -1617958.055 -1617957.632 -1617958.075 -1617957.721 -1617957.813 -1617957.576 -1617957 569 -1617957.597
y-ecef(m) 5731130.610 5731134.222 5731132.958 5731134.736 5731134.335 5731135.291 5731134.737 5731134.896 5731135.530
z-ecef(ni) 2276349.631 2276350.955 2276350.265 2276350.733 2276350.712 2276351.113 2276350 986 2276351 047 2276351 200
Tgp chi Khoa hoc Tdi nguydn vd Moi tru&ng - So 19 - ndm 2018
-.. _ _ « - ^ * . ' . - i ^ ^ . " -
Hinh 5: So lugng ve tinh trong ca do th^e nghiem C. Xii 15^ s6 liSu
Lpc Kalman dugc trien khai theo phuang phip sd dung tich phan Euler va phucmg phip Bmh phuong nhd nhat.
Cic phuang trinh Riccati nham nang cao hieu suit cua bd lpc va tdi uu hda sau mdi budc Igc dugc trien khai dudi dang da thuc. Do vay, lpc Kalman ciing dupc trien khai d dang da thdc.
Thuc hiSn lpc Kalman dang da thiic vdi cac bac lan luot la 0, 1, 2. Ddi vdi bac 0, ta cd phuomg trinh ma tran co sd ban dau nhu sau:
Ky hieu: X/^ udc lupng Kalman t^i thdi doan k; x^_, udc lugng Kahnan thdi doan k-1; K^^ hieu ich cua udc lugng Kalman bac 0 (trang thii I) tai thdi doan k; Zj tri do tai thdi doan k.
Dd lech cua lpc bic 0 dugc dinh nghia:
Res^ =?(-^t_i
Lpc bac I kem theo van tdc cd dang nhu sau:
[I 0] (13)
Ky hieu; T^ tan sd do (khoang cich giiia cac thdi doan); ffi^ hipu ich Kalman doi vdi vj tri dilm; Kz^ hieu ich Kalman ddi vdi van tdc chuyen ddng ciia diem; xj, udc lugng van t i e dilm Kalman tai thdi doan k; x^ udc lugng gia tdc diSm Kalman tai thdi doan k;
Xfc-i udc lupng van tdc cua diSm tai thdi doan k-1.
Dp lech cua lpc bac 1 dugc dinh nghia:
Res,=z^-x^_^-Ti^_^
Hieu ich cua lpc Kalman bac I dupc tinh theo phucmg phap binh phuang nhd nhit de quy:
K„ l(lk-\)
k(k + \) ' k(k + \)T, k=l, 2, 3 n.
Lpc bjc 2 kem theo gia t6c va van t6c CO dang nhu sau:
24 Tgp chi Khoa hoc Tdi nguyen va Moi truang -So 19- nam 2018
x^_,
^ . - 1
L4-ij +
X i l Kn A:,,
J
1 7; 0.57;;
0 I 7;
0 0 1
Ky hieu: x^ udc lugng gia tic diem Kalman tai thdi doan k; -V;_i udc lupng gia tdc cua diem tai thdi doan k-l; Kji_
hieu ich Kalman ddi vdi gia tde chuyen ddng cua diem.
Dp lech cua lpc bac 2 dugc dinh nghTa:
Res, - z ^ -i^_, -T^ik-^ -0.5T;:\_^ (15)
1 7; 0.57"/" ^k'\
1
-[1 0 0] 0 1 r, i._, (14) 0 0 1 i , L JL%-iJJ HiSu ich cxia lpc Kahnan bac 2 dugc tinh:
''-"^^'.k^U + i)(k-H)
\%(2k-\) k(k+l)(k+2)T, '
60 k(k + l)(k + 2 J . 2 .
• (16) (17) (18)
Hinh 6: Gid tri tga do Y (net liin) vd lgc Kalman (net dut) theo thai gian Breu dd X
2323360 QOOO
232E350XHKIO
232E3400QDa 232S33D OODO
2328320.0000
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5 0 1
--¥4
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0 150
^:__z
,i<^^;^^fzSs»
Vung udc lucmg
-t- -1
Hinh 7: Gia tri tga do X(net lien) vh lgc Kalman (net dirt) theo thdiglan Luu y, khoang 7 phiit cudi may 100961 d trang thai tTnh. Theo dd thi 6 v i 7, tri do X v i Y cua m i y biln ddng rit ldn, thim chi hon 10 met. Sd dung bd Ipc Kalman cho ta mdt udc lugng tdt hon.
