Modelling and Analysis of Mutually Adaptive Vehicle Maneuvers During the Inserting Process of Lane Changing Vehicle in Urban Driving Context

This study developed a pair of lane changing (LC) model and car following (CF) model. They are used to investigate mutually adaptive maneuver of LC vehicle (LCV) and its immediate CF vehicle during the LCV’s inserting process. The two models are developed based on a multianticipative CF model to simulate both of the LC and LC-adaptive CF maneuvers. A group of pairwise parameters are applied in the models to reflect the stimuli perceived by the two vehicle drivers simultaneously. Based on the united model framework, the features of the two vehicle maneuvers which are mutually and intensively influenced can be compared and analyzed. Vehicle trajectories collected on the urban arterial are applied to calibrating and validating the two models. Results show that the developed models can fit the trajectories in a higher accuracy than the previous models. The estimates of the model parameters revealed that dynamics of lateral moving vehicle influence the LC and CV maneuvers in different ways. It is found that the lateral influence has the heaviest influence on the vehicle maneuvers than other stimuli. The vehicles also adjust their maneuvers along with the change of traffic signal or LC target.


I. INTRODUCTION
A number of inserting lane changing vehicles (LCVs) disturb highway traffic, reduce roadway capacity, and threaten driving safety [1], [2]. The heaviest impact of a LCV on its adjacent car following vehicles (CFVs) appears when it changes from a lane to adjacent lane. Unfortunately, the inserting maneuver of the LCV and the reactions of the CFVs during the inserting process have not been simulated well by most of microscopic traffic flow simulation software. Currently the simulated LCV often jumps from the original lane to the The associate editor coordinating the review of this manuscript and approving it for publication was Fabrizio Messina . target lane during its lane insertion. This could result in its immediate CFVs on the two lanes accelerates or decelerates abruptly [3], as it leaves little time for the CF model to adopt the disturbance caused by the LCV [4]. Such deficiency directly reduces the accuracy of traffic flow simulation, which is usually applied to design optimal traffic control algorithm, especially in the context involving the connected and automated vehicles [5]. The reason of this kind of simulation error is that the adaptability of the LCV and CFV driver in the inserting process are neglected in the LC and CF models [6], [7]. Hence, it is urgent to build new models to present mutually adaptive maneuvers of the LCV and CFV during the inserting process.
In the observation of real traffic, it is easy to find that the LCV's lane insertion is a continuous and time consuming maneuver, but this feature is often deliberately omitted by the LC model of many traffic simulation software. This is owing to that the main concern of most existing LC models is to describe the LCV's inserting gap seeking process, so the insertion execution is assumed to be fast to reduce simulation complexity [8]. In contrast, the CF maneuver is often modelled as the process that the CFV successively adjusts position and speed [9]. However, the assumption of the LCV's fast inserting maneuver is not always reasonable, as previous study had found that duration of the LC movement largely depends on traffic flow state [10]. In the congested traffic, the LCV's inserting process could last over 5 seconds and its inserting trajectory could be twisty [11]. Obviously, such maneuver is hard to be captured by the LC models which assume the maneuver executing fast. In fact, the studies of LCV's inserting maneuver have just emerged in recent years [12]- [15], but the obtained knowledge cannot directly adopted by the simulation model, as the analyses only revealed the statistical characteristics of the LCV's inserting maneuver. For the same reason, it is not surprising to find that the CFV driver's adaptive response to the LCV's maneuver during its inserting process is hardly reproduced by existing CF models. Instead, most of the related studies analyzed and modeled the adaptive CF maneuver before or after the LCV's insertion [16], [17].
To fill the research gaps mentioned above, this study develop a pair of LC and LC-adaptive CF models based on a multianticipative CF model to simulate the maneuvers of the LCV and its immediate CFV on target lane during the LCV's inserting process. The LC and CF models maintain favorable properties of original model meanwhile integrate in additional variables to reflect the complex circumstance of urban arterial on the vehicles' maneuvers. The vehicle trajectories extracted from urban traffic video were applied in the model calibration and validation. The model performances were evaluated in terms of predicting accuracy of the vehicle kinematics calculated from the trajectory data.
The contributions of this study are as follows: 1) A method is proposed to model mutually adaptive maneuvers of the LCV and its immediate CFV on the target lane during the LCV's inserting process. The developed LC and CF models have the higher accuracy in prediction of the vehicle acceleration than the models proposed in other study. Meanwhile, since the LC and LC-adaptive CF models have the developed based on the same model and have the same model structure, the complexity of traffic simulation software could be reduced, and the speed and accuracy of LC traffic simulation could be increased.
2) It reveals how the LCV and CFV adjust their kinematics respectively as responses to the same stimulus during the LCV's inserting process. The adjustments are found to deeply depend on the drivers' observing views and incentives during this process. The rest of this paper is organized as follows. Section II analyses the maneuver characteristics of three major LC participants, and describes the related modelling methods can be referred by this study. Section III introduces the modelling work of the LC and LC-adaptive CF maneuver. Section IV reports the data collection work. Section V proposes the model calibration and validation methods. Section VI presents and discusses the study results. Section VII concludes main findings of this study and points out some directions for future improvements.

