FORMATIONS NEAR THE LIBRATION POINTS: DESIGN STRATEGIES USING NATURAL AND NON-NATURAL ARCS K. C. Howell and B. G. Marchand

Transcription

1 1 FORMATIONS NEAR THE LIBRATION POINTS: DESIGN STRATEGIES USING NATURAL AND NON-NATURAL ARCS K. C. Howell and B. G. Marchand

2 Formations Near the Libration Points ŷ ˆx Moon L 2 Deputy S/C (Orbit Chief Vehicle) L 1 Chief S/C Path (Lissajous Orbit) B θ Xˆ ( inertial) 2 EPHEM = Sun + Earth + Moon Motion From Ephemeris w/ SRP CR3BP = Sun + Earth/Moon barycenter Motion Assumed Circular w/o SRP

3 3 Control Methodologies Considered in both the CR3BP and EPHEM Models Continuous Control Linear Control State Feedback with Control Input Lower Bounds Optimal Control Linear Quadratic Regulator (LQR) Nonlinear Control Input Feedback Linearization (State Tracking) Output Feedback Linearization (Constraint Tracking) Spherical + Aspherical Formations (i.e. Parabolic, Hyperbolic, etc.) Discrete Control Nonlinear Optimal Control Impulsive Constant Thrust Arcs Impulsive Targeter Schemes State and Range+Range Rate Natural Formations Impulsive Deployment Hybrid Formations Blending Natural and Non-Natural Motions

4 4 IMPULSIVE FORMATION KEEPING: TARGETER METHODS

5 State Targeter: Impulsive Control Law Formulation V 3 D(t 2 ) V 2 D(t 3 ) Segment of Nominal Deputy Path δ r 3 C(t 3 ) Segment of Chief S/C Path δ r 2 δ v 0 V 0 δ v + 0 V 1 D(t 1 ) δ r 1 C(t 1 ) C(t 2 ) STM δr A k 1 k B + k δr k+ 1 = v C k 1 k D δ + k δvk+ 1+ Vk ( ) V = B δr Aδr δv 1 k k k+ 1 k k k 5

6 State Targeter: Radial Distance Error Distance Error Relative to Nominal (cm) ρ = ( ) ˆ 10 m Y Nominal Separation Max.Deviation 6 Time (days)

7 State Targeter: Achievable Accuracy Maximum Deviation from Nominal (cm) r = ryˆ Formation Distance (meters) 7

8 8 State Targeter: Maneuver Schedule

9 Range + Range Rate Targeter V 3 Range + Range Rate Constraint: V 2 r Deputy S/C T ( r ) 1/2 f rf r = = rf f g T f r rf r f f Chief S/C V 1 9 r r STM r r δ r0 (, 0) δ 0 Λ ( t) (, 0) r r δ v0 + V 0 dgf = Φ tt x = Φ tt r r State Relationship Matrix

10 Comparison of Range and State Targeters Deputy S/C Path Chief S/C at Origin 10

11 11 DESIGN OF NON-NATURAL FORMATIONS USING NATURAL SOLUTION ARCS

12 CR3BP Analysis of Phase Space Eigenstructure Near Halo Orbit Reference Halo Orbit Deputy S/C Chief S/C (,0) δ ( 0) Jt 1 ( ) ( 0) δ x( 0) δx = Φ t x = E t e E Solution to Variational Eqn. in terms of Floquet Modes: 6 6 ( ) = δ ( ) = ( ) e ( t) = ( ) ( t) δx t x t c t E t c j j= 1 j= 1 j j Floquet Modes Mode 1 Mode 2 Modes (3,4) and (5,6) 1-D Unstable Subspace 1-D Stable Subspace 4-D Center Subspace 12

13 Natural Solutions: Periodic Halo Orbits Near Libration Points Earth Sun ẑ ẑ ˆx ŷ ˆx 13

14 Natural Formations: Quasi-Periodic Relative Orbits 2-D Torus Chief S/C Centered View (RLP Frame) ẑ ŷ ˆx ŷ ˆx ẑ ẑ ẑ ˆx ŷ ˆx ŷ 14

