Car Following Models

Transcription

1 Car Followig Models Lecture Notes i Trasportatio Systems Egieerig Prof. Tom V. Mathew Cotets 1 Overview Car followig models Notatio Pipe s model Forbes model Geeral Motors model Optimal velocity model Geeral motor s car followig model Basic Philosophy Follow-the-leader model Derivatio of Greeberg s formula (uder developmet) 8 5 Summary 10 Exercises 10 1 Overview-1 Logitudial spacig of vehicles are of particular importace from the poits of view of safety, capacity ad level of service. The logitudial space occupied by a vehicle deped o the physical dimesios of the vehicles as well as the gaps betwee vehicles. For measurig this logitudial space, two microscopic measures are used- distace headway ad distace gap. Distace headway is defied as the distace from a selected poit (usually frot bumper) o the lead vehicle to the correspodig poit o the followig vehicles. Hece, it icludes the legth of the lead vehicle ad the gap legth betwee the lead ad the followig vehicles. IIT Bombay (tvm@civil.iitb.ac.i) September 19,

2 Directio of traffic v Follower v Leader x y +1 x x +1 2 Car followig models Figure 1: Notatio for car followig model Car followig theories describe how oe vehicle follows aother vehicle i a uiterrupted flow. Various models were formulated to represet how a driver reacts to the chages i the relative positios of the vehicle ahead. Models like Pipes, Forbes, Geeral Motors ad Optimal velocity model are worth discussig. 2.1 Notatio Before goig i to the details, various otatios used i car-followig models are discussed here with the help of figure 1. The leader vehicle is deoted asad the followig vehicle as ( + 1). Two characteristics at a istat t are of importace; locatio ad speed. Locatio ad speed of the lead vehicle at time istat t are represeted by x t ad vt respectively. Similarly, the locatio ad speed of the follower are deoted by x t +1 ad vt +1 respectively. The followig vehicle is assumed to accelerate at time t+ T ad ot at t, where T is the iterval of time required for a driver to react to a chagig situatio. The gap betwee the leader ad the follower vehicle is therefore x t x t Pipe s model The basic assumptio of this model is A good rule for followig aother vehicle at a safe distace is to allow yourself at least the legth of a car betwee your vehicle ad the vehicle ahead for every te miles per hour of speed at which you are travelig Accordig to Pipe s car-followig model, the miimum safe distace headway icreases liearly with speed. A disadvatage of this model is that at low speeds, the miimum headways proposed by the theory are cosiderably less tha the correspodig field measuremets. 2

3 2.3 Forbes model I this model, the reactio time eeded for the followig vehicle to perceive the eed to decelerate ad apply the brakes is cosidered. That is, the time gap betwee the rear of the leader ad the frot of the follower should always be equal to or greater tha the reactio time. Therefore, the miimum time headway is equal to the reactio time (miimum time gap) ad the time required for the lead vehicle to traverse a distace equivalet to its legth. A disadvatage of this model is that, similar to Pipe s model, there is a wide differece i the miimum distace headway at low ad high speeds. 2.4 Geeral Motors model The Geeral Motors model is the most popular of the car-followig theories because of the followig reasos: 1. Agreemet with field data; the simulatio models developed based o Geeral motors car followig models shows good correlatio to the field data. 2. Mathematical relatio to macroscopic model; Greeberg s logarithmic model for speeddesity relatioship ca be derived from Geeral motors car followig model. I car followig models, the motio of idividual vehicle is govered by a equatio, which is aalogous to the Newto s Laws of motio. I Newtoia mechaics, acceleratio ca be regarded as the respose of the particle to stimulus it receives i the form of force which icludes both the exteral force as well as those arisig from the iteractio with all other particles i the system. This model is the widely used ad will be discussed i detail later. 2.5 Optimal velocity model The cocept of this model is that each driver tries to achieve a optimal velocity based o the distace to the precedig vehicle ad the speed differece betwee the vehicles. This was a alterative possibility explored recetly i car-followig models. The formulatio is based o the assumptio that the desired speed v desired depeds o the distace headway of the th vehicle. i.e.v t desired = v opt ( x t ) where v opt is the optimal velocity fuctio which is a fuctio of the istataeous distace headway x t. Therefore at is give by a t = [1/τ][V opt ( x t ) v] t (1) where 1 is called as sesitivity coefficiet. I short, the drivig strategy of τ th vehicle is that, it tries to maitai a safe speed which i tur depeds o the relative positio, rather tha relative speed. 3

