Interpreting quaternions. Inertial navigation systems

[PieterP]

@Romuald_VII

how to interpret these four values

You don't. If you're very close to (1,0,0,0), q1 will change linearly with roll/2, q2 will change linearly with pitch/2 and q3 with yaw/2. q0 will always be the square root of 1 minus the sum of squares of the other components, because ‖q‖₂ = 1 at all times.
Unless your angles are really small and you only have roll, pitch or yaw, you cannot really interpret the quaternion components on sight.

@Romuald_VII

How to get roll, yaw and pitch angles from this

src/AH/Math/Quaternion.hpp

@Romuald_VII

or, better, a radius vector of direction

Quaternions encode rotations, not an absolute angle or direction. If you want a vector to point in a direction, you have to start with a reference vector (e.g. (0 0 1), i.e. pointing up) and rotate this vector by the quaternion.

src/AH/Math/Quaternion.hpp

@Romuald_VII

On the Internet, it is proposed to calculate Euler angles, calling them roll, yaw and pitch, but there is a huge problem in this - when any angle reaches a value of 90 degrees, the angle responsible, for example, for roll, begins to show yaw, and so on. I don’t understand why this happens, frankly

Are you referring to Gimbal lock - Wikipedia? This is exactly the problem you're trying to avoid by using quaternions in the first place. Converting the quaternion to Euler angles is not an issue, as long as you don't manipulate them. For your sensor fusion algorithms and state estimation, you should use quaternions, not Euler angles.