# Computing Mass Displacement for Motion Capture Sequence, fail.

1. Sep 29, 2011

### Mikipedia

Hello!

1. The problem statement, all variables and given/known data
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.

2. Relevant 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;
$$E_k=\iiint_{i}\mu_i(\mathbf{p}_i-\mathbf{\bar{p}}_i)^{2}dx dy dz$$
pi is a point in the body part and μi is its infinitesmal mass. Note that all things "bar"-ed relates to "the other frame". Furthermore,
$$\mathbf{p'}_i=\mathbf{R}_0\mathbf{R}_1\cdots \mathbf{R}_{j-1}\mathbf{x}_i$$
where xi 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;
$$\mathbf{p}_i=\mathbf{R}_1\cdots \mathbf{R}_{j-1}\mathbf{x}_i=\mathbf{W}_j\mathbf{x}_i$$
Ultimately Ek is given by;
$$E_k=tr(\mathbf{W}_i\mathbf{M}_i(\mathbf{W}_i-2\mathbf{\overline{W}}_i)^T)$$
Here tr() is the trace operator - the sum of the diagonals of a matrice and Mi 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 R1...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 Wi really?

The paper I am getting this from is available http://www.cs.washington.edu/homes/zoran/sigg99/preprint.pdf" [Broken], 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.

3. The attempt at a solution
My attempted solution was to define Wi as the 3x3 rotation matrice that transforms a point by rotation, from its local coordinate frame to its position in a global coordinate frame, pi.
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

Last edited by a moderator: May 5, 2017