Citation
Orbital Debris Mitigation Using Solar Sail Trajectory Optimization

Material Information

Title:
Orbital Debris Mitigation Using Solar Sail Trajectory Optimization
Creator:
Kelly, Patrick W
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (182 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:
CONKLIN,JOHN
Committee Members:
RAO,ANIL
FRY,JAMES N

Subjects

Subjects / Keywords:
astrodynamics -- optimization
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:
Solar-sailing satellites offer unique propulsive advantages for orbit manipulation and propellant mass requirements. When compared with traditional, propellant-based satellites, solar-sailing spacecraft are not limited by propellant consumption or $\Delta V$ availability. By harnessing the existing elements of a star-based system, solar-sails take advantage of ``free'' propulsion by means of momentum exchange with incoming photons. Given the current trend of satellite miniaturization, solar-sails may provide a feasible propulsive solution for small satellites in the near future. A major challenge facing solar-sailing mission concepts is that of satellite guidance and control. Trajectory design for a solar-sailing satellite poses a sophisticated optimal control problem, ripe with nonlinearities and uniquely defined constraints. Planetocentric applications in particular are especially challenging, explaining the focus on mostly heliocentric applications in the literature. In an effort to advance planetocentric solar-sailing mission concepts, two applications will be investigated for this doctoral study. First, formulation of a large debris orbit transfer orbit will be determined, using a high-performance solar-sailing satellite launched from geostationary orbit. Additionally, a geostationary return orbit will be executed, placing a solar sailing satellite within a desired geostationary longitude. Both maneuvers are encapsulated within the ``TugSat'' mission concept. This application will not only explore solutions to satellite orbit transfer and rephasing, but will additionally examine the capabilities of solar-sailing satellites in a perturbed orbital environment. Solution methods encompass eclipsing phenomena, incorporating an alternating combination of coast-phases and thrust phases. Various optimal control techniques are utilized, from transcribing the problem to a nonlinear programming problem to formulation of a suitable problem statement to be solved using particle swarm optimization. This work showcases a combination of numerical methods, optimization, and orbital mechanics to produce effective trajectories for solar-sailing satellite concepts. ( 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, 2018.
Local:
Adviser: BEVILACQUA,RICCARDO.
Local:
Co-adviser: CONKLIN,JOHN.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2019-06-30
Statement of Responsibility:
by Patrick W Kelly.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
6/30/2019
Classification:
LD1780 2018 ( lcc )

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Full Text

PAGE 8

kSRP=0:50 kSRP=1:00 kSRP=1:50 kSRP=2:00 1m2 kg

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0 0 0 90 90 180 180 180 180 270 270 270

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V

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V

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4:5N1AU m2m2

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++)]TJ /F4 11.9552 Tf 5.479 -9.684 Td[(1000CRA m CR11:5A mm2 kg 35;786rGEO)]TJ /F4 11.9552 Tf 11.956 0 Td[(200rGEO+20015

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nx f(x)=266664f1(x)fm(x)377775=0f(x)mf(x)x?f(x?)=0f0=df dxf(x)x f(x)=f(x)+f0(x)(x)]TJ /F2 11.9552 Tf 11.955 0 Td[(x)xf(x)=0x x=x)]TJ /F2 11.9552 Tf 14.549 8.088 Td[(f(x) f0(x)f(x)6=0 x?

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x x(i+1)=x(i))]TJ /F2 11.9552 Tf 14.548 8.088 Td[(f(x(i)) f0(x(i))x(i+1))]TJ /F2 11.9552 Tf 11.955 0 Td[(x(i)< x(i+1)=x(i)+d)]TJ /F12 11.9552 Tf 5.48 -9.684 Td[(x(i)d(x) )]TJ /F12 11.9552 Tf 11.955 0 Td[(x=J(x)d(x)J(x) J(x)=rxf(x)=266666664@f1 @x1@f1 @x2:::@f1 @xn@f2 @x1@f2 @x2@f1 @xn@fm @x1@fm @x2@fm @xn377777775 J(x) y=f(x)nthPn(x)y=[y0y1:::yn]x=[x0x1:::xn] Pn(x)=nXi=0`i(x)yi

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`i(x)=nYj=0j6=i(x)]TJ /F2 11.9552 Tf 11.955 0 Td[(xj) (xi)]TJ /F2 11.9552 Tf 11.955 0 Td[(xj)xf(x)Pn(x)f(x)8xf(x)>(n+1)x(n+1)Pn(x)n Pn)]TJ /F3 7.9701 Tf 6.587 0 Td[(1(x)+Pn(x)Pnnth Pn=1 2nn!dn(x2)]TJ /F4 11.9552 Tf 11.955 0 Td[(1)n dxn

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xi+1 xi+1=xi+hRK4h RK4=1 6(k1+2k2+2k3+k4) k1=f(xi;ti) k2=f(xi+h 2k1;ti+h 2) k3=f(xi+h 2k2;ti+h 2) k4=f(xi+hk3;;ti+h) _x=f(x;t)

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h4x(t)x(t) xi+1ti+hxi+1(fi;fi)]TJ /F3 7.9701 Tf 6.586 0 Td[(1;fi)]TJ /F3 7.9701 Tf 6.586 0 Td[(2;fi)]TJ /F3 7.9701 Tf 6.586 0 Td[(3)fi xi+1=xi+hAB4 AB4=1 24(55fi)]TJ /F4 11.9552 Tf 11.955 0 Td[(59fi)]TJ /F3 7.9701 Tf 6.586 0 Td[(1+37fi)]TJ /F3 7.9701 Tf 6.587 0 Td[(2)]TJ /F4 11.9552 Tf 11.955 0 Td[(9fi)]TJ /F3 7.9701 Tf 6.586 0 Td[(3) fi=f(xi;ti) xi+1=xi+hAM4 AM4=1 24(9fi+1+19fi)]TJ /F4 11.9552 Tf 11.955 0 Td[(5fi)]TJ /F3 7.9701 Tf 6.587 0 Td[(1+fi)]TJ /F3 7.9701 Tf 6.587 0 Td[(2)

