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Active magnetic attitude control system providing three-axis inertial attitude. M. Ovchinnikov, V. Penkov , D. Roldugin , A. Guerman Keldysh Institute of Applied Mathematics of RAS University of Beira Interior. Contents. Main goal Problem statement Geomagnetic field vectors - PowerPoint PPT Presentation
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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
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems4
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
5Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems
Analysis methods
• System in arbitrary motion is often analyzed using numerical methods. Not enough assumptions can be made to simplify the system
• Floquet theory
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems6
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
7Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems
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
8Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems
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
9Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems
Floquet theory: more accurate results
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems10
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
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems11
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
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems12
2
2 2
14 sin 2 2 cos2
2 4 2 4u u
Floquet theory: stability area change
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems13
2(1.0255, 1.5393, 1.8172) , 30 , 600·kg mdiag i km J o
Floquet theory: stability area change
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems14
Floquet theory: stability area change
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems15
2(1.0255, 1.5393, 1.8172) , 60 , 600·kg mdiag i km J o
Floquet theory: stability area change
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems16
Periodic solutions amplitude
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems17
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
Mar 24-26, 20142nd IAA Conference on Dynamics and
Control of Space Systems18
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
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