Modeling Gap-Maintenance in Heterogeneous No Lane-Discipline Traffic

1 st International Conference on Transportation Infrastructure and Materials (ICTIM 2016) ISBN: 978-1-60595-367-0 Modeling Gap-Maintenance in Heterogeneous No Lane-Discipline Traffic Anuj Kishor Budhkar 1 and Akhilesh Kumar Maurya 2 1 Research Scholar, Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam-781039, India; anuj.budhkar@iitg.ernet.in 2 Associate Professor, Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam-781039, India; maurya@iitg.ernet.in. ABSTRACT Traffic stream of developing countries have a characteristic heterogeneous traffic stream moving in a weak or no lane-discipline. Due to these reasons, state-of-the-art carfollowing models cannot replicate vehicular interactions in developing countries. Carfollowing behavior is in a staggered fashion, with following gap changing as per amount of stagger, vehicle type of leading and following vehicles as also their speeds. The aim of this paper is to determine the variation in longitudinal gap as staggering or centerline separation, and speed changes, for different leading and following vehicle pairs moving in a heterogeneous traffic stream. A top view video of moving traffic is recorded with accurate field markings. Twelve mid-block sections in cities of India are selected for this purpose; each having ideal roadway conditions and varying widths and flow levels. Camera calibration technique of vanishing point method is adopted for extracting vehicle pairwise trajectory data from different sections. From these data, leading and following vehicle sizes, their speeds, types are noted along with longitudinal gap, centerline separation between them. Data are processed for finding relationships, and model is calibrated and validated. It is seen that longitudinal gap increases with decrease of centerline separation and increase of speed. The graphs change for varying vehicle types and speed ranges. Established relationships can be used for modeling gap-maintenance characteristics in heterogeneous traffic. INTRODUCTION Traffic stream of developing countries have two characteristic features- 1. Lack of lane discipline and 2. Heterogeneous traffic. Vehicles overtake as and when sufficient gaps are available to pass. Moreover, several vehicle types are generally observed on Indian roads which vary significantly in their sizes and operating characteristics. These different vehicles are generally classified in various classes like Car, Truck, Light Commercial Vehicles (L.C.V), Bus, Auto-rickshaw, Motorized Two wheelers and Non-motorized vehicles. Traffic stream behavior is an outcome of human driving process, which is a complex, natural phenomenon. In order to analyze such a stream, a study of interaction (longitudinal as well as lateral) between vehicles is necessary. In such a traffic stream, vehicles do not follow each other in demarcated paths. This is because the response of driver of a following vehicle to any perturbation made by the leading vehicle is not just in 590

the form of acceleration or braking, but also in the form of steering. Car-following behavior is in a staggered fashion. It is thus important to study the relationship of staggering with car-following parameters such as distance headway, velocities and also vehicle types of a pair of vehicles in such a traffic stream. Few work, such as that by Gunay (2007) has been done in this regard. The objective of this paper is to establish relationships of longitudinal gap with stagger and velocities for a given vehicle pair. From mid-block, near-ideal traffic streams, videos are captured and trajectory points of an interacting pair are extracted. The scope of this paper is limited to only steady-state car-following. PREVIOUS WORK Car-following models describe the processes by which drivers follow each other in the traffic stream. They have been studied for more than half century (e.g., Pipes, 1953). A detailed description of all such car-following models can be obtained in Brackstone s (1999) review paper. But unlike the basic car-following behavior, the staggered car-following behavior widely exists in the real world (Gunay, 2007). It is a psychological tendency of drivers that they refrain from driving side by side with other vehicles for long periods of time. They either pass the neighbouring vehicle or lag behind it by maintaining a staggered headway. Hence, in modelling, consideration of an adjustment exercise for the calculation of the longitudinal positions of vehicles by taking the effect from the adjacent lane vehicles is important. Gunay (2003) defined the term lane-based driving discipline as the tendency to drive within a lane by keeping to the centre as closely as possible. Jin et al. (2011) developed a theory for staggered car-following based upon geometric analysis and proposed a general equation for time to collision based upon non-lane based traffic. This was obtained by using visual angle information that can be perceived by drivers directly. This approach is novel but data-collection from external source is difficult. Models are also proposed by using least distance between the vehicles (or pores) as a criterion. These include the continuum model by Nair et al. (2011), or pore-space model by Ambarwati et. al (2013). However, there has been no consideration of speed of interacting vehicles while considering this study. For considering vehicular heterogeneity, contributions such as Arasan and Koshy (2005) for transverse clearance between vehicles, and Ravishankar and Mathews (2011) for evaluating parameters of Gipp s model (1981) between various vehicle pairs in heterogeneous traffic; are important. Grid-based (Gundaliya et al., 2010) or strip based (Mathew et al., 2013) approaches are also important. Mallikarjuna and Rao (2006) have modeled heterogeneous traffic which includes governing of sub-lanes based upon the dimensions of the smallest vehicle in traffic that is the two-wheeler. Chakraborty et al. (2004) developed comprehensive model which describe both lateral and longitudinal control of vehicles based on a force field (potential field) analogy of the driving environment. Gunay (2009) attempted to explore the issue of two-dimensional headway analysis (lateral and longitudinal) in detail for better realism in traffic flow modelling. From the literature review we can conclude that there has been limited study to model no lane-discipline traffic combined with vehicle heterogeneity. This analysis is a crucial stage before going to modeling of traffic especially in developing countries like India. 591

