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The phenomenon of car-following is special in traffic operations. Traditional car-following models can well describe the reactions of the movements between two concessive vehicles in the same lane within a certain distance. With the invention of connected vehicle technologies, more and more advisory messages are in development and applied in our daily lives, some of which are related to the measures and warnings of speed and headway distance between the two concessive vehicles. Such warnings may change the conventional car-following mechanisms. This paper intends to consider the possible impacts of in-vehicle warning messages to improve the traditional car-following models, including the General Motor (GM) Model and the Linear (Helly) Model, by calibrating model parameters using field data from an arterial road in Houston, Texas, U.S.A. The safety messages were provided by a tablet/smartphone application. One exponent was applied to the GM model, while another one applied to the Linear (Helly) model, both were on the stimuli term “difference in velocity between two concessive vehicles”. The calibration and validation were separately conducted for deceleration and acceleration conditions. Results showed that, the parameters of the traditional GM model failed to be properly calibrated with the interference of in-vehicle safety messages, and the parameters calibrated from the traditional Linear (Helly) Model with no in-vehicle messages could not be directly used in the case with such messages. However, both updated models can be well calibrated even if those messages were provided. The entire research process, as well as the calibrated models and parameters could be a reference in the on-going connected vehicle program and micro/macroscopic traffic simulations.

Car-following is a special process in traffic operation where a following vehicle adjusts its accelerations based on the performance of the leading vehicle(s) and current status of the following vehicle. Car-following behaviors have been carefully studied with the first generation of models initiated in earlier 1950s [

Car-following models are one of the core processes of almost all microscopic traffic simulation models, including FRESIM, NETSIM, INTEGRATION, and CARSIM [

The headway distance is an important variable in car-following models, the inverse of which is density. With this and other linkages, the car-following models can be used to bridge macroscopic with microscopic traffic flow analyses as well to characterize not only the behaviors of individual vehicle, but also the overall relationships among traffic flow variables, such as volume, speed, and density [

Factors that may impact car-following behaviors include driver’s psychological and physical status, the level of service of the roadway, and vehicle’s performance [

Recently, with the invention of many innovative technologies, The United States Department of Transportation (USDOT) initiated a Connected Vehicle (CV) program to develop a platform combining well-defined technologies, interfaces, and processes to minimize risks and enhance the overall performance of traffic operations. It includes the Connected Vehicle Human Factors Research focusing on “understanding, assessing, planning for, and counteracting the effects of signals or system-generated messages that take the driver’s eyes off the road (visual distraction), the driver’s mind off the driving task (cognitive distraction), and the driver’s hands off the steering wheel (manual distraction)” [

Before drivers’ hands are “fully off”, the prompted messages from the CV system are actually additional stimuli to drivers, which would definitely affect driving behaviors as well as the resulted car-following behaviors. This creates a challenge on how to incorporate the impacts of such supplementary warning messages into traditional car-following models.

The objective of this paper is to improve the traditional car-following models by considering the impacts of in- vehicle warning messages from tablet/smartphone applications within a CV system. The parameters of the improved models would be re-calibrated using field test data.

A CV system relies on wireless communications and/or other innovative technologies to detect/identify the presence of surrounding vehicles, pedestrians, bicycles, and other objects nearby, and provide drivers with corresponding messages to enhance the mobility, safety, and air quality [

The effects of the corresponding messages on driving behaviors have been studied in different situations, such as smart warning messages at a work zone advance warning area [

These in-vehicle warning messages are usually in either audio or visual forms to notify drivers on driving speed [

a safe message, a careful message, and a warning message in

In a car-following situation illustrated in

From the engineering view, there are basically five types of car-following models [

From the view of statistical physics, vehicles can be regarded as a moving particle [

Among the above mentioned models, the GM models and the linear (Helly) models are probably one of the most traditional models. In the rest of this paper, these two types of models are further updated by incorporating the impacts of advance warning messages from a tablet/smartphone application.

GM models are perhaps the most famous class of car-following models with its first version dated more than 60 years ago. It is based on an intuitive hypothesis of “stimuli-reaction” philosophy that a driver receives a “stimulus” (the difference of velocity

where

where

In Equation (2), the sensitivity factor is expressed as

The prevalent use of in-vehicle messages could be additional “stimuli” to drivers, which possibly impacts drivers’ reaction and should be involved into the “stimuli” portion

The additional parameter k as well as the coefficient c, could be calibrated using observed data in the case that safety messages are provided.

