a few thing, gaining energy in one axis is correct. because as it loses spin energy along the principal spinning axis the precision rotation gain energy only to keep conservation of both energy and angular momentum, you can see a demonstration on thsi video.
https://www.youtube.com/watch?v=fv_AinDLHJYThsi is what made me excited and knew the derivatives are correct, when I run the Gyroscope an saw that gradually over time they try to align to vertically, while the top drops down because teh friction drain a long more energy when the top drift want around over the surface.
this is also part of whe it call teh middle axis there a spinning object will gradually align to one the principal axis with the larger angular momentum.
The problem with is selecting the axis of rotation, it is clear that the joint on the right because we get the twist axis, and the the other two are form by rotating the other tow around teh twist only wan of the tow axis align with he cone. So far this seem to be the only method to select axis system that does not suffer from gimbal lock and is orthogonal.
but he penalty is that you can no get a Cartesian de compotation of the Euler for defining limits.
teh huge but I discovered is fact made sense worse. it actually show me how wrong my method was so I am abandoning the small approximation method.
I am still going to give a shot and the limit. withe a simple diffrent method. and cna de decipoe liek this.
1-gettion the twist angle for the quaternion is very eassy, and we had that working fine.
we sa it is the motor joint and that does ever fail,
so basically step one is getting the twist angle and check limit on tha angle the isseu a row use the pin and the cpil angle.
he is teh hard part.
2-Get the cone angle and it is perpendicular axis.
here we can issue one or tow row along thso angle to correct any angle sine teh are orthognal.
how evet we cna do better. since the matrix formed by the twist, teh cone and it perpemdicular axis is ortogulan, we can calcular the roathion of teh two secun axis aroing the twist axis only the align wi ith the Yaw and Roll angle. the tow componend of that rotation and the cosine and the angel the we need to rotate the frame to satfi the clip error.
itis like say you wna to correct the linera position, and you knwo one is vertical, bu the ohorzonal is variable.
we has not problem using the three is any direction as long as there are mutually permendicular thjsi si whe we wnat to do with euler.
but we can do it in a different way. we can use teh vertical, and teh normalized the horizontal alone one direction and issue only one row, thsi has proven to be unstable sine it always happen that the object can move along the side, but can be made more stable by issuing a their row with zero acceleration. however is not always as good as using the Cartesian decomposition to clip linear error.
It seem the using angle teh method is teh other way around we nee to get the cone angle as teh reference frame. and here we can issue one row the correct error and the oerehis usually zero.
bu I claim we can do the oethe way fodn out what roation long teh towist match the yw and rall angles and these cosing diorection fo these angle are the clip error that reco e teh error along the cone axis system.
so basically it come down to get teh cone and do corewntion along teh cone and also along teh side of the cone.
I will try that tomorrow maybe that's is a better solution, at least we know that tah work when doing only teh cone correction.