Citation
A Drag Device and Control Algorithm for Spacecraft Attitude Stabilization and De-Orbit Point Targeting Using Aerodynamic Drag

Material Information

Title:
A Drag Device and Control Algorithm for Spacecraft Attitude Stabilization and De-Orbit Point Targeting Using Aerodynamic Drag
Creator:
Omar, Sanny R
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (164 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering
Mechanical and Aerospace Engineering
Committee Chair:
Bevilacqua,Riccardo
Committee Co-Chair:
Fitz-Coy,Norman G
Committee Members:
Conklin,John
Matcheva,Katia Ivandva
Orr,Jeb
Graduation Date:
12/13/2019

Subjects

Subjects / Keywords:
control -- cubesat -- drag -- hardware -- reentry
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Aerospace Engineering thesis, Ph.D.

Notes

Abstract:
The ability to re-enter the atmosphere at a desired location is important for spacecraft containing components that may survive re-entry. This work discusses the use of solely aerodynamic drag forces to perform this targeting, which is especially useful for smaller spacecraft that do not contain thrusters capable of facilitating a controlled re-entry. It is shown that by varying the ballistic coefficient of a spacecraft over time, any desired longitude and latitude can be targeted provided that the maneuvering begins early enough and the latitude is less than the inclination of the orbit. An analytical solution based on perturbations from a numerically propagated trajectory is developed to estimate the ballistic coefficient profile necessary to reach a given target point assuming small deviations from the numerical trajectory. An iterative process whereby analytical solutions are tested and refined can be utilized to determine the ballistic coefficient necessary for re-entry point targeting. Monte Carlo simulations are conducted to verify the algorithm performance. Desired de-orbit points were reached with an error under 106 km in all tested scenarios. Guidance trajectory tracking methods based on feedback control are also investigated to ensure that the spacecraft has a means of following the guidance trajectory in the face of uncertainties in the drag force estimate. The design of a physical drag device capable of modulating the ballistic coefficient of a host spacecraft is also discussed along with the design of a CubeSat intended to test the drag device. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2019.
Local:
Adviser: Bevilacqua,Riccardo.
Local:
Co-adviser: Fitz-Coy,Norman G.
Statement of Responsibility:
by Sanny R Omar.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
LD1780 2019 ( lcc )

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Full Text

PAGE 1

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PAGE 9

Cb tswap tswap tswap CbCb10=:025Cb20=:012 Cb10=:025Cb20=:012 Cb2 rad des=:01

PAGE 10

Cb Cb Cb Cb Cb Cb Cb Cb

PAGE 12

aCb2eiJ2KnQRRe~rsdTttcu

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~vWd22323!!e

PAGE 18

Cb1aswapCb2 Cb=CdA 2mCdAmCb1Cb2aswap

PAGE 19

J2 ft(a)=p Herfi(p a=H) )]TJ /F2 11.9552 Tf 9.299 0 Td[(2Cbp 0ea0=H Cb Cb1=Cbavg(1+Cb) Cb2=Cbavg(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(Cb)Cbaswap

PAGE 20

Cb Cbavg=2tswapaswapCb1Cb2 tswap

PAGE 22

QR

PAGE 25

Cb1tswapCb2ttermCbterm

PAGE 26

Cb1Cb2tswaptgtg

PAGE 27

J2QR

PAGE 28

QR

PAGE 29

x x=266666664~r~vq~!377777775

PAGE 30

~r=rxryrzT~v=vxvyvzTq=q1q2q3q4T~!=!x!y!zT _x=266666664~v~a_q~377777775~a~a ~a=P~F mq q=266666664cos(=2)nxsin(=2)nysin(=2)nzsin(=2)377777775^n _q=1 2266666664q1)]TJ /F3 11.9552 Tf 9.298 0 Td[(q2)]TJ /F3 11.9552 Tf 9.298 0 Td[(q3)]TJ /F3 11.9552 Tf 9.299 0 Td[(q4q2q1)]TJ /F3 11.9552 Tf 9.298 0 Td[(q4q3q3q4q1)]TJ /F3 11.9552 Tf 9.299 0 Td[(q2q4)]TJ /F3 11.9552 Tf 9.298 0 Td[(q3q2q13777777752666666640!x!y!z377777775

