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The authors have declared that no competing interests exist.

This study aims to develop a microscopic pedestrian behavior model considering various interactions on pedestrian dynamics at crosswalks. Particularly, we take into account the evasion behavior with counter-flow pedestrians, the following behavior with leader pedestrians, and the collision avoidance behavior with vehicles. Aerial video data at one intersection in Beijing, China are extracted for model calibration. A microscopic calibration approach based on maximum likelihood estimation is applied to estimate the parameters of a modified social force model. Finally, we validate step-wise speed, step-wise acceleration, step-wise direction change, crossing time and lane formation phenomenon by comparing the real data and simulation outputs.

Studying the self-organization phenomena of pedestrian crowd is an active subject in transportation science. To date, pedestrian behavior modeling has attracted considerable attentions [

Generally, existing pedestrian behavior models can be classified into three categories: macroscopic, mesoscopic and microscopic models. In the last decades, mesoscopic and microscopic models have attracted much attention because they enable to offer a more detailed analysis on pedestrian behavior.

The mesoscopic models are usually based on kinetic theory and game theory, in which the microscopic interaction can be represented through a statistical distribution of the microscopic position and velocity. One of the motivations of applying kinetic theory is to model the complexity issues of living systems such as crowds. Following the kinetic theory, Bellomo et al developed a simulation model representing the dynamics of collective behaviors [

The family of microscopic models includes Cellular Automata [

Pedestrian microscopic simulation has gathered a lot of interest in the modeling of safety analysis of pedestrian infrastructures and crowd evacuation. Lian et al. analyzed collective movement characteristics on a four-directional intersecting flow and they found that putting an obstacle in the center of cross area will improve traffic stability [

Crossing behaviors at unsigalized and signalized intersections are critical factors that may result in safety problems. A case study in China showed that the rates of compliance with traffic rules at signalized intersections are influenced by crossing distance, signal timing, and pedestrian volume [

In addition, many research focused on the model calibration and validation. Isenhour et al. developed a pedestrian simulation tool for fire evacuation analysis and they recommended seventeen verification tests according to the United States’ National Institute of Standards and Technology [

To develop and calibrate microscopic pedestrian models usually requires accurate and comprehensive trajectory data on individual pedestrian movement. The traditional data acquisition method is to shoot video and apply tools to detect and track the individual movement of each pedestrian. Both manual and automated tracking approaches can be found. Multiple cameras were usually used to shot videos at intersections and manually extracted trajectories of individual pedestrians and turning vehicles [

In summary, there is growing interest in developing a simulation model for pedestrian dynamic behavior in various scenarios. However, only limited studies shed light on crossing behavior such as interaction with counter-flow pedestrians, turning vehicles, and traffic regulations. For practical applications such as traffic safety assessment at crosswalks, the current pedestrian models for crowd simulation are inadequate. This study aims to fill this gap by developing a microscopic model considering various interaction behaviors among road users at crosswalk. Furthermore, an estimation approach for calibrating the microscopic model based on real trajectory data is proposed. Last, the validation is conducted to confirm the model performance on step-wise location, fundamental diagram and lane formation phenomenon.

As shown in

The microscopic model includes two layers, i.e., a tactical layer and an operational layer. The desired direction of movement is determined in the tactical layer. The operational layer determines the microscopic behavior when pedestrians interact with other agents.

As shown in

The desired direction is assumed to be determined by the desired exit position of the crosswalk. The desired exit position is defined as the intersecting point of the curve of crosswalk edge and the desired walking trajectory.

The concept of truncated normal distribution is able to represent a distribution with arbitrary skewness and a specified range.

We assumed that the distribution of exit position is influenced by the OD direction, crosswalk length _{l}, crosswalk width _{w} and pedestrian density _{0}, _{1}, and _{2} are the dummy variables representing eight directions of the OD on a crosswalk; _{0},…,_{6},_{0},…,_{6} are the model coefficients to be estimated.

The step-wise decision process of movement is assumed to include two steps. First, the pedestrian selects the velocity direction based on the desired direction and subsequently the desired speed. Second, the pedestrian adjusts the speed to avoid the conflict with other pedestrians and vehicles.

Pedestrians are assumed to move with individual desired speed _{α}.

According to the empirical analysis [_{w} and pedestrian density _{0}, _{1}, and _{2} are the model coefficients to be estimated;

According to the social force theory [

(a) It is usually assumed that the magnitude of the repulsive force increases monotonically as the relative distance decreases. (b) It shows the case of valid and invalid conflicts.

The relative time to collision (RTTC) is defined as the time difference between the first road user arriving at the potential conflicting location and the second road user arriving at this location if they keep their current speeds.

