**9.4.4 Gimbal Attitude Implementations**

The primary function of gimbals is to isolate the ISA from vehicle rotations, but
they are also used for other INS functions.

**9.4.4.1 Accelerometer Recalibration **Navigation accuracy is very sensitive
to accelerometer biases, which can shift as a result of thermal transients in
turnon/turnoff cycles, and can also drift randomly over time. Fortunately, the
gimbals can be used to calibrate accelerometer biases in a stationary 1-g environment.
In fact, both bias and scale factor can be determined by using the
gimbals to point the accelerometer input axis straight up and straight down,
and recording the respective accelerometer outputs *a*_{up} and *a*_{down}. Then the bias
*a*_{bias} =
(*a*_{up} + *a*_{down}) /2 and scale factor s =
(*a*_{up} - *a*_{down}) /2*g*_{local}, where *g*_{local}
is the local gravitational acceleration.

**9.4.4.2 Gyrocompass Alignment **This is the process of determining the orientation
of the ISA with respect to locally level coordinates (e.g., NED or ENU).
Gyrocompassing allows the ISA to be oriented with its sensor axes aligned parallel
to the north, east, and vertical directions. It is accomplished using three
servo loops. The two "leveling" loops slew the ISA until the outputs of the nominally
"north" and "east" accelerometer outputs are zeroed, and the "heading"
loop slews the ISA around the third orthogonal axis (i.e., the vertical one) until
the output of the nominally "east-pointing" gyro is zeroed.

**9.4.4.3 Vehicle Attitude Determination** The gimbal angles determine the vehicle
attitude with respect to the ISA, which has a controlled orientation with respect
to locally level coordinates. Each gimbal angle encoder output determines the relative
rotation of the structure outside gimbal axis relative to the structure inside
the gimbal axis, the effect of each rotation can be represented by a 3 x 3 rotation
matrix, and the coordinate transformation matrix representing the attitude of
vehicle with respect to the ISA will be the ordered product of these matrices.

For example, in the gimbal structure shown in Fig. 2.6, each gimbal angle represents
an Euler angle for vehicle rotations about the vehicle roll, pitch and yaw
axes. Then the transformation matrix from vehicle roll-pitch-yaw coordinates
to locally level east-north-up coordinates will be

where

**9.4.4.4 ISA Attitude Control** The primary purpose of gimbals is to stabilize
the ISA in its intended orientation. This is a 3-degree-of-freedom problem, and
the solution is unique for three gimbals. That is, there are three attitude-control
loops with (at least) three sensors (the gyroscopes) and three torquers. Each
control loop can use a PID controller, with the commanded torque distributed
to the three torquers according to the direction of the torquer/gimbal axis with
respect to the gyro input axis, somewhat as illustrated in Fig. 9.22, where

DISTURBANCES includes the sum of all torque disturbances on the individual gimbals

and the ISA, including those due to mass unbalance and acceleration,

air currents, torque motor errors, etc.

GIMBAL DYNAMICS is actually quite a bit more complicated than the rigid-body

torque equation

which is the torque analog of **F** = *m***a**, where **M**_{inertia} is the moment of
inertia matrix. The IMU is not a rigid body, and the gimbal torque motors
apply torques *between *the gimbal elements (i.e., ISA, gimbal rings and
host vehicle).

DESIRED RATES refers to the rates required to keep the ISA aligned to a moving

coordinate frame (e.g., locally level).

RESOLVE TO GIMBALS is where the required torques are apportioned among the

individual torquer motors on the gimbal axes.

The actual control loop is more complicated than that shown in the figure, but it
does illustrate in general terms how the sensors and actuators are used.

For systems using four gimbals to avoid gimbal lock, the added gimbal adds
another degree of freedom to be controlled. In this case, the control law usually
adds a fourth constraint (e.g., maximize the minimum angle between gimble axes)
to avoid gimbal lock.

**9.4.4 Gimbal Attitude Implementations**

The primary function of gimbals is to isolate the ISA from vehicle rotations, but
they are also used for other INS functions.

**9.4.4.1 Accelerometer Recalibration **Navigation accuracy is very sensitive
to accelerometer biases, which can shift as a result of thermal transients in
turnon/turnoff cycles, and can also drift randomly over time. Fortunately, the
gimbals can be used to calibrate accelerometer biases in a stationary 1-g environment.
In fact, both bias and scale factor can be determined by using the
gimbals to point the accelerometer input axis straight up and straight down,
and recording the respective accelerometer outputs *a*_{up} and *a*_{down}. Then the bias
*a*_{bias} =
(*a*_{up} + *a*_{down}) /2 and scale factor s =
(*a*_{up} - *a*_{down}) /2*g*_{local}, where *g*_{local}
is the local gravitational acceleration.

**9.4.4.2 Gyrocompass Alignment **This is the process of determining the orientation
of the ISA with respect to locally level coordinates (e.g., NED or ENU).
Gyrocompassing allows the ISA to be oriented with its sensor axes aligned parallel
to the north, east, and vertical directions. It is accomplished using three
servo loops. The two "leveling"

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