PAGE 31

q=266666664nxsin(=2)nysin(=2)nzsin(=2)cos(=2)377777775=266666664q1q2q3q4377777775 _q=1 2266666664q4)]TJ /F3 11.9552 Tf 9.299 0 Td[(q3q2q1q3q4)]TJ /F3 11.9552 Tf 9.299 0 Td[(q1q2)]TJ /F3 11.9552 Tf 9.298 0 Td[(q2q1q4q3)]TJ /F3 11.9552 Tf 9.298 0 Td[(q1)]TJ /F3 11.9552 Tf 9.299 0 Td[(q2)]TJ /F3 11.9552 Tf 9.299 0 Td[(q3q4377777775266666664!x!y!z0377777775~H ~Tnet=X~T=J~+~!(J~!)J~~Tnet~!~J ~ ~=J)]TJ /F5 7.9701 Tf 6.586 0 Td[(1h~Tnet)]TJ /F3 11.9552 Tf 11.968 0 Td[(~!(J~!)i

PAGE 32

~r ~ag=)]TJ /F3 11.9552 Tf 10.494 8.087 Td[(e~r r3e~ag U=e Re1Xn=onXm=0(CnmVnm+SnmWnm)CnmSnmnmVnmWnm Vmm=(2m)]TJ /F2 11.9552 Tf 11.955 0 Td[(1)xRe r2Vm)]TJ /F5 7.9701 Tf 6.586 0 Td[(1;m)]TJ /F5 7.9701 Tf 6.587 0 Td[(1)]TJ /F3 11.9552 Tf 13.151 8.088 Td[(yRe r2Wm)]TJ /F5 7.9701 Tf 6.587 0 Td[(1;m)]TJ /F5 7.9701 Tf 6.586 0 Td[(1Wmm=(2m)]TJ /F2 11.9552 Tf 11.955 0 Td[(1)xRe r2Wm)]TJ /F5 7.9701 Tf 6.587 0 Td[(1;m)]TJ /F5 7.9701 Tf 6.586 0 Td[(1+yRe r2Vm)]TJ /F5 7.9701 Tf 6.586 0 Td[(1;m)]TJ /F5 7.9701 Tf 6.587 0 Td[(1

PAGE 33

Vnm=2n)]TJ /F2 11.9552 Tf 11.955 0 Td[(1 n)]TJ /F3 11.9552 Tf 11.955 0 Td[(mzRe r2Vn)]TJ /F5 7.9701 Tf 6.587 0 Td[(1;m)]TJ /F9 11.9552 Tf 11.956 16.857 Td[(n+m)]TJ /F2 11.9552 Tf 11.955 0 Td[(1 n)]TJ /F3 11.9552 Tf 11.955 0 Td[(mR2e r2Vn)]TJ /F5 7.9701 Tf 6.586 0 Td[(2;mWnm=2n)]TJ /F2 11.9552 Tf 11.955 0 Td[(1 n)]TJ /F3 11.9552 Tf 11.955 0 Td[(mzRe r2Wn)]TJ /F5 7.9701 Tf 6.587 0 Td[(1;m)]TJ /F9 11.9552 Tf 11.955 16.857 Td[(n+m)]TJ /F2 11.9552 Tf 11.955 0 Td[(1 n)]TJ /F3 11.9552 Tf 11.956 0 Td[(mR2e r2Wn)]TJ /F5 7.9701 Tf 6.587 0 Td[(2;m x=Xn;mxnm;y=Xn;mynm;z=Xn;mznm xnm(m=0)= R2e[)]TJ /F3 11.9552 Tf 9.298 0 Td[(Cn0Vn+1;1]xnm(m>0)= 2R2e[)]TJ /F3 11.9552 Tf 9.298 0 Td[(CnmVn+1;m+1)]TJ /F3 11.9552 Tf 11.955 0 Td[(SnmWn+1;m+1+(n)]TJ /F3 11.9552 Tf 11.955 0 Td[(m+2)! (n)]TJ /F3 11.9552 Tf 11.955 0 Td[(m)!(CnmVn+1;m)]TJ /F5 7.9701 Tf 6.587 0 Td[(1+SnmWn+1;m)]TJ /F5 7.9701 Tf 6.586 0 Td[(1)] ynm(m=0)= R2e[)]TJ /F3 11.9552 Tf 9.298 0 Td[(Cn0Wn+1;1]ynm(m>0)= 2R2e[)]TJ /F3 11.9552 Tf 9.299 0 Td[(CnmWn+1;m+1+SnmVn+1;m+1+(n)]TJ /F3 11.9552 Tf 11.955 0 Td[(m+2)! (n)]TJ /F3 11.9552 Tf 11.955 0 Td[(m)!()]TJ /F3 11.9552 Tf 9.298 0 Td[(CnmWn+1;m)]TJ /F5 7.9701 Tf 6.586 0 Td[(1+SnmVn+1;m)]TJ /F5 7.9701 Tf 6.586 0 Td[(1)] znm= R2e[(n)]TJ /F3 11.9552 Tf 11.955 0 Td[(m+1)()]TJ /F3 11.9552 Tf 9.299 0 Td[(CnmVn+1;m)]TJ /F3 11.9552 Tf 11.956 0 Td[(SnmWn+1;m)]C00=1 ~agC20J2J2

