Formations Near The Libration Points: Design Strategies Using Natural And Non-Natural Arcs
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The document discusses formation strategies for spacecraft near libration points, employing both natural and non-natural trajectory arcs. It covers various control methodologies including continuous and discrete control techniques, and explores targeter methods for impulsive formation keeping. The document emphasizes the hybridization of natural and non-natural motions to enhance formation keeping and targeter effectiveness.
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X [au]
Y[au]
Controlled Deputy S/C Motion (Example 2):
Formation Fixed in the Inertial Frame
ˆYρ ρ=Chief S/C
Deputy S/C
C(t1)
C(t2)C(t3)
C(t4)
D(t1)
D(t2)D(t3)
D(t4)](https://image.slidesharecdn.com/gsfc2004-140321091736-phpapp02/85/Formations-Near-The-Libration-Points-Design-Strategies-Using-Natural-And-Non-Natural-Arcs-32-320.jpg)
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Deployment into Torus
(Remove Modes 1, 5, and 6)
Origin = Chief S/C
Deputy S/C
( ) [ ]
( ) [ ]
0 5 00 0 m
0 1 1 1 m/sec
r
r
=
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Deployment into Natural Orbits
(Remove Modes 1, 3, and 4)
Origin = Chief S/C
3 Deputies
( ) [ ]
( ) [ ]
00 0 0 m
0 1 1 1 m/sec
r r
r
=
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Formations Near The Libration Points: Design Strategies Using Natural And Non-Natural Arcs
- 1. 1 FORMATIONS NEAR THE LIBRATION POINTS: DESIGN STRATEGIES USING NATURAL AND NON-NATURAL ARCS K. C. Howell and B. G. Marchand
- 2. 2 Formations Near the Libration Points Moon 1L 2L ˆx ( )ˆ inertialX B θ ˆy Chief S/C Path (Lissajous Orbit) EPHEM = Sun + Earth + Moon Motion From Ephemeris w/ SRP CR3BP = Sun + Earth/Moon barycenter Motion Assumed Circular w/o SRP Deputy S/C (Orbit Chief Vehicle)
- 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. 5 State Targeter: Impulsive Control Law Formulation 1V∆ 2V∆ 3V∆ Segment of Nominal Deputy Path 0vδ − 0vδ + 0V∆ 1rδ 2rδ 3rδ STM 1 1 1 1 k kk k k kk k k A Br r C Dv v V + + − − + + = + ∆ δ δ δ δ ( )1 1k k k k k kV B r A r v− − +∆= − −δ δ δ Segment of Chief S/C Path C(t1) C(t2) C(t3) D(t1) D(t2) D(t3)
- 6. 6 State Targeter: Radial Distance Error ( ) ˆ10 m Yρ = DistanceErrorRelativetoNominal(cm) Time (days) Max.Deviation Nominal Separation
- 7. 7 State Targeter: Achievable Accuracy MaximumDeviationfromNominal(cm) Formation Distance (meters) ˆr rY=
- 9. 9 Range + Range Rate Targeter Chief S/C Deputy S/C r 1V∆ 2V∆ 3V∆ ( ) 1/2 Range + Range Rate Constraint: T f f f T f f f f f r r r g r rr r = = ( ) ( ) ( ) STM 0 0 0 0 0 0 State Relationship Matrix , ,f r r rr r d g t t x t t v V t r r r r − ∂ ∂ ∂ ∂ = Φ = Φ + ∆∂ ∂ Λ ∂ ∂ δ δ δ
