1. Joint Angles Calculation
Lei Zhou and Xiaolin Li
October 28, 2014

2. Outline Joint Coordinate System (JCS) Euler’s Angle Helical Method
Concept
Example
Euler’s Angle
Helical Method

3. Joint Coordinate System ----concept
The joint coordinate system(JCS) is defined by two independent body-fixed axes and the common perpendicular.

4. Joint Coordinate System ----concept
The Joint Coordinate System (JCS) was proposed by Grood and Suntay (1983) to encourage the use of clinically relevant models.

5. Joint Coordinate System ----concept
Joint Coordinate System is composed of the two body fixed axes, e1 and e3 and their mutual perpendicular, e2.
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6. Example---- Hip joint Anatomical landmarks used
ASIS: anterior superior iliac spine (Nomina anatomica: Spina iliaca anterior superior).
PSIS: posterior superior iliac spine (Spina iliaca posterior superior).
FE: femoral epicondyle (Epicondylus femoris medialis, Epicondylus femoris lateralis).

7. Example Where is the “O”
The common origin of both axis systems is the point of reference for the linear translation occurring in the joint, at its initial neutral position.

8. Example Pelvic coordinate system—XYZ
O: The origin coincident with the right hip center of rotation.
Z: The line parallel to a line connecting the right and left ASISs, and pointing to the right.
X: The line parallel to a line lying in the plane defined by the two ASISs and the midpoint of the two PSISs, orthogonal to the Z-axis, and pointing anteriorly.
Y: The line perpendicular to both X and Z, pointing cranially.

10. Example Femoral coordinate system—xyz
o: The origin coincident with the right hip center of rotation, coincident with that of the pelvic coordinate system (O) in the neutral configuration.
y: The line joining the midpoint between the medial and lateral FEs and the origin, and pointing cranially.
z: The line perpendicular to the y-axis, lying in the plane defined by the origin and the two FEs, pointing to the right.
x: The line perpendicular to both y- and z-axis, pointing anteriorly

11. Example JCS and motion for the right hip joint
e1: The axis fixed to the pelvis and coincident with the Z-axis of the pelvic coordinate system.
Rotation (a): flexion or extension.
Displacement (q1): mediolateral translation.
e3: The axis fixed to the femur and coincident with the y-axis of the right femur coordinate system.
Rotation (g): internal or external rotation.
Displacement (q3): proximo-distal translation.

13. Example JCS and motion for the right hip joint
e2: The floating axis, the common axis perpendicular to e1 and e3.
Rotation (b): adduction or abduction.
Displacement (q2): antero-posterior translation.

14. Euler’s Angle Definition and Function
Describe the orientation of a rigid body
Used to represent the orientation of a frame of reference relative to another.
Represent a sequence of three elemental rotations
The elemental rotations can either occur about the axes of the fixed coordinate system (global coordinate system) or about the axes of a rotating coordinate system
(local coordinate system).

15. Euler’s Angles Characters
Specific orders, the sequence is not actual path of motion taken to arrive at that position
Different rotation orders can lead to different orientations
ex: book on a desk; The book will no doubt be oriented in space differently after each set of rotation
Twelve possible sequences of rotation axes
Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y)
Cardan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z)

16. Euler angles in biomechanics
Purpose
A method used to describe three dimension motion of a joint
General sequence
The first rotation is defined relative to an axis oriented in the global coordinate system.
The third is defined with regard to an axis fixed within the rotating body. (Local coordinate system).
The second is performed relative to the floating axis, which is always orthogonal to both the first and third axis.

17. A gymnast performation
Precession
The first rotation takes place relative to an axis
defined in the global reference system.
Tilt
Floating axis: The axis of tilt is not fixed with
regard to both the global reference frame and
the local reference frame.
Spin
Rotates around its longitudinal axis:
fixed in the body.

18. Common Sequence in biomechanics studies
XYZ sequence
X is the flexion/extension in sagittal plane
Y is the abduction/adduction
Z is the axial (internal/external) rotation
The second and the third rotations are about local axes transformed by
previous rotations. Ex. Xy’x’’
Why do we use this sequence?
XYZ sequence is associated with minimal planar crosstalk and as such its
use is encouraged.
Sinclair J, Taylor P J, Edmundson C J, et al. Influence of the helical and six available Cardan sequences on 3D ankle joint kinematic parameters[J]. Sports Biomechanics, 2012, 11(3):

19. About an axis of GCS
flexion
About an axis of LCS
abduction
About an axis of LCS
External rotation
a) 45° about the z-axis b) -30°about the x-axis c) -45°about the y-axis
The leg in the final position with Euler angles (45° -30°-45°)

20. Helical angles Also a method to describe three dimension motions
A screw axis/helical axis: a line that is simultaneously the axis of rotation and the line along which translation of a body occurs.
Chasles’ Theorem: any rigid-body motion can be obtained as the rotation around an axis, and a translation parallel to the screw.

21. Six parameters to define a helical motion:
Two coordinates of the piercing point of
the helical axis with any one of the three
coordinate planes
Two direction cosines of the helical axis
The translation along and the rotation
about the helical axis

22. When using helical method
Large displacement of the
glenohumeral joint
Arm translation-arm rotation
Euler angles are not specific

23. Thank You