PAGE 34

J2J2J2 aosc=amean+asp asp=J2R2e aa r3)]TJ /F2 11.9552 Tf 36.689 8.088 Td[(1 (1)]TJ /F3 11.9552 Tf 11.955 0 Td[(e2)3=2+)]TJ /F9 11.9552 Tf 11.291 13.27 Td[(a r3+1 (1)]TJ /F3 11.9552 Tf 11.955 0 Td[(e2)3=2+a r3cos(2!+2)3sin2i 2J2C20

PAGE 35

J2 ~Fd=)]TJ /F2 11.9552 Tf 10.494 8.088 Td[(1 2CdAvrel~vrelCdA~vrel

PAGE 36

~ad=)]TJ /F3 11.9552 Tf 9.298 0 Td[(Cbvrel~vrelCd~vreld~r dt!e !e !e ~vrel=d~r dt)]TJ /F3 11.9552 Tf 11.969 0 Td[(~!e~r~r~vrel 1:2!e

PAGE 39

~Fp=)]TJ /F2 11.9552 Tf 10.494 8.088 Td[(1 2CdAv?~v?CdA~v?~vrel^np~v1^np~v?~vrel~v? ~vrel=~v)]TJ /F3 11.9552 Tf 11.969 0 Td[(~!e~r ~v?=max(~vrel~np;0)~!e~rp ~p=~rp~Fp

PAGE 40

~gg=3e r3^n(J^n)r^nJ ~t~t ~Tmag=~t~BEarth~BEarth ~t

PAGE 41

HBBsHHcBHBr

PAGE 42

HHcB)]TJ /F3 11.9552 Tf 9.299 0 Td[(BcHHcHrBsHBH BBH B=0H0 arctan

PAGE 43

B=2 Bstan)]TJ /F5 7.9701 Tf 6.587 0 Td[(1[k(H+Hc)]BsBrHc k=1 HctanBr 2Bs B=2 Bstan)]TJ /F5 7.9701 Tf 6.587 0 Td[(1[k(H)]TJ /F3 11.9552 Tf 11.955 0 Td[(Hc)] HHdB=dH B0=dB dHbound=2 kBscos2B 2BsdB=dHBBB=0HBBsdB=dHB0f dB dH=[q0+(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(q0)fp]B0q0p f=jH)]TJ /F3 11.9552 Tf 11.956 0 Td[(HLj 2HcdH=dtHr dH=dtHL dB=dHHB

PAGE 44

dB dt=dB dHdH dtdH=dtHBdB=dt ~=BV~l 0V~l0~bt ~Thyst=~bt~BEarth

PAGE 46

(a;e;;!;;i)=(6778;0;50;40;10;70)Cb=:1122

PAGE 48

Cb1tswapCb2atermttermCbtermCb1Cb2

PAGE 49

tswapCbtermtgCb

PAGE 50

Cbtgtgtg tg Cb

PAGE 52

da dt=2 np 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(e2hesinFR+p rFsipFRFse=0p=rFs=ad da dt=2ad n n=r a3 ad=)]TJ /F3 11.9552 Tf 9.299 0 Td[(Cbv2

