ACCELERATION AND DECELERATION MODELS OF MOTORCYCLE AT SIGNALIZED INTERSECTIONS Chu Cong MINH Lecturer School of Civil Engineering Hochiminh city University of Technology 268 Ly Thuong Kiet street, District 10 Hochiminh city, Vietnam Fax: +84-8-8637003 Email:
[email protected]
Kazushi SANO Associate Professor Civil and Environmental Engineering Nagaoka University of Technology Kamitomioka, 1603-1, Nagaoka, Niigata 940-2188, Japan Fax: +81-258-47-9650 Email:
[email protected]
Shoji MATSUMOTO Professor Civil and Environmental Engineering Nagaoka University of Technology Kamitomioka, 1603-1, Nagaoka, Niigata 940-2188, Japan Fax: +81-258-47-9650 Email:
[email protected]
Nguyen Cao Y Graduate Student Civil and Environmental Engineering Nagaoka University of Technology Kamitomioka, 1603-1, Nagaoka, Niigata 940-2188, Japan Fax: +81–258–47–9650 Email:
[email protected]
Abstract: The knowledge of traffic operation of motorcycles is too little compared to the development of motorcycles in Southeast Asia. This study addresses a model framework of motorcycles’ acceleration/deceleration at signalized intersections. Different from previous researches, which are associated with the static strip-based approach for referring motorcycle positions, a new technique, so-called dynamic motorcycle lane, is developed. Then, motorcycles’ deceleration and acceleration are treated similarly to cars’ by applying deceleration and acceleration models with some modifications. Two regimes in the deceleration and acceleration models are proposed: free-regime and following-regime. At every observed interval, the particular regime is specified by the longitudinal threshold distance. A case study is introduced to estimate the parameters of proposed models using microscopic traffic data collected at some intersections in Vietnam. The finding here can be applied to develop a motorcycle simulation model, which is very necessary for Asian countries. Key Words: Motorcycle traffic, Motorcyclist behavior, Traffic operation 1. INTRODUCTION An distinct characteristic of the rapid motorization in Asian is the predominance of two wheelers, which have made Asian cities significantly different from other regions. In Hanoi and Hochiminh city, Vietnam, for example, two-wheelers are more than 80% of the total transportation means. Motorcycle ownership and usage in Asia are influenced by several agents different from those of western contries. Agents may include the weather, the climate, the economic and infrastructure development, the population density and the culture background. In that area, the term “motorcycle dependent city” has been used to indicate a city with low income, high density land use and motorcycles’ domination in traffic flow. Behavior of motorcyclists is different from drivers of four-wheeled vehicles, especially at intersections. For example, due to slender shape and small size, following motorcyclists may respond to the actions of the leader at a closer distance than the distance between the following and leading
four wheelers. The acceleration, deceleration, speed and the maneuverability of motorcycles are also different from those of four-wheeled vehicles. Moreover, motorcycles may not travel in a single line in a lane and may not follow the lane discipline as four wheelers do. Hence, the following model is different from four-wheelers’. However, very few researches have been conducted on behavior of motorcyclists in following. In order to obtain a better understanding about behavior of motorcyclists at intersections, this study aims to model motorcycle decelerations/accelerations at signalized intersections. The general model captures the deceleration/acceleration applied by motorcyclists while traveling in queues. The model incorporates both situations, with and without the constraint from leading vehicles. It is necessary to note that the zig-zag maneuverability of motorcycles in queues is not taken into consideration in this study.
2. METHODOLOGY The terminology “motorcycle” used in this research refers to motorized two-wheelers. In Vietnam, the engine capacity of motorcycles mostly ranges from 50cc to 150cc, including mopeds, scooters and normal motorcycles. Figure 1 presents the structure of the deceleration model in the present study.
