Performance in a discrete aiming task was compared under several transformed visual/motor mappings: rotations by 45, 90, 135, and 180 deg and reflections about the horizontal and the vertical midlines. Eight aiming targets were used, corresponding to eight directions of movement: up, down, right, left, up-right, down-left, up-left, and down-right. Direction of movements were characterized in terms of separable visual and motor components, and two kinds of direction of movement effects were considered. First, a direction of movement effect paralleling that seen in rapid aiming under the usual nontransformed mapping. Second, because rotations, but not reflections, are physically realizable 2-D transformations, a visual/motor control system which is sensitive to physical constraints should perform reflections, but not rotations, in a piecemeal fashion. Results supported the hypothesis that a motor factor having to do with complexity of limb movement accounts for differences in movement accuracy between right and left oblique directions. Direction of movement effects were more evident in reflections than in rotations, and were consistent with the hypothesis that the visual/motor-control system seeks a physically realizable 2-D rotation solution to reflections. Results also suggested that reversal of two orthogonal basis dimensions is far less difficult than reversing only one and leaving the other intact

Cunningham, H. A.

Pavel, M.

English

NASA Technical Reports Server

\DIRECTION
N90-22947
OF MOVEMENT EFFECTS UNDER TRANSFORMED
VISUAL/MOTOR MAPPINGS
H. A. Cunningham and M. Pavel
Stanford University
Stanford, California
SUMMARY
Performance in a discrete aiming task was compared under several transformed visual/motor
mappings: rotations by 45 ° , 90 °, 135 °, and 180 ° and reflections about the horizontal and the verti-
cal midlines. Eight aiming targets were used, corresponding to eight directions of movement: up,
down, right, left, up-right, down-left, up-left, and down-right. Direction of movement effects
were characterized in terms of separable visual and motor direction components, and two kinds of
direction of movement effects were considered. First, a direction of movement effect paralleling
that seen in rapid aiming under the usual nontransformed mapping might be seen. If it is seen for
motor directions, but not visual directions, then this supports a motor factor hypothesis for the
effects seen under the nontransformed mapping. Second, because rotations, but not reflections,
are physically realizable two-dimensional (2-D) transformations, a visual/motor control system
which is sensitive to physical constraints should perform reflections, but not rotations, in a piece-
meal fashion. Results supported the hypothesis that a motor factor having to do with complexity
of limb movement accounts for differences in movement accuracy between right and left oblique
directions. Direction of movement effects were more evident in reflections than in rotations, and
were consistent with the hypothesis that the visual/motor-control system seeks a physically realiz-
able 2-D rotation solution to reflections. Results also suggested that reversal of two orthogonal
basis dimensions is far less difficult than reversing only one and leaving the other intact.
INTRODUCTION
This research investigates directional nonuniformities in the performance of a 2-D discrete
aiming task, under transformed mappings between visual and motor spaces. Various rearrange-
ments of the visual/motor map have been studied over the years (see Howard, 1982, for an excel-
lent review). This work has focused primarily on the process of adaptation to visual/motor trans-
formations. The present research, in contrast, compares the effects of different transformations
and examines direction of movement effects within and between different transformations.
Direction of movement effects (DMEs) have important implications for our understanding of
human visual/motor control. If there is nonuniformity in performance under physically uniform
conditions, this reveals something about the organization of the internal representation of external
space and about the mechanisms involved in visual/motor control. DMEs also are of practical
importance because they can lead to biases in an operator's input to a system. Such biases are not
easily detected when evaluating overall performance of the task, because they involve only a subset
of the inputs. Understanding this source of bias would allow the development of systems that
prevent biases or correct for them during operation.
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https://ntrs.nasa.gov/search.jsp?R=19900013631 2020-03-19T22:07:40+00:00Z
In thisresearch,visuallyguidedaiminghasbeenstudiedundertwokindsof transformation
of theusualdirectionalmappingbetweenahorizontalinputsurface(motorspace)andavertical
displayscreen(visualspace),which is suchthat:
Right-> Right
Left-> Left
Forward-> Up
Backward-> Down
