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Effects of Basic Path Propertieson Human Path IntegrationXiaoang Wan a , Ranxiao Frances Wang b & James A.Crowell ba Tsinghua University, Beijing, Chinab University of Illinois at Urbana-Champaign,Champaign, Illinois, USAAccepted author version posted online: 23 Apr2012.Version of record first published: 14 Jan 2013.
To cite this article: Xiaoang Wan , Ranxiao Frances Wang & James A. Crowell (2013):Effects of Basic Path Properties on Human Path Integration, Spatial Cognition &Computation: An Interdisciplinary Journal, 13:1, 79-101
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Spatial Cognition & Computation, 13:79–101, 2013
Copyright © Taylor & Francis Group, LLC
ISSN: 1387-5868 print/1542-7633 online
DOI: 10.1080/13875868.2012.678521
Effects of Basic Path Properties on
Human Path Integration
Xiaoang Wan,1 Ranxiao Frances Wang,2 and James A. Crowell2
1Tsinghua University, Beijing, China2University of Illinois at Urbana-Champaign, Champaign, Illinois, USA
Abstract: We investigated how path integration performance can be influenced by
five basic path properties in a Virtual Reality Cube. Participants performed path-
completion tasks in hallway paths with up to 12 segments. Distance information was
visual, whereas turning angles were specified through vision and body senses. The
ridge regression analyses suggested that path integration was affected by the number
of segments, overall path length/turning angles, and the correct homing distance.
Moreover, an un-correlation paradigm showed that path completion performance might
be affected by participants’ expectations for the correct homing distance of different
paths. Implications on models of path integration were discussed.
Keywords: path integration, virtual reality, path properties, spatial updating
1. INTRODUCTION
Moving animals can integrate information regarding self-motion, such as
velocity and acceleration information, to estimate their current position and
orientation relative to the starting point of their travel, a phenomenon re-
ferred to as path integration (Etienne, 1992; Gallistel, 1990; Mittelstaedt
& Mittelstaedt, 1982). A variety of species have been observed to show
path integration ability, including insects (Collett & Collett, 2000; Müller
& Wehner, 1988; Wehner & Srinivasan, 1981), birds (Regolin, Vallortigara,
& Zanforlin, 1995; von Saint Paul, 1982), and mammals (Etienne, 1992;
Mittelstaedt & Mittelstaedt, 1982) including humans (Klatzky et al., 1990;
Loomis et al., 1993).
Correspondence concerning this article should be addressed to Xiaoang Wan,
Department of Psychology, School of Social Sciences, Tsinghua University, Beijing,
China 100084. E-mail: [email protected]
79
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80 X. Wan et al.
Humans’ path integration ability has been directly measured in path
completion tasks (e.g., Klatzky et al., 1990; Loomis et al., 1993), or in-
directly assessed in other spatial tasks (e.g., Allen, Kirasic, Dobson, Long,
& Beck, 1996; Passini, Proulx, & Rainville, 1990). In the path completion
tasks, participants are asked to directly return to the starting point of their
travel without the aid of direct perceptual cues of the paths after they have
traveled along several segments, made several turns at the intersections of
the segments, and arrived at the end of the paths. When there are only two
segments in the outbound paths, this task can be also referred to as a triangle
completion task.
Humans can use different types of information regarding self-motion to
perform path completion tasks, including purely internal information from
the vestibular, proprioceptive and efferent systems (Klatzky et al., 1990;
Loomis et al., 1993; Wiener, Berthoz, & Wolbers, 2011), purely external
information such as optic flow (Ellmore & McNaughton, 2004; Péruch, May,
& Wartenberg, 1997; Riecke, van Veen, & Bülthoff, 2002; Wiener & Mallot,
2006), or a mixture of both (Kearns, Warren, Duchon, & Tarr, 2002; Wan,
Wang, & Crowell, 2010).
Previous studies have suggested that human path completion performance
can be affected by at least five basic properties of the outbound paths, includ-
ing the number of segments, overall path length and overall turning angle,
and the correct distance and direction responses participants need to make to
return to the origin. First, Klatzky and colleagues (1990) demonstrated that
blind or blindfolded sight participants showed longer reaction times (RTs) and
greater turning errors when the number of segment within each outbound path
increased from one to three. However, In Wiener and Mallot (2006)’s study,
participants were passively and visually guided along 2-, 3-, 4-, and 5-segment
paths in virtual environments and then pointed to the direction of the origin,
with the overall travel distance and overall turning angles being constant for
every outbound path. Interestingly, participants showed longer RTs to point
to the origin for 2- and 3-segment trials than for 4- and 5-segment trials. They
also showed larger unsigned pointing errors for 2-segment paths than for 3-,
4-, and 5-segment paths, although this effect was eliminated when researchers
excluded those 2-segment trials in which the correct turning angle was 140
degrees. In short, participants showed longer RTs for paths containing fewer
segments than paths containing more segments. These results suggested that
the number of segments, overall path length, and overall turning angles might
interact to influence human path integration.
In addition, Wiener and colleagues (2011) showed that the influence of
overall path length on path completion may be modulated by task demands.