Where

The positive values of TTCPs of both pedestrians indicate that the conflict exists, while the negative values indicate that one pedestrian had passed the conflict point and no potential collision will occur.

Accordingly, TTCP can be formulated as follows.

The relative time (_{αβ}) to the conflict point instead of the relative distance is considered as the influential factors to the repulsive force. The repulsive force (_{i} to

We observed that pedestrians preferred to follow the leading pedestrians and joined the group with similar walking directions to avoid intensive interaction with the counter-flow pedestrians. In crowded situations, pedestrians can keep a stable speed and move smoothly by following the leading pedestrians without interactions with the counter-flow pedestrians frequently. Such leader-follower behavior naturally caused the lane formation phenomenon especially when the pedestrian flow became crowded. Even though the lane formation could be also generated without explicitly considering the leader-follower behavior, the generated “lanes” were weak and were easy to breakdown in our original model. To present the collective behavior that result in the swarming effect when the pedestrian density become high at signalized intersections, the attractive force from the leading pedestrians is also formulated.

As shown in

The subject pedestrian will be attracted by the “footprints” of the pedestrians ahead with the same movement direction.

_{i}at time (

_{i}at time (

Pedestrian-vehicle conflicts frequently occur due to the shared signal phase and risk-taking behavior. Drivers are usually required to yield to crossing pedestrians in a right-turn-permitted signal phase (right-hand traffic). However, in developing countries such as China, certain drivers might undertake risky behavior such as entering the crosswalk even if pedestrians are approaching. On the other hand, some risk-taking pedestrians enter the crosswalk before the green light or at the beginning of the red light, which also increase the severe conflict with the vehicles. Since this study mainly focuses on the crossing behavior of pedestrian, the behavior of vehicle is not discussed here. To model the pedestrian behavior during the pedestrian-vehicle interaction process, a three-layer strategy is implemented. In the upper layer, the pedestrian can choose to wait or cross to avoid the conflict when a vehicle is approaching. If the pedestrian chooses to cross, he/she will plan a detour route to avoid the potential collision in the middle strategy layer. Then, a repulsive force from the turning vehicle further acts on the pedestrian moving behavior in the lower layer.

In the upper layer of decision making, we assume that there are two types of pedestrian giving-way maneuvers when the pedestrian-vehicle conflict occurs: waiting until the vehicle passes by and crossing before the vehicle passes by. We expect that the probability of choosing crossing is lower if the pedestrian reaches the potential conflict point later than the vehicle. Therefore, a binary logit model can be utilized to describe this behavior, in which the utility function can be formulated by the relative time to the potential conflict point as shown in

There are two types of pedestrian giving-way maneuvers when the pedestrian-vehicle conflict occurs: waiting until the vehicle passes by and crossing before the vehicle passes by.

_{r}(

_{ttc}is the relative time to the potential conflict point,

_{veh}is the time to the potential conflict point for the vehicle,

_{ped}is the time to the potential conflict point for the pedestrian,

_{0}and

_{1}are the coefficients to be estimated.

In the middle strategy layer, we assume that pedestrians can identify possible routes to avoid the collision with vehicles based on the occupied location of conflicting vehicles. Pedestrians move through the desired route from their origin to destination via intermediate destinations. A link-node network as shown in

The walking space is divided into separate cells where each cell is connected to each other by directed links.

As shown in ^{2} [

In the lower strategy layer, the microscopic moving behavior is assumed to be affected by the vehicle force field once the vehicle enters the personal interaction range. Similar to the force from an obstacle, the repulsive force is determined by the pedestrian position and the direction of speed. Since the size of a vehicle is much larger than the pedestrian, it cannot be regarded as a point. As shown in _{Vα}) which depends on the angle between the moving direction of the vehicle and the moving direction of a close-by pedestrian. The radius can be formulated as follows.

A vehicle is now represented by an ellipse with the radius which depends on the angle between the moving direction of the vehicle and the moving direction of a close-by pedestrian.

Accordingly, the repulsive force from a conflicting vehicle can be presented as follows.
_{Vα}) is the distance between the central of pedestrian

The sum of the force terms exerted to pedestrian _{k}, _{k}, _{k}, _{k}, _{k},

The step-wise speed and position can be expressed as follows:

_{k};

_{k−1};

Δ

_{k};

_{k−1}.

As shown in

Empirical data were extracted using aerial videos captured by an optical camera with a 1920 × 1080 resolution mounted on a quadrotor with the flight altitude of about 40m-60m above the ground.

To reproduce reasonable pedestrian trajectories in simulation, we calibrate the regression models (Eqs (_{k} of a single prediction step is directly related to the probability density function of the normal distribution as follows.
_{k}, _{p} the model parameters to be estimated.