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xi+1 x13 fi+1 2[)]TJ /F4 11.9552 Tf 9.299 0 Td[(1;1) Z1)]TJ /F3 7.9701 Tf 6.587 0 Td[(1f()dNXk=1wif(i)winf()2n)]TJ /F4 11.9552 Tf 11.955 0 Td[(1 Zabf(x)dx=b)]TJ /F2 11.9552 Tf 11.955 0 Td[(a 2Z1)]TJ /F3 7.9701 Tf 6.587 0 Td[(1f()dx x=1 2[b+a+(b)]TJ /F2 11.9552 Tf 11.955 0 Td[(a)]

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Pn(ti)=x(ti) _x(i)=f(x(i);i);8i2f1;:::;ng i=_P(i))]TJ /F12 11.9552 Tf 11.955 0 Td[(f(x();)wii=0;8i

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f(x)=1 1+x2n=f10;2030g

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u(t)2Rnux(t)2Rnxt2[t0;tf]J2R J=M(x(t0);t0;x(tf);tf)+Ztft0L(x(t);u(t);t)dtM2RL2R a2Rnx a(x(t);u(t);t)=_x(t)b2Rnb b(x(t0);t0;x(tf);tf)=0c2Rnc c(x(t);u(t);t)0

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Ja2R(t)2Rnx2Rnb(t)2Rnc(t)2RnxJa Ja=M)]TJ /F12 11.9552 Tf 5.479 -9.683 Td[(x(t0);t0;x(tf);tf+Tb(x(t0);t0;x(tf);tf)+Ztft0L(x(t);u(t);t)+(t)T)]TJ /F12 11.9552 Tf 5.479 -9.684 Td[(a(x(t);u(t);t))]TJ /F4 11.9552 Tf 13.878 .166 Td[(_x(t)+(t)Tc(x(t);u(t);t)dt HH(x(t);u(t);(t);(t);t)=L(x(t);u(t);t)+(t)Ta(x(t);u(t);t)+(t)Tc(x(t);u(t);t) Ja=M)]TJ /F12 11.9552 Tf 5.479 -9.684 Td[(x(t0);t0;x(tf);tf+Tb(x(t0);t0;x(tf);tf)+Ztft0H(x(t);u(t);(t);(t);t))]TJ /F16 11.9552 Tf 11.955 0 Td[((t)T_x(t)dt

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Ja Ja=@M @x(t0)x(t0)+@M @x(t0)_x(t0)t0+@M @t0t0+@M @x(tf)x(tf)+@M @x(tf)_x(tf)tf+@M @tftf+Tb+T@b @x(t0)x(t0)+@b @x(t0)_x(t0)t0+@b @t0t0+@b @x(tf)x(tf)+@b @x(tf)_x(tf)tf+@b @tftf+H(tf)tf)]TJ /F16 11.9552 Tf 11.956 0 Td[((tf)T_x(tf)tf)-222(H(t0)t0+(t0)T_x(t0)t0+Ztft0@H @x(t)x(t)+@H @u(t)u(t)+@H @(t)(t)+@H @(t)(t))]TJ /F2 11.9552 Tf 11.955 0 Td[((t)T_x(t))]TJ /F16 11.9552 Tf 11.955 0 Td[((t)T_x(t)dt _x(t) Ztft0(t)T_x(t)=(tf)Tx(tf))]TJ /F16 11.9552 Tf 11.955 0 Td[((t0)Tx(t0))]TJ /F10 11.9552 Tf 11.955 16.272 Td[(Ztft0_(t)Tx(t)dt Ja=@M @x(t0)x(t0)+@M @x(t0)_x(t0)t0+@M @t0t0+@M @x(tf)x(tf)+@M @x(tf)_x(tf)tf+@M @tftf+Tb+T@b @x(t0)x(t0)+@b @x(t0)_x(t0)t0+@b @t0t0+@b @x(tf)x(tf)+@b @x(tf)_x(tf)tf+@b @tftf+H(tf)tf)]TJ /F16 11.9552 Tf 11.956 0 Td[((tf)T_x(tf)tf)-222(H(t0)t0+(t0)T_x(t0)t0)]TJ /F16 11.9552 Tf 11.955 0 Td[((tf)Tx(tf)+(t0)Tx(t0)+Ztft0@H @x(t)x(t)+@H @u(t)u(t)+@H @(t)(t)+@H @(t)(t))]TJ /F2 11.9552 Tf 11.955 0 Td[((t)T_x(t)+_(t)Tx(t)dt

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x x=x(t)+_x(t)t x0xf Ja=@M @x(t0)x0+@M @t0t0+@M @x(tf)xf+@M @tftf+Tb+T@b @x(t0)x0+@b @t0t0+@b @x(tf)xf+@b @tftf+H(tf)tf)-222(H(t0)t0)]TJ /F16 11.9552 Tf 11.955 0 Td[((tf)Txf+(t0)Tx0+Ztft0@H @x(t)x(t)+@H @u(t)u(t)+@H @(t)(t)+@H @(t)(t))]TJ /F2 11.9552 Tf 11.955 0 Td[((t)T_x(t)+_(t)Tx(t)dt Ja=@M @x(t0)+T@b @x(t0)+(t0)Tx0+@M @t0+T@b @t0)-222(H(t0)t0+@M @x(tf)+T@b @x(tf))]TJ /F16 11.9552 Tf 11.955 .001 Td[((tf)Txf+@M @tf+T@b @tf+H(tf)tf+bT+Ztft0@H @x(t)+_(t)Tx(t)+@H @(t))]TJ /F4 11.9552 Tf 13.878 .166 Td[(_x(t)T(t)+@H @u(t)u(t)+@H @(t)(t)dtJa=0

