Direct Methods of Optimal Control for Space Trajectory Optimization
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1 Direc Mehods o Opimal Conrol or Space Trajecory Opimiaion Rober G. Melon Deparmen o Aerospace Engineering Compuaional Science Seminar February 6, 7
2 Trajecory Opimiaion A B (min. ime and/or propellan) subjec o consrains Venus Earh Sun
3 Solar Sail Propulsion Use phoon pressure on large relecive sail n F ( + r) PA c ( r n) n releciviy r r ideal sail spacecra
4 Realisic sail
5 A simple approach Discreie eqs o moion in ime ( j, j,n) Unknowns: conrols u j a j ( known?) Esablish perormance measure J arge, sae (pos., vel.) Inegrae eqs o moion; use shooing mehod and LP o adjus he u j : minimie J LP nonlinear programming minimie (u) subjec o consrains Too sensiive o small changes in u j
6 Opimal Conrol Problem subjec o: minimie J h(,,, ) + g( ( ), u ( ), ) d ( ) ( ( ), u( ), ) b L ( ) g ( ), ( ),, ) b L ( g( ( ), u( ), ) b U b U and simple bounds on he saes and conrols
7 Indirec Mehods Form Hamilionian H g(, u, ) + p T ecessary condiions or opimaliy o u: opimal )* ( ) * ( ) *, *, *, ( ) *( ) *, ( ) *, *, *, ( cosae ) *, *, *, ( * sae ) *, *, *, ( * + + T h H h H H H δ δ p u p p u u p p u p p u p Diicul o ormulae necessary condiions Soln o eqs requires muliple ieraions; sensiive o i.c. s
8 Direc Collocaion wih LP (DCLP) Use implici soln o eqs o moion o ormulaion o necessary condiions rqd. Bu require soln o large-scale LP wih large number o equaliy consrains Discreie eqs o moion in ime ( j, j,n n number o nodes) Unknowns: conrols u j and saes j a nodes ( known?) Use splines or j, inerpolaion or u j min J ( c) c [ subjec o : Δ( c) M deecs Δ ( c) n w w( c) w L L ( c,, L, n ), u U U, L, u, inerior consrains erminal consrains and simple bounds on he conrols n ]
9 Calculaion o he Deecs slope ds/d Δ j d i /d -ds/d i slope d i /d i (,u,) cener poin node j node j+ spline S() j c j+
10 A Pseudospecral Mehod (also a collocaion mehod) Epands and u as inie sums o Lagrange polynomials evaluaed a Chebyshe-Gauss- Lobao (CGL) poins k cos(πk/) k,, [-,], scaled ime True imeτ [(τ -τ ) + (τ +τ )]/ ( ) φ ( ) u ( ) u φ ( ) j j j j j j where φ j Lagrange inerpolaing polynomial
11 ow solve he LP (variables: k, u k, and possibly τ ) ) (, ),,, ) ( /( subjec o /,,, cos 4 4 ),, ( ), ( " / k k U k k k k L j s s k k k k k s js j w w w w w g h J d u d u L + τ τ τ π τ τ τ τ
12 Solar Sail Problems Conrol auhoriy srong uncion o sail orienaion (angles α, δ ) D case easily solved by several direc opimiaion mehods (and including realisic orce models) 3D case -- plane change creaes challenges or some mehods Eample: compare perormance o wo mehods: DCLP and pseudospecral (DIDO)
13 Eample Cases Ideal sail a T PA mc ( ) n r n A 3 m, m 5 kg a T.5 mm/s a au rom Sun Hohmann-like ransers (no deparure or capure deails) Planeary orbis assumed circular D cases (Earh and Venus in same plane) 3D cases (plane change required)
14 Sail angles: Consrains π α, δ π OK or DCLP, no or DIDO Trig. repres. o crls: cosα, sinα and cosδ, sinδ), wih addiional consrain cos ( ) + sin ( ) Simple bounds cos( ), sin( ).
15 3D Final Consrains + r V ( ) r V μ θ + + ( ) i V cos θ θ θ ( ) ( ) Ω V cos sin cos + + θ θ θ θ ( ) ( ) Ω V sin cos sin + θ θ θ θ
16 D case wih nodes -5 - DIDO DCLP -5-3 Saii angle α (deg.) Time (years)
17 D case wih nodes (DIDO) Venus' orbi
18 D case wih nodes (DCLP) Venus' orbi
19 D case wih nodes -5 - DIDO DCLP -5-3 Saii angle α (deg.) Time (years)
20 D case wih nodes (DIDO) Venus' orbi
21 D case wih nodes (DCLP) Venus' orbi
22 Sail Angles or 3D case wih nodes 3 α (in-plane) δ (ou-o-plane) Saii angles (deg.) Time (years)
23 Trajecory or 3D case wih nodes.5 Earh Venus
24 Sail angles or 3D case wih nodes 3 α (in-plane) δ (ou-o-plane) Saii angles (deg.) Time (years)
25 Trajecory or 3D case wih nodes.5 Earh Venus
26 Processing Time Eac comparison no possible (dieren processors) D cases DCLP: 4 DIDO:.67 (sparse mari algorihms) 3D cases DCLP: eremely sensiive o iniial guess -- had o use perurbed DIDO soluion as irs guess o obain convergence DIDO:.9
27 Conclusions DIDO less sensiive o iniial guess or o addiional inal consrains in 3D problem eed a horough benchmark sudy suie o problems wih variey o consrains run under idenical condiions useul guide or selecing mehod appropriae or given ype o rajecory problem
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