Available online at Physics of Life Reviews10(2013)168–190/locate/plrevReviewThe emergence of design in pedestrian dynamics:Locomotion, self-organization,walking paths and constructal lawAntonio F.Miguel a,b,∗a Department of Physics,Evora University,Rua Romão Ramalho59,7000-671Evora,Portugalb Geophysics Center of Evora(CGE),PO Box84,7002-554Evora,PortugalReceived25February2013;accepted7March2013Available online26March2013Communicated by J.FontanariAbstractGait is inherent to human life and hence its importance is often overlooked.But walking remains the most basic form of transportation and almost all journeys begin and end with a walk,regardless of the modes used in-between.Gaining a good under-standing of pedestrian’s dynamics is thus a crucial step in meeting the mobility and accessibility needs of people by providing safe and quick walkingflows.This paper presents a critical and integrative review of research on pedestrian’s dynamics and associated topics.The review focuses on comprehensive theories and models,with an emphasis on the advances made possible by the application of the con-structal law.Constructal law points out that the emergence and evolution of design in pedestrian dynamics is analogous to that of animateflow systems.Most importantly,it also highlights that the basic features of pedestrian dynamics and supportive walking infrastructures can be optimally envisaged with the help of a few fundamental physics laws.©2013Elsevier B.V.All rights reserved.Keywords:Human gait;Walk–run transition speed;Fundamental pedestrian diagram;Self-organization;Walking paths;Constructal law1.IntroductionAlthough there are over250species of primates,only one moves primarily on two legs.Bipedalism developed approximately4–5million years ago and it was thefirst major adaptation that separated hominids from other primates [1–4].Human anatomy is built on a body planned for bipedal locomotion.The two distinct gait modes of humans are walking and running,which make use of strikingly different mechanics and energetics.Currently,the fastest human being on Earth,Usain Bolt,is able to run at a speed greater than10.44m/s.Regular pedestrians,however,tend to walk rather than run,and do this mainly at a comfort walking speed of around1.3m/s[5,6].The different speeds at which humans walk or run translate themselves into varying energy costs.Keeping energy spending low is highly desirable and leads to greater mobility,given that humans must carry their energy supply with them[7,8].*Correspondence to:Department of Physics,Evora University,Rua Romão Ramalho59,7000-671Evora,Portugal.Tel.:+351266745372;fax: +351266745394.E-mail address:afm@uevora.pt.1571-0645/$–see front matter©2013Elsevier B.V.All rights reserved./10.1016/j.plrev.2013.03.007A.F.Miguel/Physics of Life Reviews10(2013)168–190169Since the middle of last century,the study of the pedestrian dynamics has become an active subject of research in science[5–8].Pedestrian dynamics has theoretical importance and a multitude of practical applications.Primary questions such as why and how human walks,or how crowd dynamics occurs based on individual interactions,along with some more practical aspects like the design of pedestrian optimized facilities,have puzzled many researchers.In-situ observational studies and time-lapsefilms,together with the modeling have revealed important quantitative details and have led to a unique understanding of pedestrian dynamics with important practical implications[5,6].Despite the considerable progress in understanding various aspects of pedestrian dynamics,there has been no systematic attempt to integrate what remain disconnected,because of the wide scope of the subject.Fitting the“pieces”together will contributes decisively to a unified view of the topic.This paper starts with a brief description of basic mechanics of human locomotion and its energetic cost.Factor such as gender,age and environment are important in the walking speed choice.Pedestrians react to other pedestrians and obstacles,and can only move freely when there is enough free space in front of them.Then,the relation between spatial headway(distance to the predecessor)and speed of pedestrians is analyzed.Empirical data(i.e.,the fundamental diagram)available in the literature is presented and models of pedestrian interactions are introduced.The calibration and the validation of the models using empirical data are also discussed.Finally,I turn the attention to the Constructal law which constitutes the thermodynamics law of nonequilibrium (flow)systems with configuration.For these systems,this law provides new physical insights,and a unified view on domains apparently foreign to each other.The Constructal law is introduced and,then,the Constructal view of pedestrian dynamics is presented