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(t):@H @(t)=_x(t)Tx(t):@H @x(t)=)]TJ /F4 11.9552 Tf 11.682 3.155 Td[(_(t)Tu(t):@H @u(t)=0(t):@H @(t)=0:b=0x0:(t0)T=)]TJ /F2 11.9552 Tf 16.788 8.088 Td[(@M @x(t0))]TJ /F16 11.9552 Tf 11.956 0 Td[(T@b @x(t0)xf:(tf)T=@M @x(tf)+T@b @x(tf)t0:H(t0)=@M @t0+T@b @t0tf:H(tf)=)]TJ /F2 11.9552 Tf 10.494 8.087 Td[(@M @tf)]TJ /F16 11.9552 Tf 11.956 0 Td[(T@b @tfi(t)8i2f1;:::;ncg i(t)8>>>>>><>>>>>>:0;ci(x(t);u(t);t)=0(minimality)0;ci(x(t);u(t);t)=0(maximality)=0;ci(x(t);u(t);t)<0Hx(t)H ci(t)<0i(t)=0 Ja

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u(t)2Uu(t)U @H @u(t)=0 @2H @u(t)2>0u(t) u(t)UHu(t) u(t)=argminu(t)2UH(x(t);u(t);(t);(t);t) H(x(t);u(t);(t);(t);t)
PAGE 38

argminzF(z) s:t:g(z)=0 h(z)0 z2RnzF2Rg2Rngh2Rnh L L(z;;)=F(z))]TJ /F16 11.9552 Tf 11.955 0 Td[(Tg(z))]TJ /F16 11.9552 Tf 11.955 0 Td[(Th(z)

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rzL(z?;?;?)=0 g(z?)=0 h(z?)0 ?0 ?iwi(z?)=0 w(z?)2Rnw w(z?)T=hg(z?)T~h(z?)Ti=0?2Rnww(z?) ?T=?T~?T~h(z?)h(z?)hi(z?)=0~?A?iwi(z?)=0 wi(z?)?i=0 h(z?) rzL(z?;?)=rF(z?))]TJ /F10 11.9552 Tf 11.955 11.358 Td[(Xi2A?irwi(z?)=0

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N)]TJ /F4 11.9552 Tf 12.047 0 Td[(1 J=M(x(1);t0;x(N);tf)+tf)]TJ /F2 11.9552 Tf 11.955 0 Td[(t0 2Z1)]TJ /F3 7.9701 Tf 6.586 0 Td[(1L(x();u();;t0;tf)d dx d)]TJ /F2 11.9552 Tf 13.151 8.088 Td[(tf)]TJ /F2 11.9552 Tf 11.955 0 Td[(t0 2a(x();u();;t0;tf)=0 b(x(1);t0;x(N);tf)=0 c(x();u();;t0;tf)0x()=[x1()x2():::xn()]Xi=iUiN)]TJ /F4 11.9552 Tf 13.051 0 Td[(1(N)]TJ /F4 11.9552 Tf 11.209 0 Td[(1)nXLGRXiXN=1XNn2[)]TJ /F4 11.9552 Tf 9.299 0 Td[(1;1]i x()NXi=1Xi`i()`i `i()=NYj=1j6=i)]TJ /F2 11.9552 Tf 11.955 0 Td[(j i)]TJ /F2 11.9552 Tf 11.955 0 Td[(j

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i _x()NXi=1Xi_`i()=NXi=1DkiXi(N)]TJ /F4 11.9552 Tf 12.435 0 Td[(1)NDDki=_`k(i)k2f1;2;:::;N)]TJ /F4 11.9552 Tf 12.262 0 Td[(1gN)]TJ /F4 11.9552 Tf 12.263 0 Td[(1 NXi=1DkiXi)]TJ /F2 11.9552 Tf 13.151 8.088 Td[(tf)]TJ /F2 11.9552 Tf 11.955 0 Td[(t0 2a(Xk;Uk;k;t0;tf) JM(X1;t0;XN;tf)+tf)]TJ /F2 11.9552 Tf 11.956 0 Td[(t0 2N)]TJ /F3 7.9701 Tf 6.587 0 Td[(1Xk=1wkL(Xk;Uk;k;t0;tf)wk b(X1;t0;XN;tf)=0 c(Xk;Uk;k;t0;tf)0 D2Rn

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J NDn(i)(i=1;:::;N) (i),1(i)2(i):::n(i)T;min(i)maxminmaxn(i)(i=1;:::;N) (i),1(i)2(i):::n(i)T;min(i)maxminmax min,min)]TJ /F16 11.9552 Tf 11.956 0 Td[(max max,max)]TJ /F16 11.9552 Tf 11.955 0 Td[(min jJ(j)(i)(j)(i)ij j?=argmin`=1;:::;jJ(`)(i) J?(j)(i)=J(j?)(i)

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(j)(i)=(j?)(i)N i?=argmin`=1;:::;iJ?(j)(`) (j)=(j)(i?)(j+1)thk(k=1;:::;n)(j)(i) (j+1)k(i)=(j)k(i)+(j+1)k(i) (j+1)k(i)=cI(j)k(i)+cCh(j)k(i))]TJ /F2 11.9552 Tf 11.955 0 Td[((j)k(i)i+cSh(j)k)]TJ /F2 11.9552 Tf 11.955 0 Td[((j)k(i)icIcCcS cI=1+r1(0;1) 2 cC=1:49445r2(0;1) cS=1:49445r3(0;1) r()(0;1)

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? (j+1)(i) (j)(i) (j)]TJ /F3 7.9701 Tf 6.587 0 Td[(1)(i) (j)(i) (j) (j+1)(i) ij

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U U= rr r=rU=)]TJ /F2 11.9552 Tf 12.139 8.088 Td[( r3rrr r+ r3r=adad

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f^x;^y;^zg^x^y^x^z^y90^x ^i^j^k^z^i^j90

