Table 3.

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

		Metric	Description	Equation	Eqn
Velocity profile based metrics		Speed [80, 81]	Ratio of mean speed to peak speed	${\eta }_{\mathit{speed}}=v_{\mathit{mean}}/v_{\mathit{peak}}$	(1)
		Speed arc length [79]	Temporal length of the velocity profile	${\eta }_{\mathit{spal}}=-ln\left({\int }_{t_1}^{t_2}\sqrt{{\left(\frac{1}{t_2-t_1}\right)}^2+{\left(\frac{d\widehat{v}}{\mathit{dt}}\right)}^2}dt\right)$	(2)
		Local minima of velocity [10]	Number of minimums in the velocity profile	${\eta }_{\mathit{minima}}=\sum Min\left(v\left(t\right)\right)$	(3)
		Velocity peaks [75, 76, 81]	Number of maximums in the velocity profile	${\eta }_{\mathit{peaks}}=\sum Max\left(v\left(t\right)\right)$	(4)
		Tent [80]	Ratio of area under the entire velocity profile to area under a single-peak velocity profile	${\eta }_{\mathit{tent}}={\int }_{t_1}^{t_2}v\left(t\right)dt/{\int }_{t_1}^{t_2}{v}_{\mathit{speak}}\left(t\right)dt$	(5)
		Spectral [25]	Summation of maxima Fourier transformed velocity vector	$Smoothness=-\sum Maxim{a}_{\overline{v}\left(\omega \right)}$	(6)
		Spectral arc length [25, 81]	Vector norm of the frequency spectrum of the fast Fourier transformed velocity profile	$SAL=-\underset{0}{\overset{{\omega }_c}{\int }}\sqrt{{\left(\frac{1}{{\omega }_c}\right)}^2+{\left(\frac{d\widehat{V}\left(\omega \right)}{d\omega }\right)}^2}d\omega$	(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	$\text{Eqn}\text{.7 w/}{\omega }_c=min\{{\omega }_c^{max},min\left\{\omega ,\widehat{V}\left(r\right)〈\overline{V}\forall r〉\omega \right\}\}$	(8)
		Mean arrest period ratio [76]	Time portion that movement speed exceeds a given percentage of peak speed	${\eta }_{\mathit{MAPR}}=\frac{t_{\mathit{vc}}}{t_{\mathit{total}}},vc\ge .1v_{\mathit{peak}}$	(9)
Jerk profile based metrics		Root mean square jerk [82]	Root-mean-square of the jerk normalized by the movement duration	${\eta }_{\mathit{rmsj}}=-\sqrt{\frac{1}{t_2-t_1}\underset{t_1}{\overset{t_2}{\int }}{\left|\frac{d^{2}v}{d{t}^2}\right|}^{2}dt}$	(10)
		Normalized mean absolute jerk [80, 82]	Mean of the magnitude jerk normalized or divided by the peak velocity	${\eta }_{\mathit{nmaJ}}=-\frac{1}{v_{\mathit{peak}}\left(t_2-t_1\right)}{\int }_{t_1}^{t_2}\left|\frac{d^{2}v}{d{t}^2}\right|dt$	(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	${\eta }_{\mathit{dj}}=-\frac{{\left(t_2-t_1\right)}^3}{v_{\mathit{peak}}^2}\underset{t_1}{\overset{t_2}{\int }}{\left|\frac{d^{2}v}{\mathit{dt}}\right|}^{2}dt$	(12)
		Log dimensionless jerk [81]	Logarithm of normalized jerk defined in equation	${\eta }_{\mathit{ldj}}=-ln\left(\frac{{\left(t_2-t_1\right)}^3}{v_{\mathit{peak}}^2}\underset{t_1}{\overset{t_2}{\int }}{\left|\frac{d^{2}v}{\mathit{dt}}\right|}^{2}dt\right)$	(13)

Equations have been rewritten using consistent variables where: $v\left(\omega \right)$ is the frequency domain of the limb’s velocity; $t_i$ is time at instant i; $v_{\mathit{peak}}$ is the peak velocity of the end-effector; ${\omega }_c$ is the cutoff frequency; $v_{\mathit{speak}}$ is the single peak velocity profile; $\widehat{v}$ is the normalized velocity vector; $\overline{v}$ is the normalized zero-padded velocity vector; and $\overline{V}$ is a given threshold of the velocity in the frequency domain