A gyro-stabilized platform system, using restrained gyros, is well suited for automatic leveling because of the characteristics of the gyro-platform-servo combination.
The restrained gyro-stabilized platform with reasonable response characteristics operates with an approximate equation of motion, neglecting transient effects, as follows: Af where U is a torque applied about the output axis of the controlling gyro.

The platform angle **yf is the angle about which the gyro is controlling.
This is normally termed the gyro input axis, 90-degrees away from the gyro output or **yj axis.
The gyro angular momentum is defined by H.

Thus if the gyro and platform-controller combination maintains the platform with zero angular deviation about the **yf axis, the system can be rotated with an angular velocity Af if a torque is supplied to the gyro output axis Aj.
It is assumed that the gyros are designed with electrical torquers so that a torque can be applied about their output axes.

In the system shown in Fig. 7-1, the accelerometer output is amplified and the resulting voltage is applied to the gyro output-axis torquer.
This torque causes the entire system to rotate about the **yf axis, since the response to Af.
If the polarities are correct, the platform rotates in such a direction as to reduce the accelerometer output to zero.
As the accelerometer output is decreasing, the torque applied to the gyro output axis decreases and, therefore, the rate decreases.
Finally, when the accelerometer output is zero, the entire system remains stationary, and the platform is, by definition, leveled.

A mathematical block diagram for the leveling system is shown in Fig. 7-2.
The platform is initially off level by the angle Aj.
The angle generated by the platform servo **yf multiplied by G is the effective acceleration acting on the accelerometer.
Af is the scale factor of the accelerometer (Af).
The voltage Af is amplified by Af and applied to the gyro torquer with scale factor Af.
Finally, the gyro-stabilized platform characteristic is represented by Af.
The system as indicated in Fig. 7-2 is fundamental and simple because the transient effects of both the platform servo and the accelerometer have been neglected.
With these factors included, an upper limit is placed on the allowable loop gain by stability considerations.
In this type of system, a high loop gain is desirable because it provides a fast response time.

When the frequency response characteristics of practical components are considered, their effect on stability does not present the most serious limit on the system loop gain.
The time required for the system to reach a level position is approximately inversely proportional to the servo loop gain.
In addition, the cutoff frequency for input accelerations is approximately proportional to the servo loop gain;
i.e., high loop gain causes the system to respond to horizontal components of accelerations.
This problem usually determines the lower limit of loop gain rather than response time.

It must be noticed in Fig. 7-1 that the accelerometer responds to any input acceleration.
The equation relating input acceleration to output platform angle is Af.
In practice, the preflight leveling process takes place with the system mounted in the airframe.
When the system is arranged for automatic leveling, the platform angles respond to any horizontal components of acceleration acting on the accelerometers.
There are many such components of acceleration present due to the effect of wind gusts, engine noise, turbulence around the vehicle, etc.
One of the greatest problems associated with automatic leveling is establishing a true level in the presence of high-level acceleration noise.
One solution to the problem is to operate with a low loop gain and to include low-pass filters.
This technique causes the system to respond only to low frequency acceleration components such as the platform tilt.
Since a lower loop gain and low-pass filtering increases the response time, a practical compromise must be reached.

One of the most desirable solutions is achieved by the use of a non-linear amplifier for Af.
The amplifier is designed so that its gain is large for accelerometer signals above a certain threshold level.
Below this level, the amplifier gain Af is proportional and is of small value, in order to provide adequate noise filtering.
The effect is that the platform returns from an off-level position at a rapid rate until it is nearly level, at which point the platform is controlled by a proportional servo with low enough frequency response so that the noise has little effect on the leveling process.

When the system is on automatic leveling, the gyro drift is canceled by the output of the leveling system (amplifier Af).
The platform actually tilts off level so that the accelerometer output, when amplified by Af, will supply the correct current to the gyro torquer to cancel the gyro drift.
The amount of platform dip required depends upon the scale factors of the system.

7-3.
Practical leveling considerations.

The automatic leveling system described in this section is readily adaptable to a gyro-stabilized platform consisting of three integrating gyros.
The system requires some switching of flight equipment circuits.
However, the leveling operation can be maintained and controlled remotely with no mechanical or optical contact with the platform.

This leveling system will hold the platform on-level, automatically, as long as the system is actuated.
A useful by-product of this system is that the information necessary to set the gyro drift biases is available from the currents necessary to hold the system in level.

The leveling process can be accomplished manually, and the results are as satisfactory as those obtained with automatic equipment.
The process consists in turning the platform manually until the outputs of both accelerometers are zero.
The turning is accomplished by applying voltage to the gyro torquers described above.
In brief, the human replaces amplifier Af in Figs. 7-1 and 7-2.

Manual leveling requires an appropriate display of the accelerometer outputs.
If high accuracy is required in preflight leveling, it is usually necessary to integrate or doubly integrate the accelerometer outputs (this also minimizes the noise problem).
With integration, the effect of a small acceleration (or small platform tilt angle) can be seen after a time.
However, skill is required on the part of an operator to level a platform to any degree of accuracy.
Also, it requires more time as compared to the automatic approach.