- 10. 10 Comparison of Range and State Targeters Chief S/C at Origin Deputy S/C Path
- 11. 11 DESIGN OF NON-NATURAL FORMATIONS USING NATURAL SOLUTION ARCS
- 12. 12 CR3BP Analysis of Phase Space Eigenstructure Near Halo Orbit ( ) ( ) ( ) ( ) ( )1 ,0 0 0 0Jt x t x E t e E x− = Φ = δ δ δ Reference Halo Orbit Chief S/C Deputy S/C ( ) ( ) ( ) ( ) ( ) ( ) 6 6 1 1 :Solution to Variational Eqn. in terms of Floquet Modes j j j j jx t x t c t E t ce t t = = = = =∑ ∑δ δ Mode 1 1-D Unstable Subspace Mode 2 1-D Stable Subspace Modes (3,4) and (5,6) 4-D Center Subspace Floquet Modes
- 13. 13 Natural Solutions: Periodic Halo Orbits Near Libration Points ˆx ˆz ˆy Earth ˆx ˆz Sun
- 14. 14 Natural Formations: Quasi-Periodic Relative Orbits → 2-D Torus ˆy ˆx ˆz Chief S/C Centered View (RLP Frame) ˆy ˆx ˆx ˆz ˆy ˆz ˆz ˆx ˆy
- 15. 15 Floquet Controller (Remove Unstable + 2 Center Modes) ( ) 1 2 5 6 3 1 3 4 2 5 6 3 Remove Modes 1, 3, and 4: 0r r r v v v x x x x x x x x x IV − + + −∆ δ δ δα δ δ δ δ δ δ ( ) 1 2 3 4 3 1 5 6 2 3 4 3 Remove Modes 1, 5, and 6: 0r r r v v v x x x x x x x x x IV − + + −∆ δ δ δα δ δ δ δ δ δ ( )3 2,3,43 or 2,5 6 6 1 , 0 1 :Find that removes undesired response modes j j j j j j V x I x V δ α δ = = = + ∆= + ∆ ∑ ∑
- 16. 16 Sample Deployment into Relative Orbits: 1-∆V at Injection Origin = Chief S/C Deputy S/C Quasi-Periodic Origin = Chief S/C 3 Deputies “Periodic” 1800 days
- 17. 17 Natural Formations: Nearly Periodic and Drifting Relative Orbits Chief S/C @ Origin 1800 days
- 18. 18 Expansion of Drifting Vertical Orbit Origin = Chief S/C ( )0r ( )fr t 18,000 days = 100 Revolutions
- 19. 19 Transitioning Natural Motions into Non-Natural Arcs: Targeter Approach STEP 1: Identify a suitable initial guess Target Orbital Drift Control
- 20. 20 Application of Two Level Corrector ( )qTΓ ( )0r t ( )mr t ( ) ( )0mr t r t= ( )qV T∆ STEP 2: Apply 2-level corrector (Howell and Wilson:1996) w/ end-state constraint STEP 3: Shift converged patch states forward by 1 period STEP 4: Reconverge Solution
- 21. 21 Drift Controlled Vertical Orbit (6 Revs) ( ) 3 5 m/sec 1 maneuver/year V∆ = −
- 22. 22 Geometry of Natural Solutions in the Ephemeris Model w/ SRP Inertial Frame Perspective: Rotating Frame Perspective:
- 23. 23 Transitioning Natural Motions into Non-Natural Arcs: IFL Example (1) Consider 1st 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 S/C @ Origin Sphere for Visualization Only
- 24. 24 Hybrid Control: Natural Motions + Continuous Control ½ Period Natural Arc ½ Period IFL Control
- 25. 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.
