Global Spacecraft Attitude Control Using Magnetic Actuators

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Global Spacecraft Attitude ControlUsing Magnetic Actuators

Marco Lovera and Alessandro AstolfiDipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, Italy

Department of Electrical and Electronic Engineering, Imperial College, London, England

The problem of inertial pointing for a spacecraft with magnetic actuators is addressed. Itis shown that a global solution to the problem can be obtained by means of (static) attitude

and rate feedback and (dynamic) attitude feedback. Simulation results demonstrate the

feasibility of the proposed approach.

1 INTRODUCTION

The problem of (global) regulation of the attitude of rigid spacecraft, i.e., spacecraft mod-

eled by the Eulers equations and by a suitable parameterization of the attitude, has been

widely studied in the recent years.

If the spacecraft is equipped with three independent actuators a complete solution to

the set point and tracking control problems is available. In [5, 11] these problems have been

solved by means of PD-like control laws, i.e., control laws that make use of the angular

velocities and of the attitude, whereas [1], building on the general results developed in [3],

has solved the same problems using dynamic output feedback control laws. It is worth

noting that, if only two independent actuators are available, as discussed in detail in [4], the

problem of attitude regulation is not solvable by means of continuous (static or dynamic)

time-invariant control laws, whereas a time-varying control law, achieving local asymptotic

(nonexponential) stability, has been proposed in [10].

The above results, however, are not directly applicable if the spacecraft is equipped

with magnetic coils as attitude actuators. As a matter of fact, it is not possible by means

of magnetic actuators to provide three independent torques at each time instant, yet as

the control mechanism hinges on the variations of the Earths magnetic field along the

spacecraft orbit, on average the system possesses strong controllability properties. In [2, 7,

8, 13] the regulation problem has been addressed exploiting the (almost) periodic behavior

of the system, hence resorting to classical tools of linear periodic systems, if local results

are sought after, or to standard passivity arguments, if (global) asymptotic stabilization of

open-loop stable equilibria is considered.

However, several problems remain open. In particular, if only inertial pointing is

considered, the global stabilization problems by means of full- (or partial-) state feedback

is still theoretically unsolved. Note, however, that from a practical point of view these

2004 by CRC Press LLC


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problems have an engineering solution, as demonstrated by the increasing number of ap-

plications of this approach to attitude control.

The aim of this chapter is to show how control laws achieving global1 inertial pointing

for magnetically actuated spacecraft can be designed by means of arguments similar to

those in [1, 11], provided that time-varying feedback laws are used and that the control

gains satisfy certain scaling properties. In particular, while previous work ([9]) dealt with

the case of state feedback control for a magnetically actuated, isoinertial spacecraft, thischapter deals with the more general problems of full (attitude and rate) and partial (attitude

only) state feedback for a generic magnetically actuated satellite.

The chapter is organized as follows. In Section 2 the model of the system is presented,

while in Section 3 the model of the geomagnetic field used in this study is described. In Sec-

tion 4 a general result on the stabilization of magnetically actuated spacecraft is presented.

Namely, using the theory of generalized averaging, it is shown how stabilizing control laws

designed for spacecraft with three independent control torques have to be modified to con-

struct stabilizing laws in the presence of magnetic actuators. In Section 5 the general theory

is used to design control laws using only attitude information, so avoiding the need for ratemeasurements in the control system. Finally, Sections 6 and 7 present some simulation

results and concluding remarks.

2 THE MODEL

The model of a rigid spacecraft with magnetic actuation can be described in various refer-

ence frames [12]. For the purpose of the present analysis, the following reference systems

are adopted.

Earth Centered Inertial reference axes (ECI). The origin of these axes is in the Earths

center. The X-axis is parallel to the line of nodes, that is the intersection between the

Earths equatorial plane and the plane of the ecliptic, and is positive in the Vernal

equinox direction (Aries point). The Z-axis is defined as being parallel to the Earths

geographic northsouth axis and pointing north. The Y-axis completes the right-

handed orthogonal triad.

PitchRollYaw axes. The origin of these axes is in the satellite center of mass. TheX-axis is defined as being parallel to the vector joining the actual satellite center of

gravity to the Earths center and positive in the same direction. The Y-axis points in

the direction of the orbital velocity vector. The Z-axis is normal to the satellite orbit

plane and completes the right-handed orthogonal triad.

Satellite body axes. The origin of these axes is in the satellite center of mass; the

axes are assumed to coincide with the bodys principal inertia axes.

The attitude dynamics can be expressed by the well-known Eulers equations: [12]:

I = S()I + Tcoils + Tdist, (1)

1To be precise, the control laws guarantee that almost all trajectories of the closed-loop system converge to the

desired equilibrium.



