A Learning-based Stochastic Driving Model for Autonomous Vehicle Testing
AA Learning-based Stochastic Driving Model for Autonomous Vehicle TestingLin Liu
School of Automotive StudiesTongji University, Shanghai, China, 201804Department of Civil and Environmental EngineeringUniversity of Michigan, Ann Arbor, MI, USA, [email protected]
Shuo Feng, Ph.D., Corresponding Author
Department of Civil and Environmental EngineeringUniversity of Michigan, Ann Arbor, MI, USA, [email protected]
Yiheng Feng, Ph.D.
University of Michigan Transportation Institute, Ann Arbor, MI, USA, [email protected]
Xichan Zhu, Ph.D.
School of Automotive StudiesTongji University, Shanghai, China, [email protected]
Henry X. Liu, Ph.D.
Department of Civil and Environmental EngineeringUniversity of Michigan Transportation InstituteUniversity of Michigan, Ann Arbor, MI, USA, [email protected] a r X i v : . [ ee ss . S Y ] F e b iu, Feng, Feng, Zhu and Liu 2 ABSTRACT
In the simulation-based testing and evaluation of autonomous vehicles (AVs), how backgroundvehicles (BVs) drive directly influences the AV’s driving behavior and further impacts the testingresult. Existing simulation platforms use either pre-determined trajectories or deterministic drivingmodels to model the BVs’ behaviors. However, pre-determined BV trajectories can not reactto the AV’s maneuvers, and deterministic models are different from real human drivers due tothe lack of stochastic components and errors. Both methods lead to unrealistic traffic scenarios.This paper presents a learning-based stochastic driving model that meets the unique needs of AVtesting, i.e. interactive and human-like. The model is built based on the long-short-term-memory(LSTM) architecture. By incorporating the concept of quantile-regression to the loss function ofthe model, the stochastic behaviors are reproduced without any prior assumption of human drivers.The model is trained with the large-scale naturalistic driving data (NDD) from the Safety PilotModel Deployment (SPMD) project and then compared with a stochastic intelligent driving model(IDM). Analysis of individual trajectories shows that the proposed model can reproduce moresimilar trajectories to human drivers than IDM. To validate the ability of the proposed model ingenerating a naturalistic driving environment, traffic simulation experiments are implemented. Theresults show that the traffic flow parameters such as speed, range, and headway distribution matchclosely with the NDD, which is of significant importance for AV testing and evaluation.
Keywords : autonomous vehicle; testing and evaluation; stochastic driving behavior; LSTM; quan-tile regressioniu, Feng, Feng, Zhu and Liu 3
INTRODUCTION
Testing and evaluation of autonomous vehicles (AVs) become an active research topic in the pastfew years ( ),( ),( ),( ). Among three major testing methods (simulation, test track, and on-road)( ), simulation is the most cost-effective, efficient, and safe method, which attracts significantattention especially in the early development stage of AVs ( ), ( ), ( ), ( ). To evaluate the perfor-mance of AV models in a simulation environment, background vehicles (BVs) need to be generatedto interact with the AV models in different testing scenarios ( ). To model driver behaviors forthe purpose of AV testing, the following features should be included:1. Interactive. The BVs should react to the AV’s behavior in real-time.2. Human-like. The BVs should act like a human driver with stochastic components anderrors. For example, different driving styles or mental states of a driver may lead to the variationof driving behavior even in the same traffic environment.In the past few years, two main methods have been proposed to model BV behaviors forAV testing. In the first method, the BVs behaviors are pre-defined before the testing. The methodis firstly used for testing of advanced driver assistant systems (ADAS) like adaptive cruise control(ACC) and autonomous emergency brake (AEB). The speed profile of the leading vehicle is pre-defined to generate the testing matrix ( ), ( ), ( ). However, the pre-defined BV trajectorycannot reflect the real human driving patterns. Another commonly used approach to apply pre-defined trajectories is to utilize the real-world data collected by vehicles equipped with multiplesensors to replicate the testing scenarios ( ). Although the behaviors of human drivers can beprecisely captured, the main problem of using pre-defined BV trajectories is that the testing