Figure 1 Structure of acceleration/deceleration models 2.1 Dynamic Motorcycle’s Lane Different from four-wheelers, two-wheeled vehicles do not follow lane disciplines. Even in the same lane, riders would not experience being constrained in maneuverability, if the neighboring or front vehicles are a little far. In the literature, previous studies usually applied the static strip-based approach for modeling this dynamic characteristic, such as Botma et al. (1991), Oketch (2000) and Lan et al. (2003 and 2005). In the static strip-based approach, the roadway width is divided into sub-lanes (strips) and every vehicle occupied a certain of sub-lanes based on its width. However,
because of the dynamic characteristic, two-wheelers can occupy any lateral position across the roadway, and therefore this method cannot model the rider behavior correctly. In order to overcome this shortcoming, the term “dynamic motorcycle’s lane with respect to the subject motorcycle” is introduced in this study. This concept is used for only straight roads, not for curves. The dynamic motorcycle’s lane is not stable as previous studies but flexible according to the subject motorcycle’s position. The width of motorcycle’s lane is defined as the width of the operating zone of the subject motorcycle. In other words, a motorcyclist rides freely if no other vehicle appears in his/her area. That occupied area is used to determine the width of the motorcycle’s dynamic lane regarding to the subject motorcycle. According to Hussain et al. (2005), the physical width of a static motorcycle and the width of the operating space are 0.8 (m) and 1.3 (m), respectively. However, the authors neglected the fact that those values mainly depend on speeds of motorcycles. In the other words, the width of the operating space is larger for a higher speed motorcycle. In the present study, to obtain the width of dynamic motorcycle’s lane and to present the relationship between that value and speed, the minimum lateral distance between two motorcycles in paired riding was used. The paired riding of motorcycles is defined when a couple of motorcycles travel abreast together as a pair over 10 (m), while maintaining a maximum 1.5 (m) of longitudinal distance and without any effort to pass over.
hn ( t − τ n ) > S na ( t )
Figure 2 Dynamic’s motorcycle lane 2.2 Longitudinal Threshold Distance According to the behavior of motorcyclists, when the distance to the front vehicle is considerably far, the motorcyclist rides without any constraint from the leader. When the distance to the front vehicle is close sufficiently, the subject rider accelerates/ decelerates with constraints from the leader. The longitudinal threshold distance of a motorcycle is defined as the threshold distance between the subject motorcycle and the leader or the stop line, at which a motorcyclist starts an action to change the speed with the purpose of preventing collision or stopping safely at the stop line. That threshold determines the driving regime. If the distance between subject motorcycle and its leader is less than the longitudinal threshold distance, the motorcyclist is assumed to ride with some constraints from the leader and therefore, the following-regime is applied. Otherwise, the motorcyclist rides freely from the leader and he/she is in the free-regime. Previously, for passenger-car analysis, Leutzbach (1988) proposed the psycho-physical spacing model and introduced the term “perceptual threshold” to identify the relationship between the relative speed threshold and the space headway. Leutzbach, however, didnot provide any mathematical formulation of the proposed model. Subramanian (1996) assumed the distance threshold to be constant for a particular driver but varies across drivers according to a truncated shifted lognormal distribution. The distribution of the headway threshold is assumed to be skewed to the left since the probability of a driver being aggressive is much less than the probability of a driver being
conservative. Unlike Subramanian, Ahmed (1999) assumed that the time headway threshold follows a truncated normal distribution with truncation on both sides. The truncation is applied because time headway threshold must be positive and finite. Lemessi (2001) defined critical distance for a generic vehicle is a function of (i) braking space for that vehicle; (ii) leader advancement while the follower is braking; (iii) stochastic following distance; and (iv) leader’s vehicle length. In this study, this threshold is assumed to vary from individual to individual but be constant for each one. It is computed based on the kinematic formula, which describes the correlation among distance, speed and acceleration. Sn = α
Vn an
β γ
×υ p
δ np
×υg
δg n
(1)
