MEGN 536 – Computational Biomechanics

Euler Angles

Prof. Anthony J. Petrella

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

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

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:

1. after the first rotation → x’’, y’’, z’’

2. after the second rotation → x’, y’, z’

3. after the third rotation, the final configuration → x, y, z

Using Euler Angles

Moving 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

Z, z”seg

X

Y

x”seg

y”seg

f

Z

X

Y

x’seg

y’seg

fq z’seg

Z

X

Y

xseg fq

zseg

yseg

y

1

2

3

Using Euler Angles

If we use Type I Euler angles,1. Rotation f around z-axis (z’’)

2. 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’)

3. Rotation y around x-axis

Then the local segment frame moves as shown at right

From part c of the figure at theright, the Line of Nodes can be computed as:

Finding Euler Angles

From the figure below we can easily compute the three Euler angles as…

Where positive values areshown in the figure andcorrespond to…

Z

X

Y

xseg fq

zseg

yseg

y

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)

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

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

Joint Coordinate System (JCS) for Knee

Figure below shows the left knee, but the right knee is the same

The JCS is definedas shown

Note: you can seethat the commonfloating axis L is theperpendicular to bothZ and x

Note that Z and x arenot necessarily perpendicular to eachother…so this is not an orthogonal ref frame

X

Y

Z

x

y

z

Z

x

Adduction and abduction

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

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]