IEEE Robotics & Automation Magazine - March 2022 - 88

extensions. Overall, regarding variables influencing MT, rotational
variables a and ~ were significant in both cases, although
some inconsistencies were observed between pointing and
manipulation. Contrary to experiments 1 and 2, none of the
adjusted models fitted this rotational task well, where all formulations
were below
r 068 for pointing and manipulation.
2 # .
Experiment 4: Combined Translational and
Rotational Tasks With Directions and Inclinations
For experiment 4, we combined all possible variations of
experiments 1-3, creating a fully combined movement task.
We merged translational and rotational task variations with
changing directional and inclination gains to describe a full
3D task. To limit the number of variations, we restricted the
directional movements z toward the front and right (0 and
90º), still investigating view and lateral directional influences.
We also kept the object size fixed at F = 4 cm. These
decisions were made to limit the number of tasks, enabling
us to control our already exhaustive experiment.
Design
For the final experiment, 26 within-subjects design combinations
were used, in which the independent variables were two target
sizes (W = 4 and 8 cm), two target separations (A = 12 and 24
cm), two direction angles (z = 0 and 90º), two inclination angles
(i = 15 and 30º), two target rotations (a = 30 and 45º), and two
rotational tolerances (~ = 7.5 and 15º). The dependent variable
was MT. Representing 64 tasks and with four repetitions, we conducted
256 trials for pointing and manipulation tasks, i.e., 512 trials
per participant, resulting in 10,240 trials.
Pointing Results
An ART was applied, as normality was violated (p = 0.016).
The mean MT for pointing tasks was 2.71 ± 0.55 s. A total of
120 error trials out of 4,800, or 2.5% of the data, were
excluded from the analysis. Sphericity was met in all cases
since all tested variations were two levels. There was no significant
effect of target width W on MT: p > 0.05. On the
other hand, the target separation A, consistent with experiments
1 and 2, significantly increased MT with higher separation
values (p < 0.001). Neither directions z nor
inclinations i had a significant effect on MT: p = 0.141 and
(p = 0.2). As for rotation, a significantly affected MT
(p < 0.01), and ω had the highest influence (p < 0.001), consistent
with experiment 3.
Manipulation Results
The data were normally distributed (p = 0.091). The mean
MT for manipulation tasks was 3.10 ± 0.59 s. A total of 82 trials
out of 4,800, or 1.7% of the data, were excluded from the
analysis, due to error. Sphericity was met in all cases. Contrary
to the pointing task, target width W had a significant
effect on MT (p < 0.001). Target separation A significantly
increased MT with higher separation values (p < 0.001).
Directional angles z and inclination angles i followed the
trend in the pointing equivalent, not affecting MT much:
88 * IEEE ROBOTICS & AUTOMATION MAGAZINE * MARCH 2022
Remarks
For experiment 4, we added the IDs of translation and rotation
of all existing approaches. The rotation was adjusted as
described in experiment 3 and Table 3. Despite these models,
in principle, being incompatible for combined movements,
we added these two IDs to explore the potential of summing
these separate concepts of motion and conduct a fair comparison.
For example, the combined ID for Fitts's model would be
lo (/ )
ID =
t
g AW for translation, plus ID = log ()/
2 2
r
2 a~
2
for rotation, with the other formulations following the same
fashion. Overall, neither directional z nor inclination i
angles had any significant effect on MT (p > 0.05) in pointing
and manipulation. For combined movements, adding IDs of
translation and rotation, Fitts's, Shannon's, Welford's, and even
Murata and Iwase's directional formulations proved insufficient
when fitted for pointing (. )
better results for manipulation (. ). Hoffmann's had
r 052
2
r 077
1 but yielded slightly
1
a significantly higher fitting toward pointing, with an approximately
10% increase. Consistent with experiment 1, this suggests
the necessity of adding the object size F in a model. Cha
and Myung's model had the highest fitting on the data overall
but only marginally across the rest, primarily due to the incorporation
of object size F, as also seen in Hoffmann's.
Model Derivation
In most prior work related to extending Fitts's law, a clear definition
of distance is missing [4]. For example, the distance
from the object to the target (A) can be defined from the
object center to the target center (Acc), the object center to the
target edge (Ace), and the object edge to the target edge (Aee).
These definitions influence the formula
An
Z
[
\
]
]
AA
AA
]AA WF,
1
2
3
=+
=+ +
cc
ec
ee
2
2
From the preceding, a total of three different target separations
are defined as An, with 22 The Euclidean
AA .A
cc
ec
ee
distance between two points in 3D space is represented as dn,
with n = 3. At each stage of the experiments, our results show
that 3D translational movement closely follows Fitts's formulation.
Target separation A and target width W are an integral
part of the formulation. However, another variable with a
determining effect in translational tasks was the object size F,
as part of Hoffmann's model but not of Fitts's. Hence, the
most appropriate definition of effective target separation
would be A3 from (8), which would include W and F. By
incorporating this definition into Fitts's model and more specifically
the ID in (1), we have
=
,
W, dx y 2
n
ni i
i
==
/^h
(8)
1
.
p = 0.099 and p = 0.227, respectively. Rotational separation a
did significantly affect the increase of MT as the degrees of
separation rose (p < 0.01). Rotational tolerance ~ again had
the biggest influence on MT, consistent with the pointing
equivalent and experiment 3 (p < 0.001).
IEEE Robotics & Automation Magazine - March 2022

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2022
