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 v i = (s i+1 – s i ) / (∆t) Work out formula for a i in terms of s Backward Difference : good for the last frame of data v n = (s n – s n-1 ) / (∆t) Work out formula for a n 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 A B C a b c A 2 = B 2 + C 2 – 2BC cos (a) Therefore, cos (a) = (B 2 + C 2 – A 2 ) / (2BC) a = acos ( (B 2 + C 2 – A 2 ) / (2BC) ) Where a = knee flexion angle

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 Y Θ 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 = (a 1, a 2 ) IXI = [ (a 1 ) 2 + (a 2 ) 2 ] 0.5 Ex. Given X = (-2, 6), what is IXI? Dot Product X ∙ Y = (a 1 b 1 ) + (a 2 b 2 ) 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. (-6.1, 92.4) (-3.8, 50.2) (-8.6, 13.7) HIP KNEE ANKLE

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