active magnetic attitude control system providing three-axis inertial attitude

Active magnetic attitude control system providing three-axis inertial attitude

M. Ovchinnikov, V. Penkov, D. Roldugin, A. Guerman

Keldysh Institute of Applied Mathematics of RASUniversity of Beira Interior

Contents• Main goal• Problem statement

– Geomagnetic field vectors– Analysis methods

• Analysis– Fast and moderate rotations– Slow rotation without gravity– Optimal control gains– Gravity influence

• Conclusion

Mar 24-26, 20142nd IAA Conference on Dynamics and

Control of Space Systems2

Main goal• To overcome two principal MACS problems

– Underactuation, that is the torque is perpendicular to the geomagnetic induction vector

– Torque value limit due to the limited power and dimension capabilities

• To assess gravity effect on the inertial attitude

3Mar 24-26, 20142nd IAA Conference on Dynamics and

Control of Space Systems

Problem statement

• Euler and osculating equations are used• Satellite is equipped with three magnetorquers• Attitude is known• Control torque or control and gravitational torques

are taken into account• Orbit is a Keplerian one• Averaged and IGRF models are used• Control torque is based on the PD-controller



23 32 31 13 12 21, , ,T

ak k a a a a a a m B ω a a

• Averaged: Geomagnetic induction vector evenly rotates on the cone surface

Geomagnetic field models

• IGRF



Analysis methods

• System in arbitrary motion is often analyzed using numerical methods. Not enough assumptions can be made to simplify the system

• Floquet theory



Transient motionMultiple scales

method

Arbitrary motionNumerical analysis

Steady-state motionPoincare method

Fast/moderate rotation

• Control gains are comparable• Mean influence of the positional control part

and gravitational torque is equal to zero• The control is identical to the “-Bdot” one and

to the eddy currents• The angular velocity is exponentially damped

to the orbital velocity value



Slow rotation. Averaged equations• Averaged linearized equations without gravity

• Damped oscillations for each angle• Linearized averaged equations are asymptotically stable, so

initial equations allow asymptotically stable limit cycle. The solution is in the vicinity of the averaged equations equilibrium, so the averaging is valid on the unlimited time interval

1 1

1 1

1 1

2 0,

2 0,

2 4 0.

p q p q

A Ap q p q

B BA A

p pC C



Control gains

2 1 1: 2

12

18

2aK K K

2 22 28 8a aK K K K K

22 1 28 8 ,a aK K K

22 2

1 22

8

2 1 aK K

21,2

23,4 12

1 1

25,6 22

2 2

2 20 02 20 0

1 2

18 ,

2

1 18 ,

2

1 18 ,

2

, ,

, 2

a

a

a

a a

K K K

KK K

KK K

B BK p q k K p q k

A A

B A C p q pA



Floquet theory: more accurate results



Poincare method: damping effect

• Three-axis control with gravity, orbital plane motion

• Oscillations without damping

• Damping-driven modulus change for discrete frequency values

• Approximate solution



2 sin 0

0

0

2arcsin sn ,

2 cn ,

k u

k u

2E 1 KK

k k k kk k

0 exp 2k k u

Periodic solutions

• Periodic solutions are found numerically• Approximate formulas for the amplitude

depending on satellite parameters are found• Simple formula may be used in the vicinity of

necessary attitude



2

2 2

14 sin 2 2 cos2

2 4 2 4u u

Floquet theory: stability area change



2(1.0255, 1.5393, 1.8172) , 30 , 600·kg mdiag i km J o




2(1.0255, 1.5393, 1.8172) , 60 , 600·kg mdiag i km J o

Periodic solutions amplitude



Results

• Three-axis attitude is proven to be accessible• Simple formulas for optimal control gains are

found• More accurate Floquet-based optimal

parameters are found• Gravity effect is found to be controversial,

acting as control or disturbing torque in general



Acknowledgment

2nd IAA Conference on Dynamics and Control of Space Systems

19

The work was supported by RFBR grants №№ 12-01-33045, 13-01-00665, 14-01-31313project Odyssea

Mar 24-26, 2014

active magnetic attitude control system providing three-axis inertial attitude

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