IEEE Robotics & Automation Magazine - March 2022 - 86

Manipulation Results
The data were normally distributed (p = 0.083), and the mean
MT for manipulation tasks was 2.11 ± 0.25 s. A total of 60 trials
out of 4,800, or 1.25% of the data, were excluded from the analysis,
due to error. In the trial, MT significantly decreased as target
width W increased (p < 0.001). Consistent with experiment 1, an
increase in target separation A significantly grew MT (p < 0.001).
Contrary to pointing tasks, directional angles z did not have an
apparent effect on MT (p = 0.476). Finally, increased incline
angles θ significantly affected MT (p < 0.05), as in [12].
Remarks
All variables had significant effects on MT, except directional
angles
z , in manipulation. Contrary to experiment 1, we
observed that Fitts's, Shannon's, and Welford's formulations
had a relatively higher fitting in pointing and manipulation
than Hoffmann's model, by approximately 5%. This is confirmed
with Cha and Myung's model, which is based on Hoffmann's,
having a very similar fitting for pointing and
manipulation. Only Murata and Iwase's directional model
presented a marginally higher fitting than the rest in both
pointing and manipulation types of tasks.
Experiment 3: Purely Rotational Tasks
To this point, only translational tasks were investigated; however,
rotation is an inseparable and fundamental part [2], [10],
[11]. Here, translation is eliminated. as the object and the target
will overlap in the center, with only their rotation differing.
Consequently, this task is an almost natural equivalent of
Fitts's original experiment, only for rotation.
Design
The study used a 2 × 2 × 3 × 4 within-subjects design, in
which the independent variables were two object sizes (F = 4
and 5 cm), two target sizes (W = 5 and 10 cm), three target
rotations (a = 15, 30, and 45º) and four rotational tolerances
(~ = 2.5, 5, 7.5, and 10º). The dependent variable was MT,
and 9,600 trials were recorded.
Pointing Results
The data deviated from normality (p < 0.001), and an ART
was applied; the mean MT was 3.37 ± 1.51 s. A total of 225 trials
out of 4,800, or 4.6% of the data, were excluded from the
analysis, due to error. An increase in object size F and target
width W showed a slight decrease in MT but not at a significant
level: p = 0.226 and p = 0.59, respectively. However, we
observed a significant correlation between the rotational separation
a and MT (p < 0.001). The largest effect occurred with
the rotational tolerance ,~ showing an almost inverse exponential
relationship with MT (p < 0.001). We can infer that the
effect of ~ was so significant that it had almost the same
inverse influence on MT as the equivalent of W and F for
translational tasks, although to a nonlinear fashion.
Manipulation Results
The data violated normality (p < 0.01), and an ART was thus
used. The mean MT for was 2.79 ± 0.67 s. A total of 64 trials
out of 4,800, or 1.3% of the data, were excluded due to error.
In line with pointing, no significant effect was observed on
MT from the object size F (p = 0.286). In contrast to pointing,
an increase in target width W did significantly decrease MT
(p < 0.001). Increasing rotational separation a also increased
MT significantly (p < 0.001). Finally, as with pointing, increasing
rotational tolerance ~ significantly decreased MT
(p < 0.001) in a nonlinear fashion. With the exception of target
width W, we observed the same relationship of the tested
variables on MT for pointing and manipulation. Rotational
tolerance ~ predominately affected MT in a purely rotational
setting. This was further investigated in the final experiment
to increase our understanding of the underlying factors.
Remarks
We fit all existing equations to accommodate the rotational
nature of this experiment, as detailed in Table 3. By doing so, we
are able to apply these methods to the rotational nature of the
trial and investigate any significant effects for comparison. We
further extended the work of [11] and covered all known
Table 3. An overview of the models and the r2 fitting results.
Experiment 2
Experiment 1
(Translation)
Model Fit (r2)
Fitts [5]
Hoffmann [8]
Welford [17]
Shannon [7]
0.862
0.852
0.859
Murata and Iwase [13] 0.859
Cha and Myung [12]
Our model
0.862
0.887
0.778
0.869
0.776
0.773
0.773
0.869
0.877
(Translation With
Directions)
0.832
0.789
0.832
0.831
0.845
0.833
0.788
0.579
0.511
0.577
0.575
0.585
0.574
0.509
0.663
0.659
0.673
0.68
0.68
Experiment 3
(Rotation)
0.604
0.6
Experiment 4
(Translation With Rotation
and Directions)
Pointing Manipulation Pointing Manipulation Pointing Manipulation Pointing Manipulation
0.838
0.61
0.659
0.901
0.613
0.613
0.6
0.804
0.459
0.569
0.465
0.469
0.476
0.578
0.805
0.748
0.762
0.755
0.758
0.759
0.77
0.803
For experiments 3 and 4, existing models are, in theory, incompatible, but to facilitate a fair comparison, as in [11], they were adjusted to accommodate rotation;
e.g., Fitts's IDt = log2(2A/W) becomes IDr = log2(2a/~) and both IDs are added for experiment 4. Bold represents the best model in the column group.
86 * IEEE ROBOTICS & AUTOMATION MAGAZINE * MARCH 2022
IEEE Robotics & Automation Magazine - March 2022

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