4 3 Geeral motor s car followig model 3.1 Basic Philosophy The basic philosophy of car followig model is from Newtoia mechaics, where the acceleratio may be regarded as the respose of a matter to the stimulus it receives i the form of the force it receives from the iteractio with other particles i the system. Hece, the basic philosophy of car-followig theories ca be summarized by the followig equatio [Respose] α [Stimulus] (2) for the th vehicle (=1, 2,...). Each driver ca respod to the surroudig traffic coditios oly by acceleratig or deceleratig the vehicle. As metioed earlier, differet theories o car-followig have arise because of the differece i views regardig the ature of the stimulus. The stimulus may be composed of the speed of the vehicle, relative speeds, distace headway etc, ad hece, it is ot a sigle variable, but a fuctio ad ca be represeted as, a t = f sti (v, x, v ) (3) where f sti is the stimulus fuctio that depeds o the speed of the curret vehicle, relative positio ad speed with the frot vehicle. 3.2 Follow-the-leader model The car followig model proposed by Geeral motors is based o follow-the leader cocept. This is based o two assumptios; (a) higher the speed of the vehicle, higher will be the spacig betwee the vehicles ad (b) to avoid collisio, driver must maitai a safe distace with the vehicle ahead. Let x t +1 is the gap available for ( + 1)th vehicle, ad let x safe is the safe distace, v+1 t ad vt are the velocities, the gap required is give by, x t +1 = x safe +τv+1 t (4) where τ is a sesitivity coefficiet. The above equatio ca be writte as x x t +1 = x safe +τv+1 t (5) Differetiatig the above equatio with respect to time, we get v t vt +1 = τ.at +1 a t +1 = 1 τ [vt vt +1 ] 4

5 Geeral Motors has proposed various forms of sesitivity coefficiet term resultig i five geeratios of models. The most geeral model has the form, [ ] a t αl,m (v+1 t +1 = )m [v ] t (x t xt +1) l v+1 t where l is a distace headway expoet ad ca take values from +4 to -1, m is a speed expoet ad ca take values from -2 to +2, ad α is a sesitivity coefficiet. These parameters are to be calibrated usig field data. This equatio is the core of traffic simulatio models. I computer, implemetatio of the simulatio models, three thigs eed to be remembered: 1. A driver will react to the chage i speed of the frot vehicle after a time gap called the reactio time durig which the follower perceives the chage i speed ad react to it. 2. The vehicle positio, speed ad acceleratio will be updated at certai time itervals depedig o the accuracy required. Lower the time iterval, higher the accuracy. 3. Vehicle positio ad speed is govered by Newto s laws of motio, ad the acceleratio is govered by the car followig model. Therefore, the goverig equatios of a traffic flow ca be developed as below. Let T is the reactio time, ad t is the updatio time, the goverig equatios ca be writte as, v t = v t δt x t = x t δt [ a t +1 = (6) +a t δt δt (7) +v t δt α l,m (v t δt +1) m (x t T δt+ 1 2 at δt δt 2 ] (8) (v t T v+1 t T (9) x t T +1 ) l The equatio 7 is a simulatio versio of the Newto s simple law of motio v = u+at ad equatio 8 is the simulatio versio of the Newto s aother equatio s = ut at2. The acceleratio of the follower vehicle depeds upo the relative velocity of the leader ad the follower vehicle, sesitivity coefficiet ad the gap betwee the vehicles Numerical Example Let a leader vehicle is movig with zero acceleratio for two secods from time zero. The he accelerates by 1 m/s 2 for 2 secods, the decelerates by 1m/s 2 for 2 secods. The iitial speed is 16 m/s ad iitial locatio is 28 m from datum. A vehicle is followig this vehicle with iitial speed 16 m/s, ad positio zero. Simulate the behavior of the followig vehicle usig Geeral Motors Car followig model (acceleratio, speed ad positio) for 7.5 secods. Assume the parameters l=1, m=0, sesitivity coefficiet (α l,m ) = 13, reactio time as 1 secod ad sca iterval as 0.5 secods. 5