They asked their participants to perform triangle completion tasks but giving
them different instructions. Specifically, they asked some participants to re-
member the configuration of the outbound path and to calculate the homing
vectors on the basis of their representation of the path, so these partici-
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Path Integration & Path Properties 81
pants were instructed to perform configural updating. Other participants were
asked to continuously update the location of the starting point, so they were
instructed to perform continuous updating. Also, although all the outbound
paths were 2-segment paths, the overall length of the paths were manipulated
as short or long (8.3 or 15.3 m on average, respectively). Accordingly,
participants performing continuous updating showed shorter RTs to turn to
face the origin than those performing configural updating. On the other hand,
participants performing configural updating showed more accurate homing
performance than those performing continuous updating for long paths, but
the two groups showed comparable performance for short paths.
Human path completion performance has also been shown to depend on
the Euclidean distance between the starting and ending points of the paths,
which is referred to as the correct homing distance, and the turn participants
need to make to return to the origin, which is referred to as the correct homing
angle (Fujita, Klatzky, Loomis, & Golledge, 1993; Loomis et al., 1993;
Klatzsky et al., 1997, 1999). For example, Loomis and colleagues (1993)
guided blind or blindfolded sighted participants to perform the triangle path
completion task. They varied the length of each segment (2, 4, or 6 m) and the
inner angle between the two segments (60, 90, or 120 degrees) to obtain 27
different path configurations with varied correct homing distance and correct
homing angles. They found that, when the correct homing distance/angle
increased, the observed response distance/angle also increased, but with a
slope smaller than 1 and a positive intercept. These results suggested that
participants overestimated the small values and underestimated the large
values for both distance and direction responses.
However, two issues remained unsolved and more studies are needed
to examine the influence of path properties on human path completion per-
formance. First, to understand the mechanism of human path integration,
one needs to have a comprehensive picture on which factors affect people’s
performance. The number of segments, overall path length, overall turning
angle, correct homing distance, and correct turning angle may all influence
path completion performance. These factors are generally correlated with
each other, and therefore can confound the analyses unless all of them
are taken into consideration simultaneously. However, in previous studies,
researchers primarily focused on the effects of the number of segments
without controlling both overall path length/turning angle and correct homing
distance/turning angle at the same time (Klatzky et al., 1990; Wiener &
Mallot, 2006). In the current study, we included all five factors in our analyses
to examine how they affect path integration. Second, most of previous studies
focus on relatively short outbound paths which contained no more than
5 segments, whereas path integration is quite common with long paths in
real situations. Our goal is to examine the effects of basic path properties
on human path integration under more general situations by using paths
containing 2 to 12 segments.
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82 X. Wan et al.
2. EXPERIMENT 1
In this experiment, we manipulated the number of segments to examine how
path completion can be influenced by the number of segments, overall path
length, overall turning angle, correct homing distance, and correct turning
angle. Participants traveled through 2-, 4-, 8-, and 12-segment paths and then
attempted to directly return to the starting point of each outbound path.
2.1. Methods
2.1.1. Participants. Fifteen students (4 female) from the University of Illinois
at Urbana-Champaign (UIUC) participated in this experiment. Participants ei-
ther were paid or received course credits to fulfill an introductory psychology
class requirement. They all reported to have normal or corrected-to-normal
visual acuity and normal stereo vision.
2.1.2. Apparatus. The experiment took place in a Virtual Reality (VR) Cube
at the UIUC. Driven by a cluster of computers, the Cube produces stereo-
scopic imagery at 60 frames per second per eye on all six surfaces, each
of which is a 3 m � 3 m rear-projection screen. In other words, this VR
Cube is a 3 m � 3 m � 3 m cubic room in which participants could
make body rotations freely, but their translational movements are restricted
by the size of the room. Participants’ position and orientation in the VR
Cube were tracked by Ascension MotionStar Wireless tracking system. A
Stereographics LCD shutter-glass system was used to provide active stereo,
and binocular images were produced on the basis of each individual’s inter-
ocular distance measured before the experiment started. An Intel Wireless
Series Gamepad was used by the participants to interact with the virtual
environments.
2.1.3. The Virtual Environment. Hallway mazes were used, and each maze
consisted of 2, 4, 8, or 12 segments. As shown in Figure 1, outbound and
homing paths were specified by brown-and-white ceramic pattern and sandy
yellow rocky pattern, respectively. These patterns were selected to make the
seams of the VR Cube invisible. Each hallway was 1 m wide, 2.2 m high, 2
or 3 m long. Both the beginning and the end of each hallway were in circular
shapes with 0.6-m radius, and the circular end of the N hallway was also
the circular beginning of the .N C 1/ hallway. The turning angle at each
intersection of two connecting hallways was clockwise 60 deg, clockwise
120 deg, counterclockwise 60 deg, or counterclockwise 120 deg.
When an outbound path was generated by the computer, the length of
each hallway and the turning angle at each intersection was randomly chosen
from all the options mentioned above with the constraint that there should
be no cross-over between any two hallways. When participants traveled in
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Path Integration & Path Properties 83
Figure 1. Examples of virtual displays used in this study. (A) During the outbound
travel, a hallway appeared, and the participants pressed a button to drive along it. (B)
During the response stage, a 1,000-m long hallway appeared in the direction that the
participants had selected, and they pressed a button to drive along it until they judged
to be at the origin of the outbound path (color figure available online).
these mazes, they obtained information about distance purely through optic
flow, as they kept their bodies still and pressed a button on the game pad to
drive along each hallway at a constant speed of 1.5 m per second. In contrast,
they obtained information about rotation through both optic flow and body
senses, as they actually rotated their bodies at their own speed when they
needed to make a turn. It should also be noted that only one segment was
presented at a time. It is possible that participants attempted to track localized
features to help with their self-motion estimation when they traveled along
one segment, but no localized features could function as landmarks to allow
for landmark-based navigation in the current study.
