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'

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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,

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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.

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since
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where r length of the vector, a the angle r1
makes with the X axis.
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Expanding 5 to 3-dimension
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Similarly,
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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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