Practice effects on local and global dynamics of the ski-simulator task

Abstract

This experiment examined the acquisition of the global and local dynamics of the changes in the total body center of mass–platform and inter-limb coordination motions over the course of practice (20, 30 s trials each day for 7 days) in the ski-simulator task. Four blocks of trials, representative of early, moderate, and extensive practice were analyzed through power spectrum and coherence analyses. The oscillation frequencies of the knee joints became tuned to that of the platform–performer system and there were changes due to practice in the lower inter-limb coordination dynamics independent of the center of mass and platform coordination pattern. Acquisition of global level dynamics occurs to achieve a stable task solution that can allow for degenerate frequency- and phase-locking of the mechanical degrees of freedom at both the local intra- and inter-limb levels.

Appendix

Derivation of knee joint angles

The knee joint angles were derived from the positions of the joint centers in 3 dimensions as follows (subscripts denote axis, X = medio-lateral, Y = anterior-posterior, Z = superior-inferior):

$$ \begin{aligned} {\text{AB}}_{X} & {\text{ = Ankle}}_{X} {\text{ - Knee}}_{X} \\ {\text{AB}}_{Y} & {\text{ = Ankle}}_{Y} {\text{ - Knee}}_{Y} \\ {\text{AB}}_{Z} & {\text{ = Ankle}}_{Z} {\text{ - Knee}}_{Z} {\text{,}} \\ \end{aligned} $$

$$ \begin{aligned} {\text{AC}}_{X} & {\text{ = Hip}}_{X} {\text{ - Knee}}_{X} \\ {\text{AC}}_{Y} & {\text{ = Hip}}_{Y} {\text{ - Knee}}_{Y} \\ {\text{AC}}_{Z} & {\text{ = Hip}}_{Z} {\text{ - Knee}}_{Z} {\text{.}} \\ \end{aligned} $$

The dot products of each line were then obtained:

$$ \begin{aligned} {\text{product}}_{X} & {\text{ = AB}}_{X} {\text{ $ \times $ AB}}_{X} \\ {\text{product}}_{Y} & {\text{ = AB}}_{Y} {\text{ $ \times $ AB}}_{Y} \\ {\text{product}}_{Z} & {\text{ = AB}}_{Z} {\text{ $ \times $ AB}}_{Z} {\text{.}} \\ \end{aligned} $$

The knee angles were then determined as:

$$ a{\text{cos}}{\left( {\frac{{{\text{product}}_{X} {\text{ + product}}_{Y} {\text{ + product}}_{Z} }} {{{\sqrt {{\text{(AB}}_{X} ^{{\text{2}}} {\text{ + AB}}_{Y} ^{{\text{2}}} {\text{ + AB}}_{Z} ^{{\text{2}}} {\text{)}}} }{\text{ $ \times $ }}{\sqrt {{\text{(AC}}_{X} ^{{\text{2}}} {\text{ + AC}}_{Y} ^{{\text{2}}} {\text{ + AC}}_{Z} ^{{\text{2}}} {\text{)}}} }}}} \right)}{\text{.}} $$

Coherence

Coherence analysis tests the similarity of two power spectra obtained from the FFT for any two sampled signals. The coherence, Cxy is computed as:

$$ Cxy(f) = \frac{{{\left| {Pxy(f)^{2} } \right|}}} {{Pxx(f)Pyy(f)}}, $$

Pxy represents the cross spectrum of signals x and y, while Pxx and Pyy represent the individual spectra for the aforementioned signals. The cross spectrum, Pxy is the fast Fourier transform of the cross-correlations between the two signals/time series, which transforms the cross-correlation values from the time domain to the frequency domain, with a coherence value (ranging from 0 to 1) assigned to each frequency bin.