II. LC PARTICIPANTS AND THEIR MANEUVERS
A. LC PARTICIPANTS Fig. 1 illustrates five vehicles who possible involve in a LC scenario. The x coordinate measures their movements in the horizontal direction, while the y coordinate measures their movements in the vertical direction. When the LCV seeks an acceptable inserting gap, it could be influenced by the movements of the rest four vehicles. If the gap is found, the LCV's inserting maneuver is mostly affected by the following vehicle (FV) and the leading vehicle (LV) on the target lane, so this study focused on the movements of the two vehicles together with the LCV. The LCV's inserting process ends at the moment (t =n) when the LCV finishes insertion and parallels with the target lane. The start of the process (t = 1) is defined according to specific LC context. If the LCV overtakes the FV from behind (namely scenario1), the start is defined as the moment when the LCV's front bumper aligns with the FV's. If the LCV retreats from the leading position and lets the LV overtake it (namely scenario 2), the start is defined as the moment when the LCV's back bumper aligns with the LV's. If the LCV continuously runs between the FV and the LV on the original lane (namely scenario3), the start is the moment when the LCV's turning light is illuminated or the LCV head begins tilting.
According to the definition of the LCV's inserting process, the LV always travels ahead of the LCV and the FV in this period, so the LV driver hardly observes the two following vehicles while the following drivers can observe its maneuver. It assumed that LV's maneuver affects the followers' but the LV is not affected by them. In the two followers, the LCV is the active executor of the inserting maneuver, while the FV is the negative adaptor of the insertion. But both of them can determine the success of the LCV's insertion [18]. This is the reality foundation that the two vehicles are treated equally in this study and their maneuvers are modelled in a united framework. From Fig. 1, it is known that both of the LCV and the FV involve in two pairs of driving relationships with the other vehicles. They are illustrated in Fig. 2, including: 1) the FV and the LCV compose a LC couple; 2) the FV longitudinally follows the LV, which moves in x direction of the figure; and 3) the LCV laterally follows the LV, which moves in both of x and y directions. The two drivers perceive the influences from longitudinally and lateral moving vehicles, so they have to adjust their vehicles' kinematics according to their anticipations of the movements of the other vehicles. Such driving abilities are modelled based on a multianticipative CF model, which will be discussed in next section.

