**Abstract**:

**Abstract**
**When people walk together in groups or crowds they must coordinate their walking speed and direction with their neighbors. This paper investigates how a pedestrian visually controls speed when following a leader on a straight path (one-dimensional following). To model the behavioral dynamics of following, participants in Experiment 1 walked behind a confederate who randomly increased or decreased his walking speed. The data were used to test six models of speed control that used the leader's speed, distance, or combinations of both to regulate the follower's acceleration. To test the optical information used to control speed, participants in Experiment 2 walked behind a virtual moving pole, whose visual angle and binocular disparity were independently manipulated. The results indicate the followers match the speed of the leader, and do so using a visual control law that primarily nulls the leader's optical expansion (change in visual angle), with little influence of change in disparity. This finding has direct applications to understanding the coordination among neighbors in human crowds.**

*behavioral dynamics,*a physical description of the observed behavior in terms of physical variables (i.e., what agents are doing), and the

*control laws*that characterize how the behavior is regulated by perceptual information (i.e., how agents do it). This dual modeling approach has two advantages. First, modeling the behavioral dynamics simplifies the problem of identifying a control law, because a physical description of the behavior constrains the possible optical variables that might govern it. Second, the physical description is more general, because the same behavior may be governed by different information in different contexts. Here we introduce several candidate behavioral models of following and several hypotheses about the information used for visual control.

*ẋ*is the leader's speed,

_{l}*ẋ*is the follower's speed, and

_{f}*c*is a free parameter. This can be equivalently stated in terms of the relative speed Δ

*ẋ*, which is the difference in speed between the leader and follower, or the speed of the leader in the follower's reference frame: Thus, acceleration goes to zero as the follower's speed approaches the leader's speed; that is, as the relative speed between them goes to zero. One advantage of this speed-matching strategy is that it does not assume a fixed distance between leader and follower; this is an advantage both for the follower (who does not need to store a reference value) and for the model (which does not require an additional parameter).

*fixed distance*behind the leader, as proposed for car-following by Kometani and Sasaki (1958). When the distance between leader and follower is above a reference value, the follower should accelerate; when it is below that value, the follower should decelerate. Formally, the follower's acceleration

*ẍ*at each time step is given by: where Δ

_{f}*x*is the current distance (difference in position) between the leader and follower, Δ

*x*

_{0}is the fixed reference distance, and

*c*is a free parameter. Thus, the follower's acceleration goes to zero as the current distance approaches the reference distance. From a modeling perspective, the reference distance might be chosen in several ways: it can be derived from observational data, such as the initial distance between leader and follower (we refer to this as the

*initial distance*model), or it can be a free parameter that represents the “preferred” distance (the

*free parameter distance*model).

*distance that depends on the current velocity*, rather than the constant distance described above. This strategy was proposed in the car-following literature by Pipes (1953) and Herman, Montroll, Potts, and Rothery (1959), and is the basis for the “one-car length for every 10 mph” rule of thumb taught in driving schools. It can be formalized by the expression: where

*ẋ*is the follower's speed, Δ

_{f}*x*is the current distance between leader and follower, and

*α*,

*β*, and

*c*are free parameters. Acceleration goes to zero as the distance between leader and follower approaches the desired velocity-based distance determined by

*α*and

*β*.

*T*, provides an estimate of the time before one collides with an object (or vice versa), given the distance to the object and one's speed, assuming a constant velocity. For following, it provides an estimate of the time before a follower collides with a leader, given the distance between them, Δ

_{C}*x*, and their relative speed, Δ

*ẋ*: A negative value of

*T*specifies that the follower will collide with the leader at some time in the future, and thus is gaining ground; a positive value implies that the leader is getting away from the follower and, if both maintain their current speeds, the two will not collide. Thus, followers might maintain a value of

_{C}*T*that is neither positive nor negative, to avoid either colliding with the leader or letting him get away. If Δ

_{C}*x*is zero then

*T*will be zero, but that means a collision has occurred. Alternatively, if Δ

_{C}*ẋ*is zero then

*T*will be undefined, but this applies regardless of the value of Δ

_{C}*x*and is thus a reformulation of the speed-matching strategy, which aims to bring Δ

*ẋ*to zero (Equation 2). Therefore, a time to contact strategy will not be considered further, but the next strategy is based upon its inverse, the “immanence” of collision.

*ẋ*is the follower's speed, Δ

_{f}*ẋ*and Δ

*x*are the relative speed and distance, respectively, between leader and follower, and

*c, M,*and

*L*are free parameters. Like the speed-matching strategy, acceleration will go to zero as the relative speed between leader and follower goes to zero, but it is modulated by both the follower's current speed and the relative distance between leader and follower.