isnot interactive, because the BVs cannot adjust their maneuvers dynamically based on the AV’sbehaviors.The second method models the BVs’ behaviors with microscopic traffic flow models, whichis an interactive approach. Based on certain driver behavior rules, the motion of each BV at eachsimulation time step can be updated according to the current traffic state. However, most existingdriving behavior models are deterministic such as Newell’s model ( ), Intelligent Driving Model(IDM, ( )), Gipps’ model ( ), etc., which cannot capture the real human driver behaviors. Con-sequently, the AV may pass the test by just ‘remembering’ the experienced scenarios.Recently, several studies have been focused on modeling stochastic driving behaviors byadding noise to existing deterministic models. Based on the Newell’s model, Laval et al. ( )proposed a stochastic desired acceleration model and added white noise to the driver’s desiredacceleration. Treiber et al. ( ) extended the IDM model ( ), optimal-velocity model ( ) andvelocity-difference model ( ) by adding Gaussian white noise to the driver’s desired time gap.However, adding the Gaussian distributed noise to a pre-determined diver model may not reflectreal human stochastic driving behaviors, and it is difficult to fit a predefined parametric distributionunder different traffic flow conditions (e.g., free flow verse congested).In this paper, we propose a learning-based stochastic driving model to meet the uniqueneeds (i.e., interactive and human-like) for the assessment of AVs. The goal of the model is togenerate the action distribution that is consistent with the naturalistic driving data, given currentvehicle states. Then by sampling the action from the distribution at each time step, the model caninteract with the AV in a stochastic and realistic way. To achieve this goal, the quantile-regression(QR,( )) method is incorporated into the learning process. Instead of one single output (e.g.,expected acceleration) as commonly designed in existing methods, our method provides a series ofoutputs, which are designed as the different quantiles of actions. Correspondingly, the pinball lossiu, Feng, Feng, Zhu and Liu 4function ( ) is applied to calculate the loss. By decreasing the loss function, the output actionslearn to be the quantiles, which can fit the naturalistic driving data via the kernel density estimation(KDE,( )). To further capture the temporal dependency of the behavior model, the long-short-term-memory (LSTM, ( ), ( )) recurrent neural network (RNN) architecture is utilized. Theproposed method is referred to as QRLSTM hereafter in this paper. To validate the effectivenessof the proposed method, a large-scale naturalistic driving database is utilized from the Safety PilotModel Deployment (SPMD) project ( ). Simulation results show that the proposed QRLSTMmodel can represent human driving behaviors at both microscopic and macroscopic levels, whichgreatly enhances the previous studies by providing a human-like interactive driving environmentfor AV testing and evaluation.The new model has two significant advantages. First, it does not apply any assumption ofthe distribution of human driving error or requires any prior knowledge of human drivers. Withthe quantile-regression model structure and KDE, the stochasticity of human driving is obtained ina data-driven way. Second, the model has the ability to generate a realistic driving environment,which is of significant value for AV testing and evaluation.The rest of this paper is organized as follows. Section 2 formulates the modeling problemand describes the structure of the QRLSTM model, and Section 3 introduces the model trainingprocess. After that, simulation experiments are presented, following by the results and discussions.Conclusions and further research are provided in the final section. METHODProblem formulation
In the introduction section, two features, i.e. interactive and human-like stochasticity, are proposedfor the background vehicles to generate a realistic traffic environment for the AV testing. Thebehavior modeling problem of the background vehicles is formulated as follows.A microscopic driving model can realize the interaction of the BV with AV as well as otherBVs. The goal of a microscopic driver behavior model f d , is to calculate or predict the action (e.g.,speed or acceleration) of the BV ˆ y t + at the time step t +
