Introduction to Sports Biomechanics: Analysing Human Movement ...

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Quintic spline curve fitting Many techniques used for the smoothing and differentiation of data in sports biomechanics involve the use of spline functions. These are a series of polynomial curves joined – or pieced – together at points called knots. This smoothing technique, which is performed in the time domain, can be considered to be the numerical equivalent of drawing a smooth curve through the data points. Indeed, the name ‘spline’ is derived from the flexible strip of rubber or wood used by draftsmen for drawing curves. Splines are claimed to represent the smoothness of human movement while rejecting the normally-distributed random noise in the digitised coordinates. Many spline techniques have a knot at each data point, obviating the need for the user to choose optimal knot positions. The user has simply to specify a weighting factor for each data point and select the value of the smoothing parameter, which controls the extent of the smoothing; generally the weighting factor should be the inverse of the estimate of the variance of the data point. This is easily established in sports biomechanics by repeated digitisation of a film or video sequence. The use of different weighting factors for different points can be useful, particularly if points are obscured from the camera and, therefore, have a greater variance than ones that can be seen clearly. Inappropriate choices of the smoothing parameter can cause problems of over-smoothing (Figures 4.19(a) and (b)) or under-smoothing (Figures 4.19(c) and (d)). The optimum smoothing is shown in Figures 4.19(e) and (f). Generalised cross-validated quintic splines do not require the user to specify the error in the data to be smoothed, but instead automatically select an optimum smoothing parameter. Computer programs for spline smoothing are available in various software packages, such as MATLAB, and on the Internet (for example, http:// isbweb.org.software/sigproc/gcvspl/gcvspl.fortran) and allow a choice of automatic or user-defined smoothing parameters. Generalised cross-validation can accommodate data points sampled at unequal time intervals. Splines can be differentiated analytically. Quintic splines are continuous up to the fourth derivative, which is a series of interconnected straight lines; this allows accurate generation of the second derivative, acceleration. APPENDIX 4.2 BASIC VECTOR ALGEBRA QUANTITATIVE ANALYSIS OF MOVEMENT The vector in Figure 4.20(a) can be designated F (magnitude F) or OP (magnitude OP ). Vectors can often be moved in space parallel to their original position, although some caution is necessary for force vectors (see Chapter 5). This allows easy graphical addition and subtraction, as in Figures 4.21(a) to (f). Note that the vector F in Figure 4.20(a) is equal in magnitude but opposite in direction to the vector G in Figure 4.20(b); if the direction of a vector is changed by 180°, the sign of the vector changes. 157
INTRODUCTION TO SPORTS BIOMECHANICS Figure 4.20 Vector representation: (a) vector F and (b) vector G = − F. 158 Vector addition and subtraction When two or more vector quantities are added together the process is called ‘vector composition’. Most vector quantities, including force, can be treated in this way. The single vector resulting from vector composition is known as the resultant vector or simply the resultant. In Figure 4.21 the vectors added are shown in black, the resultants in blue, and the graphical solutions for vector addition are shown between dashed vertical lines. The addition of two or more vectors having the same direction results in a vector that has the same direction as the original vectors and a magnitude equal to the sum of the magnitudes of the vectors being added, as shown in Figure 4.21(a). If vectors directed in exactly opposite directions are added, the resultant has the direction of the longer vector and a magnitude that is equal to the difference in the magnitudes of the two original vectors, as in Figure 4.21(b). When the vectors to be added lie in the same plane but not in the same or opposite directions, the resultant can be found using the vector triangle approach. The tail of the second vector is placed on the tip of the first vector. The resultant is then drawn from the tail of the first vector to the tip of the second, as in Figure 4.21(c). An alternative approach is to use a vector parallelogram, as in Figure 4.21(d). The vector triangle is more useful as it easily generalises to the vector polygon of Figure 4.21(e). Although graphical addition of vectors is very easy for two-dimensional problems, the same is not true for three-dimensional ones. Component addition of vectors, using trigonometry, can be more easily generalised to the three-dimensional case. This technique is introduced for a two-dimensional example on pages 160–1. The subtraction of one vector from another can be tackled graphically simply by treating the problem as one of addition, as in Figure 4.21(f); i.e., J = F − G is the same as J = F + (−G). Vector subtraction is often used to find the relative motion between two objects.
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Introduction to Sports Biomechanics
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First edition published 1997 This e
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Contents List of figures x List of
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Study tasks 216 Glossary of importa
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FIGURES 1.26 Underarm throw - femal
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FIGURES 5.9 Typical path of a swimm
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Tables 2.1 Examples of slowest sati
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Preface Why have I changed the cove
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Introduction MISSING TEXT The first
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INTRODUCTION principal movements in
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INTRODUCTION TO SPORTS BIOMECHANICS
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Figure 2.2 ‘Principles’ approac
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Figure 2.13 Identifying critical fe
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Figure 6.8 Main skeletal muscles: (
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INDEX 282 balls air flow past 173 b
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INDEX 284 energy 219 kinetic 219 mi
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INDEX 286 isokinetic contraction 24
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INDEX 288 muscle length-tension rel
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INDEX 290 three-dimensional 146-7,
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INDEX 292 validity 220 vantage poin
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