B. MULTIANTICIPATIVE CF BEHAVIOUR
This section introduces previous efforts of modelling the longitudinal multianticipative driving ability in the CF context. A multianticipative CF model, which is selected as the prototype model of this study will also be introduced here.
The longitudinal multianticipative CF model is used to simulate the CF maneuver which is impelled by the CF driver's anticipation of the moving trends of multiple leading vehicles in the same lane. The model usually stems from the basic CF model that only involves two vehicles. Hoogendoorn et al. extended a linear basic CF model to the multianticipative CF model [19]. They found that it performs better than the basic model in stable CF scenario, but it did not perform well when simulating fluctuated lane traffic. This could be owing to its linear response feature deriving from the basic CF model. To increase nonlinear capability of multianticipative CF model, Lenz et al. [20] extended the optimal velocity CF model to its multianticipative version: where a (t) is the CFV's horizontal acceleration in its lane at the moment t; v (t) is its horizontal velocity; x j (t) is its horizontal spacing to the jth leader; J is the number of the CFV's leader considered in the model; V [.] is the CFV's optimal velocity; b j (j = 1,. . . , J ) is the weight of the jth leader's influence on the CFV driver, and all weights sum up to 1. Hoogendoorn and Ossen [21] further introduced the driver's reaction time into the model: where τ is the CFV driver's reaction time. This model succeeds the nonlinear feature of its basic model. But the optimal velocity CF model could output abrupt acceleration if the following vehicle is much slower than its leader, so the multianticipative CF model derived from it is not suitable to simulate the condition that the velocity of the LCV or the LV is much faster than the FV. To improve the defect of optimal velocity CF model, Jiang et al. [22] proposed an improved optimal velocity CF model, namely the full velocity difference CF model: where v (t) is the horizontal velocity difference of the CFV to its immediate leader; λ is the CFV driver's sensitivity of v (t). The model was extended to its multianticipative version in [23]: where ψ j is the parameter measuring the CFV driver's sensitivity of x j (t), and J j ψ j = 1; v j (t) is the horizontal velocity difference of the CFV to its jth leader; λ j is the parameter measuring the CFV driver's sensitivity of v j (t), and J j λ j = 1. The model in (4) has fully considered the kinematical relationships among the vehicle platoon, and its string stability have been tested in theory. So the model is further transformed to meet the modelling needs of this study which will be detailed in Section 3. Note that the CF models mentioned in this section cannot describe all situations that may arise in single-lane traffic. Instead, they require 1) all vehicle interactions are of finite reach; 2) the following vehicles are not ''dragged along''; 3) an equilibrium speed exists [24]. Hence, the applications of the models are restricted to non-free traffic flow.

C. LATERAL CF BEHAVIOUR
LCV interacts with the FV mutually during its inserting process, meanwhile it is influenced by the leading vehicle LV. Hence, similar to the longitudinal multianticipative CF driver, it is reasonable to assume that the LCV driver adjusts maneuvers based on the prediction of lateral moving trends of the FV and LV. Similar to the longitudinal anticipation, the lateral anticipation can also be modelling based on some fundamental CF models.
Gunay [25] developed a model based on Gipps CF model to analyze the CFV's reactions to the lateral leading vehicle. The influences of lane width and malposition of the lateral leader on the CFV driver were considered in the model. Jin et al. [26] extended the optimal velocity model to a lateral CF model with the variable that measures the lateral spacing of two vehicles. Choudhury and Islam [27] studied the maneuvers of a CFV who follows multiple lateral leaders in the condition of weak lane discipline. Besides lateral spacing, the angle change between a vehicle and its lateral leader can also be considered in the CF model. Wang et al. [12] applied turning angle and horizontal velocity of the LCV to compute its velocity in the vertical direction of the lane traffic, which was a major component of the time-variant lateral influential parameter, µ (t). Then µ (t) was integrated to the optimal velocity CF model to simulate the LCV's inserting maneuver as follows: where µ (t) reflects the influence of the horizontal movement of the vehicle (i−1) in Fig. 1 on the driver of vehicles i, while on the driver. This model was extended to the one considering the influences of more vehicles on the driver [14]: where ϕ (t) measures the degree of the influence of the horizontal movement of the vehicle (i + 2) on the driver of the vehicles i. In [14], µ (t) and ϕ (t) used dynamic variance of the vertical spacing between the vehicle i and other vehicles to reflect their lateral influence on the inserting execution of the LCV. Since vehicle spacing variance is easier to observe than the angle variance for the LCV driver, so this modelling way is much closer to the driver's habit in real traffic.
To sum up, the previous studies reviewed in this section reveal a fact that a driver's multianticipative ability work in both of horizontal and vertical directions of the lane traffic. The method used in these studies will be adopted in the modelling work of the LC and LC-adaptive CF models in this study that will be introduced in next section.

