MAGNETIC ATTITUDE CONTROL FOR GRACE-LIKE MISSIONS

Yaroslav Mashtakov1

Mikhail Ovchinnikov1 Benny Rievers2

Meike List2 Florian Wöske2

Keldysh Institute of Applied Mathematics of RAS1 Center of Applied Space Technology and Microgravity (ZARM) 2

International Workshop on Satellite Constellation & Formation Flying 16-19 July 2019, Glasgow, Scotland

Introduction

GRACE mission:

• Launched 17 March 2002

• Two spacecrafts at the same polar orbit

• Conduct measurements of the Earth gravitational field

• Obtained data is used in oceanology, ice glaciers observation etc.

• Enhanced mission GRACE-FO was launched 22 May 2018

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Introduction

• Microwave ranging device provide measurements of relative distance

• It is necessary to maintain line of sight between the satellites

• RW produce too much noise

• We can use only magnetorquers

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Problem statement

We know:

• Orbital motion of each satellite

• Orbits are near circular and almost identical

• Distance between spacecrafts is about 200 km

• Spacecraft parameters

We have to:

• Maintain line of sight between the satellites

• Solar panels should always look away from the Earth

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Reference motion

• Line of sight:

• Solar panels:

• Reference motions is given as a DCM:

• Reference angular velocity from Poisson kinematics:

1 21

1 2

re

r r

r

1 1 2 2

3

2 1 1 2

,

,

e r r

ee

r e r

e

1 2 3

TD e e e

ω DD

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Orbital Frame

• It is described as:

23

2

2 3 3 2

2 3 3 2

1

2 3 1

,

,

rj

r

v j j v

j

v j j v

j

j

j

Difference between Orbital and Reference Frames

• Orbital and reference frame are almost the same • The problem is similar to stabilization of the spacecraft in

Orbital Frame

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Equations of motion

• Only GG and magnetic torque are considered

• Satellites are on the circular orbits with orbital angular velocity

• Motion is near the equilibrium in Orbital Frame

• Note: equilibrium is unstable (due to mission requirements – we have to minimize atmospheric drag)

53

1

2

Eabs abs abs

abs

r

ω ω Jω r Jr m B

Q Q

J

ω

0

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Linearization

• Equations of relative motion

• – skew symmetric cross product matrix

1

α ω

ω A α A ω J B m

B

2 2 2

0 0 0

0

0

diag 4 3

0 0

0 0 0

0 0

, ,C B C A A B

A B C

C A B

A

B C A

C

A

A

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Control

Lyapunov-based control Advantages: • Simple • Ensures asymptotic stability

Disadvantages: • Does not take into account

magnetic control specifics • Depends on two control

parameters which are must be chosen adequately

ideal abs abs ext

ref ref

sk k

M Mω J

J J

s

ω

ω ω ω

ω

refω — reference angular velocity — external torque — relative angular velocity — vector part of relative quaternion

ω

extM

s

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Control

• Using magnetorquers create as mush torque as possible

• Provided control torque

• Linearized equations of motion

• Is it possible to make it asymptotically stable?

2

idealMB

mB

, ,ctrl ideal B B ideal B B ideal B

BM M e e M e e M e

B

1

B B kk

α ω

ω A α A ω J JA α JA ω α ωe e

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How to choose coefficients?

• System is linear and time dependent

• We will use simple direct dipole model

• In this model system is periodical

• Floquet theory might be applied for stability analysis of the system

• It allow us to choose appropriately

• For better tuning they can be replaced by diagonal matrices

,k k

1

B B kk

α ω

ω A α A ω J JA α JA ω α ωe e

,1 ,2 ,3

,1 ,2 ,3

dia , ,

,

g

di g ,a

k k k

k k

k

k k

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Floquet theory

• Obtain monodromy matrix

• Find its eigenvalues

• For asymptotic stability

• corresponds to the convergence rate

• We get optimization problem

• Solve it numerically

k

max Re ln 0k

k

max Re ln mink

k

max Re lnk

k

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Gas thrusters

• Magnetic coils cannot provide necessary accuracy

• We will use cold gas thrusters

• They are turned on only when satellite reaches allowed accuracy borders

• Denote

•

, 1 1, ,des k k k k k k k k ks t f t t t

1

1

1

1,

, 1 ,

1 ,

k k

k k k k

k k

if t t T

f t t a T t t if firing in the same direction

a T t t if firing in the opposite direction

, , ,

1thr

ctrl k ii des k rel k

thr

Jt

M

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Simulation results

• Orbital motion corresponds to the real GRACE mission (polar orbit, height about 500 km)

• Atmospheric drag, solar radiation pressure, gravity gradient and magnetic torques are all included in the simulation

• Sensor’s noise also considered (Kalman filtration for startracker and gyros)

• Different mission scenarios: pure magnetic control, GRACE accuracy requirements, GRACE-FO accuracy requirements

• Simulation is carried out using eXtended High Performance satellite dynamics Simulator (XHPS) developed by ZARM and DLR

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Results

Purely magnetic attitude control

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Results

Magnetic coils + gas thrusters GRACE requirements: 0.46 deg along line of sight 0.17 deg along other axes

Amount of firings per day: 20, 0, 50

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Original GRACE required ~300 firings per day

Results

Magnetic coils + gas thrusters GRACE-FO requirements: 0.14 deg along line of sight 0.014 deg along other axes Hand tuning of the coefficients is necessary

Amount of firings per day: 60, 0, 150

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Summary

• Problem of attitude control for GRACE-like mission is considered

• Lyapunov based attitude control is suggested

• Method of control coefficients selection is proposed (works for pure magnetic control and low accuracy requirements)

• For an extreme accuracy maintaining “hand tuning" is necessary

This work is supported by RFBR grant no. 18-31-20014

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