This paper discusses several improvements to the computational model of MakkSim, with the aim of allowing simulation of aged people as well as persons with mobility impairments. In particular, a method for modelling heterogeneity in speed is discussed and two special objects of the environment (i.e., stairs and seats), have been de ned; in addition, a proposal for modelling the presence of a caretaker is described as a particular type of group of pedestrians. Finally, the paper presents a way of computing social costs implied by the environment taking into account the characteristics of pedestrians moving throughout the related facilities. The overall objective is to achieve a system usable for the evaluation of the usability and accessibility of planned environments and facilities by means of simulation, by also taking into account this category of people.
Urbanization is currently one of the most signi cant tendencies of the world
population: it has been forecast that by 2025 the 58% of the global population
will live in cities and urban agglomerates [
The concept of Age-Friendly city, de ned by the World Health Organization
[
Nowadays, the usage of computer models for simulating the pedestrian dynamics can help designers to perform a deep and dynamical analysis of their projects, allowing them to populate and simulate environments by con guring the so-called what-if scenarios. These tests support, therefore, the improving of the overall security and perceived comfort of the plans. Given the importance of this kind of analysis, several commercial simulators have been recently developed and can be currently found on the market1. These tools provide simulation frameworks whereby it is possible to con gure su ciently heterogeneous populations of pedestrians (e.g. with a di erent walking speed) and, in some cases, even to simulate the presence of groups of people (modelled with an attractive force among members, although this feature is generally not systematically documented and evaluated). However, although some signi cant results have been achieved, the overall issue of simulating large and heterogeneous crowds of pedestrians still presents open challenges, since the crowd is a complex system and all of these mathematical/computational models can be improved for obtaining more microscopic and realistic simulations.
The work described in this paper is focussed on the realization of a
computational model speci cally tailored to simulate the presence of pedestrians with
restricted mobility (elderly people as well as persons with physical impairments),
basically characterized by a lower walking speed [
Finally, in fact, the paper presents a proposal for the calculation of the socalled \social costs"2 which can emerge by the actual usage of the environment by pedestrians. The presence of not comfortable elements (e.g. stairs) or situations (e.g. waiting in a queue) in the navigable space, in fact, can increase stress and fatigue of people and especially fragile ones. In addition to this, this kind of facility and the need to employ it can lead to falls that can have signi cant negative e ects for aged people and, therefore, imply healthcare costs sustained by the society (in a di erent way according to local policies) therapies.
The paper will, rst of all, introduce the modelling approach to show how it can be applied to simulate the presence of particular categories of pedestrians like elderlies and people with mobility impairments, sometimes moving with an accompanying person. Then, a proposal for a comprehensive way to evaluate social costs implied by the plan of an environment will be introduced and discussed. 1 see http://www.evacmod.net/?q=node/5 for a signi cant although not necessarily comprehensive list of simulation platforms. 2 Currently, several de nitions of the social costs can be found in the literature and they are often built-in commercial simulation platforms, but an extension of this concept with more speci c variables for this category of people can be very useful for the designing of more Age-Friendly facilities.
Computational Model: Baseline and Improvements
This section aims to explain the improvements introduced in MakkSim, an
agentbased model employing a discrete environment and the oor eld approach [
Baseline
Baseline model is described through discussion of its main features: environment,
time, agents and social interactions.
2.1.1 Environment The environment is modelled in a discrete way and
represented as a grid of squared cells with 40 cm2 size (according to the average area
occupied by a pedestrian [
The information related to the scenario4 of the simulation is represented by means of spatial markers, special sets of cells that describe relevant elements in the environment. The model baseline contains three kinds of spatial markers: (i) start areas, that are, generation points of agents in the scenario; (ii) destination areas, possible targets of the pedestrians in the environment; (iii) obstacles.
A oor eld -like approach[
One path eld for each destination point is generated in each scenario; { obstacles eld, that indicates for every cell the distance from neighbour obstacles or walls (static). Only one obstacles eld is generated in each simulation scenario; 3 in which the density is higher than 6:25 m2 (i.e. the maximum density reachable by our discetisation). 4 It represents both the structure of the environment and all the information required for the realization of a speci c simulation, such as crowd management demands (pedestrians generation pro le, origin-destination matrices) and spatial constraints. { density eld, that indicates for each cell the pedestrian density in the surroundings at the current time-step (dynamic). Like the previous one, the density eld is unique for each scenario.