15 Floquet Controller (Remove Unstable + 2 Center Modes) Find V that removes undesired response modes: 0 Remove Modes 1, 3, and 4: 1 α δx2r δx5r δx6r 03 = V δx2v δx5v δx6v I 3 Remove Modes 1, 5, and 6: ( 1 j) 6 3 δ x j + V j 1 I = = 3 j= 2,3,4 or + α δ j= 2,5, 6 ( δx δx δx ) 1 α δx2r δx3r δx4r 03 = V δx2v δx3v δx4v I 3 x j ( δx δx δx ) 15

16 Sample Deployment into Relative Orbits: 1- V at Injection 1800 days Deputy S/C 3 Deputies Quasi-Periodic Origin = Chief S/C Periodic Origin = Chief S/C 16

17 Natural Formations: Nearly Periodic and Drifting Relative Orbits Chief Origin 1800 days 17

18 Expansion of Drifting Vertical Orbit 18,000 days = 100 Revolutions r ( 0) r ( t f ) Origin = Chief S/C 18

19 Transitioning Natural Motions into Non-Natural Arcs: Targeter Approach STEP 1: Identify a suitable initial guess Target Orbital Drift Control 19

20 Application of Two Level Corrector STEP 2: Apply 2-level corrector (Howell and Wilson:1996) w/ end-state constraint r( t 0 ) r( t m ) Γ( T q ) V ( T q ) ( ) = r( t ) r t m 0 STEP 3: Shift converged patch states forward by 1 period STEP 4: Reconverge Solution 20

21 Drift Controlled Vertical Orbit (6 Revs) V = 3 5 m/sec ( 1 maneuver/year ) 21

22 Geometry of Natural Solutions in the Ephemeris Model Rotating Frame Perspective: Inertial Frame Perspective: w/ SRP 22

23 Transitioning Natural Motions into Non-Natural Arcs: IFL Example (1) Consider 1 st Rev Along Orbit #4 as initial guess to simple targeter. (2) Choose initial state on xz-plane (3) Target next plane crossing to be (4) Use resulting arc as half of the reference motion. (5) Numerically mirror solution about xz-plane and store as nominal. (6) Use IFL control to enforce a closed orbit using stored nominal. Chief Origin Sphere for Visualization Only 23

24 Hybrid Control: Natural Motions + Continuous Control ½ Period Natural Arc ½ Period IFL Control 24

25 Concluding Remarks Precise Formation Keeping Continuous Control Is it possible? Depends on hardware capabilities and nominal motion specified Not if thruster On/Off sequences are required & tolerances too high Precise Navigation Natural Formations Targeter Methods Natural motions can be forced to follow non-natural paths Success depends on non-natural motion specified Hybrid Methods (Natural Arcs + Continuous Thrust Arcs) May prove beneficial for non-natural inertial formation design. 25

26 26 BACKUP SLIDES

27 27 Hybrid Control: Accelerations

28 Range + Range Rate Constraint: T ( r ) 1/2 f rf r = = rf f g T f r rf r f f Desired Range + Range Rate: g * f * r f = 0 Radial Targeter: Control Law Formulation First Order Approximation: * g g f = g f + x x δ r r STM * g x r r dgf = ( g f g f ) = δx = Φ ( tt, ) δx = Λ ( t) Φ( tt, ) δx r r x x0 r r State Relationship Matrix

29 Dynamical Model Generalized Dynamical Model for Each S/C: Control Input I r P P 2 s ( t) = µ N PP PP s j 2 j P2 + µ P PP 2 s j s j 2 j 3 3 r j= 1, j 2, s r r r r ( PP ) ( PP ) f ( P ) s + + srp ( ) u t Gravity Terms Solar Radiation Pressure Assumptions: Chief S/C Evolves Along Natural Solution u = 0 (Nominal) Deputy S/C Evolves Along Non-Natural Solution u 0 c ( t) d ( t) 29