2.1.4. Procedure. At the beginning of each trial, a hallway was presented in
a random direction, and participants were instructed to physically remain still
and to press a button to drive along. When they arrived at the end of this
hallway, the next hallway was presented. A red arrow pointing to the left or
right appeared in front of the participants to indicate the direction of the next
hallway, and they were instructed to physically turn around their bodies to
face it. This process was repeated until they arrived at the end of the outbound
path. At that time, the response stage started immediately accompanied by the
wallpaper change, and participants were asked to directly return to the origin
of the path. They first made a direction response by turning around their
bodies to face the origin and pressed a button. Then a 1000-m long hallway
appeared in the selected direction, and they made a distance response by
pressing a button to drive and stopping when they judged to be at the origin.
Each participant was encouraged to complete as many trials as possible in 30
minutes. For every four trials they needed to finish one of each 2-, 4-, 8-, and
12-segment path, while the order of these trials was randomly determined.
As a result, participants completed 27 to 60 trials.
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84 X. Wan et al.
Figure 2. An illustration of the path completion task. Participants start from H ,
travel along four segments to arrive at A, and then attempt to directly return to H .
To successfully return to H , they should turn ˇ degrees (unsigned correct turning
angle) and travel for distance c (correct distance). However, their actual responses
might be to turn ˇr degrees, travel for distance cr , and arrive at a new location
Hr . We assessed the accuracy of their path completion by calculating position errors
(Euclidean distances between H and Hr ), unsigned distance errors .jcr � cj/, signed
distance errors .cr � c/, unsigned direction errors .jˇr � ˇj/, and signed direction
errors .ˇr � ˇ/.
2.1.5. Data Analysis. Participants’ RTs to make the direction responses and
their direction responses as well as distance responses were recorded. We used
five parameters to characterize each outbound path, including the number of
segments, overall path length, overall turning angle, correct distance, and
correct turning angles. As shown in Figure 2, we also used six measures
to assess path completion performance. We assessed overall path completion
performance by the RTs and position errors (Euclidean distances between the
actual origins and observed endpoints of the distance responses). Longer RTs
and/or greater position errors indicate poorer path completion performance,
whereas shorter RTs and/or smaller position errors indicate better path com-
pletion performance.1 The accuracies of direction responses were assessed by
unsigned and signed difference between the response angles and the correct
angles, referred to as the unsigned direction errors and the signed direction
1Given the potential tradeoffs between RT and accuracy, the difference in
performance may manifest itself in either RT or accuracy or both depending on
whether the participants emphasized on speed, or accuracy, or in-between, therefore
the exact theoretical meaning of the effects in these two types of measures is difficult
to assess.
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Path Integration & Path Properties 85
errors, respectively. The accuracies of distance responses were assessed by
unsigned and signed difference between the response distance and correct
distance, referred to as the unsigned distance errors and the signed distance
errors, respectively. For the unsigned direction and distance errors, greater
values indicate lower accuracies and smaller values indicate higher accuracies.
For the signed distance and direction errors, positive values indicate overes-
timation and negative values indicate underestimation. Note that the signed
direction and distance errors measure the systematic biases and are reflected
in the accuracy measures (unsigned direction and distance errors), which
include contributions from both these systematic errors (constant errors) and
variable errors.
2.2. Results
The results of this experiment are shown in Figure 3. To examine which
factors have influence on the path completion performance, we used five
independent variables, including the number of segments, overall path length,
overall turning angle, correct homing distance, and correct turning angles, to
predict the RTs and different types of path completion errors. As can be seen
in Table 1, these five independent variables were highly correlated with each
other. To minimize the influence of multicollinearity on the results in ordinary
multivariate regression, we performed ridge regressions (Hoerl & Kennard,
1970), examined the ridge trace and chose the smallest biasing constant k for
each model when all the regression coefficients stopped changing signs from
Table 1. Intercorrelations between number of segments, overall path length, overall
turning angle, correct homing distance, and correct turning angle in Experiments 1
and 2
1. 2. 3. 4. 5.
Experiment 1 .n D 15/
1. Number of segments — .999** .998** .976** �.738**
2. Overall path length — .997** .976** �.724**
3. Overall turning angle — .969** �.737**
4. Correct homing distance — �.674**
5. Correct turning angle —
Experiment 2 .n D 15/
1. Number of segments — .994** .985** 0 �.723**
2. Overall path length — .976** .032 �.718**
3. Overall turning angle — �.025 �.674**
4. Correct homing distance — .165
5. Correct turning angle —
Note. **indicates significance at the ˛ D 0:01 level.
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86 X. Wan et al.
Figure 3. The results of Experiments 1. The RTs and position errors are shown in
Panels A and B, respectively. Unsigned and signed direction errors are shown in Panels
C and D, respectively. Unsigned and signed distance errors are shown in Panels E
and F, respectively.
positive to negative or vice versa and ridge trace became smooth and steady.2
The data showed that the number of segments, overall path length, and overall
turning angle could predict the position errors, R2 D :61, F.5; 54/ D 17:18,
p < :001, and unsigned direction errors, R2 D :22, F.5; 54/ D 3:01, p <
:05. Moreover, the number of segments, overall path length, overall turning
2Given the high correlations among these factors, their independent contributions
may only be separated partially, even with the ridge regression analysis.
