On Solving the Perturbed Multi- Revolution Lambert Problem: Applications in Enhanced SSA

Transcription

1 On Solving he Perurbed Muli- Revoluion Lamber Problem: Applicaions in Enhanced SSA John L. Junkins and Robyn M. Woollands Texas A&M Universiy Presened o Sacie Williams (AFOSR/RT) AFOSR REMOTE SENSING PORTFOLIO REVIEW 2 DECEMBER

2 Ouline Two Moivaions Advanced SSA Daa Associaion for Radar Sensing Exremal Field Maps For Muli-Revoluion Orbi Transfer Impulsive, Low-Thrus, and Hybrid Recen Advances in Solving he Fundamenal Iniial and Two-Poin Boundary Value Problem of Asrodynamics: Inegral Pah Ieraion Mehods Comparisons o Sae-of-Pracice Algorihms Regularizaion, Insighs & Consequences Thereof KS Regularizaion Regularized Orbi Elemens Some Examples & Impacs 2

3 Daa Associaion 101 for Radar Daa Radar Daa Associaion Problem has >N 2 complexiy, if we es preliminary orbi hypoheses using all possible observaion pairs. For shor arc case (fracion of an orbi), he problem is much easier han muli-orbi case. Compuaion ime for each hypohesis es is dominaed by orbi propagaion cos implici in Lamber soluion process. Three Coupled Imporan Challenges: For longer arcs, approximaion errors in he Keplerian Lamber soluion are larger han measuremen errors => no good! Lamber algorihms using sae-of-pracice numerical propagaion & high fideliy force models leads o SSA compuing boleneck. For muli-rev case, in general, more han one orbi soluion saisfies wo posiions and imes. Wish Lis for Research o Mee Challenges: Means for much more efficien soluion process for perurbed Lamber problem. Means o resolve ambiguiies due o uncerainy, and especially, due o mulipliciy of soluions for muli-revoluion case. Desire a parallelizable & scalable approach Seek higher fideliy hypohesis es wih ~ 100x speedup 3/18

4 Acceleraed Picard Ieraion Successive pah approximaion mehod for solving nonlinear differenial eqns of he form: x( ) f (, x( )), x 0 x0 Can be rearranged wihou approximaion o he following inegral: x( ) x( ) f ( s, x( s)) ds, 0 0 Sequence of rajecory approximaions (Picard Ieraion) generaed by: Charles Émile Picard x ( ) x( ) f (, x ( )), 1,2,...; x ( )) "warm sar" ( ) n1 n 0 0 s s ds n 0 ( ), n Picard proved he general circumsances under which: x ( ) x( ). Large domain of convergence, wih a geomeric convergence rae. In discussions wih Aluri, we found dramaic erminal convergence speedup using an inegral eqn error feedback erm: n1 n x x 0 f x ( ) ( ) ( s, ( s)) ds, 0 s n 1 n 1 n n n n ( ) ( ) {[ ( s, ( s))][ ( 0) ( ( ), ) d ( s)]} ds, [ (, ( ))] x x J x x f x x J x x n 1 ( s) n1 n1 n n1 n x ( ) x ( ) {[ J( s, x ( s))][ x ( s) x ( s)]} ds n h ieraion equaion error f( x, ) x n x () very signigicanly acceleraes Picard Ieraion 4/18

5 X X max Impac of Inegral Error Feedback Inermediae Fideliy Model (degree & order 40 graviy) e = Engineering accuracy Ieraion Number 5/18

