Examining he limiaions for visual anglecar following models Al Obaedi, JTS and Yousif, S Tile Auhors Type URL Published Dae 009 Examining he limiaions for visual angle car following models Al Obaedi, JTS and Yousif, S Conference or Workshop Iem This version is available a: hp://usir.salford.ac.uk/9687/ USIR is a digial collecion of he research oupu of he Universiy of Salford. Where copyrigh permis, full ex maerial held in he reposiory is made freely available online and can be read, downloaded and copied for non commercial privae sudy or research purposes. Please check he manuscrip for any furher copyrigh resricions. For more informaion, including our policy and submission procedure, please conac he Reposiory Team a: usir@salford.ac.uk.
EXAMINING THE LIMITATIONS FOR VISUAL ANGLE-CAR FOLLOWING MODELS Jalal Al-Obaedi PhD suden Universiy of Salford Saad Yousif Senior Lecurer Universiy of Salford Absrac Visual angle car following models assume fixed values (hresholds) for he angular velociy ( Ө/ ) over which he driver of he following vehicle is affeced by he leading vehicle. This paper explains he advanages and limiaions of such models and provides possible soluions for hese limiaions. Sensiiviy analysis has been carried ou for hese purposes as a par of his paper. I was found ha he main advanages of using visual angle models are heir simpliciy and abiliy o represen he effec of he widhs of differen ypes of vehicles (e.g. heavy goods vehicles, cars and moorbikes) on space headway (i.e. he clear disance beween leading and following vehicles). Also, hese models are simply capable of including he effec of driver s reacion ime as a funcion of raffic densiy, relaive speeds beween leading and following vehicles and he acion/behaviour of he leading vehicle (i.e. acceleraing/deceleraing following a change in local raffic condiions). However, he main concern in using such models is relaed o he appropriae choice of he hreshold values. The paper will illusrae ha using a fixed hreshold value for he angular velociy will be illogical since i eliminaes he effec of variaions in speed of he following vehicle for he same given relaive speed beween he leader and follower ( V). Moreover, oher limiaions are relaed o he effec of leader s widh on he acceleraion/deceleraion raes of he follower. Finally, and in order o deal wih he above limiaions and advanages ha visual angle models have, an alernaive car following model has been proposed. The proposed model will hen be esed agains daa from sies o examine is applicabiliy. Keywords: Traffic micro-simulaion models, car following, visual angle models 1. Background and lieraure review 1.1 Inroducion Car following models describe he reacion of he driver of he following vehicle ravelling close o he vehicle ahead. This reacion is represened by his/her acceleraion/deceleraion rae based on he differences in speeds and spacing beween hese wo vehicles. From he middle of he wenieh cenury and unil now, several car following models were developed. Gazis-Herman-Rohery (1960) model (GHR) represened he earlier car following model which was formulaed in 1958 a he General Moors Research labs in Deroi. According o he model, he acceleraion of he following vehicle is based on relaive velociy, relaive spacing and he following vehicle s velociy as shown in he following equaion: a f v l, m, cv f,. Equaion 1 l x l, v f, x f, This paper is produced and circulaed privaely and is inclusion in he conference does no consiue publicaion. 5A1.1
w Where: a f, is acceleraion of he follower is driver s reacion ime (for he following vehicle) v l,, x l, is he velociy and posiion of he leading vehicle a ime, respecively v f,, x f, is he velociy and posiion of he following vehicle a ime, respecively and c, m and l are model parameers. 