This mapping is a natural one that humans as young as 3 yr of age can do immediately without any
period of adaptation. This mapping will be referred to as the "usual" or "nontransformed"
mapping.
The transformations that were studied constitute a subset of the linear orthogonal transforma-
tions. They were 1) rotations about the center of the space and 2) reflections about axes in the
space. Rotations and reflections both preserve line length, angles, and parallelism of points in the
original space when mapped into corresponding points in the image space. In general, the
expression:
TX_-X t
describes a transformation, T, of points X = [x y] in the original space into points X' = [x' y']
in the image space. In this research, T took one of the following forms:
Ices(O)
TRO T = i_sin(0)
-sin(0) 1
cos(e) J
These transformations represent, respectively, rotation about the center of the 2-D space by angle
q, reflection about the horizontal midline of the space, reflection about the vertical midline, and
reflection about a 45 ° line going through the center of the space.
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METHODS
Six right-handedsubjectsperformedadiscreteaimingtaskwith multiplepossibletarget
positions.ThevisualdisplaywasaverticalCRTscreenandthemotor inputwasmovementof a
hand-heldstylusona horizontaldigitizingtablet. Therewereeightpossibletargetpositions,
arrangedat45° intervalsaroundthecenter.An aimingtrial consistedof 1) thesubjectaligningthe
cursorwith amarkeratthecenterof thedisplayscreen;2) acueingtonesounding;3) afteravari-
ableforeperiod(250to 750msec),a targetappearingin oneof thetargetpositions;4) thesubject
capturingthetargetby movingthecursorintoalignmentwith it; and5) thetargetextinguishing.
Subjectswereinstructedto emphasizeaccuracyoverspeedandto executeasstraightatrajectoryas
possibleoneverytrial.
Eachexperimentalsessionconsistedof 32baselinetrialsundertheusualmapping,followed
by 128trialsunderoneof thesix transformedmappings:rotationof 45°, 90°, 135°, or 180 °, or
reflection about the vertical midline or about the horizontal midline. Transformations of the motor
space relative to the visual space were effected using a combination of software manipulation and
physical rotation of the digitizing tablet.
Root-mean-squared error (RMS ERROR) measured the deviation of a trajectory from a
straight line and is reported here as the measure of difficulty experienced by subjects under the
various visual/motor mappings. Reaction time and angular error of the initial segment of the tra-
jectory were also obtained on each aiming trial, and are reported elsewhere (Cunningham, 1987a).
HYPOTHESES
Two aspects of DMEs were considered, and they correspond to two specific questions that
were asked. First, can DMEs observed under transformed mappings help us to understand DMEs
observed under nontransformed mappings? Previous work by this author (Cunningham, 1987b)
has shown that under the nontransformed mapping, movement in some directions produces more
error than movement in others. Specifically, among right-handed subjects movement along the left
oblique produces more error than movement along the right oblique, and horizontal movement
produces more error than vertical movement. Are these directional nonuniformities due to proper-
ties of the motor system or to nonmotor properties of visual or cognitive processes? Under the
nontransformed visual/motor mapping, motor direction and nonmotor direction are congruent (i.e.,
confounded). Testing left-handed subjects will not disconfound them because it is possible that
left-handers have reversed lateralization of information processing at many levels, not just in the
motor system. Transformation of the visual/motor mapping, however, allows us to disconfound
motor and nonmotor factors because directions of movement are no longer aligned in the usual
way. Under a 90 ° rotation, for example, the visual right oblique becomes the motor left oblique,
and vice versa. Under a 135 ° rotation, visual vertical corresponds to motor right oblique, and so
forth. Thus it was asked: Will the expected pattern of DMEs be observed under transformed
visual/motor mappings and, if so, will it be observed in display directions only (visual), in tablet
directions only (motor), or in both?
The second question arises from considerations of the properties of the two kinds of trans-
formations studied in this research: rotations and reflections. These are both linear orthogonal
31-3
transformations,andsoaremathematicallysimilar. Theydiffer, however,in oneimportant
respect:theyarenotequallyphysicallyrealizableoperations.A rotationof pointsona2-Dsurface
is arigid motionthatcanberealizedin twodimensions.A reflectionof pointsona2-D surface,
however,is neitherrigid norphysicallyrealizablein twodimensions.Are themechanisms
responsiblefor visual/motorcontrolsensitivetothisdifference?If so,performanceunderreflec-
tionsshouldbequalitativelydifferentfrom thatunderrotations.Specifically,it wasasked:Is
