IEEE Robotics & Automation Magazine - March 2022 - 79

different evaluation metrics with a single unit in 3D space.
The derivation of a model extension based on Fitts's law for
full 3D space is expected to increase the overall comparability
of the results of different studies attempting to capture 3D
human motion [2], [3], particularly attributed to the significant
focus on the law in current work. Hence, relying on multiple
types of metrics to assess human performance in VR and
teleoperation should be avoided, as comparability between
study results is rendered challenging [2], [16]. Earlier works
on teleoperation [16] and recent ones with anthropomorphic
robotic hands (19 DoF) leveraging MR technologies [15],
unfortunately, do not make use of metrics such as Fitts's law
and instead rely on multiple time-/spatial-/behavior-based
metrics. As a result, comparability, even between similar
works, is harder to achieve since the studies need to have
identical evaluation metrics.
2D Extensions
While originally developed for 1D tasks, Fitts's predictive
model has been widely applied to 2D pointing tasks [7], [8],
[17]. Hoffmann [8] conducted a series of discrete tapping
tasks, using participants' fingers as pointing probes, with the
width of a finger added to ID in (1) as
ID log ` + j ,
=
2
WF
2A
(2)
where F represents the index finger pad, which can be interpreted
as the size of the object to be transported to the target
W as a natural extension toward pick-and-place tasks [18].
Welford [17] proposed another variant of (1):
ID log2=+`
W
A
05j
.,
(3)
which has been demonstrated to do well in 2D task settings.
Mackenzie [7] extended Fitts's original law, known as the
Shannon equation, yielding a better fit and formulated as
ID log2=+`
W
A
1j .
(4)
The robustness and linear fit of the Shannon model has
been demonstrated for translational [11], [13] and rotational
tasks [11]. The resulting IDs of Mackenzie's model are one bit
less than with Fitts's formulation. Contrary to Fitts's model, the
Shannon formulation is limited in terms of mathematical
expressiveness. Although Mackenzie argued that the addition
of the +1 term avoids negative IDs in (1), this is not entirely
true. A negative ID in Fitts's model would mean that the cursor
or probe is already within the target area. This limits the theoretical
justification for the purpose of higher model fitting [4].
Nevertheless, adopting Mackenzie's model facilitates the comparability
of numerous extensions, as this has become the norm
for most of the subsequent models presented in the article.
3D Extensions
While Fitts's law has been applied to 3D pointing tasks [12],
[13], it does not accurately represent 3D movement [2], [11].
Murata and Iwase [13] introduced an extension of Fitts's law
to 3D pointing tasks, taking into account directional angles i
between an origin and a target. Their model is based on (4):
ID log `
2
=+ i ,
W
A
1j+ c $ sin
(5)
with i being the azimuth angle and effectively added to the
ID, following an almost sinusoidal relationship with MT [13].
A later study by Cha and Myung [12] extended the previous
work by adding inclination angles, representing higher
dimensions in finger-aimed pointing tasks in the spherical
coordinate system. Based on (2), it is formulated as
MT ab cd ` + j ,
sin
=+ $$ $ii log12 2
++
WF
2A
(6)
where 1i and 2i represent the inclination and azimuth angles
from the starting point to the target, respectively. Directional
angles again followed an almost sinusoidal relationship with
MT, as shown in Murata and Iwase's work [13]. The constants
a, b, c, and d are empirically determined through linear
regression. However, in the work of [12], the authors limited
their investigation to forward motions between 30! and
! 60c
azimuth angles. Recent work by Barrera Machuca and
Stuerzlinger [1] accounted for the preceding by introducing
pointing tasks with the use of 3D displays to indicate targets
ranging from azimuth angles of 90to
90c and 0 to 18 .0c
Their work confirmed that left-to-right movements are easier
than those away from and toward a user.
While this investigation covered a wider range of azimuth
angles, it still disregarded inclination angles, as the height of the
objects was adjusted to the viewing height of each participant.
However, human motor skills vary significantly with directions;
for example, upward movements appear to be more demanding
than downward ones [12]. Hence, it is important to study the
effects of directions and inclinations. So far in our analysis, no
model has investigated rotational variations or even combined
rotational with translational movements all in one setting. Additionally,
the identified spatial arrangements and factors in this
section should ideally be included, as well. Our work aims to
address these by effectively combining them in one setting.
Combining Translation and Rotation
To this point, we analyzed the most widely used extensions of
Fitts's law, limited to purely translational tasks. However, while
performing almost any type of manipulation or pointing task,
we attempt to match the rotation of the object, as well, to satisfy
certain spatial criteria [2], [16], [19]. In addition, to effectively
describe general human movement, the simultaneous
presence of translation and rotation needs to be accounted
for. Both are essential and inseparable parts when manipulating
real and virtual objects in the 2D and 3D domains [10],
[11], [14]. One early study showed that Fitts's law can be
adjusted and applied to purely 2D rotational tasks but did not
investigate combined movements [9].
The combination of translational and rotational tasks in
2D space is visually depicted in Figure 2. Stoelen and Akin
MARCH 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
79
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