Computing Mass Displacement for Motion Capture Sequence, fail.

[Mikipedia]

Hello!

Homework Statement

As the title says I am attempting to calculate the mass displacement of a motion capture sequence, however I am not seeing the results I expect which is why I am posting here to make sure I have understood everything correctly. The data that is known is the transformation hierarchy of the character over time which encodes the position and orientation of each body part.

Homework Equations

Calculating the mass displacement for the whole sequence comes down to calculating the displacement over one frame to another by summing the displacement for each body part.

The displaced mass of body part k is given by;
[tex]E_k=\iiint_{i}\mu_i(\mathbf{p}_i-\mathbf{\bar{p}}_i)^{2}dx dy dz[/tex]
p_i is a point in the body part and μ_i is its infinitesmal mass. Note that all things "bar"-ed relates to "the other frame". Furthermore,
[tex]\mathbf{p'}_i=\mathbf{R}_0\mathbf{R}_1\cdots \mathbf{R}_{j-1}\mathbf{x}_i[/tex]
where x_i is a point in the local space of a body part, and R₀ contain the global rotation and translation, but since we are interested in the relative mass displacement we define;
[tex]\mathbf{p}_i=\mathbf{R}_1\cdots \mathbf{R}_{j-1}\mathbf{x}_i=\mathbf{W}_j\mathbf{x}_i[/tex]
Ultimately E_k is given by;
[tex]E_k=tr(\mathbf{W}_i\mathbf{M}_i(\mathbf{W}_i-2\mathbf{\overline{W}}_i)^T)[/tex]
Here tr() is the trace operator - the sum of the diagonals of a matrice and M_i is the inertia tensor if the body part i.
The thing that confuses me at the moment is that since R₀ contain the global rotation and translation all the matrices R_1...j-1 should be 4x4 matrices. However, as far as I have understood it is not possible to use anything but 3x3 matrices with inertia tensors in R³, so what are these W_i really?

The paper I am getting this from is available http://www.cs.washington.edu/homes/zoran/sigg99/preprint.pdf" , and the equations are at page 6. The section they are in should be understandable out of context, so no need to read too much.

The Attempt at a Solution

My attempted solution was to define W_i as the 3x3 rotation matrice that transforms a point by rotation, from its local coordinate frame to its position in a global coordinate frame, p_i.
Also, I should say that I use the inertia tensor to compute the angular momentum for the motion and that seems work fine.

Anyhow, I suppose my problem could be anything, but for now I'll stick to my question regarding the 4x4/3x3 matrice mixup. Any help would be beyond awesome :-)

Cheers,

- Miki

 

[KataKoniK]

Hello Miki,

Thank you for reaching out to the forum for help with your calculations. It seems like you have a good understanding of the equations and variables involved, but you are unsure about the use of 4x4 and 3x3 matrices in this context.

Firstly, let me clarify that the Wi matrices are indeed 3x3 rotation matrices. The reason for this is that in this particular scenario, we are only interested in the rotational displacement of the body parts, not the translational displacement. Therefore, we do not need to consider the translation component of the transformation, which is why we only use the rotational component (represented by a 3x3 matrix).

As for the confusion about using 4x4 matrices in R0, this is because R0 contains the global rotation and translation information, which is necessary for calculating the relative mass displacement of each body part. However, when we are calculating the actual displacement of a body part, we only need to consider the rotation component, which is why we use 3x3 matrices for the Wi matrices.

I hope this clarifies your confusion. If you have any further questions or need additional help with your calculations, please don't hesitate to reach out.