Limited pedestrian microcosmic simulation models focus on the interactions between pedestrians and vehicles at unmarked roadways. Pedestrians tend to head to the destinations directly through the shortest path. So, pedestrians have inclined trajectories pointing destinations. Few simulation models have been established to describe the mechanisms underlying the inclined trajectories when pedestrians cross unmarked roadways. To overcome these shortcomings, achieve solutions for optimal design features before implementation, and help to make the design more rational, the paper establishes a modified social force model for interactions between pedestrians and vehicles at unmarked roadways. To achieve this goal, stop/go decision-making model based on gap acceptance theory and conflict avoidance models were developed to make social force model more appropriate in simulating pedestrian crossing behaviors at unmarked roadways. The extended model enables the understanding and judgment ability of pedestrians about the traffic environment and guides pedestrians to take the best behavior to avoid conflict and keep themselves safe. The comparison results of observed pedestrians’ trajectories and simulated pedestrians’ trajectories at one unmarked roadway indicate that the proposed model can be used to simulate pedestrian crossing behaviors at unmarked roadways effectively. The proposed model can be used to explore pedestrians’ trajectories variation at unmarked roadways and improve pedestrian safety facilities.
Unlike pedestrians at the marked and signalized roadways, pedestrians at unmarked roadways are not restricted by the crosswalk boundary, so their trajectories have a wider variation, which results in a wider range distribution of conflict points. Pedestrians will take more complex stop/go decision-making and conflict avoidance mechanisms to cross the road quickly and safely. Pedestrians can wait at the lane line and the curb or bypass one vehicle from the front or behind the vehicle. Besides, pedestrians are attracted by traffic attraction point, such as bus station and market. Pedestrian prefer to choose the shortest path to the destination, which results in inclined crossing trajectory at unmarked roadways.
Many models have been put forward to model the crossing process of pedestrians, such as social force model, discrete choice model, cellular automata model, and gap-acceptance-based models. Microsimulation model can reproduce many complex traffic phenomena, and it is an effective method to assess new traffic design scheme. Cellular automata (CA) and social force model are two methods used frequently, and their effectiveness has been fully verified. Existing CA models related to pedestrians focus on the roadway conditions [
Considering more realistic behaviors, existing force-based models have been developed by modifying the SFM established by Helbing and Molnár [
Based on the SFM, a modified and extended social force model of pedestrian-car mixed traffic flow at unmarked roadways which is capable of reproducing the behaviors and trajectories of pedestrians at unmarked road is established and it can be implemented in traffic design scheme assessment. The essential mutual interferences of cars and pedestrians are analyzed. The paper effectively combines the stop/go decision-making model and conflict avoidance mechanism of pedestrians into social force model. In this paper, SFMs of single pedestrian flow, single vehicle flow, and pedestrian-car mixed traffic flow are presented in Section
The basic social force model contains four basic forces: driving force
In (
Driving force drives pedestrian
The interaction force between pedestrian
Interaction force between pedestrians and obstacles also contains sociopsychological force
In the initial social force model, Helbing indicated that all pedestrians, boundaries, and barriers in all directions have the same influences on the moving pedestrians. However, pedestrians will adjust their behaviors according to the surrounding traffic environment and the locations of pedestrians and obstacles. The pedestrians and obstacles in the sight of the moving pedestrians have greater influences on the moving pedestrians’ behaviors. To explain this behavioral mechanism, they introduced anisotropic factor
Schematic diagram of different anisotropic coefficients.
Besides,
Possible speed directions and
As expressed in (
Generally speaking, when vehicles arrive at the crossing area of pedestrians, pedestrians choose to cross the unmarked roadway without signal control by waiting the proper time gap in the vehicle traffic flow according to the complex traffic environment. However, when pedestrians have waited a long time, they tend to take adventure to cross the dense vehicle traffic flow. The interactions between pedestrians and vehicles become very severe. A stop/go decision-making model can enhance pedestrian capabilities of choosing proper crossing behaviors. Pedestrian crossing behaviors have been well described by some models, such as gap acceptance theory model. Brewer et al. [
Based on the gap acceptance theory, a stop/go decision-making model is established. At unsignalized and unmarked roadways, pedestrians change their crossing behaviors according to the time gap of cars, their locations, and group size. So, we try to develop a stop/go decision-making model by combining all these influence factors. The proposed stop/go decision-making model can describe the interactions between pedestrians and cars. Pedestrians will decide to go or not by weighing threats from vehicles, their locations, and group size. If the time needed by pedestrians to pass the conflict point is smaller than the critical gap time, they will cross. Otherwise, pedestrians will choose to wait until the proper time gap arises.
