Simulating Pedestrian Navigation Behavior Using a Probabilistic Model

Transcription

1 Proceedings of the 2009 Industrial Engineering Research Conference Simulating Pedestrian Navigation Behavior Using a Probabilistic Model John M. Usher, Eric Kolstad, and Lesley Strawderman Department of Industrial & Systems Engineering Mississippi State University, MS 39762, USA Abstract The Intermodal Simulator for the Analysis of Pedestrian Traffic (ISAPT) is being developed for the purpose of modeling pedestrian traffic within intermodal facilities, such that designers may evaluate the impact of building design on the Level of Service provided to pedestrians. The ISAPT system models each pedestrian s behavior individually and the collective behavior of the crowd is allowed to emerge from the interactions between individuals. This paper presents a probabilistic approach for pedestrian navigation and demonstrates its application in example navigational scenarios along a pedestrian corridor. Keywords Pedestrian simulation, behavior simulation, probabilistic navigation, discrete choice model, utility factors 1. Background Three major interaction schemes are often seen in simulating crowds of agents: cellular automata, social forces and rule-based systems each of which has certain tradeoffs [1]. The first of these relies on a grid cell-based division of space for pedestrian travel, occupancy and consideration of movement alternatives [2]. Using a force strategy attempts to distribute motion via physically motivating forces such as forward movement, steering and avoidance of obstacles. Rule-based systems enact logical choices when a certain series of conditions or overall criteria have been met. System models are somewhat divided as to whether choices are examined in terms of a continuous space of motion alternatives (as with social forces systems modeled as repulsive forces [3, 4] or these are focused on several discrete combinations of base navigation factors during a given time step for example, discrete spatial regions in cellular automata [2, 5] or representative combinations of direction and speed [6, 7]. When simulated movement choices are not viable or have already resulted in an undesirable state, however, it is up to the individual system to allow for braking force, stop-and-wait conditions, and other actions to enable individual pedestrians (and the system overall) to re-adjust, for example, after one or more collisions have been encountered. Another primary distinction is whether a simulation model examines the pedestrians as a higher-level process (e.g. with large-scale crowd and flock-influenced behavior, where each agent is influenced by system-wide information), or opts to focus more so on representation of individual agents decision-making based on their available knowledge and objectives. A more encompassing scale model is often utilized in computer-generated animations [8]; crowds of insects in Indiana Jones and the Kingdom of the Crystal Skull [9] and urban visualization, where system-level effects may take precedence over micro-scale realism [3, 10]. The latter, however, better allows for imperfect awareness of the environment along with individual preferences and agendas. 2. Pedestrian Navigation The pedestrian s overall objectives determine their higher-level behaviors, e.g. focus on accomplishing certain tasks and planning routes to reach an eventual destination. More immediate in any system, however, are the basic concerns encountered in the process of moving from one desired location to another. Thus at the core of any navigational approach are the evaluation and choice of a movement option, which generally requires examining such base factors as collision avoidance with other pedestrians and/or stationary or moving objects, preference for less crowded areas, and maintenance of a desired speed and route/direction towards a goal. 1634