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=tan)]TJ /F3 7.9701 Tf 6.587 0 Td[(1rj ri =sin)]TJ /F3 7.9701 Tf 6.586 0 Td[(1rk r rxx=fi;j;kgGMST xyz IJK=266664cosGMSTsinGMST0)]TJ /F4 11.9552 Tf 11.291 0 Td[(sinGMSTcosGMST0001377775 IJK xyz=xyz IJKT f^r;^s;^wg^r^w^s^s

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^r=r r ^w=rv krvk ^s=^w^r IJK rsw=^r^s^w rsw IJK=IJK rswT s x=264r_r375=rIrJrKvIvJvKT _x=264_rr375=vIvJvKaIaJaKT

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rxvxaxx2fx;y;zgr_rxr x=aei!Taei! =v2 2)]TJ /F2 11.9552 Tf 13.151 8.088 Td[( r a=)]TJ /F2 11.9552 Tf 12.744 8.088 Td[( 2

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e=)]TJ /F2 11.9552 Tf 5.48 -9.683 Td[(v2)]TJ /F11 7.9701 Tf 13.151 5.255 Td[( rr)]TJ /F4 11.9552 Tf 11.955 0 Td[((rv)v e=kek cosi=hK h h=rvh=khk!n n=^Kh cos=nI n;if(nJ<0)then=2)]TJ /F4 11.9552 Tf 11.956 0 Td[( cos!=ne ne;if(eK<0)then!=2)]TJ /F2 11.9552 Tf 11.955 0 Td[(! cos=er er;if(rv<0)then=2)]TJ /F2 11.9552 Tf 11.956 0 Td[(

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rI=r(coscos)]TJ /F4 11.9552 Tf 11.956 0 Td[(sinsincosi) rJ=r(sincos+cossincosi) rK=rsinsini r=p (1+ecos) vI=)]TJ /F2 11.9552 Tf 10.494 8.088 Td[( h[cos(sin+esin!)+sin(cos+ecos!)cosi] vJ=)]TJ /F2 11.9552 Tf 10.494 8.088 Td[( h[sin(sin+esin!))]TJ /F4 11.9552 Tf 11.955 0 Td[(cos(cos+ecos!)cosi] vK= h(cos+esin!)sini

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da dt=2 np 1)]TJ /F2 11.9552 Tf 11.955 0 Td[(e2hesinFr+p rFsi de dt=p 1)]TJ /F2 11.9552 Tf 11.956 0 Td[(e2 nasinFr+cos+e+cos 1+ecosFs di dt=rcosu na2p 1)]TJ /F2 11.9552 Tf 11.956 0 Td[(e2Fw d dt=rsinu na2p 1)]TJ /F2 11.9552 Tf 11.956 0 Td[(e2siniFw d! dt=p 1)]TJ /F2 11.9552 Tf 11.956 0 Td[(e2 nae)]TJ /F4 11.9552 Tf 11.291 0 Td[(cosFr+sin1+r pFs)]TJ /F2 11.9552 Tf 13.151 8.088 Td[(rcotisinu hFw d dt=pcos ehFr)]TJ /F4 11.9552 Tf 11.955 0 Td[((p+r)sinFs+h r2 u=!+n F=rsw IJKad=FrFsFwT dx dt=TxF+bxxx2fa;e;i;;!;gbx e=0i=0!

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=pfg~kLT p=a)]TJ /F4 11.9552 Tf 5.479 -9.684 Td[(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(e2 f=ecos(+!) g=esin(+!) ~=tani 2cos k=tani 2sin L=+!+

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pL a=p p 1)]TJ /F2 11.9552 Tf 11.955 0 Td[(f2)]TJ /F2 11.9552 Tf 11.955 0 Td[(g2 e=p f2+g2 i=2tan)]TJ /F3 7.9701 Tf 6.586 0 Td[(1p ~2+k2 =tan)]TJ /F3 7.9701 Tf 6.587 -.001 Td[(1k ~ !=tan)]TJ /F3 7.9701 Tf 6.587 0 Td[(1g f)]TJ /F4 11.9552 Tf 11.955 0 Td[( =L)]TJ /F4 11.9552 Tf 11.955 0 Td[(tan)]TJ /F3 7.9701 Tf 6.586 0 Td[(1g f rI=r &2)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(cosL+%2cosL+2~ksinL rJ=r &2)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(sinL)]TJ /F2 11.9552 Tf 11.955 0 Td[(%2sinL+2~kcosL rK=2r &2(hsinL)]TJ /F2 11.9552 Tf 11.955 0 Td[(kcosL)

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vI=)]TJ /F4 11.9552 Tf 12.532 8.088 Td[(1 &2r p)]TJ /F4 11.9552 Tf 5.479 -9.684 Td[(sinL+%2sinL)]TJ /F4 11.9552 Tf 11.955 0 Td[(2~kcosL+g)]TJ /F4 11.9552 Tf 11.955 0 Td[(2f~k+%2g vJ=)]TJ /F4 11.9552 Tf 12.532 8.088 Td[(1 &2r p)]TJ /F9 11.9552 Tf 5.479 -9.684 Td[()]TJ /F4 11.9552 Tf 11.291 0 Td[(cosL+%2cosL+2~ksinL)]TJ /F2 11.9552 Tf 11.955 0 Td[(f+2g~k+%2f vK=2 &2r p(~cosL+ksinL+f~+gk) w,1+fcosL+gsinL %2,~2)]TJ /F2 11.9552 Tf 11.955 0 Td[(k2 &2,1+~2+k2 r=p w _x=Z(x)TF+b=a(x;u;t) Z(x)T=1 wr p 266666666666666402p w0wsinL(w+1)cosL+f)]TJ /F2 11.9552 Tf 9.299 0 Td[(g(~sinL)]TJ /F2 11.9552 Tf 11.955 0 Td[(kcosL))]TJ /F2 11.9552 Tf 9.299 0 Td[(wcosL(w+1)sinL+gf(~sinL)]TJ /F2 11.9552 Tf 11.955 0 Td[(kcosL)00&2cosL 200&2sinL 200~sinL)]TJ /F2 11.9552 Tf 11.955 0 Td[(kcosL3777777777777775 b=00000p pw pT

