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A collision model for safety evaluation of autonomous intelligent cruise control
Accident Analysis and Prevention 31 (1999) 567 – 578 www.elsevier.com/locate/aap
A collision model for safety evaluation of autonomous intelligent cruise control Ali Touran a,*, Mark A. Brackstone b, Mike McDonald b a
Department of Ci6il and En6ironmental Engineering, Northeastern Uni6ersity, 420 Snell Engineering Centre, Boston, MA 02115, USA b Transportation Research Group, Uni6ersity of Southampton, Southampton, SO17 1BJ, UK Received 14 June 1998; received in revised form 14 January 1999; accepted 19 January 1999
Abstract This paper describes a general framework for safety evaluation of autonomous intelligent cruise control in rear-end collisions. Using data and specifications from prototype devices, two collision models are developed. One model considers a train of four cars, one of which is equipped with autonomous intelligent cruise control. This model considers the car in front and two cars following the equipped car. In the second model, none of the cars is equipped with the device. Each model can predict the possibility of rear-end collision between cars under various conditions by calculating the remaining distance between cars after the front car brakes. Comparing the two collision models allows one to evaluate the effectiveness of autonomous intelligent cruise control in preventing collisions. The models are then subjected to Monte Carlo simulation to calculate the probability of collision. Based on crash probabilities, an expected value is calculated for the number of cars involved in any collision. It is found that given the model assumptions, while equipping a car with autonomous intelligent cruise control can significantly reduce the probability of the collision with the car ahead, it may adversely affect the situation for the following cars. © 1999 Elsevier Science Ltd. All rights reserved. Keywords: Autonomous intelligent cruise control (AICC); Collision; Monte Carlo simulation; Risk assessment; Human behaviour
1. Introduction Autonomous intelligent cruise control (AICC) is one of the devices that has been introduced under the broad field of intelligent transportation systems (ITS) with the objectives of increasing the capacity and improving the safety of existing highway systems. AICC, in the context of this work, is defined as a device that exerts control on the car’s throttle and brakes to obtain a set headway. It is one of the few advanced vehicle control systems (AVCS) devices that may be implemented in the next 5–10 years; working prototypes are being tested extensively at this time (Tribe et al., 1994). One reason for imminent use of AICC is that no road infrastructure modifications are needed for the device to work effectively (Chira-Chavala and Yoo, 1994). In contrast to AICC, many ATT systems such as convoy
* Corresponding author. Tel.: +1-617-3732444; fax: + 1-6173734419. E-mail address: [email protected] (A. Touran)
driving require considerable investment in road infrastructure (DIATS, 1996). Furthermore, current cruise control technology is well tested and many drivers, especially in the U.S., are familiar with its function. Several researchers have worked on various aspects of AICC safety. These aspects include the effect of AICC (or similar systems such as adaptive cruise control) on regulating the traffic flow and reducing smaller headways (Benz, 1994), its effectiveness in avoiding crashes in critical traffic conditions (Nilsson, 1994), and safety benefits derived from its implementation (ChiraChavala and Yoo, 1994; Najm and Burgett, 1997; Godbole et al., 1998). Most of the work carried out in the area of AICC safety, especially the earlier ones, has been based on hypothetical specifications for the device. As mentioned before, several AICC prototypes are now in the final development stages by various manufacturers. Detailed technical information for the working prototypes is available. The work reported in this paper uses this information to conceive a realistic AICC system and develop a framework for safety evaluation of AICC
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under extreme driving conditions. This is accomplished by developing a rear-end collision model and evaluating the effects of AICC on the car ahead and on cars following the equipped vehicle. This is important because most of the work done so far only considers the effects of the car ahead on the equipped vehicle, disregarding what might happen to followers as a consequence of equipping a car with AICC. The model developed here does not consider some important cases such as large differences in following cars’ speeds, possible driver inattentiveness because of his dependence on AICC, or the effect of queues in highway driving. Also, we are not considering device reliability and its probability of failure here; that is the subject of another research project currently under way. The results of the models need to be verified with real world testing but the models developed can provide a foundation for these testing programs.
2. Model development In order to assess AICC’s potential impact on the number of rear-end collisions, the following steps were taken. First, a set of specifications was developed for the device. Then a car collision model was developed by faithfully incorporating device specifications. Then input data for the collision model were chosen and model parameters were refined using sensitivity analysis. Finally, we ran several simulation experiments to calculate the probability of collision under various conditions. The model considers the probability of collision of the equipped car with the car ahead under extreme braking conditions by the lead car. Also, it evaluates the collision probability for the cars following the equipped car. These steps are described in detail in the following sections.
3. System specification Using available technical data and literature, a set of specifications was developed for the AICC device (for example see Alvisi et al., 1991). The authors were in contact with various European car and device manufacturers and benefited from their advice and consultation. The AICC device envisaged for this research, although hypothetical, is similar to several working prototypes and has the following characteristics: it detects the car ahead at a distance of up to 150 m, and can compute relative speed and distance at 120 m. It maintains a target headway of 1.4 s by exerting control over the throttle (from − 0.8 to +1 m/s2) and if distances between cars fall below a minimum, it can activate the brakes (up to − 3 m/s2). If this braking rate is not sufficient to maintain minimum safe distance, the device
warns the driver. When the driver intervenes by braking, the control is transferred to him. Also, when the driver accelerates, for example in case of overtaking another car, the controls are transferred to him. All the details of the device braking mechanism and system delays are carefully specified based on current working prototypes such as those being produced by the industry.
4. Car collision model Two car collision models were developed. In the first model, in order to evaluate the effectiveness of the AICC device in avoiding rear-end collisions, a group of four vehicles was considered where the second vehicle from the front was assumed to be an AICC-equipped car. The second model consisted of four vehicles where none were equipped with AICC. The results obtained from these two models were compared in order to evaluate the effect of using AICC in a car group.
4.1. AICC-equipped car model Except for platoons, most of the safety research done in the past focused on the performance of AICC with respect to the car moving in front of the equipped vehicle. This model considers not only what happens to the car in front of the equipped vehicle but also what may happen to the cars following it. The model for the third car from front describes the behavior of a car following a car with AICC, while the model for the last car that will behave approximately similar to cars following it (not modeled here), is modeled as an unequipped car following unequipped vehicles. The main characteristics of the model are given in Fig. 1. The graph shows car decelerations vs time. The heavy line represents the equipped car. Assume that the lead car breaks hard in an emergency situation at t=0 with the rate of aL. The AICC device in the following car reacts by applying the brakes after a reaction time of tR. The jerk limit, J, is used to describe the transition between the instant that the system activates the brakes and the time of full braking by AICC at the rate of aF. As the driver of the equipped car realizes that the lead car deceleration is far sharper than can be safely accommodated by automatic brakes, he intervenes by braking harder with a deceleration rate of a *; F tA is the time delay between the application of brakes by AICC and the harder application of brakes by driver (the time when the driver overrides AICC and brakes hard to avoid collision). The following parameters describe model characteristics: tR, aF,
AICC reaction time (s) AICC brake levels (m/s2)
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Fig. 1. Car following model for the car equipped with AICC.
a *, F J, Dt = tA, VL, VF, xL(t), xF(t),
driver brake rate after deciding AICC deceleration is not sufficient (m/s2) jerk limit (m/s2) time delay for the jerk limit (s) time delay after AICC braking and before driver Intervention (s) lead car’s original speed (m/s) AICC-equipped car’s original speed (m/s) distance travelled by lead car after t seconds since the start of braking by lead car (m) distance travelled by AICC car after t seconds since the start of braking by lead car (m)
After the equipped car brakes, there is a delay of T seconds before the following car breaks. T, or the perception/reaction time (PRT), is the time that it takes the following driver to see the brake light of the car in front and to react. A two step deceleration is considered in this model: an initial milder deceleration rate, a, and a final harder deceleration rate, a*. This, in effect, says that followers imitate the behaviour of drivers ahead. The following parameters describe cars that are following the AICC-equipped car. The first car following the equipped car is denoted with i and the car following that by i +1. ai, a *, i Ti ,
Initial brake level (m/s2) final brake level (m/s2) time delay after the driver brakes the first time with rate ai and before braking harder (s)
Vi, Vi+1, xi (t),
xi+1(t),
original speed of the first car following AICC-equipped car (m/s) Original speed of the car following the ith car (m/s) distance travelled by the first car following AICC car after t seconds since the start of braking by lead car (m) distance travelled by the follower to the first car following AICC car after t seconds since the start of braking by lead car (m)
The car following the ith car is the i+ 1th car with parameters ai + 1, a *i + 1, and Ti + 1. It is further assumed that this is a series system (Glimm and Fenton, 1980), wherein each vehicle must sense the braking of its nearest front vehicle before starting to brake. This is conservative because in most cases, the following cars can see the brake lights of several vehicles ahead and take action appropriately. In a series system the total delay Ttot before the ith car brakes would be the sum of individual delays: ttot =tR = nT, where n is the nth vehicle following the AICC-equipped car. Equations of distance and speed as functions of time, i.e. x(t) and 6(t), were derived for each vehicle using basic Newtonian motion principles.
