# pitch, yaw and roll

Imagine a spacecraft in space, which is required to dock with other. Presume that it needs to "pitch" by X degrees, "yaw" by Y degrees, and "roll" by Z degrees (although "roll" may not have any importance w.r.t. docking, presuming that the mating components would be aligning axially) so as to align perfectly for docking.

The question is: are all the three maneuvers done simultaneously - with each motion taking place in its own closed loop and the entire process supervised by an external closed loop to correct eventual errors, or sequentially, with the supervisory loop still being active?

• It's not directly my field, so my comments aren't authoritative enough for an answer, but these days, spacecraft tend to use quaternions for their attitude state instead of Tait-Bryan angles, as the quaternion representation avoids singularities, and the math works out cleaner. In general attitude maneuvers tend to be a "shortest path" approach -- it can be proven that the difference between any two attitudes is a single rotation about a single axis. Aug 11, 2020 at 14:36
• @Tristan. Honestly, I could not understand what you have said. May be that is why, I feel my question is not answered. My question was asking about the correct method to achieve perfect alignment. Perhaps the two words - quaternions and Tait-Bryan angles talk about the same, which is beyond my wisdom. But thanks any way for responding. Aug 11, 2020 at 14:49
• The Apollo spacecrafts had to avoid a gimbal lock when doing an atittude maneuver. Therefore not all combinations of pitch, yaw and roll were possible.
– Uwe
Aug 11, 2020 at 14:57
• FYI, roll does matter for docking. Exactly how much it matters depends on what docking system is being used, but I don't think I've heard of having even 10 degrees of roll tolerance. Aug 11, 2020 at 22:48
• even probe-and-drogue doesn't eliminate roll requirements. Take a look at the Soyuz docking interface in spaceflight.nasa.gov/gallery/images/station/crew-17/hires/… See the pin on the lower left of the Soyuz' front and the socket opposite it? Those align with features on the space-station side of the docking interface. The International Docking Adapters also have roll requirements, both for petal engagement and for power and data hookups around the rim of the docking adapter. Aug 12, 2020 at 8:24

With modern computers and software models of spacecraft, there is no reason not to take the "shortest path" approach when adjusting a spacecraft's attitude outside of certain edge scenarios. This is because, for a computer, adjusting all three axis simultaneously and handling the potentially complex interplay/second order effects resulting from using multiple attitude adjustment systems simultaneously is rather easy.

Some of the potential effects the computer needs to deal with include:

• Shifting center of mass due to shifting passengers or fuel depletion
• Plume interactions between firing RCS thrusters
• Gyroscopic effects resulting from gyroscopes
• Magnetic field effects from magnetorquers
• Mechanical backlash for valves and thrusters

The sequential approach would only be useful if you're working under some limitation, for example:

• A human pilot who can only handle one or two axis at a time (This is why the sequential approach is recommended to people playing KSP or other space-sim games)
• An RCS system which can only fire a limited amount of thrusters simultaniously
• I agree. Computers are certainly accurate and faster in taking an "overview" and deciding corrective action and its quantum. Further they can look over all 3 axis simultaneously unlike humans. I had missed out on the effects of shifting C.G. while attempting to dock. Aug 12, 2020 at 7:27

The order in which the rotations is performed always important in an Euler rotation sequence. You can show this to yourself by picking up a book and applying a roll / pitch sequence versus a pitch / roll sequence. This is very unlike translation, where first going 1 km north and then 1 km east brings you to the same point as does first going 1 km east and then 1 km north. Translations commute; rotations do not.

We humans use Euler-like rotation sequences because it's hard for even the best of us to fully understand the weirdness of rotations in three dimensional space. The mathematics is well understood, making the difficulty of visualization a non-problem for computers. For any given multiple axis rotation sequence, there is always a single axis rotation that will arrive at the same orientation.

• First of all thanks for your answer, and secondly apologies for my delayed response. I will surely gather more knowledge on Euler rotation sequence. As of date, I have not come across the same. Thank you. Sep 3, 2020 at 4:36
• David is right, but the order of rotations matters only for large rotations (larger than ~10 deg). For small rotations, you could technically shift their order or do them simultaneously with a small error (which a feedback controller would easily correct for). Moreover, it is possible to decompose the angular velocity vector into components (since these do not depend on XYZ order) and integrate those to obtain "simultaneous" XYZ rotations to pass as errors a feedback controller IF the individual rotations are small. NASA used this approach on Saturn V and there is a paper on the subject.
– user36480
Jan 8, 2021 at 19:56
• You can check that the order of rotations becomes less and less important as the individual rotations in the sequence become smaller... Take a pencil and rotate it 90 degrees about X, then Y, then Z... then again in reverse. Very different results, right? Now reduce the rotation to just 10 degrees. Now so different anymore, huh? Now reduce it to just 3 degrees. If the attitude error about any one axis is over 10 degrees at any one time, then I think you have bigger trouble then any error you might have introduced in ignoring rotation sequence...
– user36480
Jan 8, 2021 at 20:00