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Optimal Low-Thrust Deorbiting of Passively Stabilized LEO Satellites

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### 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

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

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

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

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

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

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

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

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

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

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

Circular deorbiting: PMS case

N=700

N=800

N=900

N=1000

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

Circular deorbiting: SS case

N=700

N=800

N=900

N=1000

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

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

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

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

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

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

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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