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where R3 is the vector of spacecraft angular rates, expressed in body frame, I R33

is the inertia matrix, S() is given by

S() =

0 z yz 0 x

y x 0

, (2)

Tcoils R3 is the vector of external torques induced by the magnetic coils and Tdist R3 isthe vector of external disturbance torques, which will be neglected in what follows.

In turn, the attitude kinematics can be described by means of a number of possible

parameterizations (see, e.g., [12]). The most common parameterization is given by the

four Euler parameters (or quaternions), which lead to the following representation for the

attitude kinematics:

q = W()q, (3)

where q = q1 q2 q3 q4T

= qTr q4T

is the vector of unit norm Euler parameters

and

W() =1

2

0 z y xz 0 x yy x 0 z

x y z 0

. (4)

It is useful to point out that Eq. (3) can be equivalently written as

q = W(q), (5)

where

W(q) =1

2

q4 q3 q2q3 q4 q1

q2 q1 q4q1 q2 q3

. (6)

Note that the attitude of inertially pointing spacecraft is usually referred to the ECI refer-

ence frame.

The magnetic attitude control torques are generated by a set of three magnetic coils,

aligned with the spacecraft principal inertia axes, which generate torques according to the

law:

Tcoils = mcoils b(t),

where mcoils R3 is the vector of magnetic dipoles for the three coils (which represent

the actual control variables for the coils) and b(t) R3 is the vector formed with thecomponents of the Earths magnetic field in the body frame of reference. Note that the

vector b(t) can be expressed in terms of the attitude matrix A(q) (see [12] for details) andof the magnetic field vector expressed in the ECI coordinates, namely b0(t), as

b(t) = A(q)b0

(t).The dynamics of the magnetic coils reduce to a very short electrical transient and can be

neglected. The cross-product in the above equation can be expressed more simply as a

matrix-vector product as

Tcoils = B(b(t))mcoils, (7)



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where

B(b(t)) =

0 bz(t) by(t)bz(t) 0 bx(t)

by(t) bx(t) 0

(8)

is a skew symmetric matrix, the elements of which are constituted by instantaneous meas-

urements of the magnetic field vector.

As a result, the overall dynamics, after application of the preliminary feedback,

mcoils = BT(b(t))u,

can be written asq = W(q)

I = S()I + (t)u,(9)

where (t) = B(b(t))BT(b(t)) 0.

3 MAGNETIC FIELD MODEL

A time history of the International Geomagnetic Reference Field (IGRF) model for the

Earths magnetic field [12] along five orbits in PitchRollYaw coordinates for a near-polar

orbit (87 inclination) is shown in Fig. 1.As can be seen, bx(t), by(t) have a very regular and almost periodic behavior, while

the bz(t) component is much less regular. This behavior can be easily interpreted by notic-ing that the x and y axes of the Pitch-Roll-Yaw coordinate frame lie in the orbit plane while

the z axis is normal to it. As a consequence, the x and y components of b(t) are affectedonly by the variation of the magnetic field due to the orbital motion of the coordinate frame(period equal to the orbit period) while the z component is affected by the variation ofb(t)due to the rotation of the Earth (period of 24 h).

When one deals with the problem of inertial pointing, however, it is more appropriate

to consider a representation of the magnetic field vector in Earth centered inertial coordi-

nates to be more convenient, as shown in Fig. 2.

4 STATE FEEDBACK STABILIZATION

In this section, a general stabilization result for a spacecraft with magnetic actuators is

given, in the case of full-state feedback (attitude and rate). For, let q =

0 0 0 1T

and consider the system

q = W(q)I = S()I +

(10)

and the control law

= kpqr kv. (11)

In the light of Theorem 1 in [11], the control law (11) guarantees that qr 0 and 0as t for the closed-loop system (10) and (11). Also, an analysis of the Lyapunovfunction used in the same reference shows that the equilibrium (q, 0) of the closed-loopsystem (10) and (11) is asymptotically stable, while the other possible equilibrium (q, 0)is unstable.



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Figure 1 Geomagnetic field in PitchRollYaw coordinates, 87 inclination orbit, 450km altitude.

Proposition 1 Consider the system (9) and the control law

u = 2kpqr kv. (12)

Then, there exists > 0 such that for any 0 < < the control law (12) ensuresthat (q, 0) is a locally exponentially stable equilibrium of the closed-loop system (912).

Moreover, almost all trajectories of (912) converge to (q, 0).

Proof. In order to prove the first claim, introduce the coordinates transformation

z1 = q z2 =

. (13)

In the new coordinates, the system (9) is described by the equations:

z1 = W(z1)z2Iz2 = S(z2)Iz2 + (t)(kpz1r kvz2).

(14)

System (14) satisfies all the hypotheses for the application of generalized averaging theory

([6, Theorem 7.5]). Moreover, using the Lyapunov function of Theorem 1 in [11] one can

conclude that the system obtained applying the generalized averaging procedure has (q, 0)as locally asymptotically stable equilibrium provided that

= limT

1

T

T

0

B(t)BT(t)dt > 0.



