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Optimal Low-Thrust Deorbiting of Passively Stabilized LEO Satellites

Optimal Low-Thrust Deorbiting of Passively Stabilized LEO Satellites. Sergey Trofimov Keldysh Institute of Applied Mathematics, RAS Moscow Institute of Physics and Technology Michael Ovchinnikov Keldysh Institute of Applied Mathematics, RAS. Contents.

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Optimal Low-Thrust Deorbiting of Passively Stabilized LEO Satellites

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  1. Optimal Low-Thrust Deorbiting ofPassively Stabilized LEO Satellites Sergey Trofimov Keldysh Institute of Applied Mathematics, RAS Moscow Institute of Physics and Technology Michael Ovchinnikov Keldysh Institute of Applied Mathematics, RAS

  2. Contents • Deorbiting of nano- and picosatellites • Orbital control of passively stabilized satellites • Two-time-scale approach to low-thrust optimization • Reduction to the nonlinear programming problem • Numerical solution and results • Conclusions and future work 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  3. Deorbiting of nano- and picosatellites Propellantless propulsion • Drag sails – only for orbits with altitudes < 800 km • Electrodynamic tethers – dynamic instability issues Conventional propulsion • Chemical propulsion – large thruster + propellant mass (low specific impulse) • Electric propulsion – large power consumption Electrospray propulsion is a promising solution: • Specific impulse > 2500 s • Power 1-5 W • Thrust 0.1-5 mN Courtesy: MIT Space Propulsion Lab 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  4. Passive stabilization and orbital control Kinds of passive stabilization techniques: • Passive magnetic stabilization (PMS) • Spin stabilization (SS) These techniques • do not require massive and bulky actuators • are well suited for nano- and picosatellites but • provide one-axis stabilization • at most two orbital control thrusters can be installed along the sole stabilized axis 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  5. Two-time-scale optimization Two-time-scale approach to low-thrust optimization: • Over one orbit, five slowly changing orbital elements are considered constant; optimal control is obtained (in parametric form) by using Pontryagin’s maximum principle • Discrete slow-time-scale problem is formulated as a nonlinear programming problem (NLP) with respect to unknown optimal control parameters 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  6. Gaussian variational equations where are respectively the thrust and perturbing accelerations, , u is the argument of latitude We use the averaged equations (i.e., for mean elements) with J2 + no drag environment model For mean semimajor axis (all the overbars are omitted): 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  7. Thrust direction in PMS and SS cases Suppose two oppositely directed thrusters are installed onboard the spacecraft along the sole stabilized axis • In the case of PMS, the axial dipole model of the geomagnetic field is used • In the case of SS, the spin axis direction is defined in inertial space by two slowly changing spherical angles Spin axis direction in the ascending node: 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  8. Two modes of deorbiting Circular mode: • The orbit keeps being near-circular, with a gradually decreasing radius • Both thrusters are used in the deorbiting operation Elliptic mode: • The perigee distance is decreased while the apogee distance is almost not changed • Just one thruster is used for deorbiting 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  9. Fast-time-scale optimal control In the near-circular orbit approximation (i.e., with on the right side of GVE): From Pontryagin’s maximum principle: • optimal control is of a bang-bang type • for the k-th orbit, the central points of the two thrust arcs are defined by formula (PMS) or (SS) • the thrust arc lengths are to be determined in the case of PMS in the case of SS 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  10. Reduction to the NLP problem • Objective function • Equality constraint • Bound constraint in the case of PMS in the case of SS 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  11. Auxiliary expressions Fuel depletion: Change in inclination: RAAN drift: Tsiolkovsky’s rocket equation in the case of PMS in the case of SS 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  12. Circular deorbiting: PMS case N=700 N=800 N=900 N=1000 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  13. Circular deorbiting: SS case N=700 N=800 N=900 N=1000 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  14. Deorbit maneuver performance Case of spin stabilization (spacecraft’s spin axis points towards the Sun) Case of passive magnetic stabilization Spacecraft and thruster parameters: m0= 5 kg, Isp = 2500 s, Tmax = 1 mN Orbit: a0= R + 900 km, e = 0, i0= 51.6,0 = 30,af = R +300 km Initial Sun’s ecliptic longitude: 0 = 90 For reference: Hohmann transfer requires 330.9 m/s 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  15. Performance sensitivity to changes in orbit plane orientation of SS spacecraft Spacecraft’s spin axis points towards the Sun Spacecraft and thruster parameters: m0= 5 kg, Isp = 2500 s, Tmax = 1 mN Orbit: a0= R + 900 km, e = 0, i0= 51.6,af = R +300 km 0 = 90, N = 900 (left table) and 0 = 30, N = 800 (right table) 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  16. Verificationof models used The actual altitude evolution is in close agreement with the results of solving the NLP problem, except for the last stage when the drag force becomes dominant 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  17. Elliptic deorbiting: SS case The optimal control obtained earlier for the circular mode appears to be quasi-optimal for the elliptic mode as well (with one thrust arc dropped): • the eccentricity of a low-Earth orbit cannot exceed 0.05  near-circular approximation has lower accuracy but is still valid • at the start of deorbiting, the center of the sole thrust arc is at the apogee; the optimal control is the same since Orbit: a0= R + 900 km, i0= 51.6,0 = 30,r, f = R +200 km N = 750 V = 326.2 m/s, mprop = 66.0 g 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  18. Conclusions and future work • It is possible to deorbit passively stabilized satellites using a propulsion system such as the iEPS • The increase in maneuver cost (in comparison with the full attitude controllability case) is not dramatic (15-50%) and depends on the passive stabilization technique used • Optimal control problem is analytically reduced to the nonlinear programming problem • For the same deorbit time, the elliptic mode of deorbiting requires about 60% less fuel (besides, one of the thrusters is no longer needed) • Influence of attitude stabilization errors on the maneuver performance is worth being analyzed 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

  19. Acknowledgments • Russian Ministry of Science and Education, Agreement No. 8182 of July 27, 2012 • Russian Foundation for Basic Research (RFBR), Grant No. 13-01-00665 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

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