Appendix 2
Asymmetries, Priorities, and Controllability of Multiple Degrees of Freedom:
A Dimensional Analysis of 6 DOF Tracking
This chapter is a more detailed analysis of Experiment 3. Based on the measure of integrated RMS error, Chapter 3 has analysed the relative performance between the isometric controller and the elastic controller. This appendix takes the approach of decomposing subjectsÌ 6 DOF tracking process into separate dimensions, so as to gain more insight into how users handle manipulation tasks with multiple degrees of freedom. Of particular interest to this chapter are differences between performance in different axes of 3D space and how well humans can handle all 6 degrees of freedom simultaneously.
With regards to human performance in X, Y, Z dimensions in 3D space, it can be expected that performance in the depth dimension will not be the same as that in the horizontal or vertical dimensions even with multiple sources of depth cues (see Chapter 5 for a review of depth cues and 3D display techniques). Unknown, however, is the magnitude of this difference: 10 percent, 50 percent, or order of magnitude? This appendix will evaluate subjectsÌ performance in the Z dimension in comparison to the X and Y dimensions, with the presence of the depth cues that are easily available with todayÌs technology.
As indicated in the human visual performance literature, performance differences between the horizontal and the vertical dimensions are also conceivable. For instance, in a task that required nursery school children to reproduce lines on a circular background, Berman, Cunningham, and Harkulich (1974) found that the reproductions of the vertical lines were significantly more accurate than the reproductions of the horizontal and the oblique lines, as measured by orientation differences. Gottsdanker and Tietz (1992) found that humans tend to be more sensitive in judging relative length in the horizontal dimension than in the vertical direction. It is therefore valuable not only to compare performance in the Z dimension with the X and Y dimensions but also to evaluate the performance difference between the X and the Y dimensions. In other words, the general objective here is to examine the (an)isotropies or asymmetries in X, Y, and Z dimensions.
The second goal of this chapter is to examine the simultaneous controllability of all 6 DOF. There have been some concerns in the teleoperation literature with regard to the controllability of all 6 degrees of freedom with one hand. Rice, Yorchak, and Hartley (1986) observed that controlling 6 DOF with one hand was difficult, due to unwanted cross coupling between axes. Some practical teleoperation systems, such as the Shuttle Remote Manipulator, also known as the "CANADARM", require two-handed operation, one for rotation control and the other for translation control. On the other hand, O'Hara (1987) contradicted Rice et alÌs observation and found no difference between two 3 DOF controllers versus one 6 DOF control. McKinnon and King (1988) further argued that a one hand 6 DOF controller should be preferable to distributed control among separate controllers. Detailed empirical evidence has not been found on this issue, however.
A2.2.1 Review of Experiment 3
The analyses in this appendix are based on Experiment 3. In that
experiment, subjects were asked to continuously control a 3D cursor
and align it, as closely as possible in both location and orientation,
with a 3D target which moved unpredictably in 6 DOF within a virtual
environment. Both the tracking target and the controlled cursor
were tetrahedral in shape. The target in the experiment was driven
by six independent forcing functions for each degree of freedom.
The cursor was controlled by the subjects through elastic or isometric
rate controllers. Four types of depth cues were implemented in
the experimental display: stereoscopic disparity, perspective,
interposition (edge occlusion), and partial occlusion through
semi-transparency (see Chapter 5).
A2.2.2 The Decomposed Performance Measures
At sampling step 
(where i
is the step number and 
is the sampling
period), the vector from the centre of the cursor to the centre
of the target is defined as translation vector 
.
The translation error in 3D is the norm of
, i.e. the Euclidean distance from the centre of the cursor to
the centre to the target. For each entire trial, the translation
root-mean-square (RMS) error is:
(A2.1)
where N is the last step.
consists of three components in the horizontal
(xi), vertical (yi) and
depth (zi) dimensions respectively, i.e.:
(A2.2)
For each trial the decomposed RMS tracking errors in X, Y and Z dimensions are:
, 
and 
(A2.3)
respectively. Xrms, Yrms,Zrms are the decomposed translation measures used later in the dimensional analysis of translation.