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Path Integration & Path Properties 87
angle, and correct homing distance could predict the unsigned distance errors,
R2 D :37, F.5; 54/ D 6:35, p < :001. The regression coefficients in these
ridge regression models are shown in Table 2.
The significant positive regression coefficients indicate that position er-
rors, unsigned direction errors, and unsigned distance errors increased when
the number of segments, overall path length, or overall turning angle in-
creased. Moreover, the unsigned distance errors also increased when the cor-
Table 2. Ridge regression models chosen for Experiment 1
Dependent
variable
Independent
variable k b ˇ t p
RTs Number of segments .35 �.002 �.01 .38 .71
Overall path length �.002 .04 .97 .34
Overall turning angle 0 0 — —
Correct homing distance �.01 �.09 1.37 .18
Correct homing angle .002 .07 .71 .48
Position errors Number of segments .10 .20 .15 3.37 **
Overall path length .14 .26 5.15 ***
Overall turning angle .003 .19 3.16 **
Correct homing distance .16 .12 1.15 .26
Correct homing angle �.03 �.08 .82 .41
Unsigned direction Number of segments .45 .53 .10 3.25 **
errors Overall path length .27 .13 3.94 ***
Overall turning angle .007 .11 3.26 **
Correct homing distance .004 .001 .01 .99
Correct homing angle �.13 �.10 1.17 .25
Unsigned distance Number of segments .20 .08 .10 2.72 **
errors Overall path length .04 .14 3.26 **
Overall turning angle .001 .10 2.17 *
Correct homing distance .13 .17 2.04 *
Correct homing angle �.02 �.10 .95 .34
Signed direction Number of segments .20 .63 .10 2.23 *
errors Overall path length .34 .14 2.72 **
Overall turning angle .01 .09 1.53 .13
Correct homing distance �1.25 �.21 2.13 *
Correct homing angle �.05 .03 .24 .81
Signed distance Number of segments .15 �.10 �.08 1.59 .12
errors Overall path length �.01 �.02 .36 .71
Overall turning angle 0 �.004 .08 .94
Correct homing distance �.21 �.19 1.55 .13
Correct homing angle �.04 �.14 .94 .15
Note. *indicates significance at the ˛ D 0:05 level.
**indicates significance at the ˛ D 0:01 level.
***indicates significance at the ˛ D 0:001 level.
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88 X. Wan et al.
rect homing distance increased. In contrast, the RTs, signed direction errors,
and signed distance errors could not be well predicted by our multivariate
regression models, all R2 < :07, all F s < 1.3 However, as can be seen in
Table 2, we found significant individual regression coefficients for the signed
direction errors (positive for the number of segments and overall path length
but negative for the correct homing distance), although the overall model
was not significant. Furthermore, note the mean unsigned direction errors for
2-, 4-, 8-, and 12-segment paths were 28 deg, 46 deg, 52 deg, and 56 deg,
respectively, all of which were smaller than the unsigned direction errors at
the chance level (90 deg), all ts > 5, suggesting that participants’ performance
was above chance level.
2.3. Discussion
In this experiment, we varied the number of segments to be 2, 4, 8, or 12 and
examined how path completion were influenced by the number of segments,
overall path length, overall turning angle, correct homing distance, and correct
turning angle. First, we found that position errors, unsigned direction errors,
and unsigned distance errors increased when the number of segments, overall
path length, or overall turning angle increased. These results suggest that
path integration was impaired by the increase in the number of segments
and the length of the trip (overall path length and overall turning angle).
Second, participants’ RTs were not affected by any of the five variables.
Third, we found that unsigned distance errors increased when the correct
homing distance increased. Specifically, when participants were further away
from the origin of the paths, they showed less accurate distance responses,
but we did not find such patterns for position errors or unsigned direction
errors.4 Last, participants’ systematic biases (signed distance and direction
errors) could not be well predicted by the ridge regression models.
Note that the five basic properties of the outbound path were quite highly
correlated, as they do in some real life situations such as city-block navigation.
Experiment 1 addressed this intrinsic problem by taking the correlation as is
and using statistical tools (e.g., ridge regression) to disentangle their effects.
This statistical approach has the advantage of preserving the characteristics
of the task in its natural condition, but has the disadvantage of limited power
3There was a marginally significant negative correlation between the signed
distance errors and the correct homing distance .p D :06/, as shown in Figure 3.
However the ridge regression data suggested that the correct homing distance did not
have independent contribution to the signed distance errors.4Note that in this experiment, position errors could not be well predicted by
the correct homing distance. Position errors are affected by both distance errors and
angular errors, but the relationship is not simple addition. Thus, an increase in either
distance errors or in direction errors, or in both, does not always result in an increase
in position errors.
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Path Integration & Path Properties 89
when the correlation is high. Thus, Experiment 2 was conducted by taking
the alternative approach of un-correlating the number of segment and the
correct homing distance in the experimental design. This approach examined
only a subset of special trial types, but with much higher power. Together
the two approaches may provide complimentary and converging evidence on
the effects of various basic path properties on path integration performance.
3. EXPERIMENT 2
In this experiment, we manipulated the number of hallways and correct
homing distance to make these two variables uncorrelated, and then examined
how the number of segments, overall path length, overall turning angle,
correct homing distance, and correct turning angle influence path completion.
Participants traveled through 4- or 8-segment paths and then attempted to
directly return to the starting point of each outbound path, while the Euclidean
distance between the starting and ending points of each path might be 3, 6,
or 9 meters.