directionalnonuniformitymorelikely to occurunderreflectionthanunderrotationasthesystem
seeksaphysicallyrealizablesolutiontothetransformation?
RESULTS
Transformation condition exerted an important influence on aiming error. On average, the
four rotations differed both from one another and from the reflections. The condition which pro-
duced the highest average RMS ERROR was the 90 ° rotation. This was followed by the two
reflections (which were the same) and the 135 ° rotation. The 45 ° and 180 ° rotations produced the
least error and were similar to one another. These averages are for all movement directions under a
particular transformation, and they are consistent with results obtained by other investigators in a
three-dimensional tracking task under transformed visual/motor mappings (Kim et al., 1987).
DMEs were also seen under both kinds of transformation, but they were qualitatively different
under rotation and reflection. In figure 1, RMS ERROR under the four rotation conditions is
plotted against axis of movement: horizontal (right and left), vertical (up and down), right oblique
(up-right and down-left), and left oblique (up-left and down-right). Axes of movement correspond
to directions of movement on the tablet (motor direction), irrespective of display direction. The
vertical offset of the curves for each condition indicates the overall effect of the transformation
condition. The expected right oblique/left oblique difference is seen for the 90 ° and 135 ° rotations.
This is also true for the 45 ° condition, although the scale of this plot makes the difference less
obvious. The horizontal-vertical difference seen under nontransformed mapping was not pre-
served in either visual or motor coordinate systems under rotation.
An interesting and very different pattern of DMEs emerges under the reflection conditions.
Figure 2 shows RMS ERROR under a reflection about the horizontal midline. Note that under
this transformation, the horizontal axis (axis of reflection) is preserved: direction of travel along
the axis is the same as under the nontransformed mapping. The vertical axis is reversed. The right
and left obliques are exchanged, which is equivalent to rotating each of them by 90 ° . The surpris-
ing result shown in this figure is that the axis along which sign is preserved (right and left) has
considerably higher aiming error than that along which sign is reversed (up and down). The axes
corresponding to 90 ° rotations also exhibit high error.
Figure 3 demonstrates that this effect is also seen under the reflection about the vertical axis.
Here, vertical axis movement is preserved as in the nontransformed mapping, and the error for
movements along that axis is high. The horizontal axis is reversed, and error for movements along
that axis is low. Again, error for movements along the other two axes is also high. The signifi-
cance of direction of movement under reflections appears to relate not to the orientation of a move-
ment axis in external space, but rather to its orientation with respect to the transformation per-
formed on the space.
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PRELIMINARY DISCUSSION
DMEs were observed under both rotation and reflection transformations. Under rotation, the
pattern of results for right versus left oblique confirmed a probable motor locus for the right
oblique advantage. This was seen in three out of the four rotation transformations and was espe-
cially strong in those where the overall error is high (90 ° and 135 ° rotations). This motor effect is
consistent with the fact that movement along the right oblique can be done with arm movements
from the elbow, whereas movement along the left oblique requires movement from the shoulder.
Movement from the shoulder involves more joints and the control of more mass than does move-
ment from the elbow.
The DMEs seen under reflection are qualitatively different from those seen under rotation.
They are also large. Under reflection, the reversed axis has the lowest aiming error, and the two
oblique axes have the highest. The error along the axis of reflection was surprisingly high, con-
sidering that the reflection transformation preserves that axis entirely. To what may we attribute
these directional nonuniformities seen under reflection? There are two separate questions to
answer:
1. Why do the oblique axes exhibit higher error than the nonoblique axes? Is it because they
are oblique or because they are transformed by the equivalent of a 90 ° rotation?
2. Why do the preserved axes exhibit greater error than the reversed axes?
Another Transformation: Oblique Reflection
In order to answer the first question, an additional condition was run: reflection about an
oblique axis. Under this reflection, the right oblique was the axis of reflection and so was pre-
served. The left oblique was thus reversed. The horizontal and vertical axes were exchanged for