1, given the traffic state x t at the time step t , i.e. ˆ y t + = f d ( x t ) , (1)where traffic state x t refers to the dynamic state of BV and all vehicles around it, including AV. Inan AV testing simulation, the action of the AV will change the traffic state and BV will calculatethe next action accordingly. In this way, the microscopic driving model realizes the interactionbetween BVs and the AV.In the car-following situation, the traffic state x typically only considers the velocity of theBV v , the velocity of the AV v l , the range between the BV and the AV r , and the range rate betweenthe BV and the AV rr , i.e. x = [ v , v l , r , rr ] . The action could be either the velocity or accelerationof the BV, denoted as v or a . Therefore a car-following model could be written as v t + = f d ( v t , v lt , r t , rr t ) or a t + = f d ( v t , v lt , r t , rr t ) . (2)For example, the IDM model has the following form: a t + = a t [ − ( v t v ) − ( s + v t T + v t rr t √ ab r t ) ] , (3)where v , s , a , b , T are constants ( ). Therefore, the IDM model could be simplified as a t + = f IDM ( a t , v t , r t , rr t ) . (4)However, in actual driving scenarios, human drivers do not behave deterministically, re-iu, Feng, Feng, Zhu and Liu 5sulting in the need of a stochastic microscopic driving model. With this model, the action of thevehicle might be various even given the same traffic state as input. Several studies have tried tobuild such a model by adding simple noise (e.g., Gaussian noise) to existing deterministic models.For example, the output of the IDM model is added with a white noise term in ( ). The modifiedIDM model could be written as a t + = f IDM ( a t , v t , r t , rr t ) + (cid:112) Q ξ α ( t ) , (5)where Q is the fluctuation strength and ξ α ( t ) is the white noise. The method of adding white noiseto an analytic deterministic model has two disadvantages: the assumption of Gaussian distributionmay not reflect real human stochastic driving behaviors, and, the analytic form will disenable themodel to fit the changing driving behavior.To release the unrealistic assumption of the Gaussian distribution of randomness, a newmodel structure is proposed in this paper. Instead of calculating the action, the model directlyoutputs the distribution of action, and the final action is sampled accordingly. The model is definedas F ( ˆ y t + ) = f d ( x t ) , (6)where F () denotes the distribution function. To achieve such a model structure, we modify theLSTM with quantile-regression loss and kernel density estimation (KDE). The structure of themodel is described in detail in the following sections. Model framework
Figure 1 shows the overall framework of the proposed QRLSTM model, which contains threecomponents: the QRLSTM, KDE, and a sampler. Given the traffic state x t , the QRLSTM modeloutputs a set of predicted actions S t according to the quantile definition P . Then, these actions willbe used in KDE to estimate the continuous action distribution F . Finally, action ˆ y t + is sampledfrom the F distribution. The integration of QRLSTM, KDE, and the sampling process forms astochastic microscopic driving model. The action ˆ y t + will be used to update the traffic state x t + at time t +
1, and the loop runs repeatedly.
FIGURE 1 The overall model framework. iu, Feng, Feng, Zhu and Liu 6
LSTM structure
A significant trend in driver behavior modeling is utilizing machine learning techniques to takeadvantage of real-world driving data. Neural networks were introduced to model car-following in( ) and improved by adding human factors by Khodayari et al. ( ). Different learning-basedmodel structures further improve the modeling performance, such as the deep neural networksmodel built by Wang et al. ( ) and the reinforcement learning model built by Zhu et al. ( ).In ( ) and ( ), the RNN model, which can take the historical state into consideration, shows abetter performance in terms of the mean square error in speed prediction.In this paper, we apply the LSTM neuron network, a widely used neuron network structure,as the base model to calculate the BV action, though our framework is applicable for genericneuron network structures. A simple illustration of LSTM is shown in Figure 2. The detailedneuron structure can be found in ( ). To calculate ˆ y t + , LSTM considers both the current input x t and the hidden state h t , which is calculated based on x t − and h t − . With this structure, the LSTMcan learn both the corresponding output and hidden sequence patterns with sequential training data. FIGURE 2 Illustration of the LSTM schema.