- 26. 26 BACKUP SLIDES
- 28. 28 ( ) 1/2 Range + Range Rate Constraint: T f f f T f f f f f r r r g r rr r = = ( ) ( ) ( ) ( ) STM * 0 0 0 0 0 0 State Relationship Matrix , ,f f f r r g x r r d g g g x t t x t t x r rx r t x r δ δ δ ∂ ∂ ∂ ∂ ∂ ∂ = − = = Φ = Φ ∂ ∂∂ ∂ ∂ Λ ∂ * 0 0 First Order Approximation: f f g g g x x δ ∂ = + ∂ Radial Targeter: Control Law Formulation * * Desired Range + Range Rate: 0 f f r g =
- 29. 29 Dynamical Model ( ) ( ) ( ) ( ) ( ) 2 22 2 2 3 3 1, 2, s j j s js s j j s P P P P PP N P PP P P P P P sr PI j j s p r r u tr r r r ft µ µ = ≠ − + − = + +∑ Gravity Terms Solar Radiation Pressure Control InputGeneralized Dynamical Model for Each S/C: ( ) ( ) Assumptions: Chief S/C Evolves Along Natural Solution 0 (Nominal) Deputy S/C Evolves Along Non-Natural Solution 0 c d u t u t → ∴ = → ∴ ≠
- 30. 30 Chief S/C Motion: Natural Solutions Near L1 and L2 “Halo”OrbitsNearLi LissajousTrajectoriesNearLi Sun Sun
- 31. 31 Controlled Deputy S/C Motion (Example 1): Formation Fixed in the Rotating Frame Chief S/C Deputy S/C ˆyρ ρ= C(t1) C(t2) C(t3) C(t4) D(t1) D(t2) D(t3) D(t4)
- 32. 32 X [au] Y[au] Controlled Deputy S/C Motion (Example 2): Formation Fixed in the Inertial Frame ˆYρ ρ=Chief S/C Deputy S/C C(t1) C(t2)C(t3) C(t4) D(t1) D(t2)D(t3) D(t4)
- 33. 33 MAXIM: APPLICATIONS OF IFL AND OFL
- 36. 36 Free Flyer Configuration FF2 FF1 FF3 FF4 FF5 FF6 Hub
- 37. 37 Radial Error wrt. Hub S/C Thrusters off = 100,000 sec W Wr r− Nominal Radial Vector in UVW Coordinates Actual Radial Vector in UVW Coordinates
- 38. 38 Free Flyers UV-Plane Angular Drift (DEG) ν ν− ε ε− Nominal Actual Nominal Actual
- 39. 39 Thrust Profile Thrusters Off Between t1 = 1 day & t2 = t1 + 100,000 sec. A B C ( )O 0.05 Nµ ( )O 0.05 Nµ 3 mN≈ 3 mN≈
- 41. 41 Target Reconfiguration ˆX ˆZ ˆY Detector Target #11 ˆw 1 ˆu 2 ˆw 2 ˆu Target #2 1 ˆv 2 ˆv Hub (t1): α = 0°, δ = 0° Hub (t2): α = 0°, δ = 0°
- 42. 42 Graphical Representation of Reconfiguration for Free Flyers 1 ˆˆ ||w X 2 ˆ ˆ||w Y INITIAL ORIENTATION OF UV-PLANE FINAL ORIENTATION OF UV-PLANE
- 43. 43 Thrust to Reconfigure From α = 0o to α = 90o with δ = 0o 1 | ˆˆ |w X 2 ˆ ˆTarget: ||w Y Reconfiguration Time Increased to 7 days to reduce Detector S/C Control Thrust
- 44. 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
- 46. 46 Natural Formations: String of Pearls ˆx ˆy ˆz ˆz ˆy ˆx
- 47. 47 Deployment into Torus (Remove Modes 1, 5, and 6) Origin = Chief S/C Deputy S/C ( ) [ ] ( ) [ ] 0 5 00 0 m 0 1 1 1 m/sec r r = = −
- 48. 48 Deployment into Natural Orbits (Remove Modes 1, 3, and 4) Origin = Chief S/C 3 Deputies ( ) [ ] ( ) [ ] 00 0 0 m 0 1 1 1 m/sec r r r = = −
- 50. 50 OFL Controlled Response of Deputy S/C Radial Distance Tracking 2 ( ) ( ) ( )2 , Tg r r r r r u t r r f r r rr = − + − ∆ 3 4 ( ) ( ) ( )2 2 ,1 2 Tg r r r r u t r f r r r = − − ∆ ( ) ( ) ( )2 , 3 T r r r u t rg r r r r f r rr = − − + − ∆ Control Law ( ) ( ), ˆ H r r u t r r = Geometric Approach: Radial inputs only1 1u 3u 2u 4u Chief S/C @ Origin (Inside Sphere)