30 Chief S/C Motion: Natural Solutions Near L 1 and L 2 Halo Orbits Near L i Sun Lissajous Trajectories Near L i Sun 30

31 Controlled Deputy S/C Motion (Example 1): Formation Fixed in the Rotating Frame Deputy S/C Chief S/C C(t 1 ) D(t 1 ) C(t 4 ) D(t 4 ) C(t 2 ) D(t 2 ) C(t 3 ) D(t 3 ) ρ = ρŷ 31

32 Controlled Deputy S/C Motion (Example 2): Formation Fixed in the Inertial Frame D(t 3 ) D(t 2 ) C(t 3 ) C(t 2 ) D(t 4 ) D(t 1 ) Y [au] C(t 4 ) C(t 1 ) Deputy S/C Chief S/C ρ = ρyˆ 32 X [au]

33 MAXIM: APPLICATIONS OF IFL AND OFL 33

34 34 MAXIM Mission Sequence

35 MAXIM: THRUSTER ON-OFF SEQUENCE 35

36 Free Flyer Configuration FF 3 FF 2 FF 4 Hub FF 1 FF 5 FF 6 36

37 Radial Error wrt. Hub S/C r W r W Nominal Radial Vector in UVW Coordinates Actual Radial Vector in UVW Coordinates Thrusters off = 100,000 sec 37

38 Free Flyers UV-Plane Angular Drift (DEG) ν ν Actual Nominal ε ε Actual Nominal 38

39 Thrust Profile Thrusters Off Between t 1 = 1 day & t 2 = t ,000 sec. 3 mn 3 mn A O( 0.05 µ N ) O( 0.05 µ N ) B C 39

40 MAXIM: FORMATION RECONFIGURATION 40

41 Target Reconfiguration Target #2 Ẑ ˆv 2 ŵ 2 û 2 Hub (t 2 ): α = 0, δ = 0 ˆv 1 û 1 ŵ Detector 1 Target #1 Yˆ ˆX Hub (t 1 ): α = 0, δ = 0 41

42 Graphical Representation of Reconfiguration for Free Flyers INITIAL ORIENTATION OF UV-PLANE wˆ 1 Xˆ FINAL ORIENTATION OF UV-PLANE wˆ 2 Yˆ 42

43 Thrust to Reconfigure From α = 0 o to α = 90 o with δ = 0 o wˆ ˆ 1 X Target: wˆ 2 Yˆ Reconfiguration Time Increased to 7 days to reduce Detector S/C Control Thrust 43

44 Mission Specifications Hold periscope positions to within 15-µm Detector pointing accuracy arcminutes Periscope-Detector-Target alignment µas Phase 1 1 Target /week Phase 2 1 Target/ 3 weeks Hub inter. comm. port between detector & freeflyers Reconfiguration (Formation Slewing) Times: 1 Day for Phase 1 1 Week for Phase 2 Propulsion Formation Slewing 0.02 N (Hydrazine) Formation keeping 0.03 mn (PPTs) Frequent Reconfigurations 44

45 45 NATURAL FORMATIONS

46 Natural Formations: String of Pearls ŷ ˆx ẑ ẑ ˆx ŷ 46

47 Deployment into Torus (Remove Modes 1, 5, and 6) r ( ) = [ ] Deputy S/C r ( ) = [ ] m m/sec Origin = Chief S/C 47

48 Deployment into Natural Orbits (Remove Modes 1, 3, and 4) 3 Deputies r r ( ) = [ r0 ] ( ) = [ ] m m/sec Origin = Chief S/C 48

49 49 SPHERICAL FORMATIONS

50 OFL Controlled Response of Deputy S/C Radial Distance Tracking 1 2 ( ) 3 ( ) u t Control Law (, ) H r r = r ( ) rˆ g r, r T r r r u t = r + r f r 2 r r r (, ) Geometric Approach: Radial inputs only T 1 g r r r r u( t) = r f r r r ( ) ( ) u 2 u 1 u 4 u 3 Chief Origin (Inside Sphere) 4 T r r r u t rg r r r r f r 2 r r ( ) = (, ) + 3 ( ) 50