QRLSTM structure
Due to the randomness and error of human drivers, the action is various even in the same trafficstate. Although the driver behavior is recorded as state-action pairs in real-world data, there is anaction distribution under a certain traffic state. However, the LSTM is a deterministic model thatoutputs one action given a traffic state. As shown in Figure 3, given traffic state x t , LSTM outputsthe action ˆ y t + . The error is then calculated in terms of mean squared error (MSE) for adjustingmodel weights. Training the model with the MSE cost function will indeed lead the model toestimate the median of actions in a traffic state, which will loss the stochastic information of thereal-world data.To capture the stochasticity of driver behaviors, we propose to apply the concept of quantile-regression ( ). As shown in Figure 3, the main differences between QRLSTM and LSTM are theoutput forms of the models and the loss functions used for model training. Specifically, a QRL-STM model outputs a set of N actions ˆ y t + , , ˆ y t + , ... ˆ y t + , N . The error is calculated as the pinballerror and used for adjusting the model weights. By applying the pinball loss as loss function ofLSTM, the target of training is changed to estimate action quantiles in a traffic state, instead of theaction median.iu, Feng, Feng, Zhu and Liu 7 FIGURE 3 Differences between LSTM and QRLSTM.
The pinball function is designed to calculate the error of a quantile to the real value. Pinballfunction is defined as L p , y t = (cid:40) p ( y t − ˆ y t , p ) i f ( y t − ˆ y t , p ) ≥ ( p − )( y t − ˆ y t , p ) i f ( y t − ˆ y t , p ) < , (7)where 0 < p < y t is the observed output from data, ˆ y ( t , p ) is the pre-diction of p − quantile, and L ( p , y t ) is the loss of the predicted p − quantile for y t . The loss of theQRLSTM model to estimate the p − quantile is then defined as L p = N N + ∑ t = L p , y t , (8)where N is the total number of y t .Then, QRLSTM is designed to calculate a action matrix corresponding to a set of quantileprobabilities, p ∈ P . P = { | p |− , | P |− , ..., − | P |− } , where, | P | is the length of P , i.e. the numberof quantile probabilities p . The Loss function of QRLSTM model is set as L = N | P | N + ∑ t = ∑ ∀ p ∈ P L p , y t . (9)By training with real-world data, the action set will converge to the action quantiles. Kernel density estimation
Given a traffic state x t , QRLSTM predicts a | P | × S t = { y ( t , p ) , y ( t , p ) , . . . , y ( t , p | P | ) } .To obtain a continuous prediction distribution, KDE ( ) is applied. As a classic non-parametricestimation method, KDE does not require a prior assumption of distribution form, which is suitablefor modeling the driver behaviors. The KDE estimation of S t is calculated by F ( ˆ y t ) = B | P | ∑ ∀ p ∈ P K ( ˆ y t , p − y t B ) , (10)where B > K is a kernel function. MODEL TRAINING
In this paper, we focus on one of the most common driver behaviors, car-following behavior, todemonstrate the proposed model. To capture the human driving patterns, the QRLSTM model istrained with real driving data from a naturalistic driving study. As in the real traffic environment,different types of drivers have diverse driving behavior styles, a model is trained for individualiu, Feng, Feng, Zhu and Liu 8driver respectively to capture each driver’s behaviors.
Data description
We adopt the naturalistic driving data (NDD) from the Safety Pilot Model Deployment (SPMD)project ( ) to train and test the proposed model. With 2 ,
842 participating vehicles, the SPMDproject collected NDD of over 34 . = highway2 Speed is larger than 20 m / s (72 km / h )3 A leading vehicle is identified by the DASFinally, a total number of 24 ,
816 trajectories are extracted with a total travel time of1 , ,
955 seconds (around 390 hours). Each trajectory lasts for 1 second to 1 ,
768 seconds ran-domly.