1. Visual angle car following models One of he earlier car-following models is he visual angle model. The visual angle as shown in Figure 1 is given by he following equaion: Where: an 1 w H w is a widh of he leading vehicle.. Equaion H is he spacing beween he leading and he following vehicles. Follower ө Leader 5A1. Figure 1 Illusraion of he visual angle (Ө) Michaels (1963) observed ha he deecion of he relaive velociy depends on he rae of change of angular moion (angular velociy) of an image across he reina of he eye of he follower driver (Fox and Lehman, 1967). The angular velociy is found by differeniaing Equaion wih respec o he ime () Where: VL VF w ( X L lengh( l) X F ). Equaion 3 V L and V F are speeds of leading and following vehicles respecively. w is he widh of leading vehicle. X L and X F are posiions of leading and following vehicles respecively. Visual angle models are described by previous researchers, such as Bracksone and McDonald (1999), and Panwai and Dia (005), as one ype of psychophysical or acion poin models since hese models define he nex vehicle s acion on wheher or no he follower exceeds cerain hresholds. These assume fixed values (hresholds) for angular velociy. Once he absolue value of he angular velociy exceeds he hreshold, he follower will accelerae or decelerae opposie o he sign of he relaive angular velociy. 1.3 Angular velociy hresholds for visual angle models H Pipes (1967) repored ha he mos carefully conrolled sudy by Michaels and Cozan (1963) of he absolue hreshold of he angular velociy indicaes ha i is reasonable o use 0.0006 rad/sec as a mean value for he hreshold. This means ha if he relaive speed is 10 km/h and he widh of he
leading vehicle is 1.8m, he angular velociy can be deeced if he relaive spacing is less han 91 m (based on Equaion 3 above). Fox and Lehman (1967) described a car following model based on visual angle concep. According o he model, when he hreshold value is exceeded, he driver is considered o be in a velociy-deecion mode. Below ha hreshold, he driver is assumed o be in a disancedeecion mode. If he driver of he following vehicle is in a velociy-deecion mode, GHR model is used. In a disance-deecion mode, he acceleraion of he following vehicle is considered o be mainly depends on he relaive spacing beween he leading and he following vehicles. A base value of 0.0006 rad/sec (similar o ha used by Michaels) was chosen. Ferrari (1989) used a car following model wih differen hreshold value which is 0.0003 rad/sec. The model based on he assumpion ha he follower will choose o decelerae or accelerae in case he hreshold is exceeded. A minimum ime gap of 1 second is assumed beween he wo successive vehicles. Based on experimens carried ou o scale he relaive velociy beween vehicles, Hoffmann and Morimer (1994 and 1996) suggesed a differen hreshold value of 0.003 rad/sec. They concluded ha only when he subended angular velociy of he lead vehicles exceeded ha hreshold were he subjecs able o scale he relaive velociy. Table 1 presens a brief summary of he values used for angular velociy hresholds by various researchers. I is clear ha he assumed range of values used was differen in mos cases. Therefore, his paper examines some of he limiaions in using hese visual angle hresholds for use in car following models. Table 1 Summary of angular velociy hresholds used by various researchers Researcher(s) Threshold value ( Ө/ ) (rad/sec) Michaels and Cozan (1963) 0.0003 0.001 Fox and Lehman (1967) 0.0006 Ferrari (1989) 0.0003 Hoffman and Morimer (1994 and 1996) 0.003 1.4 Possible applicaions of visual angle models According o Traffic Technology Today (008), researchers in Germany (i.e. Karlsruhe Universiy wih Fraunhofer Insiue for Informaion and Daa Processing) are carrying ou insrumenal work on co-ordinaing manoeuvres of cars following each oher wihou he need o drivers inervenion. This is done in an emergency siuaion by equipping cars ravelling on he same lane wih car-ocar communicaion and inegraed sensors o enable hem recognise heir surroundings and avoid poenial obsacles (such as a sopping vehicle in fron). The selecion of appropriae angular velociy hreshold values which suis such safe manoeuvres could help in providing he basis for he seings of such sensors used in he car-o-car communicaion sysem.. Visual angle model assumpions This secion presens he main assumpions for he visual angle model which is used in his heoreical sudy o show he advanages and limiaions of such models. The main assumpion of he model is based on wheher or no he angular velociy calculaed from Equaion 3 exceeds he assumed angular velociy hreshold values. The effec of he visual angle model s parameers (i.e. relaive velociy and relaive spacing) on he angular velociy values is shown in Figure. A value of 1.8 m is assumed as he widh of he leader. This paper is produced and circulaed privaely and is inclusion in he conference does no consiue publicaion. 5A1.3