*ẋ*and Δ

*x*are the relative speed and distance, respectively, between leader and follower,

*ẋ*is the follower's speed, and

_{f}*c*

_{1},

*c*

_{2},

*α*,

*β*, and

*c*are free parameters. In general, acceleration goes to zero when the relative speed is zero (i.e., speed is matched),

*and*the difference between the current distance and the velocity-based reference distance is zero (i.e., distance is maintained).

*α*, is a function of the distance between leader and follower, Δ

*x*, and the leader's size,

*w*. Assuming the latter is fixed, visual angle depends only on the distance between leader and follower, so maintaining a constant distance behind the leader (the distance strategy) can be achieved by maintaining a constant visual angle of the leader. The change in visual angle, or optical expansion,

*α̇*, is a function of the relative speed and distance between the leader and follower, but a constant relative speed (the speed-matching strategy) can be achieved by cancelling changes in the leader's visual angle (i.e., nulling optical expansion). Anderson and Sauer (2007) proposed that drivers use a weighted sum of these two variables to follow a lead vehicle, which is similar to the linear strategy (Equation 7).

*both*optical expansion and binocular disparity in a one-handed catching task, relying on whichever cue specified the earliest time of arrival.

*x,y*plane were analyzed. Each time series was filtered, using a forward and backward fourth-order low-pass Butterworth filter with a cutoff frequency of 1 Hz, to reduce error due to the position tracker and attenuate anterior–posterior accelerations due to the step cycle. To eliminate edge effects from filtering at the end of the trial (endpoint error), the position time series were extended by 2 s using linear extrapolation based on the last 0.5 s of data (Howarth & Callaghan, 2009; Vint & Hinrichs, 1996). The extrapolated data were only used to extend the time series during filtering, and were not used for any subsequent analysis. The filtered position data were differentiated to produce a time series of speed, and differentiated again, to produce a time series of acceleration. Due to tracking errors, 78 trials (14%) were excluded from further analysis.

*r*) between the simulated follower acceleration produced by each model with the observed follower acceleration. The Broyden–Fletcher–Goldfarb–Shanno (Shanno, 1985) method for numerical optimization was used to find the set of parameter values that maximized the mean value of

*r*for each model across all trials using a least-squares criterion. The same parameter values were used for all participants to avoid overfitting and to yield a model that generalizes to novel (untested) pedestrians. For statistical comparisons, mean

*r*values for each participant were computed using Fisher's

*z*′ transform to correct for nonnormality (Martin & Bateson, 1986); the mean

*z*′ values were transformed back into the mean

*r*values reported below. The root-mean-squared-error (RMSE) between the two time series was also analyzed.

*t*(303) = 6.11,

*p*> 0.05, so they were combined; histograms for the resulting speed change condition and the constant speed condition appear in Figure 3.

*r*= 0.68, median

*r*= 0.67) and slightly lower in the constant speed condition (mean

*r*= 0.53, median

*r*= 0.50), indicating a strong temporal coupling between follower and leader. The mean optimal delay in the speed change condition (

*M*= 420 ms,

*Mdn*= 417 ms,

*SD*= 373 ms) was significantly greater than that in the constant speed condition (

*M*= 25 ms,

*Mdn*= 0 ms,

*SD*= 530 ms),

*t*(435) = 8.87,

*p*< 0.001, which in turn was not significantly different from zero,

*t*(131) = 0.557. By design, there was little variation in leader speed during constant speed trials, yielding lower correlations and poorer estimates of the delay. Therefore, we take the mean optimal delay of 420 ms in the speed change condition as an estimate of the follower's visual–motor delay. This value is similar to estimates from other locomotor tasks (e.g., Benguigui, Baurés, & Le Runigo, 2008; Cinelli & Warren, 2012; Le Runigo, Benguigui, & Bardy, 2010). In sum, the leader produced marked changes in speed, and the follower responded with closely coordinated speed changes after a short delay.

Model | Mean r | Mean RMSE (m/s^{2}) | Number of parameters | Parameter values | Duncan grouping |