Model settings
In the car-following scenario, the traffic state x includes the velocity of the BV v , the velocity of theleading vehicle v l , range between the BV and the AV r , and range rate rr . The action is the accel-eration of the BV a . The memory time of the LSTM is set as 10, i.e., the input of the model is thetraffic states from the previous 1 seconds, namely, x t = { v t − , v t − ... v t ; v lt − , v lt − ... v lt ; r t − , r t − ... r t ; rr t − , rr t − ... rr t } . The QRLSTM model has one hidden layer with 32 LSTM neurons. The quan-tile probability P is set as { . , . ... . } . The bandwidth B in KDE is 0 .
75, and K is theGaussian kernel. This model setting is implemented on all QRLSTM models in this paper. Model training
Although there are data from 86 drivers available from SPMD, not all of them have enough datato train a QRLSTM model. The total driving time of each driver is shown in Figure 4. There is alarge variance in travel time among different drivers. To confirm the required data for the trainingprocess, the QRLSTM model is trained with 0 . . . . . .
91% models outperformthe mixed model. The underperformance of the mixed driver against individual models owes tovarious driving styles and habits of different drivers.iu, Feng, Feng, Zhu and Liu 9
FIGURE 4 Distribution of total travel time of 86 drivers.FIGURE 5 Validation error v.s. size of training data. iu, Feng, Feng, Zhu and Liu 10
FIGURE 6 Validation loss.SIMULATION EXPERIMENTSExperiments Setup
To verify the performance of the proposed method, simulation experiments are designed and con-ducted with Python 3.7, in a work-station equipped with Intel i7-10700K CPU and 16G RAM.Since only car-following behavior is considered, a 3-mile single-lane highway is built. The trafficdemand varies from 500-2000 vehicles/hour in different experiments. Each experiment lasts for 1hour and the simulation resolution is 10 Hz. The workflow of the simulation is shown in Figure 7.
FIGURE 7 Workflow of mixed driver simulation experiment.
When the simulation starts, the traffic initializer generates the first two vehicles in thenetwork, by randomly sampling the speed of the leading vehicle v l , range r , and the speed ofthe following vehicle v from the SPMD, and then calculating the corresponding range rate rr . Itforms the traffic state at time step 0, i.e., x = v l , v , r , rr . Since the LSTM model structure needsthe traffic state of the last 10-time steps as the input, the IDM is used to generate the following 9time-steps traffic state, to obtain the traffic state of the first 10 time-steps.After the initialization, given traffic state at time t for each vehicle model, s t = x t − , x t − ... x t ,the vehicle model predicts its action a t . Then with action a t , the state updater calculates the traf-fic state for time t + Result and Discussion
As discussed in the introduction, to act as background vehicles in AV testing scenarios, the drivermodel should interact with the AV, which is already satisfied with the QRLSTM structure andthe simulation workflow. Another requirement is that the model should consist with stochastichuman driving behaviors to construct a realistic driving environment. Therefore, performance onboth trajectory level and traffic level need to be examined to see whether the vehicle trajectoriesgenerated from the proposed model matches the human drivers in the NDD. For comparison, themodel proposed by ( ) is replicated as the baseline, which is an extension of the commonly usedIDM by adding Gaussian white noise to the driver’s acceleration. The model is defined in Eq(3) and Eq (4) and the model parameters are also calibrated with the SPMD data set for a faircomparison, shown in Table 1. TABLE 1 Parameters of the modified IDM parameter valuev0 desired velocities 34.99 m / s s0 minimum gap 1.70 m a acceleration 0.15 m / s b comfortable deceleration 0.66 m / s T desired time headway 0.73 s Q fluctuation strength 0 . m / s Trajectory reproducing accuracy
Figure 8 shows the comparison at the individual trajectory level among the NDD, QRLSTM, andmodified IDM models. The figure represents a continuous action profile of 36s, given the initialcondition (i.e., traffic states for the first second). The red, yellow, and blue lines stand for thereal vehicle action profile obtained from SPMD, generated by the proposed QRLSTM model, andthe modified IDM model, respectively. The upper and lower figures show the action in terms ofspeed and acceleration respectively. The upper figure shows that the QRLSTM model outperformsthe modified IDM model in terms of the speed error, especially for a longer prediction horizon.The lower figure proves that the QRLSTM model is able to capture the frequent variations ofacceleration while the modified IDM shows a less sensitive pattern. These frequent changes areindeed from human driving behaviors, which is difficult to be captured by analytical car-followingmodels like IDM.
