Transcript Angular Kinematics - University of Ottawa

Angular Kinematics
D. Gordon E. Robertson, PhD, FCSB
School of Human Kinetics
University of Ottawa
Angular Kinematics
Differences vs. Linear Kinematics
Three acceptable SI units of measure
– revolutions (abbreviated r)
– degrees (deg or º, 360º = 1 r)
– radians (rad, 2p rad = 1 r, 1 rad ≈ 57.3 deg)
Angles are discontinuous after one cycle
Common to use both absolute and relative
frames of reference
In three dimensions angular
displacements are not vectors because
they do not add commutatively
(i.e., a + b ≠ b + a)
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absolute angles
for segments
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relative angles
for joints
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Absolute or Segment Angles
Uses Newtonian or inertial frame of reference
Used to define angles of segments
Frame of reference is stationary with
respect to the ground, i.e., fixed, not
moving
In two-dimensional analyses, zero is a
right, horizontal axis from the proximal
end
Positive direction follows right-hand rule
Magnitudes range from 0 to 360 or
0 to +/–180 (preferably 0 to +/–180) deg
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Angle of Foot
right horizontal axis
from proximal end
angle of foot is –60 deg
or 300 deg
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Angle of Leg
right horizontal axis
from proximal end
angle of leg is –75 deg
or 285 deg
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Relative or Joint Angles
Uses Cardinal or anatomical frame of reference
Used to define angles of joints, therefore
easy to visualize and functional
Requires three or four markers or two
absolute angles
Frame of reference is nonstationary, i.e.,
can be moving
“Origin” is arbitrary depends on system
used, i.e., zero can mean “neutral”
position (medical) or closed joint
(biomechanical)
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Angle of Ankle
ankle angle is
+110 deg
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Angle of Knee
knee angle is
–120 deg
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Absolute vs. Relative
knee angle =
[thigh angle
– leg angle] –180
=[–60–(–120)]–180
= –120
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angle of thigh
is –60 deg
knee angle is
–120 deg
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angle of leg
is –120 deg
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Joint Angles in 2D or 3D
q = cos–1[(a∙b)/ab]
a and b are vectors
representing two
segments
b
knee angle is
–120 deg
ab = product of
segment lengths
a
a∙b= dot product
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Angular Kinematics
Finite Difference Calculus
Assuming the data have been smoothed, finite
differences may be taken to determine velocity
and acceleration. I.e.,
Angular velocity
– omegai = wi = (qi+1 – qi-1) / (2 Dt)
where Dt = time between adjacent samples
Angular acceleration:
– alphai = ai = (wi+1 – wi-1) / Dt = (qi+2 –2qi +qi-2) / 4(Dt)2
– or ai = (qi+1 –2qi +qi-1) / (Dt)2
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3D Angles
Euler Angles
Ordered set of rotations:
a, b, g
Start with x, y, z axes
rotate about z (a) to N
rotate about N (b) to Z
rotate about Z (g) to X
Finishes as X, Y, Z axes
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Visual3D Angles
Segment Angles
Segment angle is angle
of a segment relative to
the laboratory
coordinate system
x, y, z vs. X, Y, Z
z-axis: longitudinal axis
y-axis: perpendicular to
plane of joint markers
(red)
x-axis: orthogonal to
y-z plane
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Visual3D Angles
Joint Cardan Angles
Joint angle is the angle of a
segment relative to a
second segment
x1, y1, z1 vs. x2, y2, z2
order is x, y, z
x-axis: is flexion/extension
y-axis: is varus/valgus,
abduction/adduction
z-axis: is internal/external
rotation
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Computerize the Process
Visual3D, MATLAB,
Vicon, or SIMI etc.
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