where Sn (m): Longitudinal threshold distance of motorcycle n; Vn (m/sec): Speed of motorcycle n; an (m/sec2): Deceleration of motorcycle n; α, β, γ, υnp ,υng : Estimated parameters; If only one person is on the motorcycle 0 δ np = Otherwise 1
0 1
δ ng =
If the rider is male
(2)
(3)
Otherwise
2.3 Reaction Time Because the response of a motorcycle is a function of reaction time, the reaction time should be measured before estimating the parameters of models. The reaction time of a motorcyclist is expected to be a function of age, gender, number of people on a motorcycle, weather condition, traffic condition, speed, roadway geometry, vehicle’s type, etc. For a particular motorcyclist, the reaction time tends to vary as the traffic condition or environment change. To quantify the parameter value for the reaction time, field experiments were conducted on the General Motors test track. The driver of the lead vehicle was instructed to follow pre-specified speed pattern, while the driver in the following vehicle was unaware of the pre-specified speed pattern and instructed to maintain a safe minimum distance behind the lead vehicle. (May, 1990). From the observed traffic data, Ozaki (1993) developed models, in which reaction time is the function of relative headway and subject vehicle’s acceleration. In this research, reaction time is assumed to be a constant for a motorcyclist across observations and be varied over motorcyclists. From several set of historical data about speed, acceleration, distance, the reaction time value is estimated by matching distance with speed and acceleration. It means that for each person, among several assumed reaction times, the one with the highest correlation among distance, speed and acceleration is defined as the reaction time for that person. 2.4 Acceleration/deceleration Models After the dynamic lane and the longitudinal threshold distance of motorcycles are identified, the behavior of motorcycles is modeled upon the knowledge from passenger cars. Intuitively, when the distance headway between the subject motorcycle and the leader is large (or no leading vehicle is case of some first coming motorcycles), the motorcyclist of the subject motorcycle will accelerate/decelerate without any constraint from the leader. Otherwise, the rider will travel under
constraint from the leader. Thus, the acceleration/deceleration applied by a motorcyclist belongs to two regimes (i) free-regime or (ii) following-regime. The boundary between free-regime and following-regime is determined upon the space headway between the subject motorcycle and the leading vehicle. The longitudinal threshold distance is used to determine the regime, to which a particular motorcycle belongs at a given instance. fra frd an (t) or an (t) an ( t ) = foa fod an (t) or an (t)
If hn(t – τn) ≥ Sn
(4)
Otherwise
where anfra(t) or anfrd(t) (m/s2): Free-acceleration and free-deceleration, respectively; anfoa(t) or anfod(t) (m/s2): Following-acceleration and following-deceleration, respectively; τn (sec): Reaction time of motorcyclist n; Sn (m): Longitudinal threshold distance; hn(t– τn) (m): Space headway at time (t– τn). 2.4.1 Free-deceleration Model A motorcycle is in the free-deceleration model if the motorcyclist decelerates with no leader or the leader space headway is larger than the longitudinal threshold distance, and so the motorcyclist is not affected by the leader’s behavior. This model applies for two cases: (i) Some first motorcycles coming to the intersection during red periods, being aware of the red signal, the riders decelerate freely until stopping at the stop line; (ii) Motorcycles come to the back of the queue, and then speed down without any constraint from their leader. The model assumes that the decelerations of a given motorcyclist over time are captured by the reaction time. The free-deceleration of a motorcyclist applies:
(
)
a nfrd ( t ) = sfrd X nfrd ( t − τ n ) + ε nfrd ( t )
(5)
where, sfrd(Xnfrd(t - τn)) is the sensitivity function for free-deceleration. Xnfrd(t - τn) is the vector of explanatory variables describing the sensitivity function. εnfrd(t) is the random term associated with the free-deceleration of the motorcyclist n at time t. This term captures unobserved effects on free-deceleration model. It is assumed to follow an independent and identical normal distribution. It varies for different motorcyclists and it is the same for each motorcyclist over time: εnfrd(t) ∼ N[0, (σ frd 2 ) ] which (σ frd)2 is the variance of the free-deceleration error term. Under these assumptions, the probability density function of the free-deceleration is given by: f(anfrd(t)) =
1
σ
frd
(
)
a nfrd ( t ) − s frd X nfrd ( t − τ n ) σ frd
φ
(6)