Traffic parameters comparison
To compare the driving environment generated by the proposed model and the real driving en-vironment, the distributions of speed, range, and time headway are selected as measurements todescribe the traffic flow. A total of 2,386 km of vehicle trajectories are generated by simulation,and 100,000 data points are randomly selected from SPMD to form the distributions. As shown iniu, Feng, Feng, Zhu and Liu 12
FIGURE 8 Upper: the continuous speed profile. Lower: the continuous acceleration profile. iu, Feng, Feng, Zhu and Liu 13Fig 9, the blue distributions represent the proposed QRLSTM model and the yellow distributionsrepresent the real driving data from SPMD.Speed and range are two major parameters that can describe the car-following behavior.The upper two sub-figures in Figure 9 present the speed and range distributions respectively. Bothdistributions of QRLSTM are similar to the SPMD, indicating the similarity between these twotraffic environments. Moreover, the Time headway (THW) is also compared, which is critical forthe safety test of autonomous vehicles. As shown in the lower figure in Figure 9, the THW distri-bution of the QRLSTM model could also capture the trend of the real-world driving environment.
FIGURE 9 Traffic parameters comparisons between QRLSTM and NDD.
The modified IDM model is also implemented on the same simulation platform as a base-line. To compare the performance, cross-entropy is introduced as a numerical measurement of thesimilarity between two distributions. The cross-entropy of the distribution q relative to a distribu-tion p within the same sample space X is defined as follows: H ( p , q ) = − ∑ x ∈ X p ( x ) logq ( x ) . (11)The smaller the cross-entropy is, the more similar the two distributions are. As shown inTable 2, the cross-entropies of all three distributions of the QRLSTM model and NDD are signif-icantly smaller than that of the modified IDM and NDD. It means that the distributions of speed,iu, Feng, Feng, Zhu and Liu 14range, and THW of the QRLSTM model are more similar to real-world distributions. Therefore,the proposed QRLSTM model outperforms IDM in terms of generating a more realistic trafficenvironment. TABLE 2 Cross entropy of QRLSTM and IDM with NDD model speed range THWQRLSTM& NDD 0.09500 0.44776 0.44828IDM& NDD 0.11708 0.92446 0.76247
CONCLUSION AND FURTHER RESEARCH
This paper proposed a learning-based stochastic driving model structure for generating a realisticdriving environment for AV testing and evaluation purpose. Starting from the well-studied LSTMmodel structure, the model introduces stochasticity from the quantile-regression-based loss func-tion without any assumptions on the distribution of human driver behavior. The model is trainedwith real driving data from SPMD and compared with a modified IDM model, showing its su-periority over traditional car-following models such as IDM. Microscopic comparison betweenindividual trajectories shows the proposed model is able to capture frequent variations of humandriving. Moreover, the comparison at the macroscopic level shows the speed, range, and THWdistributions of the proposed model match the NDD distributions well. The results indicate thatthe traffic environment generated by the proposed model can reflect human driving behaviors.This study has several limitations. First, this model is only applied to the car-followingscenario in this paper and needs to be extended to scenarios that include lateral vehicle maneuverssuch as lane changing and cut-in. Second, as the crash and near-crash scenarios are of great valuefor AV testing, whether the traffic environment generated with the model can conduct these criticalscenarios is another important feature in the naturalistic driving environment modeling. Third, howto integrate physical knowledge into the learning methods deserves more investigation. Recently,the concept of physics regularized machine learning has been proposed for macroscopic trafficflow modeling ( ) ( ), which is also a promising direction for microscopic behavior modeling. ACKNOWLEDGEMENT
The authors would like to thank the US Department of Transportation (USDOT) Region 5 Univer-sity Transportation Center: Center for Connected and Automated Transportation (CCAT) of theUniversity of Michigan for funding the research. The views presented in this paper are those of theauthors alone.
