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This answer to a related question suggests that the order in which rotations around three principal axes are applied (in order to estimate the conversion of an attitude to a set of roll, pitch and yaw figures) is important.

This leads to another question: How exactly were CSM maneuvers for the purpose of changing its attitude performed?

  1. Were there three separate consecutive sets of firings of the RCS thrusters (i.e. one firing to rotate around one axis, then firing to come to stop, then another firing around second axis, come to stop, then third firing around third axis, come to stop), or was there a multiple simultaneous firing of various RCS thrusters at once (i.e. to get to the the new required attitude in one go, so to speak)?

  2. If it is the former, was there a rule for a specific order to be followed, for example, first Roll, then Pitch, after that Yaw, or maybe first Pitch, then Yaw after that Roll?


Let's consider a particular example from Apollo 11 Flight Journal. Capcom Bruce McCandless at 025:49:20 into the flight gives PAD for Midcourse Correction burn number 2. The Journal editors give the following interpretation for the spacecraft attitude:

Spacecraft attitude: Roll, 277°; Pitch, 355°; Yaw, 15°. This is with respect to the attitude of the guidance platform, itself aligned to the PTC REFSMMAT.

If I understand the quote correctly, then regardless of current CSM attitude, at the end of the maneuver, the spacecraft will have to be positioned at Roll 277°; Pitch, 355° and Yaw, 15° with respect to the corresponding axes of the current alignment of the guidance platform.

I assume the astronauts just plug in the numbers in DSKY, and computer does the calculation to get them from whatever their current attitude is to the required one.

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  • $\begingroup$ In a perfect world all masses of the CSM would be balanced with respect to all three axes of the spacecraft and all RCS thrusters would be perfectly oriented and would deliver equal thrusts. But in real world there are errors of balance and orientation and thrust forces. I wonder if the computer acheived an acceptable orientation error with only a pair of firings for each axis. Rotating around one axis may cause unwanted rotations around the other axis. $\endgroup$ – Uwe Jan 31 at 16:26
  • $\begingroup$ @Uwe Maybe there was a procedure to "trim the residuals" after maneuver finishes (like a very fine final positioning after the spacecraft comes to a stop, I don't know). The focus of my question though is on the bulk of the attitude change maneuver. How exactly was it performed? $\endgroup$ – Sergiy Lenzion Jan 31 at 16:39
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    $\begingroup$ It's not that the order of spacecraft rotations matters, it's that the order of rotations matters. This is because the group of rotations in 3d-space, $\mathrm{SO(3)}$, is not abelian: it's a fundamental property of the space we live in. $\endgroup$ – tfb Jan 31 at 18:10
  • $\begingroup$ In practice, the order of rotations matters only for large rotation angles. For small angles, all rotation sequences give approximately the same results, and the exact sequence can be ignored. So if you can get the spacecraft close to the target orientation, then you can accurately finish off by simultaneously controlling all axes, it seems... And if simultaneous axis control is enough to get you from the large-angle region to the small-angle region, then you might get away with always applying simultaneous control about all axes... $\endgroup$ – Alex Aug 18 at 18:53
  • $\begingroup$ @Alex sounds like you are confused about what a single axis rotation is. $\endgroup$ – Organic Marble Aug 18 at 19:30
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A single axis rotation (SAR) was performed from the initial to final attitude, unless they were avoiding gimbal lock, in which case it was split into two rotations.

It has been shown in MIT/IL Report E-1832 that it is convenient to perform attitude maneuvers by simultaneous maneuvers in three axes. However, under certain circumstances this leads to maneuvers through the area of gimbal lock warning on the Inertial Measurement Unit. In this event the maneuver is split into two parts such that the gimbal lock area is avoided.

The inputs to the attitude maneuver computation are the three gimbal angles desired as the final orientation of the spacecraft with respect to the IMU stable member. In order to convert from spacecraft axes to control axes, all outer gimbal angles are modified by the 7. 25 degree reaction jet offset.The rotational rate to be used by the maneuver is also required.

From the present gimbal angles and the required final gimbal angles a rotation matrix is computed which describes the transformation from the initial to the final attitude. From this matrix the eigenvector giving the direction of required rotation is derived by partitioning the matrix into its symmetric and antisymmetric components. The equivalent angle of maneuver is obtained,and using the magnitude of maneuver rate, the time of maneuver is computed. In addition, the rotation vector of the maneuver rate is resolved into the three control axes of the spacecraft.

The inputs from the maneuver computation program to the Reaction Control System being described are these three spacecraft rates, and the time of maneuver. Where gimbal lock is to be avoided, the two component maneuvers are sent to the control system separately.

Apollo GNC paragraph 3.6

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    $\begingroup$ Thank you for the prompt and definitive answer! $\endgroup$ – Sergiy Lenzion Feb 1 at 0:31
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    $\begingroup$ So it seems the GNC takes the specified Euler angles relative to the IMU gimbals, converts the Euler angles to a rotation matrix, then converts the rotation matrix to an axis-angle parameterization, before resolving the axis into the spacecraft's local coordinate system (or frame) in order to partition the reaction control command among the local x, y, and z axes? Thanks for the quote and reference, Organic Marble. Very useful! $\endgroup$ – Alex Aug 18 at 18:40

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