Planning a Trip to the Moon? And Bac k?

Planning a Trip to the Moon? And Bac k?

35 Pages · 2006 · 593 KB · English

Planning a Trip to the Moon? ..And Bac k? John T. Betts Boeing isatrademar kofBoeing Management Compan y Cop yright!c 2006 Boeing allrights reser ved

Planning a Trip to the Moon? And Bac k? free download

Planning a T r ip to the Moon? . . . . . . And Bac k? John T . Betts Boeing is a tr ademar k of Boeing Management Compan y Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 T ypical T w o Bur n Orbit T r ansf er Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Par kOrbit: 150 nm circular ,28.5 deg inclination Mission Orbit: Geosynchronous Equator ial Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Orbit Mechanics 101 Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology . ................................................. ................................................ ............................................... .............................................. ............................................. ............................................ ........................................... ........................................... ............................................ ............................................. .............................................. ............................................... ................................................ ................................................. ! . ................................................. ................................................ ............................................... .............................................. ............................................. ............................................ ........................................... ........................................... ............................................ ............................................. .............................................. ............................................... ................................................ ................................................. ! ! ! v1 ! ! vm " ! v2 tI tm tF Par k Orbit Mission Orbit ? Plane change most ef?cient at high altitude ? Three Bur n Transf er Does This " Eliminate Middle Bur n and Use Moon F or High Altitude ! v "John T.Betts ,?Optimal Three-Bur n Orbit Transf er ,?AIAA Jour nal, June ,1977 Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Lunar Swingb y to Geosynchronous Orbit Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology #!V1# #!V2# Total (fps) Hohmann 8056.67 5851.44 13908.12 Swingb y 10201.39 3518.72 13720.11 Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Lunar Swingb y to P olar 24-hr Orbit Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology #!V1# #!V2# Total (fps) Hohmann 8113.34 8610.82 16724.17 Swingb y 10225.16 3440.44 13665.61 Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Another Swingb y to P olar 24-hr Orbit Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology #!V1# #!V2# Total (fps) Hohmann 8113.33 8610.77 16724.10 Swingb y 10251.24 3459.66 13710.90 Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Swingb y to Molniy a Orbit ( i = 116.6) Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology #!V1# #!V2# Total (fps) Hohmann 546.99 39682.76 40229.76 Swingb y 10392.39 4826.32 15218.71 Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Equations of Motion Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology ECI spacecr aft state (r,v) and lunar state (rm ,vm ) ?r = v ?v = $ ?e r3r+ gm ?rm = vm ?vm = $ ?% r3m rm where lunar gravitational per turbations on S/C are gm = $ ?m # 1 d3d + 1 r3m rm $ with d = r$ rm Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Analytic T w o-Body Propagation Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Propagate from state (r%,v%) through angle ! E to state (r,v). "%= 1 & ?rT%v% 1 a = 2 #r%#$ # vT%v% ? $ C = a(1$ cos !E) S = & asin !E F = 1$ C #r%# G = 1 & ?(#r%#S+ "%C) # = 1$ #r%# a r= #r%#+ #C + "%S Ft= $ & ? r#r%#S Gt= 1$ C r r= Fr%+ Gv% v= Ftr%+ Gtv% !t= % a3 ? # !E + "%C a& a$ # S& a $ State propagation is explicit ,i.e . r = hr(r%,v%,!E) v = hv(r%,v%,!E) Time change ! t is implicit ,from K epler? s Equation . !t= ht(r%,v%,t%,!E) Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Equations of Motion?D AE F or m ulation Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Treat ! E (t) as an algebr aic (control) var iab le Dynamics de?ned by the diff erential-algebr aic (D AE) system ?r = v ?v = $ ?e r3r+ gm 0 = !t$ ht(r%,v%,t%,! E ) where lunar gravitational per turbations on S/C are gm = $ ?m # 1 d3d + 1 r3m rm $ with d = r$ hr(r%,v%,! E ) Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Ho w T o Solv e A Hard Prob lem Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Break A Har d Pr ob lem Into A Sequence of Easy Subpr ob lems ? Ne wton? s Method Solv e a nonlinear constr aint by solving a sequence of linear appro ximations; ? Nonlinear Pr ogramming Solv e a nonlinear optimization prob lem by solving a sequence of: % quadr atic prog ramming subprob lems (an SQP Method) or % unconstr ained subprob lems (a Barr ier Method) ? Optimal Contr ol Solv e a sequence of NLP subprob lems . ? Optimal Lunar Swingb y Solv e a sequence of ?Easier? subprob lems . Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 F our Step Solution T echnique Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Step 1: Three Impulse ,Conic Solution Solv e small NLP with analytic propagation ignor ing lunar gravity; Step 2: Three-Bod y Appr o ximation to Conic Solution Solv e ?in verse prob lem? to ?t three-body dynamics to conic solution; Step 3: Optimal Three-Bod y Solution with Fix ed Swingb y Time Use solution from step 2 to initializ e optimal solution. Step 4: Optimal Three-Bod y Solution Compute solution with free swingb y time ,using step 3 as an initial guess . Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Step 1: Three Impulse , Conic Solution Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Optimization V ariab les (24) (ro,vo,! v1,! E o) : State at Par k Orbit Depar ture (ri,vi,! v2,! E i) : State at Mission Orbit Arr iv al (! vL,! E L) : Velocity Increment and Transf er Angle at Lunar Intercept P ark Orbit Conditions rp = ro Position Contin uity vp = vo$ ! v1 Impulsiv e Velocity Change $p(rp,vp) = 0 Par k Orbit Constr aints Mission Orbit Conditions rm = ri Position Contin uity vm = vi+ ! v2 Impulsiv e Velocity Change $m (rm,vm) = 0 Mission Orbit Constr aints Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Step 1: Three Impulse , Conic Solution Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Lunar Conditions hr(ro,vo,! E o) = hr(ri,vi,! E i) Outbound/Inbound Position hr(ro,vo,! E o) = hr(rL%,vL%,! E L) Outbound/Lunar Position hv(ri,vi,! E i) = hv(ro,vo,! E o)+ hv(rL%,vL%,! E L)+ ! vL Velocity Change Objective Minimiz e F = #! v1# + #! v2# + #! vL# . ................................................................... ................................................................ ............................................................. .......................................................... ........................................................ ....................................................... ...................................................... .................................................... ................................................... .................................................. ................................................. ................................................ ............................................... .............................................. """# . .............................................. ............................................... ................................................ ................................................. .................................................. ................................................... .................................................... ...................................................... ....................................................... ........................................................ .......................................................... ............................................................. ................................................................ ................................................................... $$$% &&&&' (ro,vo,!v1) (rL%,vL%) (ri,vi,!v2) !vL !Eo !Ei !EL Par kOrbit Mission Orbit Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Step 2: Three-Body Appro ximation Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Conic solution solv es ?r = $ ?e r3r ?rm = $ ?% r3m rm not ?r = $ ?e r3r+ gm ?rm = $ ?% r3m rm ?Fit? Three-Body Trajector y to Conic i.e . minimiz e F = % k #rk$ &rk#2 subject to ?r = $ ?e r3r+ gm ?rm = $ ?% r3m rm rLmin ' #r$ rm# where &rk is S/C position at k points on conic tr ajector y Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Step 3: Fix ed Swingb y Time Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Tw o Phases with Three-Body Dynamics (ODE or D AE) ?r = $ ?e r3r+ gm ?rm = $ ?% r3m rm . ................................................................... ................................................................ ............................................................. .......................................................... ........................................................ ....................................................... ...................................................... .................................................... ................................................... .................................................. ................................................. ................................................ ............................................... .............................................. """# . .............................................. ............................................... ................................................ ................................................. .................................................. ................................................... .................................................... ...................................................... ....................................................... ........................................................ .......................................................... ............................................................. ................................................................ ................................................................... $$$% t= tL &&&&' (ro,vo,!v1,tI) (rL%,vL%) (ri,vi,!v2,tF) Par kOrbit Mission Orbit Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Step 3: Fix ed Swingb y Time Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Phase 1: Outbound Transf er At (free) tISatisfy Par k Orbit Conditions rp = ro Position Contin uity vp = vo$ ! v1 Impulsiv e Velocity Change $p(rp,vp) = 0 Par k Orbit Constr aints At (?x ed) tL Satisfy Lunar Conditions #r$ rm# ( rLmin Closest Approach (v $ vm)T(r$ rm) = 0 Lunar Flight Path Angle Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Step 3: Fix ed Swingb y Time Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Phase 2: Inbound Transf er At (?x ed) tL (r,v,rm,vm)($)= (r,v,rm,vm)(+) State Contin uity (rm,vm) = (rm,vm) Lunar State At (free) tF Satisfy Mission Orbit Conditions rm = ri Position Contin uity vm = vi+ ! v2 Impulsiv e Velocity Change tmax ( tF $ tI Mission Dur ation $m (rm,vm) = 0 Mission Orbit Constr aints Objective Minimiz e F = #! v1# + #! v2# Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Step 4: Optimal Three-Body Solution Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Form ulation as in Step 3 except: (a) F ree Lunar Swingb y Time tL (b) Lunar State Speci?ed at (free) tI rm = h r(r%,v%,! E %) vm = h v(r%,v%,! E %) Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 The Swingb y Subprob lems Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Ho w To Ef?cientl y Solve The Subpr ob lems? ? Optimal Contr ol .................................................Step 3 and 4 ? P arameter Estimation (In ver se Pr ob lem) ..............................Step 2 ? Nonlinear Pr ogramming ...............................................Step 1 Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Good Softw are/Algor ithms to Solv e NLP Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Find V ar iab les xT = (x1,...,xn) to minimiz e the Objectiv e F (x) subject to Constr aints cL ' c(x) ' cU . Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Prob lems W e W ant to Solv e Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Find Control Functions u (t) and/or par ameters p to minimiz e J = Z tF tI w [y(t),u (t),p ,t]d t or J = % k #y(tk) $ &y(tk)#2 subject to constr aints over the domain tI ' t ' tF ?y = f[y(t),u (t),p ,t] 0 ' g[y(t),u (t),p ,t] and boundar y conditions Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 So What? s the Rub? Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology The NLP W or ks with a Finite Set of V ar iab les x and Functions F (x),c(x) ... But Optimal Control/Estimation is an In?nite Dimensional Prob lem; i.e . the functions u (t) and y(t) Ho w do w e for m ulate the prob lem? Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Shooting Methods Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology ?Eliminate? In?nite Dimensional Prob lem by solving ?y = f[y(t),u (t),t] and/or ?y = f[y(t),u (t),t] 0 = g[y(t),u (t),t] The NLP in volv es the Finite Set of Boundar y V alues BVP is ver y nonlinear? ! v at boundar y,lunar gravity in ?middle? ODE or D AE can be ver y unstab le ODE error control at suboptimal points?inef?cient P ath inequalities cumbersome (impr actical?) Shooting for Control ) * GRG for NLP Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Discretization Methods Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology V ar iab les ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!! [y(t),u (t)] x = [y1,u 1,...,yM ,u M ]+ . Constr aints ?y = f[y(t),u (t),t] yk+ 1 = yk + hk 2 (fk + fk+ 1) Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 An Optimal Control Algor ithm Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Direct Transcription Transcr ibe the optimal control prob lem into a nonlinear pro- gramming (NLP) prob lem by discretization; Spar se Nonlinear Pr ogram Solv e the sparse (SQP or Barr ier) NLP Mesh Re?nement Assess the accur acy of the appro ximation (i.e . the ?nite dimen- sional prob lem), and if necessar y re?ne the discretization, and then repeat the optimization steps . SNLP :Sequential Nonlinear Prog ramming Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Barr ier or SQP? Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Step 1?Small, Dense NLP Subpr ob lems Mission Equato rial P olar Molniy a SQP-Ne wton (10,4) (16,10) (135,44) SQP-BFGS (24,19) (36,31) (186,96) Barr ier-Ne wton (24,22) (57,55) (70,68) ? Barr ier-BFGS (242,241) ? (58,56) (286,284) Key: (Gr adient Ev al., Hessian Ev al.) ? No Solution Some Sweeping Generalizations SQP Most Ef?cient and Rob ust Quasi-Ne wton Hessian Too Slo w for Large/Sparse Barr ier Method Lac ks Rob ustness ,Speed Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Sequential Nonlinear Prog r amming Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Is an SQP better than Barr ier for Sequential Nonlinear Prog ramming? Coarse Gr id xc = [y1,u 1,...,ym ,u m ]+ Fine Gr id x f = [y1,u 1,...,yM ,u M ]+ NLP prob lem siz e gro ws?typically M > m Question: Ho w do w e ef?ciently solv e a sequence of NLP? s? Ans w er : Use coarse grid inf or mation to ?Hot Star t? ?ne grid NLP Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Estimating V ar iab les for SNLP Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology SQP Algorithm high order inter polation of coarse grid solution consistent with discretization for m ula (e .g. collocation polynomial) ver y good guess Interior P oint Algorithm m ust be feasib le ) * barr ier algor ithm per turbs guess not consistent with coarse grid discretization for m ula Barrier Algorithm Cannot Exploit a Good Guess! Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Barr ier vs SQP? Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Step 3 Subpr ob lem, P olar Mission Step 2 Trajector y De?nes: Smooth Three-Body Trajector y Guess Propagate Trajector y Using V ar iab le Step Integ rator To De?ne Initial Gr id SQP k M n m NGC NHC NFE & Time (sec) 1 594 7136 7715 18 10 3794 1, 10 $4 30.1 2 881 10580 11446 4 1 454 4, 10 $7 7.8 3 1113 13364 14462 4 1 454 1, 10 $8 10.6 Total 26 12 4702 48.6 Barrier ? k M n m NGC NHC NFE & Time (sec) 1 594 7720 7715 328 319 112475 1, 10 $5 749.3 2 881 12985 12980 57 49 17637 2, 10 $7 233.7 3 1113 14090 14085 6 2 881 1, 10 $8 18.5 Total 391 370 130993 1001.6 ? Diff erent Local Solution than SQP Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Is Mesh Re?nement Needed? Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Step 2 In ver se Pr ob lem Solution For P olar Mission Step 1 Conic Trajector y De?nes: 1800 Residuals ? 600 Equal ! E increments Initial Trajector y Guess ? Omit P oint at Moon k M n m NGC NHC NFE & Time (sec) 1 599 7188 7775 12 2 144 6, 10 $1 3.5 2 606 7272 7866 21 17 525 9, 10 $5 1.1 3 606 7272 7866 4 2 724 1, 10 $6 7.6 4 742 8904 9634 4 1 448 1, 10 $8 5.6 Total 41 22 1841 27.7 k Re?nement No M Gr id Pts n NLP vars m NLP cons . NGC Gr ad Ev al NHC Hess Ev al NFE Func Ev al & Disc. Error Time CPU Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 V elocity Discontin uity Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Mesh Re?nement Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Mesh Re?nement ?Smooths? Out V elocity Discontin uity In verse Prob lem Approach K eeps Process Stab le Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 D AE or ODE F or m ulation? Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Steps 3 & 4, Optimal Solution for Molniy a Mission M n NGC NHC Time (sec) 630 7568 26 7 32.02 807 9692 7 2 12.30 940 11288 4 1 8.66 1190 14288 4 1 11.22 1190 14291 10 4 45.49 ODE Form ulation Total Time = 109.69 sec M n NGC NHC Time (sec) 1097 8782 30 14 45.27 1530 12246 4 1 7.98 2056 16454 4 1 11.82 2056 16456 7 2 31.02 DAE Form ulation Total Time = 96.09 sec No clearcut diff erence in speed or accur acy Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06 Summar y and Conclusions Engineer ing and Oper ations Technology |Phantom W or ks E&IT |Mathematics and Computing Tec hnology Optimal Lunar Swingb y Signi?cant P erf or mance Bene?ts for Ear th Orbital Missions with Large Plane Change Swingb y Trajector y V er y nonlinear boundar y value prob lem SQP More Rob ust, Ef?cient Barr ier algor ithm cannot exploit ?good guess?. Mesh Re?nement Cr itical for stab le solution. Ov er all Approach Applicab le to Man y n-body Prob lems Cop yr ight c! 2006 Boeing all rights reser ved Planning a Trip to the Moon?...And Bac k? 10/06

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