AUTHOR CONTRIBUTIONS
The authors confirm contribution to the paper as follows: study conception and design: Lin Liu,Henry X. Liu, Yiheng Feng, Shuo Feng, Xichan Zhu; data collection: Yiheng Feng, Shuo Feng;analysis and interpretation of results: Lin Liu; draft manuscript preparation: Lin Liu, Shuo Feng,Yiheng Feng, Henry X. Liu. All authors reviewed the results and approved the final version of themanuscript.iu, Feng, Feng, Zhu and Liu 15
REFERENCES
1. Kalra, N. and S. M. Paddock, Driving to safety: How many miles of driving would it taketo demonstrate autonomous vehicle reliability?
Transportation Research Part A: Policyand Practice , Vol. 94, 2016, pp. 182–193.2. Zhao, D., X. Huang, H. Peng, H. Lam, and D. J. LeBlanc, Accelerated evaluation ofautomated vehicles in car-following maneuvers.
IEEE Transactions on Intelligent Trans-portation Systems , Vol. 19, No. 3, 2017, pp. 733–744.3. Li, L., X. Wang, K. Wang, Y. Lin, J. Xin, L. Chen, L. Xu, B. Tian, Y. Ai, J. Wang, et al.,Parallel testing of vehicle intelligence via virtual-real interaction, 2019.4. Li, L., N. Zheng, and F.-Y. Wang, A Theoretical Foundation of Intelligence Testing andIts Application for Intelligent Vehicles.
IEEE Transactions on Intelligent TransportationSystems , 2020.5. Thorn, E., S. C. Kimmel, M. Chaka, B. A. Hamilton, et al.,
A framework for automateddriving system testable cases and scenarios . United States, Department of Transportation,National Highway Traffic Safety Administration, 2018.6. Feng, S., Y. Feng, C. Yu, Y. Zhang, and H. X. Liu, Testing scenario library generation forconnected and automated vehicles, part I: Methodology.
IEEE Transactions on IntelligentTransportation Systems , 2020.7. Feng, S., Y. Feng, H. Sun, S. Bao, Y. Zhang, and H. X. Liu, Testing scenario librarygeneration for connected and automated vehicles, part II: Case studies.
IEEE Transactionson Intelligent Transportation Systems , 2020.8. Feng, S., Y. Feng, H. Sun, Y. Zhang, and H. X. Liu, Testing scenario library generationfor connected and automated vehicles: An adaptive framework.
IEEE Transactions onIntelligent Transportation Systems , 2020.9. Feng, S., Y. Feng, X. Yan, S. Shen, S. Xu, and H. X. Liu, Safety assessment of highly auto-mated driving systems in test tracks: A new framework.
Accident Analysis & Prevention ,Vol. 144, 2020, p. 105664.10. Feng, S., X. Yan, H. Sun, Y. Feng, and H. X. Liu, Intelligent driving intelligence test forautonomous vehicles with naturalistic and adversarial environment.
Nature Communica-tions , Vol. 12, 2021, p. 748.11. Moon, S., I. Moon, and K. Yi, Design, tuning, and evaluation of a full-range adaptivecruise control system with collision avoidance.
Control Engineering Practice , Vol. 17,No. 4, 2009, pp. 442–455.12. Euro, N.,
European New Car Assessment Programme (Euro NCAP)—Test Protocol—AEBVRU systems . Technical Report, 2017.13. Arcidiacono, A.,
ADAS virtual validation: ACC and AEB case study with IPG CarMaker .Ph.D. thesis, Politecnico di Torino, 2018.14. Li, W., C. Pan, R. Zhang, J. Ren, Y. Ma, J. Fang, F. Yan, Q. Geng, X. Huang, H. Gong,et al., AADS: Augmented autonomous driving simulation using data-driven algorithms. arXiv preprint arXiv:1901.07849 , 2019.15. Newell, G. F., A simplified car-following theory: a lower order model.