2 1.5x speed zone ( x) 1.0x speed zone ( x) 0.5x speed zone ( x) stop option pedestrian 85 o 60 o 60 o 40 o 40 o 25 o 15 o 5 o 5 o 15 o 25 o In our initial development work we examined the possibility of using rule-based navigation criteria in conjunction with navigation primarily via steering forces, as implemented using the OpenSteer framework [8]. For our purposes in simulating large-scale facilities with dense traffic (akin to findings by others, e.g. Pelechano et al. [1]), it was apparent that casebased logic was not sufficient in itself to describe the range of possible conditions in these environments, and basing motion on force-steered adjustments did not permit the range of options necessary for reasonable navigation. We felt it was important to represent a variety of pedestrian behaviors that varied based on individual choice, while allowing an acceptably realistic range of movement options that could be evaluated based on navigation factors, and a system whose criteria would correspond to real-world observations. We have opted to model pedestrian navigation as a series of choices made based on current environmental conditions. This model is based on the concepts developed by Antonini et al. [7] and Robin et al. [11], where a continuous set of combined speed and direction movement choices is deconstructed into a discrete set of radial and angular sector divisions. 85 o The basic idea of the Intermodal Simulator for the Analysis of Figure 1: Speed and direction alternatives. Pedestrian Traffic ISAPT navigational system is to provide each pedestrian agent with a series of alternative decisions they can choose from for their next step. There are 33 alternative movement decisions (plus stop) that represent potential combinations of direction and speed. The pedestrian s field of view (FOV) is split into 11 angular divisions whose sizes differ based on their position. Similarly, their travel range is divided into three zones centered at 0.5, 1.0 and 1.5 times their current speed. Thus for each sector there are three possible choices: deceleration (0.5), no change (1.0), or acceleration (1.5). Each of the resulting regions represents a continuous space (as shown in the diagram) for purposes of computing collision avoidance and other factors involving other pedestrians. The actual speed/direction outcome, however, will be one of 33 discrete combinations of direction and speed at a region s center (see Figure 1). A decision to stop may be arrived at based on evaluation of the 33 alternatives utility values. Given the availability of alternatives that a pedestrian can choose from, a utility function is employed to indicate the value of each alternative based on the pedestrian s current environmental situation. The environment they consider takes into account the location of stationary obstacles (building architecture, furniture, etc.), moving obstacles (carts, baggage, etc.) and other pedestrians (both standing and in motion). The utility function used in this project is composed of a weighted sum of individual utility factors that each focus on some aspect that influences the navigational decisions of a pedestrian. The overall utility function is: Utility = w 1 U KD + w 2 U TD + w 3 U FF + w 4 U PA + w 5 U OA (1) The individual utility factors include collision avoidance with pedestrians (PA) and obstacles (OA), free-flow acceleration based on crowd density (FF), maintenance of current travel direction (KD), and movement towards a goal (TD). An overall utility value is computed for each sector/speed combination corresponding to a decision alternative. At each simulation frame the maximum utility represents the best sector and speed that results in beneficial movement. Many of these utility factors are an outgrowth of the work by Antonini et al. [7] and Robin et al. [11]. Each of these factors is discussed in the sections below. 2.1 Keep Direction Utility Factor This factor is motivated by the desire for the pedestrian to maintain their current direction and minimize the need for angular displacements. To account for this, the utility measures the angular deviation between the direction of the possible alternative, d alt, and their current direction, d n. U KD = ( (d alt d n ) ) (2) 1635

3 The initial values employed for the equation parameters in this and the other factors to follow were estimated based on first principles to obtain a desired utility (disutility) based on values of the variables. For example, given a maximum possible deviation of π/2 radians to the left or right of the current direction, a coefficient of (1/1.5707) results in a utility value that ranges linearly from 0 to -1. A value of zero indicates no change in direction and a value of -1 represents a disutility measure due to the maximum possible deviation from the current direction. A study employing formal parameter estimation techniques using observational data is part of our future plans. 2.2 Toward Destination Utility Factor This factor is motivated by the observed behavior where once a trajectory change has been made, the pedestrian will tend to return to a trajectory in the direction of their original path. The desire of the pedestrian is to not deviate too far from their destination throughout their journey, but to make continual progress toward their goal. This factor supports the basic concept of efficiency of movement. Two terms are included in its utility equation. The first considers the distance between the destination and the future position for the corresponding alternative (d alt-dest ). The objective is to determine how much improvement in moving toward their destination the proposed alternative provides with respect to the distance between the current position and the destination (d cur-dest ). The second term relates how much the proposed direction of the alternative (d alt ) deviates from the direction to the destination (d dest ). U TD = (d cur-dest - d alt-dest ) - ( * (d dest d alt ) ) 0.5 (3) 2.3 Free Flow Utility Factor The free-flow speed for a pedestrian is that at which they normally travel when not faced with any imposing traffic. Their preferred speed takes into account the need for them to adjust their speed to fit the overall situation; this is a pedestrian s desired speed for the given density independent of the need to adjust speed in the face of impending collisions. The utility equation for this factor rewards alternatives where a pedestrian makes a speed adjustment that brings them closer to their preferred speed. However, one option for avoiding collisions is for a pedestrian to change their speed; such speed changes have been observed to be favored over making directional changes to avoid collision [7]. Since the benefit of accelerating or decelerating is a function of the pedestrian s current speed, four terms are considered in the factor equation. These include the pedestrian s current speed, the speed recommended by decision alternative being evaluated, and their preferred speed. The overall equation used for this factor is: U FF = 2 I bps I dec v s ( v d /v max ) + I aps I dec ( v s )( v d /v max ) I bps I acc v s ( v d /v max ) I aps I acc ( v s )( v d /v max )) 0.4 (4) where: v n = current speed v s = current speed - preferred speed v d = speed of alternative - preferred speed v m = max speed - current speed v max = maximum speed of pedestrian I acc = 1 if decision is to accelerate I dec = 1 if decision is to decelerate I bps = 1 if pedestrian s current speed is below the preferred speed I aps = 1 if pedestrian s current speed is equal to or above the preferred speed Each I indicator variable will take on a value of zero or one depending on whether the condition for that specific variable is met. This has the effect of there being at most one non-zero term in the equation at any one time. The first term in the equation becomes non-zero when an alternative prescribes that the pedestrian decelerate further when they are already traveling below their preferred speed. In like manner, the third provides an encouragement for the pedestrian to accelerate when they are below their preferred speed. The second term becomes active when a pedestrian is currently traveling above their preferred speed and the alternative prescribes they decelerate. This action to decelerate is favored since it will bring them closer to their preferred speed. Subtracting 0.05 from v s in this second term is required since I aps includes the case where current speed is equal to preferred speed. In that case everything comes out zero for all terms, yet we still want to penalize the choice to slow down since the pedestrian is at their preferred speed, and any change from that speed should be discouraged in proportion to the size of the 1636

