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0066_frame_C31.fm Page 9 Wednesday, January 9, 2002 7:49 PM

31.6 Spacecraft Adaptive Attitude Regulation Example

Consider the problem of a rigid spacecraft with an initial nonzero attitude and body angular velocity vector
that has to be brought to rest at a zero attitude vector. This rigid body adaptive attitude regulation problem
13
based on the feedback linearization approach has been derived by Schaub, Akella, and Junkins. The
governing equations are described by Euler’s rotational equations of motion and the desired linear closed-
loop dynamics (LCLD) can be of either PD or PID form. 13, 14 Only a crude estimate of the moment of
inertia matrix is assumed to be known. An adaptive control law is presented, which includes an integral
feedback term in the desired closed-loop dynamics and achieves asymptotic stability even in the presence
of unmodeled external disturbances.
The resulting simulation is illustrated in Fig. 31.3. The attitude vector is specified in terms of the
modified Rodrigues parameter (MRP) whose components s i are shown in Fig. 31.3(a). Without any
adaptation, the open-loop control is still asymptotically stable. However, the transient attitude errors
don’t match those of the desired LCLD well at all. With adaptation turned on, the performance matches
that of the ideal LCLD very closely.

0.3
10 -1
0.2
10 -2
0.1
10 -3
0.0
10 -4
Ideal LCLD
-0.1 -5
10 No Adaptation
Ideal LCLD
-0.2 -6 Adaptation without
No Adaptation 10
. Disturbance Learning
Adaptation with
-0.3 10 -7 Adaptation with
. Disturbance Learning
. Disturbance Learning
-0.4 10 -8
0 10 20 30 40 0 20 40 60 80 100
time [s] time [s]
(a) MRP attitude vector s (b) MRP attitude vector magnitude |s |
4
Ideal LCLD
5.0 3 No Adaptation
Adaptation with
. Disturbance Learning
2
0.0
1
Ideal LCLD
-5.0 No Adaptation 0
Adaptation without
. Disturbance Learning
-1
-10.0 Adaptation with
. Disturbance Learning
-2
0 10 20 30 40
0 20 40 60 80 100
time [s] time [s]
(c) Control vector u (N m) (d) Adaptive external torque estimate (N m)
10 -1 No Adaptation 10 -2 No Adaptation
Adaptation without Adaptation without
. Disturbance Learning . Disturbance Learning
10 -2 10 -3
Adaptation with Adaptation with
. Disturbance Learning . Disturbance Learning
10 -3 10 -4
10 -4 10 -5
10 -5 10 -6
10 -6 10 -7
0 20 40 60 80 100 0 20 40 60 80 100
time [s] time [s]
.
.
(e) Performance error  s − s r  (f) Performance error  s − s r 
FIGURE 31.3 Rigid body stabilization while enforcing LCLD in the presence of large inertia and external distur-
bance ignorance.
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