Traditionally, the so-called linear car-following model is referred to the Helly model [

Here,

where, θ reflects the impacts of warning messages and should be re-calibrated together with the associated parameter

In order to obtain the real data for parameter calibration of the aforementioned revised models, field tests were conducted along an arterial road in Houston, Texas, USA. in June, 2015 during the afternoon non-peak hours (12:00 pm to 4:00 pm). In the test route shown in

A and B is 6.9 km (4.3 miles), namely 13.8 km (8.6 miles) long for each complete test round.

There are 10 traffic signals between intersections A and B in

The leading vehicle was a 2014 Toyota Corolla and the following vehicle was a 2012 Toyota Camry. The same drivers were employed for the leading vehicle and following vehicle, respectively throughout the test. Dedicated test devices (see

A tablet application, which can also function in any smartphones, was used to provide an audio and visual alarm for “Careful” and “Warning” distance as shown in

Next to the velocity display there was a reading of the rear-front distance z between the leading vehicle and the following vehicle, which was measured by the tablet application through the image from the back camera of the tablet. Such a distance was displayed with black text within a colored box on the screen. Still, the color represents relevant safety status.

Once the tablet application detects that the time to collision is within 4.0 seconds, the full screen of the tablet would become all yellow with a text message “Careful” for one second together with a short sound alarm. If the time to collision estimation is within 3.0 seconds, the tablet screen would become all red with a text “Warning” for one second also with a sound alarm prompted. The readings of velocity would change their corresponding colors after the full screen “Careful” or “Warning” messages disappeared, but the time to collision could still be within the mentioned margins above.

There were three types of information recorded during the on-road test: 1) the velocity of leading vehicle from its OBD II scanner; 2) the GPS data recording both vehicles’ geo-locations at a sampling rate of 10 Hz (10 records per second); and 3) recorded video of tablet images from the camera inside the following vehicle. For the test rounds with no safety messages, the tablet kept silent and placed at a location where the driver could neither see nor hear any safety message. However, the following vehicle’s speed and rear-front distance displayed on tablet screen were still recorded by the camera.

After the field test, the collected raw information was post-processed in the lab. The speed of following vehicle, rear-front distance, and instant “Careful (Yellow)” and “Warning (Red)” messages were retrieved from the videos of tablet interface at a sampling rate of 1 Hz.

All types of recorded data for both the leading and following vehicles were synchronized based on the time from both GPSs. Data quality was controlled to eliminate the outlier in each data set. The velocity of the leading vehicle from OBD II scanner, which was collected at an uneven sampling rate of a little bit more than 1 second, was interpolated into the evenly distributed second-by-second data.

Each prepared data pair included: 1) acceleration rate of the following vehicle; 2) velocity of the following vehicle; 3) rear-front distance between the leading vehicle and the following vehicle; 4) velocity of the leading vehicle; and 5) time recorded in second.

The recorded data were sorted into four major groups: 1) deceleration period when approaching intersection without messages, 2) deceleration period when approaching intersection with safety messages, 3) acceleration period when leaving intersection without messages, and 4) acceleration period when leaving intersection with safety messages.

A total of 226 sets of data pairs were prepared from the field observations. For deceleration scenario, there were 40 data pairs for without and 68 pairs for with in-vehicle messages; while for acceleration scenario, there were 60 and 58 data pairs prepared for without and with messages, respectively.

(

The single factor ANOVA test indicated that, the difference in deceleration rates with and without safety messages were significant (F (1, 68) = 13.3, p = 5.2E−4), and the difference in acceleration rates with and without safety messages were significant as well (F (1, 30) = 21.2, p = 7.1E−5). The statistically significant differences are consistent with the mentioned assumption of the change in car-following behaviors.

The revised GM model and Linear (Helly) model would be calibrated using the collected field data. The calibration and examination process is composed of the following steps.

Step 1. Applying parameters from literature. The collected field data, including both without message and with messages, were first used to calculate the acceleration of leading vehicles using parameters from literatures based on existing car-following models in Equations ((2) and (4)).