- 51. 51 OFL Controlled Response of Deputy S/C Radial Distance + Rotation Rate Tracking * 5 kmr = 1 rev / 6 hrs 1 rev / 1 day Chief S/C @ Origin (Inside Sphere)
- 52. 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 106.50 mN → → → → 700 kgsm =
- 54. 54 Parameterization of Parabolic Formation pa ph pu ν ( )3 ˆ focal linee 1 ˆe Nadir q 1 2 3 ˆ ˆ ˆiCD Er xe ye ze= + + Chief S/C (C) iDeputy S/C (D ) { } i i i i CD CD TI EE I CD CDE I E I r r C r r = : inertially fixed focal frame : inertially fixed ephemeris frame E I / cos / sin p p p p p p p x a u h y a u h z u q ν ν = = = + Zenith Transform State from Focal to Ephemeris Frame
- 55. 55 Controller Development ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 0 , 2 2 2 1 , 0 2 u x y x y q x y z z h h h x y g u u x y x f y f a a au h h h x y u g q q x y x f y f f a a a u x y xx yy xy yx gx y x y x y ν ν − + + ∆ + ∆ − − = + + + ∆ + ∆ − ∆ + −− + ++ + + ( ) ( )2 2 x yy f x f x y ∆ − ∆ + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * 2 * * * * * 2 * , 2 , 2 u p p p n p p n p p q p p n n n g u g u u u u u u g u u q q q q k q ν ν ν ω ν ω ω ω ω ν = − − − − = − − − − =− − Desired Response for , , and :u q ν Solve for Control Law: , critically dampedu qδ δ → exponential decayδθ →
- 56. 56 OFL Controlled Parabolic Formation ˆX ˆZ ˆY 0t 0 1.6 hrst + 0 3.8 hrst + 0 8 hrst + 0 5 dayst + 0 6 dayst + 3 ˆe 10 km 1 rev/day 500 m 500 m Phase I: 200 m Phase II: 300 m /1 day Phase III: 500 m p p p p p q h a u u u ν = = = = = = =
- 57. 57 OFL Thrust Profile 700 kgsm =
- 58. 58 New Maxim Pathfinder Science Phase #2 High Resolution (100 nas) 20,000 km 1 km http://maxim.gsfc.nasa.gov/documents/SPIE-2002/spie2002.ppt
- 59. 59 Maxim Configuration Example 19,500km 1 rev/day 500 m 1 km 1 km p p p q u h a ν = = = = = 700 kgsm =
- 61. 61 Formulation ( ) ( ) ( ) ( ) 11 1 0 0 min , , , , i i tN N N i i i N i i t J x L t x u x L t x u d tφ φ +− − = = =+ =+∑ ∑ ∫ Subject to: ( ) ( )1 0, , ; Subject to 0i i i ix F t x u x x+ = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 0 0 min , , , , ; , , Subject to 0 N n N N i i i i i i i i n i n i i i i J x x t x x F t x u x F t x u x t x t L t x u x x φ φ+ + + + + + = + = = = = + = Equivalent Representation as Augmented Nonlinear System:
- 62. 62 Euler-Lagrange Optimality Conditions (Based on Calculus of Variations) ( ) 1 1 Condition #1: 1 Condition #2: 0 ; 0, , 1 NT T Ti i i N i N Ti i i i i i i xH x x F x F N u H i u φ λ λ λ λ + + ∂ ∂ ∂ ∂ ∂ = = → = ∂ ∂ ∂ ∂ = = = − ∂ ( )1 , ,T i i i i iH F t x uλ += Identify and from augmented linear system.i i i i F F x u ∂ ∂ ∂ ∂