3.1. Methods
Fifteen UIUC students (9 female) participated in this experiment. None of
them participated in Experiment 1. The methods of this experiment were the
same as those of Experiment 1 except the following. First, a 2 (number of
segments, 4 or 8) � 3 (correct homing distance, 3, 6, or 9 m) orthogonal
design was used in this experiment. That is, we only used 4- and 8-segment
outbound paths for which the correct distance was fixed to 3, 6, or 9 m.
Although the length of each hallway was 2 or 3 m and there was no cross-
over between any two hallways as in Experiment 1, the turning angles at each
intersection in this experiment ranged from 30 to 150 degrees, clockwise or
counterclockwise, to make it possible to generate the desired path types.
A pair of 4-segment and 8-segment paths with constant correct distance is
shown in Figure 4. Second, a random order of the 6 trial types was generated,
and this process repeated for as many times as needed to produce a random
sequence of trials for each participant. All participants were encouraged to
complete as many trials as possible in 40 minutes, and eventually completed
22 to 95 trials.
3.2. Results
The results of this experiment are shown in Figure 5. As can be seen in
Table 1, the number of segments, overall path length, overall turning angle,
and correct turning angle were highly correlated with each other, whereas
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90 X. Wan et al.
Figure 4. An illustration of a pair of 4-segment and 8-segment outbound paths in
Experiment 2. In the former case, participants start from H , travel along four segments
to arrive at A4; whereas in the latter case, they start from H , travel along eight
segments to arrive at A8. These two paths generate the same correct distance, that is,
c4 (Euclidean distance between H and A4) is equal to c8 (Euclidean distance between
H and A8).
none of them was correlated with correct homing distance. To minimize
the influence of multicollinearity on the results in ordinary multivariate re-
gression, we performed ridge regressions on all the measures as we did in
Experiment 1. The data showed that position errors could be predicted by the
number of segments, overall path length, and overall turning angle, R2 D :22,
F.5; 84/ D 4:76, p < :001. Unsigned direction errors could be predicted by
the number of segments, overall path length, overall turning angle, and correct
homing distance, R2 D :33, F.5; 84/ D 8:31, p < :001. Unsigned distance
errors could be predicted by the number of segments and correct homing
distance, R2 D :11, F.5; 84/ D 1:99, p D :088. Moreover, signed distance
errors could also be predicted by the number of segments and correct homing
distance, R2 D :25, F.5; 84/ D 5:55, p < :001. In contrast, the RTs and
signed direction errors could not be well predicted by our ridge regression
models, both R2 < :08, both F s < 1.45, p > :21, although some of the
individual coefficients were significant as shown in Table 3. Taken together,
the significant positive regression coefficients indicate that position errors and
unsigned direction errors increased when the number of segments, overall path
length, or overall turning angle increased; the significant negative regression
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Path Integration & Path Properties 91
Figure 5. The results of Experiments 1. The RTs and position errors are shown in
Panels A and B, respectively. Unsigned and signed direction errors are shown in
Panels C and D. Unsigned and signed distance errors are shown in Panels E and F.
Error bars show standard errors of the mean.
coefficients indicate that unsigned direction errors, unsigned distance errors,
and signed distance errors decreased when correct homing distance increased.
To examine whether participants were sensitive to the correct homing
distance, we further performed a 2 (number of segments, 4 or 8) � 3 (correct
distance, 3, 6, or 9 m) Analyses of Variances (ANOVAs) on the observed
distance responses. The main effect of the number of segments was signif-
icant, F.1; 14/ D 11:20, p < :01, suggesting that participants made shorter
distance responses for 4-segment paths (7.93 m) than for the 8-segment paths
(9.87 m). The main effect of correct homing distance was also significant,
F.2; 28/ D 6:92, p < :01. Planned pair-wise comparisons with Bonferroni
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92 X. Wan et al.
Table 3. Ridge regression models chosen for Experiment 2
Dependent
variable
Independent
variable k b ˇ t p
RTs Number of segments .20 �.004 �.01 .23 .82
Overall path length �.007 �.05 .89 .38
Overall turning angle �.001 �.13 1.88 .06
Correct homing distance .03 .08 .94 .35
Correct homing angle �.002 �.05 .49 .62
Position errors Number of segments .25 .27 .17 4.34 ***
Overall path length .11 .17 3.78 ***
Overall turning angle .003 .13 2.40 *
Correct homing distance .001 .0004 .01 .996
Correct homing angle .01 .07 .84 .40
Unsigned direction Number of segments .50 1.17 .09 3.37 **
errors Overall path length .64 .13 4.36 ***
Overall turning angle .02 .10 2.83 **
Correct homing distance �3.18 �.31 5.24 ***
Correct homing angle �.001 �.001 .02 .98
Unsigned distance Number of segments .50 .14 .10 2.88 **
errors Overall path length .04 .07 1.90 .06
Overall turning angle .001 .06 1.40 .17
Correct homing distance �.19 �.15 2.18 *
Correct homing angle .00 �.001 .01 .99
Signed direction Number of segments .35 .80 .05 1.23 .22
errors Overall path length .26 .04 .93 .36
Overall turning angle .01 .03 .66 .51
Correct homing distance �2.75 �.19 2.48 *
Correct homing angle �.003 .002 .02 .98
Signed distance Number of segments .10 .30 .15 2.27 *
errors Overall path length .03 .03 .40 .69
Overall turning angle .001 .04 .39 .69
Correct homing distance �.68 �.40 4.63 ***
Correct homing angle 0 �.005 .04 .96
Note. *indicates significance at the ˛ D 0:05 level.
