Robust three-axis attitude stabilization for inertial pointing spacecraft using magnetorquers Fabio Celani Department of Astronautical, Electrical, and Energy Engineering
Outline • • • • •
three-axis inertial pointing model for spacecraft and geomagnetic field spacecraft controllability attitude and attitude rate feedback attitude only feedback
Fabio Celani Sapienza University of Rome – DyCoSS 2014
Page 2
Three-axis inertial pointing Z
spacecraft on a circular Low Earth Orbit
Earth Centered Inertial frame
body frame X, !
Y
attitude of body frame with respect to inertial frame parametrized by quaternion vector part of attitude matrix three-axis inertial pointing Fabio Celani Sapienza University of Rome – DyCoSS 2014
Page 3
Spacecraft model relative kinematics dynamics is uncertain but bounds are known
and
on principal moments of inertia
spacecraft is equipped with three coils aligned with body axis vector of coils’ magnetic moments
geomagnetic field in body frame
geomagnetic field in inertial frame spacecraft model
Fabio Celani Sapienza University of Rome – DyCoSS 2014
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Dipole model of geomagnetic field geomagnetic field at spacecraft in inertial frame
orbit
spacecraft model
Fabio Celani Sapienza University of Rome – DyCoSS 2014
is the sum of sinusouidal functions having different frequencies (vector of almost periodic functions)
nonlinear almost periodic system Page 5
Control design nonlinear almost periodic system
spacecraft model
objective: design control law for
so that
and
preliminary control can be measured (magnetometers)
new control vector
preliminary control
spacecraft model
q˙ J !˙
= =
W (q)! ! ⇥ J! + A(q) i (t)A(q)T u
almost periodic spacecraft is not controllable at each t Fabio Celani Sapienza University of Rome – DyCoSS 2014
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Average controllability spacecraft model almost periodic system is not controllable
average of average controllability assumption - spacecraft’s orbit satisfies
average controllability assumption is satisfied for all circular orbits except equatorial orbit
Fabio Celani Sapienza University of Rome – DyCoSS 2014
Page 7
Robust attitude plus attitude rate feedback spacecraft model
attitude plus attitude rate feedback (PD-like feedback)
locally exponentially stable for all ‘s with principal moments of inertia in the range [Jmin Jmax ]
Fabio Celani Sapienza University of Rome – DyCoSS 2014
Page 8
Case study
circular orbit
rest-to-rest maneuver
inclination 87°
altitude 450 km
average controllabilty €
• initial attitude • desired final attitude
Fabio Celani Sapienza University of Rome – DyCoSS 2014
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Case study – state feedback
−3
0.4
coils magnetic moments ( A m2)
0.3 Euler angles (rad)
3
roll pitch yaw
0.2 0.1 0 −0.1 0
2
4 orbit
6
Fabio Celani Sapienza University of Rome – DyCoSS 2014
8
x 10
mcoils x mcoils y
2
m
coils z
1 0 −1 −2 −3 0
2
4 orbit
6
8
Page 10
Case study - robustness
principal moments of inertia of
robust state feedback
Fabio Celani Sapienza University of Rome – DyCoSS 2014
Page 11
Case study – attitude and attitude rate feedback
0.4
0.2 0.1
0.2 0.1 0
0 −0.1 0
2
4 orbit
6
8
−0.1 0
−3
3 coils magnetic moments ( A m2)
mcoils x mcoils y
2
mcoils z 1 0 −1 −2 −3 0
2
4 orbit
6
8
−3
x 10
2
coils magnetic moments ( A m )
3
roll pitch yaw
0.3 Euler angles (rad)
0.3 Euler angles (rad)
0.4
roll pitch yaw
2
4 orbit
6
8
Fabio Celani Sapienza University of Rome – DyCoSS 2014
x 10
mcoils x mcoils y
2
mcoils z 1 0 −1 −2 −3 0
2
4 orbit
6
8
Page 12
Robust attitude only feedback spacecraft model
attitude only feedback - no rate gyros
locally exponentially stable for all
‘s with principal moments of inertia in the range [Jmin Jmax ]
Fabio Celani Sapienza University of Rome – DyCoSS 2014
Page 13
Case study – attitude only feedback
0.4
0.2 0.1
0.2 0.1 0
0 −0.1 0
2
4 orbit
6
8
−0.1 0
−3
3
mcoils x
2
mcoils y
1
mcoils z
0 −1 −2 −3 −4 0
2
4 orbit
6
8
−3
x 10
coils magnetic moments ( A m2)
2
coils magnetic moments ( A m )
3
roll pitch yaw
0.3 Euler angles (rad)
0.3 Euler angles (rad)
0.4
roll pitch yaw
2
4 orbit
6
8
Fabio Celani Sapienza University of Rome – DyCoSS 2014
x 10
mcoils x
2
mcoils y
1
mcoils z
0 −1 −2 −3 −4 0
2
4 orbit
6
8
Page 14
Conclusions and way forward • three-axis stabilization for inertial pointing spacecraft using magnetorquers • attitude and attitude rate feedback • attitude only feedback • local exponential stabilization and robustness with respect to uncertainty on inertia matrix • Earth-pointing case • global stabilization
Fabio Celani Sapienza University of Rome – DyCoSS 2014
Robust three-axis attitude stabilization for inertial ...
Sapienza University of Rome â DyCoSS 2014. Page 2. ⢠three-axis inertial pointing. ⢠model for spacecraft and geomagnetic field. ⢠spacecraft controllability.
determined by length of measurement process (not a control design parameter) ... closed-loop dynamics given by sampled-data nonlinear time-varying system.
attitude control problem. ⢠model of spacecraft with magnetorquers. ⢠separation between measurement and actuation. ⢠design of piecewise-constant control. ⢠case study. Page 3. Attitude control problem. Fabio Celani. Sapienza University of R
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