where φ is the probability density function of a normal distribution. 2.4.2 Free-acceleration Model A motorcycle is in the free-acceleration model if the rider accelerates with no leading vehicle or the lead gap is greater than the longitudinal threshold distance. This model applies for motorcycles in front of queues. Usually, when the signal light changes from red to green, those motorcyclists accelerate freely according to their ability and the motorcycles’ capability until attaining the desired speed. If the desired speed is higher than the current speed, the riders tend to accelerate. Otherwise, they are expected to decelerate. The free-acceleration model is given by:
(
)(
)
a nfra ( t ) = sfra X nfra ( t − τ n ) Vnds ( t − τ n ) − Vn ( t − τ n ) + ε nfra ( t )
(7)
where, sfra(Xnfra(t - τn)) is the sensitivity function for the free-acceleration model. Xnfra(t - τn) is the vector of explanatory variables describing the sensitivity function. (Vnds(t - τn)-Vn(t - τn)) is the stimulus function. εnfra(t) is the random term associated with the free-acceleration of the motorcyclist n at time t. It is assumed to follow an independent and identical normal distribution. It varies for different motorcyclists, and it is the same for each motorcyclist over time: εnfra(t) ∼ N[0, (σ fra)2] which, (σ fra)2 is the variance of the free-acceleration error term. Under these assumptions, the probability density function of the following deceleration is given by: f(anfra(t)) =
σ
(
)(
)
a nfra ( t ) − s fra X nfra ( t − τ n ) Vnds ( t − τ n ) − Vn ( t − τ n ) fra σ
1
φ fra
(8)
where φ is the probability density function of a normal distribution. 2.4.3 Following-acceleration and Following-deceleration Models When distance headway is less than the longitudinal threshold distance, the following-regime is applied and motorcyclists are assumed to accelerate/decelerate with the constraint from the leaders. In the following regime, if the relative speed to the leader is greater than 0 ((∆Vn(t - τn)>0), the subject rider is assumed to be in the following-deceleration model. Otherwise, the rider is assumed to be in the following-acceleration model. The following-acceleration and following-deceleration models are given by:
(
)
(9)
(
)
(10)
anfoa ( t ) = sfoa X nfoa ( t − τ n ) .f foa (∆Vn ( t − τ n )) + ε nfoa ( t ) anfod ( t ) = sfod X nfod ( t − τ n ) .f fod (∆Vn ( t − τ n )) + ε nfod ( t )
where, Xnfoa(t - τn) and Xnfod(t - τn) are the vector of explanatory variables describing the sensitivity function. sfoa(Xnfoa(t - τn)) and sfod(Xnfod(t - τn)) are the sensitivity function of the following-acceleration and following-deceleration models, respectively. ∆Vn(t – τn) is the speed relative to the leader at time (t – τn). f foa(∆Vn(t - τn)) and f fod(∆Vn(t - τn)) are the respective stimulus function. εnfoa(t) and εnfod(t) are the random term associated with the following-acceleration and following-deceleration of the motorcyclist n at time t. It is assumed to be i.i.d distributed. It varies for different motorcyclists, and it is the same for each motorcyclist over time. εn foa∼ N[0, (σ foa)2] and εn fod ∼ N[0, (σ fod)2],which (σ foa)2 and (σ fod)2 are the variance of the following error term. Under these assumptions, the probability density functions of the following-acceleration and following-deceleration models are given by: f(anfoa(t)) =
f(anfod(t)) =
1
σ
fod
)
σ
1
σ
(
anfoa ( t ) − s foa X nfoa ( t − τ n ) f
φ foa
(
foa
)
anfod ( t ) − s fod X nfod ( t − τ n ) f φ σ fod
foa
(∆Vn ( t − τ n ))
fod
(∆Vn ( t − τ n ))
(11)
(12)
where φ is the probability density function of a normal distribution. 2.4.4 Likelihood Function For passenger car analysis, according to Subramanian (1996), all parameters of the car-following models and the distributions of reaction time, headway threshold were estimated jointly by using a
maximum likelihood estimator and detailed vehicle trajectory data. In the present study, the model parameters are treated separately after obtaining the reaction time distribution and longitudinal threshold distance. The acceleration/deceleration applied by a motorcyclist is given by formula (6), (8), (11), and (12), where error term is assumed to be distributed i.i.d normal. Therefore, the probability density function of the deceleration of motorcyclist n across In observations is given by: In
f(an1, an2, .., anIn) =
1
∏σ t =1
ε nr ( t ) r σ
φ r
(13)
The log-likelihood is given by the product across all N motorcyclists: In 1 ∑ ∑ ln σ r n =1 t =1 N
L=
ε nr ( t ) r σ
φ
(14)
where ani is the acceleration/deceleration of the motorcycle n at the observation i. εnr(t), σr are error term and its standard deviation for r = free-acceleration, free-deceleration, following-acceleration or following-deceleration. The maximum likelihood estimates of the model parameters are obtained by maximizing function L.
