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Apollo Guidance and Navigation
Considerations of Apollo IMU Gimbal Lock

MIT Instrumentation Laboratory Document E-1344

David Hoag

April 1963


This report was scanned and formated by Marv Hein.


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E -1344

CONSIDERATION OF APOLLO

IMU GIMBAL LOCK

by David Hoag

April 1963

 

 

ABSTRACT

The Apollo inertial measurement unit provides specific force measurements within the guidance system as well as orientation signals to the control system and the pilot's attitude display. The proper operation of the IMU requires that the gyros mounted on the stable member - the "platform" - generate signals to the gimbal drive servos so that the stable member is kept non-rotating independent of any vehicle rotations. This memo will discuss the limitations of the IMU in maintaining this stabilizing function and the resulting operational and emergency constraints imposed both from mission success and crew safety points of view.

 

 

 

CONSIDERATIONS OF APOLLO IMU GIMBAL LOCK

1. Introduction

The Apollo inertial measurement unit provides specific force measurements within the guidance system as well as orientation signals to the control system and the pilot's attitude display. The proper operation of the IMU requires that the gyros mounted on the stable member - the "platform" - generate signals to the gimbal drive servos so that the stable member is kept non-rotating independent of any vehicle rotations. This memo will discuss the limitations of the IMU in maintaining this stabilizing function and the resulting operational and emergency constraints imposed both from mission success and crew safety points of view.

2. Existing IMU Characteristics

The IMU now reaching design completion for use in Apollo is a three degree of freedom gimballed structure shown in "stick and wire" schematic form in Figure 1. Structural features of the gimbals can be seen in Figures 2 and 3 which show various degrees of assembly of a display model.

 

Each gimbal axis of this IMU has the servo torque motors and electromagnetic data transducers directly coupled to the two adjacent members without operating through a gear train. This operation without gears has a multifold advantage. Most obvious is the elimination of concern about gear wear and accuracy of mesh which are critical factors in making a reliable servo of the necessary high performance. Equally important is elimination of the requirement for the servo to provide torque to accelerate gear train inertia when no angular velocity is desired of the driven gimbal. Without gear trains the inertia of the stable member would hold itself stationary without any help from the servo against any magnitude of base rotations except for the effects of (1) bearing friction, (2) motor "back emf", and (3) outer gimbal inertia. Discounting these last two effects, the servo need only overcome the small bearing friction of the Apollo IMU gimbals, even for extremely violent base or vehicle rotations. There are none of the limitations due to high angular rates or accelerations of the vehicle imposed by the usual gimbal axis gear trains.

The second effect mentioned above, back emf, concerns the voltage generator action of the motor, the servo amplifier drive impedance, a corresponding lag term in the servo loop, and a base motion coupling. The level of the output current feedback in the Apollo IMU servo controls these effects adequately so that any concern of base motion angular velocity coupling due to motor voltage is trivial within any conceivable uncontrolled vehicle maneuvers from which recovery is possible.

The third effect - that due to outer gimbal inertia - comes into play only with large middle gimbal angles away from the zero orientation shown in Figure 1. In the extreme, large middle gimbal angles cause an effect called "gimbal lock". Gimbal lock occurs when the outer gimbal axis is carried around by vehicle motion to be parallel to the inner gimbal axis. At this trivial point the three gimbal axes lie in a single plane. No gimbal freedom now exists to "unwind" base motion about an axis normal to this plane. Even though any vehicle orientation with respect to the stable member can be accommodated by particular sets of the three gimbal angles the condition at gimbal lock prevents accommodation of a particular orientation change from the locked condition.

