Confusion about how inverse bind pose is actually calculated and used?

Question

I am trying to do skeletal animation using Assimp and the inverse bind pose matrix just trips me up. I will give a little example to illustrate my point.

Root
    Bone A
        Bone B
            Bone C

Let's say we have a hierarchy like above. To get a vertex in root space into C space, I would have to do (CBA) * v. The inverse bind pose, which is supposed to get a vertex from bone space to root space, should be the inverse of this. Therefore, the way to calculate inverse bind matrix should be (CBA)^-1 or (A^-1)(B^-1)(C^-1). The formula then is (A^-1)(B^-1)(C^-1) * v, which makes sense. However, I am not able to reproduce the result of Assimp's mOffsetMatrix (inverse bind matrix caculated by Assimp) with (CBA)^-1. (ABC)^-1, however, surprisingly produces the correct result, albeit a little marginal error here and there which I attribute to floating point error.

Answer 1

To get a vertex in root space into C space, I would have to do (CBA) * v.

Well, yes. But that's not actually what you want to do in skeletal animation. You have it reverse.

It's the other way around. You assume you have a point in C space and want to compute its position in root/global space, in order to pump it further through the transformation pipeline, which gets increasingly more global with each transformation until your point ends up in eye-space (and further). That's why A doesn't transform from root space into A space, it transforms from A space into root space. So $ABC$ transforms from C space into root space.

Now your mesh vertices already are in root space, that's where the inverse bind pose comes into play, which transforms a point from root space into C space. So you basically transform your point from root space to C (or whatever joint) space with the inverse bind matrix that's only dependent on the joint and constant over the whole animation and then from that joint space back into root space with the animation-dependent joint transformation.

So C's transformation matrix for frame $t$, that transforms a vertex from unanimated root space, which is how your mesh is defined, into animated (with respect to C's joint angle) root space would be $A(t)B(t)C(t)(A_RB_RC_R)^{-1}$, where $M(t)$ is the local joint matrix at animation time $t$ and $M_R$ is the local joint matrix corresponding to the rest pose, the pose in which the mesh is defined, and thus $(A_RB_RC_R)^{-1}$ is C's inverse bind pose. You can also see that using the rest pose for both transformations makes them basically cancel each other out, because the vertex would already be at the proper position with respect to the rest pose animation frame.