3. DATA COLLECTION Three locations in Hanoi and Hochiminh city, which are (i) mixed traffic with sufficient motorcycle volumes; (ii) not near bus stop, petrol station, etc.; and (iii) easy to observe motorcycles discretely, were used for data collection. The entire data was collected under conditions of clear weather, dry pavement, low magnitude of wind, non-congested traffic environment, in November 2004 and January 2005. The first location in Hanoi has three lanes per each direction with the raised median. The lane widths are 3.50 (m), 3.64 (m), and 3.37 (m) for the inner, medium and outer lanes, respectively. The fixed-time signalized control has been installed with 130 (sec) of a cycle time, including 37 (sec), 3 (sec), and 90 (sec) for green, yellow, and red times, respectively. There is no all-red time for this signalized intersection. The average queue length is 30 (m). The second location in Hochiminh city has two lanes per each direction and no hard median. The inner lane and outer lane are 3.00 (m) and 3.95 (m) in width, respectively. Most of four-wheeled vehicles use the outer lane. The motorcycle proportion is approximately 80%. The queue length is about 25 (m). It is the fixed-time signalized control with 30 (sec), 3 (sec) and 40 (sec) for green, yellow and red times, respectively. It is also no all-red time. The third location in Hochiminh city is the one-way street with three lanes. The lane widths for the inner, middle and outer lanes are 5.05 (m), 3.85 (m) and 3.15 (m), respectively. This location is the most crowded among three. The average queue length is approximately 43 (m). The traffic is mixed among non-motorized vehicles, motorcycles, cars, vans, buses, etc., in which the motorcycle proportion is more than 85%. The signalized control time is 33 (sec), 3 (sec), and 37 (sec) for green, yellow, and red times, respectively. The digital video recorder was set up overhead the study sites, right angle to the direction of vehicle traffic. Motorcycle’s positions were identified from image video file every one-fifth of a second
interval. These positions regarding time events were calculated according screen coordinates then converted into roadway coordinates by using image processing software SEV, developed in the traffic lab for specific purposes. The methodology for converting from screen coordinates to roadway coordinates was adopted from Khan et al. (2001). For every screen coordinate pair xs, ys and roadway coordinate pair xr, yr, the following expression may be derived:
xr =
C1 + C 2 x s + C 3 y s C4 xs + C5 y s + 1
(15)
yr =
C6 + C7 x s + C 8 y s C4 x s + C5 y s + 1
(16)
Where xr, yr : Roadway x and y coordinates; xs, ys : Screen x and y coordinates; C1, .., C8 : Coefficients. In order to match the coordinate between screen and roadway, all coefficients (C1, .., C8) would be computed by solving the above equations. This process requires at least four points, so-called base points, from which, all screen and roadway coordinates were determined. From that, the trajectory function (space-time) of each motorcycle is developed. Then, speed, deceleration, acceleration profiles are computed by taking first and second derivatives across time of each motorcycle’s trajectory function. A motorcycle was observed for over 5 seconds. It is worth to note that the advantage of this research is it can capture the behavior of motorcycles according to both time series and motorcycle population. For each motorcycle entity, it is not only one observation but also 20 observations at different times were used for estimating model parameters.
4. ESTIMATION RESULTS The parameters of acceleration and deceleration models are estimated by the statistical estimation software GAUSS. Estimation results of the proposed models are shown in Table 1. 4.1 Dynamic Motorcycle’s Lane In the present study, the minimum lateral distance between two motorcycles riding abreast was applied to determine the width of dynamic motorcycle’s lane. That value varies according to different traffic conditions, such as motorcycle’s speed, number of people in the motorcycle, traffic density, etc. However, according to the purpose of this study, the width of dynamic motorcycle’s lane of a given motorcycle is just used to identify the leader. Therefore, only the most significant variable affecting the dynamic lane-width, average speed, is considered. The relationship between motorcycle’s average speed (V) (m/sec) and motorcycle’s dynamic lane width (lw) (m) may be expressed as: lw = 0.07V + 0.80 (17) Figure 3 shows the direct proportion between average speed and motorcycle’s dynamic lane width. The sample size used to calibrate Equation 17 includes 150 motorcycle entities. Because the motorcycle speed limit is 11.11 (m/sec) (40 Km/h), the range of speeds in Figure 3 is from 0 (m/sec) to 11.11 (m/sec). The lane width is used to identify the leader of the subject motorcycle in the following regime.
