MEGN 536Computational Biomechanics

Rotations for Rigid Body Kinematics

Prof. Anthony Petrella

Knee (or any joint) Biomechanics

You are all MATLAB ninja or sous-ninja now

So, let’s start thinking about knee biomechanics…

Joint motion (kinematics) and forces (kinetics) are the foundation for defining design requirements

How do we express these quantities?

Let’s consider an example…

Introducing Abaqus Knee Simulator (AKS)

AKS has automation to help designer… Test knee implant laxity Test knee implant contact mechanics Test knee implant wear performance

To run it… Create a working folder in C:\scratch Right click in folder and “open command prompt here” At command prompt… >> abaqus kneeapp

Run Abaqus Knee Model

Download Abaqus knee model from project page, unzip (on hard drive!)

Follow PDF instructions to run solution

View an animation of a gait cycle…1. Color Code by Part instances

Run Abaqus Knee Model

Download Abaqus knee model from project page, unzip (on hard drive!)

Follow PDF instructions to run solution

View an animation of a gait cycle…1. Color Code by Part instances

2. Options Common… Feature edges

Run Abaqus Knee Model

Download Abaqus knee model from project page, unzip (on hard drive!)

Follow PDF instructions to run solution

View an animation of a gait cycle…1. Color Code by Part instances

2. Options Common… Feature edges

3. Result Active Steps/Frames… deactivate “compress”

Run Abaqus Knee Model

Download Abaqus knee model from project page, unzip (on hard drive!)

Follow PDF instructions to run solution

View an animation of a gait cycle…1. Color Code by Part instances

2. Options Common… Feature edges

3. Result Active Steps/Frames… deactivate “compress”

4. Options Animation… make Frame Rate about 75%

Run Abaqus Knee Model

Download Abaqus knee model from project page, unzip (on hard drive!)

Follow PDF instructions to run solution

View an animation of a gait cycle…1. Color Code by Part instances

2. Options Common… Feature edges

3. Result Active Steps/Frames… deactivate “compress”

4. Options Animation… make Frame Rate about 75%

5. Use Ctrl+Alt+Left mouse to rotate view

6. Animate Time History

Question: What’s the best way to express 6 DOF kinematics and forces for this knee?

Rigid Body Motion

It is standard to describe rigid body motion by rotations and translations of a reference frame fixed in body – relative to a fixed global frame

For the knee, we will measure motion of the tibia relative to the femur

Femur will act like fixed global and tibia will be moving

(Frey et al., 2006)

Right Leg

X

Z

Y

zy

x

Rigid Body Rotations

We will begin to consider rigid body motion with rotations only (translations are easy)

We’ll define the global fixed frame with capital letters and the moving local frame with lowercase letters

Rigid Body Motion

Rigid Body Motion

Collecting the three dot products into a single expression gives…

Rotation matrix Rz is called an “orthonormal” rotation matrix because the rows are orthogonal unit vectors

Notice the rows of Rz are the unit vectors defining the local frame L

Rz also has theproperty…

Rigid Body Motion

Rigid Body Motion