Please, somebody explain why gimbal lock happens properly. Not a single correct solution out there, people always seem to gloss over the important details and if you dig into the common explanations, they all fall to pieces under closer examination.
The common explanation that is given goes something like this...
A plane pitches upwards 90 degrees, now its forward axis "collapses into" its up axis, and suddenly, yaw looks like roll, and it "loses a degree of freedom, blah blah blah."
It's abhorrent that this is the commonly accepted explanation, here's why that is wrong... the controls of the plane are LOCAL to its perspective, so it doesn't matter WHICH way the plane is currently oriented, pitch will always pitch, yaw will always yaw, roll will always roll, and the pilot will be none the wiser, right?
Alternatively, you can think of the world frame and local aircraft frame as two different data sets, and one does not "overwrite the other" ( same in games, you have the world coordinate system and the player local coordinate system, and its not like you discard one in favor of the other; you can get the player world up or local up at any time, and use it to apply the data you need, and yet people always as the copout explanation, "just use squats, don't use Euler angles..." ).
in the example where the plane is now looking up at the sky after it pitched 90 degrees, yes, yawing and rolling at that point will now have different meanings WITH RESPECT to the world coordinate system!
a plane that is pitched up 90 degrees, looking up at the sky, who thinks it is rolling locally, will look like it's YAWing from an outside observer who has the world frame of reference.
a plane that is pitched up 90 degrees, looking up at the sky, who thinks it is YAWING locally, will look like it's ROLLING from that same outside observer who has the world frame of reference.
note that (1) and (2) are STILL DISTINCT ROTATIONS, they DID NOT collapse into the same behavior in any way, shape, or form, as is commonly claimed in gimbal lock!
I obviously know that gimbal lock happens, I BELIEVE it happens, and have seen it plenty of times. I know that the "cop out" solution is just use quaternions, which are obviously better, however, this does NOT address the problem - why does it actually happen? Can somebody please explain how axes are "collapsing into each other?"
*Please do not post a gif of a toy plane rotating within 3 nested gimbals - yes, that is sufficient to prove that gimbals can align... AND???
...its not like they physically get "locked" after that! To wash your hands of the issue and say, "welp that explains it" is intellectually dishonest and lazy, please don't do that. Also, gimbals are an extremely bad and unintuitive way of grokking rotations, please use forward, up, and right axes ( or whatever coordinate conventions u prefer ).
( and one more thing, it's my understanding that Euler rotations are intrinsic, each defined with respect to the latest state of the rotational frame axes, and this "accumulation" effect seems to have something to do with it and maybe that's the missing link here, but I guess my math is just too damn weak to see it, can somebody please help! Thank you. )
Edit: Updated the diagram to try and make it easier to visualise.
This is not a mathematical answer. It is just demonstrating that your claim that "its not like they physically get "locked" after that" is not correct.
Take a look at my very crudely drawn set of nested gimbals:
The rectangular object in the middle represents the aircraft's gyroscope instruments mounted on gimbals. The red gimbal is physically locked to the aircraft by the red blocks but the red gimbal can rotate relative to the red blocks. It is clear the rectangle can yaw around the vertical axis because the blue and red gimbals are free to rotate about the vertical axis. The rectangle can also roll around the tail to the nose axis, because the green gimbal is free to rotate around that axis. However it is not possible to pitch the rectangle around the wing tip to wing tip axis (rotate it in the plane of the screen) because there is no gimbal axis perpendicular to the screen. The rectangular instrument is literally physically locked relative to the red gimbal as far as pitching is concerned. (Under normal circumstance the green gimbal would be at 90 degrees to the red gimbal and there would be no gimbal lock.)
If an aircraft has gyroscope instruments, the gyroscopes are supposed to stay in a fixed orientation relative to the universe, no matter how the aircraft manoeuvres and provide a fixed reference to calculate the orientation of the aircraft, (which is useful when there is zero visibility such as when flying in cloud). If the gimbals are aligned as in the diagram and the aircraft pitches, the gyroscope will be physically forced to pitch with the aircraft (and due to gyroscopic torque reaction it will precess around another axis also) and the gyroscope will no longer be providing a reliable fixed reference to calculate the aircraft's orientation. If due to gimbal lock, the gyroscopic instrument is forced to pitch with the aircraft, the instruments could think the aircraft is level when in fact the aircraft is nose down and heading for the ground, which in foggy conditions, is not a good thing.