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THREE-DIMENSIONAL ROTATIONS
While working for a biodynamics company, you are given the opportunity to demonstrate your knowledge of Autolev, Matlab, motion analysis, human anatomy, intermediate reference frames, and three-dimensional (3D) rotations. An orthopedic surgeon with whom you work wants to know a patient’s left hip flexion-extension, internal-external rotation, and abduction-adduction during gait movement recorded with a marker-based motion analysis system. A schematic of the left hip region is shown below along with the motion capture marker locations and recommended body segment coordinate systems.
For this assignment, your goal is to create the pelvis and thigh segment coordinate systems, define a 3D rotation matrix between the pelvis and thigh, and use inverse trigonometric calculations to compute two distinct sets of three Euler angles from the rotation matrix. Before proceeding to the following pages you will need to download and save the 3D rotations homework files from the course web page.
To complete the assignment, you need to perform the series of steps indicated below.
Body 3 1-2-3 Rotation Matrix
- Using Autolev (or by hand), derive the rotation matrix resulting from a series of three simple rotations ( q1 , q2 , and q3 ) about body-fixed axes using the rotation sequence 1-2-3.
- Record the rotation matrix below (feel free to use shorthand such as: cos ( q1 ) = c 1 or sin ( q 2 ) = s 2 ).
R (^) Body 31 − 2 − 3 =
Body 3 3-1-2 Rotation Matrix
- Using Autolev (or by hand), derive the rotation matrix resulting from a series of three simple rotations ( q1 , q2 , and q3 ) about body-fixed axes using the rotation sequence 3-1-2.
- Record the rotation matrix below (feel free to use shorthand such as: cos ( q1 ) = c 1 or sin ( q 2 ) = s 2 ).
R (^) Body 33 − 1 − 2 =
Matlab and Data Files
Place the Matlab file (Rotations_3D.m) and the six input data files (r_asis.txt, l_asis.txt, sacral.txt, thigh_upper.txt, thigh_front.txt, and thigh_rear.txt) in your Matlab working directory.
Pelvis Segment Coordinate System
In the appropriate section of Rotations_3D.m, define the pelvis segment coordinate system for the current time frame (f).
- Define a vector (pelvis_v1) from the l_asis marker to the r_asis marker.
- Normalize the vector to define the pelvis z -axis unit vector (pelvis_z).
- In the appropriate section of Rotations_3D.m, assuming a Body 3 1-2-3 rotation sequence, determine the hip joint angles from the rotation matrix using inverse trigonometric calculations for the current time frame (f). a. Compute the hip abduction-adduction angle (ab_adduction_123), or q1 about the 1 -axis. b. Compute the hip internal-external rotation angle (in_external_123), or q 2 about the 2 -axis. c. Compute the hip flexion-extension angle (flex_extension_123), or q 3 about the 3 -axis.
- In the appropriate section of Rotations_3D.m, assuming a Body 3 3-1-2 rotation sequence, determine the hip joint angles from the rotation matrix using inverse trigonometric calculations for the current time frame (f). a. Compute the hip flexion-extension angle (flex_extension_312), or q 1 about the 3 -axis. b. Compute the hip abduction-adduction angle (ab_adduction_312), or q 2 about the 1 -axis. c. Compute the hip internal-external rotation angle (in_external_312), or q 3 about the 2 -axis.
Generating Results
- After saving your work, run the Matlab file Rotations_3D.m by entering the file name at the Matlab command prompt.
- Following the animation of the markers within the laboratory coordinate system, Matlab will automatically generate three plots that you must print and include with your assignment.
Discussion
Considering your goal of quantifying a patient’s hip joint movement, which rotation sequence (i.e., 1-2-3 or 3-1-2) would you choose? Why?