 

Problems with a three degree-of-freedom system like the Apollo IMU can occur not only directly at the trivial gimbal locked situation. As the locked configuration is approached the stabilization capabilities of the assembly become more and more marginal depending on the design. With proper gyro error signal resolution and gain control the locked configuration can be very closely approached without undesirable effects. However, as gimbal lock is approached higher and higher angular accelerations are required of the outer gimbal to hold the inner member fixed against particular components of base angular velocity. By their own tendency to stay fixed, the inertias of the inner gimbals can create much of the necessary reaction to help cause the required acceleration over a limited range of the outer gimbal. This would be perfect without accelerating torque from the outer servo motor if either the inner structures were of infinite inertia or the outer structure was zero inertia. Lacking an infinite ratio of the two, the remaining burden of providing the accelerating torque must be taken by the servo and outer gimbal motor. In Apollo; the outer gimbal structure achieves necessary stiffness through the thin section spherical shape (Fig. 3) with a relatively small inertia. The inner stable member (Fig. 2) carries all the mass of the inertial components and necessary thermal sink mounting block.

Besides favorable inertia ratios, much of the consistent capability of the Apollo IMU to handle near gimbal lock conditions can be attributed to the use of a small angular accelerometer (ADA) as a servo stabilization feedback element on each axis. This permits very high torque gains over all frequencies and allows specification operation over a wide gain margin. No critical adjustments are necessary. Transient loss of attitude due to gimbal lock effects, or any other disturbance, does not necessarily mean a permanent loss of orientation unless the gyro gimbal limit stops are reached. Within the integration range of the gyro the attitude is recovered as the gyro error is brought back to zero.

Firm groundrules on how close to gimbal lock the Apollo IMU could operate satisfactorily depend upon experimental results with the actual flight configuration IMU. Within the last month these data have started to be collected using IMU #3 with breadboard electronics. It appears that gimbal lock can be approached as close as 10 degrees without risk and even much closer with some possibility of loss of stable member attitude. Stated more dramatically, the inner axis of the system was aligned within 10 degrees of a base motion axis perpendicular to the output axis.

Base motion angular velocity then caused gimbal lock to be passed within 10 degrees. Stable member attitude was held consistently for this configuration with base angular velocities of 60 degrees per second.

A tentative listing of acceptable vehicle rates and accelerations follows:

Vehicle Angular Velocity Allowable

About inner gimbal axis (continuous)

720 deg/sec*

About middle gimbal axis (80 deg)

720 deg/sec*

About outer gimbal axis (continuous)

720 deg/sec*

About any arbitrary body or inertial axis 
resulting in passing up to 10 deg of 
gimbal lock (continuous)					

60 deg/sec

Vehicle Angular Acceleration Allowable

About any axis and within above rate limits

360 deg/sec2*

*Values marked with an asterisk have much higher limits, but are as yet undetermined.

 

3. IMU Operation to Avoid Gimbal Lock

Although the allowable vehicle motions described above to avoid effects near gimbal lock are less constricting than might be expected, the area near gimbal lock should be avoided. The Apollo IMU will normally be shut down during all long periods not requiring its use. This is done primarily to save power and corresponding fuel cell battery reactant (estimated saving of 43 pounds of reactant in a 200 hour command module lunar landing mission). For this reason, and because of drift over long time intervals, the guidance system provides for in-flight IMU alignment against star references before the start of each accelerated phase of the mission. This allows the IMU stable member alignment to be chosen for each use to the most logical orientation. Simplifications result in the computer generation of steering commands if the "X" accelerometer axis on the stable member is aligned in some direction near parallel to the coming thrust (or entry atmospheric drag). This happens also to be optimum with respect to IMU measurement effects causing errors in velocity measurement. Since the X accelerometer is perpendicular to the IMU inner gimbal axis, the direction of this inner axis can be chosen as required. For each mission phase involving rocket burning or atmospheric drag, the trajectory and the thrust or drag lie fairly close to some fixed plane. The inner gimbal axis is then aligned somewhere nearly perpendicular to this plane. All required maneuvers result mostly in inner gimbal motion, thus avoiding the difficulty of approaching gimbal lock associated with large middle gimbal angles. Finally because large roll maneuvers are desirable (for instance during entry) the outer gimbal axis is mounted to the spacecraft along or near the roll axis so that no restriction on roll maneuver ever exists.

