Page 222 - Modern Control Systems

196 Chapter 3 State Variable Models

attitude motion. The aerodynamic torque acting on the space station is generated by
the atmospheric drag force that acts through the center of pressure. In general, the cen-
ter of pressure and the center of mass do not coincide, so aerodynamic torques develop.
In low earth orbit, the aerodynamic torque is a sinusoidal function that tends to oscil-
late around a small bias. The oscillation in the torque is primarily a result of the earth's
diurnal atmospheric bulge. Due to heating, the atmosphere closest to the sun extends
further into space than the atmosphere on the side of the earth away from the sun. As
the space station travels around the earth (once every 90 minutes or so), it moves
through varying air densities, thus causing a cyclic aerodynamic torque. Also, the space
station solar panels rotate as they track the sun. This results in another cyclic compo-
nent of aerodynamic torque. The aerodynamic torque is generally much smaller than
the gravity gradient torque. Therefore, for design purposes we can ignore the atmos-
pheric drag torque and view it as a disturbance torque. We would like the controller to
minimize the effects of the aerodynamic disturbance on the spacecraft attitude.
Torques caused by the gravitation of other planetary bodies, magnetic fields,
solar radiation and wind, and other less significant phenomena are much smaller
than the earth's gravity-induced torque and aerodynamic torque. We ignore these
torques in the dynamic model and view them as disturbances.
Finally, we need to discuss the control moment gyros themselves. First, we will
lump all the control moment gyros together and view them as a single source of torque.
We represent the total control moment gyro momentum with the variable h. We need
to know and understand the dynamics in the design phase to manage the angular mo-
mentum. But since the time constants associated with these dynamics are much shorter
than for attitude dynamics, we can ignore the dynamics and assume that the control
moment gyros can produce precisely and without a time delay the torque demanded by
the control system.
Based on the above discussion, a simplified nonlinear model that we can use as
the basis for the control design is

6 = RO + n, (3.96)
2
-
i n = ft x Ifl + 3n c x Ic - u, (3.97)
h = ft x h + u, (3.98)
-
where

cos 0 3 -cos $i sin 0 3 sin 0! sin 0 3
R(0) = —!— 0 cos 0i -sin 0i
v
' cos 0 3 0 sin 0i cos 0 3 cos 0i cos 0 3 _
o" o>i ~*r «1
n n , ft = Oil , © = 0 2 , u = " 2
_0_ _&>3 _03_ _ " 3 _
where u is the control moment gyro input torque, ft in the angular velocity, I is the
moment of inertia matrix, and n is the orbital angular velocity. Two good references
that describe the fundamentals of spacecraft dynamic modeling are [26] and [27].
There have been many papers dealing with space station control and momentum