- 63. 63 Identification of Gradients From the Linearized Model ( ) ( ) ( ) ( ) ( )x t A t x t B t u tδ δ δ= + ( ) ( ) 0 0 A t A t L x = ∂ ∂ ( ) 3 3 0 0T B t I = ( ) ( ) ( ) ( ) 0 11 0, , ; 0 0, , nn x xf t x ux xL t x ux ++ = = Augmented Nonlinear System: Augmented Linear System:
- 64. 64 Solution to Linearized Equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 7 , , , , ; , t t x t t t x t B u d t t A t t t t t I δ δ τ τ δ τ τ=Φ + Φ Φ = Φ Φ = ∫ i F x ∂ ∂ Relation to Gradients in E-L Optimality Conditions: ( ) ( ) ( ) ( ) 1 1 1 1, , i i t i i i i i t x t t x t B u dδ δ τ τ δ τ τ + + + +=Φ + Φ∫
- 65. 65 Control Gradient for Impulsive Control ( ) ( )( ) ( ) ( ) 1 1 1 1 1 , , , , i i i i i i i i i i i i i i x t t x t t x B V t t x t t B V δ δ δ δ − + + + − + − + + = Φ =Φ + ∆ = Φ + Φ ∆ i F u ∂ ∂ ( )1,i i i F t t B u + ∂ = Φ ∂
- 66. 66 Control Gradient for Constant Thrust Arcs ( ) ( ) ( ) 1 1 1 1, , i i t i i i i i i t x t t x t B d uδ δ τ τ τ δ + + + + =Φ + Φ ∫ i F u ∂ ∂ ( ) ( ) ( ) 1 1 1, , i i t i i i i t F t t t B d u τ τ τ + − + ∂ =Φ Φ ∂ ∫ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1* , , , , ,, ,, n i i ii x f t x u x L t x u A t t tt t t t Bt t + − = ΦΦ ΦΦ Equations to Integrate Numerically
- 67. 67 Numerical Solution Process: Global Approach ( ) 0(1) ; ( 0,1,..., 1) (2)1-Scalar Equation to Optimize in 3 1 Control Variables Use optimizer to identify optimal Input , , and initia given . During each iteratio l guess for n i i N i i i N N H u x u u t = − − ∂ ∂ ( ) ( ) ( ) Sequence forward and store of the optimizer, the following steps are followed: ( ) Evaluate cost functional, ( ) Evaluate 1 ; 1 ( , , 1 N NT N N N i N b J x x c x x a x i N d φ φ φ λ = … − = ∂ ∂ = = ∂ ∂ ( ) ) Sequence backward and compute ; 1, ,1 and used in next update of control input. i i i i i H i N u H e J u λ ∂ = − ∂ ∂ ∂
- 68. 68 Formulation ( ) ( ) ( ) ( ) 11 1 0 0 min , , , , i i tN N N i i i N i i t J x L t x u x L t x u d tφ φ +− − = = =+ =+∑ ∑ ∫ Subject to: ( ) ( )1 0, , ; Subject to 0i i i ix F t x u x x+ = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 0 0 min , , , , ; , , Subject to 0 N n N N i i i i i i i i n i n i i i i J x x t x x F t x u x F t x u x t x t L t x u x x φ φ+ + + + + + = + = = = = + = Equivalent Representation as Augmented Nonlinear System:
- 69. 69 Impulsive Optimal Control Minimize State Error with End-State Weighting ( ) ( ) ( ) ( ) 11 0 1 1 min 2 2 i i tN T T N N N N i t J x x W x x x x Q x x d t +− = = − − + − −∑ ∫
- 70. 70 State Corrector vs. Nonlinear Optimal Control: Cost Function
- 71. 71 State Corrector vs. Nonlinear Optimal Control: Impulsive Maneuver Differences
- 72. 72 Impulsive Radial Optimal Control ( ) 11 2 0 1 min 2 i i tN i t J q r r dt +− = = −∑ ∫
- 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. 77 Range Targeter: Spatial Behavior of Corrected Solution Chief S/C at Origin Deputy S/C Path