4.2. Headways We have used the concept of time headways to describe inter-vehicle distances in this research. Defini-
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Fig. 2. Car following model with no AICC.
tion of time headway is compatible with Drew (1968), where the speed of the following vehicle is applied to the time gap to calculate the distance between vehicles. To avoid collision, distance travelled by the lead vehicle plus the follower’s headway tH, should be greater than the distance travelled by the AICC-equipped car plus a minimum distance, x0.
cars. Because there is no intermediate braking level by AICC, only one level of braking is assumed for each of the four cars resulting in a set of simple motion equations.
XL + tHVF ]XF +x0
Input data for the collision model were selected from a wealth of information supplied by industry, research conducted at the Transportation Research Group (TRG) of the University of Southampton on behavioural aspects of driving in the past few years, and published reports and papers. We also benefited from data collected by an instrumented vehicle that has been assembled in the past 2 years by the TRG. This car is equipped with a radar, optical speedometer, and videoaudio monitoring system (McDonald et al. 1998) and can detect other cars, their distance and relative speed. A base case scenario was developed for an extreme case where the lead vehicle brakes hard to cope with an unexpected event. The equipped car reacts to this situation utilizing AICC capabilities and driver intervention. Following cars react according to drivers’ perception/ reaction times and the behaviour of the car immediately preceding them. The base case scenario considered here models an emergency brake by the lead vehicle with a rate of −7.5 m/s2. Several sources report maximum braking rates of between 0.8 g and g (Glimm and
(1)
In the above equation, XL is the distance to stop for the lead vehicle and XF is the distance to stop for the following vehicle. The same equation can be applied to any pair of following cars. The headway used for AICC and the followers can be chosen independently. In this way, one can evaluate the difference between the headways of an AICC-equipped car and a non-equipped car. The equations developed were used in constructing a computer model using Excel™ spreadsheet. The same spreadsheet structure was later used for performing probabilistic Monte Carlo simulation using Crystal Ball™ 1996 software package.
5. Model with no AICC Fig. 2 shows the model used for the case where no AICC was present. This model also consists of four
6. Data for the collision model
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Table 1 Input data for the model parameters Parameter
Lead car
Equipped car
First follower
Second follower
Initial velocity (m/s) Initial braking rate (m/s2) Final braking rate (m/s2) Delay time for braking (s) Initial brake duration (s)
VL = 30 – aL = −7.5 – –
VF =30 aF =−3.0 a *F =−7.5 tR =0.45 tA =1
Vi =30 ai =−4.0 a *i =−7.5 T =1.1 Ti =1
Vi+1 =30 ai+1 =−5.0 a *i+1 =−7.5 T= 1.1 Ti+1 =1
Fenton 1980; Tsao and Hall 1994; Godbole et al. 1998). Mannering and Kilareski (1990) report a maximum coefficient of road adhesion in good, dry pavement of 1 g. The effective braking rate will obviously be smaller than this coefficient depending on several factors. In our case, it is assumed that braking takes place on good dry pavement. The car which is equipped with AICC, is travelling with an initial speed similar to the lead car’s speed. As the AICC warns the driver at 1.4 s headway, the case base scenario provides a worst case situation where emergency braking by lead vehicle coincides with the minimum distance available between cars. AICC alerts the driver of the equipped car while starting to apply the brakes automatically up to a deceleration of − 3 m/s2. Maximum braking level for AICC has been studied in several sources and the value chosen is representative of those reported. This level seems to be the maximum braking rate at which the driver does not feel discomfort and panic (Touran 1998). The driver applies the brakes 1 s after application of brakes by AICC, thus taking over the control of the brakes (tA =1 s). This value is analogous to the perception/reaction time of the driver; we have assumed that activation of brakes by AICC has already alerted the driver, hence a relatively short perception/reaction time is justified. The followers imitate the same pattern by initially braking softer and then harder. The input values used in the base case scenario are presented in Table 1.
6.1. AICC characteristics To specify AICC response time, we contacted device manufacturers. The following is a typical specification given the current state of technology. AICC’s brake controls system has to detect the object and actuate the brakes. Object detection consists of tracking and selecting the object (to identify the vehicle ahead and distinguish it from other vehicles in other lanes, etc). Object detection takes from 100 to 400 ms. Actuation has two components: processing delay of 40 ms and application of brakes (for a typical −3 m/s2 deceleration) of 250
ms1. With these values in mind, tR (detection=processing delay) was chosen as 0.45 s, and transition time to achieve full braking rate t, was chosen as 0.25 s. From here by referring to Fig. 1, jerk limit J, is calculated as 12 m/s3 (aF = − J×Dt = − 12×0.25= −3 m/s2).
6.2. Safe dri6ing distance A collision happens when at any point during a brake by the lead car, the distance travelled by the lead car plus the headway distance between the follower and lead car are smaller than the distance travelled by the follower. Various time headways were selected for the cars in the model and each situation was evaluated. Fig. 3 gives the results of analysis. In each case, distances travelled after braking by the lead car and follower, xL(t) and xF(t), are plotted against time following the format suggested by Tsao and Hall (1994). Similar graphs were generated for each pair of cars. Also, various other graphs such as vehicles’ relative speeds vs time and time to collision (TTC) 6s time were developed and are reported in Touran (1998). In each case, a minimum distance of 5 m was assumed to be necessary at the time when cars are stopped. This should be sufficient to account for the length of cars, especially if the headway gap is measured between the front of the follower and the front of the preceding car. It is evident that a time headway of 1.4 s is sufficient to avoid collision under assumed conditions (Fig. 3). Assuming wet conditions, we ran the model with reduced deceleration rates, where the lead car and the followers brake with a rate of −6.5 m/s2. This did not create a more critical situation and still a headway distance of 1.4 s was sufficient to avoid collision. The calculated headway seems consistent with the results of research on traffic conflicts, where drivers seem to use time gaps of
1
The data reported is the result of discussion with Mr Mark Basten of Lucas/Varity (Lucas, undated; Lucas Varity, undated). Some published sources, especially from U.S., report much shorter response times such as 0.1 s. These values could not be substantiated at this point, although it might be achievable in the future. The emphasis of the paper is a realistic model given current state of technology.
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Fig. 3. AICC model output for each pair of following cars.
1.5 s as a natural limit, although in many situations the following times on motorways are much shorter.
7. Sensitivity analysis In order to ensure that safe headway distances are established, a deterministic sensitivity analysis was conducted to evaluate the impact of each major parameter on the possibility of a collision. Parameters considered included initial speeds and decelerations, perception– reaction times, and initial braking durations. In each case, the value of parameter of interest was incrementally changed while keeping other parameters constant. Collision was investigated for each scenario. By far, the perception–reaction time T, had the greatest impact on the required headway to avoid collision for the car following the equipped car (Table 2). The model was far less sensitive to changes in initial speeds or initial braking rates in the deterministic analysis. We did not consider some of worst possible cases such as traffic queues or large speed differences. While the model can Table 2 Impact of PRT on the required headway PRT [T] (s)
Headway for 1st follower (s)
Headway for 2nd follower (s)
0.7 1.1 1.5 2.0
0.7 1.1 1.5 2.0
0.8 1.2 1.5 2.0
easily evaluate the impact of speed differences between cars, it was assumed that original car speeds were equal. This is partially justified because AICC tends to adapt the car speed to the speed of the lead car and under normal driving conditions, manages to achieve a DV of zero.
8. Monte Carlo simulation Several parameters used in the model are a function of driver behaviour and his or her interaction with car and AICC. It would be unrealistic to model such parameters only with deterministic variables. In order to assess the inherent variability of such parameters, and also to incorporate our lack of precise knowledge of some parameter values, a Monte Carlo simulation was performed. Several model parameters including braking rates and durations, and perception–reaction times were randomized according to selected distributions and the collision model was simulated. Simulation results gave the probability of collision between any of the two following cars. Based on crash probabilities, an expected value was calculated for the number of cars involved in any crash. One can use this parameter as a comparison measure for assessing the safety of any given combination of equipped and non-equipped cars. The same procedure was repeated for the model with no AICC. Similar input values were used for this model and probabilities of collision were simulated for any two following cars. Results were compared to assess the effect of AICC on safety.
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8.1. Car distances after collision
8.2. Simulation 6ariables
The model described earlier in this paper needed to be modified for probabilistic analysis. This is because when a collision occurs, distances travelled by cars will be restricted. As an example, if the AICCequipped car collides with the lead car, maximum distance travelled by the equipped car will be limited to the distance up to the collision point plus the distance to stop for both cars after collision. This may have an effect on the probability of collision between the car following the equipped car with the equipped car. In order to consider this limitation, a headway distance of 1.4 s is chosen for all cars; in each simulation iteration, the model checks to see if a collision has occurred between cars ahead of a specific car. If there is a collision, the model updates distances available to that specific car and checks for collision between its followers. Vehicle motion after collision has been modelled by several authors in the past. A comprehensive paper by Grime and Jones (1969) gives mathematical equations for motion after collision. When two vehicles collide, given certain mechanics assumptions, the combination will move at a common velocity of V3 = (M1V1 +M2V2)/(M1 +M2), where M and V are mass and velocity of the two vehicles before the crash. If the preceding vehicle is stopped (V1 = 0 ), then the speed of the combination would be roughly half of the speed of the follower, assuming equal vehicle weights. Deceleration of cars during collision have been studied in the past; various ranges are suggested. Glimm and Fenton (1980) report on experiments conducted with a car colliding with a fixed barrier, a moving car, and a stopped car. In all cases, collision deceleration is much higher than pre-crash deceleration of the follower. Glimm and Fenton (1980) use a collision deceleration of 10 g. Furthermore, they suggest that motion of cars after collision may be calculated using collision deceleration and the new combined speed. Distance to stop after collision, Xcoll, may be found from Xcoll =V 23/2acoll, where acoll is the rate of deceleration in collision. In our case, V3 is sufficiently small to make Xcoll negligible, especially since acoll is so large. In cases where the collision happens before the preceding car has come to a stop, it is assumed that the distance to stop for the combination is equal to distance to stop for the preceding car. This means that the effect of collision would be to limit the distance available to the following car. In this research, all possible cases of collision between various cars were considered and distances available to followers after collision between cars in front were calculated and incorporated into the simulation models (Touran 1998).