PAGE 53

Cb=CdA 2m )]TJ /F9 11.9552 Tf 11.955 17.507 Td[(r a31 2Cbv2da=dt v=r a v )]TJ /F3 11.9552 Tf 30.805 8.088 Td[(da 2p aCb=dta0af t=Zafa0)]TJ /F3 11.9552 Tf 28.149 8.087 Td[(da 2p aCbCb tCb=Zafa0)]TJ /F3 11.9552 Tf 21.9 8.087 Td[(da 2p aa aG(a) tCb=G(af))]TJ /F3 11.9552 Tf 11.955 0 Td[(G(a0)a0afCbAa0aftAtBCbB

PAGE 54

tB=CbAtA CbBu=+! n=du dtu du da=)]TJ /F3 11.9552 Tf 30.769 8.087 Td[(n 2p aCb n du da=)]TJ /F2 11.9552 Tf 25.199 8.087 Td[(1 2a2Cb Cb(da) uCb=Zafa0da 2a2a aP(a) uCb=P(af))]TJ /F3 11.9552 Tf 11.955 0 Td[(P(a0)a0afCb ACbAuAa0afa0afB

PAGE 55

CbB uB=CbAuA CbBa0af !avg=u t=P(af))]TJ /F3 11.9552 Tf 11.955 0 Td[(P(a0) G(af))]TJ /F3 11.9552 Tf 11.955 0 Td[(G(a0) Cb!avga0afaJ2 _avg=)]TJ /F9 11.9552 Tf 11.291 20.922 Td[(3p J2R2e 2(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(e2)2a(7=2)cosiReJ2_avg =_avgta a0af_avg

PAGE 56

CbCb Cbu02u uA=100 J2J2

PAGE 57

_avgCb a Cb iCbi0iti0ui0i0 tiuiiiCbi ti=Cbi0ti0 Cbi ui=Cbi0ui0 Cbi i=i0 ti0ti t0tsoldtsoldteqoldteqoldttermold

PAGE 58

ttermold utii ti0ui0i0 t20teqoldt2=tsnew)]TJ /F3 11.9552 Tf 12.444 0 Td[(teqnewt1=teqnew t0 tf=t0+4Xi=1Cbi0ti0 Cbi uf=uinit+4Xi=1Cbi0ui0 Cbi f=init+4Xi=1i0 ti0ti !f=ef=0 if=iinittfCb

PAGE 59

J2 ae!i

PAGE 60

Cb=:0252

PAGE 61

xzyz rsin(lat)=rsin(!)sin(i)cos(!)sin(i)cos(i)266664cos()sin()0377775

PAGE 62

sin(u)=sin(!+)=(sin(!)cos()+cos(!)sin()) u1=sin)]TJ /F5 7.9701 Tf 6.586 0 Td[(1sin(lat) sin(i)sin)]TJ /F5 7.9701 Tf 6.586 0 Td[(1)]TJ /F3 11.9552 Tf 9.299 0 Td[(=2=2u1u u2=)]TJ /F3 11.9552 Tf 11.955 0 Td[(u1ut0 uf=u)]TJ /F3 11.9552 Tf 11.955 0 Td[(u0+2kku0kkufuf0ufuf e=imp)]TJ /F3 11.9552 Tf 11.955 0 Td[(des

PAGE 63

td=e !e)]TJ /F2 11.9552 Tf 14.556 3.022 Td[(_avg!e_avg!e)]TJ /F2 11.9552 Tf 14.557 3.022 Td[(_avgCbtswap t0ttermt0tswaptswapttermt4iu4ittut ut=uf)]TJ /F2 11.9552 Tf 11.955 0 Td[(u4i=ut0+ud tt=tt0+tdtt0ut0ud Cb1Cb2tswaputttutttCb1Cb2 ut=Cb10u10 Cb1+Cb20u20 Cb2 tt=Cb10t10 Cb1+Cb20t20 Cb2

