PART I The Force-Motion Relationship
Describing
Motion
Photo reprinted from Marey, 1889.
X velocity-Time
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Movement is Motion – Motion is Movement
Laboratory Movement
Small Movement
Review of Math Review
Systeme Internationale = Metric System
Fundamental Units: mass in kg, linear distance in m,
angular distance in rad, time in s
All other physical measurements are derived from these
variables:
Force = N = kg*m / s2
Energy = J = kg*m2 / s2
Website for conversions
http://catcode.com/trig/trig08.html
More review of Math Review
Radian – the angle created by the arc on a circle with the
length of the radius of the circle (~ 57.3 degrees)
Arc length = 1 radius
Math Review
Trigonometry – sine, cosine, tangent, and inverse
functions
sin a = A/C, cos a= B/C, tan a= A/B
C
sin-1 A/C = a, cos-1 B/C = a,
tan-1 A/B = a
A
a
B
Math Application: important in signal
processing
Sine function – continuous wave over angular position
+1
0
-1
0
180
360 degrees
Math Application: important in signal
processing
Cosine function – continuous wave over angular position
+1
0
-1
Math Review
Website for sine and cosine waves
http://catcode.com/trig/trig08.html
Describing Motion = Kinematics
Kinematics describes the
Time – Geometry of Motion
or the
Movement Pattern
during static or dynamic activity
Two Fundamental Movement Patterns
Translation – Linear Movement – displacement
from one point to another in either:
Straight lines – rectilinear translation or
Curved lines – curvilinear translation
Animals can do both but curvilinear motion more
common
Curvilinear Translation During Walking
Two Fundamental Movement Patterns
Rotation – Angular Movement – displacement
around an axis
Principle means of
animal motion
Translation Through Rotation
Animals translate by skillfully combining joint
rotations
A person stands up by
rotating the hip, knee, and
ankle joints
Animals rotate to translate
Animals are rotating
machines
Translation Related to Rotation
Linear displacement and
velocity related to the
angular kinematics:
s = r
v = r
Calculate Arc Length when radius = 1 cm and  = 90°
Four Kinematic Variables or Motion
Descriptors
Position – location within the environment
Displacement – the change in position with movement
Velocity – rate of change of position
Acceleration – rate of change of velocity
(All variables are vectors)
Biomechanics Laboratories
Position in a Linear 2D Reference Frame
Heel Strike:
Shoulder=1.01,1.34
Knee
= 1.11, 0.47
Toe Off:
Shoulder=1.87,1.35
Knee
= 1.78, 0.44
Position in an Angular Reference Frame
Segment Angles –
Angle between a body
segment and the right
horizontal from distal end of
segment
Trunk = 85° or 1.48 rad
Arm = 95° or 1.66 rad
Position in an Angular Reference Frame
Joint Angles –
Angle between two body
segments
Shoulder = 20° or 0.35 rad
Knee = ???
Generate Angular Position Data
1) Identify location of skeletal joints
2) Define joint angles
3) Calculate segment angles
4) Combine segment angles to calculate joint angles
Position in an Angular Reference Frame
Acromion
1.10, 1.34
Greater Trochanter
1.05, 0.8
Lateral Knee
1.18, 0.5
Lateral Malleolus
1.23, 0.1
Heel
1.20, 0.02
5th Met
1.35, 0.08
Position in an Angular Reference Frame
Joint angular position for
obese and lean subjects
while walking
Obese less flexed at hip
and knee and less
dorsiflexed at ankle
Obese walk in a more
erect pattern
Displacement
Displacement = difference between final and initial
positions
Linear displacement (d) = Pf – Pi (m)
Angular displacement () = f - i ( or rad)
Displacement does not necessarily equal distance (the
length of the path traveled)
Displacement in a Linear Reference Frame
Horizontal displacement:
heel strike to toe off
Shoulder = 0.86 m
Met Head = 0.09 m
Total displ. Shoulder =
1.87,1.35
-1.01,1.34
0.86,0.01
Displacement in a Linear Reference Frame
Magnitude Result. Displ. = (Hor disp2 + Vert disp2)1/2
Resultant displacement
between heel strike
and toe off for:
Shoulder = 0.87 m
Met head = 0.10 m
Linear Displacement During Walking
Step length – forward displacement of one foot
during swing phase
Stride length – combined forward displacement of
both feet during left and right swing phases
Linear Displacement During Walking
Step length – mean value ~ 0.75 m for healthy adults,
less for shorter, older, ill, or injured people
Left and right step length symmetry
Stride length – mean value ~1.5 m for healthy adults,
less for shorter, older, ill, or injured people
Velocity
Velocity = rate of change of position
= amount of displacement per unit time
“rate of change” = calculus concept of the
derivative or slope
Linear velocity (v) = (Pf – Pi) / time (m/s)
Angular velocity () = (f - i) / time (/s or rad/s)
Johnson vs Lewis
100m, Seoul 1988
Gross body
movement
More information with shorter measurement intervals
Newsweek, 7-29-96
Average vs. Instantaneous Velocity
Velocity
Velocity = rate of change of position
= amount of displacement per unit time
“rate of change” = calculus concept of the
derivative or slope
Linear velocity (v) = (Pf – Pi) / time (m/s)
Angular velocity () = (f - i) / time (/s or rad/s)
Simple Finite Difference Technique
Velocity: displacement / time
vector
•magnitude: how fast
•direction: specification of “which way”
•This is motion
Cyclic Movement – Angular Kinematics
Angular Position
2.00
0.00
Time
Positive & negative
slopes on position
curve have positive
and negative phases
on the velocity
curve
Cyclic Movement – Angular Kinematics
Angular Position
2.00
Increasing +
0.00
Time
Positive & negative
slopes on position
curve have positive
and negative phases
on the velocity
curve
