Chapter 9.4.4: INERTIAL SYSTEMS TECHNOLOGIES: Gimbal Attitude Implementations

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) /2g_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 = ma, 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.

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