But these additions are not rotations applied to rotations... they are the sum of impulses that result in an acceleration or velocity (or accumulate into a position)paladin17 wrote: ↑Sat Sep 19, 2020 6:07 pmYou have to keep in mind though that 3D rotations (e.g. Lie group SO(3) - rotations around 3 orthonormal vectors) are non-Abelian. I.e. the order of the application of rotations matter. Rotate around X to x degrees and then rotate around Y to y degrees is not the same as in the reverse (in the general case anyways).
I would like to know more about these assignments, and why they think that that actually works... I can certainty generate the result of a rotation with 2 more rotations applied to it (and yes the order matters, A then B then C is not the same when 'applying' when 'adding' a+b+c IS actually the same).This actually is very important in quantum field theory, where the fundamental interactions are represented by abstract rotations in the abstract space (see principle of local gauge invariance) and some of these rotational groups are Abelian (e.g. U(1) which represents electromagnetism) while others aren't (e.g. SU(3) which represents the strong nuclear interaction), which leads to radically different properties of the resulting theories.
There are a few places where additive rotations are better 1) it's an accurate SLERP operation that's cheaper...(these days) it's the pure rotation axis between one rotation frame and another... the vector between them is the rotation axis also.. the distance between them is how much around that axis the space is rotated.... and again, there is a place and time where X+Y+Z is the same no matter the order; however, due to physical limitations, the ability to USE that is limited.
Spatial position can be measured absolutely from one point to another, pick one on the floor, and one on the roof across the street, changes are high that most of those direct paths are not actually paths you can take to get to (x,y,z) target... it's the same with rotations;
If you're in a rocket with fixed point engines, the rotation of that body affects the directions it can apply thrust to change its axis of rotation; but all the thrusters at once, simply composite with addition.
Through experimentation and stubborn pursuit of 'what exactly ARE addable rotations' I can identify where they do work, but that's a very limited circumstance.... and unfortunately it's still a complex system of transformations to values that have to happen to rotate axis-angle around axis-angle .
(axis-angle is just angle-angle-angle separated ) like linear velocity can be speed*(x,y,z) or (speedX,speedY,speedZ) and the angles just add like normal linear x/y/z points (in some cases)
But absolutely, if you have two positions, you can immediately identify an axis of rotation and amount of rotation around that axis to get to that target point. (and compose a matrix to apply to get there even)
A physical system though has limitations, and a robot arm has 'yaw/pitch and roll' settings that are fix - and usually fixed as a result of each other... so a specific mounting order of 3 motors on a robot arm segment have to be applied separately - and that becomes a complex system of 3 equations that mutually affect each other.
And let me finish with reiterating
I would like to know more about these assignments, and why they think that that actually works... (how it's mapped) And I'm not certain WHY or the mechanism for it, but I suspect gravity is actually a spin axis; but rotates x/y/z at once - so it's more of a rotation around 'w'? but barring that I'd like to know what the SO(1) things actually imply, since I can directly visualize such things now.