Topic one: Kinematic quantities and measurement
Scalar vs vector quantities
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SCALAR: includes magnitude and unit
o E.g. temperature of 32°C – magnitude is 32° and the unit is Celcius
VECTOR: includes magnitude, unit and DIRECTION
o E.g. take two steps north – magnitude is 2 , unit is steps and direction is north
 Vectors are drawn using arrows and must show;
 Direction (the way the arrow head is facing)
 Point of application (where it started, depicted by a dot at the start of the arrow)
 Magnitude (represented by the length of the arrow)
 Line of action (dotted li e depi ti g the di e tio the e to ould o ti ue alo g if it did t stop/
change direction
 Vector quantities receive an underline under their symbol e.g. X a d a e ead as the e to X
Linear kinematic quantities
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POSITION: refers to a location in space, always in relation to origin
o Denoted by X and usually in metres
o A vector quantity
DISPLACMENT: the change in position i.e. the difference between two points
o Δx usually in metres
 Δ (delta) is the symbol for change. A larger delta symbol means a larger change
o Xi = initial position , Xf = final position, thus
 Δx = Xf – Xi
VECTOR ADDITION: polygon method
o E.g. take 2 steps east, then two steps north, then two steps west
 Scaled and directed arrows are used which join head to tail
 DISPLACEMENT = Δx = 2 steps north
DISTANCE (SCALAR ADDITION)
o A.k.a path length
 E.g. take 2 steps east, then two steps north, then two steps west
 Distance = 6 steps
Coordinate system
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CARTESIAN COORDINATE SYSTEM:
o Right handed
 X-axis rotates in towards y axis to produce the Z axis
 If you grab the X axis with your right hand and rotate
you ha d so it s gi i g the thu s up – your thumb is
running along the Z axis
 X – runs medial to lateral
 Y – runs anterior to posterior
 Z – runs superior to inferior
COORDINATE SYSTEM ORIGIN:
o Laboratory based/ global coordinate system
 Reference frame is based on the origin (which is a designated point in the laboratory)
o Segment based/ international/local coordinate system
 Reference frame is based on the way a body part moves relative to another body part
 E.g. how the tibia moves related to the femur
LINEAR Kinematic quantities
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- VELOCTIY (V) ho ui kly you e o i g a d i hat di e tio you e goi g (thus a vector quantity)
o Indicates the rate of change indisplacement
o Calculated using t (time in seconds) and x (displacement)
 V = Δdispla e e t/Δti e = Δx/Δt = xf – xi /tf – ti at time t= (tf + ti) / 2
o Expressed as m.s-1
ACCELERATION (a) the rate of change in velocity
 a = Δ elo ity/Δti e = Δv/Δt = vf – vi /tf – ti at time t= (tf + ti) / 2
*** NOTE, THESE EQUATIONS GIVE THE AVERAGE ACCELERATION OR VELOCITY***
ROTATIONAL/ANGULAR Kinematic quantities
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POSITION
o Denoted by Θ , units is radians
o Location is with respect to reference point
o Vector quantity
 Anticlockwise is positive
DISPLACEMENT
o De oted y ΔΘ
o Cha ged i positio ΔΘ= Θf – Θi
UNITS
o 2π (radians) = 360°
o 1 radian = 57.3°
VELOCITY
o Denoted by ω , units being radians.s-1 – a.k.a hz (pronounced herz)
 Is the rate of change of displacement
 ω = ΔΘ/Δti e = ΔΘ/Δt = Θf – Θi /tf – ti at time t= (tf + ti) / 2
ACCELERATION
o Denoted by α , units being radians.s-2 –
 Usually dropped tp
 Is the rate of change of velocity
 ω = ΔΘ/Δti e = ΔΘ/Δt = Θf – Θi /tf – ti at time t= (tf + ti) / 2
Often radians is
dropped from the
u its, so it s just
displayed as s-1
(velocity) or s-2
(acceleration)
Kinematic measures
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Eadweard Muybridge fathered biomechanics using sequential photographs to analyse equine gait
o Then moved to human gait
Biomechanics began using cinematography, then moved to single video cameras (like siliconcoach) and now
multiple camera systems
o Only 2 perpendicular cameras are required for a 3D position, but more cameras give better detail
Vicon reflective markers are now used
o Placed on landmarks of the body
o Are retroreflective and special cameras pick up on this reflection
PBM tute
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ONE gait cycle is generally around 1.0 second with or without AFO
One frame is taken every 0.04 seconds  0.04/1.0 x 100 = 4% - each frame is thus worth 4% of the gait cycle
Perspective: things that look further away will look smaller even though they may be the same size
For linear movement, velocity and speed are used interchangeably
LENGTH
SCALAR (s.d)
Path length a.k.a DISTACE (m)
VECTOR
Straight line change in direction
a.k.a (x) DISPLACEMENT (metres)
Speed and velocity
Acceleration (a)
SPEED = Δdista e/Δti es
= path length/ time
a = Δspeed/ Δt
(V) VELOCITY= Δdispla e e t/Δti e
a = Δv/ Δt
Kinematic analysis
Differentiation
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We use NUMERICAL DIFFERENTIATION (not calculus methods) to determine velocity and acceleration from
displacement
o Specifically we use the CENTRAL DIFFERENCES METHOD
Central differences method
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used to smooth the data
o Sampling must be taken at equal time intervals , e.g. every 2 seconds
o The fi st ti e i te al o t ha e a y data (i.e. t1)
o In order to obtain the smoothed data at a given time, the
position/displacement for the time period immediately BEFORE and AFTER it
must be known
 We can then use the equation V = (X3 – X1) / (t3 – t1) to obtain a smoothed
data point for t2
Usain Bolt
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Ran 100m in under ten seconds
o Was seen to reduce his efforts
before crossing the line though
 This results in graphs for
velocity and acceleration vs
time being very jagged and not smooth
 So how fast could he have really gone?
Smoothing
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We k o all easu e e ts o tai e o a.k.a oise
Every time we differentiate, we enhance the noice
o Hence the need to smooth it
We can smooth by
o Drawing a smooth line through the data, and trying to re-estimate the points
o Using numerical techniques to filter and smooth the data
o Fit an equation
o Reconstruct the data from low frequency components of a Fourier series
FOURIER ANALYSIS: calculates components using sinusoids **NOT REALLY USED
ANYMORE**
o Lo f e ue ies a e the t ue sig als
o Highe f e ue ies a e o side ed to e oise
 Signal is reconstructed using only lower frequencies
EQUATION: can develop an equation, from which we can then pick time periods of
equal distribution and use this to determine the displacement/velocity at that time
Integration of kinematic quantities (not assessed)
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Use the equations to determine velocity and displacement from the acceleration data
o Vf =Vi + a(tf – ti)
o Xf = Xi +V(tf – ti)
PBM tute
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The Polygon method for determining displacement wants you to do it GRAPHICALLY
o i.e. try and draw it to scale on some graph paper and use your ruler to estimate the distances