1. MEGN 536 – Computational Biomechanics Euler Angles
Prof. Anthony J. Petrella

2. Rotational Transformations
Recall from our discussion last time…
Global ref is denoted by uppercase letters, X, Y, Z
Body-fixed rotations can be computed for x, y, z axes
Any combination can be applied in any order
The combined (total) rotation is computed by a simple product of individual rotation matrices
Order of body-fixed rotations is read right-to-left
The combined (total) rotation transforms vectors from the global reference frame to the local frame
Rows of the total rotation matrix are the unit vectors of the local frame

3. Rotational Transformations
What are common combinations / orders of rotations? Euler was one of the first to propose…
Type I angles are used for the joint coordinate system (JCS)
These are body-fixed rotations
Greenwood, Principles of Dynamics, 2nd ed., Prentice Hall, 1988

4. Euler Angles
We will define the orientation of a moving body segment relative to a global reference using three rotations around z, y, and x body-fixed axes
Let the local segment frame be initially coincident with the global ref (fixed in another part of body)
There will then be three rotated configurations of the local segment frame:
after the first rotation → x’’, y’’, z’’
after the second rotation → x’, y’, z’
after the third rotation, the final configuration → x, y, z

5. Using Euler Angles Moving reference frame Global reference frame
Lowercase axis labels: x, y, z
Lowercase unit vectors also: i, j, k
Fixed in a body segment (such as tibia) and used to define motion of segment based on rotation and translation of ref frame
Global reference frame
Uppercase axis labels: X, Y, Z
Uppercase unit vectors: I, J, K
Fixed in another portion of the anatomy (such as femur) and used as a foundation from which to measure motion of moving ref frame

6. Using Euler Angles If we use Type I Euler angles,
Rotation f around z-axis (z’’)
Rotation q around Line of Nodes (common perpendicular to Z and x, this is the intermediate y-axis, which is the same as y’’ and y’)
Rotation y around x-axis
Then the local segment frame moves as shown at right
From part c of the figure at the right, the Line of Nodes can be computed as:

7. Finding Euler Angles
From the figure below we can easily compute the three Euler angles as…
Where positive values are shown in the figure and correspond to…

8. Joint Coordinate System (JCS) for Knee
Purpose: express rotation angles and translations in clinically / anatomically meaningful ways
Joint angles referenced to JCS allow us to quantify…
Flexion/Extension (F/E)
Adduction/Abduction (Ad/Ab) also referred to sometimes as Varus/Valgus (V/V) …remember, valgus is knock-knee’d
Internal/External Rotation (I/E)
Joint translations allow us to quantify…
Superior/Inferior translation (S/I)
Anterior/Posterior translation (A/P)
Medial/Lateral translation (M/L)

9. Joint Coordinate System (JCS) for Knee
Let the femur represent the global reference
Tibia moves relative to the femur
Let us define the reference frames as shown
Uppercase letters on femur
Lowercase letters on tibia
X-axis is the long axis, S/I
Y-axis is A/P
Z-axis is M/L
Same for lowercase letters X Z Y x y z

10. Joint Coordinate System (JCS) for Knee
We adopt some conventions (Grood & Suntay, 1983)
F/E (f) is rotation of tibia around M/L axis of femur (Z-axis)
Ad/Ab (q) is rotation of tibia around common floating axis that is at right angles to F/E and I/E axes (L-axis)
I/E (y) is rotation of tibia around its own long axis (x-axis)
Floating axis always defined by cross product of femur M/L (Z-axis) with tibia longitudinal axis (x-axis)
Translations expressed as distances along JCS axes, this is done with simple dot products
Other quantities (forces or moments) may also be expressed in terms of components along JCS axes

11. Joint Coordinate System (JCS) for Knee
Figure below shows the left knee, but the right knee is the same
The JCS is defined as shown
Note: you can see that the common floating axis L is the perpendicular to both Z and x
Note that Z and x are not necessarily perpendicular to each other…so this is not an orthogonal ref frame x Z X Z Y x z y
Adduction and abduction

12. Joint Coordinate System (JCS) for Knee
Clinical translations are expressed as distances along JCS axes
Find the total displacement vector that defines the tibial origin location relative to the femoral origin (D)
M/L translation is measured along the F/E axis fixed in the femur (Z-axis)
A/P translation is measured along the floating axis (L-axis)
S/I translation is also called compression/distraction and is measured along the long axis of the tibia (x-axis)
Use simple dot products to find the components of the displacement vector along the anatomical axes…
DM/L = D  K ; DA/P = D  L ; DS/I = D  i

13. Joint Coordinate System (JCS) for Knee
We have developed JCS equations in a completely general way
X, Y, Z are the global ref axes (femur)
x, y, z are the moving ref axes (tibia)
Note that JCS equations depend on coordinates of all the above unit vectors
If global ref is actually moving (like the femur), then we simply write X, Y, Z and x, y, z both in terms of a fixed inertial ref frame that is fixed to the earth / lab
If global ref is truly fixed (like today’s worksheet), then X = [1,0,0]; Y = [0,1,0]; Z = [0,0,1]