JoeJ wrote:How does the order you provide rows finally matter in the solver
I do not know if they get solved 'at once' or in given / random order.
ah that's the trick, the point is the order of the angle does not matter, as long as the angles are small.
That's the Rodrigues theorem, is hard to find the proof anywhere, I have in a book Dynamics of Multibody Systems by Ahmed A, Shabana page 42.
it represents the rotation as complex number by expanding the sin and cos in using Taylor
leading to a expansion of the for I + v * sin A + 2 V^2 sin2(A/2)
the product of these expansion shows that anything beyond the third term can be neglected.
then the product of these three terms index shows that angular rotation do not commute.
because of the effect of the imaginary part,
then on page 49, it shows that for infinitesimal angular rotations, the third term of the expansion is quadratic and can also be neglected and the product of these two expression (I + v sin (A)) * (I + v sin(B)) do commute. It is an interesting there but is a long, long time I do not read those book,
I just know the theorem was in one of my books.
but I found a segment in Wikipedia that does a simple demonstration.
https://en.wikipedia.org/wiki/Rotation_group_SO(3)#Infinitesimal_rotationsThey do not proof the theorem they do a verification to show that in fact concatenating small angular rotation, the rotation matrices commuted on the first order term plus there is a second order error term that vanish for really small angles. (the definition of small angle is the part I am abusing).
here is the conclusion:
Since dθ dφ is second order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is commutative. In fact,
d A x d A y = d A y d A x
again to first order. In other words, the order in which infinitesimal rotations are applied is irrelevant.
I am abusing the concept, because when they talk about infinitesimal angles the really mean small angles and I am declaring the error angle small regardless of their magnitude, therefore my second order error is not so negligible.
But that is my claim, I claim the we can manage the error even for relative small error angles like under 5 to 10 degrees.
The Theorem is 100% true, in fact Quaterion rotation is a Corollary of the theorem.
It is I who is stretching to the breaking point where it fail.
However no matter how good the theory is, it is not working so is back to the drawing board and see if I can find some error on my side.
I will later write a agebraic demostartion the in fact the order of the angles is irrelevant in the newton system, maybe I find my mistake.
This is very important for what I want to do, because quaternion is not going to do it. a quat assume the the rotation is predetermined so it can only do a motors along the quaternion axis, but the axis is not known.