Table 2 Details of various measures used to quantify jumping jack movements

Efficiency

K₁

Highest wrist position normalized by height: Maximum difference wrists and shoulders heights, divided by height

\(D_{w/h} = E_{t} \left[ {\frac{{\left( {b_{1} , \ldots ,b_{4} } \right) }}{{\left( {a_{1} ,a_{2} } \right) }}} \right]\)

K₂

Widest leg split normalized by height: Widest distance between the ankles, divided by height

\(D_{b/h} = E_{t} \left[ {\left( {\frac{d}{{\left( {a_{1} ,a_{2} } \right) }}} \right) } \right]\)

Synchrony

H₁

Dominant frequency variance: Variance of dominant frequencies of articulated figure angles \({\theta }_{1},\dots ,{\theta }_{4}\) where dominant frequency \({f}_{i}^{max}\) is the frequency on the fast Fourier transform spectrum of \({\theta }_{i}\) with highest magnitude

\(\sigma_{{f^{max} }}^{2} = \frac{1}{3}\sum\nolimits_{i = 1}^{4} {\left( {f_{i}^{max} - \mu_{{f^{max} }} } \right)^{2} }\)

\(\mu_{{f^{max} }} = \frac{1}{4}\sum\nolimits_{j = 1}^{4} {f_{j}^{max} }\)

H₂

Mean absolute relative phase: Average difference in absolute value of instantaneous phase angles (PAs) of two signals

\(MARP_{{s_{1} ,s_{2} }} = E_{t} \left[ {\left| {PA_{{s_{1} }} - PA_{{s_{2} }} } \right|} \right]\)

\(PA_{s} = \left( {\frac{{s^{\prime}}}{s}} \right)\)

H₃

Continuous relative phase standard deviation: Standard deviation of continuous relative phase of two signals

\(CRPSD_{{s_{1} ,s_{2} }} = \sqrt {var_{t} \left[ {PA_{{s_{1} }} - PA_{{s_{2} }} } \right]}\)

H₄

Average of hand stop differences: Average difference in absolute value of time instants at which left and right arms stop moving, e.g. at the apex of JJ

\(A_{w} = \left[ {\frac{1}{{\left( {n_{R} ,n_{L} } \right) }}\sum\nolimits_{k = 1}^{{\left( {n_{R} ,n_{L} } \right) }} {\left| {T_{i + k}^{{w_{R} }} - T_{j + k}^{{w_{L} }} } \right|} } \right]\)

H₅

Average of leg stop differences: Average difference in absolute value of time instants at which left and right legs stop moving, e.g. at the apex of JJ

\(A_{l} = \left[ {\frac{1}{{\left( {m_{R} ,m_{L} } \right) }}\sum\nolimits_{k = 1}^{{\left( {m_{R} ,m_{L} } \right) }} {\left| {T_{i + k}^{{l_{R} }} - T_{j + k}^{{l_{L} }} } \right|} } \right]\)

Symmetry

M₁

Average hands bilateral symmetry: Average difference in absolute value of horizontal distance between the hands

\(\mu_{{X^{w} }} = E_{t} \left[ {\left| {c_{1} - c_{2} } \right|} \right]\)

M₂

Standard deviation of hands bilateral symmetry: Standard deviation of difference in absolute value of horizontal distance between the hands

\(\sigma_{{X^{w} }} = var_{t} \left[ {\left| {c_{1} - c_{2} } \right|} \right]\)

M₃

Horizontal hand velocities bilateral symmetry: Average difference in absolute value of horizontal velocities of left and right hands

\(\mu_{{V_{ \to }^{w} }} = E_{t} \left[ {\left| {V_{ \to }^{{w_{R} }} - V_{ \to }^{{w_{L} }} } \right|} \right]\)

M₄

Vertical hand velocities bilateral symmetry: Average difference in absolute value of vertical velocities of left and right hands

\(\mu_{{V_{ \downarrow }^{w} }} = E_{t} \left[ {\left| {V_{ \downarrow }^{{w_{R} }} - V_{ \downarrow }^{{w_{L} }} } \right|} \right]\)