one another, which is equivalent to a 90 ° rotation of each of them. Figure 4 shows the result of
this reflection. Observed DMEs are consistent with those found under horizontal- and vertical-axis
reflection. The reversed axis exhibits low error and the preserved axis exhibits high error. The
axes whose transformation is equivalent to a 90 ° rotation also exhibit high error.
GENERAL DISCUSSION
DMEs were observed under several different transformations of the usual mapping between
visual (display) space and motor (input) space. Two types of DMEs were seen. First, aiming
error was lower for fight oblique motor directions than for left oblique motor directions, irrespec-
tive of visual direction. This supports the hypothesis that the right oblique "advantage" seen under
nontransformed visual/motor mapping is due to motor factors. A tendency for vertical error to be
lower than horizontal error under the nontransformed mapping was not seen in either the motor or
the visual directions under the transformed mappings.
DMEs also differed qualitatively between rotations, on the one hand, and reflections, on the
other. Under reflection, DMEs are related to an axis of movement's orientation with respect to the
31-5
axisof reflection,notwith respecto extemalspace.Thefact thathumanperformancexhibitsthis
particularkind of directionalnonuniformityunderreflection,butnotunderrotation,isconsistent
with thehypothesisthatthehumanrepresentationof 2-Dspaceis constrainedbyphysicalrealiz-
ability. Thepatternof DMEsunderreflectionsuggeststhatthehumanimposesa2-D rotation
solutionon thereflectioncondition.Axeswhosetransformationis equivalento a 180° rotation
exhibit lesserrorthanthosewhosetransformationisequivalento a90° rotation,just asa 180°
rotationof theentirespaceproduceslesserror,in all directions,thana90° rotation of the entire
space.
Another interesting aspect of the DMEs found under reflection (and one which complicates
somewhat the 2-D solution hypothesis) is the strong tendency for the reversed axes to exhibit
lower error than the nonreversed axes. This was seen in every reflection. This is probably due to
error correction during movement execution. During execution of a movement, subtle corrections
are required to keep the trajectory on a straight path toward the target. For a straight-line trajectory,
corrective movements will have a large vector component in the dimension orthogonal to the
straight-line path. Under reflection, when moving along the axis of reflection (the preserved axis),
the orthogonal dimension is reversed. The small, quick, and largely automatic corrections made
during movement execution will initially be in the wrong direction. As the error is detected, further
automatic attempts to correct it may result in enhancing it instead. This is equivalent to reversing
the sign of a feedback loop and the result is similar: error "blows up." In the case of movement
along the reversed dimension, the orthogonal dimension (dimension of correction) is preserved and
so automatic corrections reduce the error as they should.
In summary, DMEs are intrinsic to discrete aiming on a 2-D surface. The mechanisms
responsible for visual/motor control are sensitive to motor factors having to do with the number of
joints involved in movement in a given direction. They also appear to be constrained to find 2-D
physically realizable solutions to visual/motor transformations, even when these solutions do not
exist.
REFERENCES
Cunningham, H. A. (1987a). Development of visual-motor control. Submitted for publication to
J. Exp. Psychol: Human Percept. Perf.
Cunningham, H. A. (1987b). Human motor control under transformed visual-motor mappings.
Submitted for publication to Biol. Cybernet.
Howard, Ian P. (1982) Human Visual Orientation. Toronto: John Wiley & Sons.
Kim, W. S., Ellis, S. R., Tyler, M. E., Hannaford, B., and Stark, L. W. (1987) Quantitative
evaluation of perspective and stereoscopic displays in three-axis manual tracking tasks. IEEE
Trans. Syst., Man, and Cybernetics, 17, 61-72.
31-6
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movement on the tablet, irrespective of display direction). Axes are horizontal, vertical, right
oblique, and left oblique. Note that the right oblique/left oblique difference seen under the
usual mapping is preserved in motor coordinates and so is probably motor in origin. The
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tal axis (right and left) is preserved and the vertical axis (up and down) is reversed. Oblique
axes correspond to a 90 ° rotation. Note that the oblique axes have the highest error, the
reversed axis the least, the preserved axis intermediate. Note also that the "motor oblique
effect" is present (right and left obliques are exchanged).
31-7
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served axis and 90 ° rotation axes exhibit high error.
31-8


Direction of movement effects under transformed visual/motor mappings

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