Step 2. Self-calibrating parameters using field data. The parameters in the existing car-following models Equations ((2) and (4)) were calibrated using the set of data for both with and without safety messages (i.e. calibrating parameters C, m, l in Equation (2), and parameters C_{1}, C_{2},

Step 3. Calibrating new parameters with message. The parameters k and C in the revised GM model in Equation (2), and the parameters θ and C_{1} in the revised Linear (Helly) model in Equation (4) were calibrated using the data set with safety messages. The other parameters in these two equations were from the calibration using data with no message in step 2.

Step 4. Validating revised models using additional data with messages. Additional set of field data with safety messages were used to validate the revised models with additional parameters from Step 3 and the rest parameters from Step 2.

Step 5. Examining modeling and validation errors from all above steps. The accelerations from all above steps were compared with the field observations. The Normalized Route Mean Square Error (NRMSE) and the Pearson correlation coefficients were used to compare the goodness of models.

The NRMSE in Step 5 is calculated based on Equation (6).

where,

At the first step, the parameters of the Ozaki’s GM car-following model [

In

For the scenarios with warning messages, 28 of 60 data pairs were reserved for validation, while 40 data pairs were used for calibration.

Situation | Parameter | No Messages | With Messages | ||||
---|---|---|---|---|---|---|---|

Ozaki’s Parameters^{a } (a) | Self-Calibration (b) | Ozaki’s Parameters^{ b } (c) | Self-Calibration (d) | Parameters^{c } (e) | Calibrated K^{d} (f) | ||

Deceleration | Sample size | 40 | 40 | ||||

m | 0.90 | −0.54 | 0.90 | 4.32 | −0.54 | −0.54^{c} | |

l | 1.00 | 1.48 | 1.00 | 13.04 | 1.48 | 1.48 ^{c} | |

k | N/A | 0.04 | |||||

NRMSE | 38.2% | 10.6% | 63.1% | 318.3% | 554.3% | 10.4% | |

R | 0.51 | 0.95 | 0.24 | 0.16 | 0.76 | 0.86 | |

p-value | N/A | 1.24E−11 | N/A | 6.50E−03 | N/A | 6.46E−05 | |

Acceleration | Sample Size | 60 | 30 | ||||

m | −0.20 | 0.11 | −0.20 | 1.39 | 0.11 | 0.11 | |

l | 0.20 | 0.49 | 0.20 | 0.94 | 0.49 | 0.49 | |

k | N/A | 0.36 | |||||

NRMSE | 18.5% | 4.2% | 57.8% | 39.5% | 57.8% | 8.9% | |

R | 0.98 | 0.99 | 0.82 | 0.85 | 0.86 | 0.93 | |

p-value | 2.85E−09 | 3.40E−01 | 1.93E−13 |

a. Ozaki, 1993; b. The results that direct use Ozaki’s parameters; c. Calibrated parameters results from the scenarios with no message; d. Calibrated k results using revised Equation (3).

with safety messages. The resulted NRMSE (63.1%) and R (0.24) is not the indicator of a good fit to the observed data. The deceleration rates from this set of parameters are indicated as the magenta line in

Column (d) in

The calibration results in column (f) are for the revised GM model in Equation (3). The added parameter k was calibrated together with C_{1} based on the field data with messages. Other parameters were still from column (b) with m (−0.54) and l (1.48). Magically, the NRMSE is 10.4% and R is 0.86 for deceleration, both are in the acceptable ranges. This tells that, the revised GM model could possibly reflect the car-following mechanism with the presence of safety messages in a deceleration situation. This is also reflected as the red line in

The calculation and calibration for the process of acceleration is pretty similar to the one for deceleration. The calibrated parameters and associated indexes are listed in

In column (a) and (b) and

In columns (c) to (f) and

The calculation and calibration of the revised linear (Helly) model for the process of deceleration and acceleration are listed in

For the deceleration cases with no messages, the NRMSE14.3% and R0.83 for self-calibrated parameters in column (b) are better than the direct use of Helly’s parameters in column (a) (NRMSE 41.0% and R 0.08). For the deceleration cases with safety messages, both the self-calibrated from Helly’s model in column (d) and calibrated from revised Helly’s model in column (f) provides better fits of deceleration rates with NRMSEs being 8.6% and 11.98%, respectively, and R values being 0.90 and 0.83, respectively. They are much better than the other two situations in columns (c) and (e). The greatest NRMSE 341.9% is in column (e), which further indicates the significant impacts of safety messages on deceleration rates as having been discussed for the GM model. These are also reflected in the multiple plots of deceleration rates in