PAGE 64

Cb1 Cb1=u10Cb10Cb2 utCb2)]TJ /F2 11.9552 Tf 11.955 0 Td[(u20Cb20 Cb1 Cb2 Cb2=Cb20(t20u10)]TJ /F2 11.9552 Tf 11.955 0 Td[(t10u20) ttu10)]TJ /F2 11.9552 Tf 11.955 0 Td[(t10uttswap tsnew=tsoldCb10 Cb1 utttutttut ttut tt tt=Cb20(t20u10)]TJ /F2 11.9552 Tf 11.955 0 Td[(t10u20)+Cb2t10ut Cb2u10ttCb2Cb1utuCbmaxCb2 Cb1utCb2=CbmaxCb1Cbmin

PAGE 65

Cb1Cb2Cb2Cb2Cb1 Cb2CbminCb1 Cb2=Cbminu20Cb20 Cbminut)]TJ /F2 11.9552 Tf 11.955 0 Td[(u10Cb10ttCb1Cb2 ttCb1Cb2utCbCbminCb2 Cb2Cbmin
PAGE 66

ai Tl=2s R3e Tu=2s a3i utut td0 ut[uti+2floortd Tl)]TJ /F2 11.9552 Tf 11.955 0 Td[(1;uti+2ceiltd Tu+1]td>0 ut[uti+2floortd Tu)]TJ /F2 11.9552 Tf 11.955 0 Td[(1;uti+2ceiltd Tl+1]floorceilut ututCb1=Cb2=Cbmin utmax=Cb10u10+Cb20u20 CbminutCb1=Cb2=Cbmax utmin=Cb10u10+Cb20u20 Cbmaxtt tt=tti+ 2 !e)]TJ /F2 11.9552 Tf 14.557 3.022 Td[(_avg!q

PAGE 67

qut tt ttmax=Cb10t10+Cb20t20 Cbmin ttmin=Cb10t10+Cb20t20 Cbmax ttuttctcttminttmaxutttdestt ttdesut tc=)]TJ /F3 11.9552 Tf 9.298 0 Td[(min(jttmin)]TJ /F2 11.9552 Tf 11.955 0 Td[(ttdesj;jttmax)]TJ /F2 11.9552 Tf 11.955 0 Td[(ttdesj) ttdesut tc=min(jttmin)]TJ /F2 11.9552 Tf 11.956 0 Td[(ttdesj;jttmax)]TJ /F2 11.9552 Tf 11.955 0 Td[(ttdesj)tswaptswaptitswaptswapCb1 tsmax=t10+Cb20t20 Cb10t1t2u1u2tswap

PAGE 68

tut10;t20;u10u20tswaptcCb1Cb2tswap tg_Wd _Wd=~ad~v=)]TJ /F3 11.9552 Tf 9.298 0 Td[(Cbvrel~vrel~v WdtwCbCb titi0 twtwiitw tiw=tw)]TJ /F4 7.9701 Tf 13.737 14.944 Td[(i)]TJ /F5 7.9701 Tf 6.586 0 Td[(1Xk=1ti

PAGE 69

tiwiti0wtw ti0w=Cbitiw Cbi0t0tw tweq=ti0w+i)]TJ /F5 7.9701 Tf 6.586 0 Td[(1Xk=1ti0WdtweqWdtwtgCbWdtgCbWdsdWdtg sd=Wddes WdactCbt0tgsdCbWdWdCbCbminCbmaxsdCbCbmaxCbCb

PAGE 70

Cb tgtctctgtgtg CbCb

PAGE 71

tmodCbCbCbudCbuinittmodu r=uinit uinit+ududCbtmodrtswap CbCbCbtermCbCbudr CbtmodCbtmodCbCb10Cb20u10u20t10

PAGE 72

t20tmodCbCbtmodttermt10=u10=0Cb10=Cb20Cb1CbmaxCb1Cbmax Cb2 udtswap Cb1Cb2CbmaxCbCbmaxudCb2>CbmaxCb2CbmaxCb1udCbCbmin u u+2kkT u+2kk[)]TJ /F2 11.9552 Tf 9.298 0 Td[(9;9]uCbterm

