Route selection in barrier avoidance

Abstract

Background:

Fajen and Warren’s steering dynamics model can reproduce human paths around an extended barrier by adding ‘waypoints’ at each end – if one waypoint is selected to minimize the global path curvature (Gérin-Lajoie and Warren, 2008). We propose that waypoint selection behaves like a choice between two competing goals, in which the smaller distance (d) and deviation angle (β) is preferred (Ulrich and Borenstein, 1998). Here we manipulate these two variables to test the determinants of route selection.

Research question:

How does route selection in barrier avoidance depend on the local distance (d) and deviation angle (β) of each end, and on global path length (P) and curvature (C)? Methods: Participants (Exp1 N = 19; Exp2 N = 15) walked around a barrier to a visible goal in a virtual environment. Barrier orientation and lateral position were manipulated to vary the difference in distance (Δd) and in deviation angle (Δβ) between the left and right ends of the barrier. Left/Right route data were analyzed using a mixed-effects logistic regression model, with Δβ, Δd, and observed ΔP and ΔC as predictors.

Results:

The main effects of Δβ and Δd significantly predicted Rightward responses (p < .001), more strongly than ΔP and ΔC (ΔBIC = 29.5). When Δβ and Δd agreed, responses were toward the smaller distance and deviation (88% overall); when they conflicted, responses were in between (65% toward smaller β). The 75% choice threshold for Δβ was ± 1.65°, and for Δd was 0.75 m, from the 50% chance level.

Significance:

During barrier avoidance, participants select a route that minimizes the local distance (d) and deviation angle (β) of the waypoint, rather than the global path length (P) or path curvature (C). These findings support the hypothesis that route selection is governed by competing waypoints, instead of comparing planned paths to the final goal.

1. Introduction

When walking through the environment, our paths often twist and wind, detouring around obstacles and short-cutting through open spaces to arrive at the intended destination. On these daily walks, we also circumvent extended barriers such as construction barricades and park benches. How are these elongated obstacles avoided, and how might barrier circumvention be modelled?

One approach to obstacle avoidance is Fajen and Warren’s [1] steering dynamics model. A pedestrian’s heading (direction of travel) is attracted toward the direction of the goal and repelled away from the directions of obstacles, thereby reducing the deviation from the goal while avoiding collisions. The attraction increases with larger deviation angles (β) and smaller distances (d) from the goal; conversely, the repulsion increases with smaller deviation angles from an obstacle, as well as smaller distances. As the agent weaves through the environment, the evolving attractors and repellers determine an emergent trajectory, which successfully predicts human paths [2]. The model treats obstacles as points, however, and it is not clear how to generalize it to elongated barriers. In the Social Force Model [4], the pedestrian is repelled from the nearest point on the barrier. However, this model generates wall-following behaviour rather than circumvention [6].

In unpublished work, Gérin-Lajoie and Warren [3] studied participants walking around a barrier to a goal, and manipulated the size, position, and orientation of the barrier. They simulated the data by extending the steering dynamics model in three ways. First, treating the barrier as a set of point-obstacles yielded model paths that were in poor agreement with the human data and sometimes even passed through the barrier (Fig. 1A). Second, making repulsion proportional to the visual angle of the barrier [5] also produced a poor match to the data, generating exaggerated or insufficient deviations around the barrier (Fig. 1B).

Fig. 1.

Adapted from Gérin-Lajoie and Warren [3]. (A) Barrier modelled as a set of points. Some simulations produced paths where the agent pass between these points to reach the goal. (B) Repulsion term proportional to the angular width of the barrier tended to produce exaggerated avoidance trajectories. (C) Waypoint goal specified at the end of the barrier that the agent first passes through. The waypoint that produced the straighter path was selected. (D) Example of the human data.

The most successful model treated the barrier endpoints as obstacles and added a waypoint just off each end of the barrier [3] (Fig. 1C). A waypoint was selected to minimize the cumulative amount of turning (total curvature) on the planned path to the goal, which agreed with human routes on ~96% of trials. However, this measure was confounded with waypoint distance (d) and deviation angle (β): in the experiment, the path with less total curvature always coincided with the end of the barrier that was closer to the starting position and had a smaller deviation angle from the goal direction (Fig. 1D). This suggests two alternative explanations for waypoint selection.

First, participants might have selected the end of the barrier with the smaller deviation angle from the goal (β_L < β_R in Fig. 1D). Deviation angle is a local variable that is visually specified at the starting position, in contrast to the global variable of total path curvature. Fajen and Warren [1] observed that the deviation angle from the goal tends to be reduced during steering and obstacle avoidance, which is also a property of robot path-planning algorithms [7,8]. This simple principle could explain Gérin-Lajoie and Warren’s [3] data.

