IEEE Robotics & Automation Magazine - September 2014 - 119

Analysis of Movement Speed Variability
In the literature, various measures are used to quantify movement intermittency or variability. Usually, a number of significant peaks in the speed profile quantifies movement
intermittency [6], [7], [20]. Movement variability measures,
on the other hand, are most commonly defined as end-point
error or variability [8], [10], quantifying only trial-to-trial
variability [14]. In contrast, within-trial variability measures
are commonly used for force variability, such as SD of force,
and most importantly a normalized version of SD, coefficient
of variation (CV) of force. CV facilitates comparing the results
of different studies [22] and is defined as SD of force normalized by the mean level of force.
Although trial-to-trial variability measures are well suited
to discrete movement tasks, such as reaching, a within-trial
variability measure is much better suited to continuous movement tasks, such as maintaining a constant speed during
movement. Hence, we use the CV of speed ^CVspeed h as the
measure of movement variability in this article. For each trial
in the experimental protocol described in the previous section,
CVspeed during the last 3 s of the trial quantified the speed
variability. The speed was obtained from encoder readings via
Euler's forward difference method and was bidirectionally filtered offline (for zero-phase shift) with a second-order lowpass Butterworth filter with 20-Hz cutoff frequency.
Statistical Analysis
We used a repeated measures analysis of variance (ANOVA)
with no between-subjects factors and with subject, trial,
speed, and torque within-subjects factors. CVspeed constituted

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Speed (°/s)

as always making a constant speed flexion movement to
match the target constant speed level as closely as possible.
The subjects were instructed to always check the mean speed
indicator after every trial and adjust their speed in the following trials accordingly. As a secondary goal, they were also
instructed to observe the speed profile plots to not only match
the mean speed but also to keep their speed constant
throughout the trial and avoid increasing or decreasing trends
in this plot. The instructions asked them to avoid slowing
down or stopping toward the end of the trial but rather to
keep a constant speed until the trial ended. After the subjects
read the written instructions, example speed profile plots
depicting successful and unsuccessful trials (in terms of satisfying target speed levels) were shown to them and explained
by the experimenter.
At the beginning of each session, the subjects were allowed
to practice as many trials as they wanted until they were convinced that they were able to successfully and consistently
complete the constant speed movement task. Only the last 20
trials out of 40 for each block was included in data analysis.
Also, the last 3 s of each 4-s trial was used in the analyses to
avoid the sudden jerks that occasionally occurred at movement initiation and during movements near the joint limits.
Note that the experiment's focus was on the sustained constant
speed movements rather than the initiation of the movements.

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Target Speed = 5°/s
Target Speed = 10°/s

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0.5
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Resistive Torque Level (Nm)

Figure 3. The subjects were successful in matching the target
speed level on average. The error bars denote the SD of speed.

the dependent measure. The trial had 20 levels, speed had two
levels (5 and 10°/s), and torque level had five levels (0-2 Nm
with 0.5-Nm increments). The subject (eight levels) is treated
as a random factor. Out of 1,600 total observations, three data
points were not included in the statistical analysis. In these
three trials, the subject mistakenly thought that the trial did
not initiate properly and quit moving at before the midpoint
of the trial. We used the Kenward-Rogers adjusted degrees of
freedom method to account for Type I error risk. The alpha
level was set at 0.05 for all significance tests. Since the trial did
not lead to any significant results when included as a factor
main or interaction effects, we report only the main and interaction effects of speed and torque on CVspeed. Tukey-Kramer's
post hoc analysis test was used for pairwise comparisons of
the main and interaction effects of torque. We used Statistical
Analysis System (SAS) software by SAS Institute Inc. for conducting the statistical analyses. We used a MIXED procedure
(PROC MIXED) design (due to both random and fixed effects), with the trial treated as a repeated measure and with a
compound symmetry structure for the covariance matrix.
This design allows for the incorporation of all available observations, excluding only missing individual observations, without having to drop a group or condition of data points [26],
[27], and therefore provides higher statistical power for data
sets with missing data points.
Results
Figure 3 shows the mean speed values achieved by the subjects
in the experiment with error bars depicting the SD of speed.
The subjects were able to perform the constant speed flexion
task reasonably well but with high variability, which is an
expected observation for slow movements. Increasing resistive
september 2014

IEEE ROBOTICS & AUTOMATION MAGAZINE

119

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2014