PAGE 73

Cb1=(Cbmax+Cbmin)=2Cb=Cbterm ufut0u4i ud(ut) Cb10 Cb1=Cb10ut0 utu(k[9;9]) Cb1uCb1 Cb1 Errmax1 2!eTRecos(lat)!eTRelat

PAGE 74

tswaptswapCb1Cb2tswaptsoldtsnewCb1Cb2 t20t2 t2=Cb2t20 Cb1Cb2Cb1a0aftswap td=t2)]TJ /F3 11.9552 Tf 11.955 0 Td[(t20=tswap)]TJ /F3 11.9552 Tf 11.955 0 Td[(t20 td=tswap)]TJ /F2 11.9552 Tf 11.956 0 Td[(tswapCb1 Cb2=tswap1)]TJ /F3 11.9552 Tf 13.151 8.088 Td[(Cb1 Cb2Cb1Cb2tswap

PAGE 75

tswaptswaptswapt2t2 td d=!a2td!a2 tswaptswap Cb1tswaptswap tsmax=tsold+(ttermold)]TJ /F3 11.9552 Tf 11.955 -.001 Td[(tsold)Cb2 Cb1 tswaptswap2 tswapCb22

PAGE 76

tswap Cb1Cb2tswaputut Cb2 Cb2uttd =)]TJ /F2 11.9552 Tf 9.299 0 Td[(td!e)]TJ /F2 11.9552 Tf 14.557 3.022 Td[(_avg!e_avg Cb

PAGE 77

tswap tswap

PAGE 78

CbCb10=:025Cb20=:012 tswaptdCb1Cb2

PAGE 79

Cb10=:025Cb20=:012 tswapCb10=:025Cb20=:01Cb2 Cb1Cb2tswap

PAGE 80

Cb2 Cb10=:012Cb20=:0252ts0=150;000 Cb2

PAGE 82

Cb xzy rgvg

PAGE 83

E^i=Erg rg E^k=ErgEvg jErgEvgj E^j=E^kE^i RE2L=266664E^iTE^jTE^kT377775 r=rsc)]TJ /F6 11.9552 Tf 11.955 0 Td[(rg v=vsc)]TJ /F6 11.9552 Tf 11.955 0 Td[(vg)]TJ /F9 11.9552 Tf 11.955 16.857 Td[(rgvg r2gr RE2LJ2r<
PAGE 84

x;y;_x;_yyCb 266666664_x_yxy377777775=26666666400100001b00d00)]TJ /F3 11.9552 Tf 9.298 0 Td[(d0377777775266666664xy_x_y377777775+266666664000)]TJ /F3 11.9552 Tf 9.298 0 Td[(vg2377777775Cb Cb=)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(Cbsc)]TJ /F3 11.9552 Tf 11.955 0 Td[(Cbg

PAGE 85

n=r a3;c=r 1+3J2Re2 8a2[1+3cos(2i)];d=2nc;b=)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(5c2)]TJ /F2 11.9552 Tf 11.955 0 Td[(2n2 _x=Ax+Bu K u=)]TJ /F6 11.9552 Tf 9.298 0 Td[(Kx J=Z10)]TJ /F6 11.9552 Tf 5.479 -9.684 Td[(xTQx+uTRudtQRQRK Cb=)]TJ /F9 11.9552 Tf 11.291 16.856 Td[(K1K2K3K4266666664xy_x_y377777775Jt!1QRQR

PAGE 86

QR Q=2666666640000010000000000377777775RQ Qyx_x;_yyx_x_yyyQRrsatyCbmax)]TJ /F3 11.9552 Tf 12.669 0 Td[(Cbminy=rsatKRR=10;000RKy=rsatR Rnew=R00BBB@Cbmax)]TJ /F3 11.9552 Tf 11.955 0 Td[(Cbmin K00rsat00T1CCCA2

PAGE 87

QQRABKABnewoldp p=new oldp=1:2 CbCbminCbCbminCbCbmaxCbCbmax Cb

PAGE 88

CbCbminCbmax Cb u=)]TJ /F6 11.9552 Tf 9.299 0 Td[(Kx(A)]TJ /F6 11.9552 Tf 11.955 0 Td[(BK) =e(A)]TJ /F14 7.9701 Tf 6.587 0 Td[(BK)ttx)]TJ /F4 7.9701 Tf 0 -8.012 Td[(ix)]TJ /F4 7.9701 Tf 0 -8.011 Td[(ix+i)]TJ /F5 7.9701 Tf 6.586 0 Td[(1