The following is a brief description of random input variables.
8.2.1. Initial braking rates Initial braking rate for each car was modelled as a linear function of the braking rate of the car in front. The AICC brakes with a rate of −3 m/s2; initial braking rate for each follower was set equal to initial braking rate of the car in front plus a uniform distribution with mean −0.5 m/s2 (ranged between 0 and − 1.0). Assuming a larger braking rate for the follower is consistent with findings of technical literature on driver behaviour. For example, within the SISTM model (Wootton Jeffreys 1990) it is assumed that if a driver sees a vehicle ahead decelerating at a rate harder than − 3 kph/s, at a speed of greater than 30 kph, he will decelerate at − 6 kph/s. Najm and Burgett (1997) use a follower’s deceleration rate of between −0.5 and − 0.7 g in response to a lead car deceleration with a mean of − 0.31 g. Use of uniform distribution denotes our lack of precise data and knowledge regarding the nature of this variable. 8.2.2. Final braking rates The same logic was used for modeling final braking rates. Godbole et al. (1998) collected data on maximum deceleration capabilities of new light duty passenger cars on dry pavement. They fitted a truncated normal distribution with a mean value of − 7.01 m/s2 to the data. We have assumed that the lead car brakes with a rate of −7.5 m/s2.The final braking rate for each following car was modelled as the final braking rate of the car in front plus a uniform distribution with mean − 0.5 m/s2 (ranged between 0 and − 1.0). In no case this value was allowed to go below − 10 m/s2. 8.2.3. Braking duration for AICC Braking duration for AICC was modelled using a log-normal distribution with mean 1.0 and standard deviation 0.2 s ranging between 0.5 and 1.92 s. In this case, the braking duration is somewhat analogous to the driver’s reaction time. Perception reaction times have been modelled using a log-normal distribution (Taoka 1989; Najm and Burgett 1997). 8.2.4. Initial braking duration for followers Initial braking duration for follower of the equipped car was set equal to AICC braking duration plus a triangular distribution with the following parameters: minimum= − 0.5, mode= 0, maximum= 1 s. The braking duration for the last car was set equal to the preceding car’s braking duration plus
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a triangular distribution with the following parameters: minimum = − 0.5, mode=0, maximum= 1 s.
8.2.5. Perception/reaction time for followers of the equipped car A truncated log-normal distribution (mean= 1.1, standard deviation=0.22, ranged between 0.8 and 2.5 s) was used for modelling PRT for both drivers. The mean value is consistent with values suggested by Najm and Burgett (1997). Also, McGehee et al. (1994) report that in a study by Olson and Sivak, a range between 0.81 and 1.76 s was found for PRT. Studies suggest that PRT may be shorter for more intimidating conditions. In our case, given relatively high speeds assumed, we feel that values chosen are realistic. 8.2.6. Input 6alues for the model with no AICC Input values for the model with no AICC were identical to the data used for the other model described thus far. The only difference was the second vehicle from the front, where in this model, was not equipped with AICC. Its braking rate was modeled as − 7.5 m/s2 plus a uniform random variable between 0 and − 1.0 m/s2. Perception/reaction times were modeled with distributions similar to the other followers. Gap between cars was assumed to be 1.4 s. Also, because there was only one level of braking for each car, there were no initial braking rates and durations (refer to Fig. 2). 8.3. Simulation results The model was run using Excel™ spreadsheet and Crystal Ball™ simulation package. Simulation was run for 5000 iterations using a Latin Hypercube sampling technique. It was noted that as the number of simulation iterations was increased beyond 5000, the outcome remained practically unchanged. In each iteration, three objective functions were calculated verifying the possibility of collision between each pair of following cars. For each pair of cars, the objective function was set equal to the distance travelled by the preceding vehicle plus headway distance minus the distance travelled by the follower plus the minimum distance. These functions in effect, give inter-vehicle distances after the lead car starts to brake until cars come to a stop. If this value was positive, no crash was assumed. The probability of crash between AICC-equipped car and the lead car was 2.76%. This probability was much higher for the cars following the equipped car. The probability of crash between the car following the equipped car and the AICC car was 18.38%. The probability of crash between equipped car and all others (the lead and the followers) was 0.52%. In each case a headway distance of 1.4 s (distance of 301.4 = 42m) was assumed between cars. Using basic axiom of
probability, one can calculate probability of crash for the equipped car as follows: P(crash)= P(crash between lead car and equipped car)+P(crash between equipped car and its followers)− P(crash between equipped car, lead car and followers)= 0.0276+0.1838−0.0052= 0.2062.
9. AICC’s impact on the followers Most of the work describing the effect of AICC on safety consider the equipped car and the car ahead (for examples, see Zhang and Benz, 1993; Farber, 1996). It is also important to study what happens to the following cars when the equipped car reacts to an emergency situation. Researchers working on vehicle convoys have considered this issue (Glimm and Fenton, 1980), but in convoy driving, unlike our situation, all vehicles are equipped with warning and control devices. Takubo (1995) considered the possibility of crash between several vehicles, one of which was equipped with ACC, in his simulation study. The study predicted the number of fatalities and injuries in case of accidents, but the hypothetical ACC system used was capable of full braking, something that is not conceivable under current industry trends.
9.1. Number of cars in6ol6ed in a collision We suggest that in order to make a fair assessment of the impact of AICC on safety, one should look at an expected value of the number of cars involved in a collision rather than just focusing on the equipped car and the car ahead. A routine was developed to calculate the number of cars involved in each of the crash situations realized in the simulation. The results of the simulation reported earlier (5000 iterations) was used to compile these ‘accident’ data. Data provided in Table 3 gives the outcome of the analysis. For the case where the second car is equipped with AICC, the expected number of cars involved in a collision in an emergency situation is calculated as 0.622× 0+ 0.297× 2+0.079× 3+0.003× 4= 0.843 (see Table 3 for collision statistics). Note that most of the time (62.2% in this scenario), a collision does not occur in an emergency situation. The expected number of cars involved in a collision, given that a collision has occurred, is calculated as 2.22. These values can be used as measures of system safety in comparative analyses. For the case where no car is equipped with AICC, the expected number of cars involved in a collision in an emergency situation is 0.935 (0.601× 0 + 0.285× 2+ 0.095×3 +0.020× 4= 0.935; see Table 3). The expected number of cars involved in a collision, given that a collision has occurred, is 2.34.
A. Touran et al. / Accident Analysis and Pre6ention 31 (1999) 567–578 Table 3 Multi-car collision statistics Number of cars involved in crash (1)
Frequency of crash (2)
Probability of crash 2} 5000 (3)
2nd Car equipped w/ AICC No crash 2 Cars 3 Cars 4 Cars
3108 1485 394 13
0.622 0.297 0.079 0.003
No car equipped w/AICC No crash 3005 2 Cars 1424 3 Cars 473 4 Cars 98
0.601 0.285 0.095 0.020
9.2. Safety impact of AICC Using simulation outputs, one can calculate probabilities of avoiding a collision under assumed conditions. Table 4 gives these probabilities for various values of braking rates. In each case, the simulation was run for 2000 iterations. As can be observed from Table 4, the probability of avoiding a collision is increased as the braking rate is decreased. This is partly due to modeling assumptions. Assuming that the parameter values mirror realistic behavior, in case of lead car’s lower braking rates, the followers have a better chance of stopping in time by reacting with higher deceleration rates. For braking rates softer than −6.5 m/s2, the cause of a rear-end accident would not likely be limited to perception/reaction time, but includes other factors such as driver inattention; there is sufficient braking capacity to avoid the collision under assumed headways. While AICC has improved the chances of avoiding a collision between the equipped car and the car ahead, its impact on the followers is much less clear. It appears that having AICC has slightly increased the probability
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of collision for the followers. Furthermore, while AICC alerts the driver if the gap falls below the safe distance, the followers have no warning system and are likely to drive much closer than the assumed headway of 1.4 s. This will cause an increase in the probability of collision for the cars following the equipped car. We kept the headway distance for the followers equal to 1.4 s so that a more direct comparison with the effect of AICC can be accomplished. Collision probability values are very sensitive to perception/reaction time and to initial brake duration (duration of soft braking). Increasing perception/reaction times or initial braking durations greatly increases the probability of collision behind the equipped car. In order to investigate the share of each input parameter in variance of the outcome, a sensitivity analysis was conducted on the results of simulation. In case of followers, the variability in PRT and initial braking durations contributed greatly to the probability of collision. PRT values have been extensively reported in published literature and the values used in the model should be reliable. Initial braking durations on the other hand, are based on perceived behaviour of drivers, and parameters used in triangular distributions are educated guesses at best and should be verified in the field. Unfortunately, simulation results are highly sensitive to parameters of the triangular distributions used. An increase in the high tail of the distribution, would indicate a sharp increase in the probability of collision for cars following the equipped car. Table 5 summarizes the results of sensitivity analysis. For each pair of cars, the contribution to variance of the outcome by the most sensitive parameters is presented. The percentage points are relative percentage of variance or uncertainty in the forecast (here the forecast is the final distance between cars after stopping, where a negative distance is an indication of collision) attributable to each input parameter. The higher the percentage point reported in Table 5, the more influential the parameter on the final outcome (crash or no crash). Input parameters not reported in this table had negligible effect on the probability of collision.