and discussed.Furthermore,the design of improved pedestrian facilities,using the Constructal law,is also addressed.The objective of this Constructal section is to forge a more structured and unified view that engage a set of issues which are addressed under different frameworks and disciplinary spaces.2.Modern human anatomy and locomotion:mechanics,energetics and gait speedBipedal locomotion is the sole form of locomotion in all healthy modern humans and it sets them apart from all other living primates[1–4].A large number of anatomical features are functionally related to this type of locomotion. The human pelvis is pronouncedly bowl-shaped,with an adapted musculature which allows the thighs to be angled in, the lower limbs are lengthened and have an enlarged joint surface area to properly support body weight,the vertebral column is shortened and S-shaped,for rigidity and balance,lining up head and trunk vertically above the feet,the hole through which the column cord enters the skull is situated near the center of the cranium which allows the head to balance easily atop the column,the feet have an arched shape with the enlarged great toe brought in line with the other toes,to absorb shock and a more efficient walk,are among other important evolutionary features[9,10].From an evolutionary perspective,the global impression of the human anatomy is that of a bio-cylinder shape[7,10].Although bipedalism is not the most stable and fast way of locomotion,it must clearly represent an evolutionary advantage to humans,because we still walk on two legs.Some of the advantages associated to bipedalism include the ability to carry things over longer distances,the freeing of arms and hands for other tasks(foraging and protection,for instance),the acquisition of improved long-distance perception and the improvement of body thermoregulation[1,9, 11–13].It is not uncommon to see apes such as chimpanzees walking on two legs in order to carry things.Octopuses have been also documented walking bipedally in order to camouflage themselves from predators[14].Placing six limbs close to their head,the octopus assumes the shape of a drifting plant,and then uses its two remaining limbs to walk away from predators.Another possible explanation for bipedalism is its lower energetic cost compared to other forms of locomotion. In a study by Taylor and Rowntree[15],chimpanzees(Pan troglodytes)and capuchin monkeys(Cebus capucinus) were trained to run on a treadmill on either two or four legs.Results showed that the energy spent by the animals on this exercise was fairly independent of the mode of locomotion they adopted.Fedak and Seeherman[16]also reported negligible differences of the scaling of energy requirements for locomotion between bipedal and quadrupedal behavior. On the other hand,Alexander[7]stated that gait of humans is distinct from the occasional bipedalism of apes because the patterns of force exerted on the ground are different,due to biomechanical differences in anatomy mentioned above.Therefore,the trajectory of the center of mass is completely different for gaits in humans and other animals [7,17].Sockol et al.[18]reported that the energy expended in human walking is approximately75%less than that expend in both quadrupedal and bipedal walking in chimpanzees.The study of Carrier et al.[8]also supported the170 A.F.Miguel/Physics of Life Reviews10(2013)168–190idea that humans are more economical when they walk.Besides,Alexander[7]stated that while the walk spends less energy than the gait pattern of quadrupedal mammals of the same body mass,running actually spends more.2.1.Forms of human gait:walking and runningPeople move across the Earth’s surface(are not be attached to one site),and this capability is called locomotion. Walking and running are the two most common forms of human gait.The increase of speed intuitively triggers the switch from a walk to a run.Walking gait is characterized by two distinct phases[19]:the stance phase,when the leg is on the ground,and the swing phase,when it is off the ground.The stance phase begins with heel-strike,as the foot strikes the ground.Most of the time just one foot is on the ground,with brief periods where both feet come in contact with the walking surface.The body vaults up and over each stiff leg in an arc.This gait pattern can be well approximated by a linear“inverted pendulum”[7,17,19,20].An ideal inverted pendulum system exhibits an optimal conversion of gravitational potential energy to kinetic energy.The energy converted by such an“inverted pendulum”mechanism can reduce the mechanical work required from the muscular system by up to70%[17].Running,in contrast,involves longer aerial phases,in which neither foot is in contact with the ground.In this case,the pendulum mechanism