51 OFL Controlled Response of Deputy S/C Radial Distance + Rotation Rate Tracking * r = 5 km 1 rev / 1 day 1 rev / 6 hrs Chief Origin (Inside Sphere) 51

52 Impact Commanded Rotation Rate on Cost 1 rev /24 hrs 0.19 mn 1 rev /12 hrs 0.76 mn 1 rev / 6 hrs 6.40 mn 1 rev / 1 hrs mn m s = 700 kg 52

53 53 ASPHERICAL FORMATIONS

54 Parameterization of Parabolic Formation Zenith h p u p a p 3 ( ) eˆ focal line ν Deputy S/C (D i) ê 1 E : inertially fixed focal frame I : inertially fixed ephemeris frame r = xe ˆ + ye ˆ + ze ˆ CD i E x = a u / h cosν p p p y = a u / h sinν p p p z = u + q p 54 Nadir Chief S/C (C) q Transform State from Focal to Ephemeris Frame CDi CDi T re I E ri = E CD { C } I CD r r i i E I

55 Controller Development Desired Response for u, q, and ν : g ( u, u * ) = u 2ω ( u * 2 u ) ω ( u * u ) ( * * 2, ) = 2ω ( q ) ωn ( *) * * gν ( ν) = ν kωn ( ν ν ) u p p p n p p n p p g u u q q q q q p p n δu, δq critically damped δθ exponential decay Solve for Control Law: 2h 2h 2h 2 2 x y gu ( uu, ) 2 ( x + y + x f x + y f y) a a u a x 2h 2h 2h x y 1 u g qq, x y x f y f f a a = a uz x y 0 xx + yy xy yx ( x + y ) ( x + y ) g ν ( ν ) ( ) ( 2 2 ) ( )( ) ( y f x x f y) ( x y ) ( x + y ) 2 2 y q 2 x y z 55

56 OFL Controlled Parabolic Formation t days t days ê 3 56 Ẑ q = 10 km ν = 1 rev/day hp = 500 m ap = 500 m ˆX Phase I: u = 200 m p Phase II: u p Phase III: u p t 0 = 300 m /1 day = 500 m t hrs t hrs Yˆ t hrs

57 m s = 700 kg OFL Thrust Profile 57

58 New Maxim Pathfinder Science Phase #2 High Resolution (100 nas) 1 km 20,000 km 58

59 Maxim Configuration Example m s = 700 kg q = 19,500km ν = 1 rev/day u h a p p p = 500 m = 1 km = 1 km 59

60 60 NONLINEAR OPTIMAL CONTROL

61 Formulation N 1 N 1t ( ) ( ) φ( ) ( ) min J= φ x + Lt, xu, = x + Ltxu,, dt Subject to: x N i i i N i= 0 i= 0 t (,, ); Subject to ( 0) i+ 1 x = + F t x u x = x x i 1 i i i 0 Equivalent Representation as Augmented Nonlinear System: min ( ) ( ) N n 1 N φ( + N) F( t, x, u ) F ( t x u ) ( ) + (,, ) J = φ x + x t = x i+ 1 i i i i+ 1 = i i i xn 1( t = = + i+ 1) xn+ 1 ti Lti xu i i Subject to x 0 x 0 0 = i,, ; 61

62 Euler-Lagrange Optimality Conditions (Based on Calculus of Variations) H ( x ) T H T F Condition #1: T φ N i i λi = = λi+ 1 λn = 1 x i x i xn Hi T F i Condition #2: 0 = = λ i+ 1 ; i = 0,, N 1 u u i ( ) T = λ 1 F + t, x, u i i i i i F i F i Identify and from augmented linear system. x u i i i 62

