Table 3 Velocity profile and jerk profile based smoothness metrics found in reviewed papers

Velocity profile based metrics

Speed [80, 81]

Ratio of mean speed to peak speed

\(\eta_{speed} = v_{mean} /v_{peak}\)

(1)

Speed arc length [79]

Temporal length of the velocity profile

\(\eta_{spal} = - \ln \left( {\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} \sqrt {\left( {\frac{1}{{t_{2} - t_{1} }}} \right)^{2} + \left( {\frac{{d\hat{v}}}{dt}} \right)^{2} } dt} \right)\)

(2)

Local minima of velocity [10]

Number of minimums in the velocity profile

\(\eta_{minima} = \sum Min\left( {v\left( t \right)} \right)\)

(3)

Velocity peaks [75, 76, 81]

Number of maximums in the velocity profile

\(\eta_{peaks} = \sum Max\left( {v\left( t \right)} \right)\)

(4)

Tent [80]

Ratio of area under the entire velocity profile to area under a single-peak velocity profile

\(\eta_{tent} = {{\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} v\left( t \right)dt} \mathord{\left/ {\vphantom {{\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} v\left( t \right)dt} {\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} v_{speak} \left( t \right)dt}}} \right. \kern-0pt} {\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} v_{speak} \left( t \right)dt}}\)

(5)

Spectral [25]

Summation of maxima Fourier transformed velocity vector

\(Smoothness = - \sum Maxima_{{\overline{v}\left( \omega \right)}}\)

(6)

Spectral arc length [25, 81]

Vector norm of the frequency spectrum of the fast Fourier transformed velocity profile

\(SAL = - \mathop \smallint \limits_{0}^{{\omega_{c} }} \sqrt {\left( {\frac{1}{{\omega_{c} }}} \right)^{2} + \left( {\frac{{d\hat{V}\left( \omega \right)}}{d\omega }} \right)^{2} } d\omega\)

(7)

Modified spectral arc length [79]

Spectral Arc Length with the cutoff frequency modified to a given threshold velocity and an upper-bound cutoff frequency

\({\text{Eqn}}{\text{. 7 w/ }}\omega_{c} = \min \{ \omega_{c}^{\max } ,\min \left\{ {\omega ,\hat{V}\left( r \right)\left\langle {\overline{V} \forall r } \right\rangle \omega } \right\}\}\)

(8)

Mean arrest period ratio [76]

Time portion that movement speed exceeds a given percentage of peak speed

\(\eta_{MAPR} = \frac{{t_{vc} }}{{t_{total} }}, vc \ge .1v_{peak}\)

(9)

Jerk profile based metrics

Root mean square jerk [82]

Root-mean-square of the jerk normalized by the movement duration

\(\eta_{rmsj} = - \sqrt {\frac{1}{{t_{2} - t_{1} }}\mathop \smallint \limits_{{t_{1} }}^{{t_{2} }} \left| {\frac{{d^{2} v}}{{dt^{2} }}} \right|^{2} dt}\)

(10)

Normalized mean absolute jerk [80, 82]

Mean of the magnitude jerk normalized or divided by the peak velocity

\(\eta_{nmaJ} = - \frac{1}{{v_{peak} \left( {t_{2} - t_{1} } \right)}}\mathop \smallint \nolimits_{{t_{1} }}^{{t_{2} }} \left| {\frac{{d^{2} v}}{{dt^{2} }}} \right|dt\)

(11)

Dimensionless-squared jerk [80]

Square root of the integral of the square of the jerk times the duration of the movement to the fifth power over the length squared

\(\eta_{dj} = - \frac{{\left( {t_{2} - t_{1} } \right)^{3} }}{{v_{peak}^{2} }}\mathop \smallint \limits_{{t_{1} }}^{{t_{2} }} \left| {\frac{{d^{2} v}}{dt}} \right|^{2} dt\)

(12)

Log dimensionless jerk [81]

Logarithm of normalized jerk defined in equation

\(\eta_{ldj} = - \ln \left( {\frac{{\left( {t_{2} - t_{1} } \right)^{3} }}{{v_{peak}^{2} }}\mathop \smallint \limits_{{t_{1} }}^{{t_{2} }} \left| {\frac{{d^{2} v}}{dt}} \right|^{2} dt} \right)\)

(13)