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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
10
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11
Bryant (Cardan) Angle
12
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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
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
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