Modeling of inter-vehicular gaps
4.1 Analysis and modeling of lateral clearance between vehicles
4.1.5 Analysis of simultaneous multiple lateral interactions
If vehicles interact simultaneously on both sides of the test vehicle, there is simultaneous influence of both these vehicles on test vehicle equipped with sensors. Interaction of vehicles are decided based on vehicle’s detection by the sensors fitted on both sides of the vehicles (refer to Fig. 4.2).
The necessary condition for multiple vehicle interactions is that a vehicle should be detected in at least one sensor of one side of the test vehicles, while at least one sensor on opposite side also detects another vehicle, at the same time-step, as shown in Fig. 4.9. This illustration is shown in Fig. 4.9.
100 150 200 250
0 30 60 90
Lateral clerance (cm)
Speed (km/h)
(a) Auto-Auto
Auto-Bike Bike-Bike
Average (AA & BB) 3W-3W
3W-2W 2W-2W
Average (AA&BB)
100 150 200 250
0 30 60 90
Lateral clearance (cm)
Speed (km/h)
(b) Auto-Auto
Car-Auto Car-Car
Average (AA & CC)
100 150 200 250
0 30 60 90
Lateral clearance (cm)
Speed (km/h)
(c) Bike-Bike
Car-Bike Car-Car
Average(CC & BB) 2W-2W
Car-2W Car-Car
Average (CC&BB)
3W-3W Car-3W Car-Car
Average (AA&CC)
Fig. 4.9 Necessary condition of detecting interacting vehicles for multiple vehicle interactions 4.1.5.1 Conditions for constrained interaction
Four conditions may arise when vehicle interacts laterally with one or more vehicles- 1. Interaction is only on one side.
2. Interaction is on both sides, and test vehicle is overtaking both the vehicles (Fig. 4.9, v1 >
v2, v3).
3. Interaction is on both sides, and test vehicle is being overtaken by other two vehicles. (v1 <
v2, v3)
4. Interaction is on both sides, and test vehicle is overtaking one vehicle and being overtaken by other vehicle. (v2> v1 >v3).
Case 1 is considered as unconstrained lateral interaction for analysis. Case 2 is considered as constrained lateral interaction. Cases 3 and 4 are ambiguous and not considered for comparison, since it is difficult to conclude whether test vehicle is in constrained or unconstrained condition.
The situation of test vehicle overtaking both vehicles by moving in the gap between them, is a definite indicator of constrained lateral interaction. If Case 2 is observed, then interaction with both the vehicles are assigned as constrained interactions. Data of different vehicle pairs from Cases 1 and 2 are compared with each other. To avoid constraining due to median or road edges, the data obtained from carriageways with less than three lanes width is removed. Data for both the cases are modeled as per Equation 4.1 and 4.2.
4.1.5.2 Modeling and comparison of constrained behavior
Regression equations (slopes and intercepts) and Beta-distributed residuals are calculated for some vehicle-pairs (car-car, car-3W, car-2W, and 3W-2W) with significant data for both the conditions.
Both the regression equations are plotted in Fig. 4.10 for each vehicle pair. The comparisons of slopes and intercepts of constrained conditions with unconstrained conditions for various pairs are
Interacting vehicle-2
Test vehicle
Interacting vehicle-1
Direction of traffic flow
𝑉 3
𝑉 1
𝑉 2
Sensors
shown in Table 4.8. Last coloumn in this table presents p-statistic of fit of beta distribution with actual residuals from field data.
Fig. 4.10 (a) Plots of speed-lateral clearance relationships for constrained, unconstrained cases for Car-car pairs
Fig. 4.10 (b) Plots of speed-lateral clearance relationships for constrained, unconstrained cases for Car-3W pairs
y = 0.3805x + 125.19 y = 0.5189x + 120.65
0 50 100 150 200 250
0 10 20 30 40 50 60 70 80
Lateral clearance (cm)
Speed km/h
Car-car constrained Car-car Unconstrained
Linear (Car-car constrained) Linear (Car-car Unconstrained)
y = 0.2286x + 123.24 y = 0.3258x + 134.04
0 50 100 150 200 250
0 10 20 30 40 50 60 70 80
Lateral clearance (cm)
Speed km/h
Car-Auto constrained Car-Auto Unconstrained
Linear (Car-Auto constrained) Linear (Car-Auto Unconstrained)
Fig. 4.10 (c) Plots of speed-lateral clearance relationships for constrained, unconstrained cases for Car-2W pairs.