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Figure 2 Geomagnetic field in Earth-centered inertial coordinates, 87 inclination orbit.

To conclude the proof of the second claim it is necessary to prove that the matrix isgenerically positive definite. For, note that the matrix is obtained by integration of a three-by-three square (symmetric) matrix of rank two, namely the matrix B(t)BT(t). However,the kernel of the matrix B(t)BT(t) is not generically a constant vector, which implies > 0 generically. The set of bad trajectories, i.e., the trajectories for which the matrix is singular, is described by the simple relation

K = Ker(B(t)) = Im(b),

for some constant vector b. However, by a trivial property of the vector product one has

K = Im(b(t)) = Im(A(q)b0(t)),

hence all badtrajectories are such that, for all t,

A(q)b0(t) = (t)b,

for some scalar function (t), which is obviously a nongeneric condition.

5 STABILIZATION WITHOUT RATE FEEDBACK

The ability of ensuring attitude tracking without rate feedback is of great importance from

a practical point of view. The problem of attitude stabilization without rate feedback has



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been recently given an interesting solution in [1] for the case of a fully actuated spacecraft.

In this section a similar approach is followed in the development of a dynamic control law

that solves the problem for a magnetically actuated satellite. First, notice that the system

(10) and the control law (which is similar in spirit to the one proposed in [1]):

z = q z

= kpqr WT(q)(q z),(15)

(where > 0 and > 0) give rise to a closed-loop system having (q, 0, q/) as a locallyasymptotically stable equilibrium and qr 0 and 0 as t . On the basis of thisconsideration, which can be proved by means of the Lyapunov function

V = kp[(q4 1)2 + qTr qr] +

1

2TI +

1

2(q z)T(q z), (16)

it is possible to give a solution to the magnetic attitude control problem without rate feed-back.

Proposition 2 Consider the system (9) and the control law

z = q z

u = 2(kpqr + WT(q)(q z)).

(17)

Then there exists > 0 such that for any 0 < < the control law renders the equilib-

rium (q, 0, q/) of the closed-loop system (917) locally asymptotically stable. Moreover,almost all trajectories of the closed-loop system converge to this equilibrium.

Proof. As in the case of the state feedback control law we now introduce the coordinates

transformation

1 = q 2 =

3 = z . (18)

In the new coordinates, the system (9) is described by the equations:

1 = W(1)2I2 = S(2)I2 (t)(kp1r + W

T(1)(1 3))3 = (1 3).

(19)

Again, system (19) satisfies all the hypotheses for the application of [6, Theorem 7.5] and

using the Lyapunov function given in Eq. (16) one can conclude that the system obtained

applying the generalized averaging procedure has (q, 0, q/) as a locally asymptoticallystable equilibrium provided that

= limT

1

T

T

0

B(t)BT(t)dt > 0,

and this holds nongenerically as demonstrated in the proof of Proposition 1.



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Figure 3 Quaternion and angular rates for the attitude acquisition: state feedback controller.



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Figure 4 Quaternion and angular rates for the attitude acquisition: output feedback controller.



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Figure 5 Quaternion and angular rates for the attitude maneuver: state feedback controller.



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Figure 6 Quaternion and angular rates for the attitude maneuver: output feedback controller.



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6 SIMULATION RESULTS

To assess the performance of the proposed control laws the following simulation test case

has been analyzed. The considered spacecraft has an inertia matrix given by I = diag[27,17, 25] kg m2 and it is operating in a near-polar (87 inclination) orbit with an altitude of450 km and a corresponding orbit period of about 5600 s. For such a spacecraft, two sets of

simulations have been carried out; the first is related to the acquisition of the target attitudeq =

0 0 0 1

Tfrom an initial condition characterized by a high initial angular rate;

the second is related to a point-to-point attitude maneuver from the initial attitude given by

q0 =

0 0 0 1T

to the target attitude q = 12

1 0 0 1

T.

In all cases, both the full-state feedback control law and the control law without rate

feedback have been applied.

The results of the simulations are displayed in Figs. 3 and 4 for the attitude acquisi-

tion, and Figs. 5 and 6 for the attitude maneuver, from which the good performance of the

proposed control laws can be seen.

7 CONCLUDING REMARKS

The problem of inertial attitude regulation for a small spacecraft using only magnetic coils

as actuators has been analyzed and it has been shown that a nonlinear low-gain PD-like

control law yields (almost) global asymptotic attitude regulation even in the absence of

additional active or passive attitude control actuators such as momentum wheels or gravity

gradient booms.

Acknowledgments

The work for this chapter was partially supported by the European network Nonlinear and

Adaptive Control and by the MURST project, Identification and Control of Industrial

Systems.

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Global Spacecraft Attitude Control Using Magnetic Actuators