Chessboard metric with p2 variation over corners [
Regarding the update mechanism, three di erent strategies are usually
applied in this context [
In the baseline model we chose to adopt the shu ed sequential strategy, updating the list of agents using a dynamic list of priority that is randomly generated every step. 2.1.3 Pedestrians and Movement Formally, our agents are de ned by the following triple:
with their own numerical identi er, their group (if any) and their internal state, that de nes the current position of the agent, the previous movement and the nal destination, associated to the relative path eld.
Before describing agent behavioural speci cation, it is necessary to introduce the formal representation of the nature and structure of the groups they can belong to, since this is an in uential factor for movement decisions.
Social Interactions { In order to deeper model social relationships, two
kinds of groups have been de ned in the model: the simple group, that indicates
a family or a restricted group of friends, or any other small group in which there
are a strong and simply recognizable cohesion; the structured group, generally
a big one (e.g. a group of team supporters or a touristic group), that shows a
slight cohesion and a natural fragmentation into subgroups, in which the cohesion
gets stronger. In particular, between members of a simple group like a family
it is possible to identify an evident tendency to stay quite close, in order to
guarantee the possibility to perform interactions by means of verbal or
nonverbal communication [
In particular, if the group is simple, it will have an empty set of subgroups, otherwise it will not contain any direct references to pedestrians inside it, which will be stored in the respective leafs of its three structure. Di erences on the modelled behavioural mechanism in simple/structured groups will be analysed in the following section, with the description of the utility function.
Agent Behaviour { In order to perform the agent behaviour, its life-cycle has been de ned on four steps: perception, utility calculation, action choice and movement. The perception step provides all the information needed for choosing its destination cell to the agent. In particular, if an agent does not belong to a group (hereafter individual ), in this phase it will only extract values from the oor elds, while in the other case it will perceive also the positions of the other group members within a con gurable distance, for the calculation of the cohesion parameter. The choice of each action is based on an utility value, that is assigned to every possible movement according to the following function: U (c) = gG(c) + obOb(c) + sS(c) + cC(c) + iI(c) + dD(c) + ovOv(c) d
Function U (c) takes into account the behavioural component considered relevant for pedestrian movement. For each function has been introduced a coe cient for its calibration. The purpose of d is to constrain the diagonal movements, in which the agents cover a greater distance (0:4 p2 instead of 0.4) and assume an higher speed than the non-diagonal ones.
The rst three functions combine information derived by local oor elds and they model the basic factors considered in the pedestrian behaviour: goal attraction (i), geometric (ii) and social repulsion (iii). The fourth and fth elements aggregate the perceived positions of members of agent group, both simple (iv) and structured (v), to calculate the level of attractiveness of each neighbour cell, relating to cohesion phenomenon. Moreover, two factors represent preferences with respect to movement, helping the model to reproduce more realistic simulations both in qualitative and quantitative perspective: (vi) adds a bonus to the utility of the cell next to the agent according to his/her previous direction, while (vii) describes the overlapping mechanism, a method used to allow our model the possibility to treat high density situations, allowing two pedestrians temporarily occupying the same cell at the same step.
As previously explained, the main di erence between simple and structured groups resides in the cohesion intensity, which is signi cantly stronger in the simple ones. Functions C(c) and I(c) have been de ned to correctly model this di erence. Nonetheless, various preliminary tests on benchmark scenarios show us that, used singularly, function C(c) is not able to reproduce realistic simulations. Human behaviour, in fact, is very complex and people can react di erently even in simple situation, for example by allowing temporary fragmentation of simple groups in front of several constraints (obstacles or opposite ows). Acting statically on the calibration weight, it is not possible to con gure this dynamic behaviour: with a small cohesion parameter several permanent fragmentations have been reproduced, while with an increase of it we obtained no group dispersions, but also an excessive and unrealistic compactness of them.