Fig. 4.10 (d) Plots of speed-lateral clearance relationships for constrained, unconstrained cases for 3W-2W pairs.
y = 0.6323x + 127.35
y = 0.2676x + 134.77
0 50 100 150 200 250
0 10 20 30 40 50 60 70 80
Lateral clearance (cm)
Speed (km/h)
Car- Bike Unconstrained Car- Bike constrained
Linear (Car- Bike Unconstrained) Linear (Car- Bike constrained)
y = 0.8635x + 126.46
y = 0.1711x + 136.6
0 50 100 150 200 250
0 10 20 30 40 50 60 70 80
Lateral clearance (cm)
Speed (km/h)
Auto-Bike Unconstrained Auto-Bike Constrained
Linear (Auto-Bike Unconstrained) Linear (Auto-Bike Constrained)
Table 4.8 Comparison of lateral clearance with average speed models for constrained and unconstrained conditions
Vehicle pair Condition Sample size
Regression line Parameters of beta distribution of residuals Slope Intercept
(cm)
𝛼1 𝛼2 a b p-
value Car-Car Unconstrained 585 0.519 120.65 3.016 1.854 -72.05 118.31 0.034
Constrained 122 0.381 125.19 2.154 1.379 -67.96 106.96 0.054 3W-Car Unconstrained 243 0.326 134.04 7.668 4.634 -118.72 196.68 0.045 Constrained 95 0.229 123.24 2.712 2.085 -96.63 125.81 0.059 Car-2W Unconstrained 553 0.632 127.35 2.875 2.058 -79.31 112.31 0.051 Constrained 141 0.267 134.78 2.193 1.339 -82.83 58.63 0.055 3W-2W Unconstrained 151 0.864 126.46 1.276 1.322 -87.57 86.59 0.075 Constrained 67 0.171 136.6 2.063 1.868 -91.84 103.12 0.071 Validation
data Car-car
Unconstrained 414 0.51 121.03 4.164 2.412 -81.69 141.44 0.028 Constrained 75 0.312 125.12 1.864 1.325 -74.86 107.4 0.083
From Table 4.8, following inferences can be observed-
There is an increase of lateral clearance with increase in average speed for both the cases- constrained as well as unconstrained and for all vehicle pairs.
There is a reduction in slopes by 26.67%, 29.75%, 57.77% and 80.19% for car-car, 3W- car, car-2W and 3W-2W pairs. It may mean that at moderate and higher speeds, vehicles maintain lesser gap during constrained overtaking than unconstrained overtaking. Slopes of data from constrained and unconstrained cases are compared with each other using ANCOVA comparison and they are not significantly similar except for car-car case at 5%
significance levels. (p=0.088, 0.046, 0.035 and 0.041 between slopes of constrained and unconstrained behavior for car-car, 3W-car, car-2W and 3W-2W pairs). This indicates that drivers behave significantly different at higher speeds, under constrained and unconstrained overtaking conditions.
At lower speeds (<20 km/h), the value of LC maintained by vehicles at constrained condition is higher than unconstrained conditions. This information is derived only from best-fit lines. However, in Fig. 4.10, one can observe that at lower speeds, there is no significant change in lateral clearance-maintenance. An ANCOVA test on intercepts also confirms the result. (p=0.067, 0.348, 0.219 and 0.078 between intercepts of constrained and unconstrained interactions, for car-car, 3W-car, car-2W and 3W-2W pairs).
Two-wheelers have a higher capability to veer, thus may maintain lesser gaps during constrained cases. Moreover, when vehicles of larger size like three-wheeler or car overtake the two-wheelers, there is not much risk due to collision with a two-wheeler. So, lesser gaps maybe maintained with two-wheelers during constrained overtaking. This property is reflected from the coefficients of slope for constrained conditions of car-2W and 3W-2W pairs. There is a high compromise (58% for car-2W pair and 80% for 3W-2W pair) when constrained relationship is compared with unconstrained relationship (Fig. 4.10
(c), (d)). On the other hand, for cars maintaining gap with other cars, there is not a significant change in relationships as observed from Fig. 4.10 (a).