For a normal mission the only time where star fix IMU realignment might not be desirable between uses of the IMU is during the LEM rendezvous midcourse correction maneuver. There are a number of possible procedures to handle this situation if a thrusting is required too close to gimbal lock direction which is likely to occur on only 1 1/2% of the missions.

One or more of the following will be implemented as a result of current studies.

  1. Two thrust. A desired thrust in a lock direction can be broken into two components lying sufficiently away from the inner axis. A rendezvous midcourse correction maneuver of about 50 feet per second could be made up from two 26. 5 feet per second burns 20 degrees from the inner axis causing a penalty of 3 feet per second or about 1 pound of fuel.
  2. Offset outer axis. The gimbal outer axis could be mounted 33 degrees (for instance) away from the thrust axis as it is in the command module. This would permit thrusting in any direction, the only constraint being in how thrusting orientation is approached. This will be discussed later.
  3. Thrust component in non-sensitive direction. At the point of midcourse correction, there is a direction in which thrusting will cause no effect on accomplishing rendezvous ... only an effect on the time it occurs. This direction lies near the orbital planes and is well away from the lock direction. Thus a 50 ft/sec correction in the lock direction could be replaced by a single correction no larger than 53 feet per second in a direction 20 degrees away from gimbal lock. If a small delay in rendezvous could be tolerated the possibility of a smaller than 50 ft/sec correction might result. The fuel penalties determined in these cases must recognize the effect of changing the closing velocity and the corresponding braking maneuver.
  4. Planned velocity change. If the takeoff trajectory is cut off slightly less than 160 ft/sec early, then a planned correction in the "easy" direction of 160 ft/sec would be required to be added to (at worst) to a 150 ft/second (3 sigma magnitude) midcourse correction in a random direction. The result could get no closer than 20 degrees towards gimbal lock where only 55 ft/sec would be required. (In fact the rms total correction magnitude due to the delayed application of the 160 ft/sec and the 150 ft/sec 3 sigma correction is only 168 ft/sec. The "worst case" of 160 + 150 = 310 ft/sec would only rarely occur.)
  5. Open loop realignment. The accuracies required for the rendezvous maneuvers are such that simple realignment by the computer of the IMU is sufficiently precise without requiring star fixes and astronaut activity. The probability of needing realignment is small (as will be seen) but if it is required then it can be accomplished automatically by the computer at a rate of about 1 degree per second to accuracies better than 500 parts per million.

Either of the last two methods are judged at present to be the most likely procedures to use.

Avoidance of any gimbal lock problem during the actual thrusting and entry phases in Apollo seems to offer no difficulty. The astronauts should be concerned, however, of what might be required in their manual attitude change procedures during any phase the IMU is on and aligned to avoid trouble. First will be explained just what attitudes and angles are permitted and what are forbidden. The situation will be described in terms of the LEM. The operations in the command module differ slightly due to the 33 degree offset of the IMU outer axis from the thrust axis.

The naming of the LEM body axis must be emphasized. Since the crew sits with their backbone along the thrust axis in the LEM (unlike an airplane where they look towards the thrust axis) there is possibility of confusion until the convention is firmly fixed in mind: In an airplane, in the command module, and in the LEM the roll axis is defined parallel to the thrust axis.* Thus in the stationary hover orientation, the LEM roll axis is vertical with respect to the local surface of the moon. The crew looks forward in the direction of the yaw axis.

*This definition of LEM roll axis was given to the writer by APO at MSC.

The IMU will probably be mounted with the outer gimbal axis parallel to the LEM roll axis. The inner gimbal axis will be aligned and inertially stabilized in a direction perpendicular to the landing or takeoff trajectory plane. The vehicle is at zero roll when the yaw axis is in this plane.

