- Email: [email protected]
The human element in car following models
Transportation Research Part F 2 (1999) 207±211
www.elsevier.com/locate/trf
The human element in car following models Wim van Winsum * Human Factors Research Institute, TNO-TM, Kampweg 5, P.O. Box 23, 3769 ZG Soesterberg, Netherlands
Abstract This commentary distinguishes between the engineering-inspired formalisms used by Brackstone, M., and McDonald, M. (Transportation Research Part F (1999)) and the psychological knowledge of, and assumptions about human behaviour when car following. It outlines a preliminary mathematical model, based on psychological knowledge about car following behaviour by human drivers, which could be applied in trac engineering, as a means of bringing these two ways of studying car following closer together. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Close following
1. Introduction In the paper, ÔCar following: A historical reviewÕ by Brackstone and McDonald (1999), a clear and thorough overview of car following models from an engineering point of view is presented. According to the authors it is the intent of the paper to brie¯y assess the range of options available in the choice of car following models. In the concluding remarks they state that it is Ôhighly tempting to attempt to increase the realism of a chosen model by attempting to incorporate motivational or attitudinal factors that may be able to explain the dierences between driversÕ, but that there is Ôlittle evidence to relate such features to observable dynamic behaviourÕ. In this commentary, it is argued that there is a substantial psychological literature on car following that has not been covered by Brackstone and McDonald (1999). The models they describe have been developed starting from assumptions that are based on what appears to be common sense but for which there is little evidence. Having speci®ed the models, parameters are selected such that they ®t the observed data. The models fail to clarify why the relationships between variables are speci®ed as they are. A number of assumptions appear to be based on the idea that the human
*
Tel.: +31-3463-563977; fax: +31-3463-56252. E-mail address: [email protected] (W. van Winsum).
1369-8478/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 9 - 8 4 7 8 ( 0 0 ) 0 0 0 0 8 - 5
208
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
driverÕs behaviour is completely rational and that he or she is perfectly able to perceive distance, speed and acceleration. There is little evidence for this in the psychological literature. A lot of research on driver behaviour and car following has been done from a psychological point of view and this research has generated a substantial amount of knowledge that is applied in models of car following. What this research makes clear is that human drivers apply heuristics to cope with their limitations in an adaptive and ¯exible manner. However, a problem with this literature is that only certain elements of car following behaviour have been studied and that a consistent and an elaborated model is still lacking. In this commentary I would like to show how psychological knowledge about car following behaviour by human drivers can be applied in a mathematical model that can be used in trac engineering. It is hoped that this helps in bringing two dierent ways of studying car following behaviour a bit closer together. In Section 2, a number of results from psychological studies on car following are integrated into one preliminary model. This should not be considered as a validated and complete model but only as an example of how a model can be built that is rooted in known psychological facts and ®gures. 2. Psychological factors in car following There is substantial evidence that human drivers regulate available time as a control mechanism. The following distance that the driver attempts to maintain is based on a time headway that is constant in given similar circumstances. Van Winsum and Heino (1996) found that individual drivers follow a time headway that is independent of vehicle speed. This high level of stability of time headway within drivers is also reported by Van Winsum and Brouwer (1997). Thus, it is asserted here that the following distance Dp that drivers try to maintain is Dp tp vi ;
1
where Dp is preferred following distance in m, tp preferred time headway in s and vi is the speed of the driverÕs vehicle in m/s. The rationale behind this rule is that drivers use time headway as a safety margin. Controlling the distance to the lead vehicle then basically consists of controlling available time in case the lead vehicle decelerates. There are substantial dierences in the value of tp between drivers, and Van Winsum (1998) has reported experimental evidence that this is related to perceptual-motor skills related to the transformation of visual information to an action in driving. This means that drivers who are less skilled in adjusting the braking response to the Timeto-Collision (TTC) with the lead vehicle generally choose to drive at a larger time headway. This may be considered as a mechanism in which drivers compensate for dierences in skill level. When drivers are required to follow at a smaller time headway than they prefer, this results in substantial cost in terms of mental eort. For example, Heino (1996) found a large increase in mental eort, measured as reduction in heart rate variability, when drivers were asked to follow at a smaller headway than preferred. However, there are circumstances under which drivers are willing to temporarily invest more eort in the following task, resulting in paying closer attention to the lead vehicle. This may be the case when driving at high speed in the left lane, while overtaking vehicles that are driving in the right lane, or just before the driver overtakes a lead vehicle and is waiting for an opportunity to safely do so. Also, when the driver is under considerable time pressure the willingness to increase attention and eort may result in choosing a smaller time headway than tp .