6 KS Regularizing Transformaion KS Coordinaes ODEs (a rigorous linearizaion of 2-body prob) : Caresian Coordinaes ODEs: 1 1 T T T new form of KS u u r 4u uri 4 uu [ L( u)] ad,, where x x a 4 4 eqs of moion 3 dx r d ( ) ( ) [ L4 ( u d )] u 0 is an exac inegral, even for generally perurbed moion, and y y a, 3 d y r r x y z Iniial condiions mus saisfy: 1/2 z z a d [ ( (0))] (0) (0) of (0) soluions 3 dz r [ 4 T L u u r u r r ] de u u 21 1 T u(0) [ L( u(0))] r(0) The KS Transformaion is: 2 (0) x x x u1 u1 u2 u3 u4 y y y u 2 u2 u1 u4 u 3 r L( u) u r, r, r, u, L( u). z z z u 3 u d 1 3 u4 u1 u 2 r u 4 u4 u3 u2 u1 de, where T 2 r r 1 = T 1 T Imporan Properies : L ( u) rl ( u ), r = u u. r a Noe he zero 4h elemen of r in he KS ransformaion r Lu gives he ideniy T 1 2[ r 4 u u] L4 ( u) u [ u4 u3 u2 u1] u u4u1 u3u2 u2u3 u1u 4 0. If we have a vecor v ha saisfies L4 ( u) v 0, hen you can show L( u) v L( v) u. Quesion: Will Picard Ieraion (MCPI) converge beer using he KS equaions of moion? Ans: Yes, MCPI convergences faser and over a ~3x larger ime inerval for boh IVP & BVP. 6

7 KS Uniqueness Challenges (for he perurbed Lamber problem) Kusaanheimo - Siefel Transformaion Uniqueness Theorem For any given { x, y, z, x, y, z}, here is an infiniy of geomerically feasible u-vecors ha mus lie on a 4D feasible u-space curve known as a fiber. Once a paricular feasible u-poin is seleced from some poin on he feasible fiber, hen he ransformed velociy Feasible fiber The 4-D u-sphere is ime-varying. The insananeous radius is 1/2 T u(1/ 2)( / a) L ( u) r is unique and saisfies L4 ( u) u0. u r. 1/2 2( / a) Imporan : The inverse ransformaion { r( ) L( u) u, r( ) L( u) u} from all infiniy of r feasible rajecories { u( E), u( E )} ha ensue from a feasible fiber iniial sae{ u(0), u (0) } give he same unique Caresian space rajecory { r( ), r( )}. Wha does i mean for solving iniial value problems? 1/2 I means { u( E), u( E)} from any geomerically feasible iniial u-posiion, wih u (1/ 2)( / a) L T ( u) r will generae (upon inverse ransformaion) he correc physical soluion of a general iniial value problem. => This ruh is well-known in he lieraure and is he main use of he KS ransformaion. However, o solve a TPBVP Lamber problem u-space There are an infiniy of feasible choices of geomerically compaible erminal (boundary) posiion coordinaes on he wo fibers in u-space. Quesion: When I selec one geomerically feasible poin on he lef (iniial) fiber, which one do we selec on he righ (final) fiber such ha boh of he specified iniial and final u vecors lie on he same dynamical pah in u-space? 7/18

8 Numerical MCPI Validaions IVP MCPI BVP Algorihm Has ~3x Increased Domain of Convergence using he KS ODEs Compared o Caresian ODEs Convergence is Independen of Eccenriciy Theoreical Convergence for IVP Caresian coordinaes Theoreical Convergence for BVP Caresian coordinaes 8/18

9 Efficiency of KS Lamber MCPI vs Sae of he Pracice for Fracional Orbi Case (40,40 Spherical Harmonic Graviy) KS MCPI more efficien in serial compuaion. Imporan: ~50x addiional speedup Due o parallel srucure of MCPI. (GJ8 and RKN(12)10 no amenable o parallelizaion) 9

10 A Specific Orbi Transfer. 10/18

11 0 ime of fligh (minues from iniial orbi deparure o arrival a arge orbi sae ) 450 Fracion of one orbi of orbi ransfer vehicle Exremal Field Maps (EFMs): Muliple Soluions of he Perurbed Lamber Problem => Visibiliy of Reachabiliy (40,40 Spherical Harmonic Graviy Field) 1 < laps < 2 orbis of orbi ransfer vehicle 2 < laps < 3 orbis of orbi ransfer vehicle 3 < laps < 4 orbis of orbi ransfer vehicle 4 < laps < 5 orbis of orbi ransfer vehicle Pink Regions are Unreachable for V < 3 km/s Amenable Near Real Time o Massive EFMs Parallelizaion Have Imporan Uiliy: => A Near Quick Real Look Time over Mission large Planning domain can ( be on-board?) near-insanly => generaed Obain Quick using SSA Keplerian Answers Lamber Quesions Solver Such As Can Who hem can zoom reach on whom? ROIs & include For how full much force V? model When? minues pas epoch minues pas epoch minues pas epoch minues pas epoch minues pas epoch 11

