1. Joint Coordinate System

2. Coordinate Systems Coordinate systems are generally: Purpose:
Cartesian
Orthogonal
Right-Handed
Purpose:
To quantitatively define the position of a particular point or rigid body

3. Cartesian Coordinate Systems
Purpose: used to establish a Frame of Reference
Generally, this system is defined by 2 things:
An origin: 2-D coordinates (0,0) or 3-D location in space (0,0,0)
A set of 2 or 3 mutually perpendicular lines with a common intersection point
Example of coordinates:
2-D: (3,4) – along the x and y axes
3-D: (3,2,5) – along all 3 axes

4. Orthogonal Definition:
Refers to axes that are perpendicular (at 90°) to one another at the point of intersection

5. Right-Handed Rule
Coordinate systems tend to follow the right-hand rule
This rule creates an orientation for a coordinate system
Thumb, index finger, and middle finger
X-axis = principal horizontal direction (thumb)
Y-axis = orthogonal to x-axis
(index)
Z-axis = right orthogonal to
the xy plane (middle)

6. Global Coordinate Systems
A reference system for an entire system. When labelling the axes of the system, upper case (X, Y, Z) may be useful in a GCS
Example – a landmark from a joint in the body (lateral condyle of the femur for the knee joint)
Within a global coordinate system, the origin is of utmost importance
Using a global coordinate system, the relative orientation and position of a rigid body can be defined. Not only a single point.

7. Local Coordinate Systems
A reference system within the larger reference system (i.e. LCS is within the GCS)
This system holds its own origin and axes, which are attached to the body in question
Additional information:
Must define a specific point on or within the body
Must define the orientation to the global system
Origin and orientation= secondary frame of reference (or LCS)

8. Joint Coordinate Systems
A reference system for joints of the body in relation to larger GCS(the whole body) and to other body segments (LCS)
Purpose
To be able to define the relative position between 2 bodies.
Relative position change = description of motion
Orientation
Origin
Could be the centre of mass of a
body segment (ex. The thigh)
Could be the distal and proximal
ends of bones

9. Human Movement Biomechanics Lab
Joint Angles
Methods Used Within Biomechanics
Euler/Cardan Angles
Joint Coordinate System
Helical Axes
Each method has specific advantages and disadvantages and the best method to use for a project depends on numerous factors
Human Movement Biomechanics Lab

10. Human Movement Biomechanics Lab
Euler’s Angles
Leonhard Euler ( )
3D finite rotations are non-commutative
They must be performed in specific ORDER
Ex: book on desk
The order of rotations is precisely described in biomechanics depending on the application
12 possible sequences of rotations
First rotation defined relative to a GLOBAL axis
Third rotation defined about an axis in rotating body (LOCAL)
Second rotation defined about a floating axis in the second body
Ex: (Xglobal, Ylocal, Xlocal)
When the terminal rotation is the same it is known as an EULER ROTATIONS (6)
When the terminal rotations are NOT the same these are considered CARDAN ROTATIONS (6)
Human Movement Biomechanics Lab

11. Euler Angles
Purpose:
A method used to describe 3-dimensional motion of a joint
`Represent three sequential rotations about anatomical axes`
Important to note about Euler angles is that they are dependent upon sequence of rotation
Classified into two or three axes

12. Standard Euler Angles and Euler Angle of JCS
Sequence dependency differs depending on which system is being looked at in order to describe 3-dimensional rotation about axes
Standard Euler Angles:
Dependent upon the order in which rotations occur
Classified into rotations about 2 or 3 axes
Euler Angle in a Joint Coordinate Systems:
Independent upon the order in which rotations occur
All angles are due to rotations about all 3 axes

13. Common Cardan Sequence in biomechanics studies
Xyz sequence
Rotation about medially-directed X axis (Global CS)
Rotation about anteriorly-directed y axis (Local CS)
Rotation about vertical axis (Local CS)
See Fig 2.12 in text
This sequence chosen to represent joint angles and recommended within biomechanics (Cole et al., 1993)
Rotations occur about: flexion-extension axis, ab/adduction axis, and axial rotation
Major Disadvantage: Gimbal Lock  when middle rotation equals π/2 it results in mathematical singularity and causes computational problems
Human Movement Biomechanics Lab

14. Cardan Sequence Application
Movement of a joint is defined as the motion of the distal (far) segment to the proximal segment (near)
Ex (knee):
thigh (proximal segment)
Shank (distal segment)
Find TTS
Decompose rotation matrix into the three Cardan angles of flexion-extension, ab-adduction, axial rotation
Human Movement Biomechanics Lab

15. Joint Coordinate System (JCS)
Grood & Suntay (1983)
Describe the motion of the knee joint
Purpose: to insure that all three rotations had functional meaning for the knee
How is it different than an Euler/Cardan rotation?
NOT an orthogonal system
Two segment-fixed axes and a FLOATING axis
Essentially we must define the anatomical axes of interest from bony markers, the clinical axes of rotation, and the origin of the joint coordinate system for a complete analysis of motion
Human Movement Biomechanics Lab

16. Human Movement Biomechanics Lab
Helical Angles
Woltring (1985, 1991)
Another method to describe the orientation (both rotation & translation) between two reference systems
Any two reference systems can be “matched” up through a single rotation and a translation about a single axi
This axis does not necessarily have to line up with one of the axis of the local CS
Good for joints that are hinge-like
i.e. talocrural joint
Human Movement Biomechanics Lab