1. Wednesday March 1st Dr. Moran
Lecture 6
Wednesday March 1st
Dr. Moran

2. Lecture Outline Review Sheet for Midterm Recap of 3D Kinematics
Where we left off
Matrix Method
Joint Angle Computation
Euler Angles vs Cardan Angles
Joint Coordinate System
Finite Helical Angles

3. 4x4 Matrix Applications
Anatomical Calibration: location of anatomical axes of rotation can be determined to global marker locations through accurate calibration
Reflective Marker Wand (dimensions known)
Joint Rotation: Rotation of the knee, for example, can be described by the knee joint center plus the motion of the shank relative to motion of the thigh.
Virtual Points: It may be impossible to place markers at all the key locations (e.g. the hip joint center), therefore a calibration procedure facilitates hidden landmark identification.

4. Rotation Matrix
Recall this is the 3x3 inner matrix (lower right elements) of the 4x4 Tmatrix
To generate the 3x1 vectors comprising the rotation matrix the unit vector of the local CS axes in the global CS are used.
Dividing each vector by its length (to get the unit vector) gives the cosine of the angle that the vector makes with each axes of the global CS
Thus these are known as DIRECTION ANGLES and the DIRECTION COSINES

5. Rotation Matrix (Continued)
cosXx cosXy cosXz
[R] = cosYx cosYy cosYz
cosZx cosZy cosZz
What do the elements mean?
Ex: cosXy means the cosine of the angle formed by the X-axis of global CS and the y-axis of local-CS
Why are direction cosines useful?
If the rotation matrix is known for a local CS, then it is possible to determine the angles between the local and global axes

6. Pure Rotation A Simple Example (rotation about the Z-axis)Y X Y
(Px,Py)
You have point P’s coordinate in the local CS, how can you get it coordinates in the global CS?
alpha
90-alpha
First the DIRECTION COSINES:
Local x WRT Global X: cos (alpha)
Local x WRT Global Y: cos (90 -alpha)
Local y WRT Global X: cos (90 + alpha)
Local y WRT Global Y: cos (alpha)
Black = global
Red = local
Pglobal = cos (alpha) cos (90+alpha) Px = [R] [Plocal]
cos (90-alpha) cos (alpha) Py

7. Pure Rotation (continued)
Ex: What would be the global coordinates of P if the local coordinates are [3,1] and the local CS is rotated about the z-axis 25 degrees?
Pglobal = cos (25) cos (90+25) 3
cos (90-25) cos (25) =
2.17
Pglobal =

8. Translation & Rotation
To convert a point’s coordinate from one CS to another, a similar principle is applied except that the 4x4 transformation matrix is multiplied by the 4x1 point. A “1” is element 1 for the above the x,y,z point coordinates
The 4x4 transformation matrix is known as a HOMOGENEOUS TRANSFORM

9. Manipulation of Transformation Matrices
The general goal of transformation algorithms is to convert the motion of global 3D coordinates to meaningful relative rotations of two bodies. Some tools are needed to ease the manipulation of the transformation matrices:
Position Matrix: a transformation from local (body 1 or 2) to global coordinates [ TG1 ], [ TG2 ],
Local Transformation Matrix: a transformation in local coordinates from one body to another [ T12 ]
Displacement Matrix: a transformation in global coordinates from one body to another
[ D12 ]

10. Common Problems Transformation Matrices
1.) Given global coordinates of two bodies, find relative position in local reference frame
Given: [ TG1 ], [ TG2 ]
Wanted: [ T12 ]
Solution: [ T12 ] = [ TG1 ]-1 [ TG2 ]
TG2
TG1
BODY 1
T12
BODY 2
GLOBAL

11. Common Problems Transformation Matrices
2.) Given global coordinates of one body and its relative position to another body, find global coordinates of second body
Given: [ TG1 ], [T12]
Wanted: [ TG2]
Solution: [ TG2] = [ TG1 ] [T12]-1
3.) Given global coordinates of two bodies, find displacement matrix between bodies (assume it is the SAME body but at 2 different points in time)
Given: [ TG1 ], [TG2]
Wanted: [ D12]

12. Common Problems Transformation Matrices
3.) con’t
Body 1 – time 2
Body 1 – time 1
Consider point P ( ):
[PG1] = [TG1] PB1
[PG2] = [TG2] PB1
[TG1]-1 [PG1] = [TG2]-1 [PG2]
[TG2] [TG1]-1 [PG1] = [PG2]
[D12] = [TG2] [TG1]-1
GLOBAL
NOTE: this is different than [T12] which relates LOCAL points b/c this relates GLOBAL points

13. 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

14. Euler’s Angles Leonhard Euler (1707-1783)
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)

15. Y X . Z

16. 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

17. 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

18. 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

19. 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