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Reference Frame

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The gymnast's perspective is a moving (rotating), non-inertial, and local reference frame. The term 'coordinate system' is slightly different from 'reference frame' ... – PowerPoint PPT presentation

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Title: Reference Frame


1
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.

2
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
3
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
2
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.

6
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.
7
Expanding 5 to 3-dimension
7
8
Similarly,
8
9
From 2 - 4 and 7 - 9
10
  • 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.

9
Euler Angle
1st rotation about
2st rotation about
3rd rotation about
10
(No Transcript)
11
Bryant (Cardan) Angle
1st rotation about
2st rotation about
3rd rotation about
12
(No Transcript)
13
  • 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

14
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
15
By the multiplication of the rotation matrices
2
16
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.

17
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.

18
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.

19
Computation of the Orientation Angles The
transformation matrices based on the global
coordinates of the markers Let
4
From 3 and 4
   5
20
Therefore
6
 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.

21
A special case (Gimbal Locks)
From 4 5
8
Therefore
  9
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.
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