1. 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

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 2/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

3. Deorbiting of nano- and picosatellites Conventional propulsion Chemical propulsion – large thruster + propellant mass (low specific impulse) Electric propulsion – large power consumption 3/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Propellantless propulsion Drag sails – only for orbits with altitudes < 800 km Electrodynamic tethers – dynamic instability issues Electrospray propulsion is a promising solution: Specific impulse> 2500 s Power1-5 W Thrust0.1-5 mN Courtesy: MIT Space Propulsion Lab

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 4/19 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 5/19 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): 6/19 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 7/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Spin axis direction in the ascending node:

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 8/19 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 9/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China in the case of PMS in the case of SS

10. Reduction to the NLP problem 10/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Objective function Equality constraint Bound constraint in the case of PMS in the case of SS

11. Auxiliary expressions Fuel depletion: Change in inclination: RAAN drift: 11/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Tsiolkovsky’s rocket equation in the case of PMS in the case of SS

12. Circular deorbiting: PMS case 12/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China N=700 N=900 N=800 N=1000

13. Circular deorbiting: SS case 13/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China N=700 N=900 N=800 N=1000

14. Deorbit maneuver performance 14/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Deorbit time, revolutions Total  V budget, m/s Propellant mass, g 700700.2140.6 800571.5115.0 900525.9106.0 1000500.3100.3 Deorbit time, revolutions Total  V budget, m/s Propellant mass, g 700472.295.3 800425.385.9 900395.279.8 1000385.177.8 Case of passive magnetic stabilization Case of spin stabilization (spacecraft’s spin axis points towards the Sun) Orbit: a 0 = R  + 900 km, e = 0, i 0 = 51.6 ,  0 = 30 , a f = R  + 300 km Initial Sun’s ecliptic longitude: 0 = 90  For reference: Hohmann transfer requires 330.9 m/s Spacecraft and thruster parameters: m 0 = 5 kg, I sp = 2500 s, T max = 1 mN

15. Performance sensitivity to changes in orbit plane orientation of SS spacecraft 15/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Ecliptic longitude of the Sun at t=0, deg Total  V budget, m/s Propellant mass, g 0402.981.4 90425.385.9 180402.981.4 270425.385.9 RAAN at t=0, deg Total  V budget, m/s Propellant mass, g 30395.279.8 120387.878.4 210393.879.5 300500.3100.3 Spacecraft’s spin axis points towards the Sun Orbit: a 0 = R  + 900 km, e = 0, i 0 = 51.6 , a f = R  + 300 km Spacecraft and thruster parameters: m 0 = 5 kg, I sp = 2500 s, T max = 1 mN 0 = 90 , N = 900 (left table) and  0 = 30 , N = 800 (right table)

16. Verification of models used 16/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China 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

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 17/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Orbit: a 0 = R  + 900 km, i 0 = 51.6 ,  0 = 30 , r , f = R  + 200 km  V = 326.2 m/s, m prop = 66.0 g N = 750

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 18/19 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 19/19 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China