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F L dL dt=aL(x;u;t)t0 t0=dt dL=1 aL(x;u;t)Lx0 x0=p0f0g0~0k0t0T=1 aL(x;u;t)_p_f_g_~_k1T=a(x;u;L)

PAGE 57

Unm= r1Xn=0nXm=0Rn rnPnm)]TJ /F4 11.9552 Tf 7.472 -9.684 Td[(sin()(Cnmcos(m)+Snmsin(m))RPnmCnmSnmnm r=rUnmCnmSnm UPA= rR r23)]TJ /F3 7.9701 Tf 13.167 -1.794 Td[(22r2x)]TJ /F2 11.9552 Tf 11.955 0 Td[(r2y r2)]TJ /F4 11.9552 Tf 13.151 8.087 Td[(1 2)]TJ /F3 7.9701 Tf 7.314 -1.794 Td[(201)]TJ /F4 11.9552 Tf 11.955 0 Td[(3r2z r2r=[rxryrz]T^z^z^x^y^z=)]TJ /F4 11.9552 Tf 9.299 0 Td[(14:93

PAGE 58

)]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(20)]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(22 )]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(20'C20 )]TJ /F3 7.9701 Tf 7.314 -1.793 Td[(22'q C222+S222 a=rUPARC20C22 nn r=)]TJ /F2 11.9552 Tf 10.494 8.088 Td[(1 r3r+nXj=3jr2;j r2;j)]TJ /F16 11.9552 Tf 13.151 8.088 Td[(r1;j r31;jrx;y=ry)]TJ /F16 11.9552 Tf 12.164 0 Td[(rxj=1j=2 n r2;n r2;n)]TJ /F16 11.9552 Tf 13.15 8.087 Td[(r1;n r31;n=Qr2;n)]TJ /F16 11.9552 Tf 13.151 8.087 Td[(r1;2 r31;n

PAGE 59

Q=)]TJ /F2 11.9552 Tf 5.48 -9.683 Td[(r21;2+2r1;2r2;n)]TJ /F2 11.9552 Tf 12.951 -9.683 Td[(r21;n+r1;nr2;n+r2n;n r31;nr32;n(r1;n+r2;n)n an=nXj=3jQr2;j)]TJ /F16 11.9552 Tf 13.15 8.088 Td[(r1;2 r31;j rSRP=)]TJ /F2 11.9552 Tf 9.298 0 Td[(PAU2 r2A mcos[(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(")^e+2"cos^n]PAUA m"^e cos,^e^n;20; 2^n ("=1) dP dt=0 )]TJ /F11 7.9701 Tf 7.249 -4.977 Td[(A m

PAGE 60

aSRP=)]TJ /F2 11.9552 Tf 9.298 0 Td[(a0cos2^n a0,2PAU2 r2A m^n

PAGE 61

^I ^J ^K;^z Equator ^x ^y GMST PrimeMeridian r SatelliteOrbit ^r ^s ^w

PAGE 62

^I ^J ^K n e 2a r h i ^x ^y ^z r rxy

PAGE 63

^x ^y ^z;^z ^x ^y ^x;^y;^zf^x;^y;^zg A FSRP ^n ^e FSRP

PAGE 64

"=1)]TJ /F4 11.9552 Tf 10.257 .256 Td[(^=0= 2 2

PAGE 65

RARBdp dp=RBdA RA+RBdAp p=arcsinRB dp=arcsinRA+RB dAshs^eA s=(s^eA)^eAds ds,ksk hs=ks)]TJ /F4 11.9552 Tf 12.205 .166 Td[(skhp hp=(dp+ds)tanp

PAGE 66

s^eA<0 hshp dshphs p=arctanhp ds;2h0; 2i s=arctanhs s^eA;2[0;] sps=p 4647141days133days

PAGE 67

23:4 ^ef^e1;^e2;^e3g ^e1=^e ^e3=^e k^ek ^e2=^e3^e1 ^e1^n12^n^e1^e2^e3^n12^n^e1 (;)2)]TJ /F11 7.9701 Tf 10.494 4.707 Td[( 2; 22[0;] ^n=coscos^e1+sincos^e2+sin^e3

PAGE 68

aSRP=)]TJ /F2 11.9552 Tf 9.298 0 Td[(a0(coscos)2^n u2RN ,q)]TJ /F12 11.9552 Tf 11.955 0 Td[(qd=q _=_qq2RNNqdd dtqd=0qV2R V=1 2TW _V=_TWW2RNNw1wN W=266664w1wN377775;kWk=1q()2q _()=T()F

PAGE 69

_V=eTFe2R3 eT,NXj=1wjjTj ad=aSRP _V=)]TJ /F2 11.9552 Tf 9.298 0 Td[(a0(coscos)2eT^n=)]TJ /F2 11.9552 Tf 9.298 0 Td[(a0e(coscos)2(coscoscos+sinsincos)=)]TJ /F2 11.9552 Tf 9.298 0 Td[(a0ecos3()cos2()cos()]TJ /F2 11.9552 Tf 11.955 0 Td[()kek=e_V0 _V a0e>0;8tJV JV=cos3()cos2()cos()]TJ /F2 11.9552 Tf 11.955 0 Td[()