PAGE 89

tf)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(ti)]TJ /F5 7.9701 Tf 6.586 0 Td[(1;ti;x+i)]TJ /F5 7.9701 Tf 6.586 0 Td[(1P)]TJ /F4 7.9701 Tf 0 -8.012 Td[(i x)]TJ /F4 7.9701 Tf 0 -8.012 Td[(i=f)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(ti)]TJ /F5 7.9701 Tf 6.587 0 Td[(1;ti;x+i)]TJ /F5 7.9701 Tf 6.587 0 Td[(1P)]TJ /F4 7.9701 Tf 0 -8.012 Td[(i=iP+i)]TJ /F5 7.9701 Tf 6.587 0 Td[(1i+QQzi Ki=PiGT)]TJ /F6 11.9552 Tf 5.48 -9.684 Td[(GP)]TJ /F4 7.9701 Tf 0 -8.012 Td[(iGT+W)]TJ /F5 7.9701 Tf 6.587 0 Td[(1x+i=x)]TJ /F4 7.9701 Tf 0 -8.012 Td[(i+Ki)]TJ /F6 11.9552 Tf 5.479 -9.684 Td[(zi)]TJ /F6 11.9552 Tf 11.955 0 Td[(Gx)]TJ /F4 7.9701 Tf 0 -8.012 Td[(iP+i=(I)]TJ /F6 11.9552 Tf 11.955 0 Td[(KiG)PiW=1:02P G z=GxzG MnMn

PAGE 90

J2 ! M=2tan)]TJ /F5 7.9701 Tf 6.587 0 Td[(1 r 1)]TJ /F3 11.9552 Tf 11.956 0 Td[(e 1+etan 2!)]TJ /F3 11.9552 Tf 13.151 8.088 Td[(ep 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(e2sin 1+ecosa n=r e a3 =Mg)]TJ /F3 11.9552 Tf 11.955 0 Td[(Ma _=d dt=ng)]TJ /F3 11.9552 Tf 11.955 0 Td[(naga Cba=Cbg+Kd_+Kp()]TJ /F3 11.9552 Tf 11.955 0 Td[(des)

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CbaCbgKdKpdes_aaag _=r e a3g)]TJ /F9 11.9552 Tf 11.956 16.029 Td[(r e a3a=p e 1 a3=2g)]TJ /F2 11.9552 Tf 19.897 8.087 Td[(1 a3=2a! =aa)]TJ /F3 11.9552 Tf 11.955 0 Td[(ag _=p ea)]TJ /F5 7.9701 Tf 6.586 0 Td[(3=2g)]TJ /F2 11.9552 Tf 11.955 0 Td[((ag+))]TJ /F5 7.9701 Tf 6.587 0 Td[(3=2 (1+x)n=nXk=0n! k!(n)]TJ /F3 11.9552 Tf 11.956 0 Td[(k)!xkx<<1 (1+x)n1+nx <
PAGE 92

=3p e 2a5=2g__Cb E=)]TJ /F3 11.9552 Tf 10.758 8.087 Td[(e 2a dE dt= 2a2da dt dE dt=)]TJ /F3 11.9552 Tf 9.298 0 Td[(advCbv2Cbe a3=2adv=p e a da dt da dt=2Cbp ea _=daa dt)]TJ /F3 11.9552 Tf 13.151 8.088 Td[(dag dt=)]TJ /F2 11.9552 Tf 9.299 0 Td[(2p ea(Cba)]TJ /F3 11.9552 Tf 11.955 0 Td[(Cbg) =)]TJ /F2 11.9552 Tf 10.494 8.088 Td[(3e a2(Cba)]TJ /F3 11.9552 Tf 11.955 0 Td[(Cbg)(Cba)]TJ /F3 11.9552 Tf 11.956 0 Td[(Cbg) =)]TJ /F2 11.9552 Tf 10.494 8.088 Td[(3e a2Kd_+Kp()]TJ /F3 11.9552 Tf 11.955 0 Td[(des)