Second, participants might have selected the closer end of the barrier (d_L < d_R in Fig. 1D). Egocentric distance is visually specified at the starting position, for example, by declination angle from the horizon [9]. The closer end of the barrier always coincided with the less curved path, and could also explain the data.

We were struck by the similarity of these observations with a separate unpublished study of competing goals. Cohen and Warren [10] manipulated the initial distance and deviation angle of two goals, and asked participants to walk to their preferred goal. Participants preferred the closer goal overall, but when distances were equal they tended to prefer the smaller deviation angle. A nonlinear competition term was introduced into the steering dynamics model so the attraction of one goal increased as its distance or deviation angle decreased, while the other goal was suppressed. The model successfully simulated human goal selection on 91% of trials.

Here we combine these insights by proposing that barrier waypoints act like competing goals en route to the final destination. On this hypothesis, participants should prefer the end of the barrier that is closer (smaller d) and requires deviating less from the final goal (smaller β). When d and β agree, participants should prefer that end of the barrier, as previously observed [3]. When we place d and β in conflict, the competing waypoints hypothesis predicts that the influence of the two variables should trade off. Finally, the hypothesis predicts that these local variables should account for route selection better than global path curvature or path length.

To vary the difference in distance (Δd = d_L - d_R) and difference in deviation angle (Δβ=β_L - β_R) between the left and right ends of the barrier, we manipulated barrier orientation and lateral position. Shifting the barrier from left to right, while holding the orientation constant, primarily affected Δβ; conversely, changing the orientation of the barrier while holding its position constant primarily affected Δd. Experiment 1 tested 6 lateral positions and 3 orientations; to probe intermediate values of Δd, Experiment 2 tested 4 lateral positions and 4 orientations.

The contributions of the present work are two-fold: theoretically, it applies the concept of competing goals to barrier avoidance, and experimentally, it tests new conditions in which waypoint distance and deviation angle are in conflict. We found that waypoint distance and deviation angle trade off, and that these local variables predict waypoint selection better than global path length or curvature, consistent with the competing waypoints hypothesis.

2. Methods

2.1. Participants

Separate groups of volunteers participated in Experiment 1 (19 adults; 10 F, 9 M) and Experiment 2 (15 adults; 9 F, 6 M). None reported having any visual or motor impairment, and they were paid for their participation. The protocol was approved by Brown University’s Institutional Review Board, and was in accordance with the Declaration of Helsinki.

2.2. Apparatus

Research was conducted in the Virtual Environment Navigation Laboratory at Brown University. A participant walked diagonally across a 12 × 14 m tracking area while viewing a virtual environment in a head-mounted display (HMD; Oculus Rift DK1, Irvine CA; 90° horizontal × 65° vertical field of view; 640 × 800 pixels per eye; 60 Hz). Stereoscopic images of a virtual environment were generated in Vizard (WorldViz, Santa Monica CA). Head position (4 mm RMS) and orientation (0.1° RMS) were measured with an inertial/ultrasonic tracker (IS-900, InterSense, Billerica, MA) at 60 Hz and used to update the visual display (latency ~ 50 ms).

2.3. Displays

The virtual environment (Fig. 2) contained a blue ‘start’ pole (0.24 m radius, 1.05 m tall), a green ‘goal’ pole (0.24 m radius, 2.1 m tall) 11.5 m away, and a blue textured barrier (3 m wide × 1.2 m tall) between them, its center 7.5 m from the start pole. They rested on a ground plane having a greyscale random noise texture, with a black sky.

Fig. 2.

Experimental set-up. (A) Illustration of all barrier orientations at the leftmost lateral position relative to the direct path from start position to goal (dotted line) in Experiment 1 (left) and Experiment 2 (right). (B) HMD worn by participant to freely walk in VENLab (left), and participant ‘s view of the goal pole obstructed by the barrier (right). (C) Illustration of difference scores for deviation angle (Δβ = βL - βR) and distance (Δd = dL - dR).

2.4. Design and procedure

Participants were instructed to walk naturally to the goal while avoiding the barrier. On each trial, the participant stood at the start pole, faced the goal pole, and started walking upon a recorded verbal command. After they walked 1.5 m, keeping the goal near the center of their field of view (± 7.5°), the barrier appeared with the goal pole visible above it (Fig. 2B). When the participant neared the goal (0.95 m radius) it disappeared, a new start pole appeared (1.5 m away), and the next trial began.