63 Identification of Gradients From the Linearized Model Augmented Nonlinear System: Augmented Linear System: (,, ) ( ) ( 0) ( ) x f txu x x0 = ; = x 1,, xn 1 0 n+ Ltxu + 0 ( ) = ( ) ( ) + ( ) ( ) δx t A t δx t B t δu t At ( ) 0 ( ) B ( t) At = L x 0 03 = I 3 0 T 63

64 Solution to Linearized Equations ( ) =Φ (, ) + Φ(, ) ( ) ( ) 0 0 (, ) ( ) (, ); (, ) t t δxt tt δx tτ Bτ δu τ dτ Φ tt = At Φ tt Φ t t = I Relation to Gradients in E-L Optimality Conditions: F x i i+ 1 0 (, ) (, ) ( ) ( ) i+ 1 i+ 1 i i i+ 1 t t δx =Φ t t δx + Φ t τ B τ δu τ dτ i 64

65 Control Gradient for Impulsive Control (, ) δx = Φ t t δx + i+ 1 i+ 1 i i ( )( t ) i 1, ti δ x i B + Vi =Φ + (, ) δ (, ) =Φ t t x +Φ t t B V i+ 1 i i i+ 1 i i F u i F u i = Φ ( ) t t B, i+ 1 i 65

66 Control Gradient for Constant Thrust Arcs t i+ 1 δx =Φ ( t, t ) δx + Φ( t, τ) B( τ) dτ δu i i+ 1 i+ 1 i i i+ 1 i t 66 ti+ 1 F 1 (, 1 ) (, =Φ ti+ ti Φ τ ti) B( τ) dτ u i ti F u i Equations to Integrate Numerically f( txu,, ) (,, ) ( ) Φ ( tt, i ) 1 ( tt, ) B x x n+ 1 Ltxu = Φ ( tt, i ) At * Φ ( tt, i ) Φ i

67 Numerical Solution Process: Global Approach (1) Input x, t, and initial guess for u ; ( i = 0,1,..., N 1) 0 N (2)1-Scalar Equation to Optimize in 3 Use optimizer to identify optimal u i ( N ) i 1 Control Variables Hi given. u During each iteration of the optimizer, the following steps are followed: ( a) Sequence xi forward and store; i = 1,, N 1 ( b) Evaluate cost functional, J = φ x ( x ) ( ) T N N ( c) Evaluate φ φ λ N = = 1 x N xn ( d) Sequence Hi λi backward and compute ; i = N 1,,1 u ( e) J and H u i i used in next update of control input. N i i 67

68 Formulation N 1 N 1t ( ) ( ) φ( ) ( ) min J= φ x + Lt, xu, = x + Ltxu,, dt Subject to: x N i i i N i= 0 i= 0 t (,, ); Subject to ( 0) i+ 1 x = + F t x u x = x x i 1 i i i 0 Equivalent Representation as Augmented Nonlinear System: min ( ) ( ) N n 1 N φ( + N) F( t, x, u ) F ( t x u ) ( ) + (,, ) J = φ x + x t = x i+ 1 i i i i+ 1 = i i i xn 1( t = = + i+ 1) xn+ 1 ti Lti xu i i Subject to x 0 x 0 0 = i,, ; 68

69 Impulsive Optimal Control Minimize State Error with End-State Weighting 1 i 1 1 T N + 1 T min J = ( xn xn) W( xn xn) + ( x x ) Q( x x ) d 2 2 i= 0 t t i 69

70 70 State Corrector vs. Nonlinear Optimal Control: Cost Function

71 71 State Corrector vs. Nonlinear Optimal Control: Impulsive Maneuver Differences

72 Impulsive Radial Optimal Control N 1 t i min J = q ( r r ) dt 2 i= 0 t i 72

73 73 Radial Optimal Control: Maneuver History

74 74 State Corrector vs. Nonlinear Optimal Control: Cost Function

75 75 RANGE + RANGE RATE TARGETER

76 76 Comparison of Range and State Targeters

77 Range Targeter: Spatial Behavior of Corrected Solution Deputy S/C Path Chief S/C at Origin 77