Table 4 Probabilities of crash avoidance with various emergency braking rates Braking rate
Lead-followinga
Following-ith
ith–i+1th
a=−8.5 m/s2 With AICC Without AICC
0.932 0.837
0.768 0.789
0.705 0.736
a=−7.5 m/s2 With AICC Without AICC
0.975 0.860
0.828 0.820
0.745 0.788
a=−6.5 m/s2 With AICC Without AICC
0.995 0.881
0.860 0.845
0.801 0.837
a
‘Following’ is the car that may be equipped with AICC.
10. Probability of a rear-end collision In the previous section, the probability of a crash under various braking rates was calculated. It would be useful to assess the likelihood of occurrence of such decelerations. We have collected field data using an instrumented vehicle at TRG as noted earlier in this paper. Using this vehicle, 147 min of driving data was collected during 3 different days on the M3 motorway near London. Total miles driven amounted to 104. The data was sampled at 1-s intervals and a histogram of various braking rates of less than − 0.3 m/s2 was
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Table 5 Contribution of input parameters to the variance of the model’s outcome Input parameters
Lead-followinga AICC brake duration Final braking rate for the equipped car Following-ith ith driver’s PRT Initial braking duration (Ti) of i th car ith–i+1th i+1 th driver’s PRT Initial braking duration (Ti+1) of i+1th car a
Contribution to variance (%)
77.5 22.0 59.6 35.1 62.9 28.4
‘Following’ is the car equipped with AICC.
constructed. These decelerations spanned 13.8% (1220 readings out of a total of 8,866 readings) of the experiment time. An exponential distribution with a mean of − 0.7334 m/s2 (mean value for decelerations smaller than −0.3 m/s2) (distribution parameter=1.3635) was fit to the histogram data. It is understood that one needs more data to get a reliable distribution for decelerations; the field data used here is to present a general approach for solving the problem at hand. Using the total probability theorem, we have: P[accident]=P[accident a =6.5] ×P[a = 6.5] =P[accident a =7.5] ×P[a = 7.5] + P[accident a =8.5] ×P[a=8.5]. From the exponential distribution described above values of P[a] can be calculated. As an example, P[a =6.5] is set equal to P[6.0 B aB7.0]= e − 1.3635 × (6 − 0.3) −e − 1.3635 × (7 − 0.3) = 0.0004213− 0.0001077 =0.0003136. Conditional probabilities of accidents given various decelerations for the lead car and the follower (with and without AICC) came from simulation and were presented earlier in Table 4. These were P[accident a]. Using these values, P[accident] is calculated as follows: with AICC: P[accident] = 0.138× (0.005× 3.136 ×10 − 4 +0.025 × 0.802 ×10 − 4 +0.068× 0.205 ×10 − 4) =0.685 × 10 − 6 and without AICC: P[accident] = 0.138× (0.119×3.136 ×10 − 4 +0.141 × 0.802 ×10 − 4 +0.164× 0.205 ×10 − 4) =7.168 × 10 − 6.
In the above equations, 0.138 is the probability of braking with a rate harder than − 0.3 m/s2. For deceleration rates of milder than − 6.5 m/s2, it was felt that causes of collision would include other parameters such as faulty brakes or driver inattention, because under normal circumstances, followers would be able to decelerate at higher rates and thus avoid collision. Probabilities calculated above may be used to compare performance of a car equipped with AICC with respect to unequipped cars. This probability only considers the car ahead of the equipped car without regard to the following cars. This approach can be used in general to calculate the expected number of accidents with and without AICC. It is clear though, that one needs much more data to arrive at reliable results.
11. Conclusion and future work This paper described a general framework for safety evaluation of AICC in rear-end collisions. AICC is one of the few ITS devices that is close to implementation at a commercial level at this time. Because of this, realistic data and specification is available that allows accurate modelling of its performance. Using this data, a set of specifications and two collision models were developed. These models were used in studying various aspects of AICC safety by analyzing vehicles in front and following the equipped car. A measure of safety was introduced that allowed comparison between models with different input values. This measure, the expected number of cars involved in a collision, looks at the total impact of a potential accident on the traffic flow. The software platform used for model development is widely available and easy to work with. The models are sufficiently flexible to allow various kinds of safety studies. Using these models the probabilty of collision for any two cars in the car train was calculated. Also, the expected number of cars in any crash situation was calculated. It was found that the probability of a rear-end collision between the lead car and the car equipped with AICC is significantly lower compared to unequipped cars. Also, it was found that having AICC appears to slightly increase the probability of collision for the followers of the equipped car. Several future studies can be conducted using the developed system. Examples of the work ahead include: (a) Much work needs to be done in studying various human behaviour parameters that can affect the safety of the system. Variations in perception/reaction time and initial durations of braking, have profound impact on safety. Also, collecting more data will help us refine selected distributions and improve the accuracy of results.
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(b) Equipping a car with AICC may change the driver’s behaviour, such as his attentiveness, response time, and driving speed. Using the model in conjunction with relevant data, one can evaluate (and possibly quantify) potential impact of each of these behaviour changes on safety. (c) AICC limitations in detecting other vehicles and obstacles under adverse conditions such as inclement weather should be quantified. Also, false alarms can be a source of risk and irritation to drivers. Examples include AICC’s emergency braking in case of another car cutting in front of the equipped vehicle, or the radar picking up signals from cars in adjacent lanes, especially on curves, and causing brake actuation. Although these issues have been documented in several publications, a rigorous effort should be made in quantifying their impact on safety. As the technology evolves, it is expected that many of these problems will be resolved; current prototypes are far superior compared to their predecessors of a few years ago. (d) The model assumes that each driver reacts by observing the car immediately in front. In reality, given road geometry, the driver may be able to see several cars ahead and take precautionary action. The model can be modified to take this into account if sufficient field data becomes available. The field data can come from instrumented vehicles or car simulators. (e) The model depicts a situation where one car in four is equipped with AICC. It would be important to investigate other instances where different penetration rates are assumed for AICC and then evaluate their impact on safety. At this point, we have assumed that cars following the AICC-equipped car maintain a headway of 1.4 s. In reality, in a car with no AICC, the gap could be much shorter. Even for equipped cars, drivers may choose to shut off the system and override minimum safe distances. Collecting field data on these behavioural aspects will allow us to quantify their impact on safety. In conclusion, the framework proposed here, along with the models developed, will allow much more work to be done in this area of traffic safety. The results of such studies would help to arrive at a true measure of effectiveness of AICC with regard to safety.
Acknowledgements The writers would like to thank the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom for the financial support provided for this research. Also, the first author would like to thank Northeastern University for supporting him during his sabbatical leave at TRG, Southamp-
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ton. The writers would also like to thank the anonymous reviewers for their constructive comments.
References Alvisi, M., Deloof, P., Linss, W., Preti, G., Rolland, A., 1991. Anticollision radar: state of the art. Proceedings of Advanced Telematics in Road Transport, Brussels. Benz, T., 1994. Checking ICC in a realistic traffic environment, Proceedings of the 1st World Congress on Applications of Transport Telematics and Intelligent Vehicle/Highway Systems, Paris. Chira-Chavala, T., Yoo, S.M., 1994. Potential safety benefits of intelligent cruise control systems. Accident Analysis and Prevention 26 (2), 135 – 146. Crystal Ball Software Package, 1996, Ver 4.0c. Decisioneering, Denver, CO. DIATS, 1996. State of the Art Review of ATT Systems, Manufacturers and Users, TRG. University of Southampton. Drew, D.R., 1968, Traffic flow theory and control, McGraw-Hill, New York. Farber, G., 1996. Adaptive cruise control as collision-avoidance: a modelling exercise, Proceedings of the 3rd Annual World Congress on Intelligent Transport Systems, Orlando, FL. Glimm, J., Fenton, R.E., 1980. An accident-severity analysis for a uniform-spacing headway policy. IEEE Transactions on Vehicular Technology 29 (1), 96 – 103. Godbole, D., Sengupta, R., Misener, J., Kourjanskaia, N., Michael, J.B. 1998. Benefits evaluation of crash avoidance systems, TRB Paper No. 981506, 77th TRB Meeting, Washington, D.C., January. Grime, G., Jones, I.S., 1969. Car collisions — the movement of cars and their occupants in accidents. Proceedings of the Institute of Mechanical Engineers 184 (5), 87 – 125. Lucas (undated), Autonomous intelligent cruise control radar headway sensor, Information Sheet, Solihull, UK. Lucas Varity (undated), Adaptive cruise control (ACC), Information Sheet, Solihull, UK. Mannering, F.L., Kilareski, W.P., 1990. Principles of Highway Engineering and Traffic Analysis. Wiley, New York. McDonald, M., Brackstone, M., Sultan, B., 1998. Dynamic behavioral data collection using an instrumented vehicle, TRB Paper No. 980535, Washington, D.C. McGehee, D., Moolenhauer, M., Dingus, T., 1994. The composition of driver/human factors in front-to-rear automotive crashes: design implementations, Proceedings of the 1st World Congress on Applications of Transport Telematics and Intelligent Vehicle/Highway Systems, Paris, France. Najm, W.G., Burgett, A.L. 1997. Benefits estimation for selected collision avoidance systems, Proceedings of the 4th World Congress on Intelligent Transport Systems, Berlin. Nilsson, L., 1994. Safety effects of adaptive cruise controls in critical traffic conditions, Proceedings of the 2nd World Congress on Intelligent Transport Systems, Paris, 1254 – 1259. Takubo, N., 1995. Evaluation of automatic braking for collision avoidance, Proceedings of the 2nd World Congress on Intelligent Transport Systems, Yokohama, 1181 – 1186. Taoka, G.T., 1989. Brake reaction times of unalerted drivers, ITE Journal, March, 19 – 21. Touran, A., 1998. Risk and reliability in advanced transport telematics, Report submitted to EPSRC., Transportation Research Group, University of Southampton, March.