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
209
This was found in the study by Van der Hulst (1999) in which drivers failed to increase time headway as an adaptive response when preview was limited due to fog, while drivers who were not in a hurry did adapt time headway to visual conditions. In conditions of reduced visibility, such as fog, rain or night driving, drivers tend to drive at a larger time headway. This strategy may have negative eects on trac ¯ow because it reduces the capacity of the road. However, it makes sense from the viewpoint of the driver, since it may be considered as an increase of the safety margin to compensate for later detections of decelerations of lead vehicles. There is also evidence that tp diers as a function of driver state. Van der Hulst (1999) found that drivers who became more fatigued during the experiment increased their headway to a greater extent compared to drivers who reported only a small increase of fatigue. A similar result was found by Fuller (1981) who found that during prolonged driving time headways were increased, accompanied by verbal reports of performance decrements, drowsiness and exhaustion. So, although tp is associated with driver skills and thus relatively constant with the driver, it also depends on driver state (fatigue increases tp as an adaptive response to performance deteriorations), visual conditions (poorer vision results in a larger safety margin) and the mental eort and attention the driver is willing to pay to the lead vehicle (resulting in temporary situation-dependent decreases of time headway). If distance to the lead vehicle is larger than Dp , there is no safety-related reason for the driver to accelerate until Dp is reached. How fast the driver wants to accelerate in that case probably depends on other motivations. However, if distance to the lead vehicle is smaller than Dp , the driver is assumed to be motivated for a safety-related reason to decelerate until Dp is reached. This indicates that a distinction should be made between positive and negative accelerations. Only negative accelerations will be discussed here. There is evidence that when actual distance headway is smaller than Dp , the (negative) acceleration initiated by the driver is a function of the TTC as estimated by the driver: ai cTTCest d e;
2
where ai is the deceleration of the driver, TTCest the TTC as estimated by the driver, c a constant, d a constant (<0), and e is a random error term. The relation is such, that ai is larger negative with smaller TTCs. However, when TTC is greater than a certain value (say 10 s), ai will be 0. The error term is included because it is thought that ai is the result of releasing the accelerator and/or pressing the brake pedal. The resulting deceleration cannot be predicted accurately by the driver and this generates some random error. Van Winsum and Heino (1996) found that the maximum braking pressure excerted by the driver in response to a deceleration of the lead vehicle was strongly related to the TTC at the moment that the braking response was initiated by the driver. Van Winsum and Brouwer (1997) reported a high correlation between the movement time of the foot from the accelerator to the brake pedal and TTC at the moment the braking response was initiated. A smaller TTC results then in a faster movement of the foot to the brake pedal, which is associated with a larger deceleration in practice and it also results in pressing the brake harder, which is also associated with a larger deceleration. A similar reasoning was presented by Lee (1976), who stated that drivers are able to control braking based on TTC information from the optic ¯ow ®eld. This would enable the driver to judge the moment to start braking and to control the braking process.