**indicates significance at the ˛ D 0:01 level.
***indicates significance at the ˛ D 0:001 level.
corrections for multiple tests showed that participants made smaller distance
responses for 3 m condition (8.14 m) and 6 m condition (8.94 m) than for
9 m condition (9.63 m), both ts > 2.60, p < :062, although their distance
responses for 3 m and 6 m conditions were comparable, t.14/ D 1:93,
p D :23. The interaction effect of these two factors was not significant,
F.2; 28/ D 1:18, p D :32. To sum up, participants were sensitive to the
difference between short (3 m and 6 m) and long (9 m) correct homing
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Path Integration & Path Properties 93
distances, but they were not sensitive to the difference between 3 m and
6 m conditions. In addition, the mean unsigned direction errors for 4- and
8-segment trials were 54 deg and 71 deg, respectively, both of which were
smaller than the unsigned direction errors at chance level (90 deg), both ts >
4), suggesting that participants’ direction responses were well above chance
level.
3.3. Discussion
In this experiment, we systematically manipulated the number of segments (4
or 8) and correct homing distance (3, 6, or 9 m) to make them uncorrelated,
and once again examined how path integration can be influenced by the
number of segments, overall path length, overall turning angle, correct homing
distance, and correct turning angle. First, we found that position errors and
unsigned direction errors increased when the number of segments, overall
path length, and overall turning angle increased. Second, participants’ RTs
could not be well predicted by the ridge regression models which include
these five variables. Both of these results are consistent with what we found
in Experiment 1. Third, we found that unsigned distance errors increased
when the number of segments increased, but they could not be predicted by
the increase in overall path length and overall turning angle as in Experiment
1. Instead, as can be seen in Figure 5, unsigned distance errors decreased
when the correct homing distance increased. This pattern was different from
that in Experiment 1 in which unsigned distance errors increased when correct
homing distance increased. Last, we found that participants were sensitive to
the difference between short (3 m and 6 m) and long (9 m) correct homing
distances, but they were not sensitive to the difference between 3 m and 6 m
conditions. The theoretical implications of these results and the underlying
mechanisms were further discussed next.
4. GENERAL DISCUSSION
In this study, we examined the contributions of the number of segments,
overall path length, overall turning angle, correct homing distance, and correct
turning angle to human path completion performance. In Experiment 1,
participants traveled along 2-, 4-, 8-, and 12-segment paths and then attempted
to directly return to the origin of the path. The five basic path properties
were highly correlated and ridge regressions were used to partially assess the
contributions of each path property. In Experiment 2, participants traveled
along 4- and 8-segment paths and then attempted to directly return to the
origin, with the correct homing distance for each path being orthogonally
manipulated as 3, 6, or 9 m. The number of segments, overall path length
and overall turning angle still correlated with each other as in Experiment
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94 X. Wan et al.
1, but the correct homing distance was uncorrelated with the number of
segments, overall path length, or overall turning angle.
In both experiments, we consistently found two results. First, position
errors and unsigned direction errors increased when the number of segments,
overall path length, and overall turning angle increased. The effect of the
number of segments on unsigned direction errors we found for paths contain-
ing up to 12 segments appeared to be similar with that for paths containing
1 to 3 segments reported by Klatzky et al. (1990) and Ruddle, Payne, and
Jones (1998). On the other hand, Wiener and Mallot (2006) reported their
participants showed larger unsigned pointing errors for 2-segment paths than
for 3-, 4-, and 5-segment paths, although this effect was eliminated when
researchers excluded those 2-segment trials in which the correct turning angle
was 140 degrees. As Wiener and Mallot’s (2006) study indicated, the reason
for the discrepancy might be that the overall path length/turning angles were
kept constant for all the paths in their study but the correct distance/turning
angle were not controlled, whereas the number of segments was highly
correlated to overall path length/turning angle in all other studies, including
our study, and the correct distance was controlled in our Experiment 2.
Second, our results showed that participants’ RTs could not be predicted
by the increase in the number of segments, overall path length, or overall
turning angle. This finding was inconsistent with Klatzky and colleagues’
(1990) findings that their participants showed increased RTs when the number
of segments increased in nonvisual path completion tasks with short paths
containing 1 to 3 segments. Also, our finding was inconsistent with Wiener
and Mallot’s (2006) findings that their participants showed longer RTs to
point to the origin for paths containing fewer segments than paths containing
more segments in visual homing task. Note that they used 2- to 5-segment
paths with the overall path length being kept constant as 18 m, whereas we
used greater range for the number of segments (2 to 12) and for the overall
path length (4 to 36 m). The discrepancy in the RT results might be due to that
different ranges of path length, different ranges of the number of segments,
and different perceptual cues were used in different studies, which could
affect the strategies used by the participants (e.g., configural vs. continuous
updating, see discussion). Due to the large methodological difference among
these studies, the exact cause of the discrepancy is unclear and need further
investigation.