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Tribe, R., Prynne, K., Westwood, I., Clarke, N., Richardson, M., 1994. Intelligent driver support, Proceedings of the 2nd World Congress on Intelligent Transport Systems, Paris, 1187–1192. Tsao, H.-S.J., Hall, R.W., 1994. A probabilistic method for AVCS longitudinal collision/safety analysis. IVHS Journal 1 (3), 261 – 274.
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Wootton Jeffreys Consultants Ltd., 1990, SISTM, Micro simulation of traffic on motorways, Transportation Research Laboratory, UK. Zhang, X., Benz, T., 1993. Simulation and evaluation of intelligent cruise control, IVHS Journal, Vol. 1(2), Gordon and Breach Science Publishers, pp. 181 – 190.
A collision model for safety evaluation of autonomous intelligent cruise control Ali Touran a,*, Mark A. Brackstone b, Mike McDonald b a
Department of Ci6il and En6ironmental Engineering, Northeastern Uni6ersity, 420 Snell Engineering Centre, Boston, MA 02115, USA b Transportation Research Group, Uni6ersity of Southampton, Southampton, SO17 1BJ, UK Received 14 June 1998; received in revised form 14 January 1999; accepted 19 January 1999
Abstract This paper describes a general framework for safety evaluation of autonomous intelligent cruise control in rear-end collisions. Using data and specifications from prototype devices, two collision models are developed. One model considers a train of four cars, one of which is equipped with autonomous intelligent cruise control. This model considers the car in front and two cars following the equipped car. In the second model, none of the cars is equipped with the device. Each model can predict the possibility of rear-end collision between cars under various conditions by calculating the remaining distance between cars after the front car brakes. Comparing the two collision models allows one to evaluate the effectiveness of autonomous intelligent cruise control in preventing collisions. The models are then subjected to Monte Carlo simulation to calculate the probability of collision. Based on crash probabilities, an expected value is calculated for the number of cars involved in any collision. It is found that given the model assumptions, while equipping a car with autonomous intelligent cruise control can significantly reduce the probability of the collision with the car ahead, it may adversely affect the situation for the following cars. © 1999 Elsevier Science Ltd. All rights reserved. Keywords: Autonomous intelligent cruise control (AICC); Collision; Monte Carlo simulation; Risk assessment; Human behaviour
1. Introduction Autonomous intelligent cruise control (AICC) is one of the devices that has been introduced under the broad field of intelligent transportation systems (ITS) with the objectives of increasing the capacity and improving the safety of existing highway systems. AICC, in the context of this work, is defined as a device that exerts control on the car’s throttle and brakes to obtain a set headway. It is one of the few advanced vehicle control systems (AVCS) devices that may be implemented in the next 5–10 years; working prototypes are being tested extensively at this time (Tribe et al., 1994). One reason for imminent use of AICC is that no road infrastructure modifications are needed for the device to work effectively (Chira-Chavala and Yoo, 1994). In contrast to AICC, many ATT systems such as convoy
* Corresponding author. Tel.: +1-617-3732444; fax: + 1-6173734419. E-mail address: [email protected] (A. Touran)
driving require considerable investment in road infrastructure (DIATS, 1996). Furthermore, current cruise control technology is well tested and many drivers, especially in the U.S., are familiar with its function. Several researchers have worked on various aspects of AICC safety. These aspects include the effect of AICC (or similar systems such as adaptive cruise control) on regulating the traffic flow and reducing smaller headways (Benz, 1994), its effectiveness in avoiding crashes in critical traffic conditions (Nilsson, 1994), and safety benefits derived from its implementation (ChiraChavala and Yoo, 1994; Najm and Burgett, 1997; Godbole et al., 1998). Most of the work carried out in the area of AICC safety, especially the earlier ones, has been based on hypothetical specifications for the device. As mentioned before, several AICC prototypes are now in the final development stages by various manufacturers. Detailed technical information for the working prototypes is available. The work reported in this paper uses this information to conceive a realistic AICC system and develop a framework for safety evaluation of AICC
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under extreme driving conditions. This is accomplished by developing a rear-end collision model and evaluating the effects of AICC on the car ahead and on cars following the equipped vehicle. This is important because most of the work done so far only considers the effects of the car ahead on the equipped vehicle, disregarding what might happen to followers as a consequence of equipping a car with AICC. The model developed here does not consider some important cases such as large differences in following cars’ speeds, possible driver inattentiveness because of his dependence on AICC, or the effect of queues in highway driving. Also, we are not considering device reliability and its probability of failure here; that is the subject of another research project currently under way. The results of the models need to be verified with real world testing but the models developed can provide a foundation for these testing programs.
2. Model development In order to assess AICC’s potential impact on the number of rear-end collisions, the following steps were taken. First, a set of specifications was developed for the device. Then a car collision model was developed by faithfully incorporating device specifications. Then input data for the collision model were chosen and model parameters were refined using sensitivity analysis. Finally, we ran several simulation experiments to calculate the probability of collision under various conditions. The model considers the probability of collision of the equipped car with the car ahead under extreme braking conditions by the lead car. Also, it evaluates the collision probability for the cars following the equipped car. These steps are described in detail in the following sections.
3. System specification Using available technical data and literature, a set of specifications was developed for the AICC device (for example see Alvisi et al., 1991). The authors were in contact with various European car and device manufacturers and benefited from their advice and consultation. The AICC device envisaged for this research, although hypothetical, is similar to several working prototypes and has the following characteristics: it detects the car ahead at a distance of up to 150 m, and can compute relative speed and distance at 120 m. It maintains a target headway of 1.4 s by exerting control over the throttle (from − 0.8 to +1 m/s2) and if distances between cars fall below a minimum, it can activate the brakes (up to − 3 m/s2). If this braking rate is not sufficient to maintain minimum safe distance, the device
warns the driver. When the driver intervenes by braking, the control is transferred to him. Also, when the driver accelerates, for example in case of overtaking another car, the controls are transferred to him. All the details of the device braking mechanism and system delays are carefully specified based on current working prototypes such as those being produced by the industry.
4. Car collision model Two car collision models were developed. In the first model, in order to evaluate the effectiveness of the AICC device in avoiding rear-end collisions, a group of four vehicles was considered where the second vehicle from the front was assumed to be an AICC-equipped car. The second model consisted of four vehicles where none were equipped with AICC. The results obtained from these two models were compared in order to evaluate the effect of using AICC in a car group.
4.1. AICC-equipped car model Except for platoons, most of the safety research done in the past focused on the performance of AICC with respect to the car moving in front of the equipped vehicle. This model considers not only what happens to the car in front of the equipped vehicle but also what may happen to the cars following it. The model for the third car from front describes the behavior of a car following a car with AICC, while the model for the last car that will behave approximately similar to cars following it (not modeled here), is modeled as an unequipped car following unequipped vehicles. The main characteristics of the model are given in Fig. 1. The graph shows car decelerations vs time. The heavy line represents the equipped car. Assume that the lead car breaks hard in an emergency situation at t=0 with the rate of aL. The AICC device in the following car reacts by applying the brakes after a reaction time of tR. The jerk limit, J, is used to describe the transition between the instant that the system activates the brakes and the time of full braking by AICC at the rate of aF. As the driver of the equipped car realizes that the lead car deceleration is far sharper than can be safely accommodated by automatic brakes, he intervenes by braking harder with a deceleration rate of a *; F tA is the time delay between the application of brakes by AICC and the harder application of brakes by driver (the time when the driver overrides AICC and brakes hard to avoid collision). The following parameters describe model characteristics: tR, aF,
AICC reaction time (s) AICC brake levels (m/s2)
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Fig. 1. Car following model for the car equipped with AICC.