The lateral position and orientation of the barrier were varied across trials (see Fig. 2A). In Experiment 1, the barrier’s center appeared in one of six lateral positions (± 0, ± 0.19, ± 0.3 m) to the left (−) or right (+) of the direct path from the start pole to the goal pole, and its orientation was either perpendicular (90°) to the direct path, or rotated 45° clockwise or counterclockwise (45°, 90°, 135°); the resulting range of Δd was −2.37 m to 2.38 m, and of Δβ was −12.5° to 11.6°. There were thus 18 conditions (6 lateral positions × 3 orientations), with six repetitions, for a total of 108 trials presented in a random order in a one hour session.

In Experiment 2, different intermediate Δd values were tested by increasing the number of orientations. The barrier had four lateral positions (± 0.2 or ± 0.4 m) and four orientations (45°, 75°, 105°, or 135°); the resulting ranges of Δd was −2.4 m to 2.35 m, and of Δβ was −12.85° to 11.5°. This yielded 16 conditions (4 lateral positions × 4 orientations), with eight repetitions, for a total of 128 trials presented in a random order in a one hour session.

2.5. Data analysis

We analyzed head position (x,y) from the time of barrier appearance until the participant reached a 0.7 m radius from the goal pole on each trial. Due to tracker failure, 32 trials (1.6%) were excluded from Experiment 1 and 119 trials (6.2%) from Experiment 2.

For each trial, we computed the initial deviation angle β and initial distance d of the left and right ends of the barrier based on the measured head position at the moment the barrier appeared (Fig. 2C). We calculated difference scores for relative deviation (Δβ=β_L-β_R) and relative distance (Δd = d_L-d_R) of the two ends of the barrier, where a positive score indicates that the right end has a smaller deviation angle or distance. The waypoint was estimated by the distance of the head from the end of the barrier when they were collinear and classified as to the left or right of the barrier — our dependent measure.

On each trial, we also computed the path length (P) and path curvature (C), estimated by the median of the inverse of the radius of curvature (median K = 1/r) for a three point rolling window along the path. In each condition in which both leftward and rightward routes were observed (44% of all trials), we then calculated a difference score between the mean left and right path length (ΔP=P_L-P_R), and the mean left and right path curvature (ΔC=C_L-C_R), for each participant; positive scores indicate that the rightward path was shorter or less curved.

2.6. Statistical analysis

We performed a mixed-effects logistic regression on the binomial left/right route data from each experiment to assess whether the relative distance (Δd) and deviation angle (Δβ) of the ends of the barrier predicted waypoint selection [11]. We performed a similar analysis of the pooled data using mean path length (ΔP), path curvature (ΔC), distance (Δd), and deviation angle (Δβ) for each participant in each condition as predictors. The data were analyzed in Matlab using the fitglme function.

3. Results

In both experiments, we observed significant effects of the relative distance Δd and deviation angle Δβ of the ends of the barrier on route choice (Table 1). Their interactions did not contribute significantly to the logistic regression; indicating that the two variables are additive and independently contribute to route selection. Let us consider each factor in turn.

Table 1.

Fixed Effects Structure and Random Effects Structure of the Mixed Effects Logistic Regression for predicting route choice to the right of the barrier. The interaction between Δd and Δβ was not significant and did not contribute to the final model.

EXPERIMENT 1
Fixed Effects	Odds Ratio	Estimate (log odds)	SE	t-value	p -value	95 % CI Lower	95 % CI Upper
Intercept	2.06	0.60	0.25	2.43	0.015	0.12	1.08
Difference in Deviation Δβ (degree)	1.94	0.67	0.08	8.74	< .001	0.52	0.83
Difference in Distance Δd (m)	4.98	1.56	0.26	6.00	< .001	1.05	2.07
Random Effects	Standard Deviation	95 % CI Lower	95 % CI Upper
Participant	0.88	0.60	1.29
Participant*Δβ	0.30	0.20	0.46
Participant*Δd	1.05	0.70	1.56
EXPERIMENT 2
Fixed Effects	Odds Ratio	Estimate (log odds)	SE	t-value	p -value	95 % CI Lower	95 % CI Upper
Intercept	0.73	−0.31	0.31	− 0.90	0.37	− 0.99	0.37
Difference in Deviation Δβ (degree)	1.92	0.65	0.05	11.93	< .001	0.54	0.76
Difference in Distance Δd (m)	3.98	1.38	0.35	3.91	< .001	0.69	2.07
Random Effects	Standard Deviation	95 % CI Lower	95 % CI Upper
Participant	1.31	0.85	2.02
Participant*Δβ	0.18	0.10	0.30
Participant*Δd	1.33	0.84	2.11

Relative deviation angle (Δβ).

