Lecture 6
Wednesday March 1st
Dr. Moran
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
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.
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
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
Pure Rotation
A Simple Example (rotation about the Z-axis)
Y
Y
You have point P’s coordinate in the
local CS, how can you get it
coordinates in the global CS?
(Px,Py)
X
90-alpha
alpha
First the DIRECTION COSINES:
alpha
X
Black = global
Red = local
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)
Pglobal = cos (alpha)
cos (90-alpha)
cos (90+alpha)
Px
cos (alpha)
Py
= [R] [Plocal]
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)
cos (90+25)
cos (25)
3
1
Pglobal = .9063
-.4226
3
= 2.30
.4226
.9063
1
2.17
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
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 ]
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 ]
T12
TG1
GLOBAL
TG2
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]
Common Problems
Transformation Matrices
3.) con’t
Consider point P ( ):
GLOBAL
[PG1] = [TG1] PB1
[PG2] = [TG2] PB1
[TG1]-1 [PG1] = [TG2]-1 [PG2]
[TG2] [TG1]-1 [PG1] = [PG2]
[D12] = [TG2] [TG1]-1
NOTE: this is different than [T12] which relates LOCAL points b/c this relates GLOBAL points
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
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
–
http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euler.html
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)
http://www.strubi.ox.ac.uk/strubi/fuller/docs/spider2003/euler.gif
Y
.
Z
X
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
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
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
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