4 current deviation. The last term addresses the situation where the pedestrian is currently traveling above their preferred speed and the alternative is suggesting that they accelerate. This action is discouraged with a reduction in utility that grows linearly. As with the second term, the ( v s +0.05) expression in this term is needed since the indicator variable, I aps, includes the case where current speed is equal to preferred speed. In this case v s is zero and we still want to penalize a decision to speed up, since the pedestrian is at their preferred speed and any change from that should be discouraged in proportion to the size of the change. 2.4 Pedestrian Avoidance Utility Factor This utility factor is a measure of the potential for a collision if the pedestrian were to choose a specific alternative. The utility equation is: U PA = -(0.25N c ) (6/(t c +0.02)) 0.5 (5) The first term takes into account the number of potential collisions, N c, that exist for a specified movement alternative, enacting a penalty for collision density vs. pedestrians given current directions of travel of all pedestrians and a time-distance limit. ISAPT incorporates a look-ahead time (currently 2.0 sec), where collisions are assessed for movement within a lower travel distance to an upper bounded range which includes this added time. This considers what may lie ahead for a certain movement choice, allowing pre-emptive action to change course away from future obstacles. The second term considers the time until the most imminent collision, t c, given the selected speed of the corresponding decision alternative. A constant of small magnitude is added to t c in case the distance to collision becomes near zero during evaluation. 2.5 Obstacle Avoidance Utility Factor The Obstacle Avoidance (OA) factor gauges the relative threat of colliding with a non-yielding obstacle in the near future for a given movement alternative. This measure accounts for the number of potential obstacles in a sector region, the distance to the most immediate obstacle, and the time to collision. The OA equation is: U OA = -1.5 N c Σ (0.31 / D(collision) i ) (6) The first term recognizes the number of potential collisions N c that can occur in a sector within the span of a given direction and speed alternative. The region ahead of the pedestrian that is explored for collisions is defined as the immediately navigable area during the next time step of the simulation (currently 0.05 sec) plus a look ahead time period setting of the pedestrian (currently 2.0 sec). Given the presence of potential collisions, the second term of the equation then accounts for the relative immediacy of each collision threat depending on how far away it is. Both terms are negative representing strict penalties, as physical obstacles do not typically yield or move in response to pedestrians, and the pedestrian would strongly prefer not to have their path intersect with them. 3.0 Probabilistic navigation While real-world pedestrians attempt to arrive at the best working solution, it won t necessarily be a perfect or nearoptimal one. Particularly where discrete movement models are used, probabilistic navigation [11, 12] may be incorporated to simulate pedestrians incomplete knowledge or attention in decision-making. Based on their knowledge of the system and assessment of movement factors, raw values are initially computed for a group of movement alternatives. The resulting set of decision factor values and overall utility are summarized vs. the range(s) of results and used to produce an initial likelihood distribution, which can be modified via function. A random number generator then determines which option is selected. The utility equation presented earlier provides a means for computing a utility value for each of the 33 alternatives representing a specific combination of direction and speed. Each of these alternatives is assumed to be independent of one another. It is not possible to select a direction and separately consider a speed, since each specific combination of speed and direction impacts utility differently. The pedestrian is modeled as selecting the alternative with the greatest utility, taking into account the pedestrian s surrounding at that moment. The actual utility values of the alternatives, U i, represent random variables and the probability of that alternative being the best is computed using a multinomial logit model (MNL) where the probability pedestrian n chooses alternative i from among the set of alternatives C n is: P n (i) = e β U in e β U jn j Cn where 0 P n (i) 1 and P n i = 1 i C n for all i C n (7) 1637