210
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
Drivers commit systematic errors in the estimation of TTC. Van der Horst (1990) reported that the relation between estimated TTC and actual TTC can be described by the psychophysical law of Stevens (1957), according to TTCest eTTCf :
3
Based on the data from Schi and Detwiler (1979), McLeod and Ross (1983) and Cavallo, Laya and Laurent (1986), Van der Horst (1990) estimated that e 1:04 and f 0:72. Actual TTC is de®ned as the distance to the lead vehicle divided by relative speed. The question then is at what distance Dd the driver decides to decelerate. It is postulated here that Dd equals Dp minus the distance at which the driver detects a dierence in Dp , which is of course a Just Noticeable Dierence in distance. This refers to the Ôaction pointÕ as discussed by Brackstone and McDonald (1999) in their paragraph concerning psychophysical or action point models. They mention that the visual angle of the lead vehicle must change by a set percentage of about 10% on an average in order to be detected by the driver. This then depends on the distance between the eyes of the driver and the rear bumper of the lead vehicle and the width of the lead vehicle. The visual angle of the lead vehicle can be approximated as a tan(W/Dp ), where W is the width of the lead vehicle in meters. The visual angle at which the driver distinguishes that the lead vehicle is closer than the preferred headway then is 1 g a tan W =Dp , where g is the weber fraction of JND in visual angle of, say, 0.1% or 10%. The distance at which the driver then decides to decelerate is
Dd W = tan 1 g a tan W =Dp :
4
The TTC that is used by the driver in deciding how to decelerate can then be de®ned as follows: TTC Dd = RT aj ;
5
where aj is the acceleration (a negative value) of the lead vehicle and reaction time (RT) is the time it takes between the moment the deceleration of the lead vehicle started and the moment the reduction of headway is detected by the driver. RT aj is the same as relative speed vi ÿ vj . Since Dp ÿ Dd ÿaj t2 =2; RT (which is equivalent to t) can be computed as q 6 RT 2 Dp ÿ Dd = ÿ aj A larger deceleration of the lead vehicle then results in a smaller driver RT. Eq. (6) also indicates that there is no direct relation between the TTC to the lead vehicle at the moment the driver starts to brake and the RT of the driver. This has been con®rmed in experimental work. Van Winsum (1998) and Van Winsum and Brouwer (1997) found that TTC at the moment the driver initiates braking is not signi®cantly related to driver RT (de®ned as the interval between the moment the lead vehicle starts to decelerate and the moment the driver starts to release the foot from the accelerator). The model as described in Eq. (2) can now be substituted ai cTTCest d e; with d < 0; where TTCest eTTCf , with e 1:04 and f 0:72, where TTC Dd = RT aj , where Dd W = tan 1 g a tan W p =Dp , with g 0:1, W normally about 1.80 m and RT 2 Dp ÿ Dd = ÿ aj , where Dp tp vi .
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
211
3. Concluding remarks The model of car following presented here is based on principles, facts and ®gures from the literature of driver behaviour. The model as described is still too simple to cover all the variation in human behaviour that can be observed in car following. For example, the simple relation Dp tp vi covers a whole range of adaptive behaviour in which tp varies as a function of driver skills, driver state and visual conditions. Still, it may give a more realistic starting point than existing mathematical models of car following that are not based on psychology for modelling the eects of environmental and human factors on trac ¯ow, congestion and car following behaviour. Obviously, the model has not been fully tested with data and until such has been done its usefulness is limited. Also, not all relevant literature on driver behaviour during car following has been discussed here. However, since all components of which the model is constructed are based on scienti®c research, this commentary hopefully contributes to the design of realistic and useful models of car following. References Brackstone, M., & McDonald, M. (1999). Car following: a historical review. Transportation Research Part F: Trac Psychology and Behaviour, 1, 2, 181±196. Cavallo, V., Laya, O., & Laurent, M. (1986). The estimation of time-to-collision as a function of visual stimulation. In: A.G. Gale, M.H. Freeman, C.M. Haslegrave, P. Smith & S.P. Taylor, Vision in Vehicles (pp. 179±183). Amsterdam: Elsevier. Fuller, R. (1981). Determinants of time headway adopted by truck drivers. Ergonomics, 24, 463±474. Heino, A. (1996). Risk taking in car driving: Perceptions, individual dierences and eects of safety incentives. Ph.D. Thesis. University of Groningen, The Netherlands. Lee, D. N. (1976). A theory of visual control of braking based on information about time-to-collision. Perception, 5, 437±459. McLeod, R. W., & Ross, H. E. (1983). Optic ¯ow and cognitive factors in time-to-collision estimates. Perception, 12, 417±423. Schi, W., & Detwiler, M. L. (1979). Information used in judging impending collision. Perception, 8, 647±658. Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64, 153±181. Van der Horst, A. R. A. (1990). A time-based analysis of road user behaviour in normal and critical encounters. Ph.D. Thesis. Institute for perception TNO, Soesterberg: The Netherlands. Van der Hulst, M. (1999). Adaptive control of safety margins in driving. Ph.D. Thesis. University of Groningen, The Netherlands. Van Winsum, W., & Heino, A. (1996). Choice of time-headway in car-following and the role of time-to-collision information in braking. Ergonomics, 39, 579±592. Van Winsum, W., & Brouwer, W. H. (1997). Time headway in car following and operational performance during unexpected braking. Perceptual and Motor Skills, 84, 1247±1257. Van Winsum, W. (1998). Preferred time headway in car-following and individual dierences in perceptual-motor skills. Perceptual and Motor Skills, 87, 863±873.