PAGE 70

(;)2)]TJ /F11 7.9701 Tf 10.494 4.707 Td[( 2; 2 @JV @=0 @JV @=0 ? @JV @=2cos2()sin()cos2()cos()]TJ /F2 11.9552 Tf 11.955 0 Td[()0=cos2()sin()?=)]TJ /F11 7.9701 Tf 10.494 4.707 Td[( 2;0; 2?=)]TJ /F11 7.9701 Tf 10.494 4.707 Td[( 2 2?=0^e1^e2 ?=0 @JV @=)]TJ /F4 11.9552 Tf 11.955 0 Td[(2cos3()cos()sin()cos()]TJ /F2 11.9552 Tf 11.956 0 Td[())]TJ /F4 11.9552 Tf -110.87 -25.9 Td[(cos3()cos2()sin()]TJ /F2 11.9552 Tf 11.955 0 Td[()0=)]TJ /F4 11.9552 Tf 9.298 0 Td[(2cos()sin()cos()]TJ /F2 11.9552 Tf 11.956 0 Td[())]TJ /F4 11.9552 Tf 11.955 0 Td[(cos2()sin()]TJ /F2 11.9552 Tf 11.955 0 Td[()0=)]TJ /F4 11.9552 Tf 9.298 0 Td[(3cos2sincos)]TJ /F4 11.9552 Tf 11.955 0 Td[(2sin2cossin+cos3sin0=)]TJ /F4 11.9552 Tf 9.299 0 Td[(3tancos)]TJ /F4 11.9552 Tf 11.955 0 Td[(2tan2sin+sin

PAGE 71

tan tan=)]TJ /F4 11.9552 Tf 9.298 0 Td[(3cosp cos2+8 4sinJV @2JV @2=)]TJ /F4 11.9552 Tf 9.299 0 Td[(4tansin)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(1+tan2)]TJ /F4 11.9552 Tf 11.955 0 Td[(3cos)]TJ /F4 11.9552 Tf 5.479 -9.684 Td[(1+tan2=)]TJ /F4 11.9552 Tf 11.291 0 Td[((4tansin+3cos)sec2@2JV @2<0 (+):p cos2+8>0 ()]TJ /F4 11.9552 Tf 11.291 0 Td[():)]TJ /F10 11.9552 Tf 9.298 10.949 Td[(p cos2+8<0 ? tan?=)]TJ /F4 11.9552 Tf 9.299 0 Td[(3cos+p cos2+8 4sin;?2h0; 2i2[0;]?sin?cos? (?;?) _Vh)]TJ /F2 11.9552 Tf 9.298 0 Td[(a0e;0iV

PAGE 72

2p dA dp s hs hp ds RA RB

PAGE 73

^e1 ^e2 ^e3 ^n ^n12

PAGE 74

1000kg220days800m2 argmin(t)tf s:t:r)]TJ /F2 11.9552 Tf 13.15 8.088 Td[( r3r=ad [a0e0i0]T=[rGEO00]T ef0:003 rp;frGEO+250km 0(t) 2 J=tf(t)

PAGE 75

523:4 A m=kSRP10)]TJ /F3 7.9701 Tf 6.586 0 Td[(6km2 kgkSRPkSRP=f0:50;1:00;1:50;2:00gL cLf(x;u;L)=0cLf cLf(x;u;L)=min(rp)]TJ /F2 11.9552 Tf 11.955 0 Td[(rp;d;0)+min(ed)]TJ /F2 11.9552 Tf 11.955 0 Td[(e;0)

PAGE 76

cp=js)]TJ /F2 11.9552 Tf 11.955 0 Td[(pj jd0 kSRP j=0 L(0)f=L0 x(0)f=[p0f0g0~0k0t0]T x(0)fad x(j)0=x(j)]TJ /F3 7.9701 Tf 6.586 0 Td[(1)f L(j)0=L(j)]TJ /F3 7.9701 Tf 6.586 0 Td[(1)f

PAGE 77

cLf=0 cp=0x(j) sp8L(j)adaSRP sp8L(j)adaSRP j=j+1j rp;d)ed T=rpeT=[rp)]TJ /F2 11.9552 Tf 11.955 0 Td[(rp;de)]TJ /F2 11.9552 Tf 11.955 0 Td[(ed]T W=264wrp00we375

PAGE 78

_V=[_rp_e]T264wrp00we375264rpe375=wrprp_rp+wee_e=[_a(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(e))]TJ /F2 11.9552 Tf 11.955 0 Td[(a_e]wrprp+wee_e=(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(e)wrprp_a+)]TJ /F2 11.9552 Tf 5.479 -9.683 Td[(wee)]TJ /F2 11.9552 Tf 11.956 0 Td[(rpwrpa_e_rprp,a(1)]TJ /F2 11.9552 Tf 12.612 0 Td[(e) _a_e _V=(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(e)wrprpTaF+)]TJ /F2 11.9552 Tf 5.479 -9.683 Td[(wee)]TJ /F2 11.9552 Tf 11.955 0 Td[(awrprpTeF=(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(e)wrprpTa+weeTe)]TJ /F2 11.9552 Tf 11.955 0 Td[(awrprpTeF=~TF rp;dedwrpwe rp;d=rGEO+250krp ed=0:003ke k()rp;d w()

PAGE 79

1=w2rp+w2ekrp=1ke=0:9wrp=we=1 p 2250km0:0027wrp=we kSRP kSRP>1:00 25days133days141days

PAGE 80

=wrpwekrpT min=000T max=10101T 0wrp2500kmwrp1

PAGE 81

2018 56%kSRP>1:00kSRP=0:50kSRP0:508:35% kSRP=2:00tf<80days133days

PAGE 82

kSRP<2:00 1200m21200kg1m2 kg

PAGE 83

rGEOprGEO+1000km)]TJ /F4 11.9552 Tf 9.299 0 Td[(0:1~0:1)]TJ /F4 11.9552 Tf 9.299 0 Td[(0:01f0:01)]TJ /F4 11.9552 Tf 9.299 0 Td[(0:1k0:1)]TJ /F4 11.9552 Tf 9.299 0 Td[(0:01g0:010ttmaxtmaxLmax L0LLmaxL0Lmax x0=[p0f0g0~0k0t0]T=[rGEO00000]T aepfg

PAGE 84

pf(1+ef)(rGEO+250km)rp,p 1)]TJ /F11 7.9701 Tf 6.586 0 Td[(e e=0pf rGEO+250kmpfffgf )]TJ /F4 11.9552 Tf 11.955 0 Td[(0:003fff;gfg0:003~ktL ~~ u=1 k~kh~r~s~wiT=[urusuw]Tuerp^r^suw=0