a *, F J, Dt = tA, VL, VF, xL(t), xF(t),
driver brake rate after deciding AICC deceleration is not sufficient (m/s2) jerk limit (m/s2) time delay for the jerk limit (s) time delay after AICC braking and before driver Intervention (s) lead car’s original speed (m/s) AICC-equipped car’s original speed (m/s) distance travelled by lead car after t seconds since the start of braking by lead car (m) distance travelled by AICC car after t seconds since the start of braking by lead car (m)
After the equipped car brakes, there is a delay of T seconds before the following car breaks. T, or the perception/reaction time (PRT), is the time that it takes the following driver to see the brake light of the car in front and to react. A two step deceleration is considered in this model: an initial milder deceleration rate, a, and a final harder deceleration rate, a*. This, in effect, says that followers imitate the behaviour of drivers ahead. The following parameters describe cars that are following the AICC-equipped car. The first car following the equipped car is denoted with i and the car following that by i +1. ai, a *, i Ti ,
Initial brake level (m/s2) final brake level (m/s2) time delay after the driver brakes the first time with rate ai and before braking harder (s)
Vi, Vi+1, xi (t),
xi+1(t),
original speed of the first car following AICC-equipped car (m/s) Original speed of the car following the ith car (m/s) distance travelled by the first car following AICC car after t seconds since the start of braking by lead car (m) distance travelled by the follower to the first car following AICC car after t seconds since the start of braking by lead car (m)
The car following the ith car is the i+ 1th car with parameters ai + 1, a *i + 1, and Ti + 1. It is further assumed that this is a series system (Glimm and Fenton, 1980), wherein each vehicle must sense the braking of its nearest front vehicle before starting to brake. This is conservative because in most cases, the following cars can see the brake lights of several vehicles ahead and take action appropriately. In a series system the total delay Ttot before the ith car brakes would be the sum of individual delays: ttot =tR = nT, where n is the nth vehicle following the AICC-equipped car. Equations of distance and speed as functions of time, i.e. x(t) and 6(t), were derived for each vehicle using basic Newtonian motion principles.
4.2. Headways We have used the concept of time headways to describe inter-vehicle distances in this research. Defini-
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Fig. 2. Car following model with no AICC.
tion of time headway is compatible with Drew (1968), where the speed of the following vehicle is applied to the time gap to calculate the distance between vehicles. To avoid collision, distance travelled by the lead vehicle plus the follower’s headway tH, should be greater than the distance travelled by the AICC-equipped car plus a minimum distance, x0.
cars. Because there is no intermediate braking level by AICC, only one level of braking is assumed for each of the four cars resulting in a set of simple motion equations.
XL + tHVF ]XF +x0
Input data for the collision model were selected from a wealth of information supplied by industry, research conducted at the Transportation Research Group (TRG) of the University of Southampton on behavioural aspects of driving in the past few years, and published reports and papers. We also benefited from data collected by an instrumented vehicle that has been assembled in the past 2 years by the TRG. This car is equipped with a radar, optical speedometer, and videoaudio monitoring system (McDonald et al. 1998) and can detect other cars, their distance and relative speed. A base case scenario was developed for an extreme case where the lead vehicle brakes hard to cope with an unexpected event. The equipped car reacts to this situation utilizing AICC capabilities and driver intervention. Following cars react according to drivers’ perception/ reaction times and the behaviour of the car immediately preceding them. The base case scenario considered here models an emergency brake by the lead vehicle with a rate of −7.5 m/s2. Several sources report maximum braking rates of between 0.8 g and g (Glimm and
(1)
In the above equation, XL is the distance to stop for the lead vehicle and XF is the distance to stop for the following vehicle. The same equation can be applied to any pair of following cars. The headway used for AICC and the followers can be chosen independently. In this way, one can evaluate the difference between the headways of an AICC-equipped car and a non-equipped car. The equations developed were used in constructing a computer model using Excel™ spreadsheet. The same spreadsheet structure was later used for performing probabilistic Monte Carlo simulation using Crystal Ball™ 1996 software package.
5. Model with no AICC Fig. 2 shows the model used for the case where no AICC was present. This model also consists of four
6. Data for the collision model
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Table 1 Input data for the model parameters Parameter
Lead car
Equipped car
First follower
Second follower
Initial velocity (m/s) Initial braking rate (m/s2) Final braking rate (m/s2) Delay time for braking (s) Initial brake duration (s)
VL = 30 – aL = −7.5 – –
VF =30 aF =−3.0 a *F =−7.5 tR =0.45 tA =1
Vi =30 ai =−4.0 a *i =−7.5 T =1.1 Ti =1
Vi+1 =30 ai+1 =−5.0 a *i+1 =−7.5 T= 1.1 Ti+1 =1
Fenton 1980; Tsao and Hall 1994; Godbole et al. 1998). Mannering and Kilareski (1990) report a maximum coefficient of road adhesion in good, dry pavement of 1 g. The effective braking rate will obviously be smaller than this coefficient depending on several factors. In our case, it is assumed that braking takes place on good dry pavement. The car which is equipped with AICC, is travelling with an initial speed similar to the lead car’s speed. As the AICC warns the driver at 1.4 s headway, the case base scenario provides a worst case situation where emergency braking by lead vehicle coincides with the minimum distance available between cars. AICC alerts the driver of the equipped car while starting to apply the brakes automatically up to a deceleration of − 3 m/s2. Maximum braking level for AICC has been studied in several sources and the value chosen is representative of those reported. This level seems to be the maximum braking rate at which the driver does not feel discomfort and panic (Touran 1998). The driver applies the brakes 1 s after application of brakes by AICC, thus taking over the control of the brakes (tA =1 s). This value is analogous to the perception/reaction time of the driver; we have assumed that activation of brakes by AICC has already alerted the driver, hence a relatively short perception/reaction time is justified. The followers imitate the same pattern by initially braking softer and then harder. The input values used in the base case scenario are presented in Table 1.
6.1. AICC characteristics To specify AICC response time, we contacted device manufacturers. The following is a typical specification given the current state of technology. AICC’s brake controls system has to detect the object and actuate the brakes. Object detection consists of tracking and selecting the object (to identify the vehicle ahead and distinguish it from other vehicles in other lanes, etc). Object detection takes from 100 to 400 ms. Actuation has two components: processing delay of 40 ms and application of brakes (for a typical −3 m/s2 deceleration) of 250
ms1. With these values in mind, tR (detection=processing delay) was chosen as 0.45 s, and transition time to achieve full braking rate t, was chosen as 0.25 s. From here by referring to Fig. 1, jerk limit J, is calculated as 12 m/s3 (aF = − J×Dt = − 12×0.25= −3 m/s2).
6.2. Safe dri6ing distance A collision happens when at any point during a brake by the lead car, the distance travelled by the lead car plus the headway distance between the follower and lead car are smaller than the distance travelled by the follower. Various time headways were selected for the cars in the model and each situation was evaluated. Fig. 3 gives the results of analysis. In each case, distances travelled after braking by the lead car and follower, xL(t) and xF(t), are plotted against time following the format suggested by Tsao and Hall (1994). Similar graphs were generated for each pair of cars. Also, various other graphs such as vehicles’ relative speeds vs time and time to collision (TTC) 6s time were developed and are reported in Touran (1998). In each case, a minimum distance of 5 m was assumed to be necessary at the time when cars are stopped. This should be sufficient to account for the length of cars, especially if the headway gap is measured between the front of the follower and the front of the preceding car. It is evident that a time headway of 1.4 s is sufficient to avoid collision under assumed conditions (Fig. 3). Assuming wet conditions, we ran the model with reduced deceleration rates, where the lead car and the followers brake with a rate of −6.5 m/s2. This did not create a more critical situation and still a headway distance of 1.4 s was sufficient to avoid collision. The calculated headway seems consistent with the results of research on traffic conflicts, where drivers seem to use time gaps of
1
The data reported is the result of discussion with Mr Mark Basten of Lucas/Varity (Lucas, undated; Lucas Varity, undated). Some published sources, especially from U.S., report much shorter response times such as 0.1 s. These values could not be substantiated at this point, although it might be achievable in the future. The emphasis of the paper is a realistic model given current state of technology.
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Fig. 3. AICC model output for each pair of following cars.
1.5 s as a natural limit, although in many situations the following times on motorways are much shorter.
7. Sensitivity analysis In order to ensure that safe headway distances are established, a deterministic sensitivity analysis was conducted to evaluate the impact of each major parameter on the possibility of a collision. Parameters considered included initial speeds and decelerations, perception– reaction times, and initial braking durations. In each case, the value of parameter of interest was incrementally changed while keeping other parameters constant. Collision was investigated for each scenario. By far, the perception–reaction time T, had the greatest impact on the required headway to avoid collision for the car following the equipped car (Table 2). The model was far less sensitive to changes in initial speeds or initial braking rates in the deterministic analysis. We did not consider some of worst possible cases such as traffic queues or large speed differences. While the model can Table 2 Impact of PRT on the required headway PRT [T] (s)
Headway for 1st follower (s)
Headway for 2nd follower (s)
0.7 1.1 1.5 2.0
0.7 1.1 1.5 2.0
0.8 1.2 1.5 2.0
easily evaluate the impact of speed differences between cars, it was assumed that original car speeds were equal. This is partially justified because AICC tends to adapt the car speed to the speed of the lead car and under normal driving conditions, manages to achieve a DV of zero.
