Lecture 2
Advanced Biomechanics
Dr. Moran
Spring 2006
Lecture Outline
• Introduction to MS Excel
• Example Worksheets
» Sine Wave Function
» Golf Ball Example
» Joint Angle Determination
• Finite Difference Calculus
• Central Difference
• Forward/Backward Difference
• Computing 2D Joint Angles
• Law of Cosines
• Dot Product
• Collect 2D Kinematic Data
• Explain Assignment #2
Finite Difference Calculus
• The derivative is nothing more than the SLOPE at a
given point. All numerical approaches can be thought of
as a way of determining that slope
• Central Difference Method: in order to compute the
derivative (slope) for a given frame i, you will need information from
the frame before and the frame after
• Do not require information from the frame i
• This method does NOT work for frames at the beginning and at the
end of a data set. Why?
Finite Difference Calculus (con’t)
• Forward Difference: good for the first frame of data
vi = (si+1 – si) / (∆t)
Work out formula for ai in terms of s
• Backward Difference: good for the last frame of data
vn = (sn – sn-1) / (∆t)
Work out formula for an in terms of s
•
Noise in the signal will greatly affect the numerical computation of derivatives! Filtering data is
incredibly important to avoid erroneous computations. Future Lecture on Signal
Processing/Filtering
Law of Cosines
c
A2 = B2 + C2 – 2BC cos (a)
B
Therefore,
cos (a) = (B2 + C2 – A2) / (2BC)
a = acos ( (B2 + C2 – A2) / (2BC) )
A
a
Where a = knee flexion angle
C
b
Dot Product
•
The dot product can be defined for two vectors X and Y by the following:
X ∙ Y = |X| |Y| cos Θ
Where:
Θ is the angle between the vectors
|X| is the norm (length of vector)
If X ∙ Y = 0, then X is orthogonal with Y
X
Θ
Y
Remember that to construct the vector
subtract coordinates of the starting
point from the end point!
Some Definitions
• Norm: can be thought of as the magnitude or length of a vector
X = (a1, a2)
IXI = [ (a1)2 + (a2) 2 ] 0.5
• Ex. Given X = (-2, 6), what is IXI?
• Dot Product
X ∙ Y = (a1b1) + (a2b2)
Ex. Given X = (-3,1) and Y = (2,9), what X ∙ Y?
Computing Joint Angles
• Ex: Compute the relative knee flexion angle? Use both
the Law of Cosines and the Dot Product.
HIP
(-6.1, 92.4)
KNEE
(-3.8, 50.2)
ANKLE
(-8.6, 13.7)
2D Planar Kinematics
Consideration of Anatomical Planes
When deciding on a particular movement for
2D kinematic analysis, first determine the
primary anatomical plane of action.
Be sure to place the camera perpendicular to
that plane
Ex: Running – the knee flexion angle predominately
occurs in the sagittal plane
http://training.seer.cancer.gov/module_anatomy/images/illu_body_planes.jpg
http://www.footmaxx.com/clinicians/pics/planefoot.gif