Sample J80 from C. J. Savant et al., Principles of Inertial Navigation. New York: McGraw-Hill Book Company, Inc., 1961. Pp. 144-150. A part of the XML version of the Brown Corpus2,046 words 24 symbols 16 formulasJ80

CopyrightMcGraw-Hill Book Company, Inc., Used by permission. 0010-1890

C. J. Savant et al., Principles of Inertial Navigation. New York: McGraw-Hill Book Company, Inc., 1961. Pp. 144-150.

Arbitrary Hyphen: non-linear [0620]Note: The input axis...are parallel [1700]

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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 ) .