6 20 19 Leader Follower 18 Velocity Time(secods) 30 Figure 2: Velocity vz Time Solutio The first colum shows the time i secods. Colum 2, 3, ad 4 shows the acceleratio, velocity ad distace of the leader vehicle. Colum 5,6, ad 7 shows the acceleratio, velocity ad distace of the follower vehicle. Colum 8 gives the differece i velocities betwee the leader ad follower vehicle deoted as dv. Colum 9 gives the differece i displacemet betwee the leader ad follower vehicle deoted as dx. Note that the values are assumed to be the state at the begiig of that time iterval. At time t=0, leader vehicle has a velocity of 16 m/s ad located at a distace of 28 m from a datum. The follower vehicle is also havig the same velocity of 16 m/s ad located at the datum. Sice the velocity is same for both, dv = 0. At time t = 0, the leader vehicle is havig acceleratio zero, ad hece has the same speed. The locatio of the leader vehicle ca be foud out from equatio as, x = = 36 m. Similarly, the follower vehicle is ot acceleratig ad is maitaiig the same speed. The locatio of the follower vehicle is, x = = 8 m. Therefore, dx = 36-8 =28m. These steps are repeated till t = 1.5 secods. At time t = 2 secods, leader vehicle accelerates at the rate of 1 m/s 2 ad cotiues to accelerate for 2 secods. After that it decelerates for a period of two secods. At t= 2.5 secods, velocity of leader vehicle chages to 16.5 m/s. Thus dv becomes 0.5 m/s at 2.5 secods. dx also chages sice the positio of leader chages. Sice the reactio time is 1 secod, the follower will react to the leader s chage i acceleratio at 2.0 secods oly after 3 secods. Therefore, at t=3.5 secods, the follower respods to the leaders chage i acceleratio give by equatio i.e., a = = 0.23 m/s2. That is the curret acceleratio of the follower vehicle depeds o dv ad reactio time of 1 secod. The follower will chage the speed at the ext time iterval. i.e., at time t = 4 secods. The speed of the follower vehicle at t = 4 secods is give by equatio as v= = The locatio of the follower vehicle at t = 4 secods is give by equatio as x = = These steps are followed for all the cells of the table. The earliest car-followig models cosidered the differece i speeds betwee the leader ad the follower as the stimulus. It was assumed that every driver teds to move with the 6

7 Table 1: Car-followig example t a(t) v(t) x(t) a(t) v(t) x(t) dv dx (1) (2) (3) (4) (5) (6) (7) (8) (9) t a(t) v(t) x(t) a(t) v(t) x(t) dv dx

8 Acceleratio Leader Follower Time(secods) Figure 3: Acceleratio vz Time same speed as that of the correspodig leadig vehicle so that a t = 1 τ (vt+1 v t +1) (10) where τ is a parameter that sets the time scale of the model ad 1 ca be cosidered as a τ measure of the sesitivity of the driver. Accordig to such models, the drivig strategy is to follow the leader ad, therefore, such car-followig models are collectively referred to as the follow the leader model. Efforts to develop this stimulus fuctio led to five geeratios of car-followig models, ad the most geeral model is expressed mathematically as follows. a t +1 = α l,m [v t δt +1 ]m [x t T ]l(vt T x+1 t T v t T +1 ) (11) where l is a distace headway expoet ad ca take values from +4 to -1, m is a speed expoet ad ca take values from -2 to +2, ad α is a sesitivity coefficiet. These parameters are to be calibrated usig field data. 4 Derivatio of Greeberg s formula (uder developmet) The Greeberg s stream model ca be derived from Geeral Motors car follwig model. The GM model is give as: a t = α l,m [v t δt +1 ]m [x t T Puttig l = 1 ad m = 0 the model becomes: l(vt T x+1 t T ] a t = α l,m(v t T (x t T At steady state codtio, the equatio becomes: v+1 t T ) v t T +1 ) (12) x t T +1 ) (13) a = α (v 1 v ) (x 1 x ) (14) 8

9 Now itegratig both sides with respect to tm we get: α (v 1 v ) a dt = dt (15) (x 1 x ) Now, put u = x 1 x, the du dt = v 1 v, or du = (v 1 v )dt. (16) Therefore, du a dt = α u. (17) v = αlogu+c (18) = αlog(x 1 x )+c (19) = αlogs +c. (20) where, s is the spacig. Now, uder steady state coditios, there is o idetity for vehicles. Hece, the above expressio will be as follows: v = αlogs+c (21) = αlog 1 k +c. (22) To fid the costat c, apply the boudary coditio, amely, at jam desity, the speed is zero. That is: 0 = αlog 1 k j +c c = αlog 1 k j v = αlog 1 k αlog 1 k j Now, to fid α we do the followig: = αlog 1 k + αlogk j v = αlog k j k. (23) q = kv = α k log k j k (24) 9

