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

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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

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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

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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

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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

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Case study - robustness

principal moments of inertia of

robust state feedback

Fabio Celani Sapienza University of Rome – DyCoSS 2014

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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

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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

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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

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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

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