that characterizes gait switches to a spring-mass system,where bent legs bounce up between aerial phases [20].As feet hit the ground in a run,the leg springs compress,as a result of jointflexion,and the mass moves downwards.At the middle of the stance phase,the leg is maximally compressed,and the mass reaches its lowest point.The plantigrade foot posture of a runner is also very distinct of that of a walker[21],with its center of mass reaching the lowest point at the middle of the stance phase.In contrast,the center of mass of walkers is at its highest point precisely at the middle of the stance phase.Moreover,the stance limb sweeps through a larger angle during walking(∼30◦)than during running(∼19◦)[17].Finally,running involves a decrease of approximately35%in time of foot–ground contact and an increase of the peak ground reaction force by about50%,when compared to walking[17].Humans prefer changing their gait from walking to running and from running to walking at increasing and decreas-ing speeds.This switch occurs at a Froude number of0.5[22,23].For adults,it corresponds to a walking speed of about2m/s[24].Both walking and running are related to several demands that must be processed simultaneously:the propulsion of the body in the horizontal plane,the maintenance of a stable equilibrium(coordination between posture and move-ment),and a continuous and instantaneous adaptation to environmental requirements[25,26].This demands the use of the central and peripheral nervous system,the regulation of the skeletal segments and the contraction of mus-cles[27].Gait speed is therefore tightly linked to neurological development/degeneration,balance control,changes in limb length,and maturation.Studies performed by several authors[28,29]suggest that the gait speed is a quick, inexpensive,reliable measure of functional capacity.Recent studies also reported a significant association between walking speed and life span in seniors[30,31].Others[32],finally,presented evidence that successive step durations during walking actually present a typical structure over time,one that is characterized by a long-range dependence (i.e.,a scaling relationship).It is suggested that this dependence plays an important role in the adaptability and the flexibility of locomotion,since it disappears in individuals with neurodegenerative pathologies.2.2.Least energy-consuming gaitHumans carry their energy supply with them,and they are limited by the rate of production of metabolic energy in their bodies,that depends of the capacity of the cardiovascular system to absorb and distribute oxygen throughout the body.The slower energy is consumed,the less energy must be carried and greater mobility is achieved.A number of experimental studies have shown that the average metabolic energy consumption per unit distance traveled,E x,may be expressed by[33]E x=0.533M bu+0.3M b u(1)Here M b is the body mass and u is the gait speed.This equation has a minimum at a speed of∼1.33m/s,which is within the walking speed range(i.e.,below the average transition speed of∼2m/s between walking and run-ning[24]).This average speed is recurrently mentioned in the literature as the comfortable walking speed or desired walking speed[5,6,34].A.F.Miguel/Physics of Life Reviews10(2013)168–190171Fig.1.The comfortable walking speed versus decade of age:!female,"male(experimental data obtained from Bohannon[35]).Fig.2.The comfortable walking speed versus the body mass:!female,"male(experimental data obtained from[6]and[35]).Several studies[6,35]report the effect of gender and age on the comfortable walking speed of healthy humans (Fig.1).Speed is also expected to increase with the body mass due to fundamental structural reasons[36,37].The effect of the body mass on the comfortable walking speed is illustrated in Fig.2.From thisfigure,one can conclude that the comfortable walking speed scales sublinearly with body mass.Indeed,the power exponents depicted for females and males are0.182and0.156,respectively.3.Pedestrian’s guided locomotionPedestrians are attentive to details of the walking environment,especially those that are very close to them[38]. They feel uncomfortable if they get too close to other pedestrians(territorial effect or private“sphere”),as well as to physical obstacles.The vision(the eye)provides information about the environment,which is then cognitively interpreted and helps shape the way pedestrians evolve in space[39,40].This enables the selection of safer and straighter paths,and the circumvention of other pedestrians and obstacles.This guided behavior constitutes a primary and an essential feature in the perfecting of locomotion.To avoid other pedestrians or obstacles,pedestrians may adjust their comfortable speed to a lower self-selected speed.A number of