PAGE 85

u2r+u2s=1 )]TJ /F4 11.9552 Tf 11.955 0 Td[(1fur;usg1~u f?;?g ^Lf^Lf^tf^Lf^Lf^Lt?f^Lftf^Lf4^tf2days^Lf

PAGE 86

^Lf 10)]TJ /F3 7.9701 Tf 6.587 0 Td[(610)]TJ /F3 7.9701 Tf 6.587 0 Td[(6L0Lfx0^xf=h^pf^ff^gf^~f^kf^tfiT ^pf=1:003(rGEO+250km)^~=0:1^f=0:003^k=0^g=0^tfrp;f=rGEO+250kme=0:003L0Lf ^u0=[10]T ^uf=[01]Tuw=0 jd0

PAGE 87

L0 Lf=^Lf xf^Lf ^Lf ^Lf ^Lf=^Lf+52 t?f t?f=tf ^Lf=^Lf)]TJ /F4 11.9552 Tf 11.955 0 Td[(2 1200m21200kg^Lf=502 L0=0

PAGE 88

L0= 901m kg50 ^r^r^s ad=0L

PAGE 89

[pfg~k]L(i)f=[pfg~k]L(i+1)0L(i)(i)aSRP u ~ut0 a(x;~u;L))]TJ /F4 11.9552 Tf 11.955 0 Td[([p0f0g0~0k0~u]T=0

PAGE 90

~u)]TJ /F4 11.9552 Tf 15.019 8.088 Td[(dt dL=0 ~u>0t0x 100m220

PAGE 91

Lx(i)0 x(N)f (i)2f1;2;:::;NgNL(i)f)]TJ /F2 11.9552 Tf 11.955 0 Td[(L(i)0<2x(i)0x(i)fx(i)0 x(i)f x(i)0x(i)f x(i)0x(i)f ^L?f^L(i)^t(i)

PAGE 92

^L(i)f=^L(i+1)0 ^t(i)f=^t(i+1)0x(1)0 L(1)0=0x(N)f L(N)f=L?fL=0L0=L(i)fL(i)02t(i)ft(i)02days x(L(i+1)0))]TJ /F12 11.9552 Tf 11.955 0 Td[(x(L(i)f)=0 L(i+1)0)]TJ /F2 11.9552 Tf 11.956 0 Td[(L(i)f=0 i2f1;2;:::;N)]TJ /F4 11.9552 Tf 11.955 0 Td[(1g s)]TJ /F2 11.9552 Tf 11.955 0 Td[(p=0

PAGE 93

s)]TJ /F2 11.9552 Tf 11.955 0 Td[(p0 p)]TJ /F2 11.9552 Tf 11.955 0 Td[(s0 jd0 L0 ^Lf=^L?f+52 ^Lf xf^Lf ^Lf ^Lf t?f t?f=tf ^Lf=^Lf)]TJ /F4 11.9552 Tf 11.955 0 Td[(2

PAGE 94

21:03 kSRP 0:50 183:8193:1177:4191:8186:51:00 174:3185:9169:0188:8179:51:50 173:4177:1169:1179:6174:82:00 172:5170:9167:0169:6170:0

PAGE 95

kSRP 0:50 1:60791:75161:66121:72201:00 1:23991:76651:30511:73201:50 1:18351:75911:28551:74952:00 1:14591:75661:18111:7317 502505 kSRP=0:50 krp250(km)ke3(10)]TJ /F3 7.9701 Tf 6.587 0 Td[(3)wrpwe 269:492:430:4620:887 287:482:310:4280:904 266:992:700:3410:940 271:942:370:4390:899 kSRP=1:00 krp250(km)ke0:003(10)]TJ /F3 7.9701 Tf 6.587 0 Td[(4)wrpwe 484:162:300:4550:887 631:91MIN0:3130:950 MAX2:410:0700:997 MAX0:780:0630:998 kSRP=1:50 krp250(km)ke0:003(10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5)wrpwe MAX6:010:0670:997 139:641:910:1120:994 MAX3:010:0650:998 203:03MIN0:0710:996 kSRP=2:00 krp250(km)ke0:003(10)]TJ /F3 7.9701 Tf 6.587 0 Td[(5)wrpwe MAX7:540:0560:998 MAXMIN0:0640:997 MAX0:700:0600:998 MAX7:410:0580:998

PAGE 96

kSRP 0:50 169:4175:3162:2176:7170:91:00 122:7137:7119:9134:2128:61:50 93:5101:792:6100:497:02:00 78:271:476:569:073:8 kSRP 0:50 1:02101:77000:96531:69081:00 0:88551:78530:87501:74491:50 0:89961:32430:87471:44302:00 0:86650:01780:85680:0589 1m2 kg 89:123088:7810 92:451492:1154 88:866689:0322 86:696586:3645 89:284489:0755 1m2 kg 90:0789:7521:6647 93:3092:9321:2045 89:1089:3021:4745 87:8387:4719:7944 90:0089:8621:0345

PAGE 122

dt dL

PAGE 145

? =!t!tn !=n=r a3 ?

