A Quantitative Analysis

R. Smith

2. School of Exercise and Sports Science, University of Sydney

3. School of Electrical Engineering and Telecommunications,

University of New South Wales

It is widely held that in coordinated movements the many biomechanical degrees of freedom of the human body are linked together so that the dimensionality of executive control is greatly reduced. Few studies, however, have attempted to actually measure the dimensionality of the kinematic degrees of freedom in coordinated movements. Here we report the results from separate 3D kinematic studies of adult walking. The first data set comprises hip, knee and ankle flexion-extension (3 angles) during two strides from six subjects; the second data set comprises the three degrees of freedom at the midfoot, ankle, knee, hip and lumbar spine (15 angles) during five strides from 43 subjects. Time-domain analysis (Box-Jenkins modelling) was employed to compute the linear dynamic relations between each pair of joint angles from both data sets. These relations accounted for >90% of the variance of the signals, indicating that the angles were very tightly coupled. Frequency-domain analysis (cross-correlation and spectral analysis) was employed to identify the best-fit linear transfer function between each pair of joint angles from the second data set. The total coherence square between each pair of angles was derived from this analysis. The coherence square is analogous to the square of the Pearson product moment correlation (r^{2}) in that both provide a measure of the amount of variance accounted for by the estimated relationships. A principal components analysis was performed on the correlation matrix consisting of the square root of these coherence square values. This showed that the first component accounted for 80% of the variance of the signals, with the next component accounting for only 4% of the variance. Pearson product moment correlations were also calculated between each pair of angles for the second data set. A principal components analysis of the resulting correlation matrix showed that six components were necessary in order to account for 80% of the variance. This finding demonstrates the inadequacy of Pearson correlations for quantifying inter-joint coordination because, unlike the time- and frequency-domain analyses, it does not take account of possible frequency-dependent gain variations and phase shifts between the angles. Finally, a single-layer neural network with a linear transfer function was employed to model the inter-joint relations from both data sets. The results again showed that >90% of the variance of the signals could be accounted for by the relations between the angles. The different mathematical and statistical procedures employed in this study are in essential agreement that the coordinative structure of walking is very low-dimensional, perhaps as low as a single virtual degree of freedom.