OPTIMAL COOPERATIVE DEPLOYMENT OF A TWO-SATELLITE FORMATION INTO A HIGHLY ELLIPTIC ORBIT

1. AAS/AIAA AstrodynamicsSpecialistConference GIRDWOOD, ALASKA July 31 - August 4, 2011 OPTIMAL COOPERATIVE DEPLOYMENT OF A TWO-SATELLITE FORMATION INTO A HIGHLY ELLIPTIC ORBIT Alessandro Zavoli Guido Colasurdo La Sapienza University – Roma Francesco SimeoniLorenzo Casalino Politecnicodi Torino - Torino

2. Introduction / Aim This work has been inspired and sponsored by CNES, in the framework of the (now cancelled) Simbol-X project A two-satellite formation creates a new-generation distributed X-ray telescope capable to observe even very far object (such as black holes) The formation operates in HEO to reduce the mission costs

3. Statement of the Problem

4. IndirectOptimization Goal: Motivi della scelta dell’indiretto Hig precision at a reasonable effort Accademic interest

5. OptimalControlTheoryApplication Multi Point Boundary Value Problem (MPBVP)

6. Multi-BoundaryApproach Pre-assignment of the burn structure allows an easy handling of the bang-bang control This peculiar management reduces the sensitivity of the BVP to the initial values in comparison to the standard approach

7. EquationsofMotions • Each S/C is view as a point-mass object; • state variables: • Position: polar coordinate in the ECI frame; • Velocity: radial, eastward and northward components in a topocentric rotating frame; • Mass. • T = Thrustap = Perturbingacceleration

8. Phasing Constraint:Main Idea The phase condition involving the intersatellite distance can be seen as a time constraint between the two satellites at the final apogee. Time Delay = Distance / Apogee Velocity The two S/C mustbe in the sameplace, in different moment • Chaser-Target approach • Cooperative approach

9. Phasing Constraint: Chaser-TargetApproach • Two single-satellite MPBVP are solved in sequence: • Less CPU expensive – Less performing free-time optimization SAT 2 Position & time at apogee SAT 2 Overall Solution SAT 1 Position & time at apogee fixed-time optimization SAT 1

10. Phasing Constraint: Chaser-TargetApproach • Two single-satellite MPBVP are solved in sequence: • Boundary Conditions at the arrival point SAT 2 SAT 1

11. Phasing Constraint: Cooperative Approach • A two-satellite MPBVP is solved: • More CPU Expensive – More Performing

12. Results: Keplerian Environment Chaser/Target Approach Cooperative Approach CollisionAvoidance

13. MissionSpecifics Sat 1 / Sat 2 InitialOrbit FinalOrbit Constraints phasingdistancesafetydistance 10 Km >1 Km

14. Chaser/Target Approach Optimal trajectories for the 3.5 revolutions mission in the Keplerian environment ( T = 1N, ΔV = 5m/s). SAT 2: Time-free deployment • the apogee duty is equally split among all burns; • the perigee burn is delayed as much as possible. SAT 1:Time-fix deployment • unequal split to reach the phasing condition; • the perigee burn is performed sooner to reduce the orbital period and to recover time.

15. Cooperative Approach • propellant savings = 1.8 kg / 0.6 kg for the 2.5 / 3.5 revs. mission

16. ResultSummary Finalvaluesof the total mass for different thrust strategies and deployment approaches ( T = 1N, ΔV = 5m/s). Sub-optimal results are marked with a star.

17. CollisionAvoidance • performance increment high thrustlevel & lowerseparationΔV • safetydistanceviolation • Collisionmayoccurduring the last apogeeburn • when the two S/C are almostphased • AAP solutionisnaturallycollison-free APA solutionhastobefixed

18. CollisionAvoidance • In case of a safety distance constraint violation, a recovering process is employed • a purposeful condition is set on the radius of penultimate apogee of the two S/C • the corresponding adjointsλr have a free jump; • same magnitude but different sign: • plus and minus sign in the subscripts refers to the value just after and before the boundary • Even though it is not “optimal”, • it is really simple to implement and effective Intersatellite distanceforconstrained and uncostrainedthree-burn A-P-A strategy (T= 8 N, ΔV = 0,5 m/s)

19. Optimization in aPerturbedEnvironment • PerturbingForces • AnalysisofPerurnationEffects • NumericalResults

20. PerturbingForces For a realistic simulation, a more complex environment must be considered • Now, perturbation acceleration ap must be considered, to account for: • Earth asphericity (8x8 model based on EGM2008); • Lunisolar attraction (DE405 JPL ephemerides for Moon and Sun positions). • Solar radiation pressure has been neglected: • it is some order of magnitude smaller. Is it possible to exploit these forces to save propellant?

21. AnalysisofPerturbationEffects Lunisolar attraction and Earth asphericity can modify significantly the S/C trajectory

22. NumericalResults Spacecraft final masses for 3.5 revolution transfers in keplerian and perturbed environments (T = 8 N, ΔV = 0.5 m/s)

23. Conclusions • The optimal deployment of a two-spacecraft formation in HEO has been successfully carried out

24. Conclusions • Future Developments

25. ThankYouForYourAttention

26. Phasing Constraint: • Boundary Conditions at the arrival point Chaser Target Approach Cooperative Approach SAT 2 SAT 1