Tibia Coordinate System

The tibia’s fixed coordinate system is defined using data reported during experimental setup and the transform (\(T_{Im\_Sen}\)) defined during the registration for the tibia (Registration). The file data-MC-oks003/State.cfg contains the transform that is used to define the tibia’s fixed coordinate system with respect to the tibia’s position sensor. Below you can see the relevant data from data-MC-oks003/State.cfg with unused data removed:

[Knee JCS]
T_Sensor2_RB2 = "<size(s)=4 4> 0.5805398524 0.7813549675 0.2290368848 0.0748743152 0.0563481400 0.2420640867 -0.9686226639 -0.0471447919 -0.8122797344 0.5752298608 0.0964999501 -0.0152677242 0.0000000000 0.0000000000 0.0000000000 1.0000000000"

Where T_Sensor2_RB2 is converted to a 4x4 matrix. The first three rows in the last last column of T_Sensor2_RB2 are the translations for the transform, and these values are multiplied by 1000 to convert from meters to millimeters.

\[\begin{split}T_{Sensor2\_RB2} = \begin{bmatrix} 0.58053985 & 0.78135497 & 0.22903688 & 74.8743152 \\ 0.05634814 & 0.24206409 & -0.96862266 & -47.1447919 \\ -0.81227973 & 0.57522986 & 0.09649995 & -15.2677242 \\ 0. & 0. & 0. & 1. \end{bmatrix}\end{split}\]

Given that oks003 is a left knee specimen, \(T_{Sensor2\_RB2}\) is reflected about the x-axis, which is shown below. The reflection of \(T_{Sensor2\_RB2}\) is \(R_{Sensor2\_RB2}\) (note that this is not needed for a right knee specimen).

\[\begin{split}\begin{split} R_{Sensor2\_RB2} &= T_{Sensor2\_RB2}*\begin{bmatrix} 1 & -1 & -1 & -1 \\ -1 & 1 & 1 & 1 \\ -1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix} \\ &= \begin{bmatrix} 0.58053985 & -0.78135497 & -0.22903688 & -74.8743152 \\ -0.05634814 & 0.24206409 & -0.96862266 & -47.1447919 \\ 0.81227973 & 0.57522986 & 0.09649995 & -15.2677242 \\ 0. & 0. & 0. & 1. \end{bmatrix} \end{split}\end{split}\]

Next the transform from the MR image’s global coordinate system to the tibia’s fixed coordinate system in the MR image (\(T_{Im\_RB2}\)) is defined. \(T_{Im\_Sen2}\) (which is determined from registration of rigid body 2 fiducial points) is multiplied by \(R_{Sensor2\_RB2}\)

\[T_{Im\_RB2} = T_{Im\_Sen2}R_{Sensor2\_RB2}\]

\(T_{Im\_RB2}\) can be used to define the tibia’s fixed coordinate system in the knee model using the SimVitro convention, where the positive direction anterior-posterior axis points posteriorly. To match the convention used by this lab, the fixed tibial coordinate system defined by \(T_{Im\_RB2}\) is rotated 180 degrees about the fixed tibia’s z-axis. This defines the positive direction along the anterior-posterior axis as pointing anteriorly.

\[\begin{split}S_{Im\_RB2} = T_{Im\_RB2}*\begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

The last row in \(S_{Im\_RB2}\) is neglected, and the columns are used to define the axes of the fixed tibial coordinate system and its origin.

\[\begin{split}\begin{split} x_{axis} &= [S_{Im\_RB2}(0,0), S_{Im\_RB2}(1,0), S_{Im\_RB2}(2,0)]\\ y_{axis} &= [S_{Im\_RB2}(0,1), S_{Im\_RB2}(1,1), S_{Im\_RB2}(2,1)]\\ z_{axis} &= [S_{Im\_RB2}(0,2), S_{Im\_RB2}(1,2), S_{Im\_RB2}(2,2)]\\ origin &= [S_{Im\_RB2}(0,3), S_{Im\_RB2}(1,3), S_{Im\_RB2}(2,3)]\\ \end{split}\end{split}\]