Mohinder S. Grewal - Global Navigation Satellite Systems, Inertial Navigation, and Integration

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

The gimbal angles determine the vehicle attitude with respect to the ISA, which has a controlled orientation with respect to navigation 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 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.

Figure 3.20Simplified control flow diagram for three gimbals.

Gimbals control ISA orientation. This is a three‐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 Figure 3.20, where

Disturbances includes the sum of all torque disturbances on the individual gimbals and the ISA, including those due to ISA mass unbalance and acceleration, rotations of the host vehicle, 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 , where 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 gimbal axes) to avoid gimbal lock.

The INS navigation solution for altitude and altitude rate is called its vertical channel . It might have become the Achilles heel of inertial navigation if it had not been recognized (by physicist George Gamow [10]) and resolved (by Charles Stark Draper and others) early on.

The reason for this is that the vertical gradient of the gravitational acceleration is negative. Because accelerometers cannot sense gravitational accelerations, the INS must rely on Newton's universal law of gravitation to take them into account in the navigation solution. Newton's law has the downward gravitational acceleration inversely proportional to the square of the radius from the Earth's center, which then falls off with increasing altitude. Therefore an INS resting stationary on the surface of the Earth with an upward navigational error in altitude would compute a downward gravitational acceleration smaller that the (measured) upward specific force countering gravity, which would result in an upward navigational acceleration error, which only makes matters worse. This would not be a problem for surface ships, it might have been a problem for aircraft if they did not already use barometric altimeters, and similarly for submarines if they did not already use depth sensors. It became an early example of sensor integration successfully applied to inertial navigation.

This is no longer a serious issue, now that we have chip‐scale barometric altimeters.

The basic signal processing functions for strapdown INS navigation are illustrated in Figure 3.21, where

G is the estimated gravitational acceleration, computed as a function of estimated position.

is the estimated position of the host vehicle in navigation coordinates.

is the estimated velocity of the host vehicle in navigation coordinates.

is the estimated acceleration of the host vehicle in navigation coordinates, which may be used for trajectory control (i.e. vehicle guidance).

is the estimated acceleration of the host vehicle in sensor‐fixed coordinates, which may be used for vehicle steering stabilization and control.

is the coordinate transformation matrix from sensor‐fixed coordinates to navigation coordinates, representing the attitude of the sensors in navigation coordinates.

is the estimated angular velocity of the host vehicle in sensor‐fixed (ISA) coordinates, which may be used for vehicle attitude stabilization and control.

is the estimated angular velocity of the host vehicle in navigation coordinates, which may be used in a vehicle pointing and attitude control loop.

Figure 3.21Essential navigation signal processing for strapdown INS.

The essential processing functions include double integration (represented by boxes containing integration symbols) of acceleration to obtain position, and computation of (unsensed) gravitational acceleration as a function of position. The sensed angular rates also need to be integrated to maintain the knowledge of sensor attitudes. The initial values of all the integrals (i.e. position, velocity, and attitude) must also be known before integration can begin.

The position vector is the essential navigation solution. The other outputs shown are not needed for all applications, but most of them (except ) are intermediate results that are available “for free” (i.e. without requiring further processing). The velocity vector , for example, characterizes speed and heading, which are also useful for correcting the course of the host vehicle to bring it to a desired location. Most of the other outputs shown would be required for implementing control of an unmanned or autonomous host vehicle to follow a desired trajectory and/or to bring the host vehicle to a desired final position.