Known from (
The relationship between perceived risk (
Schematic diagram of pedestrian crossing (
The total perceived risk
Only
Besides threats from cars, pedestrians also receive threats from their possible locations at next simulation time step (walk at the current speed). Perceived risk from locations (
The probability of accepting a time gap is inversely proportional to perceived risk from cars and pedestrians’ locations. It means that pedestrians will go when the perceived risk from cars and pedestrians’ locations is low, and they will stop when the risk from cars and pedestrians’ locations is high, because pedestrians prefer to choose the safest ways to cross the road. At the same time, pedestrians prefer to cross in group. Pedestrians think crossing will be more safe when the group size is large. Pedestrians who have the same moving destination, where the headway of the front pedestrian and the following pedestrian is not greater than 2 s, are defined as a pedestrian group. Therefore, the total perceived risk from cars and pedestrians’ locations should be adjusted by considering the group size. The modified total perceived risk model is expressed as
In conclusion, the probability to go is expressed as
Asano et al. [
Forces on individuals avoiding conflict.
Invalid and valid conflict
Force from individual evasive behavior
The TTCP for the subject pedestrian
Once a conflict is identified as valid, subject pedestrian
The adjusted speed vector
Therefore, the giving-way maneuver can be presented as an individual force
Helbing and Molnár [
Schematic diagram of vehicle modeling.
Similar to pedestrians, the basic social force model of vehicles contains driving force
The formula of
The driving force of vehicle
Vehicles are influenced by other vehicles within its visible range. The interaction force between vehicle
Vehicles are restricted in lanes, so lateral movement is impossible. The influencing range is explained in Figure
Influence range of vehicle considering vision.
Schematic diagram of
And the moving vehicle is influenced by both the front vehicle and behind vehicle. So,
Besides, different from pedestrians, vehicles prefer to follow the front vehicle. Generally, the vehicles only move in the vehicle lane, and one vehicle lane allows one vehicle to cross at one moment. Compared with pedestrians, the vehicles are queue. So the following characteristic of vehicles is more remarkable, and it becomes more noticeable with the cars increasing. Anvari et al. (2012) indicated that the social force may cause the behind vehicle to overtake rather than follow the leading vehicle. If the following characteristic is not taken into consideration, the behind vehicle with a higher speed will attempt to overtake the leading vehicle instead of slowing down when the leading vehicle decelerates. Similarly, the behind vehicle with a lower speed will attempt to speed up instead of keeping its current speed when the leading vehicle accelerates.
To model the following behavior, Helbing et al. (1998) proposed a force
However, the force
The interaction force
The interaction force
When a pedestrian moves into the road, if the vehicle or pedestrian does not decelerate immediately, a collision will happen. Once the conflict occurs, pedestrian
Conflict avoidance mechanism of pedestrians.
So, an effective giving-way mechanism to describe such states is necessary. In the real world, the giving-way maneuvers adopted by pedestrians are flexible. The pedestrians can bypass the vehicle in front of them from the front or behind by changing moving direction. At the same time, changing moving direction depends on the relative location of pedestrian and car. This paper develops a conflict avoidance model to explain this maneuver of pedestrian. The conflict avoidance model of pedestrians and vehicles is expressed as
As shown in Figure
Schematic diagram of studied site.
The maximum log-likelihood estimation method was used by Zeng et al. [
In the study of Zeng et al. [ The parameters Those parameters that can be determined from the observed dataset, for example, relaxation time of After determining these measurable parameters, nonmeasurable parameters, such as the strength coefficients that do not have concrete physical meanings, are derived by the maximum log-likelihood estimation method.
Real trajectories are extracted from video data of studied unmarked road. The trajectory extractor software developed by Jiang Sheng was used to extract the trajectories, velocities, and acceleration of pedestrians from video. The video analysis procedure was explained by Jiang (2012), and the error and accuracy of the software were analyzed in detail. Other nonmeasurable parameters, such as
The
To simplify the process of computation, logarithms are taken of both sides of (
The calibration results of parameters are presented in Table
Parameter calibration results.