In order to face this issue, another function has been introduced in the model, with the aim to balance the calibration weight of the three attractive behavioural elements, depending from the fragmentation level of simple groups: 8 1 k + ( 2 k DispBalance) Balance(k) = >< 133 k + ( 23 k (1
3 >:k
DispBalance)) if k = kg _ k = ki otherwise if k = kc DispBalance = tanh
Disp(Group) ;
Disp(Group) =
Area(Group) jGroupj where ki, kg and kc are the weighted parameters of U (c), is the calibration parameter of this mechanism and Area(Group) calculates the area of the convex hull de ned using positions of the group members. As we will see in next section, a dynamic and adaptive behaviour of groups has been obtained with this mechanism, which relaxes the cohesion if members are su ciently compact and intensi es it with the growing of dispersion.
After the utility evaluation for all the cells in the neighbourhood, the choice of action is stochastic, with the probability to move in each cell c as (N is the normalization factor):
P (c) = N eU(c)
On the basis of P (c), agents move in the resulted cell according to their set of possible actions, de ned as list of the eight possible movements in the Moore neighbourhood, plus the action to keep the position (indicated as X): A = fN W; N; N E; W; X; E; SW; S; SEg. 2.2
Improvements This section will focus on the improvements proposed for the model, in order to obtain a simulated pedestrian behaviour similar to one of a person with restricted mobility. In particular, several elements have been taken into consideration: his/her lower speed, the possible presence of a caretaker or an accompanying person, de ning a special case of group (i.e. the two persons will walk strictly together, maintaining a constant speed) and other details of their behaviour like the general need of the handrail in the stairs. As it will be described in the following, several of them have already been developed in the simulator and are under validation, while other ones are still in the developing phase.
Each new feature will be discussed with the next subsections, starting from a
preliminary yet signi cant work about the agents update strategy and following
with the ones which directly regard the aims of this work.
2.2.1 Parallel update A preliminary work to improve the expressiveness of
the model has regarded the development of the parallel update strategy. This
mechanism, in fact, describes a parallel choice of movement by the agents and
leads to the generation of con icts, considered as another important aspect of
the crowd dynamics[
With the new update strategy, the agents life-cycle does not change, but the movement execution becomes dependent from the additional con ict resolution rules of the model. The model overall activity ow, in fact, has been modi ed with the following three step procedure: { update of choices and con icts detection for each agent of the simulation; { con icts resolution, that is the resolution of the detected con icts between agent intentions, solved employing new rules in the model; { agents movement, that is the update of agent positions exploiting the previous con icts resolution, and eld update, that is the recalculation of the density eld according to the new positions of the agents.
Since it is not the principal object of this paper, a thorough description of the rules introduced for the management and resolution of con icts will not be performed. However, it is mandatory to explain that our rules describe a stochastic resolution with three possibilities: only one agent moves, two agents move (overlapping) or no-one moves (friction). Two parameters have been introduced for managing the calibration of this mechanism. In addition, when a con ict is arisen between more than two pedestrians, the situation is simpli ed by extracting two agents, which will be able to move according to con ict resolution rules, and imposing non-movement to the others. 2.2.2 Heterogeneous Speed Pro les This feature represents the experimental and innovative part of this work since, in the current literature, discrete models for pedestrian dynamics generally assume only one speed pro le for all the population. This is also considered one of the main criticism to this approach.
We evaluated several ways to improve the expressiveness of our
computational model for this purpose, some of them taken by the literature[
The rst two methods can be e ective (especially the second one), but they both lead to complications and increases of computational costs for managing micro-interactions and con icts, as shown in Fig. 1, therefore they have not been chosen as a solution. The fourth method also su ers for the increasing of computational times of the simulations, which increases proportionally to the ratio So=Sn, with So and Sn respectively equal to the old and new size of cells (e.g. if the size is half-divided, for performing the same space agents will need a number of update cycles at least doubled). These reasons led to undertake the third method, since it does not a ect computational complexity in a relevant way and it is the simplest one to develop.