3
Lateral Distance (m)
2.5 2 1.5 1 0.5 0 0
2
4
6
8
10
12
Average Speed (m/sec)
Figure 3 Dynamic lane width of motorcycles 4.2 Longitudinal Threshold Distance The longitudinal threshold distance of a motorcycle is computed when the motorcyclist starts the action to decelerate or to change the direction in order to avoid a collision or to stop safely at the stop line. From the trajectory data, the point when the motorcyclist starts decelerating is identified, then the speed and the deceleration at this point are also obtained. Those data are applied to estimate the longitudinal threshold distance according to the proposed model shown on Equation 1. The estimated coefficient α is expected to be positive due to the absolute sign of deceleration data. Motorcyclists need a long distance to start decelerating when running with a high speed. Therefore, the coefficient β is positive. In case of more than one person on the motorcycle, because the mobility of the motorcycle is reduced, the motorcyclist is likely to be more alert. Hence, the estimate coefficient υp is expected to be more than 1. Similarly, female motorcyclists seem to ride more carefully and alertly than men. The coefficient υg is therefore more than 1. The estimated longitudinal threshold distance is given by: S n = ( 4.590 )
Vn an
0.530 p
0.092
g
× 1.118δ n × 1.018δ n
(18)
4.3 Reaction Time Distribution Because the response of a motorcycle is a function of reaction time, the reaction time should be measured before estimating the parameters of models. In this research, reaction time is assumed to be a constant for a motorcyclist across observations and be varied over motorcyclists. According to the panel data collected from observation, speed, acceleration, and distance of each motorcycle entity were captured by time interval. The reaction time value was estimated by matching acceleration with various speeds and distances regarding to time intervals. It means that for each person, among several assumed reaction times, the one with the highest correlation among distance, speed and acceleration is defined as the reaction time for that person. Totally 100 samples of motorcycles are used to estimate the reaction time. The reaction time is followed lognormal distribution at the 5% significance level and shown in Figure 3. The mean and the standard deviation of the reaction time distribution are 0.79 (s) and 0.24 (s) respectively. The mean of reaction times of motorcycles is smaller than those of passenger cars which observed at General Motor test track. It may be explained by the fact that (i) motorcyclists have more advantages than drivers in observing ambient traffic environment; and (ii) drivers need time to lift the foot from
the accelerator pedal, move it laterally to the brake and then to depress the pedal. However, motorcyclists depress the brake pedal directly and instantly. 14
Percentage (%)
12 Observed Theoretical
10 8 6 4 2 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
Reaction Time (s)
Figure 4 Reaction time distribution 4.4 Free-deceleration Model In case of free-deceleration regime, the deceleration rate of a motorcycle is free from the effect of the leader. The sensitivity part in this case may include (i) the speed of the subject motorcycle (Vn(t - τn)); (ii) the spacing between the subject motorcycle and the leader or the stop line, (∆Xn(t - τn)); (iii) the dummy variable of the number of people on the motorcycle (δn p), and the dummy variable of the motorcyclist’s gender (δng), which are defined in equation (2) and (3). The sensitivity part is given by: s
frd
(X
frd n
)
(t −τ n ) = λ
Vn ( t − τ n )η
(∆X n ( t − τ n ))ν
×υ p
δ np
×υ g
δ ng
(19)
A motorcycle decelerates freely until it reaches a complete stop at the stop line or at the end of the queue, thus the negative sign is applied. The constant parameter λ is therefore expected to be negative. Motorcyclists are likely to apply a higher deceleration at higher speeds compared to lower speeds, which implies that η should be positive. Moreover, the deceleration rate is also expected to be higher if the space headway is shorter and therefore, the corresponding parameter ν should be positive. In the other hand, different from passenger cars, motorcycles are affected by the weight they carry. Therefore, the number of people on a motorcycle contributes to the change of the deceleration rate. The deceleration rate associated with more than one person on a motorcycle is decreased because the mobility of the motorcycle is reduced. As expected, this estimate coefficient is positive and less than 1. Usually, ladies ride motorcycle more carefully than men do. Thus, the gender parameter is expected to be more than 1. After estimating the parameters, the free-deceleration model is given by: an
frd
( t ) = ( −0.924 )
εnfrd ∼ N(0, 0.3252)
Vn ( t − τ n )2.021
(∆X n ( t − τ n ))
1.256
p
g
× 0.899 δ n × 1.045 δ n + ε nfrd ( t )
(20) (21)
4.5 Following-deceleration Model The sensitivity term is a non-linear function of explanatory variables that may include the speed of
the subject vehicle (Vn(t - τn)), the spacing between the subject vehicle and its leader (∆Xn(t - τn)), the dummy variable of the number of people on the motorcycle (δn p), the dummy variable of the gender of motorcyclist (δn g), and the dummy variable for the heavy leading vehicle (δn h). The functional form for the sensitivity term is given by: sfod (X nfod ( t − τ n )) = ξ
Vn ( t − τ n )ϕ
(∆X n ( t − τ n ))
ψ
×ϑp
δ np
×ϑg
δ ng
×ϑg
δ nh
(22)
In this case, as the subject motorcycle decelerates its speed according to the deceleration of the leader, the corresponding parameter ξ is therefore expected to be negative. The deceleration at higher speeds is expected to be larger relative to lower speeds and therefore, speed parameter ϕ should be positive. Furthermore, motorcyclists are likely to apply a higher deceleration at short space headways compared to large space headways. The parameter ψ is therefore expected to be positive. On the other hand, in the present research, the respective stimulus function is assumed to be a non-linear function of the relative leader speed and shown as below: f fod (∆V n ( t − τ n )) = ∆Vn ( t − τ n )
ω
(23)
The higher relative speed between the subject motorcycle and the leader is, the higher deceleration motorcyclist must apply. The sign of the corresponding parameter ω, therefore, should be positive. The definition of dummy variable δn p and δng are defined in equation (2), (3), respectively. Similar to the free-deceleration model, the estimate coefficient of occupancy variable is positive and less than 1. The gender parameter is expected to be more than 1. In order to explain the difference between the following model between a motorcycle and a four-wheeler and the following model between two motorcycles, the dummy variable for the heavy lead vehicle is used: 0 1
If the lead vehicle is a motorcycle
δ nh =
(24)
If the lead vehicle is a four-wheeler
If the lead vehicle is a four-wheeler, the motorcyclist tends to apply a higher deceleration rate due to the safety concern. Therefore, the parameter for heavy leader is expected to be greater than 1. In summary, the following deceleration model is given by:
an
fod
εnfod
( t ) = ( -1.537 )
2
∼ N(0, 0.292 )
Vn ( t − τ n )0.398
(∆X n ( t − τ n ))
0.725
p
g
h
× ∆Vn ( t − τ n )0.519 × 0.703 δ n × 1.064 δ n × 1.027 δ n + ε nfod ( t )
(25) (26)
Figure 4a and 4b show the difference between observed and estimated deceleration data in both the free-deceleration model and the following-deceleration model.