Rotational errors were measured in a similar way in the present
work, although parameterisation of rotations in 3D space is a
relatively complex subject (see Altmann, 1986; Hughes, 1986 for
mathematical treatments of rotation parameterization). At ,
the angular displacement (rotation mismatch) between the cursor
and the target can be expressed as (Altmann, 1986, p 70)
(A2.4)
signifies that at tracking step i,
the cursor and the target are angularly mismatched about axis
by scalar angle 
.
is a unit vector specifying the direction
of the orientation mismatch and is the
amount of mismatch. and
can be combined as a single rotation vector (Figure A2.1):
(A2.5)
Since is a unit vector, the
magnitude of 
i is the amount of rotation mismatch
between the cursor and the target. 
and
are the decomposed components of the rotation
mismatch in the X, Y and Z dimensions. Note that
and are not pitch, yaw and roll angles.
They are the projections of vector i onto
the X, Y and Z axes. The values of and
relative to each other reflect the inclination
of i towards the X, Y or Z axes. For instance,
the greater 
is (relative to 
and ), the more biased the rotation vector
i is towards the horizontal axis X.
For each entire trial, the RMS rotational error is:
(A2.6)
where N is the final step in the trial.
The RMS value of and
are:
, 
,
(A2.7)
,
and , reflect the totals (from i= 0 to i = N
of the projections of rotation vector
on X, Y and Z axes respectively. They
serve as the decomposed rotation measures in the following analysis.
RMS tracking error scores, as defined earlier in equations (7),
(8), (12) and (13) were analysed for 2 (controls) x 13 (subjects)
x 5 (phases) x 4 (paths) = 520 trials. This section presents the
results of analysis on these data. Non-linear (logarithmic) transformation
was performed in order to meet the model residual distribution
requirement of ANOVA analysis. The major results are organised
into three groups, respectively addressing issues related to translation
in 3D, rotation in 3D and the controllability of all 6 degrees
of freedom with one hand.
A2.3.1 Anisotropic Performance in Translation
Repeated measure variance analysis on the translation RMS Errors
, 
, 
as defined in equation (8) was conducted with one between-subject
factor (controller type) and three within-subject factors (dimension,
learning phase, and tracking path). Significant main effects included
dimension (X, Y, Z) (F(2, 48) = 59.03, p<.0001), learning phase
(F(4, 96) = 98.9, p<.0001), and tracking path (F(3, 72) = 12.73,
p<.0001). A significant interaction was also found between
dimension and learning phase (F(8, 192) = 6.96, p<.0001).
Of particular interest for the present report are the
pairwise contrast comparisons (Howell, 1992) which
showed that tracking errors in the X, Y and Z dimensions were
significantly different from each other. The means of the X, Y,
Z RMS errors are shown in Figure A2.2. As expected, the mean error
in the Z direction was significantly greater than those in the
other two directions (X vs. Z contrast: F = 114.48, p< .0001;
Y vs. Z contrast, F=48.8, p<.0001). In terms of magnitude,
the mean of is 39.64% greater than the
mean of and 16.54% greater than the mean
of . This indicates that the 3D display
with particular depth cues implemented in this experiment was
sufficient for enabling satisfactory performance in the depth
dimension; the difference in fact is well below one level of about
400% previously reported with a monoscopic display (Massimino,
Sheridan, and Roseborough, 1989) .
Interestingly, the mean error in the Y direction
was significantly greater than the mean error in the X direction
by 19.8% (X vs. Y contrast: F = 13.79, p<0.001). This was somewhat
surprising, given that both Y and X are planar dimensions. The
first possible cause was the resolution difference between the
vertical and the horizontal dimensions in the stereo display.
On the 120 Hz CRT display used, stereoscopic presentation was
implemented by means of splitting the display memory into top
and bottom halves: one half for left view and one half for right
view. The vertical resolution of the stereo display was therefore
less than half of the horizontal resolution. However, the RMS
tracking errors were at the level of 1 graphic unit or greater
(Figure A2.2) which was an order of magnitude higher than the
vertical resolution threshold (0.04 graphic unit). This means
that the resolution difference was must likely not the cause.
The second explanation of the performance difference between the
virtual and the horizontal dimensions was possible bias in the
input controllers or human motor actions which might have made
the horizontal dimension easier to manipulate than the vertical
dimension. Such a hypothesis, however, could not be supported
by further analyses either. Two types of controllers were used
in the experiment, differing both in electronic design and in
manipulative features. The elastic controller involved hand movement
while the isometric controller required tensions only (force and
torque) and yet the same relative performance pattern in the X,
Y and Z directions ( <
< ) was found for both types of controllers
(Figure A2.3). It is therefore highly unlikely that, if some biases
had existed in either the controller used or the motor actions
required, they would have been identical for the two different
controllers.
