See, also, MIT Instrumentation Laboratory Document E-1344, 'Apollo Guidance and Navigation: Considerations of Apollo IMU Gimbal Lock 'by David Hoag, April 1963.
Examples of the IMU drawing linked below can be found in various editions of Grumman's LM Apollo Operations Handbook, such as page 2.1-47 in LM 10, Vol. 1 (37 Mb)
"The Inertial Measurement Unit (IMU) has Outer, Middle, and Inner gimbals (schematic), with the stable platform - labeled 'stable member' in the diagram - mounted on the inner gimbal axis. The Outer gimbal is mounted on the Navigation Base, which in turn is rigidly mounted to the spacecraft. The 'gimbal angles' are the Euler angles between the 'stable platform' and the 'navigation base' as measured relative to the navigation base itself. In the LM, the AOT is also mounted on the NAV base. So, for platform alignment it would be convenient to measure all angles relative to the nav base.
"As an example, for the lunar landing, the LM platform is aligned with the landing site coordinate system. If the LM touches down exactly on time, and on perfectly level ground with the z-axis aligned in the plane of the CM orbit, the Outer Gimbal Angle (OGA), Inner Gimbal Angle (IG), and Middle Gimbal Angle (MGA) would all equal zero. With all the above conditions met, if the LM were 50 feet directly above the landing site, with all three gimbal angles reading zero, and I pitch the LM backward (noseup) 30 degrees, the 8-ball would read pitch 30, roll 0, yaw 0. Would the pitch gimbal angle display as 30 degrees, or as 330 degrees?"
Journal Contributor Paul Fjeld replied:
"The LM Body coordinate system is right-handed, with the +X axis pointing up through the thrust axis, the +Y axis pointing right when facing forward which is along the +Z axis. The rotational transformation matrix is constructed by a 2-3-1 Euler sequence, that is: Pitch about Y, then Roll about Z and, finally, Yaw about X. Positive rotations are pitch up, roll right, yaw left (point your right thumb along the positive axis: your fingers curl positively)."
"The Inertial Measurement Unit case is bolted to the NAV base (along with the Alignment Optical Telescope and the Abort Sensor Assembly). Three rotating gimbal rings are mounted in sequence: the outer (X axis) to the case, the middle (Z axis) to the outer ring and the inner (Y axis) to the middle ring. Torque motors keep the stable platform inertially aligned in a useful orientation."
"For the landing, that orientation was parallel to the Platform Coordinate System whose origin was the moon's center. +X pointed up through the position of the landing site at the time of landing, +Y was perpendicular to the fictitious plane formed by the Command Module's Velocity vector and +X (Vcm cross X), and +Z completed the right-hand system (X cross Y)."
"The LM's pointing in this reference system was calculated by constructing a Stable Member to Navigation Base (AKA Platform to Body) matrix using the reported angles at each gimbal. This critical bit of mathematics related all of the accelerations, velocities, positions, etc. calculated in the analytical space of the computer to (hopefully) the real world of the moon."
"In the case of Tom's pitch up, the LM rotates counter-clockwise (looking at it from the right (+Y) side) which is a positive rotation: '30 degrees on the ball'."
"If the middle (Z) gimbal was +- 90 degrees roll, the inner gimbal axis would be coincident with the outer gimbal axis and you would loose a dimension. Near that point, in a closed stabilization loop, the torque motors could theoretically be commanded to flip the gimbal 180 degrees instantaneously. Instead, in the LM, the computer flashed a 'gimbal lock' warning at 70 degrees and froze the IMU at 85 degrees, flashing an unfriendly 'no att' light. Then it waited for a realignment."
"Mathematically, the commanded gimbal angles were extracted from a commanded axis matrix, a calculation which included dividing by the cos of Z. If Z is 90, cos is 0 and that division is illegal!"
104:59:35 Garriott: Columbia, Houston. We noticed you are maneuvering very close to gimbal lock. I suggest you move back away. Over.
104:59:43 Collins: Yeah. I am going around it, doing a CMC Auto maneuver to the Pad values of roll 270, pitch 101, yaw 45.
104:59:52 Garriott: Roger, Columbia. (Long Pause)
105:00:30 Collins: (Faint, joking) How about sending me a fourth gimbal for Christmas.
[Armstrong - "This is Mike at his best. We had a four-gimbal platform on Gemini."]The following is taken from an April 1963 MIT Instrumentation Laboratory Document, E-1344, 'Apollo Guidance and Navigation: Consideration of Apollo IMU Gimbal Lock' by David Hoag. "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. The most likely operation would use the inner three gimbals to drive the stabilizing gyro error signals to zero while the fourth if 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."
"It is really interesting that Dave Hoag would broach the 4-gimbal concept so late in the Apollo design phase. MIT waged a 'war' to convince everybody, especially the astronauts, that the simpler, three-gimbal unit was the only way to get Apollo to the moon on time. The idea was to take the Polaris system and moonize it."
"Most people think Apollo was about making technological breakthroughs when I think it was really about using as much as possible of what was on the shelf where they could get away with it. I'm beginning to believe Apollo was really about making management breakthroughs. Time - meaning the Kennedy deadline of landing 'before this decade is out' was the real driver."