MEGN 536 – Computational Biomechanics Euler Angles

1. MEGN 536 – Computational BiomechanicsEuler 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 • 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 theright, 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 areshown in the figure andcorrespond 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 z y

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 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 Z X Z Y x z Adduction and abduction y

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]