8. Monte Carlo simulation Several parameters used in the model are a function of driver behaviour and his or her interaction with car and AICC. It would be unrealistic to model such parameters only with deterministic variables. In order to assess the inherent variability of such parameters, and also to incorporate our lack of precise knowledge of some parameter values, a Monte Carlo simulation was performed. Several model parameters including braking rates and durations, and perception–reaction times were randomized according to selected distributions and the collision model was simulated. Simulation results gave the probability of collision between any of the two following cars. Based on crash probabilities, an expected value was calculated for the number of cars involved in any crash. One can use this parameter as a comparison measure for assessing the safety of any given combination of equipped and non-equipped cars. The same procedure was repeated for the model with no AICC. Similar input values were used for this model and probabilities of collision were simulated for any two following cars. Results were compared to assess the effect of AICC on safety.
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8.1. Car distances after collision
8.2. Simulation 6ariables
The model described earlier in this paper needed to be modified for probabilistic analysis. This is because when a collision occurs, distances travelled by cars will be restricted. As an example, if the AICCequipped car collides with the lead car, maximum distance travelled by the equipped car will be limited to the distance up to the collision point plus the distance to stop for both cars after collision. This may have an effect on the probability of collision between the car following the equipped car with the equipped car. In order to consider this limitation, a headway distance of 1.4 s is chosen for all cars; in each simulation iteration, the model checks to see if a collision has occurred between cars ahead of a specific car. If there is a collision, the model updates distances available to that specific car and checks for collision between its followers. Vehicle motion after collision has been modelled by several authors in the past. A comprehensive paper by Grime and Jones (1969) gives mathematical equations for motion after collision. When two vehicles collide, given certain mechanics assumptions, the combination will move at a common velocity of V3 = (M1V1 +M2V2)/(M1 +M2), where M and V are mass and velocity of the two vehicles before the crash. If the preceding vehicle is stopped (V1 = 0 ), then the speed of the combination would be roughly half of the speed of the follower, assuming equal vehicle weights. Deceleration of cars during collision have been studied in the past; various ranges are suggested. Glimm and Fenton (1980) report on experiments conducted with a car colliding with a fixed barrier, a moving car, and a stopped car. In all cases, collision deceleration is much higher than pre-crash deceleration of the follower. Glimm and Fenton (1980) use a collision deceleration of 10 g. Furthermore, they suggest that motion of cars after collision may be calculated using collision deceleration and the new combined speed. Distance to stop after collision, Xcoll, may be found from Xcoll =V 23/2acoll, where acoll is the rate of deceleration in collision. In our case, V3 is sufficiently small to make Xcoll negligible, especially since acoll is so large. In cases where the collision happens before the preceding car has come to a stop, it is assumed that the distance to stop for the combination is equal to distance to stop for the preceding car. This means that the effect of collision would be to limit the distance available to the following car. In this research, all possible cases of collision between various cars were considered and distances available to followers after collision between cars in front were calculated and incorporated into the simulation models (Touran 1998).
The following is a brief description of random input variables.
8.2.1. Initial braking rates Initial braking rate for each car was modelled as a linear function of the braking rate of the car in front. The AICC brakes with a rate of −3 m/s2; initial braking rate for each follower was set equal to initial braking rate of the car in front plus a uniform distribution with mean −0.5 m/s2 (ranged between 0 and − 1.0). Assuming a larger braking rate for the follower is consistent with findings of technical literature on driver behaviour. For example, within the SISTM model (Wootton Jeffreys 1990) it is assumed that if a driver sees a vehicle ahead decelerating at a rate harder than − 3 kph/s, at a speed of greater than 30 kph, he will decelerate at − 6 kph/s. Najm and Burgett (1997) use a follower’s deceleration rate of between −0.5 and − 0.7 g in response to a lead car deceleration with a mean of − 0.31 g. Use of uniform distribution denotes our lack of precise data and knowledge regarding the nature of this variable. 8.2.2. Final braking rates The same logic was used for modeling final braking rates. Godbole et al. (1998) collected data on maximum deceleration capabilities of new light duty passenger cars on dry pavement. They fitted a truncated normal distribution with a mean value of − 7.01 m/s2 to the data. We have assumed that the lead car brakes with a rate of −7.5 m/s2.The final braking rate for each following car was modelled as the final braking rate of the car in front plus a uniform distribution with mean − 0.5 m/s2 (ranged between 0 and − 1.0). In no case this value was allowed to go below − 10 m/s2. 8.2.3. Braking duration for AICC Braking duration for AICC was modelled using a log-normal distribution with mean 1.0 and standard deviation 0.2 s ranging between 0.5 and 1.92 s. In this case, the braking duration is somewhat analogous to the driver’s reaction time. Perception reaction times have been modelled using a log-normal distribution (Taoka 1989; Najm and Burgett 1997). 8.2.4. Initial braking duration for followers Initial braking duration for follower of the equipped car was set equal to AICC braking duration plus a triangular distribution with the following parameters: minimum= − 0.5, mode= 0, maximum= 1 s. The braking duration for the last car was set equal to the preceding car’s braking duration plus
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a triangular distribution with the following parameters: minimum = − 0.5, mode=0, maximum= 1 s.
8.2.5. Perception/reaction time for followers of the equipped car A truncated log-normal distribution (mean= 1.1, standard deviation=0.22, ranged between 0.8 and 2.5 s) was used for modelling PRT for both drivers. The mean value is consistent with values suggested by Najm and Burgett (1997). Also, McGehee et al. (1994) report that in a study by Olson and Sivak, a range between 0.81 and 1.76 s was found for PRT. Studies suggest that PRT may be shorter for more intimidating conditions. In our case, given relatively high speeds assumed, we feel that values chosen are realistic. 8.2.6. Input 6alues for the model with no AICC Input values for the model with no AICC were identical to the data used for the other model described thus far. The only difference was the second vehicle from the front, where in this model, was not equipped with AICC. Its braking rate was modeled as − 7.5 m/s2 plus a uniform random variable between 0 and − 1.0 m/s2. Perception/reaction times were modeled with distributions similar to the other followers. Gap between cars was assumed to be 1.4 s. Also, because there was only one level of braking for each car, there were no initial braking rates and durations (refer to Fig. 2). 8.3. Simulation results The model was run using Excel™ spreadsheet and Crystal Ball™ simulation package. Simulation was run for 5000 iterations using a Latin Hypercube sampling technique. It was noted that as the number of simulation iterations was increased beyond 5000, the outcome remained practically unchanged. In each iteration, three objective functions were calculated verifying the possibility of collision between each pair of following cars. For each pair of cars, the objective function was set equal to the distance travelled by the preceding vehicle plus headway distance minus the distance travelled by the follower plus the minimum distance. These functions in effect, give inter-vehicle distances after the lead car starts to brake until cars come to a stop. If this value was positive, no crash was assumed. The probability of crash between AICC-equipped car and the lead car was 2.76%. This probability was much higher for the cars following the equipped car. The probability of crash between the car following the equipped car and the AICC car was 18.38%. The probability of crash between equipped car and all others (the lead and the followers) was 0.52%. In each case a headway distance of 1.4 s (distance of 301.4 = 42m) was assumed between cars. Using basic axiom of
probability, one can calculate probability of crash for the equipped car as follows: P(crash)= P(crash between lead car and equipped car)+P(crash between equipped car and its followers)− P(crash between equipped car, lead car and followers)= 0.0276+0.1838−0.0052= 0.2062.
9. AICC’s impact on the followers Most of the work describing the effect of AICC on safety consider the equipped car and the car ahead (for examples, see Zhang and Benz, 1993; Farber, 1996). It is also important to study what happens to the following cars when the equipped car reacts to an emergency situation. Researchers working on vehicle convoys have considered this issue (Glimm and Fenton, 1980), but in convoy driving, unlike our situation, all vehicles are equipped with warning and control devices. Takubo (1995) considered the possibility of crash between several vehicles, one of which was equipped with ACC, in his simulation study. The study predicted the number of fatalities and injuries in case of accidents, but the hypothetical ACC system used was capable of full braking, something that is not conceivable under current industry trends.
9.1. Number of cars in6ol6ed in a collision We suggest that in order to make a fair assessment of the impact of AICC on safety, one should look at an expected value of the number of cars involved in a collision rather than just focusing on the equipped car and the car ahead. A routine was developed to calculate the number of cars involved in each of the crash situations realized in the simulation. The results of the simulation reported earlier (5000 iterations) was used to compile these ‘accident’ data. Data provided in Table 3 gives the outcome of the analysis. For the case where the second car is equipped with AICC, the expected number of cars involved in a collision in an emergency situation is calculated as 0.622× 0+ 0.297× 2+0.079× 3+0.003× 4= 0.843 (see Table 3 for collision statistics). Note that most of the time (62.2% in this scenario), a collision does not occur in an emergency situation. The expected number of cars involved in a collision, given that a collision has occurred, is calculated as 2.22. These values can be used as measures of system safety in comparative analyses. For the case where no car is equipped with AICC, the expected number of cars involved in a collision in an emergency situation is 0.935 (0.601× 0 + 0.285× 2+ 0.095×3 +0.020× 4= 0.935; see Table 3). The expected number of cars involved in a collision, given that a collision has occurred, is 2.34.