The computational model has been modi ed in several parts. Each agent has a new parameter Speedd in its State, describing its desired speed. For the overall simulation scenario, a parameter Speedm is introduced for indicating the maximum speed of the pedestrian in the simulation (describing, therefore, the Algorithm 1 Life-cycle update with heterogeneous speed if Random() = then updateP osition()
1 end if
1 if == 0 then
( ; ) = F rac( ) end if assumed time scale). In order to obtain the desired speed of each pedestrian during the simulation, the agent life-cycle is then activated according to the probability = SSppeeeeddmd .
By using this method, the speed pro le of each pedestrian is modelled in a fully probabilistic way and, in a su ciently high number of step, their e ective speed will be equal to the desired one. But it must be noted that in a lot of cases the speed has to be rendered in a small window of time and in a small portion of space (think about speed decreasing on stairs). In order to overtake this issue, we chose to manage this variable as an extraction without remission, updating the probability with moved or not moved events. The mechanism can be formalized as the following.
{ Let F rac(r) : R ! N2 be a function which returns the minimal couple (i; j) : ji = r. { Let Random be a pseudo-random number generator. { Given the probability to activate the life-cycle of an arbitrary agent, according to its own desired speed and the maximum speed con gured for the simulation scenario. Given ( ; ) be the result of F rac( ), the update procedure for each agent is described by the pseudo-code of Alg. 1. The method updateChoices() describes the choice of movement by the agent5.
This mechanism can be considered consistently better than using only the
probability , because it allows the synchronization between the e ective speed
of an arbitrary agent and its desired one in maximum Speedm 10 step, where
describes the number of decimal positions of Speedd (e.g. if the desired speed is
xed at 1.3 m/sec and the maximum one at 2.0 m/sec, the resulting F rac( ) =
1230 , therefore the agent will e ectively simulate its desired speed after 20 steps).
Results of the application of this approach in terms of the achieved fundamental
diagram6, have shown a good reproduction of average ows at di erent densities,
5 Remember that with the parallel update movement executions are performed
aposteriori of the con ict resolution phase.
6 It is used for analysing average pedestrian ows and velocities, at di erent values
of densities. Since a su cient amount of data can be found in the literature[
Firstly, the stairs have been considered for this purpose, whose presence in the environment has been modelled by means of an additional type of spatial marker (Stairs). In particular, in order to recognize and di erentiate the direction of movement (i.e., if agents are going upstairs or downstairs), the extremes of the marker on the top and bottom sides are signed in a special way, as shown in Fig. 2 (a).
According to empirical data from the literature [15], the agents inside the Stairs marker x their desired speed to 0.5 (0.4 for elderly people) m/sec, if they are going up, or to 0.7 (0.6 for elderly people) m/sec if they are going downstairs, until they reach the end of it.
For modelling the usage of the handrail by elderly persons, we used the obstacles eld in the reverse way inside the marker, in order to get this kind of agents attracted towards the walls surrounding the stairs. Therefore, agents of type \elderly" are changing the sign of their ob during the usage of stairs. E ects are qualitatively displayed in Fig. 2 (b).
The second special object that has been considered with this work are the seats, whose introduction described more additional elements for the model: stairs can be placed in the environment by means of another spatial marker (Seats) and, in order to design its possible perception by nearby agents, another dynamical oor eld has to be introduced. This eld is quite similar to the density eld, with di erences residing in values update: their values are spread within a xed radius of distance from each Seats cell, which de nes exactly one seat, but only when this ones are free. If an agent temporary occupies one of them, the others must not be able to perceive it.
However, this feature has not yet been completely developed in the simulator, since it is necessary to de ne also the particular behaviour of the pedestrian, that is, when he/she starts feeling the need of a seat, either in a normal situation or in a waiting situation, where additional improvements are needed even in the environment section of the model. This will regard part of the future work. 2.2.4 Presence of a Caretaker Another relevant aspect which can characterize a person with moving disabilities can be the presence of an accompanying person, which drives him/her through the environment. This situation clearly describes a group of type simple, but also a special case of it, since they will walk strictly together (see Fig.3), like they were a single entity.