For a set of _{i} time steps, the two-dimensional normal likelihood function can be formulated as follows.

To facilitate the computation, the likelihood function is usually converted into a log-likelihood function as follows.

The maximum log-likelihood estimates of model parameters are obtained such that Eq (_{p}), in Matlab program. We use the “fminunc” function in Matlab to find a minimum of the negative log-likelihood function with several variables.

Equation | Variables | Description | Parameters | Estimates | p-value |
---|---|---|---|---|---|

Eq ( |
_{0} |
A dummy denotes whether the OD direction is from N to F | _{0} |
0.68 | 0.00 |

_{1} |
A dummy denotes whether the OD direction is straight | _{1} |
-1.14 | 0.00 | |

_{2} |
A dummy denotes whether the destination is on the stop line side | _{2} |
-2.42 | 0.00 | |

_{l} |
Crosswalk length | _{3} |
-0.032 | 0.05 | |

_{w} |
Crosswalk width | _{4} |
0.11 | 0.03 | |

Pedestrian density | _{5} |
1.98 | 0.01 | ||

Constant | _{6} |
5.33 | 0.00 | ||

Eq ( |
_{0} |
A dummy denotes whether the OD direction is from N to F | _{0} |
0.22 | 0.00 |

_{1} |
A dummy denotes whether the OD direction is straight | _{1} |
-0.058 | 0.05 | |

_{2} |
A dummy denotes whether the destination is on the stop line side | _{2} |
-0.083 | 0.04 | |

_{l} |
Crosswalk length | _{3} |
0.00024 | 0.05 | |

_{w} |
Crosswalk width | _{4} |
0.062 | 0.03 | |

Pedestrian density | _{5} |
1.70 | 0.00 | ||

Constant | _{6} |
0.86 | 0.05 |

Variable | Coefficient | Estimate | p-value |
---|---|---|---|

Waiting time (_{w}) |
_{0} |
0.0027 | 0.00 |

Pedestrian density ( |
_{1} |
-0.56 | 0.00 |

Constant | _{2} |
1.35 | 0.00 |

Standard deviation of error term | 0.54 | 0.00 |

_{0} indicates that pedestrians tend to cross if the arrival time to the potential conflict point is earlier than that of the conflicting vehicle. In such a situation, the conflict with the vehicle might be alleviated if the vehicle maintains speeds or decelerates. However, the conflict becomes severe if the vehicle chooses to accelerate.

Variables | Coefficient | Estimates | p-value |
---|---|---|---|

Relative time to the potential conflicting point when the pedestrian conflict with a moving vehicle (_{ttc}) |
_{0} |
0.031 | 0.00 |

Constant | _{1} |
-0.46 | 0.00 |

Parameters | Equation | Estimates | p-value |
---|---|---|---|

_{α} |
( |
0.46 | 0.00 |

( |
0.19 | 0.00 | |

( |
1.35 | 0.00 | |

( |
0.22 | 0.01 | |

( |
0.13 | 0.02 | |

_{V} |
( |
0.93 | 0.02 |

_{V} |
( |
1.54 | 0.02 |

(a) shows the pedestrian trajectories from real data and (b) shows simulation outputs.

(a) shows the average absolute error of walking speed. (b) shows the comparison of step-wise acceleration distributions. (c) shows the distribution of the step-wise direction change between current and previous directions. (d) shows the distribution of the crossing time at crosswalk.

Since the pedestrian flow characteristic can be represented by fundamental diagram, we compare the speed-density and flow-density diagrams in real data and simulation outputs to demonstrate the model performance. The spatial mean speed and density are calculated by taking a cell 40×40 cm^{2} in the crosswalk as the measurement area. We calculate the speed and density every 1s. As shown in

The simulated fundamental diagrams are in good agreement with the observed ones.

Lane formation is one of the most interesting phenomena that characterize pedestrian flow. Such a phenomenon is caused by conflict avoidance and leader-follower behavior. To demonstrate this phenomenon, a crowded scenario is set for the simulation. The bi-direction pedestrian demand is set to be 57 pedestrians per signal cycle according to the video recording.

The evolution of the lane formation in one signal cycle.

A two-layer microscopic model is presented to simulate the interactions between pedestrians and vehicles at signalized intersections. A modified social force model considering the evasion behavior with counter-flow pedestrians, the following behavior with the leader pedestrians, and the collision avoidance behavior with vehicles was developed. The calibration is undertaken using the trajectory data (samples are given in

This simulation tool is recommended to be used by public authorities to gain more knowledge about how pedestrians and drivers interact with each other in the crosswalk. Furthermore, the cause of pedestrian-vehicle conflict can be identified in advance by simulation, which enables to provide information about potential safety problems prior to facility implementation.

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