Vi tri diem 100961 cd tpa dd (VN2000) xic dinh trong ca do tinh va do ddng la:
Bdng 2. So sdnh tga dd hai ca do tinh va ddng cda mdy 100961 STT
1 2 3
Ca do tinh Ca do dong Do lech
Tpa do X (met) 2328352 7827 2328352.7540 0.0287
Toa do Y (met) 579494.7087 579494.6767 0.0320 Tap chi Khoa hgc Tdi nguyen vd Moi tru&ng - So 19 - ndm 2018
Nghien cuu
TAI LIEU THAM KHAO [1]. R.E. Kalman (1960). A New Approach to Linear Filtering and Prediction Problems, Joumal of Basic Engineering, 82 (series D):34-45. Copyright @ 1960 by ASME.
[2]. Norman Morrison (1969). Intro to Sequential Smoothing and Prediction.
McGraw-Hil Book Company, New York.
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[4]. Phan Van Hiln, Dioh Xuin Vinh (2010). l/hg dung lgc Kalman trong phdn dch bien dgng nhd cao tdng do biec xg nhiet mdt trai. Tap chi Xay dung, so 5-2010.
ISSN 0866-0762.
[5]. De tai nghien cilu khoa hpc cap ca sd "Nghien cuu ndng cao do chinh xdc xdc dinh vi tri diim mdy thu trong do GPS dgng xu ly sau phitc vy thdnh lap bdn do ty li lan a Viet Nam ",m&s6 13.01.n.0.03. C\m nhiem de tai: TS. Dinh Xuan Vmh.
C. KET LUAN
Lpc Kalman thiic su rit hieu qua khi phin tich dii lieu van ddng theo thdi gian. Nhiing nghien ciiu budc dau nham su dung m i y thu GPS re tien, thdng dung thay the nhiing may tfiu GPS RTK dat tiln. KSt qua cd thi chap nhan dugc trong dieu kien sd lugng vS tinh rit it (5 ve tinh], chit lugng tin hieu yeu v i ddi khi mat tin hiSu. Vdi dd chinh xac
±3 cm cd the ung dung trong do ve chi tiet hoac xiy dung ludi khdng che do ve. Neu cd thi su dung loai may thu dupc cac ve tinh GPS, Glonass, Beidou thi chit lugng se nang len dang ke, thdi gian do ciing vi thS ting nhanh hem.
LM cam on: NghiSn ciiu nay dugc th\rc hipn vdi su hd trg cua de tai nghien cuu khoa hpc cap ca sd "NghiSn cdu ning cao dp chinh xac xac dinh vj tri diSm miy thu trong do GPS ddng xd ly sau phuc vu thanh lap ban dd ty IS ldn d Viet Nam", ma sd 13.01.17.O.03. Tac gia chan thinh cam an nhdm smh vien D H 5 ( ^ 9 da nhiet tinh tham gia thuc nghiem.
BBT nhan bai: 22/01/2018; Phan bien xong: 27/2/2018 KHAO SAT IfNG DUNG CfJA THUAT TOAN BJERHAMMAR... (tiep theo trang 81)
[5]. Gilad Even-Tzur, Lior Shahar (2015). Application of extended fi^e net adjustment constraints in two-step analysis of deformation network. Acta Geod GeophysDOI10.1007/s40328-015-0119-3, Springer.
[6]. Dagogo M.J. FUEARA (1973).
Geodetic numerical and statistical analysis of data. Space Program Office, Battelle Columbus Laboratories, Columbus, Ohio 43201. https://link.springer.com/
article/10.1007%2FBF02522077.
[7]. Haim B. Papo (1986). Extended free net adjustment constraints. US
Department of commerce. National Oceanic and Atmospheric Administration, National Ocean Service.
[8]. Hoang Ngpc Ha, Trucmg Quang HiSu (2003). Ca sa todn hgc xu ly sd lieu trac dia. NXB Giao thdng van tai.
26 -
[9]. Hoing Ngpc Ha (2006). Binh sai tinh todn lu&i trac dia vd GPS. NXB Khoa hoc ky thuat.
[10]. Huang Shengxiang, Yin Hui, Jiang Zheng (2004). Xie ly so lifu trong quan trac bien dgng. NXB Dai hpc Vii Han (Ban dich cua PGS. TS Phan Van Hiln.
NXB Khoa hpc va ky thuat, 2010).
[11]. Trin Khanh, Nguyen Quang Phiic (2010). Quan trac chuyin dich vd biin dgng cong trinh. NXB Giao thdng van tai.
[12]. Tao Benzao (2017). Binh sai lu&i tu do vd phdn tich bien dgng. Phan Van Hiln va Pham Qudc Khanh dich. NXB Tai nguyen - Mdi trudng va Ban dd Viet Nam.
BBT nhan bai: 11/01/2018; Phin bien xong: 20/02/2018 Tgp chi Khoa ligc Tdi nguyin vd Mdt truang -Sd 19- ndm 2018
