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```					   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)
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
(i.e., a + b ≠ b + a)
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absolute angles                   relative angles
for segments                      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 Foot

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 =                                           angle of thigh
(thigh angle                                          is –60 deg

– leg angle) –180                    knee angle is
–120 deg
= –60–(–120) – 180
= –120                                                 angle of leg
is –120 deg

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Joint Angles in 2D or 3D
q = cos–1[(a.b)/ab]

a & b are vectors                                       b
representing two
segments                              knee angle is
–120 deg

ab = product of                                         a
segment lengths

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
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 lab coordinate
system (LCS)
x, y, z vs X, Y, Z
z-axis: longitudinal axis
y-axis: perpendicular to
plane of joint markers
(red points)
x-axis: orthogonal to
y-z plane (cross-
product)
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Visual3D Angles
Joint Cardan Angles
Joint angle is the angle of a
segment relative to
another segment
x1, y1, z1 vs x2, y2, z2
order is x, y, z
x-axis: is flexion/extension
y-axis: is abduction/
x-axis: is internal/external
rotation
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Visual3D Angles
3 or 4 point angles
calculates angle between
two vectors with (3-
point) or without (4-
point) a common point
can be an angle
projected onto a plane
(XY, XZ or YZ) or a 3D
angle
limited to ranges of
motion of less than 180
degrees

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Electrogoniometry
Sensors
potentiometer
polarized light
optical fibre (e.g., Measurand)
strain gauge (e.g., Biometrics)
videography (e.g., Visual3D,
Polygon)

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Electrogoniometry
Potentiometry
can measure absolute or relative
angles
usually use one-turn “pots” for
human motions
essentially a variable resistor with
dc-power input
a “wiper” changes output voltage
depending on its angular position
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Electrogoniometry
Potentiometry
simple circuit, has
dc-input and one
or more outputs

signal condition
changes gain and
offset

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Electrogoniometry
Types
single-axis and torsional
(e.g., ShapeSensor,
Biometrics)
single-axis with four-bar
twin-axis (e.g., Biometrics)

triaxial (e.g., CARS-UBC,
ShapeTape)
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Electrogoniometry
permits linear translation of one arm
of the goniometer without causing
rotation of the potentiometer
potentiometer

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Electrogoniometry
requires 4x4 matrix
(Chao, et al. J Biomech,
3:459-71, 1970)
transformation to
estimate internal joint
motion and test jig for
calibration

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