10 Differetiatig with respect to desity ad equatig to zero: [ dq d dk = α k dk log k ] ( j +α log k ) j k k [ d = α k dk logk j d ] dk logk +α [ = α k 1 ] ( +α log k ) j k k k j k or log k j k = αlog k j k = 1, α = 0. ( log k ) j k = e, or, k 0 = k j e. (25) Now, substitutig the above result, we get the value of alpha as Now the Greeberg s stream model is: v 0 = αloge = α. (26) v = v 0 log k j k (27) 5 Summary Microscopic traffic flow modelig focuses o the miute aspects of traffic stream like vehicle to vehicle iteractio ad idividual vehicle behavior. They help to aalyze very small chages i the traffic stream over time ad space. Car followig model is oe such model where i the stimulus-respose cocept is employed. Optimal models ad simulatio models were briefly discussed. Exercises 1. Discuss the cocepts ad model formulatios of Geeralised GM model, Gipps model, ad Wiedema 74 car-followig models. 2. Discuss the evolutio of the five geeratios of Geeral Motors car followig model highlightig the drawback ad advatages of each geeratio of the model. 3. A lie of vehicles are i car followig mode ad all vehicles are travellig at 18 m/s with distace headway of 20 m. After 1.2 secods, the lead vehicle suddely decelerates at a rate of 1.2 m/s 2 util it stops completely. simulate the behaviour of first followig vehicle usig the GM fifth car followig model for the first 2.5 secods. Tabulate the results. Assume headway expoet 1.2, speed expoet 1.6, sesitivity coefficiet 0.8, reactio time 0.6 secods, ad sca iterval 0.3 secods. 10

11 4. Simulate the followig vehicle behaviour for the followig data usig Widema 74 model. (a) For the case of stad still distace 3.5m, additive part of safety distace 1.5, ad multiplicative part of safety distace 0.8. (b) For the case of stad still distace 3.5m, additive part of safety distace 1.5, ad multiplicative part of safety distace 0.8. Commet o the followig vehicle behaviour for the above two cases. 5. A car is travellig with a speed of 16 m/sec at time t=0. Aother car follows the first at a distace of 28 m with same velocity. If the first car accelerated by 1 m/sec 2 from t=1 to 2 ad decelerate by 1 m/sec 2 from t=2 to 3, fid the speed, acceleratio ad spacig of the follower at time t=3.0 sec. Assume the reactio time is 1 sec, vehicle dyamics are updated every 0.5 secods, ad the car followig model is give by Eq. 28. (Use of a tabular form is ecouraged). [ ] u (t 1) u +1 (t 1) a +1 (t) = 15 (28) x (t 1) x +1 (t 1) 6. I a simulatio experimet o a sigle lae road, oe vehicle is travellig at 18 m/s. After 1.5 secods, the vehicle suddely accelerates at a rate of 1.5 m/s 2 for the ext 1.8 secods. Simulate the behaviour of subsequet vehicle with a iitial speed of 16 m/s usig GM fifth car followig model for the first 3 secods if the iitial distace headway is 20 m. Tabulate the results. Assume headway expoet 1.2, speed expoet 1.5, sesitivity coefficiet 0.8, reactio time 0.6 secods, ad update iterval of 0.3 secods. 7. A lie of vehicles are i car followig mode ad all vehicles are travellig at 15 m/s with distace headway of 20 m. After 1.2 secods, the lead vehicle suddely decelerates at a rate of 1.2 m/s 2 util it stops completely. simulate the behaviour of first followig vehicle usig the GM fifth car followig model for the first 2.5 secods. Tabulate the results. Assume headway expoet 1.2, speed expoet 1.6, sesitivity coefficiet 0.6, reactio time 0.6 secods, ad sca iterval 0.3 secods. 8. I a simulatio experimet o a sigle lae road, oe vehicle is travellig at 16 m/s. After 0.6 secods, the vehicle suddely accelerates at a rate of 1.2 m/s 2 for the ext 0.9 secods. Simulate the behaviour of subsequet vehicle with a iitial speed of 16 m/s usig GM fifth car followig model for the first 2.1 secods if the iitial distace headway is 25 m. Tabulate the results. Assume headway expoet 1.2, speed expoet 1.4, sesitivity coefficiet 0.6, reactio time 0.6 secods, ad update iterval of 0.3 secods. 9. A lie of vehicles are i car followig mode ad all vehicles are travellig at 15 m/s with distace headway of 25 m. After 1 secod, the lead vehicle suddely decelerates at a rate of 1.2m/s 2 util it stops completely. Simulate the behaviour of first followig vehicle usig the GM fifth car followig model for the first 3 secods. Tabulate the results. Assume headway expoet 1.0, speed expoet 1.5, sesitivity coefficiet 0.5, reactio time 0.5 secods, ad sca iterval 0.25 secods. Refereces 1. L. R Kadiyali. Traffic Egieerig ad Trasportatio Plaig. Khaa Publishers, New Delhi, L J Pigataro. Traffic Egieerig: Theory ad practice. Pretice-Hall, Eglewoods Cliffs,N.J.,

12 Ackowledgmets I wish to thak several of my studets ad staff of NPTEL for their cotributio i this lecture. I also appreciate your costructive feedback which may be set to tvm@civil.iitb.ac.i Prof. Tom V. Mathew Departmet of Civil Egieerig Idia Istitute of Techology Bombay, Idia 12