TransportationResearch Part B: Methodological , Vol. 36, No. 3, 2002, pp. 195–205.16. Treiber, M., A. Hennecke, and D. Helbing, Congested traffic states in empirical observa-tions and microscopic simulations.
Physical review E , Vol. 62, No. 2, 2000, p. 1805.iu, Feng, Feng, Zhu and Liu 1617. Gipps, P. G., A behavioural car-following model for computer simulation.
TransportationResearch Part B: Methodological , Vol. 15, No. 2, 1981, pp. 105–111.18. Laval, J. A., C. S. Toth, and Y. Zhou, A parsimonious model for the formation of oscilla-tions in car-following models.
Transportation Research Part B: Methodological , Vol. 70,2014, pp. 228–238.19. Treiber, M., A. Kesting, and D. Helbing, Understanding widely scattered traffic flows, thecapacity drop, and platoons as effects of variance-driven time gaps.
Physical review E ,Vol. 74, No. 1, 2006, p. 016123.20. Bando, M., K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Dynamical model oftraffic congestion and numerical simulation.
Physical review E , Vol. 51, No. 2, 1995, p.1035.21. Jiang, R., Q. Wu, and Z. Zhu, Full velocity difference model for a car-following theory.
Physical Review E , Vol. 64, No. 1, 2001, p. 017101.22. Koenker, R. and K. F. Hallock, Quantile regression.
Journal of economic perspectives ,Vol. 15, No. 4, 2001, pp. 143–156.23. Terrell, G. R., D. W. Scott, et al., Variable kernel density estimation.
The Annals of Statis-tics , Vol. 20, No. 3, 1992, pp. 1236–1265.24. Zhou, M., X. Qu, and X. Li, A recurrent neural network based microscopic car followingmodel to predict traffic oscillation.
Transportation research part C: emerging technologies ,Vol. 84, 2017, pp. 245–264.25. Wang, X., R. Jiang, L. Li, Y.-L. Lin, and F.-Y. Wang, Long memory is important: A teststudy on deep-learning based car-following model.
Physica A: Statistical Mechanics andits Applications , Vol. 514, 2019, pp. 786–795.26. Bezzina, D. and J. Sayer, Safety pilot model deployment: Test conductor team report.
Report No. DOT HS , Vol. 812, 2014, p. 171.27. Hongfei, J., J. Zhicai, and N. Anning, Develop a car-following model using data collectedby" five-wheel system". In
Proceedings of the 2003 IEEE International Conference onIntelligent Transportation Systems , IEEE, 2003, Vol. 1, pp. 346–351.28. Khodayari, A., A. Ghaffari, R. Kazemi, and R. Braunstingl, A modified car-followingmodel based on a neural network model of the human driver effects.
IEEE Transactionson Systems, Man, and Cybernetics-Part A: Systems and Humans , Vol. 42, No. 6, 2012, pp.1440–1449.29. Zhu, M., X. Wang, and Y. Wang, Human-like autonomous car-following model with deepreinforcement learning.
Transportation research part C: emerging technologies , Vol. 97,2018, pp. 348–368.30. Hochreiter, S. and J. Schmidhuber, Long short-term memory.
Neural computation , Vol. 9,No. 8, 1997, pp. 1735–1780.31. Yuan, Y., X. T. Yang, Z. Zhang, and S. Zhe, Macroscopic traffic flow modeling withphysics regularized gaussian process: A new insight into machine learning applications. arXiv preprint arXiv:2002.02374 , 2020.32. Shi, R., Z. Mo, K. Huang, X. Di, and Q. Du, Physics-Informed Deep Learning for TrafficState Estimation. arXiv preprint arXiv:2101.06580arXiv preprint arXiv:2101.06580