www.elsevier.com/locate/trf
The human element in car following models Wim van Winsum * Human Factors Research Institute, TNO-TM, Kampweg 5, P.O. Box 23, 3769 ZG Soesterberg, Netherlands
Abstract This commentary distinguishes between the engineering-inspired formalisms used by Brackstone, M., and McDonald, M. (Transportation Research Part F (1999)) and the psychological knowledge of, and assumptions about human behaviour when car following. It outlines a preliminary mathematical model, based on psychological knowledge about car following behaviour by human drivers, which could be applied in trac engineering, as a means of bringing these two ways of studying car following closer together. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Close following
1. Introduction In the paper, ÔCar following: A historical reviewÕ by Brackstone and McDonald (1999), a clear and thorough overview of car following models from an engineering point of view is presented. According to the authors it is the intent of the paper to brie¯y assess the range of options available in the choice of car following models. In the concluding remarks they state that it is Ôhighly tempting to attempt to increase the realism of a chosen model by attempting to incorporate motivational or attitudinal factors that may be able to explain the dierences between driversÕ, but that there is Ôlittle evidence to relate such features to observable dynamic behaviourÕ. In this commentary, it is argued that there is a substantial psychological literature on car following that has not been covered by Brackstone and McDonald (1999). The models they describe have been developed starting from assumptions that are based on what appears to be common sense but for which there is little evidence. Having speci®ed the models, parameters are selected such that they ®t the observed data. The models fail to clarify why the relationships between variables are speci®ed as they are. A number of assumptions appear to be based on the idea that the human
*
Tel.: +31-3463-563977; fax: +31-3463-56252. E-mail address: [email protected] (W. van Winsum).
1369-8478/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 9 - 8 4 7 8 ( 0 0 ) 0 0 0 0 8 - 5
208
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
driverÕs behaviour is completely rational and that he or she is perfectly able to perceive distance, speed and acceleration. There is little evidence for this in the psychological literature. A lot of research on driver behaviour and car following has been done from a psychological point of view and this research has generated a substantial amount of knowledge that is applied in models of car following. What this research makes clear is that human drivers apply heuristics to cope with their limitations in an adaptive and ¯exible manner. However, a problem with this literature is that only certain elements of car following behaviour have been studied and that a consistent and an elaborated model is still lacking. In this commentary I would like to show how psychological knowledge about car following behaviour by human drivers can be applied in a mathematical model that can be used in trac engineering. It is hoped that this helps in bringing two dierent ways of studying car following behaviour a bit closer together. In Section 2, a number of results from psychological studies on car following are integrated into one preliminary model. This should not be considered as a validated and complete model but only as an example of how a model can be built that is rooted in known psychological facts and ®gures. 2. Psychological factors in car following There is substantial evidence that human drivers regulate available time as a control mechanism. The following distance that the driver attempts to maintain is based on a time headway that is constant in given similar circumstances. Van Winsum and Heino (1996) found that individual drivers follow a time headway that is independent of vehicle speed. This high level of stability of time headway within drivers is also reported by Van Winsum and Brouwer (1997). Thus, it is asserted here that the following distance Dp that drivers try to maintain is Dp tp vi ;
1
where Dp is preferred following distance in m, tp preferred time headway in s and vi is the speed of the driverÕs vehicle in m/s. The rationale behind this rule is that drivers use time headway as a safety margin. Controlling the distance to the lead vehicle then basically consists of controlling available time in case the lead vehicle decelerates. There are substantial dierences in the value of tp between drivers, and Van Winsum (1998) has reported experimental evidence that this is related to perceptual-motor skills related to the transformation of visual information to an action in driving. This means that drivers who are less skilled in adjusting the braking response to the Timeto-Collision (TTC) with the lead vehicle generally choose to drive at a larger time headway. This may be considered as a mechanism in which drivers compensate for dierences in skill level. When drivers are required to follow at a smaller time headway than they prefer, this results in substantial cost in terms of mental eort. For example, Heino (1996) found a large increase in mental eort, measured as reduction in heart rate variability, when drivers were asked to follow at a smaller headway than preferred. However, there are circumstances under which drivers are willing to temporarily invest more eort in the following task, resulting in paying closer attention to the lead vehicle. This may be the case when driving at high speed in the left lane, while overtaking vehicles that are driving in the right lane, or just before the driver overtakes a lead vehicle and is waiting for an opportunity to safely do so. Also, when the driver is under considerable time pressure the willingness to increase attention and eort may result in choosing a smaller time headway than tp .
