7.2 Tracking kinematics using 3-D marker data

You are interviewing to work in a motion capture laboratory. During the job interview, you are presented with the 3-D marker convention shown in Figure 1. Since you have taken a biomechanics class, you feel confident in your ability to understand this marker convention. You quickly suspect that, for the shank, markers 3-5 are used to establish a Tracking Coordinate System (TCS) and that markers 1, 2, 6, and 7 are used to establish the Anatomic Coordinate System (ACS). You relay your suspicions to your interviewer, who is quite impressed with your ability. (NOTE: You have access to any computational tools during your interview).

1. You are told that markers 3-5 do form the TCS. Marker 4 is used as the origin. The x-axis is directed from marker 4 towards marker 3. The y-axis is directed anteriorly and is normal to the x-axis and the vector from marker 4 towards marker 5. The z-axis is normal to both the x-axis and the y- axis such that the axes form a right-handed basis. Using the values for the location the markers in the Global Coordinate System (GCS) shown in Figure 1, please derive the 4x4 homogeneous transformation matrix that expresses the  with respect to the GCS (  ) for the conditions shown in Figure 1.

2. You are once again correct and are told that markers 1,2,6, and 7 do form the ACS (at this point, you’re feeling good about the interview). You are told the following: “The origin for the shank ACS is the midpoint of markers 1 and 2. The z-axis is directed from the origin towards the midpoint of markers 6 and 7. The y-axis is directed anteriorly and is normal to the z-axis and a temp vector from marker 1 to marker 2. The x-axis normal to both the y-axis and z-axis and is directed medially.”  Using the values for the location the markers in the Global Coordinate System (GCS) shown in Figure 1, please derive the homogeneous transformation matrix that expresses the  with respect to the GCS (  ) for the conditions shown in Figure 1.
   

3. Using your solutions to parts 1 and 2, please derive the transformation matrix that expresses the ACSshank with respect to  ( ).

We discussed different angle set conventions in class. One common convention is the set of X-Y-Z Euler Angles. With this convention, the B frame is first rotated about  by angle , then rotated about  by angle , and then finally rotated about  by angle . This angle set has the following form.

Using the examples from class as a primer, and using the atan2 function, please derive the formulas necessary to extract the values of angles , , and f rom the above rotation matrix. For simplicity, you may assume that no angles equal either 0° or 90°. Please express your answers in terms of elements of the rotation matrix (r11, r23, etc.) and clearly state any assumptions that you make.

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