Broadly speaking, the influence of Δβ is demonstrated by the change in the proportion of observed routes in Fig. 3A,B. As the barrier was shifted from left to right, responses shifted from overwhelmingly rightward (black bars) to overwhelmingly leftward (grey bars). This is consistent with a preference for the smaller deviation angle.

Fig. 3.

Effect of the barrier’s lateral position and relative deviation angle (Δβ) on route choice. (A) Trajectories of a representative participant in Experiment 2 for each lateral position (− 0.4, − 0.2, 0.2, and 0.4 m) with a barrier orientation of 45°. (B) Proportion of Right/Left responses in the corresponding position conditions, collapsed across barrier orientation. (C) Psychometric function (partial effect graph) for Δβ with predicted responses by participant for Experiment 1 and (D) Experiment 2. Negative Δβ values indicate a smaller deviation angle β to the left end of the barrier, so the predicted route is to the left.

However, Δβ is also a function of barrier orientation. Consider two barriers positioned −0.19 m to the left, one at a 90° orientation and the other at 135°: the smaller deviation angle is to the right, but the difference is Δβ = 3.5° in the first case and Δβ = 7.2° in the second. We thus plot the proportion of Rightward responses as a function of Δβ in each experiment (Fig. 3C,D), based on the logistic regression. This results in a psychometric function for Δβ. The analysis shows that Δβ significantly predicted the response in both experiments (p < 0.001) (Table 1). A slight rightward bias was observed in Experiment 1 (50% chance performance at Δβ= −0.89°), and a slight leftward bias in Experiment 2 (chance at Δβ = 0.48°). The choice threshold for 25% and 75% Rightward responses corresponds to a Δβ of ± 1.63° from the 50 % chance value in Experiment 1, and ± 1.68° in Experiment 2.

Relative endpoint distance (Δd).

The influence of Δd is suggested by the change in proportion of observed routes with barrier orientation (Fig. 4A,B). As the barrier was rotated from 45° through 90° (perpendicular) to 135°, responses tended to shift from Rightward (black bars) to Leftward (grey bars), consistent with a weak preference for the near end of the barrier.

Fig. 4.

Effect of the barrier’s orientation and relative distance (Δd) on route choice. (A) Trajectories of a representative participant in Experiment 2 for each barrier orientation (45°, 75°, 105°, 135°) with a lateral position of .2 m. (B) Proportion of Right/Left responses in the corresponding orientation conditions, collapsed across lateral position. (C) Psychometric function (partial effect graph) for Δd with by participant predicted responses for Experiment 1 and (D) Experiment 2. Negative Δd values indicate that the left end of the barrier is closer, so the predicted route is to the left.

However, Δd is also affected by the barrier’s lateral position. For example, in the 0 m position, a 135° barrier had a Δd of −2.09 m; whereas at −0.3 m, it had a smaller Δd of −1.98 m. The psychometric function for Δd in each experiment appears in Fig. 4C,D, based on the logistic regressions. The Δd significantly predicted the proportion of Rightward responses in each Experiment (p < 0.001) (Table 1). There was a slight rightward bias in Experiment 1 (chance performance at Δd= −0.38 m), and a slight leftward bias in Experiment 2 (chance at Δd = 0.23 m). The choice threshold corresponded to a Δd of ± 0.70 m from the chance value in Experiment 1, and ± 0.79 m in Experiment 2.

Trade-off between Δβ and Δd.

To analyze the trade-off, we categorized the experimental conditions as congruent right (+Δβ, +Δd), congruent left (−Δβ, −Δd), or in conflict (+Δβ, −Δd or −Δβ, +Δd). Fig. 5 plots the proportion of Rightward responses in the four categories for Experiments 1 and 2. When the variables agree, responses are overwhelmingly to the right (92%) or left (85%). The conflict trials are in between: responses tend to follow the smaller deviation angle (64% overall), but are reduced by the larger distance. A two-way repeated-measures ANOVA (congruency x experiment) on Rightward responses with a Greenhouse-Geisser correction revealed a significant effect of congruency, F(1.936, 61.957) = 71.235, p < 0.001. Post-hoc tests with a Bonferroni correction revealed that all four categories were significantly different from each other (p < .05). Experiment 1 had significantly more rightward routes than Experiment 2, F(1, 32) = 5.045, p=0.032, reflecting the observed rightward bias. There was also a significant interaction, F(1.936, 61.957) = 7.62, p = 0.001; post-hoc tests revealed fewer rightward responses on incongruent −Δβ,+Δd trials in Experiment 2 than Experiment 1 (p < 0.001).