A. Touran et al. / Accident Analysis and Pre6ention 31 (1999) 567–578 Table 3 Multi-car collision statistics Number of cars involved in crash (1)
Frequency of crash (2)
Probability of crash 2} 5000 (3)
2nd Car equipped w/ AICC No crash 2 Cars 3 Cars 4 Cars
3108 1485 394 13
0.622 0.297 0.079 0.003
No car equipped w/AICC No crash 3005 2 Cars 1424 3 Cars 473 4 Cars 98
0.601 0.285 0.095 0.020
9.2. Safety impact of AICC Using simulation outputs, one can calculate probabilities of avoiding a collision under assumed conditions. Table 4 gives these probabilities for various values of braking rates. In each case, the simulation was run for 2000 iterations. As can be observed from Table 4, the probability of avoiding a collision is increased as the braking rate is decreased. This is partly due to modeling assumptions. Assuming that the parameter values mirror realistic behavior, in case of lead car’s lower braking rates, the followers have a better chance of stopping in time by reacting with higher deceleration rates. For braking rates softer than −6.5 m/s2, the cause of a rear-end accident would not likely be limited to perception/reaction time, but includes other factors such as driver inattention; there is sufficient braking capacity to avoid the collision under assumed headways. While AICC has improved the chances of avoiding a collision between the equipped car and the car ahead, its impact on the followers is much less clear. It appears that having AICC has slightly increased the probability
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of collision for the followers. Furthermore, while AICC alerts the driver if the gap falls below the safe distance, the followers have no warning system and are likely to drive much closer than the assumed headway of 1.4 s. This will cause an increase in the probability of collision for the cars following the equipped car. We kept the headway distance for the followers equal to 1.4 s so that a more direct comparison with the effect of AICC can be accomplished. Collision probability values are very sensitive to perception/reaction time and to initial brake duration (duration of soft braking). Increasing perception/reaction times or initial braking durations greatly increases the probability of collision behind the equipped car. In order to investigate the share of each input parameter in variance of the outcome, a sensitivity analysis was conducted on the results of simulation. In case of followers, the variability in PRT and initial braking durations contributed greatly to the probability of collision. PRT values have been extensively reported in published literature and the values used in the model should be reliable. Initial braking durations on the other hand, are based on perceived behaviour of drivers, and parameters used in triangular distributions are educated guesses at best and should be verified in the field. Unfortunately, simulation results are highly sensitive to parameters of the triangular distributions used. An increase in the high tail of the distribution, would indicate a sharp increase in the probability of collision for cars following the equipped car. Table 5 summarizes the results of sensitivity analysis. For each pair of cars, the contribution to variance of the outcome by the most sensitive parameters is presented. The percentage points are relative percentage of variance or uncertainty in the forecast (here the forecast is the final distance between cars after stopping, where a negative distance is an indication of collision) attributable to each input parameter. The higher the percentage point reported in Table 5, the more influential the parameter on the final outcome (crash or no crash). Input parameters not reported in this table had negligible effect on the probability of collision.
Table 4 Probabilities of crash avoidance with various emergency braking rates Braking rate
Lead-followinga
Following-ith
ith–i+1th
a=−8.5 m/s2 With AICC Without AICC
0.932 0.837
0.768 0.789
0.705 0.736
a=−7.5 m/s2 With AICC Without AICC
0.975 0.860
0.828 0.820
0.745 0.788
a=−6.5 m/s2 With AICC Without AICC
0.995 0.881
0.860 0.845
0.801 0.837
a
‘Following’ is the car that may be equipped with AICC.
10. Probability of a rear-end collision In the previous section, the probability of a crash under various braking rates was calculated. It would be useful to assess the likelihood of occurrence of such decelerations. We have collected field data using an instrumented vehicle at TRG as noted earlier in this paper. Using this vehicle, 147 min of driving data was collected during 3 different days on the M3 motorway near London. Total miles driven amounted to 104. The data was sampled at 1-s intervals and a histogram of various braking rates of less than − 0.3 m/s2 was
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Table 5 Contribution of input parameters to the variance of the model’s outcome Input parameters
Lead-followinga AICC brake duration Final braking rate for the equipped car Following-ith ith driver’s PRT Initial braking duration (Ti) of i th car ith–i+1th i+1 th driver’s PRT Initial braking duration (Ti+1) of i+1th car a
Contribution to variance (%)
77.5 22.0 59.6 35.1 62.9 28.4
‘Following’ is the car equipped with AICC.
constructed. These decelerations spanned 13.8% (1220 readings out of a total of 8,866 readings) of the experiment time. An exponential distribution with a mean of − 0.7334 m/s2 (mean value for decelerations smaller than −0.3 m/s2) (distribution parameter=1.3635) was fit to the histogram data. It is understood that one needs more data to get a reliable distribution for decelerations; the field data used here is to present a general approach for solving the problem at hand. Using the total probability theorem, we have: P[accident]=P[accident a =6.5] ×P[a = 6.5] =P[accident a =7.5] ×P[a = 7.5] + P[accident a =8.5] ×P[a=8.5]. From the exponential distribution described above values of P[a] can be calculated. As an example, P[a =6.5] is set equal to P[6.0 B aB7.0]= e − 1.3635 × (6 − 0.3) −e − 1.3635 × (7 − 0.3) = 0.0004213− 0.0001077 =0.0003136. Conditional probabilities of accidents given various decelerations for the lead car and the follower (with and without AICC) came from simulation and were presented earlier in Table 4. These were P[accident a]. Using these values, P[accident] is calculated as follows: with AICC: P[accident] = 0.138× (0.005× 3.136 ×10 − 4 +0.025 × 0.802 ×10 − 4 +0.068× 0.205 ×10 − 4) =0.685 × 10 − 6 and without AICC: P[accident] = 0.138× (0.119×3.136 ×10 − 4 +0.141 × 0.802 ×10 − 4 +0.164× 0.205 ×10 − 4) =7.168 × 10 − 6.
In the above equations, 0.138 is the probability of braking with a rate harder than − 0.3 m/s2. For deceleration rates of milder than − 6.5 m/s2, it was felt that causes of collision would include other parameters such as faulty brakes or driver inattention, because under normal circumstances, followers would be able to decelerate at higher rates and thus avoid collision. Probabilities calculated above may be used to compare performance of a car equipped with AICC with respect to unequipped cars. This probability only considers the car ahead of the equipped car without regard to the following cars. This approach can be used in general to calculate the expected number of accidents with and without AICC. It is clear though, that one needs much more data to arrive at reliable results.
11. Conclusion and future work This paper described a general framework for safety evaluation of AICC in rear-end collisions. AICC is one of the few ITS devices that is close to implementation at a commercial level at this time. Because of this, realistic data and specification is available that allows accurate modelling of its performance. Using this data, a set of specifications and two collision models were developed. These models were used in studying various aspects of AICC safety by analyzing vehicles in front and following the equipped car. A measure of safety was introduced that allowed comparison between models with different input values. This measure, the expected number of cars involved in a collision, looks at the total impact of a potential accident on the traffic flow. The software platform used for model development is widely available and easy to work with. The models are sufficiently flexible to allow various kinds of safety studies. Using these models the probabilty of collision for any two cars in the car train was calculated. Also, the expected number of cars in any crash situation was calculated. It was found that the probability of a rear-end collision between the lead car and the car equipped with AICC is significantly lower compared to unequipped cars. Also, it was found that having AICC appears to slightly increase the probability of collision for the followers of the equipped car. Several future studies can be conducted using the developed system. Examples of the work ahead include: (a) Much work needs to be done in studying various human behaviour parameters that can affect the safety of the system. Variations in perception/reaction time and initial durations of braking, have profound impact on safety. Also, collecting more data will help us refine selected distributions and improve the accuracy of results.
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(b) Equipping a car with AICC may change the driver’s behaviour, such as his attentiveness, response time, and driving speed. Using the model in conjunction with relevant data, one can evaluate (and possibly quantify) potential impact of each of these behaviour changes on safety. (c) AICC limitations in detecting other vehicles and obstacles under adverse conditions such as inclement weather should be quantified. Also, false alarms can be a source of risk and irritation to drivers. Examples include AICC’s emergency braking in case of another car cutting in front of the equipped vehicle, or the radar picking up signals from cars in adjacent lanes, especially on curves, and causing brake actuation. Although these issues have been documented in several publications, a rigorous effort should be made in quantifying their impact on safety. As the technology evolves, it is expected that many of these problems will be resolved; current prototypes are far superior compared to their predecessors of a few years ago. (d) The model assumes that each driver reacts by observing the car immediately in front. In reality, given road geometry, the driver may be able to see several cars ahead and take precautionary action. The model can be modified to take this into account if sufficient field data becomes available. The field data can come from instrumented vehicles or car simulators. (e) The model depicts a situation where one car in four is equipped with AICC. It would be important to investigate other instances where different penetration rates are assumed for AICC and then evaluate their impact on safety. At this point, we have assumed that cars following the AICC-equipped car maintain a headway of 1.4 s. In reality, in a car with no AICC, the gap could be much shorter. Even for equipped cars, drivers may choose to shut off the system and override minimum safe distances. Collecting field data on these behavioural aspects will allow us to quantify their impact on safety. In conclusion, the framework proposed here, along with the models developed, will allow much more work to be done in this area of traffic safety. The results of such studies would help to arrive at a true measure of effectiveness of AICC with regard to safety.
Acknowledgements The writers would like to thank the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom for the financial support provided for this research. Also, the first author would like to thank Northeastern University for supporting him during his sabbatical leave at TRG, Southamp-
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ton. The writers would also like to thank the anonymous reviewers for their constructive comments.
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