Angular velociy (rad/sec) 0.005 0.004 Δv=-10km/h 0.003 0.00 Δv=-5km/h 0.001 0-0.001-0.00 Δv=-1km/h Δv=+1km/h Δv=+5 km/h -0.003-0.004 Δv=+10 km/h -0.005 0 10 0 30 40 50 60 70 80 90 100 Space headw ay (m) Figure Relaionship beween he angular velociy and he space headway for differen relaive speeds beween he leader and is follower ( V) In general, he Figure shows ha he absolue angular velociy increases wih an increase in he absolue relaive velociy beween he leader and is follower, and decreases wih an increase in he relaive spacing beween hem. If he absolue angular velociy becomes higher han a cerain seleced hreshold, he follower sars o accelerae or decelerae opposie in sign o ha of he angular velociy value. For example, if he angular velociy hreshold is + 0.003 rad/sec (as suggesed by Hoffman and Morimer (1994 and 1996), he follower will decelerae if his/her angular velociy is higher han 0.003 rad/sec and accelerae if his angular velociy is below 0.003 rad/sec. The seleced values for acceleraion or deceleraion are he minimum of he following raes (see for example Fox and Lehman, 1967 and Ferrari, 1989): he acceleraion rae which is required o reach he desired speed, he acceleraion/deceleraion rae required o reach he leader s speed, and he acceleraion/deceleraion rae o mainain he minimum desired spacing. When he angular velociy value is wihin he seleced range for he angular velociy hreshold (i.e. + 0.003 rad/sec), he acceleraion/deceleraion of he follower will no be based on facors described in Equaion 3 above. Insead, he acceleraion/deceleraion of he follower will be based on he desired spacing (or desired ime headway). The acceleraion or deceleraion raes are as shown in Equaion 4 and hey are a funcion of boh relaive disance and relaive speed beween he leader and follower. Also, hey are a funcion of he desired disance he follower wishes o mainain based on his/her speed. This equaion is derived based on he same assumpions repored by Hidas (1996). ac f x l, x f, v f, * DTHead DTHead f * f * v( f, ) lengh( l). Equaion 4 Where: 5A1.4 ac f is he acceleraion (or deceleraion) rae of he follower is he scanning ime.
Saring of followers deceleraion disance (m) DTHead f is he desired ime (spacing) for he follower. x l, is he posiion of he leader a ime (oher erms are as defined before). In his paper, he hreshold values for angular velociy were esed and compared wih hose suggesed by oher researchers. In paricular, a comparison has been made beween he values of 0.0006 rad/sec used by Michaels and Cozan (1963) and ha suggesed by Hoffman and Morimer (1994 and 1996) of 0.003 rad/sec as will be discussed in Secion 3. 3. Advanages of visual angle models 3.1 Represening he effec of he widh of leading vehicle Based on real daa from UK moorway sies, Yousif (1993) repored ha some passenger car drivers ry o leave sufficien space o avoid visual problems associaed wih obsruced raffic signs or oher raffic conrol devices on he road. This could conribue o forcing drivers following heavy goods vehicles (HGVs) o leave a much larger space. Parker (1996), when sudying he effec of HGVs a hree moorway roadwork sies, repored ha he presence of HGVs in he raffic sream increases average headways, hus reducing he capaciy of he road secion. I was shown ha mos of oher car following models, such as collision avoidance and desired spacing models, could no direcly include he effec of he size of he leading vehicle. Visual angle models can ake ino consideraion he effec of he size of vehicles wihou making any furher assumpions since he widh of he leader is included in he main equaion used for hese models. Figure 3 shows he effec of having differen widhs of he leader on he saring disance for is follower o be affeced by he leader for differen angular velociy hreshold values. The Figure is based on he assumpion ha here is a 10 km/h relaive speed difference beween he wo vehicles. From he Figure, i can be shown ha he follower sars applying his/her deceleraion earlier if he leader is an HGV (i.e. widh equals o.55 m) compared wih he case when he leader is a Car (i.e. widh equals o 1.8 m). However, his direc represenaion of he widh of he leader may seem unnecessary, as will be discussed in Secion 4.1. 70 60 w =.55 m 50 w =1.8 m 40 30 w =1.5 m 0 0.00 0.005 0.003 0.0035 Angular velociy hreshold (rad/sec) Figure 3 Saring disance for he follower o be affeced by is leader This paper is produced and circulaed privaely and is inclusion in he conference does no consiue publicaion. 5A1.5