Our findings allowed us to further compare two broad classes of models
that might explain the mechanisms of human path integration. The first
class of models is the configural updating models. In general, the configural
updating models assume that path completion is performed on the basis
of detailed representation of the configuration of the paths. For example,
the Encoding-Error Model proposed that humans sense the pathways and
construct the representation of the outbound path (encoding), calculate the
homing vector (spatial reasoning), and execute the homing vector to return to
the origin (Fujita et al., 1993; Klatzky et al., 1997, 1999; Loomis, Klatzky,
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Path Integration & Path Properties 95
Golledge, & Philbeck, 1999). In addition, the Encoding-Error Model assumed
that systematic errors come solely from the encoding stage, with no systematic
errors in the spatial reasoning or response execution stages. The model pro-
vided a remarkably accurate account of the systematic errors in the triangle-
completion task of Loomis et al. (1993).5 However, the model did not do
well when the path increased to 3-legs, and Fujita et al. (1993) suggested
that additional errors sources than the encoding stage may contribute to the
path completion performance of longer trips. Other studies have suggested
that the mental spatial reasoning phase of the path completion task also
introduce systematic errors (Gramann, Müller, Eick, & Schönebeck, 2005;
Riecke, 2008; Riecke et al., 2002).
The second class of models is the nonconfigural models, for example,
the continuous path integration model (Merkle, Rost, & Alt, 2006). These
models proposed that navigators keep track of the spatial relationship between
themselves and the origin of the outbound path during their movements,
without having a detailed representation of the structure of the outbound
paths. Although there is indeed evidence that humans are able to obtain and
maintain the representation of the outbound paths as Loomis and colleagues
(1993) suggested, this evidence of path configuration memory came from
paths with only 1 to 3 segments. As the number of segments increases,
the demand on memory to reconstruct the whole path configuration also
increases, and it is not clear the human working memory capacity is capable
of supporting such a strategy for longer and more complex paths. Thus, the
nonconfigural path integration is a much more economical process, and might
be the more prevalent mode of path integration in general.
The two classes of models make similar predictions on the effects of the
five basic path properties on path completion errors. Both classes of models
predict that path completion errors may increase when the number of seg-
ments, overall path length, and overall turning angles increase. Specifically,
the configural updating models predict that more segments and increased
length of the trip (overall path length and overall turning angles) might lead
to more misperceptions of the segments and turns, more complex internal
representations, and more complex calculations in the spatial reasoning stage,
all of which might generate greater path integration errors shown in our study.
The nonconfigural updating models generally assume that errors occur at each
step of movement and accumulate over time. For example, a nonconfigural
5Their mathematical modeling actually does not require the assumption of
configural updating. For example, a moment-to-moment continuous updating model
can produce the encoding error functions for both translation and rotation equally
well, by assuming an initial overestimation when translation or rotation starts (which
determines the intercept) and a systematic underestimation in each (infinitesimal) step
of updating afterwards during a given segment of translation or angle of turning.
Therefore the modeling results are consistent with both configural and nonconfigural
updating.
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96 X. Wan et al.
model may assume that each segment or turn of the path is one “step,”
therefore as the number of segments increases, the number of updating steps
also increases, which may lead to more updating errors. Moreover, errors in
the estimation of the displacement or turning vectors of each step may depend
on the size of these vectors, therefore as the overall trip length increases, the
errors may also increase. Thus, a nonconfigural model can also predict that
the accuracies of path completion should decrease with more segments and
increased length of the trip. Finally, the response execution process is common
to both classes of models, therefore any assumptions about errors resulting
from the execution stage can be adopted by both classes of models to explain
the experimental data.
Although both classes of models can potentially accommodate similar
patterns of path completion errors, specific models within each class may
not be consistent with our data. Thus, the present findings put constraints on
the future developments of specific quantitative models of path integration.
That is, these models need to account for the influence of the four basic path
properties (i.e., the number of segments, overall path length/turning angles,
and correct homing distance) on both signed and unsigned path completion
errors. For example, the original Error-Encoding Model proposed by Fujita
and colleagues (1993) only addressed the systematic errors reflected in signed
direction errors and signed distance errors, therefore additional assumptions
regarding other error sources are needed to predict the effects with unsigned
direction errors and unsigned distance errors found in our experiments. For
example, assumptions about the size of the random errors for both distance
and angle encoding, whether they are constant or depend on the segment
length and angle of turn (and if so, is the function linear or nonlinear),
whether they depend on the length of the trip, and so on, are needed to
model the effects of unsigned errors.
Similarly, a strict, moment-by-moment vector summation model assumed
that errors occur at each moment of outbound trip and accumulate over time,
so it predicts the effects of the overall path length/turning angle, but may have
difficulty explaining the effects of the number of segments. That is, assume
a small random error (possibly with nonzero mean to allow for systematic
biases) occurs at each integration step of translation and rotation. The total
accumulated error (either from translation or rotation) is the sum of the errors
in each step during the whole trip. If the integration step is sufficiently small
and occurs numerous times during each segment of translation and each
angle of turn, then the total number of integration steps is solely determined
by the length of the trip (total path length and total rotation angle). Although
some of these errors may cancel out, in general the longer the trip, the more
integration steps, and thus the larger the errors accumulated. In contrast, it
is unclear how an increase in the number of segments would increase the
accumulated errors in this model because it does not affect the number of
integration steps, nor the size of the errors in each step. The only potential
influence of segments is by affecting the possible error-cancellation due to
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Path Integration & Path Properties 97
the configuration of the path. However, this effect is very complicated and in
general more segments don’t necessarily lead to more errors.