Manual leveling is inconvenient if the platform must be maintained accurately level for any prolonged period of time.
The operator must continually supply the correct amount of turning current to the gyro torquers so that the effect of gyro drift is canceled.
This process is especially difficult since gyro drifting is typically random.

7-4.

Platform heading.

Platform heading consists of orienting the sensitive axis of the accelerometers parallel to the desired coordinate system of the navigator.
In simpler terms, it amounts to pointing the platform in the proper direction.

For purely inertial navigators, two techniques are available to accomplish the platform heading: Use of external or surveying equipment to establish proper heading;

Use of the characteristics of the platform components for an indication of true heading.
The choice of the heading technique is dependent upon the accuracy requirements, field conditions, and the time available to accomplish the heading.

7-5.
External determination of heading -- surveying technique.

With the gyro-stabilized platform leveled, it can be headed in the proper direction by using surveying techniques.
The platform accelerometers must be slightly modified for this procedure.
Before the accelerometers are mounted on the platform, the direction of their sensitive axis must be accurately determined.
A mirror is mounted on each accelerometer so that the plane of the mirror is perpendicular to the sensitive axis of the unit.
Transit.

A precision transit is set up so that it is aligned with respect to true north.
This can be done to a high degree of accuracy by existing surveying techniques.
With the transit set up, a mirror on one of the accelerometers is sighted and the platform is turned until it is aligned.

The sighting procedure includes the use of a fixture for the transit to project a beam of light, which is darkened by crossed hairs, on the accelerometer mirror.
When the platform is aligned, the reflected image of the crossed hairs can be seen exactly superimposed upon the original crossed hairs.
The images can easily be aligned with a high degree of accuracy.
The platform is turned as required by supplying currents to the appropriate gyro torquers.

Although this technique is simple and satisfactory, one practical difficulty does exist: the direction of true north must be known for each launch point.
However, this difficulty is not too serious if it is realized that a surveying team can establish a true north base line with a few days' work.

In many installations, the inertial platform is raised off the ground a considerable height when it is mounted in the vehicle before flight.
With this situation, it is difficult to sight in on the platform with surveying equipment.
If the platform is not too high off the ground, a transit can be mounted on a stand to raise it up to the platform.
Obviously, the heading accuracy is lessened by such techniques since errors are introduced because of motion of the stand.
Autocollimator.

The transit can be replaced by an autocollimator.
This instrument provides an electrical signal proportional to the angular deviations of the platform and can be used to automatically hold the platform on true heading.
The electrical signal from the autocollimator is amplified and supplied to the Z-gyro torquer.
If the polarity is correct, the platform will turn until the heading error angle is zero.
Information is also available from this autocollimator system to set the drift bias for the Z-axis gyro.
If the Z gyro is drifting, a current generated by the autocollimator is delivered to the gyro torquer to cancel the drift.
If the drift error is systematic, it can be canceled with a bias circuit which can be arranged and adjusted to supply the required compensating current.
Electrical pickoffs.

It is possible to locate an angular electrical pickoff, which will indicate the angular deviation between the true heading direction and the platform.
Essentially, the stator or reference portion of the pickoff is established with respect to the true heading direction, and the platform is turned either manually or automatically until the angular electrical pickoff signal is reduced to zero.

7-6.

Gyrocompass heading.

Gyrocompass alignment is an automatic heading system which depends upon the characteristic of one gyro to establish true heading.
For the case of a purely inertial autonavigator consisting of three restrained gyros, a coordinate system is used where the sensitive axis of the X accelerometer is parallel to the east-west direction at the base point, and the Y accelerometer sensitive axis is parallel to the north-south direction at the base point.
The accelerometers are mounted rigidly on the platform.
Thus, if one accelerometer is properly aligned, the other is also.
The input axis of the appropriate gyros are parallel to the sensitive direction of the accelerometers.

Figure 7-3 shows a platform system with the gyro vectors arranged as described above.
The platform is leveled and properly headed, so that the X-gyro input axis is parallel to the east-west direction and the Y-gyro input axis is parallel to the north-south direction.

The input axis of the X gyro, when pointing in the east-west direction, is always perpendicular to the spin axis of earth.
If the platform is not properly headed, the X-gyro input axis will see a component of the earth's rotation.
The sensing of this rotation by the X gyro can be utilized to direct the platform into proper heading.

In Fig. 7-4, the input axis of the three-axis platform is shown at some point on the earth.
The point is at a latitude **yl, and the platform is at an error in heading east.
The earth is spinning at an angular velocity **zq equal to one revolution per 24 hr.
When the platform is level, **ye is a rotation about the Z axis of the platform Af.
Since the earth is rotating and the unleveled gyro-stabilized platform is fixed with respect to a reference in space, an observer on the earth will see the platform rotating (with respect to the earth).