12 Equinocal Variaion of Parameers Transformaion from Equinocal o classical elemens: 12/18

13 Opimal Muli-Rev Low-Thrus Orbi Transfers We seek o minimize: Subjec o:, wih, and e 0 and e f specified. The Hamilonion is. The opimal seering uni vecor ha minimizes H is u The co-sae is governed by he differenial equaion:, ( ) is unknown. The 6 unknown iniial co-saes and he unknown final ime are deermined o saisfy he erminal sae e( f ) = e f, and H( f )=0. This wo-poin boundary value problem is highly nonlinear and requires a good saring soluion. deermined by a direc opimizaion mehod. 0 13/18

14 Near Minimum Time Low hrus Orbi Transfer 14

15 Near Planar Min Time Low Thrus Orbi Transfer (much smaller plane change, shorer maneuver ime) Opimal Conrol Seering Vecor Componens (radial, ransverse, and orhogonal) 15

16 Ouline Two Moivaions Advanced SSA Daa Associaion for Radar Sensing Exremal Field Maps For Muli-Revoluion Orbi Transfer Impulsive, Low-Thrus, and Hybrid Recen Advances in Solving he Fundamenal Iniial and Two-Poin Boundary Value Problem of Asrodynamics: Inegral Pah Ieraion Mehods Comparisons o Sae-of-Pracice Algorihms Esablished significan new insighs, formulaions and compuaional mehods o accelerae fundamenal asrodynamics compuaions: orbi propagaion and soluion of Lamber s problem. >3x efficiency for serial implemenaion, ~10x o 50x furher increase feasible via parallelizaion. Regularizaion, Insighs & Consequences Thereof KS Regularizaion Found imporan insighs o resolve ambiguiies in solving he KS Lamber Problem. Regularized Orbi Elemens Regularizaion enhances Picard Ieraion for boh iniial and wo-poin boundary value problems. Some Examples & Impacs Demonsraed a parallelizable and Scalable Approach for near real ime compuing of Exremal Field Maps for enhanced SSA, Mission Planning, ec. Impulsive orbi ransfers Coninuous low hrus orbi ransfers Hybrid hrus orbi ransfers 16

17 Archival References Disseraions: Bai, X., "Modified Chebyshev-Picard Ieraion for Soluion of Iniial Value and Boundary Value Problems", PhD. Disseraion, Texas A&M, College Saion, Texas, USA, Bani Younes, A., "Orhogonal Polynomial Approximaion in Higher Dimensions: Applicaions in Asrodynamics", PhD. Disseraion, Texas A&M, College Saion, Texas, USA, Macomber, B., "Enhancemens of Chebyshev-Picard Ieraion Efficiency for Generally Perurbed Orbis and Consrained Dynamics Sysems", PhD. Disseraion, Texas A&M Universiy, College Saion, Texas, USA, Woollands, R., "Regularizaion and Compuaional Mehods for Precise Soluion of Perurbed Orbi Transfer Problems", PhD. Disseraion, Texas A&M Universiy, College Saion, Texas, USA, Read, J., "Modified Chebyshev Picard Ieraion: Inegraion of Perurbed Moion Using Modified Equinocial Elemens", PhD. Disseraion, Texas A&M Universiy, College Saion, Texas, USA, Journal Papers: Woollands, R., Bani Younes, A. and Junkins, J.L., "New Soluions for he Perurbed Lamber Problem Using Regularizaion and Picard Ieraion", Journal of Guidance, Conrol and Dynamics, Macomber, B., Probe, A., Woollands, R., Read, J. and Junkins, J., "Enhancemens for Modified Chebyshev- Picard Ieraion Efficiency for Perurbed Orbi Propagaion", Compuer Modeling in Engr Sciences, Read, J., Bani Younes, A. and Junkins, J., "Efficien Orbi Propagaion of Orbial Elemens Using Modified Chebyshev-Picard Ieraion Mehod", Compuaional Modelling in Engineering & Sciences, Woollands, R., Read. J, Probe, A. and Junkins, J., "Muliple Revoluion Soluions for he Perurbed Lamber Problem using he Mehod of Paricular Soluions and Picard Ieraion", American Asronauical Sociey, acceped