Time from he sar of leader's deceleraion (sec) 3. Modelling of driver s reacion ime 3..1 Background informaion on reacion ime Reacion ime indicaes a ime lag beween he deecion of a simuli and he applicaion of a response. O Flahery (1986) saed ha he lengh of percepion ime varies considerably since i depends upon facors such as, he disance o objec, he naural rapidiy wih which he driver reacs and he opical abiliy of he driver. The main limiaion in mos of he exising car following models is relaed o he represenaion of drivers reacion ime. Mos models assign a pre-defined single value for each driver as his/her reacion ime. Some researchers used wo values for each driver o represen he alered and surprised (unalered) cases for congesed and non-congesed raffic condiions, respecively. The majoriy of such models could no represen he follower s reacion ime o show how i varies wih raffic condiions. Table shows a summary of some of he main work in deermining driver s reacion ime. I is clear from he differen rials o esimae driver s reacion ime ha here are some difficulies in doing so accuraely. Table Summary of brake reacion ime based on previous research Researcher Median reacion ime (sec.) Siuaions Johansson and Rumer (1971) 0.73, 0.54 Surprised, Alered Lerner e al. (1995) 1.44 Surprised Maycock e al. (1999) 1. Unalered Zhang and Bham (007) 0.6 From Micro-simulaion 3.. Brake reacion ime based on visual angle models An aemp has been made o represen driver s reacion ime based on visual angle models. Figure 4 shows he relaionship beween he angular velociy and he ime for differen iniial spacings beween pairs of vehicles. Here, he ime is ha of he follower needed from he sar of deceleraion of he leader, assuming a consan deceleraion of he leader of -m/s wih an iniial speed of 90 km/h for boh leader and follower. 3.5 1.5 1 0.5 Δx=100 m Δx=70 m Δx=50 m Δx=40 m Δx=30 m 0 0 0.001 0.00 0.003 0.004 0.005 Angular velociy (rad/sec) Figure 4 Relaionship beween angular velociy and ime (assuming ha leader s consan deceleraion of -m/sec for differen iniial space headways) 5A1.6
Driver reacion ime (sec) The Figure shows ha when raffic densiy is high (i.e. spacing is relaively small), he follower will reac o he deceleraion of he leader wihin a shorer ime han for he case of lower raffic densiy. For example, when he iniial spacing is 40 m, he follower will reac o he leader s deceleraion afer 1.1 sec (assuming a hreshold value of 0.003 rad/sec). For he case of 50 m iniial spacing, he follower will reac afer 1.6 sec. Figure 5 (which is derived from Figure 4), illusraes he relaionship beween iniial spacing beween he leader and is follower and he follower s reacion ime. The Figure is based on wo seleced angular velociy hreshold values of 0.003 and 0.0006 rad/sec as suggesed by Hoffmann and Morimer (1994 and 1996) and Michaels (1963), respecively. For relaively high densiy condiions (i.e. small spacings of say 40 m) Hoffmann and Morimer s hresholds sugges ha he reacion ime is abou 1.1 sec, whereas he hreshold suggesed by Michaels yield a value of reacion ime of 0. sec. If hese wo reacion imes of 1.1 and 0. sec are compared wih hose in Table 1, he resuls show ha he hreshold value of 0.003 rad/sec (as suggesed by Hoffmann and Morimer) gives more reasonable represenaion for he reacion ime. Therefore, in order o use he hreshold values suggesed by Michaels (1963), here should be anoher parameer (namely an exra brake reacion ime) which needs o be considered in visual angle car following models. 3.5 Threshold Hoffmann and Morimer =0.003 rad/sec 1.5 1 0.5 Threshold by Michaels = 0.0006 rad/sec 0 30 40 50 60 70 Iniial spacing (m) Figure 5 Relaionship beween driver reacion ime and iniial spacing for wo values of angular velociy hresholds 4. Limiaions of visual angle models There are several limiaions involving he use of visual angle models. This secion discusses he main limiaions and how hey are deal wih. 4.1 Represening he widh of vehicles As shown in Equaion 3 and Secion 3.1 above, he widh of he leading vehicle is direcly included wihin visual angle models. Therefore he follower will reac o he vehicle ahead a a disance influenced by ha widh as shown in Figure 3. Figure 3 shows ha he follower will sar his/her deceleraion earlier if he/she is preceded by a Car having a widh of 1.8 m insead of 1.5 m. The difference in he saring disance for deceleraion beween he wo Cars in his case is abou 4 m (his is based on he assumpion ha here is a 10 km/h relaive speed beween he follower and is leader and wihou