GIMBAL LOCK OCCURS WHEN THE LEM THRUST AXIS IS POINTED ALONG THE SPACE DIRECTION OF THE INERTIALLY STABILIZED INNER GIMBAL AXIS WHICH IS ALIGNED HORIZONTAL AND PERPENDICULAR TO THE PLANE OF THE LANDING OR TAKEOFF TRAJECTORY.

Gimbal lock is avoided by a wide margin, then, by attitudes constrained as follows:

IMU GIMBAL FREEDOM IN THE LEM PERMITS:

a. ANY ROLL ANGLE

b. ANY PITCH ANGLE AT ZERO ROLL

c. ANY YAW ANGLE AT 90' OR 270' ROLL

Actually, since it was seen in the previous section that gimbal lock can be approached within 10 degrees without danger of loss of attitude the criterion becomes:

IMU PRECISION ATTITUDE WILL BE HELD AS LONG AS THE LEM THRUST AXIS IS NOT POINTED WITHIN 10 DEGREES OF THE SPACE DIRECTION OF THE INERTIALLY STABILIZED INNER GIMBAL AXIS WHICH IS ALIGNED HORIZONTAL AND PERPENDICULAR TO THE PLANE OF THE LANDING OR TAKE-OFF TRAJECTORY.

This rules out only an opposite pair of very small areas horizontally out each side of the path in which the thrust axis must not be pointed. This leads to another more general expression of the rule:

IMU PRECISION ATTITUDE IS HELD FOR:

a. ANY ROLL ANGLE ABOUT THE THRUST AXIS

b. 98.5% OF ALL POSSIBLE DIRECTIONS OF THE THRUST AXIS ACHIEVED IN ANY COMBINATION

OF PITCH AND YAW.

The manual or automatic operations also have other clear groundrules and warnings to prevent steering into the 1.5% danger areas. The flight director's attitude indicator - the "eight ball" will be constructed with a gimbal order compatible with that of the IMU. The IMU gimbal angle transducers will drive the ball attitude when the IMU is on and aligned. Two warning areas marked at each pole of the ball centered on the ball inner gimbal axis bearings give exactly the correct danger area information to the pilot required:

SPACECRAFT ATTITUDES SHOULD NOT BE PERMITTED TO PASS INTO THE TWO WARNING AREAS MARKED ABOUT EACH POLE OF THE FLIGHT ATTITUDE BALL.

The danger situation with the IMU can be expressed simply in terms of the middle gimbal angle. Middle gimbal angles within the range 80 degrees are in the safe area; outside this range is danger. The system is arranged to close a switch contact as either danger area is approached. This contact informs the computer of the fact and energizes warning lights for astronaut use:

SPACECRAFT ATTITUDES ARE APPROACHING THE GIMBAL LOCK DANGER AREA WHEN THE "GIMBAL LOCK" WARNING LIGHT GOES ON.

The computer will inhibit itself from commanding any middle CDU angle greater than that safe by the above simple criterion. This provides a direct logical rule for the computer to command safely spacecraft attitude changes.

If IMU attitude is lost for any reason including approaching too close to gimbal lock, the guidance system automatic monitor of gyro error signals indicates the fact.

LOSS OF IMU ATTITUDE DUE TO ANY CAUSE IS INDICATED IMMEDIATELY BY THE "IMU ERROR" WARNING LIGHT.

 

4. Emergency and Abort Situations

For the majority of mission aborts, the IMU gimbal lock situation puts no more constraint on successful abort than does it upon the normal phases. Two mission phases have been identified, however, which might require critical dependence upon the IMU indicated attitude to cope successfully with the emergency. The first of these is high altitude abort prior to launch escape tower jettison. If the command module tumbles during this operation there is then a possibility the outer axis might pass through the critical areas near gimbal lock causing loss of IMU attitude information. If correct entry attitude is not assumed early enough, the reaction jets might not be able to overcome and correct the improper but stable attitude "point" forward to the proper entry orientation.

