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Why the euler angles or any other methods which are being used to find out the movement of spacecraft using iterated rather than integrated to find yaw, pitch, roll of the spacecraft?

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    $\begingroup$ I’m voting to close this question because it belongs on Space Exploration $\endgroup$ Commented yesterday
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    $\begingroup$ Probably, there is no analytical solution to the equations of rotation in three dimensions, similar to the Three Body Problem. Such problems have to be solved with numerical methods. $\endgroup$ Commented yesterday
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    $\begingroup$ "iterated rather than integrated". Do you have a source for this? AFAIK, they are integrated (numerically). $\endgroup$ Commented 20 hours ago
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    $\begingroup$ In my opinion this might be better answerd on math.se. $\endgroup$ Commented 15 hours ago
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    $\begingroup$ Sanjay, if you see this, I think your question needs some clarification. Let us know if the answer from Wyck answers what you're asking. $\endgroup$ Commented 10 hours ago

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Quaternions are used internally for calculations for their numerical stability and Euler angles are used in the user interface because they can be understood intuitively.

A rotation or orientation can be represented in at least three common ways:

  • Euler angles, which is an ordered set of axis-aligned rotations, each a discrete rotation about one of the primary axes, and which can be represented by 3 scalar quantities.
  • Quaternions, which can be represented mathematically by 4 scalar quantities.
  • A transformation matrix, in 3D, often a 3x3 matrix or a 4x4 homogeneous matrix.

Euler angles suffer from at least two big problems:

  • Gimbal lock. A loss of freedom due to a topological constraint on the source and target spaces.
  • Numerical instability. The cumulative calculation of many successive small rotations often diverges from the correct answer due to floating point precision limitations. Matrix representation of rotations suffers from this too. Whereas quaternions are more stable.

Quaternions resolve both those issues. But quaternions are difficult to understand intuitively. It's much more challenging to look at the 4 components of an arbitrary quaternion and intuit what rotation it represents. Whereas Euler angles are somewhat easier to intuit as yaw, pitch and roll.

So what usually happens is that rotations and orientations are represented internally in software as quaternions so that mathematical operations on them maintain numerical stability and calculations are accurate. But user-facing representations of orientations are often converted to Euler angles for display. And user-input rotations are given as Euler angles (or as a rotation about a 3D vector) and then converted to quaternions for use in calculations internally.

A good web page on the numerical stability of quaternions representing spatial rotations will cover most of the critical material. I suggest using that as a search term for further reading.

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  • $\begingroup$ Matrices need fewer math operations if your inputs/outputs are vectors (star sightings, magnetometer observations, torques, etc.). Tradeoffs... $\endgroup$ Commented 13 hours ago
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    $\begingroup$ This doesn't seem to address the iterated vs integrated part of the question. $\endgroup$ Commented 11 hours ago
  • $\begingroup$ Nitpick: rotations don't cover all of quaternions, they cover specifically the unit quaternions. The additional constraint of unit magnitude means that unit quaternions can be represented by three quantities and the fourth determined by convention as a versor $\endgroup$ Commented 11 hours ago
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    $\begingroup$ The question is about "using iterated rather than integrated euler angles"... Where is this addressed in your answer? Why the upvotes for a not answer? $\endgroup$ Commented 8 hours ago

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