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q=3e a2 +qKd_+qKp=qKpdes T(s)=(s) des(s)=qKp s2+qKds+qKp s=)]TJ /F3 11.9552 Tf 9.298 0 Td[(qKdp q2K2d)]TJ /F2 11.9552 Tf 11.955 0 Td[(4qKp 2 !2n s2+2!ns+w2n!n=p qKp=Kd 2q q Kp!n

PAGE 94

rad =)]TJ /F2 11.9552 Tf 11.291 0 Td[(ln(%OS=100) q 2+ln2(%OS=100)TpTp!n !n= Tpp 1)]TJ /F3 11.9552 Tf 11.955 0 Td[(2!nTpKdKp Kdreq=2!n 2 Kpreq=!2n qKd=50;000

PAGE 95

des=:01 des=:01 CbCb

PAGE 96

QR_

PAGE 97

Cb

PAGE 98

kerr=k0+3Xi=1kisin2 Tit)]TJ /F3 11.9552 Tf 11.955 0 Td[(iTT1=T2=T3=5400 2k0kk1=:25k2=:1k3=:1 CbCbCbCbCbCb

PAGE 99

CbCb mcm=s rbias=266664:001)]TJ /F3 11.9552 Tf 9.299 0 Td[(:005:002377775sin2t 5400m;vbias=266664:00005:00005:000025377775sin2t 5400m=s

PAGE 101

CbmaxCbmin

PAGE 102

a=6715:97e=:000471;=214:24;!=:1790;=359:77;i=70:67

PAGE 103

Cbmin=:00592Cbmax=:03862Cbterm=:02222 CbCb Cb

PAGE 104

rsatCb Cb rsat=5

PAGE 105

J2J2

PAGE 106

Cb Cb

PAGE 107

Cb Cb

PAGE 108

Cb Cb

PAGE 110

Cb

PAGE 112

2 2 Cb=:033322

PAGE 114

2 x)]TJ /F3 11.9552 Tf 12.415 0 Td[(y

PAGE 115

x y z 2

PAGE 122

210)]TJ /F5 7.9701 Tf 6.586 0 Td[(6610)]TJ /F5 7.9701 Tf 6.586 0 Td[(5

PAGE 126

~bdot=)]TJ /F3 11.9552 Tf 9.298 0 Td[(K_^B=)]TJ /F3 11.9552 Tf 9.299 0 Td[(K^B~!)]TJ /F3 11.9552 Tf 21.918 0 Td[(K^B2)]TJ /F2 11.9552 Tf 14.752 3.022 Td[(^B1 tK_^B~bdot ~bdot=~bdot~B

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)]TJ /F3 11.9552 Tf 9.312 0 Td[(~! ~bdot=)]TJ /F3 11.9552 Tf 9.299 0 Td[(K^B~!~B~A~B~C (~A~B)~C=)]TJ /F3 11.9552 Tf 11.466 3.022 Td[(~A(~B~C)+~B(~A~C) ~bdot=K^B(!Bcos()))]TJ /F3 11.9552 Tf 11.955 0 Td[(K~!B

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~!~B ^!~bdot~!~! ~bdot~!=Kcos()(!Bcos()))]TJ /F3 11.9552 Tf 11.955 0 Td[(K!B =0bdotPmax m=ImAmnm

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ImAmnm Pm=I2mRmRm~bdot AmnmRmPreqPmax ~0bdot=~bdots Pmax Preq~0bdotPmax~bdot ~B~!~bdotx ~bdotxxxyxz

PAGE 130

xz 2x+y)]TJ /F3 11.9552 Tf 9.299 0 Td[(y+x)]TJ /F3 11.9552 Tf 9.298 0 Td[(xzx

PAGE 146

Pmax P=Pmaxcos() Pmax=4:24Pmax=2:12 )]TJ /F3 11.9552 Tf 9.298 0 Td[(x

PAGE 147

x

PAGE 148

110)]TJ /F5 7.9701 Tf 6.587 0 Td[(6

PAGE 149

2:210)]TJ /F5 7.9701 Tf 6.586 0 Td[(12

PAGE 158

th

PAGE 161

th