Actual Deceleration
Actual Deceleration
0 -4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5 0 -0.5
-4
R2 = 0.62
-3.5
-3
-2.5
-2
-1.5
-1
0 -0.5 0 -0.5
R2 = 0.59
-1.5 -2 -2.5 -3
Estimated Deceleration
Estimated Deceleration
-1
-1 -1.5 -2 -2.5 -3
-3.5
-3.5
-4
-4
Figure 4a Difference between observed and estimated data in the free-deceleration
Figure 4b Difference between observed and estimated data in the following-deceleration model
4.6 Free-acceleration Model In case of free-acceleration regime, the acceleration rate of a motorcycle is free from the effect of the leading vehicle. The parameters of the desired speed model are estimated and given by the following formula: Vnds(t - τn) = 8.585 – 1.387δnp – 1.277δng
(27)
The dummy variables δn p and δng are defined in equation (2) and (3). The desired speed associated with more than one person on the motorcycle is decreased because the mobility of the motorcycle is reduced. The estimate coefficient of this dummy variable, δn p, is therefore negative. Usually, ladies ride motorcycles more carefully than men do. Thus, the coefficient of the gender variable, δng, is more than 1. If the desired speed is higher than the current speed, the motorcyclist is expected to accelerate and vice versa. Therefore, the constant parameter in sentivity part should be positive. The free acceleration model is given by: anfra(t) = 0.150 × (Vnds(t - τn) – Vn(t - τn)) + εnfra(t)
ε
fra n
2
( t ) ∼ N(0, 0.169 )
(28) (29)
4.7 Following-acceleration Model The sensitivity term is a non-linear function of explanatory variables that includes the speed of the subject vehicle, the spacing between the subject vehicle and its leader, the dummy variable of the number of people on the motorcycle, the dummy variable of the gender of the motorcyclist, the dummy of the heavy leading vehicle. The stimulus function is assumed to be a non-linear function of the relative leader speed. As the subject motorcycle behaves according to the behavior of the leading vehicle, the constant parameter is therefore expected to be positive regarding to the acceleration. The acceleration at a higher speed, (Vn(t - τ)) (m/sec), is expected to be larger relative to a lower speed and therefore, speed parameter should be positive. In case of high relative speed between the subject motorcycle and the
leader, (∆Vn(t - τ)) (m/sec), the motorcyclist tends to highly accelerate. The sign of the corresponding parameter is expected to be positive. The dummy variables δn p, δng and δnh are defined in equation (2), (3) and (24). Similar to the free-acceleration model, the estimate coefficient of the number of people on motorcycle is positive and less than 1. The acceleration associated with more than one person on the motorcycle is reduced due to the reduction of the mobility of the motorcycle in that case. The gender parameter is expected to be less than 1. If the leading vehicle is a four-wheeler, the motorcyclist is likely to apply a lower acceleration. Therefore, this parameter is less than 1. The effect of the space headway to the leader is not clear. The motorcyclist may apply a higher acceleration rate for a larger headway to the leader due to having more available space for traveling. On the other hand, as the spacing increases the attention given to the leader and the perception of his/her behavior as a stimulus decreases, which may result in lower accelerations (Toledo, 2003). The following-acceleration model is given by: an
foa
( t ) = ( 0.385 )
Vn ( t − τ n )0.445
p
(∆X n ( t − τ n ))0.197
g
h
× ∆Vn ( t − τ n )0.841 × 0.938 δ n × 0.546 δ n × 0.899 δ n + ε nfoa ( t )
(30)
ε nfoa ( t ) ∼ N(0, 0.1232)
(31)
Figure 5a and 5b show the difference between observed and estimated data in the free-acceleration and following-acceleration model. 2.5
3 R2 = 0.76