Cyclic Movement – Angular Kinematics
Angular Position
2.00
Decreasing +
Increasing +
0.00
Time
Positive & negative
slopes on position
curve have positive
and negative phases
on the velocity
curve
Cyclic Movement – Angular Kinematics
Angular Position
2.00
Increasing Decreasing +
Increasing +
0.00
Time
Positive & negative
slopes on position
curve have positive
and negative phases
on the velocity
curve
Cyclic Movement – Angular Kinematics
Angular Position
2.00
Increasing Decreasing +
Increasing +
Decreasing -
0.00
Time
Positive & negative
slopes on position
curve have positive
and negative phases
on the velocity
curve
Cyclic Movement – Angular Kinematics
Angular Position
2.00
0.00
Time
Angular Velocity
2.00
0.00
-2.00
Time
Positive & negative
slopes on position
curve have positive
and negative phases
on the velocity
curve
Relationship Between Position and Velocity
Knee angular position
& velocity curves
during the stance
phase of running
Knee Position/Velocity in Walking
Knee Angular Position
10.0
contact
Toe off
Position (degrees)
Flexion is Negative
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Time (s)
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Knee Position
Knee Position/Velocity in Walking
Knee Angular Position and Velocity
10.0
Position (degrees)
Flexion is Negative
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0.00
Identify local minima and maxima: velocity = ??
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Time (s)
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Knee Position
Knee Position/Velocity in Walking
Knee Angular Position and Velocity
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Position (degrees)
Flexion is Negative
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What is the sign of the velocity between local min & max?
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Time (s)
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Knee Position
Knee Position/Velocity in Walking
Knee Angular Position and Velocity
10.0
Position (degrees)
Flexion is Negative
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-70.0
0.00
Identify inflection points : ?
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Time (s)
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Knee Position
Knee Position/Velocity in Walking
Knee Angular Position and Velocity
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Position (degrees)
Flexion is Negative
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Identify inflection : local minima & maxima on velocity
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Time (s)
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Knee Position
Knee Position/Velocity in Walking
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Velocity (degrees/s)
Position (degrees)
Flexion is Negative
Knee Angular Position and Velocity
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Identify local minima and maxima
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Time (s)
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Knee Position
Knee Velocity
Knee Position/Velocity in Walking
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600
0.0
500
400
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300
-20.0
200
-30.0
100
-40.0
0
-100
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-70.0
0.00
Velocity (degrees/s)
Position (degrees)
Flexion is Negative
Knee Angular Position and Velocity
-300
Identify inflection points
-400
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0.50
Time (s)
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0.70
0.80
Knee Position
Knee Velocity
Second Order Finite Differences
• Use Project to demonstrate need.
Acceleration
Acceleration = rate of change of velocity
= change in velocity per unit time
change in velocity = change in motion
Acceleration
Acceleration = rate of change of velocity
= change in velocity per unit time
“rate of change” = calculus concept of the
derivative or slope
Linear acceleration (a) = (Vf – Vi) / time (m/s2)
Angular acceleration () = (f - i) / time
(/s2 or rad/s2)
Acceleration
Acceleration due to gravity = g = 9.81 m /s2
Pos (m) or Vel (m/s)
0
0
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1
2
3
Position
Velocity
Time (s)
4
5
Describe shape of:
Position curve
Velocity curve
Acceleration?
Acceleration
Acceleration due to gravity = g = 9.81 m /s2
Pos (m) or Vel (m/s)
0
0
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-80
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-120
-140
1
2
3
Position
Velocity
Time (s)
4
5
Describe shape of:
Position curve
Velocity curve
Acceleration
Velocity – Acceleration Relation
Angular Velocity
2.00
0.00
-2.00
Time
Based on definition
of acceleration,
sketch the angular
acceleration curve
for this velocity
curve
Velocity – Acceleration Relation
Angular Velocity
2.00
0.00
-2.00
Time
Angular Acceleration
2.00
0.00
-2.00
Time
Acceleration curve
shows the slope of
the velocity curve,
including positive
and negative
directions
Acceleration is our link to kinetics
• Important to understand the concept of
positive and negative acceleration
Acceleration is our link to kinetics
• Important to understand the concept of
positive and negative acceleration
• Parabolic motion: horizontal & vertical components
• release velocity
• Peak velocity
• contact velocity
Wayne Wright
Human Cannonball
Acceleration is our link to kinetics
• Important to understand the concept of
positive and negative acceleration
• Parabolic motion: horizontal & vertical components
• release velocity
• Peak velocity
• contact velocity
Gymnast leaves ground at v v = 3m/s and v h = 4.2 m/s.