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
209
This was found in the study by Van der Hulst (1999) in which drivers failed to increase time headway as an adaptive response when preview was limited due to fog, while drivers who were not in a hurry did adapt time headway to visual conditions. In conditions of reduced visibility, such as fog, rain or night driving, drivers tend to drive at a larger time headway. This strategy may have negative eects on trac ¯ow because it reduces the capacity of the road. However, it makes sense from the viewpoint of the driver, since it may be considered as an increase of the safety margin to compensate for later detections of decelerations of lead vehicles. There is also evidence that tp diers as a function of driver state. Van der Hulst (1999) found that drivers who became more fatigued during the experiment increased their headway to a greater extent compared to drivers who reported only a small increase of fatigue. A similar result was found by Fuller (1981) who found that during prolonged driving time headways were increased, accompanied by verbal reports of performance decrements, drowsiness and exhaustion. So, although tp is associated with driver skills and thus relatively constant with the driver, it also depends on driver state (fatigue increases tp as an adaptive response to performance deteriorations), visual conditions (poorer vision results in a larger safety margin) and the mental eort and attention the driver is willing to pay to the lead vehicle (resulting in temporary situation-dependent decreases of time headway). If distance to the lead vehicle is larger than Dp , there is no safety-related reason for the driver to accelerate until Dp is reached. How fast the driver wants to accelerate in that case probably depends on other motivations. However, if distance to the lead vehicle is smaller than Dp , the driver is assumed to be motivated for a safety-related reason to decelerate until Dp is reached. This indicates that a distinction should be made between positive and negative accelerations. Only negative accelerations will be discussed here. There is evidence that when actual distance headway is smaller than Dp , the (negative) acceleration initiated by the driver is a function of the TTC as estimated by the driver: ai cTTCest d e;
2
where ai is the deceleration of the driver, TTCest the TTC as estimated by the driver, c a constant, d a constant (<0), and e is a random error term. The relation is such, that ai is larger negative with smaller TTCs. However, when TTC is greater than a certain value (say 10 s), ai will be 0. The error term is included because it is thought that ai is the result of releasing the accelerator and/or pressing the brake pedal. The resulting deceleration cannot be predicted accurately by the driver and this generates some random error. Van Winsum and Heino (1996) found that the maximum braking pressure excerted by the driver in response to a deceleration of the lead vehicle was strongly related to the TTC at the moment that the braking response was initiated by the driver. Van Winsum and Brouwer (1997) reported a high correlation between the movement time of the foot from the accelerator to the brake pedal and TTC at the moment the braking response was initiated. A smaller TTC results then in a faster movement of the foot to the brake pedal, which is associated with a larger deceleration in practice and it also results in pressing the brake harder, which is also associated with a larger deceleration. A similar reasoning was presented by Lee (1976), who stated that drivers are able to control braking based on TTC information from the optic ¯ow ®eld. This would enable the driver to judge the moment to start braking and to control the braking process.