Another possible source of variation is the particular
tracking path used. Since each tracking path was randomly generated,
there exists a probability that movement in the Y dimension might
have been more difficult than in the X dimension along a particular
path. That possibility was also rejected, however. When no input
control was applied to the cursor movement (in the baseline test),
the means of were in fact greater
than the means of in three of the four
tracking paths (Figure A2.4). However, subjectsÌ relative
performance patterns ( <
< ) were consistent across the four
distinct target trajectories (Figure A2.5), independent of the
amount of target movement in each dimension.
The puzzle of the X and Y difference was better clarified
when performance was examined in relation to experimental phase.
The X, Y and Z error components were significantly affected by
subjects' practice (dimension X phase interaction: F(8, 192) =
6.96, p<.0001). As Figure A2.6 illustrates, error in the Y
dimension initially was as great as that in the Z dimension. As
practice progressed and learning took place, however, it approached
the error level of the X dimension, with a consistent pattern
across all four distinct tracking paths (Figure A2.7) and for
the two types of input controllers (Figure A2.8). These consistencies
were confirmed by the absence of significant interactions between
dimension, phase and path (F(24, 576) = 0.867, p = .65) and between
dimension, phase and input (F(8, 192) = 1.35, p = .22).
in relation to
and as a function of experimental phase
The change of performance in Y relative to the other dimensions therefore suggests that the inferior performance in Y is due neither to perception nor to action (motor control) per se, but is a matter of attentional bias. In the early stage of learning, when subjects had difficulties in managing all of the dimensions simultaneously, they apparently gave higher attentional priority to horizontal errors than vertical errors. In the later stage of learning when their performance had improved in general and attentional resources were thus freed-up some what, subjects performed equally well in controlling errors in the X and Y dimensions.
The performance difference between the Z dimension
and the X dimension is both perceptual and attentional. In Phase
0, the mean of was 45% greater than .
In Phase 4, this difference was 35%. Although more attention might
have been paid to the Z dimension in the later stages of the experiment,
which reduced the relative difference between X and Z dimension
(from 45% to 35%), the mean was still
larger than that of due to the inherent
difficulty of perceiving in depth, even when using sophisticated
depth cues.
The hypothesis of attention priority is a plausible
one in light of evolution and daily life. In the natural world,
there are more horizontal movement to human visual stimulation
than virtual ones. Animals, including birds, move mostly in horizontal
direction. The psychological literature indicates that while there
is no acuity difference between horizontal and vertical vision,
human tend to be more sensitive to horizontal than to vertical
length differences (Berman, Cunningham, and Harkulich, 1974) ,
as indicated by the shorter reaction times in the horizontal dimension
(Gottsdanker and Tietz, 1992).
A2.3.2 Performance in Rotation
Based on the decomposed rotational tracking errors ,
and , as defined
in equation (13), a repeated measure variance analysis with one
between-subject factor (controller type) and three within-subject
factors (, and
, experimental phase, and tracking path)
showed the following significant main effects: dimensional components
, and
(F(2, 48) = 5.632, p < 0.01), experimental phase (F(4, 96)
= 48.76, p<.0001), and tracking path (F(3, 72) = 3.33, p=0.25).
The effect of dimensional components ,
and was not affected
by experimental phase (Dimension x Phase: F(8, 192)
= 2.10, p=0.66).
The differences between ,
and components
are shown in Figure A2.9. Comparison contrast tests indicated
that the rotation vector component along the Z axis ()
was significantly smaller than those along the X and Y axes (vs.
: F = 27.94, p < 0.0001;
vs. : F = 14.88, P < 0.001). On average,
was slightly smaller than ,
but this difference was not statistically significant (F = 2.04,
p = 0.16). These results are congruent with the analysis of the
translational errors: was the smallest,
because orientation mismatches about the Z axis did not involve
displacements in the Z dimension and therefore were most easily
to be perceived. Orientation mismatches about the X and Y axes,
on the other hand, both involved displacements in depth, resulting
in greater and .
was slightly smaller than
because rotation about the Y axis involves horizontal changes
while rotation about the X axis involves vertical changes. For
the given size of the target and the cursor, the dimensional differences
in rotation were less pronounced than for translations.
A2.3.3 Controlling Both Translation and Rotation with One Hand
This subsection analyses subjectsÌ performance in managing all six degrees of freedom with one hand. During the experiment, it was observed that when subjects could not do the tracking task very well, especially in the early stage of the experiment, they tended to ignore rotations and concentrated on moving the cursor to catch the target in location (translation) only. This is apparently a reasonable strategy to take, since rotation errors have a limited range (+/-180 degrees at the greatest) while translation error is theoretically unlimited. In the later stage of experiment, the majority of the subjects appeared to be able to control all six degree of freedom concurrently.