Still, the traditional Linear (Helly) Model in Equation (4) can adopt the situation with safety messages for deceleration (NRMSE = 8.6% in column (d)). In the meantime, the revised Linear Model in Equation (5) can do well also (NRMSE = 11.98% and R = 0.83 in column (f)). The best-fit parameters for Linear (Helly) model are: c_{1} = 0.01, c_{2} = 7.4E−4, α = 1.53, β = −0.06, γ = −0.14. The calibrated parameters for the revised Linear (Helly) Model are: c_{1} = 0.48, c_{2} = 0.04, α = 1.64, β = −0.02, γ = −0.11, and θ = 0.12.

For the acceleration cases, similar to the revised GM model for the acceleration situation with no message (column (a) and (b)), the self-calibrated Linear (Helly) Model provides better fit to the acceleration rates

Situation | Parameter | No Messages | With Messages | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Helly’s Parameters^{a } (a) | Self-Calibration (b) | Helly’s Parameters (c) | Self-Calibration (d) | Parameters^{b} (e) | Calibrated θ and C_{1}^{a } (f) | |||||||

Deceleration | Sample size | 40 | 40 | |||||||||

C_{1} | 0.50 | 1.17 | 0.50 | 0.01 | 1.17 | 0.48 | ||||||

C_{2} | 0.13 | 0.04 | 0.13 | 7.40E−04 | 0.04 | 0.04 | ||||||

α | 20.00 | 1.83 | 20.00 | 1.53 | 1.83 | 1.83 | ||||||

β | 1.00 | −0.02 | 1.00 | −0.06 | −0.02 | −0.02 | ||||||

γ | 0.00 | −0.11 | 0.00 | −0.14 | −0.11 | −0.11 | ||||||

θ | N/A | 0.12 | ||||||||||

NRMSE | 41.0% | 14.3% | 112.6% | 8.6% | 341.9% | 11.98% | ||||||

R | 0.08 | 0.83 | 0.46 | 0.90 | 0.51 | 0.83 | ||||||

p-value | N/A | 1.70E−08 | N/A | 1.64E−12 | N/A | 3.78E−07 | ||||||

Acceleration | Sample Size | 60 | 30 | |||||||||

C_{1} | 0.50 | 0.69 | 0.50 | 0.10 | 0.69 | 0.37 | ||||||

C_{2} | 0.13 | −0.01 | 0.13 | −0.01 | −0.01 | −0.01 | ||||||

α | 20.00 | −0.23 | 20.00 | −0.45 | −0.19 | −0.23 | ||||||

β | 1.00 | −6.20E−03 | 1.00 | −0.01 | −0.01 | −6.20E−03 | ||||||

γ | 0.00 | 0.02 | 0.00 | −0.03 | 0.02 | 0.02 | ||||||

θ | N/A | 0.37 | ||||||||||

NRMSE | 54.4% | 5.1% | 181.0% | 8.6% | 179.7% | 9.05% | ||||||

R | 0.45 | 0.98 | 0.13 | 0.94 | 0.79 | 0.93 | ||||||

p-value | N/A | 1.99E−36 | N/A | 5.18E−11 | N/A | 6.17E−14 | ||||||

a. Using Helly’s parameters directly; b. B. Calibrated parameters results from the scenarios with no message; c. Calibrated θ results using revised Equation (5).

(NRMSE 5.13% and R 0.98) than directly using the parameters from the traditional Linear (Helly) Model (NRMSE 54.4% and R 0.45). This can also be observed in

For the revised GM model for the acceleration situation with messages (column (c) to (f)), the best cases are still for the self-calibrated parameters using the Linear model (column (d) with NRMSE = 8.6%, R = 0.94), and for the use of the revised Linear Model (column (f) with NRMSE = 9.05%%, R = 0.93). Similarly, the higher NRMSE (179.7%) in column (e) indicates the significant impacts of safety messages on acceleration rates.