With closed windows and the limitations of the backup attitude system, dependency on IMU attitude information would be necessary. The probability of exceeding crew stress limits in the situation described is the product of the probability that abort is initiated at the critical altitude, the probability that the abort initiates an uncontrolled tumble, the probability that the tumble causes the IMU to pass through its critical gimbal lock areas, and the probability that the pilot cannot sense the direction of drag early enough to correct attitude or the probability that the command module enters sharp end first.* This memo will not try to evaluate these probabilities.

*As of the date of this writing the command module has aerodynamic surfaces provided to prevent wrong end entry. The above discussion is then only of academic interest.

Before discussing the second critical situation identified, it would be well to point out what would appear to be the best action in the situation of an unintentional tumble at any time in the mission. Even if the IMU gave perfect data to the attitude ball, the control of the tumble by the pilot would far better use the body rate indications.

After rate is brought to sufficiently low values the desired attitude can then be attempted. Depending upon the mission phase and other factors the pilot may use any of several attitude indications. If the IMU didn't lose orientation - assuming it was on and aligned in the first place - the pilot would know this fact by the automatic error sensing system and error indicator lights. Any loss of IMU attitude is detected and the fact displayed on the appropriate light as mentioned earlier.

Assuming the IMU attitude information is lost, the best orientation data would be obtained visually through the windows. Near the earth one would use the earth; near the moon one would use the moon. With this aid the pilot can orient the spacecraft as accurately as he can judge the proper orientation. Once this is done he can operate the "coarse align" mode control provided at the pilot's AGC keyboard. This would quickly slew the IMU to the spacecraft attitude described by the CDUs. The computer would be programmed to set the CDUs on zero as an action following emergency detection. Releasing the control sets the IMU free to indicate deviations from this attitude. This constitutes a very quick realignment procedure.

VERY FAST IMU ALIGNMENT CAN BE ACCOMPLISHED IN EMERGENCY BY THE PILOT SETTING "COARSE ALIGNMENT" MODE MOMENTARILY TO ALIGN THE IMU TO THE EXISTING SPACECRAFT ATTITUDE.

The second mission phase which has been identified as being critical with respect to IMU gimbal lock limitations is in the LEM during lunar landing. A hard over landing engine gimbal failure in the yaw direction would require positive pilot action almost immediately to avoid gimbal lock. From a vertical hover orientation a random tumble from the vertical has an 11% chance of passing within 10 degrees of the lock orientation if not stopped before 90 degrees. If the attitude information was lost by such a maneuver, the vehicle would be thrusting nearly horizontally and probably downward unless the engine were immediately shut down which, of course, would be the proper first action in the emergency. The view of the moon through windows of the LEM would be the best source of orientation to try to erect the vehicle prior to aborting with the takeoff engine. The azimuth information could be judged to a good degree by the direction lunar features pass beneath.

Once correct attitude is obtained, the IMU can be realigned to the existing vehicle orientation by use of the "IMU coarse alignment" mode mentioned above. The current LEM landing engine now has limited engine gimbaling used only to correct thrust axis direction; steering is done by separate jets. It would appear that the speed of reaction necessary for a hard over engine now is alleviated considerably.

5. Considerations of a Four Degree of Freedom IMU

The difficulties near gimbal lock can be avoided by the addition of a fourth gimbal to the IMU. This will be called here the redundant gimbal since it provides more degrees of freedom than theoretically necessary. This redundant gimbal will be considered in this memo to be mounted outside the normal outer gimbal. The order used in this description is then: inner, middle, outer, and redundant; see Figure 4. The most likely operation would use the inner three gimbals to drive the stabilizing gyro error signals to zero while the fourth is driven so as to keep the middle gimbal near zero and away from the gimbal lock orientation. This can be done by generating a redundant gimbal rate command by expressions similar to