reference o he acual operaing speed). This does no seem o be logical considering he fac ha here is only a 0.3 m difference in widh beween he wo Cars. One possible soluion o his is by assigning one specific widh for each ype of vehicles. For example, if he leader is a Car, he widh is assigned o be 1.8 m even if he acual widh of he leader is lower or higher han ha value. This is based on he assumpion ha on average he widh of Cars is equal o 1.8 m. Similarly, when he leader is an HGV, he assigned widh is.5 m for all HGVs. These seleced values could be examined furher during he calibraion and validaion process of he model. This paper is produced and circulaed privaely and is inclusion in he conference does no consiue publicaion. 5A1.7
Time (sec) There is anoher limiaion in represening he size of vehicles. The follower will no be able o reac o he effec of he widh of his/her leader if he angular velociy ( Ө/ ) is wihin he range of hreshold values of say ±0.003 rad/sec (as described in Secion 3..). This is because he follower will make his/her acceleraion/deceleraion based on he desired spacing. When he differences in speeds (i.e. relaive speeds) beween successive vehicles are low (see Figure ), i is expeced ha he angular velociy o be below he seleced hresholds. Therefore, a hese circumsances and according o his limiaion, he space headway beween he follower and he leader will be he same if he leader is a Car or an HGV. This does no seem o be logical since sudies by Yousif (1993) and Parker (1996) sugges ha his disance is influenced by he ype of vehicles in he raffic sream. One possible soluion for his limiaion is o modify he desired ime headway of he follower by including he widh of he leader as a facor. This will be discussed furher in Secion 5. 4. Ignoring he effec of he leader/follower speeds A sensiiviy analysis conduced on he visual angle model has shown ha he model is no sensiive o he leader/follower acual speeds and i is jus sensiive o he difference beween hese wo speeds. Figure 6 shows he relaionship beween angular velociy and reacion ime for any assumed speeds of he raffic sream (i.e. leader s and follower s speeds). The Figure is based on he assumpion ha he leader is assumed o apply a consan deceleraion of -3 m/sec wih iniial spacing beween he wo vehicles of 50 m..5 1.5 1 0.5 0 0 0.001 0.00 0.003 0.004 0.005 Angular velociy (Rad/sec) Figure 6 Relaionship beween angular velociy and reacion ime for any speed (assuming leader s and follower s speeds are iniially he same before he leader sars o decelerae) The Figure shows ha for an angular velociy of 0.003 rad/sec, he follower requires 1.3 seconds o reac o he deceleraion of he leader. This means ha he follower will reac o is leader regardless of he values of heir iniial speeds. This is in disagreemen wih he finding by O Flahery (1986) in which i was saed ha he percepion ime a high speeds is usually less han ha a low speeds (i.e. speed is a facor). One possible soluion for his limiaion is o make eiher he angular velociy or he angular velociy hreshold as a funcion of he leader/following s speed. Since he angular velociy value comes from a mahemaical calculaion (see Equaions and 3), i is no possible o include hese speed values ino he equaions wihou alering he whole equaion. Therefore, furher invesigaion is required o look ino how he effec of he speeds is aken ino consideraion when deermining he angular velociy hreshold values. Also, he effec of oher parameers influencing he calculaion of he angular velociy hreshold values (such as difference in spacing beween vehicles) should be included in he invesigaion. 5A1.8
4.3 Represening driver s reacion ime In addiion o he limiaions described above wih regards o he inabiliy of he visual angle model o represen driver s reacion ime as a funcion of leader/follower speeds, here is anoher limiaion in represening his facor. The model is incapable of represening how driver s reacion ime differs if he leader makes his/her deceleraion wih or wihou using he brakes (i.e. wheher or no brake lighs are obvious o he follower). However, his issue is no specific o visual angle models only. I is one of he exising limiaions in oher car following models and requires furher invesigaion. 