In order to model this phenomenon, a rst proposal meant to use the function Balance (which adapts the behaviour of group members regarding their dispersion, see Sec.2.1.3) particularly and in a very strict way for this special group, for not allowing fragmentation. The parameter of DispBalance function, therefore, was xed to values lower than 1, in order to explore how strong the e ects of this function can be. Results of this tests have shown that this method is not suitable for a good simulation of the interested phenomenon: since the probability to have a distance greater to one cell cannot be set to 0 with this mechanism, an high cohesion of the simulated group has been achieved in this way, but in several situations the simulated couple has fragmented itself anyway, even if in a very light way (around 1 meter of distance).
Since the walking path of the two persons in analysis will be practically the same, because the aged person in this situation cannot walk by him/herself, an alternative and more e ective method can be to abstract this situation by representing the couple with a single, special, agent whose shape occupies 2 cells. Naturally, this modelling assumption leads to complications in the computational model, since new rules for managing its behaviour and its interactions with other entities. A rst proposal for this method is described by the following rules: 1. the agent chooses its preferred movement according to the utility function, evaluating all the cells surrounding its shape7; 2. according to the chosen direction: (a) if the two cells next to the agent position are free, then update its position in them; (b) if only one cell is free, then move only the most fair cell of the agent shape (this will cause a rotation).
Since this method is still under development, no tests have suggested if the simulated behaviour of the group is acceptable. Therefore, even this aspect represents part of the future works. 3
The model features presented until now represent a viable approach for the simulation of the behaviour of aged people, whose presence in our society is becoming more and more important. The basic simulation outputs can already give information to decision makers about what is the perceived comfort of the planned environment: space utilisation and cumulative mean density maps are well suited for identifying critical zones of the scenario, while statistics about average travelling times and ows are able to describe their impact in its overall security, by also granting an estimation of the evacuation times.
Notwithstanding, the de nition of additional microscopic outputs in the simulator (e.g. length of stair travelled, waiting times in queues) can be useful for understanding the usability of the environment from the perspective of its \users". These additional indicators must actually be considered in a di erent way for young, healthy adults and elderlies or persons with restricted mobility or other disabilities. The de nition of parameters for estimating probabilities of injuries on one hand, like falls or faints due to stress and excessive fatigue, and for obtaining an indicator of the overall perceived comfort of the environment on the other, can be exploited for the calculation of these social costs. By analysing these costs, users and decision makers will be able to act on the planned environment in order to nd solutions which can lead to either less expenses, more bene ts, or both. For giving an example, the application of heating systems for the snow removal in side-walks is able to maintain the street clean and to allow 7 This set of cells is calculated by the union of Moore neighbourhood of each cell of its shape. persons, especially aged ones, to easily use them. In this way, possible falls are prevented, directly decreasing social expenses (for public healthcare system or for families); in addition, the accessibility of the city areas is improved, allowing aged people to normally move themselves inside the city and to maintain an active status, which implies bene ts also for the local economy (in addition to being a useful action in the direction of an implementation of active ageing policies, which in turn also aim at reducing social costs in the long run).
The overall objective of this section is to sketch a general indicator that can describe these average expenses paid by the society for the treatment of injuries happened inside pedestrian facilities, or to pedestrian in general, but also the implications of uncomfortable and non-accessible environments on the reduced activities of elderlies, which might be deciding not to go to certain places to carry out their tasks because of perceived unpleasant conditions in the environment. Two elements are composing the indicator, both dependent from the overall comfortability of the space: bad settings or situations leads to accidents, directly causing social expenses, while perceived discomfort during the navigation of the setting can lead fragile types of people to avoid its usage. The idea is to calculate them with a two steps method, rstly calculating the di erent social costs and then grouping them for obtaining an overall index of the space comfort, which can be used for estimating the lack of incomes due to a reduction of potential customers of the space.
Regarding the rst step, that is, social cost estimation, we must consider that accidents can naturally occur with di erent probabilities regarding di erent situations (walking in stairs, walking outside, etc.), di erent con gurations (e.g. temperature) and di erent types of pedestrian (i.e., younger ones will have a lower probability). Provided that statistics about these accidents were available8, the overall cost can be estimated by means of microscopic simulation with our model.