A(redundant) = k sin A(middle) / cos A(outer)

so that a negative feedback on middle angle occurs to drive middle angle towards zero.* It should be possible to make the inner three gimbals have the same dynamic performance as the simpler three degree of freedom system. Any base motion coupling, though, avoided in Apollo as described in the first section would make redundant gimbal motions appear as disturbances on the middle or inner servos requiring special attention in loop responses. The redundant gimbal must be accelerated if it is to do its job even when middle gimbal angle is near zero, In fact a situation like gimbal lock occurs for outer angles near 90 degrees as can be seen in the above equation. Close to 90 degrees outer gimbal angle, the redundant gimbal must be driven at a very fast rate to hold middle at zero. In practice this offers no real difficulty as long as the vehicle body rates are within certain limits.

*A Non-Locking Four-Gimbal Method of Isolating a Platform From a Rotating Vehicle, by Richard C. Hutchinson, MIT/IL Report R-285, April 1961, Chapter 8.

With the four degree of freedom gimbal system there are no constraints on vehicle attitude although rate limits do still exist.

To instrument the fourth gimbal requires an extra servo different from the other three servos and extra data transducers or the equivalent, on the middle and outer axes to determine the redundant gimbal commands. The expense in Apollo of adding the fourth gimbal depends upon a number of factors. The IMU alignment accuracy desired in Apollo is a critical design factor in order to achieve the required guidance system performance for the lunar mission. This means gimbal data transducers having peak errors no worse than 20 seconds of arc are required to use the optics system for star alignment references. The addition of a fourth gimbal compounds the problem in the IMU structure, in computer interface, and the program the flight computer must provide to achieve the alignment.

At present the 20 arc second data transducers dominate and determine the size of the Apollo IMU. Without these large units the IMU would be several inches smaller in diameter. A fourth gimbal using these same proven transducers would raise IMU structure weight by 15 pounds and increase the volume by 725 cubic inches. *

* Calculations by J. B. Nugent; not considering added inner element transducers to drive the redundant gimbal. Basic numbers are classified.

Certainly a smaller 4 degree of freedom IMU could be designed if the in-flight star referenced alignment was not required or if a small accurate data transducer was available. The Polaris advanced guidance system in a four degree of freedom configuration is estimated to weigh only 2/3 the Apollo three degree of freedom unit and would have equivalent performance except for the accuracy of the axis data transducers. Advances in small accurate data transducers are being studied with the goal of direct digital encoding of gimbal angles. This could lead to some simplification of the CDUs or by their replacement with a fast supplementary steering interface computer. The present computer is not configured to close satisfactorily the spacecraft attitude loop directly.

Additional complexity would result in making full use of a four degree of freedom system using current hardware. An additional CDU (3 pounds plus electronics) would be required for flexible operation with the computer. The additional gimbal would require a longer resolution chain to generate autopilot attitude error. This chain is already difficult to achieve the requirements imposed by the existing autopilot interface. Also the computer would have to assume the burden of providing the real time resolutions required to convert steering angles in accelerometer coordinates into proper CDU commands.

The addition of a fourth gimbal, other things being equal, necessarily increases the power drain on the main fuel cell batteries, besides requiring an extra set of servo electronics. Heat transfer from the gyros inside to the heat sink on the housing is made much more difficult by the additional fourth gimbal. The redundant axis would have to have slip rings to be useful which would have to carry all signals on the outer axis rings plus those extra due to the redundant axis instrumentation.

One awkward difficulty is the generation of the three degree of freedom attitude ball commands from a redundant four degree of freedom IMU. This would be solved by the development of a compatible four degree of freedom ball, but the size penalty on the panel would be severe. The electrical conversion from four degrees of freedom to the necessary three gets bound up in the equivalence of gimbal lock which the redundant gimbal was added to prevent. Although the four degree of freedom IMU avoids gimbal lock and loss of attitude, this attitude information is difficult to get at.

The advantages of the redundant gimbal seem to be outweighed by the equipment simplicity, size advantages, and corresponding implied reliability of the direct three degree of freedom unit.