The best-fit parameters for the Linear (Helly) Model for acceleration process are: c_{1} = 0.10, c_{2} = −0.01, α = −0.45, β = −0.01, and γ = −0.03, while the calibrated parameters for the revised Linear (Helly) Model are: c_{1} = 0.37, c_{2} = −0.01, α = −0.19, β = −0.01, γ = −0.02, and θ = 0.37.

The traditional and revised GM model and Linear (Helly) model were further validated with the parameters in

In

Deceleration Situation | Acceleration Situation | |||||
---|---|---|---|---|---|---|

Sample size | NRMSE | R | Sample size | NRMSE | R | |

Revised GM model with self-calibrated parameters | 20 | 10.4% | 0.93 | 28 | 5.5% | 0.92 |

Revised Linear (Helly) model with self-calibrated parameters | 15.4% | 0.87 | 10.1% | 0.92 |

In

In this paper, two types of car-following models, the GM model and the Linear (Helly) model, were re-calibrated and revised using the field data collected in Houston, Texas, U.S.A. so as to characterize the impacts of safety messages from a tablet application. Both revised GM model and revised Linear (Helly) model

Model | Deceleration | Acceleration | ||
---|---|---|---|---|

No Message | With Messages | No Message | With Messages | |

GM Model | C = 1060 m = −0.54 l = 1.48 | N/A | C = 2.68 m = 0.11 l = 0.49 | N/A |

Revised GM Model | C = 1060 m = −0.54 l = 1.48 k = 1.0 | C = 462.57 m = −0.54 l = 1.48 k = 0.043 | C = 2.68 m = 0.11 l = 0.49 k = 1.0 | C = 1.45 m = 0.11 l = 0.49 k = 0.36 |

Linear (Helly) Model | C_{1} = 1.17 C_{2} = 0.04 α = 1.83 β = −0.024 γ = −0.111 | C_{1} = 0.01 C_{2} = 0.00074 α = 1.53 β = −0.006 γ = −0.14 | C_{1} = 0.686 C_{2} = −0.01 α = −0.232 β = −0.006 γ = 0.022 | C_{1} = 0.095 C_{2} = −0.0127 α = −0.45 β = −0.014 γ = −0.028 |

Revised Linear Model | C_{1} = 1.17 C_{2} = 0.04 α = 1.83 β = −0.024 γ = −0.111 θ = 1.0 | C_{1} = 0.48 C_{2} = 0.04 α = 1.83 β = −0.024 γ = −0.111 θ = 0.123 | C_{1} = 0.686 C_{2} = −0.01 α = −0.232 β = −0.006 γ = 0.022 θ = 1.0 | C_{1} = 0.37 C_{2} = −0.01 α = −0.19 β = −0.01 γ = −0.02 θ = 0.37 |

were proposed by applying additional exponents to the stimuli term “the different of velocity between a leading vehicle and a following vehicle”. A part of the field data was used to calibrate the parameters of the traditional and revised GM model and Linear (Helly) model, while the others were for validation.

Calibration results showed that, when the safety messages from a tablet application were provided, the GM model failed to properly fit in the field car-following data, even a calibration process had been applied for both deceleration and acceleration situations. The calibrated parameters in the cases with no message for the Linear (Helly) Model should not be directly applied to the car-following data with safety message. A calibration to either the Linear (Helly) Model or the revised Linear (Helly) Model is necessary for a better fit.

Both calibration and validation results demonstrated that, the safety messages did affect the calibration of parameters of car-following models for both deceleration and acceleration situations. The entire research process, the revised car-following models, and the calibrated parameters could be good references to the on-going connected vehicle program, the development of drivers’ safety messages, as well as the traffic simulations in both microscopic and macroscopic scales.

The authors acknowledge that this research is supported in part by the Tier 1 University Transportation Center TranLIVE#DTRT12GUTC17/KLK900-SB-003, and the National Science Foundation (NSF) under grants #1137732. The opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies. The authors would like to appreciate Yijun Qiao and Jinghong Ma in shaping the initial idea and part of the data collections.

QingLi,FengxiangQiao,LeiYu, (2016) Calibration of Car-Following Models Considering the Impacts of Warning Messages from Tablet/Smartphone Application. Journal of Transportation Technologies,06,61-75. doi: 10.4236/jtts.2016.62006