researchers have studied the influence of pedestrian density(or interpersonal distances1)on1The concept“interpersonal distance”was introduced by Thompson and Marchant[41].The average interpersonal distance is obtained from the inverse of square root of pedestrians’density.172 A.F.Miguel/Physics of Life Reviews10(2013)168–190Fig.3.Experimental fundamental diagram of pedestrian movement:walking speed vs.pedestrian density(or interpersonal distance).Fig.4.Pedestrian density domains in the fundamental diagram of pedestrian movement.pedestrian speed[41–44].Several experimental studies have shown that pedestrians are able to walk at their com-fortable walking speed at densities up to0.2–0.8persons/m2.Beyond such values,an increase of pedestrian density leads to a pronounced reduction of walking speed.The walking speed–pedestrian density diagram depicted in Fig.3is known as the fundamental diagram of pedestrian movement[45–48].Seyfried et al.[45]suggested that this diagram can be divided into5density-domains.In alternative,Miguel[46]proposed that the array of points in this diagram may be divided into two major density-domains,as indicated in Fig.4:(i)Density-domain I:The interpersonal distance between the pedestrians is large enough(or the density of pedes-trians is small enough)and pedestrians are able to walk at their comfortable walking speed(0 δ δfs,whereδfs is the smaller interpersonal distance between the pedestrians that corresponds to the free walking speed).(ii)Density-domain II:Due to the reduction of the free available space,pedestrians adjust their speed,due to a natural desire of avoiding or reducing contact with other pedestrians.In this domain two sub-domains are identified which exhibit curves with different slopes:(a)the interpersonal distance between the pedestrians is not enough to walk at their comfortable walking speed,but is large enough to avoid contacts between them by a small reduction of walking speed;and(b)a more pronounced reduction in walking speed is required,since the space around each pedestrian is very small,and contacts with other pedestrians may hardly be avoided.A.F .Miguel /Physics of Life Reviews 10(2013)168–190173These pedestrian interactions can be cast into mathematical equations.Accordingly,pedestrians dynamics can be well approached within the framework of a Langevin-like equation of motion [49,50].The Langevin formulation is used to describe the Brownian particle’s motion as an Ornstein–Uhlenbeck process [51],and its position as the time integral of its velocity.Illustrative examples of this formulation for pedestrians are the social force and centrifugal force models.In the social force model [52,53],pedestrians’movement toward a destination results of the acceleration towards the desired walking speed,and the repulsive/attractive interactions with other pedestrians and obstacles.This model includes the concept of territorial effect (the private sphere)that leads to repulsive forces between pedestrians,as first suggested by Gipps and Marksjös [54].In the centrifugal force model [55,56],pedestrian movement results from the driving and repulsive forces acting on each pedestrian,with the repulsive forces being described similarly to centrifugal forces.Both models are able to capture and describe important features of pedestrian movement,such the formation of lanes in counter-flow,clogging at exit doors,oscillations at bottlenecks,among others [56].ngevin-like model for the pedestrian dynamicsA simple Langevin-like model can be derived from the above-present concept of density-domain II [46].At den-sity sub-domain IIa,pedestrians’interpersonal distances are still large enough,hence the deviation from the desired walking speed results from the necessary deceleration to adjust own speed to the speed of neighboring pedestrians.Therefore,M b d 2r dt=M b τ dr dt −dr ∗dt ,δrp δ δfs (2)and after integration of this equation,the walking speed within this domain,u IIa ,can be expressed asu IIa =u 0+1τ(δ−δfs ),δrp δ δfs (3)Here r is the position of the pedestrian,τis a relaxation time,(dr/dt −dr ∗/dt )is the mean relative speed to the pedestrians situated around,u 0is the free walking speed,δrt is the interpersonal distance that corresponds to the start of sub-domain IIb,and δfs is the smaller interpersonal distance between the pedestrians for whom free walking speed is still available.Pedestrians get too close to others at sub-domain IIb.The decrease of pedestrians’interpersonal distances leads to the presence of repulsive forces between them.These forces depend not only on the relative velocity of pedestrians,but also on the distance between them.Within this domain,M b d 2r dt=M b γ(r −r ∗) dr dt −dr ∗dt ,δmin <δ δrp (4)Integration of this equation yields the following walking speed,u IIb ,u IIb =u rp +γln δδrp ,δmin <δ δrp (5)where γis a coefficient,r −r ∗(=δ)is the mean interpersonal distance between the pedestrians,u rp is the walking speed corresponding to the minimum interpersonal distance where “repulsive forces”start to occur,and δmin is the smallest interpersonal distance possible between pedestrians in the domain.The average interpersonal distance and the pedestrians’density are related through the equation [41]δ=1√ρ(6)Substituting Eq.