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Thruster failure recovery strategies for libration point missions Maksim Shirobokov Keldysh Institute of Applied Mathematics Moscow Institute of Physics and Technology Sergey Trofimov Keldysh Institute of Applied Mathematics Moscow Institute of Physics and Technology Contents • Motivation • Problem statement • Theory background • Test case: Sun-Earth L2, planar periodic orbits • Conclusion 2/21 Missions to libration points • Successfully accomplished missions: – ISEE-3, WIND, SOHO, ACE, Genesis • Promising near-future projects – Deep Space Climate Observatory (NASA) – LISA Pathfinder (ESA/NASA) – Spektr-RG (Roscosmos/ESA) 3/21 Features related to periodic motion around collinear libration points High instability of motion requires the application of station-keeping techniques and their essential components: • Accurate trajectory determination • Regular control-law updates In average, 2-12 m/s per year is required Thus, any possible thruster (or communication) failure1 threatens a mission and can lead to a significant deviation of the spacecraft from the nominal periodic orbit 1The largest percentage of all fail occurrences relating to the control system falls on thruster failure, see Tafazoli [2009] “A Study of On-Orbit Spacecraft Failures”, Acta 4/21 Astronautica Thsuter failure issue If a thruster fails, the control is allocated to a redundant set of thrusters: • attitude control thrusters • a backup orbital thruster Most of publications are related only to collision avoidance during rendezvous and docking. The problem of libration point mission recovery has not been deeply studied yet 5/21 Problem statement Basic assumptions: • the main orbit control thruster failed and produces no thrust • the planned correction maneuver is not performed on time • with some delay, a redundant set of thrusters is used Transfer to the nominal periodic orbit may appear to be too expensive: • unstable environment leads to fast orbit decay • redundant thruster has usually less fuel than the main one Therefore, not enough fuel is left to perform station-keeping maneuvers during the planned mission lifetime 6/21 Thruster failure recovery strategies Two strategies are considered: • periodic orbit targeting (POT) • stable manifold targeting (SMT) In both cases, the aim is the same—to find the “cheapestto-get” periodic orbit for different values of correction maneuver delay (the time passed since the moment of unsuccessful correction maneuver) 7/21 Circular restricted three-body problem The planar circular restricted three-body problem (CR3BP) is studied: • a spacecraft of negligible mass moves under the gravitational influence of two masses m 1 and m 2 • the spacecraft is supposed to move in the orbital plane of the primaries Note: the proposed recovery strategies can be applied to the spatial case (for example, for halo orbits) 8/21 Reference frame Mass parameter m2 m1 m 2 Non-dimensional units: m1 1 xm1 m2 xm 2 1 0 1 For the Sun-(Earth+Moon) system 3.03939 10 6 9/21 Equations of motion In rotating frame x 2 y U x , y 2 x U y where U x, y x y 2 2 2 1 r1 r2 1 , 2 is the so called effective potential; U x and U y are the partial derivatives of with respect to the position variables. The distances between the spacecraft and the primaries equal r1 x 2 y 2 r2 x 1 y 2 2 10/21 Libration points Equilibrium (libration) points can be found from the equations Ux Uy 0 Collinear libration points x L 1 rH 1 x L 1 rH 2 xL 1 3 5 12 rH 3 1 3 1 3 2 rH rH 2 26 9 28 3 rH x L 0.989987 3 x L 1.010074 12 1 rH 9 23 49 4 Sun-(Earth+Moon) system 2 3 x L 1.000001 3 1 3 11/21 Richardson’s third-order approximation of periodic orbits The third-order approximation of periodic orbits in normalized variables x x x L and y y expressed as follows: 2 2 3 x a 2 1 A x A x c os 1 a 2 3 A x cos 2 1 a 3 1 A x cos 3 1 Aexp u x ,st exp t 2 3 y kA x sin 1 b 21 A x sin 2 1 b31 A x sin 3 1 Aexp u y ,st exp t where k p 2 c2 1 2 p 2 c 2 2 p 0 9 c 8c2 2 2 a 21 , a 23 , a 31 , b21 , b31 , s1 2 1 p t c2 x L 1 1 s1 A x 2 3 1 x L some constants 12/21 3 Periodic orbit targeting strategy J y v 1 v 2 m in y A x , T1 , T 2 , Reference periodic orbit Backup periodic orbit 13/21 Gain in delta-v for POT strategy t d , reference orbit periods 14/21 Change in amplitude for POT strategy t d , reference orbit periods 15/21 Stable manifold targeting strategy J Reference periodic orbit Stable manifold y v 1 v 2 m in y A x , T1 , T 2 , , t 16/21 Gain in delta-v for SMT strategy t d , reference orbit periods 17/21 Change in amplitude for SMT strategy t d , reference orbit periods 18/21 Conclusion • Two recovery strategies in case of possible thruster failure—periodic orbit targeting and stable manifold targeting—are proposed for collinear libration point missions • The proposed approach reduces delta-v spent by the redundant set of thrusters and increases the lifetime of the spacecraft 19/21 Future work • Targeting to periodic orbits with larger amplitudes requires higher-order approximations of these orbits • Different positions of the unsuccessful correction maneuver may bring to different results 20/21 Thank you!