METHODOLOGY For analyzing such data, initially traffic streams of mid-block sections are recorded by video camera. Since lateral position of vehicles are also important, x and y coordinates of vehicle (using roadway edge as reference point) need to be calculated. Camera calibration technique is used to obtain vehicle positions at given time stamp. Data collection and extraction Video recording for longitudinal gap analysis is carried in locations mentioned in Table 1. The sites are so chosen based upon the prerequisites for camera calibration technique as described by Fung et al. (2003). In order to extract data from video, firstly interacting vehicle pairs are identified as the video runs. Two vehicles in a stream are considered to be an interacting pair, if presence of one vehicle influences the behavior and position of another vehicle. The second vehicle either follows the first, or makes an attempt to veer (shift laterally). Image coordinates of vehicle can be marked during vehicle interaction by means of a mouse click. They get recorded in a file. Camera calibration technique devised by Fung et al. (2003) is used for converting image coordinates into actual field coordinates along and across length of road. For a particular interacting pair of vehicles in the traffic stream, vehicle size and type are noted and marked by mouse click. After this, 3-4 trajectory points for each of the pair of vehicles are marked by mouse clicking on the image. Field coordinates corresponding to these image-coordinates can be calculated using a program developed in MATLAB. The entire procedure is explained in detail by Budhkar and Maurya (2015). The user finally obtains x and y coordinate of each of two vehicles in the field, their sizes and vehicle types at different timestamps. Speed of a vehicle is calculated based upon difference in timestamp and positions of two trajectory points of a vehicle. Centerline separation (CS) is the distance between coordinates of center of front of leading vehicle (LV) and following vehicle (FV), at a given time. It is calculated by subtracting half of vehicle widths of both the vehicles from their corner point trajectory. Longitudinal gap (LG) is the gap between LV and FV, calculated by subtracting vehicle length of leading vehicle from distance headway. 592

