1. Angular Kinematics D. Gordon E. Robertson, PhD, FCSB
School of Human Kinetics
University of Ottawa

2. 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)
4/10/2017
Biomechanics Lab, University of Ottawa

3. Biomechanics Lab, University of Ottawa
4/10/2017
Biomechanics Lab, University of Ottawa

4. Biomechanics Lab, University of Ottawa
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
4/10/2017
Biomechanics Lab, University of Ottawa

5. Biomechanics Lab, University of Ottawa
Angle of Foot
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Biomechanics Lab, University of Ottawa

6. Biomechanics Lab, University of Ottawa
Angle of Leg
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Biomechanics Lab, University of Ottawa

7. Biomechanics Lab, University of Ottawa
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)
4/10/2017
Biomechanics Lab, University of Ottawa

8. Biomechanics Lab, University of Ottawa
Angle of Ankle
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Biomechanics Lab, University of Ottawa

9. Biomechanics Lab, University of Ottawa
Angle of Knee
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Biomechanics Lab, University of Ottawa

10. Biomechanics Lab, University of Ottawa
Absolute vs. Relative
knee angle =
[thigh angle
– leg angle] –180
=[–60–(–120)]–180
= –120
4/10/2017
Biomechanics Lab, University of Ottawa

11. Biomechanics Lab, University of Ottawa
Joint Angles in 2D or 3D
q = cos–1[(a∙b)/ab]
a and b are vectors representing two segments
ab = product of segment lengths
a∙b= dot product
4/10/2017
Biomechanics Lab, University of Ottawa

12. 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
4/10/2017
Biomechanics Lab, University of Ottawa

13. Biomechanics Lab, University of Ottawa
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
4/10/2017
Biomechanics Lab, University of Ottawa

14. 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
4/10/2017
Biomechanics Lab, University of Ottawa

15. 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
4/10/2017
Biomechanics Lab, University of Ottawa

16. Computerize the Process
Visual3D, MATLAB, Vicon, or SIMI etc.
4/10/2017
Biomechanics Lab, University of Ottawa