Formally, the method for the social costs analysis must consider the expenses for each type of agent and single situation/activity performed in the scenario navigation. This concept can be described as the following:
SocialCosts = X X Ci(a);
S = fC0;
; Cng a2P Ci2S with S the set of all situations considered for the social costs calculation. Each of these ones are described by an additional function able to quantify the single element:
Ci(a) = a;i AvgCosta;i 8 It is mandatory to say that studies and data actually present in the literature refer only to particular cases: for example, in [16] useful data about accidents in stairs are shown. Actually, commercial tools for pedestrian simulation are trying to ll this gap by means of several local standards provided, for instance, by transportation authorities (e.g. the London Underground Standard), but their scienti c validity is not always demonstrated. Additional studies and surveys are, therefore, needed in order to give a plausible and precise estimation of social costs.
where a;i describes the quantity, in the respective measurement unit, of the condition i (e.g. time passed in the queue) in the simulation for the agent a. AvgCosta;i represents the average cost estimation9 generated per each unit and for the category the agent a belongs to.
By means of this function, each pedestrian travel can be evaluated regarding elementary situations given by interesting components of the environment: stairs, elements which usually generate queues (e.g. ticket machines), parts of the oor which are usually wet and so on. In particular, for evaluating parts of the environment a;i will represent the covered space, while for situations like queues the measurement unit will be the time. On the other hand, complex situations can be evaluated by splitting them into simpler ones, until they reach an elementary level: for instance, waiting areas where the presence of a su cient number of seats can be signi cant, it is necessary to divide the costs evaluation regarding the time passed stood up and seated down. In addition, an extension of the analysis to crowded zones in the scenario can be very important, since they are uncomfortable and can also cause faints, especially to elderly people. A thorough costs analysis has to consider the time passed in them, weighted by means of average local density in that part of the environment.
In order to estimate the bene t of reducing social costs, we must have an approximate evaluation of missing economical activities due to avoiding of environment usage by people: a more macroscopic analysis has to be done, once calculated an index of comfort of the facility by means of the obtained social costs. Similarly to the Level of Services calculation[15], the social costs can be grouped into di erent classes for having an idea of the overall comfort and walkability of the environment. Comfort is a fuzzy concept and de ning an indicator could be not simple task, but studies from urban planners like White [17] can be exploited for this purpose. In addition, the execution of surveys and interviews to customers of di erent environments, in order to obtain additional data about comfort of their components, can help the nal de nition of a comfort index. This parameter can be, then, used with statistics about the local population composition, regarding to age and walking impairments, to evaluate the possible lack of incomes regarding this aspect. 4
This paper has discussed several improvements to the computational model of MakkSim, with the aim of allowing simulation of aged people as well as persons with mobility impairments. The overall objective is to make the system usable for evaluating planned environment by taking them into consideration, in order to understand also their fatigue assumed through the navigation of the environment, knowing if something could be done to make it more age-friendly.
In addition, an experimental method for the estimation of social costs has been proposed. This kind of analysis allows to understand the entity of the costs 9 It must refer to statistics about the average number of accidents, with respect to the measurement unit of a;i and their average cost. for the healthcare of accidents like falls or faints which can be generated by the con guration the environment, as well as lack of incomes derived by people who are not using environment for excessive discomfort.
Future works are aimed, on one hand, at improving the expressiveness of the computational model in order to consider more elements of the behaviour and more elements of the environment. On the other one, a precise and feasible estimation of social costs can be done only once having enough statistical data about accidents and perceived discomfort in the environmental setting, therefore additional studies must be performed in this direction. This line of work must also try to integrate existing approaches and results of the so called walkability analysis of the built environment [18]. 15. Fruin, J.J.: Pedestrian planning and design. Metropolitan Association of Urban
Designers and Environmental Planners (1971) 16. Health Education Authority: Older People and Accidents. (1999) 17. Whyte, W., Underhill, P.: City: Rediscovering the Center. University of Pennsylvania Press, Incorporated (2009) 18. Gori, S., Petrelli, M.N.M.: Walkability indicators for pedestrian friendly design walkability indicators for pedestrian friendly design walkability indicators for pedestrian friendly design. In: Proceedings of the 93rd Transportation Research Board annual meeting. (2014)