Reference Frame

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Reference Frame

• Reference frame
• fixed frame a reference frame that is fixed
• moving frame a reference frame that moves with
  the body.
• translate and/or rotate.
• inertial frame When a reference frame is either
  fixed or moving with a constant velocity
• non-inertial frame An accelerating reference
  frame
• global frame A fixed reference frame fixed to
  the environment, not to the moving subject
• local frame reference frames fixed to the
  moving body parts
• Among the perspectives presented in Axis
  Transformation as example, the spectators'
  perspective is an inertial, fixed, and global
  frame. The TV watcher's perspective is a moving
  (translating), non-inertial, and local reference
  frame. The gymnast's perspective is a moving
  (rotating), non-inertial, and local reference
  frame.

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The term "coordinate system" is slightly
different from "reference frame". The coordinate
system determines the way one describes/observes
the motion in each reference frame. Two types of
coordinate systems are commonly used in
biomechanics the Cartesian system and the polar
system. See Coordinate Systems for details of
these coordinate systems. One can describe a
motion differently in the same perspective
depending on the coordinate system employed.
Figure 1 shows examples of different reference
frames used to describe the human body motion.
One can easily define a local reference frame for
each body segment.
Figure 1
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Axis Rotation Matrices

• Two different reference frames
• XY vs X'Y
• Vector r in Fig. 1 can be expressed as (x, y) in
  XY system, or (x', y') in X'Y' system.
• Geometric relationships between xy and x'y'

Fig. 1
or
1
4
Expanding 1 to 3 dimensions
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the axis rotation matrix for a rotation about
the Z axis
5

• Similarly for the rotations about the X and the Y
  axis,

3
4

• Essential in developing the concept of the
  Eulerian/Cardanian angles
• See Eulerian Angles for the details. The rotation
  matrices fulfill the requirements of the
  transformation matrix.
• See Transformation Matrix for the details of the
  requirements.

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Axis Rotation vs. Vector Rotation

• In Fig. 2, the vector rather than the axes was
  rotated about the Z axis by f. This is called the
  vector rotation.
• In other words, vector r1 was rotated to r2 by
  angle f.

5
since
6
where r length of the vector, a the angle r1
makes with the X axis.
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Expanding 5 to 3-dimension
7
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Similarly,
8
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From 2 - 4 and 7 - 9
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• Vector rotation is equivalent to the axis
  rotation in the opposite direction.
• One should not be confused by the axis rotation
  and the vector rotation.
• In vector transformation, the axis rotation
  matrices should be used instead of the vector
  rotation matrices because vector transformation
  means change in the perspective.

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Euler Angle
1st rotation about
2st rotation about
3rd rotation about
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Bryant (Cardan) Angle
1st rotation about
2st rotation about
3rd rotation about
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• Orientation Angles -- Eulerian/Cardanian Angles
• Successive rotations to transform a vector from
  one reference frame to another
• The sequence of the successive rotations
• Eulerian
• Cardanian
• Eulerian type XYX, XZX, YXY, YZY, ZXZ, ZYZ
• Cardanian type
• Rotations about all three axes
• XYZ, XZY, YZX, YXZ, ZXY, ZYX
• 6 different combinations
• Very similar approach to compute the orientation
  angles
• A subset of the Eulerian

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Three successive rotations to change the
orientation of the reference frame (XYZ ? X'Y'Z'
? X''Y''Z" ? X'''Y'''Z'')
Figure 1
   where, c( ) cos, s( ) sin.
1
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By the multiplication of the rotation matrices
2
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Let the XYZ system frame A, and the
X'''Y'''Z''' system frame B, respectively. Then
, transformation matrix TB/A is
   3

• Used for any transformation matrix global to
  local or local to another local.
• These three successive rotations are mutually
  independent.
• Can treat these rotations separately to obtain
  angular velocities of the object.

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Imagine a gymnast performing a complex airborne
maneuver
A the global reference frame B the whole
body reference frame
Figure 2
X axis longitudinal, (? the inclination of the
whole body) Y axis, anteroposterior (?
somersault of the body) Z axis transverse axes
of the body, (? twist of the body)

• Let A and B in 3 be the pelvis and the right
  thigh, respectively.
• Again, the X, Y, and Z axes represent the
  longitudinal, anteroposterior, and transverse
  axes of the pelvis and the right thigh
• As a result, the three successive rotations
  represent flexion/extension, adduction/abduction,
  and medial/lateral rotation of the thigh w/t the
  pelvis.

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Two problems in interpreting the orientation
angle data

• The axis of the second rotation (rotation by
  angle q) is the Y axis of the intermediate
  frames, Y' / Y" axis in Figure 1, not that of
  either frame A or frame B. This sometimes causes
  confusion in assigning practical meanings to the
  orientation angles. In our example above (pelvis
  vs. right thigh), the thigh adduction/abduction
  occurs not in the frontal plane of the pelvis,
  but in the intermediate frontal plane of the
  right thigh. In other words, angle q in this
  example is not the same to the anatomical
  adduction/abduction angle. The anatomical
  joint-motion angles are actually the projected
  angles, not the orientation angles.
• The orientation angles are sequence-dependent so
  that you will get different sets of orientation
  angles from different sequences of successive
  rotations. In other word, type YXZ provides
  orientation angle values different from those of
  type XYZ. It is up to the analyst to choose the
  type of successive rotations, but employing
  different rotation sequences for different
  segments will complicate the analysis quite a bit.

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Computation of the Orientation Angles The
transformation matrices based on the global
coordinates of the markers Let
4
From 3 and 4
   5
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Therefore
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 and
7a b

• Must pay attention to the time history of the
  second orientation angle (q) to choose the right
  set of orientation angles.
• In many cases one may assume -p/2 lt q lt p/2, then
  a unique orientation angle
• This may not be true for the relative orientation
  angles at a ball-and-socket joint.
• In shoulder abduction, for example, the abduction
  angle continuously increases and gets larger than
  p/2 rad or smaller than -p/2 rad.

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A special case (Gimbal Locks)
From 4 5
8
Therefore
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In other words, if cos q 0, one can not compute
f and y separately, but collectively as f y or
f - y. These conditions are called the gimbal
locks.