Wednesday March 1st

1
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 RotationA Simple Example (rotation about
the Z-axis)
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 (90alpha)
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 (9025) 3
cos (90-25) cos (25) 1
Pglobal .9063 -.4226
3 .4226 .9063
1
8
Translation Rotation

• To convert a points 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 ProblemsTransformation 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

BODY 1
BODY 2
GLOBAL
11
Common ProblemsTransformation 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 ProblemsTransformation Matrices

• 3.) cont

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
Eulers 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)

http//www-history.mcs.st-andrews.ac.uk/PictDispla
y/Euler.html
http//www.strubi.ox.ac.uk/strubi/fuller/docs/spid
er2003/euler.gif
15
Y
X
Z
16
Common Cardan Sequencein 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 p/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