Parameters | Eq. | Estimates |
|
---|---|---|---|
Strength coefficients for force from other pedestrians |
( |
0.75 | 0.03 |
Reaction distance for force from other pedestrians |
( |
1.75 | 0.04 |
Strength coefficients for force from other vehicles |
( |
7 | 0.02 |
Reaction distance for force from other vehicles |
( |
10 | 0.02 |
Strength coefficients for force from obstacle |
( |
0.5 | 0.01 |
Reaction distance for force from obstacle |
( |
4.7 | 0.04 |
Strength coefficients for force from boundary |
( |
2.7 | 0.05 |
Reaction distance for force from boundary |
( |
4.9 | 0.03 |
Strength coefficients for force between vehicle and pedestrian |
( |
5.3 | 0.03 |
Reaction distance for force between vehicle and pedestrian |
( |
5.7 | 0.07 |
Reaction distance for acceleration interaction |
( |
5.59 | |
Reaction distance for braking interaction |
( |
8.62 | |
Constant |
( |
1.7 | 0.03 |
Constant |
( |
4.5 | 0.07 |
Constant |
( |
3.4 | 0.05 |
Constant |
( |
7.2 | 0.01 |
Constant |
( |
3.67 | 0.07 |
Radius of pedestrian |
/ | 0.5 | |
Relaxation time of pedestrian |
( |
0.3 | |
Relaxation time of vehicle |
( |
2.4 | |
Braking time of vehicle |
( |
0.77 | |
Simulation time step |
( |
0.06 | |
Anisotropic factor coefficient |
( |
0.3 | |
Desired speed of pedestrian |
( |
1.37 | |
Desired speed of vehicle |
( |
14.7 |
A simulation analysis of pedestrians crossing unmarked road on Yatai Street in Changchun of China is made. The width of the road segment is 27 m. In order to obtain stability simulation results, there are 573 vehicles and 713 pedestrians crossing the road segment. The simulation is repeated 100 times to eliminate the influence of random disturbance. The anisotropic characters of pedestrians and vehicles are set to 0.3 so that interactions outside of the field of view have little effect on pedestrians and vehicles.
After 100 simulations being done, all the average speeds of individuals (pedestrians or cars) in the simulation scenarios are recorded. Then, the simulation speed distribution and the observed speed distribution are compared. Figure
Speed histograms of pedestrians observed and simulation results.
Figure
Comparison of simulated and observed trajectories.
But, it is not convincing enough if we only use the similar shape and outline of trajectories to show the effectiveness of the proposed model. We should validate the proposed model from more microscopic results. Generally speaking, comparing the observed and simulated trajectory of one pedestrian every time is the best method to prove the validity of the developed model. However, it needs considerable word, and it is unreasonable. Because, pedestrian behaviors are random and unpredictable, and we cannot determine which simulated trajectory should be selected to be compared with observed trajectory of one pedestrian. We only are able to validate the proposed model by comparing the available and observed characteristics of pedestrians. So, we select the distributions of crossing positions at two sections (middle line and end line) to validate the proposed model. By comparing the probability and cumulative probability of observed and estimated crossing positions at two sections, it is obvious that the distributions of estimated crossing positions agree with observed ones well. As shown in Figure
Comparison of simulated and observed crossing positions.
The above comparison (Figure
Schematic diagram of five path types.
As shown in Figure
Examples of simulated and observed pedestrian trajectories for five path types.
Based on social force model, a modified social force model considering the interactions between pedestrians and vehicles, conflict avoidance with cars, and group evasive was established. Then, the relevant parameters of the proposed social force model were calibrated by using maximum likelihood estimation method. Finally, the effectiveness of proposed models was verified by comparing the observed and simulated pedestrian trajectories at unmarked road. The major contributions and innovations of the paper are summarized as follows. The stop/go decision-making model was developed to describe the phenomenon that pedestrians cross the vehicle traffic flow. So far, most of the studies focused on modifying the social force model for pedestrians. And the mechanisms of interactions between pedestrians and vehicles were seldom taken into consideration, let alone modeled with social force model. A conflict avoidance model of pedestrians and vehicles was modeled. The bypassing maneuver adopted by pedestrians was modeled. The model enables pedestrians to cross the vehicle traffic flow effectively when vehicles move at low speeds.
The similar shape and outline of observed and simulated pedestrian trajectories, small root-mean-square errors, and acceptable
There are no conflicts of interest regarding the publication of this paper.
This research was funded by the National Natural Science Foundation of China (nos. 51278520 and 51278220).