Optimal Low-Thrust Deorbiting of Passively Stabilized LEO Satellites Sergey Trofimov Keldysh Institute of Applied Mathematics, RAS Moscow Institute of

Optimal Low-Thrust Deorbiting ofPassively Stabilized LEO Satellites

Sergey TrofimovKeldysh Institute of Applied Mathematics, RAS

Moscow Institute of Physics and Technology

Michael OvchinnikovKeldysh 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

2/1964th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

Deorbiting of nano- and picosatellitesConventional propulsion• Chemical propulsion

– large thruster + propellant mass (low specific impulse)

• Electric propulsion– large power consumption


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• Power 1-5 W• Thrust 0.1-5 mN

Courtesy: MIT Space Propulsion Lab

for ion Electrospray Propulsion

System (iEPS) developed at MIT

Passive stabilization and orbital controlKinds 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 picosatellitesbut• provide one-axis stabilization• at most two orbital control thrusters can

be installed along the sole stabilized axis


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


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):


,d

udt

κ

A κ τ f τ f and

, , , ,a e i κ

1 2 22 sin 2r s

da du a e a p

du dt p r p

Thrust direction in PMS and SS casesSuppose 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


max max2 2

2sin sin

sin cos , ,1 3sin sin cos

i u

i ui u i

τ

cos sin

cos sinx y

y x

z

u u

u u

τ

and

, , ,T

x y z i σ σ

Spin axis direction in the ascending node:

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 deorbiting8/1964th International Astronautical Congress (IAC)

23-27 September 2013, Beijing, China

Fast-time-scale optimal controlIn 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


3

2 2

2 sin cos

1 3sin sin

da a i u

du i u

0e

32

cos siny x

da au u

du

in the case of PMS

in the case of SS

sin 0ku , ,tan k x k y ku 2 , 1...ku k N

Reduction to the NLP problem


3 2

1

Nk k

k k

a uJ

m

• Objective function

• Equality constraint

• Bound constraint

01

0N

k fk

a a a

max4T JV

0, 2ku

3

1max8sinh 3 sin sin

3k

k k k

k

a Ta i u

m

in the case of PMS

32max,

81 sink

k z k kk

a Ta u

m

in the case of SS

Auxiliary expressions

Fuel depletion:

Change in inclination:

RAAN drift:


11

0

exp kk k

sp

Vm m

I g

Tsiolkovsky’s rocket equation

00sin 1 sink

k

ai i

a

2max

, ,

4sink

k y k z k kk

a Ti u

m

in the case of PMS

in the case of SS

2

23 cosk kk

RJ i

a

Circular deorbiting: PMS case


N=700

N=900

N=800

N=1000

Circular deorbiting: SS case


N=700

N=900

N=800

N=1000

Deorbit maneuver performance


Deorbit time, revolutions

Total V budget, m/s

Propellant mass, g

700 700.2 140.6

800 571.5 115.0

900 525.9 106.0

1000 500.3 100.3

Deorbit time,

revolutions

Total V budget, m/s

Propellant mass, g

700 472.2 95.3

800 425.3 85.9

900 395.2 79.8

1000 385.1 77.8

Case of passive magnetic stabilization Case of spin stabilization (spacecraft’sspin axis points towards the Sun)

Orbit: a0 = R + 900 km, e = 0, i0 = 51.6, 0 = 30, af = R + 300 km Initial Sun’s ecliptic longitude: 0 = 90For reference: Hohmann transfer requires 330.9 m/s

Spacecraft and thruster parameters: m0 = 5 kg, Isp = 2500 s, Tmax = 1 mN

Performance sensitivity to changes in orbit plane orientation of SS spacecraft


Ecliptic longitude of the Sun at t=0, deg

Total V budget, m/s

Propellant mass, g

0 402.9 81.4

90 425.3 85.9

180 402.9 81.4

270 425.3 85.9

RAAN at t=0, deg

Total V budget, m/s

Propellant mass, g

30 395.2 79.8

120 387.8 78.4

210 393.8 79.5

300 500.3 100.3

Spacecraft’s spin axis points towards the Sun

Orbit: a0 = R + 900 km, e = 0, i0 = 51.6, af = R + 300 km

Spacecraft and thruster parameters: m0 = 5 kg, Isp = 2500 s, Tmax = 1 mN

0 = 90, N = 900 (left table) and 0 = 30, N = 800 (right table)

Verification of 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

Elliptic deorbiting: SS caseThe optimal control obtained earlierfor the circular mode appears to bequasi-optimal for the elliptic mode aswell (with one thrust arc dropped):• the eccentricity of a low-Earth orbit

cannot exceed 0.05 near-circularapproximation has lower accuracybut is still valid

• at the start of deorbiting, the centerof the sole thrust arc is at the apogee;the optimal control is the same since


2dr da

du du

Orbit: a0 = R + 900 km, i0 = 51.6, 0 = 30, r, f = R + 200 km

V = 326.2 m/s, mprop = 66.0 g

N = 750

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


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


Documents

Optimal Low-Thrust Deorbiting of Passively Stabilized LEO Satellites Sergey Trofimov Keldysh Institute of Applied Mathematics, RAS Moscow Institute of