# Introduction to Biomechanics HW #2

ENBI 3510/4510: Intro to Biomechanics HW #2

Spring 2021 Dr. Chadd Clary

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Due Date: Submit to Canvas on April 20th, 2021 by 11:59 PM.

Part 1: 3D Knee Kinematic Descriptions

Problem Statement: A subject’s femur, tibia, and fibula were segmented from a CT scan in the supine

position (laying down). The segmented bones are located in the global (CT scanner) coordinate system.

In addition, anatomic landmarks on the femur and tibia were identified that can be used to build

anatomic femoral and tibial coordinate systems using the methods described in class. Your task is to

build these coordinate systems, construct the transformation matrix describing the orientation of the

femur coordinate system in the tibial anatomic coordinate system, and calculate the Grood & Suntay

kinematics of the knee’s 3D orientation.

The raw data and additional details can be found in Canvas on the Homework #2 page:

• Homework_2_Part1_Intro.m (data introduction and description)

• KNEE.mat (raw data, keep in same folder as the .m file)

Detailed Steps:

1. Calculate the Femoral Anatomic Coordinate system using the anatomic landmarks as follows:

a. The femoral origin is located at the most distal point in the trochlear groove.

b. The S-I vector is from the femoral origin to the center of the femoral head.

c. The temporary M-L vector from the medial epicondyle to the lateral epicondyle,

pointing laterally.

d. The A-P vector is the cross product of the S-I vector and the temporary M-L vector,

pointing anteriorly.

e. The M-L vector is the cross product of the A-P and S-I vector, pointing laterally.

2. Calculate the Tibial Anatomic Coordinate system using the anatomic landmarks as follows:

a. The centers of the medial and lateral tibial plateaus should be calculated as the midpoints between the most anterior and most posterior points on each plateau,

respectively.

b. The tibial origin is the mid-point between the centers of the medial and lateral plateaus.

c. The S-I vector is from the ankle center to the tibial origin, pointing superiorly.

d. The temporary M-L vector is from the center of the medial plateau to the center of the

lateral plateau, pointing laterally.

e. The A-P vector is the cross product of the S-I vector and the temporary M-L vector,

pointing anteriorly.

f. The M-L vector is the cross product of the A-P and S-I vector, pointing laterally.

3. Transform the femoral anatomic coordinate system into the tibial anatomic coordinate system.

4. Calculate the Grood & Suntay knee kinematic from the tibia-femur transformation matrix.

5. Additional steps for the Graduate Section of the class: KNEE.mat also contains a series of tibiafemur transformation matrices that describe the knee movement during a deep knee bending

activity. Calculate the G&S kinematics for the entire movement. Generate a plot of the relative

orientation of the femur and tibia at the moment in the activity when the knee is flexed to 45°.ENBI 3510/4510: Intro to Biomechanics HW #2

Spring 2021 Dr. Chadd Clary

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Part 2: Inverse dynamics

Problem Statement: Lecture 5, we solved for the ankle

forces and moments of a foot pushing off from the

ground at toe-off during gait (see lecture notes). In this

problem, propagate the reaction forces and moments at

ankle onto the shank and solve for the reaction forces

and moments at the knee.

For the foot:

• Fgr,f = 150 N

• Fgr,n = 800 N

• mfoot = 1.4 kg

• Ifoot = 0.05 kg∙m2

• ax = 7.5 m/s2

• ay = 2 m/s2

• α = -15 rad/s2

For the shank:

• mshank = 3.4 kg

• Ishank = 0.06 kg∙m2

• ashank,x = 13 m/s2

• ashank,y = 3 m/s2

• α = -18 rad/s2ENBI 3510/4510: Intro to Biomechanics HW #2

Spring 2021 Dr. Chadd Clary

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Deliverable: Provide a summary memo using the template provided on Canvas, including…

Part 1:

• Report the tibial anatomic coordinate system

• Report the femoral anatomic coordinate system

• Include a plot of the tibia and femur in the tibial anatomic coordinate system. Include lines

(vectors) from the origin along the x-, y-, and z-axes for reference.

• Report the G&S kinematics of the knee pose in the CT Scan (ML, AP, SI, FE, AA, IE).

• Write a brief discussion of how uncertainty or error in identification of the anatomic landmarks

would influence the calculated anatomic coordinate system and the associated tibia-femoral

kinematics.

• Additional steps for the Graduate Section of the class: Include plots of the G&S Kinematics for

the flexion cycle. The first plot should show the Ad-Ab and I-E rotational kinematics versus knee

flexion. The second plot should include the translational kinematics (M-L, A-P, and S-I) versus

knee flexion. Include the plot of the relative orientation of the femur and tibia at the moment in

the activity when the knee is flexed to 45° in the tibial coordinate system.

Part 2:

• Draw the Free-body-diagram for the shank, including all externally applied loading.

• Include your complete calculations (insert a photo of neatly handwritten work).

• Report the reaction forces and moment at the knee