PAGE 146

=satellite)]TJ /F2 11.9552 Tf 11.955 0 Td[(slot _=d dt(satellite)]TJ /F2 11.9552 Tf 11.955 0 Td[(slot)slotnslot _satellite=nsatellite _=nsatellite)]TJ /F2 11.9552 Tf 11.955 0 Td[(nslot_a =_nsatellite=)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(3 2r a5_a V=1 22+1 22

PAGE 147

_V=_+_nd =nsatellite)]TJ /F2 11.9552 Tf 11.955 0 Td[(nd _=_nsatellite)]TJ /F4 11.9552 Tf 13.823 0 Td[(_nd _=nsatellite)]TJ /F2 11.9552 Tf 11.956 0 Td[(nslot+nd)]TJ /F2 11.9552 Tf 11.956 0 Td[(nd=)]TJ /F2 11.9552 Tf 11.955 0 Td[(nslot+nd _=)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(3 2r a5_a)]TJ /F4 11.9552 Tf 13.824 0 Td[(_nd _V=()]TJ /F2 11.9552 Tf 11.955 0 Td[(nslot+nd)+)]TJ /F4 11.9552 Tf 10.494 8.088 Td[(3 2r a5_a)]TJ /F4 11.9552 Tf 13.824 0 Td[(_nd nd_a nd=nslot)]TJ /F2 11.9552 Tf 11.955 0 Td[(k _a=)]TJ /F4 11.9552 Tf 10.494 8.087 Td[(2 3s a5 (_nd)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)

PAGE 148

_U=(k)]TJ /F2 11.9552 Tf 11.955 0 Td[(nslot+nslot)]TJ /F2 11.9552 Tf 11.956 0 Td[(k)+ )]TJ /F4 11.9552 Tf 10.494 8.087 Td[(3 2r a5()]TJ /F4 11.9552 Tf 10.494 8.087 Td[(2 3s a5 (_nd)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)))]TJ /F4 11.9552 Tf 13.823 0 Td[(_nd!=)]TJ /F2 11.9552 Tf 9.299 0 Td[(k2+k+(_nd)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)]TJ /F4 11.9552 Tf 13.824 0 Td[(_nd)=)]TJ /F2 11.9552 Tf 9.299 0 Td[(k2+k)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)]TJ /F2 11.9552 Tf 11.956 0 Td[(k2 _V=)]TJ /F2 11.9552 Tf 9.298 0 Td[(k2)]TJ /F2 11.9552 Tf 11.955 0 Td[(k2 )]TJ /F4 11.9552 Tf 10.494 8.087 Td[(2 3s a5 (_nd)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)=a)]TJ /F3 7.9701 Tf 8.081 -1.794 Td[(SRP=)]TJ /F2 11.9552 Tf 9.298 0 Td[(ackakcos3cos2(coscos+sinsin)=0aackak 2(_nd)]TJ /F2 11.9552 Tf 11.955 0 Td[(k)]TJ /F2 11.9552 Tf 11.955 0 Td[(k) 3as a5 =cos2()cos()]TJ /F2 11.9552 Tf 11.955 0 Td[()nd_nd=0nslot =)]TJ /F4 11.9552 Tf 10.495 8.088 Td[(2k(+) 3as a5 J J=)]TJ /F4 11.9552 Tf 11.955 0 Td[(cos2()cos()]TJ /F2 11.9552 Tf 11.955 0 Td[()J2)]TJ /F11 7.9701 Tf 10.494 4.707 Td[( 2; 2

PAGE 149

>0andaad^n=cos^e1+sin^e2 800m250kgrGEO+300km180GEO5002[0;2]%

PAGE 150

argmin(t)tf s:t:r)]TJ /F2 11.9552 Tf 13.151 8.088 Td[( r3r=ad [a0e0i0]=(rGEO+250km)00 af=rGEO ef200km rGEO if15 jfj1 0(t) 2

PAGE 151

200km ^erGEO+250km0902703601200m2 1100kg12m2 kg ^Lf x0=[p0f0g0~0k0t0]T=[(rGEO+250km)00000]T

PAGE 152

^pf=rGEO)]TJ /F4 11.9552 Tf 11.955 0 Td[(200km^~=0:08^f=200km rGEO^k=0^g=0^tf^L?f ^tfx0xf rGEO)]TJ /F4 11.9552 Tf 11.956 0 Td[(1000kmprGEO+1000km)]TJ /F4 11.9552 Tf 9.299 0 Td[(0:1~0:1)]TJ /F4 11.9552 Tf 9.298 0 Td[(0:05f0:05)]TJ /F4 11.9552 Tf 9.299 0 Td[(0:1k0:1)]TJ /F4 11.9552 Tf 9.298 0 Td[(0:05g0:050ttmaxf?;?g~^e^r^s 0

PAGE 153

a)]TJ /F3 7.9701 Tf 6.587 0 Td[(1:50f90;180;270g90circ

PAGE 154

sgnsgn(a)]TJ /F2 11.9552 Tf 11.955 0 Td[(ad)>0sgnsgn(a)]TJ /F2 11.9552 Tf 11.956 0 Td[(ad)<0 L 0 8:088:100190 38:0238:3701180 50:7951:4301270 60:4461:352:4511

PAGE 156

0

PAGE 157

0

PAGE 158

0

PAGE 159

90

PAGE 160

90

PAGE 161

180

PAGE 162

180

PAGE 163

180

PAGE 164

180

PAGE 165

270

PAGE 166

270

PAGE 167

270

PAGE 170

s=s1s2:::snsnnthsxx(t)=x1(t)x2(t):::xn(t)T2Rn1nn11mS S=266666664s11s21:::snms12s22s1msnm377777775sss s=s^s k^sk=1:k^sk=1

PAGE 171

abaTb abaxb ()Tx

PAGE 172

Rkmkm3 s2 6378:1366398600:436233$ 1737:44902:800076 695700132712440040:944000 AU 149597870:700kmTSI 1360:8W m2c 299792:458km stsol 86400stsid 86164:0905400sC2;0 )]TJ /F4 11.9552 Tf 9.298 0 Td[(1:0826261738522210)]TJ /F3 7.9701 Tf 6.586 0 Td[(3C2;2 1:5746153257229210)]TJ /F3 7.9701 Tf 6.586 0 Td[(6S2;2 )]TJ /F4 11.9552 Tf 9.298 0 Td[(9:0387278919656710)]TJ /F3 7.9701 Tf 6.586 0 Td[(7

PAGE 173

!=2 tsid rGEO=3r !2 P=TSI c !rGEOP1AUDUTUrGEO1DU=42164:169637135361km1TU=1371:344092645860sec)=1DU3 TU2