5. Proposed model 5.1 Inroducion As discussed above, he main limiaions in visual angle models are relaed o he proper selecion of suiable values for he angular velociy hresholds ( Ө/ ) for making a decision on acceleraion or deceleraion. This difficuly is due o he fac ha angular velociy hresholds have no obvious connecion wih raffic characerisics and canno be direcly measured from sie. Therefore, i is decided o propose a car-following model which should be able o deal wih he above limiaions and ake ino consideraion oher advanages for such visual angle models. The main assumpion of he proposed model is based on he use of he Jus Noiceable Difference JND hreshold in deermining he ime headway. The JND is relaed o Weber s law which saes ha any change in behaviour is no noiceable unil his change exceeds a cerain percenage from is original sae. This percenage is repored in lieraure o be beween 10 and 15%. For example, Hoffmann and Morimer (1994 and 1996) menioned 1% for he JND value, while Bracksone and McDonald (1999) repored ypical values of 10%. The proposed model depends mainly on he preferred ime headway (PTHead). This headway is a funcion of he follower s speed. If he speed of he follower is high, he disance beween he leader and follower should increase. In spie of having he widh of he leader as a facor in he equaion used for visual angle models, he effec of he heigh of he leader is no represened in hese models. This needs o be invesigaed furher since here has been an increase in he las decades in he overall size of vehicles using he road, paricularly in relaion o heigh (DETR, 1999 and Deparmen of Transpor, 007). 5. Effec of vehicle s size 5..1 Effec of vehicle s widh In order o inroduce he effec of vehicle s widh ino he proposed model, he following equaions may be used: w PTHead 1 DHead *. Equaion 5 1.8 Where: c is he consan of he equaion c DHead is he desired headway (seconds) for he case of Car following Car PTHead1 is he preferred ime headway in seconds w is he widh of leader (for w>1.8 m) 5.. Effec of vehicle s heigh Similarly, he effec of vehicle s heigh may be enered ino he model as follows: h PTHead DHead *. Equaion 6 1.5 d This paper is produced and circulaed privaely and is inclusion in he conference does no consiue publicaion. 5A1.9
Where d is he consan of he equaion h is he heigh of he leader (for h>1.5 m) PTHead is he preferred ime headway in seconds 5..3 Summary Finally, he maximum value of eiher PTHead 1 or PTHead obained from Equaions 5 and 6 will be seleced as he preferred headway PTHead for ha driver. The values for he widh (w) and he heigh (h) should be obained from sie. However, and in order o overcome such limiaions ha are discussed in Secion 4.1, hese wo values are recommended o be based on average values represening vehicle s ype (i.e. Cars or HGVs ). The wo consans in he Equaions (i.e. c and d) should be also obained based on he calibraion and validaion process. If he majoriy of drivers are no affeced by he size of he leaders, zero values for hese wo consan should be assigned. Therefore, he following Equaion is suggesed o replace Equaion 4. ac f x l, x f, 5.3 Headway hresholds and assumpions v f, * PTHead * v( f, ) PTHead * lengh( l). Equaion 7 There are wo hreshold values which can be used in he proposed model (based on Weber s law). These are he minimum and maximum preferred headway hresholds by a driver. a. Minimum headway hreshold: Once he acual headway is less han he minimum, he follower will decelerae o mainain his/her preferred headway based on Equaion 7. This is given by Equaion 8. min headway PTHead *0.88. Equaion 8 b. Maximum headway hreshold: Once he acual headway is higher han he maximum, he driver will accelerae o mainain his desired speed or o mainain his/her preferred headway based on which one of hose gives a minimum acceleraion. max headway PTHead *1.1. Equaion 9 If none of he above hresholds are exceeded, he follower will keep a consan speed (i.e. acceleraion is zero). However, if i is shown during he calibraion and validaion process ha he minimum and maximum headways are no following he above formulas, he consans values (0.88, 1.1) should be changed. Finally, furher ess are needed o show he main advanages of his proposed model. 6. Conclusions This paper presened he main advanages and limiaions in using visual angle models o simulae real raffic condiions. The main advanage is he model s abiliy o represen brake reacion ime as a funcion of raffic densiy and relaive speeds beween leading and following vehicles. Anoher advanage is in represening he effec of he widh of he leader as par of he calculaions used in his model. I was found ha if he angular velociy hreshold values ( Ө/ ) of abou ± 0.003 rad/sec were used, hese gave equivalen driver s reacion imes which were generally wihin accepable ranges. 5A1.10