From the residual coefficients a and b of constrained and unconstrained cases, one can conclude that there is less spread of data about the regression line (indicated by the difference between a and b) for constrained case as compared with the unconstrained case (except for 3W-2W pair). In the analysis, value of parameter 𝛼1 > 𝛼2 or the distributions are skewed towards the lower sides for almost all vehicle pairs.
4.1.5.3 Validation for constrained case
Validation is performed using MANOVA test, only on car-car dataset, because data for constrained situations observed for other vehicle types were insignificant (sample size < 50).
About 40% data are kept aside for validation. The speed-lateral clearance plot and regression lines are plotted similar to models described earlier. Validation exercise is performed to check whether lateral clearance-speed relationships of constrained and unconstrained conditions differ significantly with change in datasets. Fig. 4.11 compares the constrained and unconstrained conditions for validation data of car-car pairs. Table 4.9 provides various statistical tests on field and validation constrained and unconstrained data.
These results for compromise in lateral clearance in constrained overtaking conditions will be helpful in modeling traffic with weak or no lane discipline. Here, vehicles tend to squeeze in between two vehicles due to lack of proper lanes. These results, if used along with basic pairwise lateral clearance maintenance between two vehicles can reflect lateral interactions between vehicles in weak lane discipline traffic, as observed in developing countries.
Table 4.9 Validation exercise for checking constrained-unconstrained behavior on car-car pairs
Hypothesis p-value
Reject (0) or accept (1) hypothesis at 95% confidence Significant similarity between constrained
data of model, validation 0.245 1
Significant similarity between unconstrained
data of model, validation 0.314 1
Significant similarity between constrained
and unconstrained slopes of validation 0.052 1
Fig. 4.11 Constrained and unconstrained condition data for validation dataset of car-car pair 4.1.6 Effect of width of road and day/night conditions on speed-LC relationship
During manual vehicle identification of interacting vehicles through video-camera, an approximate width of road was also noted in terms of number of lanes. Paved shoulders are classified as having half-lane width. Data are segregated widthwise, and regression lines are fitted for each width group for LC with average speed.
Data are segregated widthwise, and 3d regression plane is fitted. The aim is to find width where LC is minimum for given speed. Partially differentiating obtained regression surface (Equation 4.3, second degree with N, first degree with 𝑣̅) gives N=4.59, or about 4.5 lane width. It means at this width, vehicles achieve maximum squeezing in. The graphical representation is shown in Fig.
4.12. However, it is observed that the increase or decrease in lateral clearance with number of lanes is not significant (Pearson’s coefficient of correlation is -0.043), thus width of the road do not hold significant effect in effect on lateral clearance maintenance.
LC = 149 -13.79N + 0.553𝑣̅ + 1.5N2 + 0.02𝑣̅N … 4.3
Fig. 4.12 Parameters of LC with average speed relationship varying with number of lanes.
y = 0.3116x + 125.12 y = 0.5096x + 121.03
0 50 100 150 200 250
0 10 20 30 40 50 60 70 80
Lateral clearance (cm)
Speed km/h
Validation data- Constrained Validation data- Unconstrained
Linear (Validation data- Constrained) Linear (Validation data- Unconstrained)
The obtained results are presented in Table 4.10. From the results, one can conclude that there is no significant trend of slopes and intercepts for LC-speed relationship based on road width.
However, vehicles tend to maintain least LC at four-lane road width (two lanes with paved shoulders), equivalent to 14 m road-space. LC increases roughly with increase in road width after this width.
Table 4.10 Parameters of LC with average speed relationship varying with number of lanes.
Approximate number of
lanes
Parameters of straight line relationship of speed with LC
Slope Intercept
1.5 1.00 105.34
2 0.57 130.37
2.5 0.37 129.58
3 0.56 120.81
3.5 0.72 142.95
4 0.75 122.88
5 0.87 111.03
6 0.92 105.34
7 0.88 109.83
Time of the day (day or night time) was also taken into consideration during vehicle identification.
Those vehicles which run during day as well as night were segregated. Plot of LC with average speed was calculated separately for day and night conditions. It is observed that, vehicles tend to keep significantly more LC during the daytime than night time (refer Table 4.1), as against what was expected. Linear regression trends give night time LC with average speed slopes and y- intercept as 0.371 and 119.9 (0.33 and 117 for car-car), as against 0.697 and 123.4 (0.791 and 119.69 for car-car) for the same vehicles during day time. One of the reasons for this can be the urgency of vehicles to return to the destination during night-time, due to which they take higher risks.