Table 1. Details of locations of data collection. S. no. Road (city) Location Date Time Road width Trap length 1 Guwahati bypass Games 16-Jan-15 10:30-7.6 20 m (Guwahati) Village 13:30 2 Gariahat main road (Kolkata) Gariahat 17-Nov-14 15:30-18:30 12.1 25 m 3 EM bypass (Kolkata) 4 West Chord Road (Bangalore) 5 Bellary road (Bangalore) 6 Sinhagad road (Pune) Ruby hospital Magadi underpass Mekhiri circle Nilayam bridge 7 Ganeshkhind road (Pune) Sancheti hospital 8 Pune bypass (Pune) Sus-pashan exit 9 Old Bombay- PCMC Poona Highway corporation (Pune) 10 Western Express Highway (Mumbai) 11 JV link road (Mumbai) 12 Link road (Mumbai) Ismail Yusuf College, Jogeshwari Powai garden Saidham complex (MaladWest) Calculating accuracy of data extraction 18-Nov-14 10:00-13:00 29-Nov-14 10:00-12:30 30-Nov-14 16:30-19:30 04-Dec-14 11:00-13:00 04-Dec-14 14:00-16:00 05-Dec-14 11:00-13:30 05-Dec-14 16:30-19:30 07-Jul-14 10:00-11:30 14-Jul-14 15:30-16:30 08-Jul-14 18:30-20:00 7.8 35 m 7.6 20 m 11.2 30 m 10.4 35 m 9 45 m 11.2 60 m 7.8 50 m 17.5 45 m 10.8 20 m 14 40 m Accuracy of data extraction depends upon how accurately one clicks on the exact corner of the vehicle. An extractor may make an erroneous mouse click with a fixed standard deviation on the screen (in pixels). 100 cars belonging to only a particular model of a taxi (known width) are marked for their widths by clicking on two extreme points. It is observed that there is a standard deviation corresponding to 6 pixels for one mouse click. Since length is calculated by two mouse clicks, variance of error of one mouse click will be halved. Trap lengths upto 60 m are captured. They are corresponding to a standard deviation of 0.04-0.16 m error on the field. 593

ANALYSIS Data of three locations- Bellary road in Bangalore, Old Bombay-Poona Highway in Pune, and Linking road Mumbai from table 1 are used for validation, whereas data from remaining sites are used for model development. Here, four parameters are of chief interest- speeds of LV and FV, centerline separation between them (CS); and longitudinal gap between LV and FV (LG). Since the study in this paper is focused upon steady-state car-following, only those data with relative speed of ±5kmph is used for analysis. The authors assume that relative speed magnitude less than 5 kmph between LV and FV will correspond to steady state car following. Longitudinal gap vs centerline separation and velocity Figure 1. Scatter plot of Longitudinal gap (m) with centerline separation (m) and velocity (kmph) for car-car pair. Longitudinal gap maintained between two vehicles will depend upon CS and stream speed, which is taken as the average of speeds of LV and FV. Scatter plot of 3 variablesaverage speed (v), longitudinal gap (LG) and centerline separation (CS) are extracted for every vehicle-type pair (significant data >100 points). Single degree polynomial functions are found to be reasonably fit the scatter plot. Increasing degree of equation does not improve R 2 value. General equation of curves is given in Equation 1, where a and b are coefficients of CS and v respectively, c is the constant term; and φ is residual term about the best fit curve. LG = a (CS) + b(v) + c + φ Equation (1) Longitudinal gap increases with the increase in velocity or decrease in centerline separation. As CS increases, driver of FV can be more confident of a veering maneuver on sudden stopping of LV, decreasing LG. At higher speeds, drivers are more cautious so LG increases with increase in speed. This trend is also reflected in the coefficients. A sample curve fit (snapshot from MATLAB software) for car-car plot is shown in Figure 1. In order to remove heteroscedasticity, residuals (φ) are calculated, and divided by the obtained value of LG as per best fit curve (i.e. divided by a (CS) + b(v) + c). There is no auto-correlation between CS and v. Modified residuals follow Burr distribution (4- parameter) fitting at good significance levels by K-S test. Equation (2) represents frequency distribution function f(x) of Burr distribution. The possible reason for this is, 594