The selecion of appropriae hreshold values could help in providing he seings needed for he car-o-car communicaion sysem in fuure rials on co-ordinaing car following manoeuvres wihou he need o drivers inervenion. In his paper wo main limiaions were discussed, namely he effec of a sligh change in he widh of he leader and he oher is he fac ha he equaion for visual angle models should no be used when he angular velociy is wihin he seleced hreshold values (say beween ± 0.003 rad/sec). Anoher limiaion for using such models is relaed o he appropriae choice of hreshold values for differen operaing speeds. This is due o he fac ha such models are only sensiive o relaive speeds beween he leader and is follower raher han he acual speeds ha hey are ravelling wih. The proposed model described in his paper deals wih hese limiaions. A suggesion has been made on how he size of vehicles could be represened in he model. Also, he preferred ime headway hresholds (minimum and maximum) are obained for he proposed model. Fuure work is needed in order o es he assumpions made for he proposed model agains real raffic daa. 7. References Bracksone, M. and McDonald, M. (1999). Car Following: A Hisorical Review. Transporaion Research Par F, vol., pp. 181-196. Deparmen of he Environmen, Transpor and he Regions DETR (1999) Transpor saisics Grea Briain (TSGB), 1999 Ediion, The saionary Office, 5 h Ediion,. Deparmen of Transpor (007) Transpor saisics Grea Briain (TSGB) 007 Ediion, TSO Publicaions, 33 rd Ediion,. Ferrari, P. (1989). The Effec of Driver Behaviour on Moorway Reliabiliy. Transporaion Research Par B, vol. 3B, No., pp. 139-150. Fox, P. and Lehman, F.G. (1967) Safey in car following. Newark, (N.J.), Newark college of Engineering. Gazis, D.C., Herman, R. and Rohery, W. (1960) Nonlinear Follow-he-leader Models of Traffic Flow. Research Laboraories, General Moor Corporaion, Warren, Michigan, pp. 545-567. Hidas, P. (1996) A car-following model for urban raffic simulaion. Traffic Engineering and Conrol. Volume (39) p.p. 300-305. Hoffman, E.R. and Morimer, G.R. (1994). Esimae of Time o Collision. Acciden Analysis & Prevenion, Volume 6, Issue 4, pp. 511-50. Hoffman, E.R. and Morimer, G.R. (1996). Scaling of Relaive Velociy beween Vehicles. Acciden Analysis & Prevenion, Volume 8, Issue 4, July 1996, pp. 415-41. Johansson, G. and Rumer, K. (1971) Drivers Brake Reacion Times. Human Facors, Vol.13, No. 1, Feb., pp. 3-7. Lerner N., Huey R., McGee H. and Sullivan A. (1995) Older Driver Percepion-Reacion Time for Inersecion Sigh Disance and Objec Deecion, Repor FHWA-RD-93-168, Federal Highway Adminisraion, U.S. Dep. of Transporaion, Washingon DC. Maycock, G, Brocklebank, P. and Hall, R. (1999) Road Layou Design Sandards and Driver Behaviour. Proceeding Insiuion of Civil Engineers Transpor, pp.115-1. Michaels, R.M. (1963) Percepual Facors in Car following. In Proceeding of he Second Inernaional Symposium on he Theory of Traffic Flow. pp. 44-59, Paris: OECD. Michaels, R.M. and Cozan, L.W. (1963) Percepual and Field Facors Causing Laeral Displacemens. Public Roads 3. pp. 33-40. O Flahery, C.A. (1986) Highways. Volume 1, Traffic Planning and Engineering, 3 rd Ediion, Edward Arnold,. Panwai, S. and Dia, H. (005). Comparaive Evaluaion of Microscopic Car-Following Behaviour. IEEE Transacions on Inelligen Transporaion Sysem, Vol. 6, No. 3, pp. 314-35. This paper is produced and circulaed privaely and is inclusion in he conference does no consiue publicaion. 5A1.11
Parker, M.T. (1996) The Effec of Heavy Goods Vehicles on Following Behaviour on Capaciy a Moorway Roadwork Sies. Traffic Engineering and Conrol, 37(8), pp. 54-531. Pipes, L.A. (1967) Car following Models and he Fundamenal Diagram of Road Traffic. Transporaion Research, Vol. 1, pp. 139-150. www.rafficechnologyoday.com/news.php?newsid=9008 German program coordinaes driving, new sofware/echnology from Germany published on 5 November, 008, accessed 8 November, 008. Yousif, S. (1993). Effec of Lane Changing on Traffic Operaion for Dual Carriageway Roads wih Roadworks, PhD hesis, Universiy of Wales, Cardiff. Zhang, X. and Bham, G.H. (007). Esimaion of Driver Reacion Time from Deailed Vehicle Trajecory Daa, Proceeding of he 18 h IASTED inernaional conference, (pp. 574-579). 5A1.1