(6)into Eqs.(3)and (5)gives [46]u IIa =u 0−1τ √ρ−√ρfs √ρfs ρ ,ρfs ρ ρrp (7)u IIb =u rp +γ2ln ρrp ρ ,ρrp <ρ<ρmax (8)174 A.F.Miguel/Physics of Life Reviews10(2013)168–190Empirical data available in the literature can befitted with Eqs.(7)and(8),and the corresponding empirical coeffi-cients obtained.Pedestrian traffic may be unidirectional or multidirectional by nature.An examination of the results presented in prior studies indicates that both unidirectional and multidirectional streams are consistent with these two sub-domains, but that the interpersonal distance that corresponds to the start of“repulsive”forces is different[57].Therefore, a separate analysis must be performed,which yields the following coefficients[46,57]:–For unidirectional streams:τ=3.77s,u0=1.36m/s(0.3 ρ 1.4persons/m2);andγ=1.17m/s and u rp=0.995m/s(1.5 ρ 4.1persons/m2).–For multidirectional streams:τ=5.56s,u0=1.35m/s(0.14 ρ 0.88persons/m2);γ=1.10m/s and u rp=1.03m/s(0.89 ρ 5.2persons/m2).Although these results clearly support the existence of similar velocity–density domains for unidirectional and mul-tidirectional streams,the density range corresponding to each domain is actually different.Notice that neitherγnor u rp are significantly different between both streams,because these are quantities measured in the domain with less space available to pedestrians.On the other hand,the relaxation time is higher for the multidirectional streams.As the interpersonal distances between pedestrians in this domain are still large,the existence of multidirectional stream requires that pedestrians are more attentive,and more time to adjust their motion is needed.3.2.Diffusion coefficient model for the pedestrian dynamicsEinstein’s perspective of the Langevin’s approach of motion(i.e.a Wiener process)describes the penetrants mo-bility through a diffusion coefficient[58,59]and may be approached by a single coefficient.By analogy with kinetic theory,the pedestrian diffusion coefficient is related to mean walking speed and mean free interpersonal distance via the Einstein–Smoluchowski equation.The experimental data available in the literature is bestfitted by taking[57] (a)For unidirectional streams:D=1.512δ−0.588,0.47<δ 4.47m(9) (b)For multidirectional streams:D=1.498δ−0.593,0.43<δ 4.47m(10)These results also provide direct evidence supporting the differences between uni-and multidirectionalflows of pedestrians.Additionally,these coefficients are valid for the entire range of domain II.In summary,the models based on Langevin’s and Einstein’s pioneering studies in1905and1908[49,58],respec-tively,are able tofit well experimental data that relates walking speed to pedestrian density.The analysis of pedestrian dynamics as an Ornstein–Uhlenbeck type process,however,produces more insight into the“forces”that drive pedes-trian motion,and delivers different approaches according to pedestrian density sub-ranges(Eqs.(7)and(8)).On the other hand,a Wiener-type process,which is based on a single coefficient(Eqs.(9)and(10)),provides a more compact and simples approach of pedestrian motion.Despite their different mathematical formulation,all of these approaches are physically equivalent.Gillespie[60]presents a review of the arguments that lead to the conclusion that any diffusion coefficient may be linked with quantities such asτandγ.4.Constructal theory of pedestrian dynamics4.1.A new thermodynamics insight into dissipativeflow systemsSince Anaximenes of Miletus(585–528B.C.),laws are considered operative throughout Nature[61].This consti-tutes a magnificent triumph of reason and observation:laws tell us how things operate and can guide us in the quest for news knowledge.The invariance provides a structure and coherence to the laws just as the laws provide a structure and coherence to the set of natural events.A.F .Miguel /Physics of Life Reviews 10(2013)168–190175Thermodynamics is one of the bedrocks of modern science,and is firmly grounded into laws.Different laws provide us a view of natural phenomena.The zeroth law defines a useful property “temperature”(and proposes the equality of temperature as necessary and sufficient condition for thermal equilibrium).The first law defines useful extensive property “energy”(and asserts that energy is conserved).Since it is possible to take the same amount of internal energy “forward”and “backward”,the first law expresses symmetry.Although all processes must take place in accordance to the first law,the principle of conservation of energy is,by itself,insufficient to describe preferred directions of action in time.Both the reverse flow (hot to cold and cold to hot)and the immutability of configuration are not in violation of the first law [61].The second law of thermodynamics asserts the existence of extensive property entropy and states that this property in an adiabatically isolated system never decreases in time.The entropy is a Lyapunov function of the dynamical system and the “backward”process is not allowed [61].There is a “time asymmetry“or a “direction of time”or an “arrow of time”of the state of the system.The third law defines a state known as “absolute zero”(and relates the entropy of a system to its absolute temperature).Most real systems are however not isolated and may exhibit distinct characteristics.If two mixable liquids are let to mix in a vase,diffusion takes place spontaneously and a progressive decrease of initial individual concentrations will occur with a corresponding increase of entropy.But living organisms are able to maintain the differences of liq-uid concentrations in time (by chemical reactions and active transport).Prigogine named these systems ‘dissipative structures’because they cannot exist independently of their environment [62,63].These “structures”make an effort to avoid a transition into thermodynamic equilibrium by a continuous exchange of materials and energy with the en-vironment.Therefore,they are also able to self-organize through instabilities that lead to irreversible bifurcations and new stable system states [62,63].Another of Prigogine’s contributions was the minimum entropy production principle (or Prigogine theorem),which applies to open linear nonequilibrium systems in the stationary (or approaching the stationary)state.At every instant,currents of fluid,heat,mass,or information are flowing through animate and inanimate,dissipative open systems.This involves a state of organizational structure intimately coupled with nodes and channels of supply and distribution.The ubiquitous generation of configuration (design,organization)in these nonequilibrium systems is covered by the constructal law of Adrian Bejan,which states that “For a finite-size flow system to persist in time (to live )it must evolve such that it provides greater and greater access to the currents that flow through it ”[64–70].This new law asserts that for any flow system there is a property “configuration”and relates the generation of configuration to its greater access to flow.2This is possible because systems have the freedom to morph (i.e.,freedom to change the configuration in time)to achieve their purpose of higher global performance under constraints.Flows occur against resistances (imperfections)that constantly cause energy dissipation and try to slow them down.Therefore,the emergence of configuration (organization)with a purpose,defined by the constructal law,also requires that entropy changes.For the sake of simplicity,let us consider a linear pressure–flow relation (or potential–current relation).In this case,the resistance,R ,is related with the time rate of entropy generation,d ˙S g ,by R =1T V 2d ˙Sg and R =T d ˙S g 2,where V is the potential (or pressure difference),I is the current (or flow),and T is the absolute temperature.Minimization of resistance in morphing configurations under constant I and constant V (constraints)corresponds not only to a minimization of the entropy generation rate but to a maximization of the entropy generation,respectively.In summary,configuration emerges in systems that have a purpose and are far from equilibrium,and the emergence of this organization requires that entropy changes.The constructal law is not only connected to the entropy generation rate,but it also provides the reasons for why and how design occurs in nonequilibrium systems.4.2.Constructal view of human gaitEngines (man-made,animal,geophysical)use energy to produce the work required for driving movement.In order to induce movement,they should be able to overcome internal and external resistances (i.e.,energy input is matched by energy loss).Such thermodynamic “imperfections”cannot be avoided,and the constructal improvement of functions implies the generation of a design that distributes imperfections optimally to fill the flow space.Therefore,the constructal law is about both the necessity and the evolution of design to occur.2“Maximum flow access”corresponds to minimum travel time or minimum transfer time [67].Therefore,“for a finite-size flow system to persist in time it must evolve such that it provides a minimum travel time to currents that flow through it”.。
