1) Angular kinematics describes motion that involves rotation, such as the movement of body segments. It includes concepts like angular displacement, velocity, and speed. 2) Key concepts in angular kinematics include computing angular quantities from changes in angular position over time, using degrees and radians as units of angle, and determining average versus instantaneous angular velocity. 3) Joint angles are relative angles between adjacent body segments and are important for analyzing human movement.

1. Angular Kinematics
Objectives:
• Introduce the angular concepts of absolute
and relative angles, displacement, distance,
velocity, and speed
• Learn how to compute angular displacement,
velocity, and speed
• Learn to compute and estimate instantaneous
angular velocity
Angular Kinematics
Kinematics
• The form, pattern, or sequencing of movement
with respect to time
• Forces causing the motion are not considered
Angular Motion (Rotation)
• All points in an object or system move in a circle
about a single axis of rotation. All points move
through the same angle in the same time
Angular Kinematics
• The kinematics of particles, objects, or systems
undergoing angular motion
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2. Angular Kinematics & Motion
• Most volitional movement is
performed through rotation SHOULDER
of the body segments NECK
• The body is often analyzed
ELBOW
as a collection of rigid, LUMBAR
rotating segments linked at HIP
the joint centers
• This is a rough
KNEE
approximation
ANKLE
Measuring Angles
Degrees: Radians:
90 π/2
57.3° 1 radian
180 0, 360 π 0, 2 π
270 3π/2
1 radian = 57.3° π = 3.14159
1 revolution = 360° = 2π radians Note: Excel uses
θ(degrees) = (180/π)× θ(radians) radians!
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3. Positive vs. Negative Angles
By convention, when describing angular kinematics:
• positive angles counterclockwise rotation
• negative angles clockwise rotation
Positive: Negative:
+90° -270°
+57.3°
+180° 0,+360° -180° 0,-360°
-57.3°
+270° -90°
Absolute Angle (or Angle of Inclination)
• Angular orientation of a line segment with respect
to a fixed line of reference
• Use the same reference for all absolute angles
θ
θ
Trunk angle Trunk angle
from horizontal from vertical
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4. Angular Displacement
• Change in the angular position or orientation of a
line segment
• Doesn’t depend on the path between orientations
• Has angular units (e.g. degrees, radians)
axis of rotation
initial final
orientation orientation
angular
displacement
Computing Angular Displacement
• Compute angular displacement (∆θ) by subtraction
of angular positions:
∆θ = θfinal – θinitial
∆θ
final
orientation θfinal initial orientation
θinitial
axis of rotation
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5. Angular Distance
• Sum of the magnitude of all angular changes
undergone by a rotating body
• Has angular units of length (e.g. degrees, radians)
• Distance ≥ (Magnitude of displacement)
Angular final orientation
Distance = 225°
-90°
intermediate
Angular
orientation 135° Displacement = 45°
axis of rotation initial orientation
Example Problem #1
A figure skater spins 10.5 revolutions in a
clockwise direction, pauses, then spins 60°
counterclockwise before skating away.
What were the total angular distance and angular
displacement?
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6. Relative Angle
• Angle between two line segments
• Compute relative angle by subtraction of absolute
angular positions:
θ(1→2) = θ2 – θ1
θ(1→2) segment 1
segment 2
θ2
θ1
axis of rotation
Joint Angles
• Joint angles are relative angles between longitudinal
axes of adjacent segments (or between anterior-
posterior axes for internal rotation)
θelbow
θshoulder
θhip
θknee Use a consistent sign
convention for joint angles
θankle (e.g. + = flexion)
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7. Computing Joint Angles
• Involves subtracting absolute angles of segments
• Exact formula and order of subtraction depends on
the joint and the convention chosen
θknee = θleg– θthigh θknee = 180° + θthigh– θleg
HIP θthigh HIP θthigh
θknee
θleg θleg
KNEE ANKLE KNEE
θknee ANKLE
Angular Velocity
• The rate of change in the absolute or relative angular
position or orientation of a line segment
change in angular angular
angular position displacement
velocity = =
change in time change in time
• Shorthand notation:
θfinal – θinitial ∆θ
ω= =
tfinal – tinitial ∆t
• Has units of (angular units)/time (e.g. radians/s, °/s)
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8. Angular Speed
• The angular distance traveled divided by the time
taken to cover it
• Equal to the average magnitude of the
instantaneous angular velocity over that time.
angular distance
angular speed =
change in time
• Has units of (angular units)/time (e.g. radians/s, °/s)
Angular Speed vs. Velocity
Angular end of follow-
Distance = 225° through
-90°
end of
Angular
backswing 135° Displacement = 45°
tennis player racquet at start
Assume tennis stroke shown takes 0.75 s:
225° +45°
Speed = Velocity =
0.75 s 0.75 s
= 300°/s = +60°/s
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9. Example Problem #2
The figure skater of Problem #1 completes the
first (clockwise) spin in 3 s, pauses for 1 s,
then completes the second (counterclockwise)
spin in 0.3 s.
What were her average angular velocity and
average angular speed during the first spin?
What were her average angular velocity and
average angular speed for the skill as a
whole?
Example Problem #3
A person is performing a squat exercise. She
starts from a standing (i.e. anatomical) position.
At her lowest point, 2 seconds later, her knees are
flexed to 60° and her hips are flexed to 90° from
the anatomical position.
1 second later, she has risen back to the standing
position and completed the exercise.
What were the average knee and hip angular
velocities during each phase of the exercise?
for the exercise as a whole?
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10. Average vs. Instantaneous Velocity
• Previous formulas give us the average velocity
between an initial time (t1) and a final time (t2)
• Instantaneous angular velocity is the angular
velocity at a single instant in time
• Can estimate instantaneous angular velocity using
the central difference method:
θ (at t1 + ∆t) – θ (at t1 – ∆t)
ω (at t1) =
2 ∆t
where ∆t is a very small change in time
Angular Velocity as a Slope
• Graph of angular position vs. time
slope = instantaneous
ω at t1
θ (degrees)
slope = average
ω from t1 to t2
∆θ(1→2)
∆t(1→2)
∆t
t1 t2 time (s)
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11. Estimating Angular Velocity
Identify points with
zero slope = points
θ (deg)
with zero velocity
Portions of the curve
with positive slope
time (s) have positive velocity
(i.e. velocity in the
ω (deg/s)
+ direction)
Portions of the curve
with negative slope
0
time (s) have negative velocity
(i.e. velocity in the
– direction)
Example Problem #4
A gymnast swings back and forth from the high
bar as shown below. Sketch her angular
velocity.
80
60
40
angle (deg)
20
0
0 2 4 6 8 10
-20
θ
-40
-60
time (s)
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