there is restriction on LG on one side that it cannot be negative, whereas on the other side, there is no upper limit for maintaining LG. This is the property of Burr distribution. Coefficients a, b, c of Equation 1 and parameters of Burr distribution (from Equation 2) for different LV and FV pairs are shown in Table 2. Heavy indicates heavy vehicles taken as a combined dataset of trucks and buses. Table 2. Coefficients and Residual parameters for plot of LG vs CS and velocity. Coefficients of Equation 1 (Modelled data) Burr (4P) distribution parameters of modified residuals LV FV c a b k α β γ Car Car 3.38 0.11-1.25 1.25 3.37 1.13-1.15 Auto Car 6.09 0.03-1.05 5.96 1.95 2.68-1.02 Bike Car 3.97 0.04-0.50 6.88 1.67 3.32-1.00 L.C.V. Car 4.26 0.08-1.42 3.44 1.95 1.91-1.02 Heavy Car 1.78 0.17-0.80 1.55 3.19 1.14-1.02 Car Auto 4.38 0.08-1.62 12.21 1.56 5.14-0.96 Car Bike 2.50 0.07-0.30 15.47 1.63 5.87-1.01 Car L.C.V. 2.81 0.11-1.05 3.13 2.22 1.78-1.07 Auto Auto 7.23 0.01-1.23 168.74 1.54 30.92-1.01 Auto Bike 1.73 0.09-1.01 0.90 3.29 0.75-0.94 Bike Auto 4.32 0.04-1.28 133.19 1.51 28.85-1.02 Heavy Heavy 2.66 0.26-1.52 2.80 3.59 1.84-1.34 Bike Bike 1.98 0.04 0.49 3.73 1.54 2.20-0.99 VALIDATION Equation (2) A similar curve-fitting exercise was carried out, and residuals evaluated. Since residuals are not normal, K-S test is used for comparing the residual plots with model data (Antoniou et al., 2014). Table 3 presents K-S test results conducted at 5% significance. Vehicle pairs with significant data (>100 data-points) are considered. The last column is an indicator of reject the null hypothesis (1) or accept (0) that the two samples of residuals of validation and model are drawn from the same population. 595

Table 3. Validation of Residuals using K-S test for model and validation dataset. Coefficients of equation 1 p-value LV FV Sample size (validation data) after validating residuals Reject? c a b Car Car 2331 4.449 0.085-1.05 9.89E-06 1 Car Auto 106 2.814 0.063-0.0031 0.402 0 Bike Car 118 4.627 0.077-0.9439 2.29E-04 1 Bike Bike 266 4.445 0.023-0.7803 0.165 0 Bike Auto 108 2.922 0.042 0.4626 0.179 0 Auto Car 302 2.953 0.1-0.7477 0.211 0 Car Bike 103 3.919 0.03-1.423 0.5915 0 CONCLUSION AND FUTURE SCOPE This paper attempts to establish relationships between parameters observed in steadystate car-following within a heterogeneous traffic stream not observing lane-discipline. By means of camera calibration technique, the longitudinal gap, centerline separation and speeds are calculated from trajectory data-points. Longitudinal gap increases with increase in stream speed and decrease in centerline separation. The residuals are Burrdistributed. The results are calculated and validated for a number of vehicle pairs. The paper currently focuses on only pairs of vehicles in traffic stream, i.e. one leader and one follower. In denser traffic streams, position of one vehicle may be affected by positions of more than one leading vehicles, the road edge and other factors. Also, steady state car following is seldom observed. These complex behaviors can be taken up as a future work. An entire exhaustive car-following model can be developed after considering these behaviors along with vehicle parameters such as acceleration, desired speed, braking, driver responses etc. REFERENCES Ambarwati, L., Pel, A., Verhaeghe, R., Arem, B. (2013) "Empirical Analysis of Heterogeneous Traffic Flow", Proceedings of the Eastern Asia Society for Transportation Studies, Vol.9. Antoniou, C., Gikas, V., Papathanasopoulou, V., Mpimis, T., Markou, I., Perakis, H., (2014). Towards distribution-based calibration for traffic simulation. 17th International IEEE Conference on Intelligent Transportation Systems, Qingdao, China. Arasan, V. and Koshy, R. (2005). "Methodology for modeling highly heterogeneous traffic flow" Journal of Transportation Engineering, Vol. 131(7): 544-551. Brackstone, M., and McDonald, M. (1999). "Car-following: a historical review". Transportation Research Part F, Vol. 2: 181-196. Budhkar, A. K. and Maurya, A. K. (2015). "A methodology to calculate inter-vehicular longitudinal distances in heterogeneous traffic". 3rd Conference of Transportation Research Group of India (3rd CTRG), 2015. 596

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