Calculate velocities 0.25 seconds later.
Gymnast is at peak height (v v = 0 m/s). How fast is
she traveling 0.25 m (vertical) later? What is vh?
Gymnast is falling at –1.5 m/s. How fast is she
traveling 0.25s later? What about v h?
Constant Linear Velocity
• Acceleration = 0
• what is force?
• Calculating final position of body
• Rearrange equation of velocity
• Sprinter is 70 from start. Horizontal velocity of
11.5 m/s. Maintains for 1.7s. How far from start
line?
So let’s imagine the force.
09/02/07: Labor Day occasion to commemorate lost railroad workers By Bill
Kemp Archivist/Librarian, McLean County Museum of History Advertisement
BLOOMINGTON -- On Feb. 25, 1921, "death came fantastic with horror" when
a storage tank explosion tossed Harold Downey’s whirling, lifeless body 200 feet
into the air. A Chicago & Alton Railroad boilermaker, Downey’s fatal accident
reminds us of the untold number of railroad workers who lost their lives toiling in
one of the more vital and dangerous industries in U.S. history.
On that Friday afternoon, Downey, who worked at the C&A shops on
Bloomington’s west side, was sent to repair a leaking gasoline storage tank. When
he entered through a manhole at the top, his lighted torch ignited the escaping
fumes.
"His body was propelled upwards with a force, the intensity of which can only be
imagined," the Pantagraph reported.
Forensic Biomechanics
Complete the project available from
the website.
General Kinematic Procedures
Linear data:
use position of body segments for descriptive stride
characteristics (how far, high, etc)
use position, velocity, and acceleration for complex
inverse dynamics (F = ma)
Angular data:
use joint angular position for direct comparisons
use joint angular velocity for joint power (P= T * )
General Kinematic Procedures
Other places: You name it, people use it.
Pelvic tilt
Hand displacement in reaching
Trunk vibrations in driving
Tibia accelerations in running (shock)
etc etc etc
Orthotic Effects on Kinematics
Kinematic Coordination in Running
Coordination
between knee
and subtalar
motions
Kinematic Coordination in Running
Ankle
Position:
Soft
vs
Stiff
Landings
Use angular velocity to compare
technique
Use angular velocity to compare
technique
Use angular velocity to compare
technique
Relation Between Kinematics & Kinetics
Kinetics causes kinematics
Force produces acceleration which
ultimately causes a change in position
Kinematics causes kinetics ?
Position causes force??
Is this possible???
482 Advanced Biomechanics
• How many plan to enroll in this class for Spring
2008?
• Early registration is next week, do it at that time.
• Warning: if low enrollment, class will be
cancelled, so sign up for an alternative class too.
Error in Kinematic Data
Calculations of derivatives
introduces noise into the
signal
Velocity has some error
Acceleration has more
error
Error From Data Acquisition
Digitizing process
introduces error in
position data –
Markers move or vibrate
relative to the joint center
System misses the marker
Markers are covered
Time vs Frequency
Human movement – low
frequency
Digitizing error – high
frequency
Derivatives increase high
frequency error
Pos = sin 2 
Vel = 2 cos 2 
Fourier Analysis – Frequency Content
Single frequency
Two frequencies
Many frequencies
Digital Filters to Remove Error
Low pass filter allows low
frequency signal to pass but
filters out high frequency
error
Cut off frequency ~ 5-15 Hz
Optimal filter techniques
Fourier Analysis
Optimal Filter
Finds point beyond which
filtering does not
change result
Digital Filter Demonstration
(Show digital filter program application)
Bench Press: filter the angular velocity and
acceleration curves
80% 1RM BP, Narrow vs Wide Grip
Filtered Position – Good Acceleration
Filter position data to
produce accurate velocity
and acceleration data
We filter the linear
horizontal & vertical
position vs time data
Signal Processing
Data Filtering or smoothing vs. Curve fitting
90
70
Force by Contraction Velocity
60
y = 106.02x - 108.53
R2 = 0.9502
50
40
1.50
1.60
1.70
Height
1.80
1.90
Force (N)
Mass
80
120
100
80
60
40
20
0
y = 991.57x2 - 545.25x + 101.45
R2 = 0.9871
0
0.1
0.2
Velocity (mm/s)
0.3
0.4
Signal Processing
Complex and detailed mathematical topic
We provided only the slightest introduction
Summary – Kinematics
Kinematics describes
the movement
pattern in linear and
angular reference
frames
Kinematics is the
outcome of Kinetics
(the result not the
cause)