On the other hand, different classes of models make different predictions
on the effects of the number of segments on the RTs. That is, the configural
model predicts that the RTs should increase when the number of segments
increases, because the process of finishing constructing the whole path struc-
ture and reasoning process occurs after the whole outbound trip is completed.
Thus, the more segments the path has, the longer it should take to compute
the return vector (e.g., Loomis et al., 1993).6 In contrast, the continuous
models predict that RTs should remain constant for paths of different length
and segment number, because the homing vector is calculated and updated
during each step of the outbound trip, and the only computation that needs
to be done before making a response is the updating of the last step, which
should remain largely constant. Thus, both classes of models can explain the
error data in our experiments, but our RT data are more consistent with the
nonconfigural models.
Although Wiener and colleagues (2011) suggested that participants may
use configural updating for “simple paths” and continuous updating for “com-
plex paths,” it should be noted that our participants did not know the number
of segments a path contained until arriving at the end of the outbound paths.
Therefore, it is difficult for them to rely on the number of segments to decide
beforehand which updating strategy to use in our experiments. Nonetheless,
task demand in our experiments might have biased them more toward a
representation of the homing vector without the representation of the paths.
For one, our participants knew at the beginning that they would only need
to directly return to the origin of the trip. For another, because we used
paths containing up to 12 segments, it was more economical to perform
continuous updating to complete the task, as configural updating require very
complex internal representation of the paths which may increase working
memory load. Although the current study cannot rule out the possibility that
participants might begin every path using a configural strategy and then switch
to a continuous updating strategy when their working memory capacity was
exceeded by the accumulation of segments, it is a more complex strategy
(involving two different mechanisms that need to be combined) than the
single, continuous updating strategy, thus it is not clear why participants
would use the complex strategy instead of the simpler one if they knew they
would travel along many segments (a majority of the trials in Experiment 1
and all trials in Experiment 2 had no less than 4 segments).
6Although it is conceivable to build configural representations for sub-sections
of the path along the way (e.g., every 2–3 segments), the resulting vectors of these
sub-configurations still need to be integrated in the end, so there should still be an
increase in RT (although maybe smaller than the full-configural model) as the number
of segments increase, unless these vectors are added up along the way, in which case
the model becomes very much like a large-step continuous updating model.
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98 X. Wan et al.
We also found some interesting difference between Experiments 1 and 2
regarding the independent effects of correct homing distance on unsigned and
signed distance errors. In Experiment 1, correct homing distance increased
when the number of segments increased as a consequence of random path
construction, and an increase in the correct homing distance led to an increase
in the unsigned distance errors but not the signed distance errors. However,
in Experiment 2, correct homing distance was forced to be uncorrelated with
the number of segments, and an increase in the correct homing distance led
to a decrease in both the unsigned and signed distance errors, suggesting that
participants’ systematic bias (overestimation) decreased when correct homing
distance increased. One possible reason for these results is that path comple-
tion performance might be affected by participants’ intuitive expectation on
the correct homing distance for paths of different length/number of segments.
Specifically, Experiment 1 simulated a more natural situation where correct
homing distance increased with path length, and Experiment 2 used special
paths with many segments but short correct homing distance, leading to high
systematic bias in these unusual paths. Thus, the current study may also
provide converging evidence that human path integration may be affected by
people’s nonsensory factors, including expectation as implied by the current
study, knowledge of the environment (Philbeck, Klatzky, Behrmann, Loomis,
& Goodridge, 2001; Philbeck & O’Leary, 2005; Rieser, 1999), and previous
navigation experience (Petzschner & Glasauer, 2011; Tcheang, Bülthoff, &
Burgess, 2011).
It is also very impressive how our participants completed the path com-
pletion task for 12-segment paths and continuously updated their homing
vector after such long trips, although their path integration was subject to
error accumulation, and their path integration system may not be suitable to
support long distance travels (Souman, Frissen, Sreenivasa, & Ernst, 2009).
The physical rotations our participants made during travel might make crit-
ical contributions to their path completion performance. Without physical
movements (especially rotations), people easily became disoriented in VR
navigation tasks (e.g., Klatzky, Loomis, Beall, Chance, & Golledge, 1998).
It should also be noted that the current study was conducted in the VR Cube,
and it cannot be ruled out the possibility that some participants might use
the real environment (Cube) to orient themselves in the real environment
(virtual maze), as Kelly and colleagues (2008) showed that the geometry
of the surrounding environment can prevent complete disorientation. On the
other hand, the different textures used for outbound and homing paths might
affect speed and distance perception and therefore increase the difficulty to
estimate homing distances.
In conclusion, compared to previous studies focusing on outbound paths
that contained no more than 5 segments, we examined the effects of path
properties on human path integration under more general situations with paths
containing 2 to 12 segments. We examined multiple important path properties
simultaneously, and the Virtual Reality Cube we used allowed participants
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Path Integration & Path Properties 99
to make physical rotations in both outbound and homing trips to facilitate
their path completion performance. Results of the current study suggested
that human path completion performance was impaired by increased number
of segments, increased trip length, and increased correct homing distance.
Our findings also provide preliminary evidence that human path completion
performance might be affected by people’s expectations.
AUTHOR NOTES
This work was supported by the SRF for ROCS from SEM to X. Wan and
NSF Grant BCS 03-17681 to R. F. Wang. Some of the data was presented at
the 47th Annual Meeting of the Psychonomics Society, Houston, TX, USA.
We thank Hank Kaczmarski and the Integrated Systems Laboratory at the
University of Illinois for technical support.
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