210
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
Drivers commit systematic errors in the estimation of TTC. Van der Horst (1990) reported that the relation between estimated TTC and actual TTC can be described by the psychophysical law of Stevens (1957), according to TTCest eTTCf :
3
Based on the data from Schi and Detwiler (1979), McLeod and Ross (1983) and Cavallo, Laya and Laurent (1986), Van der Horst (1990) estimated that e 1:04 and f 0:72. Actual TTC is de®ned as the distance to the lead vehicle divided by relative speed. The question then is at what distance Dd the driver decides to decelerate. It is postulated here that Dd equals Dp minus the distance at which the driver detects a dierence in Dp , which is of course a Just Noticeable Dierence in distance. This refers to the Ôaction pointÕ as discussed by Brackstone and McDonald (1999) in their paragraph concerning psychophysical or action point models. They mention that the visual angle of the lead vehicle must change by a set percentage of about 10% on an average in order to be detected by the driver. This then depends on the distance between the eyes of the driver and the rear bumper of the lead vehicle and the width of the lead vehicle. The visual angle of the lead vehicle can be approximated as a tan(W/Dp ), where W is the width of the lead vehicle in meters. The visual angle at which the driver distinguishes that the lead vehicle is closer than the preferred headway then is 1 g a tan W =Dp , where g is the weber fraction of JND in visual angle of, say, 0.1% or 10%. The distance at which the driver then decides to decelerate is
Dd W = tan 1 g a tan W =Dp :
4
The TTC that is used by the driver in deciding how to decelerate can then be de®ned as follows: TTC Dd = RT aj ;
5
where aj is the acceleration (a negative value) of the lead vehicle and reaction time (RT) is the time it takes between the moment the deceleration of the lead vehicle started and the moment the reduction of headway is detected by the driver. RT aj is the same as relative speed vi ÿ vj . Since Dp ÿ Dd ÿaj t2 =2; RT (which is equivalent to t) can be computed as q 6 RT 2 Dp ÿ Dd = ÿ aj A larger deceleration of the lead vehicle then results in a smaller driver RT. Eq. (6) also indicates that there is no direct relation between the TTC to the lead vehicle at the moment the driver starts to brake and the RT of the driver. This has been con®rmed in experimental work. Van Winsum (1998) and Van Winsum and Brouwer (1997) found that TTC at the moment the driver initiates braking is not signi®cantly related to driver RT (de®ned as the interval between the moment the lead vehicle starts to decelerate and the moment the driver starts to release the foot from the accelerator). The model as described in Eq. (2) can now be substituted ai cTTCest d e; with d < 0; where TTCest eTTCf , with e 1:04 and f 0:72, where TTC Dd = RT aj , where Dd W = tan 1 g a tan W p =Dp , with g 0:1, W normally about 1.80 m and RT 2 Dp ÿ Dd = ÿ aj , where Dp tp vi .
W. van Winsum / Transportation Research Part F 2 (1999) 207±211
211
3. Concluding remarks The model of car following presented here is based on principles, facts and ®gures from the literature of driver behaviour. The model as described is still too simple to cover all the variation in human behaviour that can be observed in car following. For example, the simple relation Dp tp vi covers a whole range of adaptive behaviour in which tp varies as a function of driver skills, driver state and visual conditions. Still, it may give a more realistic starting point than existing mathematical models of car following that are not based on psychology for modelling the eects of environmental and human factors on trac ¯ow, congestion and car following behaviour. Obviously, the model has not been fully tested with data and until such has been done its usefulness is limited. Also, not all relevant literature on driver behaviour during car following has been discussed here. However, since all components of which the model is constructed are based on scienti®c research, this commentary hopefully contributes to the design of realistic and useful models of car following. References Brackstone, M., & McDonald, M. (1999). Car following: a historical review. Transportation Research Part F: Trac Psychology and Behaviour, 1, 2, 181±196. Cavallo, V., Laya, O., & Laurent, M. (1986). The estimation of time-to-collision as a function of visual stimulation. In: A.G. Gale, M.H. Freeman, C.M. Haslegrave, P. Smith & S.P. Taylor, Vision in Vehicles (pp. 179±183). Amsterdam: Elsevier. Fuller, R. (1981). Determinants of time headway adopted by truck drivers. Ergonomics, 24, 463±474. Heino, A. (1996). Risk taking in car driving: Perceptions, individual dierences and eects of safety incentives. Ph.D. Thesis. University of Groningen, The Netherlands. Lee, D. N. (1976). A theory of visual control of braking based on information about time-to-collision. Perception, 5, 437±459. McLeod, R. W., & Ross, H. E. (1983). Optic ¯ow and cognitive factors in time-to-collision estimates. Perception, 12, 417±423. Schi, W., & Detwiler, M. L. (1979). Information used in judging impending collision. Perception, 8, 647±658. Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64, 153±181. Van der Horst, A. R. A. (1990). A time-based analysis of road user behaviour in normal and critical encounters. Ph.D. Thesis. Institute for perception TNO, Soesterberg: The Netherlands. Van der Hulst, M. (1999). Adaptive control of safety margins in driving. Ph.D. Thesis. University of Groningen, The Netherlands. Van Winsum, W., & Heino, A. (1996). Choice of time-headway in car-following and the role of time-to-collision information in braking. Ergonomics, 39, 579±592. Van Winsum, W., & Brouwer, W. H. (1997). Time headway in car following and operational performance during unexpected braking. Perceptual and Motor Skills, 84, 1247±1257. Van Winsum, W. (1998). Preferred time headway in car-following and individual dierences in perceptual-motor skills. Perceptual and Motor Skills, 87, 863±873.