Speed-matching | 0.67 | 0.21 | 1 | c = 1.87 | a, d |

Initial distance | 0.37 | 0.61 | 1 | c = 3.49 | b, e |

Free parameter distance | 0.40 | 0.82 | 3 | c = 2.69 | b |

Δx_{0}_{,1m} = 1.32 | |||||

Δx_{0}_{,4m} = 3.93 | |||||

Velocity-based distance | 0.52 | 0.80 | 3 | c = 2.44 | c |

α = 0.35 | |||||

β = 0.75 | |||||

Ratio | 0.67 | 0.21 | 3 | c = 2.09 | a |

M = 0.004 | |||||

L = 0.16 | |||||

Linear | 0.67 | 0.21 | 4 | c = 2.11_{1} | a |

c = 0.02_{2} | |||||

α = 23.31 | |||||

β = −16.91 | |||||

Speed + damping | 0.68 | 0.21 | 2 | c = 1.93 | d |

d = −0.15 | |||||

Distance + damping | 0.37 | 0.66 | 2 | c = 3.35 | e |

d = −0.07 | |||||

Speed + delay | 0.68 | 0.21 | 1 | c = 1.83 | d |

Optical expansion | 0.62 | 0.23 | 1 | c = 13.00 | f |

*z*-transformed

*r*values for each model, using the overall best fit parameters for that model. A one-way repeated measures analysis of variance showed significant differences between the models,

*F*(5,45) = 156.96,

*p*< 0.001. Post hoc comparisons for all pairwise combinations were conducted using Bonferroni adjusted alpha levels of 0.0033 (0.05/15). Results indicated that the mean

*r*values for the speed (

*M*= 0.672,

*SD*= 0.111), ratio (

*M*= 0.673,

*SD*= 0.111), and linear (

*M*= 0.673,

*SD*= 0.110) models were not significantly different from one another,

*p*> 0.05, and were all significantly greater than those for the initial distance (

*M*= 0.372,

*SD*= 0.080), free parameter distance (

*M*= 0.398,

*SD*= 0.069), and velocity-based distance (

*M*= 0.518,

*SD*= 0.081) models,

*p*< 0.001. Mean

*r*was significantly greater for the velocity-based distance model than the initial distance and free parameter distance models (

*p*< 0.01), which did not differ from one another (

*p*= 1.00). Thus the follower data were fit significantly more closely by models that contain a relative speed term than by the distance-based models.

*r*values are 0.87, 0.87, 0.87, 0.56, 0.47, and 0.45, for the speed, ratio, linear, initial distance, free parameter distance, and velocity-based distance models, respectively.

*F*(5, 45) = 164.16,

*p*< 0.001. Bonferroni-adjusted post hoc comparisons indicated that the mean RMSE values for the speed (

*M*= 0.210 m/s

^{2},

*SD*= 0.055 m/s

^{2}), ratio (

*M*= 0.210 m/s

^{2},

*SD*= 0.054 m/s

^{2}), and linear (

*M*= 0.212 m/s

^{2},

*SD*= 0.054 m/s

^{2}) models were not significantly different from one another,

*p =*1.00, but were all significantly lower than those for the initial distance (

*M*= 0.611 m/s

^{2},

*SD*= 0.092 m/s

^{2}), free parameter distance (

*M*= 0.825 m/s

^{2},

*SD*= 0.195 m/s

^{2}), and velocity-based distance (

*M*= 0.79 8 m/s

^{2},

*SD*= 0.086 m/s

^{2}) models,

*p*< 0.001. Mean RMSE was significantly lower for the initial distance model than the free parameter and velocity-based distance models (

*p*< 0.01), but they did not significantly differ from one another,

*p*= 1.00.

*d*: Mean

*r*was slightly higher for the speed-matching model with damping (

*M*= 0.678,

*SD*= 0.106,

*c*= 1.93,

*d*= −0.15) than without (

*M*= 0.672,

*SD*= 0.111,

*c*= 1.87), but this difference was not significant,

*t*(9) = 2.14,

*p*> 0.05 (paired sample

*t*test). Likewise, mean

*r*was slightly but not significantly greater for the initial distance model with damping (

*M*= 0.374,

*SD*= 0.081,

*c*= 3.35,

*d*= −0.068) than without (

*M*= 0.372,

*SD*= 0.080,

*c*= 3.49), but not significantly,

*t*(9) = 1.28,

*p*> 0.05. Furthermore, the best fit for parameter

*d*was very near zero. Taken together, these results indicate that adding a damping term does not improve performance over the simpler speed and initial distance models.

*t*to govern the follower's acceleration at the same instant

*t*. But parameter

*c*in Equations 1 and 2 modulates the follower's rate of response to a given speed difference, implicitly introducing delay into the model. To analyze the empirical adequacy of this solution, we computed the cross-correlation between time series of follower acceleration for the model and the data on each trial As shown in Figure 5, for both speed change (

*M*= 71 ms,

*Mdn*= 0 ms,

*SD*= 217 ms) and no speed change conditions (

*M*= −0.45 ms,

*Mdn*= 0 ms,

*SD*= 247 ms), the optimal delays are sharply peaked around zero, indicating that the speed-matching model implicitly accounts for visual–motor delay.