2
R = 0.66
2.5 Estimated Acceleration
Estimated Acceleration
2
1.5
1
0.5
2
1.5
1
0.5 0
0 0
0.5
1
1.5
2
2.5
Actual Acceleration
Figure 5a Difference between observed and estimated data in the free-acceleration model
0
0.5
1
1.5
2
2.5
3
Actual Acceleration
Figure 5b Difference between observed and estimated data in the following-acceleration model
Table 1 Parameter estimates for acceleration/deceleration models Variable Estimate t-statistic Free-acceleration model Constant 0.1495 39.257 Constant of the desired speed 8.5850 49.898 p -1.3870 -16.962 Occupancy dummy (δn ) g -1.2773 -14.837 Gender dummy (δn ) fra 0.1690 -95.418 σ Number of motorcyclists = 72, Number of observations = 1440 L(0) = -1887.818, L(β) = -535.961, ρ 2 = 0.716 Following-acceleration model Constant 0.3859 Subject speed (Vn(t)) (m/sec) 0.4460 0.1973 Relative leader speed (∆Vn(t)) (m/sec) 0.8407 Space headway (∆Xn(t)) (m) 0.9378 Occupancy dummy (δn p) g 0.5456 Gender dummy (δn ) h 0.8994 Heavy dummy (δn ) foa 0.1229 σ
13.141 8.412 6.205 39.130 15.291 9.804 22.277 70.149
Number of motorcyclists = 56, Number of observations = 1120 L(0) = -557.997, L(β) = -379.192, ρ 2 = 0.320 Free-deceleration model Constant Subject speed (Vn(t)) (m/sec) Space headway (∆Xn(t) (m) Occupancy dummy (δnp) Gender dummy (δng)
σ frd
-0.9240 2.0213 1.2555 0.8985 1.0447 0.3252
-25.331 57.202 60.829 60.012 69.177 -66.268
Number of motorcyclists = 87 ; Number of observations = 1740 L(0) = -3056.528; L(β) = -514.346; ρ 2 = 0.832. Following-deceleration model Constant -1.5366 Subject speed (Vn(t)) (m/sec) 0.3973 0.5186 Relative leader speed (∆Vn(t) (m/sec) 0.7253 Space headway (∆Xn(t)) (m) p 0.7025 Occupancy dummy (δn ) g 1.0639 Gender dummy (δn ) h 1.0272 Heavy dummy (δn ) fod 0.2924 σ
-30.876 10.349 38.778 27.483 36.256 47.110 27.300 -57.670
Number of motorcyclists = 55; Number of observations = 1100 L(0) = -1556.447; L(β) = -208.352; ρ 2 = 0.866.
5. CONCLUSIONS The paper proposes a methodology for estimating acceleration/deceleration behaviors of motorcycles at the signalized intersection. Firstly, since a stream of motorcycles may not be assigned well-defined lanes as that of four-wheel vehicles may, the adapted definition of the dynamic motorcycle’s lane has been introduced. The relationship between a motorcycle’s lane width and its speed is constructed based on the minimum lateral distance between two motorcycles in paired riding. The lane width is applied to identify the leader of the subject motorcycle in proposed models. In order to distinguish between acceleration and deceleration models, this study estimates model parameters separately. The general acceleration/deceleration model of motorcycles at a signalized intersection is classified into two regimes: free-regime and following-regime. In order to identify to which regime a motorcycle belongs, the longitudinal threshold distance is introduced. The model for estimating this threshold distance across the motorcyclist population is constructed based on the kinematics’ formula. The reaction time distribution, capturing the variations in perceptions among motorcyclists, is developed. It is proved to follow the log-normal function at the 5% significance level, in which mean and the standard deviation are 0.79 (sec) and 0.24 (sec), respectively. The models capture the variation across the motorcyclist population. The effect of unobserved motorcyclist/ motorcycle characteristics is identified by the motorcyclist random term. The most time consuming in behavior modeling is the data. Due to limitations of the collection equipment, available sets were collected at some intersections only. Further study requires more observations at more detail data and at several locations with different characteristics.
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