Figure A2.10 shows the means of translation error and rotation error of each individual subject in experimental phase 0 (first four trials). In the figure, translation RMS error Trms is defined by equation (7) and rotation RMS error Rrms is defined by equation (12). In order to be comparable with translation errors, Rrms has been scaled by the radius of the target tetrahedron (i.e. rRrms, where r= 3.55), which is equivalent to the distance moved by the tetrahedron vertices through the rotation.
Figure A2.10 Individual performance in translation and rotation at Phase 0 (no practice). Baseline scores indicate performance levels with no control input. Rotation scores are scaled to levels comparable to translation scores (see text)
A baseline test showed that the means of Trms and Rrms (over four trials with distinctive paths) without any control were respectively 8.62 and 6.35 (corresponding to (180/¹) *6.35 /r = 119.1_). The results in Figure A2.10 show that large individual differences exist. Two subjects, U and W, did not effectively control either translation or rotation (RMS errors were close to or even beyond the baseline levels). The other twenty four subjects controlled translations with varying degrees of success, but many of them could not manage rotations at this stage (no practice). Four of them, subject B, D, M, and P, were no more than 5% bellow the baseline rotation RMS error. These data show that without practice, 20 of 26 (76.9%) subjects were able to cope somewhat with both translation and rotation simultaneously; four of twenty six (15.3%) were only able to control translation effectively, 2 of 26 (7.7%) subjects could control neither translation nor rotation at all.
Figure A2.11 Individual performance in translation and rotation at Phase 5(40 minutes practice).Baseline scores indicate performance levels with no control input. Rotation scores are scaled to levels comparable to translation scores (see text)
Figure A2.11 shows subjectsÌ performance at
the final phase of the experiment. After 40 minutes of practice,
all subjects could control translations to a certain degree (more
than a 50% reduction from the baseline). Two subjects (B and W)
still could not effectively control rotation (less than 5% reduction
from the baseline), 3 more (R, U, and Z) had more than 5% but
less than 50% reduction from the baseline. These five subjects,
(B, W, R, U, and Z) also had larger performance disparities between
translation and rotation; their rotation errors were greater than
translation errors by 185% (B), 83% (R), 82.8% (U), 197% (W),
126% (Z) respectively. The rest of the subjects (21 of 26 = 80.7%
) controlled both rotation and translation, which required all
6 degrees of freedom, with some degree of success, as indicated
by the substantial reduction in both translation and rotation
from the baselines.
In summary, the data suggest that successful control
of all 6 degrees of freedom is individual dependent. With 40 minutes
of practice, more than 80% of the subjects could simultaneously
manage both rotations and translation. A few subjects could not
control all the degrees of freedom even by the end of the session.
These subjects tended to concentrate on fewer degrees of freedom
(translation only) of the task and ignore the others (rotation).
A2.4 Concluding Remarks
This appendix has analysed human performance in a 6 DOF pursuit tracking experiment through a dimensional decomposition method. This analysis has addressed a number of issues in 3D human machine interface evaluation with quantitative information, with respect to both the quality of depth display and on the controllability of 6 DOF inputs. With regards to display quality, the analysis showed that with the aid of interposition, perspective, binocular disparity and partial occlusion depth cues, usersÌ performance in the depth dimension was reasonably close to performance levels in the horizontal and vertical dimensions. Depending on the practice time, the mean tracking error in the depth dimension was 45% (initially) to 35% (after 40 minutes of practice) larger than that of the horizontal dimension. Subjects tended to give higher attentional priority to the horizontal dimension than to the vertical dimension. Tracking error in the vertical dimension was larger than that of the horizontal dimension in the early stage of experiment and decreased to the level of the horizontal error in the later stage of experiment.
On the input side, the analysis indicated that large individual
differences exist in the ability to simultaneously control six
degrees of freedom. After 40 minutes of practice, more than 80%
of the subjects could control both translations and rotations
effectively (requiring all 6 degrees of freedom). It appears that
in tracking 6 DOF movement, subjects tended to adopt the strategy
of allocating their attention in a certain biased order. In the
early learning stages, when they have not acquired sufficient
skills to manage all the degrees of freedom, subjects tended to
concentrate on translations and ignore rotations. Between the
three dimensions of translations, they tended to give higher attentional
priority to reducing horizontal errors over reducing vertical
or depth errors.