5 All alternative choices are assumed to have the same scale parameter, β, which has been set to 9 in our initial trials. Lowering the value of β raises the probabilities of additional alternatives being manifested, with the pedestrian switching from one decision to the next. Raising the value of β has the opposite impact, resulting in fewer alternatives with stronger probabilities, thus decreasing the likelihood that the pedestrian will select another option on successive iterations when there is little change in their environment. On each ISAPT iteration, an alternative is selected according to the computed probabilities. Compared to our previous rule-based navigation method [13], this approach permits a pedestrian to not always make the best decision, which mimics the reality of navigation as evidenced by the fact that it is not unusual to see pedestrians run into other pedestrians and obstacles on occasion. 4.0 Model Evaluation Given that the overall utility equation and individual factor equations include variables and weights, the system has the capacity to be fine-tuned to correspond more directly to certain types of crowds and/or environmental conditions observed at a certain place or time. However, in these tests initial values were chosen for the weights and parameters based on observations of the system under various scenarios. Figure 2: Two way flow in a corridor (50 pedestrians). To illustrate this navigational approach we set up two common corridor examples. In each example, the time Figure 3: Corridor with three columns (80 pedestrians). between arrivals of pedestrians follows an exponential distribution, and their start position along the width of the corridor follows a uniform distribution. An equal number of pedestrians are generated for each direction, and once a pedestrian exits the system it will be reintroduced again back at their lateral start location, with their initial position along the width of the corridor set akin to where they exited. The first example simulates a corridor 27 meters in length and 8m wide with a 4m doorway in the middle of the corridor. Even though traffic began with a random pattern, one can see in Figure 2, that in less than a minute, lanes start to form as the pedestrians begin to work out efficient traffic patterns. The second example, Figure 3, depicts a corridor of the same size, but with 3 columns located in the center. Again, starting with random placement, after roughly 735 seconds have passed, the pedestrians were observed to have organized themselves into two uniform lanes. The last example is to help illustrate the maneuvering that takes place. Approximately 3 seconds of simulation time is represented in the Figures 4a-d. Note that while the pedestrians alter their paths to avoid collision based on their own predictions, their assessments are not infallible (e.g. as in Figure 4b). 5.0 Conclusions As demonstrated in the examples above, using probabilistic navigation at the micro-level reproduces the macro-level behavior characteristics common to crowded traffic corridors. Compared with strict rule-based approaches, a probabilistic approach better emulates pedestrian behavior by permitting agents to make poor choices on occasion. Given the ability to navigate, the ISAPT system is working to simulate the tactical level planning that pedestrians employ when visiting an intermodal facility. The navigational component of ISAPT will be a lower level component of the overall system providing pedestrians the capability to navigation between points in a series defined at the tactical level for each pedestrian. 1638

6 Acknowledgements This project is sponsored by the U.S. Department of Transportation (Grant No. DTOS59-06-G-00041). References 1. Pelechano, N., Allbeck, J.M. and Badler, N.I., 2007, Controlling Individual Agents in High- Density Crowded Environments, Eurographics/ ACM SIGGRAPH Symposium on Computer Animation. 2. Kirchner, A., Namazi, A., Nishinari, K. and Schadschneider, A., 2003, Role of Conflicts in the Floor Field Cellular Automaton Model for Pedestrian Dynamics. 2nd International Conference on Pedestrians and Evacuation Dynamics, Helbing, D., Farkas, I., and Vicsek, T., 2000, Simulating Dynamical Features of Escape Panic, Nature. 407: Williams, S, and Huang, D., 2006, A Group Force Mobility Model The 9th Communications and Networking Simulation Symposium. 5. Blue, V.J., and Adler, J.L., 2001, Cellular automata microsimulation for modeling bidirectional pedestrian walkways, Trans. Res. Part B: Methodological, 35(3), Bierlaire, M., Antonini, G. and Weber, M. 2003, Behavioral dynamics for pedestrians, In Moving through nets: the physical and social dimensions of travel, Elsevier. 7. Antonini, G., Bierlaire, M., and Weber, M., (a) Time = 20.3 sec. (b) Time = 21.3 sec. (c) Time = 22.4 sec. (d) Time = 23.4 sec Figure 4: Time-lapsed navigational example. 2006, "Discrete choice models of pedestrian walking behavior". Transportation Research Part B: Methodological, 40(8): Reynolds, C.W. OpenSteer website, Robertson, Barbara, 2008, Keys To the Kingdom. Computer Graphics World, 31(6), June. 10. Loscos, C., Marchel, D., and Meyer, A., 2003, Intuitive crowd behaviour in dense urban environments using local laws, Proc. Theory and Practice of Computer Graphics, IEEE Computer Society, Robin, T., Antonini, G., Bierlaire, M., and Cruz, J., 2008, Specification, estimation and validation of a pedestrian walking behavior model, Transportation Research Part B: Methodological, 43(1), Hoogendoorn, S.P. and Bovy, P.H.L., 2005, Pedestrian Travel Behavior Modeling. Networks and Spatial Economics, 5(2), Usher, J.M. & Strawderman, L., 2008, "Emergent Crowd Behavior from the Microsimulation of Individual Pedestrians," Proc. of the 2008 Industrial Engineering Research Conference, Vancouver, BC, May, CD-ROM. 1639