*t*is a function of the speed difference at a previous time in the past,

*t – t*, where

_{d}*t*= 420 ms: As before, we fit Equation 8 using numerical optimization to maximize the mean value of

_{d}*r*across all trials. A paired-sample

*t*test revealed that this model (

*M*= 0.633,

*SD*= 0.101,

*c*= 1.52,

*t*= 301 ms) failed to perform as well as the simpler model without a delay term (

_{d}*M*= 0.672,

*SD*= 0.111,

*c*= 1.87);

*t*(9) = 2.49,

*p*< 0.05. Thus, including an explicit visual–motor delay does not improve the performance of the speed-matching model, at least over the observed range of speed differences.

*M*= 2.5 s,

*SD*= 1 s) until a “manipulation” changed the target speed specified by binocular disparity (the “disparity-specified speed”) or by visual angle (the “expansion-specified speed”) for 3 s (see Table 2).

*decrease*in walking speed, including a disparity-specified speed decrease (ΔSpeed = −0.031 m/s), constant disparity (ΔSpeed = −0.060 m/s), and, surprisingly, a disparity-specified speed increase (ΔSpeed = −0.093 m/s).

*F*(2, 144) = 221.58,

*p*< 0.001, a marginally significant effect of disparity,

*F*(2, 144) = 3.149,

*p*= 0.046, and no interaction,

*F*(4, 142) = 0.151,

*p*> 0.05. Measures of effect size indicate that optical expansion (

*ω*

^{2}= 0.690) explained a far greater proportion of the variance in follower speed than changes in disparity (

*ω*

^{2}= 0.007).

*M*= 0.31 m/s,

*SD*= 0.13 m/s) was twice that for contraction (

*M*= 0.14 m/s,

*SD*= 0.13 m/s),

*t*(100) = 6.30,

*p*< 0.001. This may reflect a fundamental asymmetry in following behavior—for example, followers may prioritize deceleration to avoid collisions in response to optical expansion (emergency braking) over acceleration in response to optical contraction.

*β*= 0.85,

*p*< 0.001) and disparity (

*β*= 0.10,

*p*= 0.015) were significant predictors of ΔSpeed (adjusted

*R*

^{2}= 0.737), but the expansion weight was 8.5 times the disparity weight. A model that included expansion alone accounted for 73% of the variance (adjusted

*R*

^{2}= 0.728); thus adding disparity to the model explained only an additional 1% of the variance. These results indicate that followers are sensitive to information from both optical expansion and binocular disparity, but rely primarily on expansion.

*following by optical expansion*can be simply stated as: where

*b*is a constant and

*α̇*is the rate of optical expansion of the leader. It can be shown that this control law is mathematically related to the speed-matching model (Equation 2) for a leader of constant size; a derivation is provided in Appendix A.

*z′-*transformed

*r*over all trials. The only input to the model was the rate of change of the leader's visual angle computed from the data, and the output was the follower's predicted acceleration. Figure 8 presents a simulation of the same sample trial as in Figure 3. A paired-sample

*t*test on the mean

*z′-*transformed

*r*values showed that the speed-matching model (

*M*= 0.672,

*SD*= 0.111) provided a slightly better fit to than the optical expansion control law (

*M*= 0.624,

*SD*= 0.086,

*c*= 13.00),

*t*(9) = 7.81,

*p*< 0.001. This result suggests that followers rely primarily, but perhaps not entirely, on optical expansion to regulate their speed, consistent with the results of Experiment 2. Binocular disparity may provide additional information necessary to perceive relative speed.

*heading alignment*model (Bonneaud & Warren, 2013; Vicsek et al., 1995), in which the difference in heading direction is nulled. One visual control law for heading alignment might be the

*constant bearing*(CB) strategy, which provides a good description of how pedestrians intercept a moving target (Fajen & Warren, 2007): steer to null change in the target's bearing direction. When the pedestrian and the target move at approximately the same speed, the CB strategy yields parallel heading directions. In a preliminary analysis, Rhea, Cohen, and Warren (2009) found that the CB model reproduced the follower's path when the follower's speed was greater than or equal to the leader's speed. However, when the follower's speed was lower than the leader's, the model often generated a mirror image of the observed path, because this solution also maintains the leader at a constant bearing. We are currently pursuing this problem experimentally.

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*α*is the visual angle that the leader subtends at the follower's eye. We can rewrite visual angle in terms of real-world variables: where

*w*is the width of the leader and Δ

*x*is the distance between leader and follower. Substituting this formula into Equation A1 yields: Taking the derivative in Equation A3, using the chain rule, yields: Simplifying and combining terms in Equation A5 yields: Thus, the expansion model (Equation A6) resembles the speed-matching model (Equation 2), except that the coefficient for the expansion model (in parentheses) is a nonlinear function of leader size

*w*and distance Δ

*x*.