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Downloaded a simple character from http://www.blendswap.com/blends/view/54875 to play with animations, moved it a little bit, but when opened the graph editor to tweak the animation I've got something scary. Axes are completely wrong and do not correspond to the world directions. Tried to recalculate the roll and everything else but it just stays all wrong. Actually I've already made a question about it but I think there was some misunderstanding because the answer that I've got (or rather understood it like this) was something like "the graph editor edits values in the local space so cope with it", but then I've found this http://www.cgmasters.net/free-tutorials/animator-tutorial-series-01-landing/ tutorial with samples and it works perfectly well, no matter how much I'm rotating the bones.

So what is the problem in the first model and how to fix it?

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The bones to wich nothing is connected (free or keep offset only) can be freely rotate in edit mode: if you select your armature and in the properties tab check the "axes", you can freely rotate those bones to align their axes with the world axes; in the photo i've changed the alignement of the hips of your model to match world's alignement. This procedure, of course, changes the pivot point of the rotations, so you should move the head of the bone to the previous location.

Sometimes this is not a good solution, even worse if another bone is connected to the one you want aligned. In those cases the solution is to add a new control bone, aligned with the world coordinates, and make it parent with the deform bone, so you can animate an aligned bone controlling a non aligned deform bone. For a better use of the graph editor you should also avoid the use of quaternion rotation, wich is useful only in the cases where a bone needs to rotate in all the three axes: if you know that a bone will rotate just a little (er even better it can be locked) in one of his axes (let's say the "z"), choose euler rotation putting that axe in the center (I.E. euler XZY): in this way the graph editor should be more usable to edit rotations.

Anyway, in my experience, lots of animators nearly do not use the graph editor, except for choosing wich type of keyframe they want (vector, bezier, etc..) and they do their editings directly in the action editor, dopesheet and directly posing the character, also because a lot of animation is done by rotating bones and quaternion is mainly used.

https://www.youtube.com/watch?v=c-0oBXJZwQE This is an amazing serie of tutorals to deeply understand some priciples of advanced rigging.

enter image description here

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