Fig. 5.

Trade-off between Δβ and Δd in Experiment 1 (black bars) and Experiment 2 (stippled bars). The proportion of rightward responses is plotted for congruent trials (Δβ and Δd have same sign) and conflict trials (opposite sign). Positive values indicate that the right end of the barrier had a smaller distance (d RIGHT < d LEFT) or deviation angle (β RIGHT < β LEFT), predicting a rightward response. The mean Δβ and Δd in each condition appear above the corresponding bar. Significant comparisons using a Bonferroni correction are indicated by the horizontal lines above the groups. * p = .033; *** p < .001. Note that Experiment 2 tested more intermediate Δd values than Experiment 1. Illustrative barrier configurations for each of the four categories appear below. The distance bar indicates the barrier end closest in distance and the triangle indicates the smaller deviation angle from the goal.

Relative path length (ΔP) and relative path curvature (ΔC).

To compare local (Δd, Δβ) and global (ΔP, ΔC) variables, we performed logistic regressions on the proportion of Rightward responses with combinations of these four variables. In a local model, Δd and Δβ were each significant predictors (p < 0.001), while in a global model, ΔP and ΔC were significant (p < 0.001), but the evidence strongly favoured the local model (ΔBIC = 29.5). In the full model, of the four variables, only Δd and Δβ were significant predictors of route choice (p < 0.001), with no significant interaction effects. Moreover, a likelihood ratio test of the local model did not significantly differ from the full model. Multiple regression revealed that the global variables were correlated with local variables: ΔP was strongly predicted by Δβ (partial r = 0.85) and Δd (partial r = 0.75), whereas ΔC was weakly predicted by Δβ (partial r = 0.16) and inversely related to Δd (partial r = −0.11) (all p < 0.001).

Waypoints.

The mean waypoint estimate was 0.6 m (SD = 0.21) from the end of the barrier. This distance did not statistically depend on barrier orientation, lateral position, or experiment.

4. Discussion

To investigate waypoint selection during barrier circumvention, we manipulated the lateral position and orientation of the barrier and tested how the initial distance (d) and the initial deviation angle (β) of each end influenced the chosen route. When the closer end of the barrier coincided with the smaller deviation angle from the goal, we found a strong preference for that end [3]. When these two variables were in conflict, they traded off with each other, as expected by the competing waypoints hypothesis.

Specifically, the relative distance (Δd) and relative deviation angle (Δβ) were each significant independent predictors of route selection, yielding well-behaved psychometric functions (Figs. 3 and 4). When the two variables agreed, responses were overwhelmingly toward the end with the smaller distance and deviation angle, consistent with Gérin-Lajoie and Warren’s [3] data. When they were in conflict, participants tended to select the waypoint with the smaller deviation angle, modulated by its distance. Choice thresholds indicated that a Δβ of 1.65° is behaviourally equivalent to a Δd of 0.75 m. Because these variables are additive, a relative deviation angle of Δβ=+1.65° to the right is cancelled by a relative distance of Δd=−0.75 m to the left. Gérin-Lajoie and Warren [3] modelled their data by adding waypoints off each end of the barrier. We observed a constant waypoint distance of 0.6 m from the end of the barrier, independent of condition, similar to a previous report [12]. This clearance might be related to repulsion from the end of the barrier [1], a constant personal space [13], or a minimal predicted distance [14].

In Gérin-Lajoie and Warren’s [3] model, a waypoint is selected by planning alternative paths to the final goal and choosing the one with least total curvature. However, we found that local waypoint variables (Δβ, Δd) were stronger predictors of path choice than global path variables (ΔP, ΔC). We conclude that waypoint selection is better explained by the visually-specified waypoint distance and deviation angle than by the planned path length or curvature. Such a strategy is adaptive because the local variables reliably predict shorter, straighter paths. This behaviour may be modelled by treating waypoints as competing goals [10]. The findings support the view that, rather than planning and evaluating alternative paths to the final goal, the path is emergent, and depends on the evolution of relative distance and deviation angle during walking.

5. Conclusion

The present experiments manipulated the distance (d) and deviation angle (β) of the ends of a barrier that blocked the way to a final goal. Participants preferred the route that minimized local variables d and β, rather than global path variables P and C. The results support the hypothesis that route selection is governed by competing waypoints, instead of comparing planned paths to the final goal.