III. MODELS A. VARIABLE DEFINITIONS
The variables and parameters used in the formulations of the LC and LC-adaptive CF models of this study are listed in Table 1. Some of them, which are stated in (7), will be explained from the views of the LCV and the FV drivers based on the coordinate system shown in Fig. 3a as follows.

1) THE DESIRED VEHICLE SPACING
It assumes that the vehicle relative positions illustrated in Fig. 3a is the desired vehicle spacing of the LCV and CFV drivers, as the LCV is unlike to execute the insertion unless it moves to this position. In this condition, the LCV's intention is relative clear and the uncertainty of the vehicle maneuver becomes low, which is desired by the drivers. Such desire is reflected by positive value of x LCV FV (t) and x LV LCV (t) in (7).

2) THE DESIRED SPEED DIFFERENCES
It assumes that the desired speed differences of the LCV and CFV drivers appear when the LCV is faster than the FV meanwhile the LV is faster than the LCV. In this case, their collision risk with other vehicle is relative low. Such desire is reflected by positive value of v LCV FV (t) and v LV LCV (t) in (7).

3) THE INFLUENCE OF RED SIGNAL
The LCV driver knows that all vehicles have to wait behind the stop line in red phase, so position of the stop line of downstream signalized intersection should be considered in the driver's estimate of the moving trends of the FV or LV in red phase. The influence of red signal on the LCV driver's decision is reflected by the definitions of x stop FV (t) VOLUME 8, 2020   (7). Similarly, such influence on the FV driver's decision is reflected by the definitions of x stop LV (t).

B. LC MODEL
The LC model developed in this section is to simulate the LCV's inserting maneuvers with the methods applied in [14] to model the influences of surrounding vehicles on the LCV driver's decision. The factors of traffic signal and LC target are also considered in the LC model.

1) LATERAL INFLUENTIAL PARAMETER
The LCV's lateral inserting maneuver can be divided into the horizontal part in x direction and the vertical part in y direction in Fig. 3a. Since the LCV aims to enter the target lane, it assumes that its horizontal movement is driven by the driver's inserting desire. Based on this assumption, the inserting maneuver can be modelled by the longitudinal CF model partially controlled by a parameter which reflects the LCV's vertical moving state. Here defines a lateral influential parameter, θ LCV (t) ∈ [0, 1]. θ LCV (t) will be used to control the dynamic influences of FV and LV on the LCV's driver during the LCV's inserting process, similar to the way used in [14]. Three forms of θ LCV (t) are defined in (8) to find the one which can reflect the influence of lateral vehicle on the LCV better. The proper form of θ LCV (t) will be selected according to calibration results of the LC model.
where k LCV is a constant parameter; p LCV (t) refers to the vertical distance of the LCV at the moment t to the finish position of its insertion at the moment n. p LCV (t) is defined as: where y h LCV (t) or y h LCV (n) is the vertical coordinate of the LCV's head bumper at the moment t or n; max y h LCV (t) or min y h LCV (t) are the maximum or minimum value of y h LCV (t).

2) BASIC LC MODEL
The LCV's maneuver is continuously influenced by the FV and the LV during its inserting process. The maneuver can be modelled under the framework of the multianticipative CF model. One of this kind of models proposed in [28], which is derived from the full velocity difference CF model, is applied in this study: To simulate the inserting maneuver of the LCV, the model in (10) is extended to the LC model in (11) with the lateral influential parameter θ j LCV (t): (12). According to the definition of p LCV (t), its value decreases when the LCV gradually approaches to its inserting ending. The values of θ LCV (t) gradually decrease in this process as well, which means that the influence of the FV on the LCV become less and less. Similar to the identification way of θ j LCV (t), the value of τ LCV in (11) will be confirmed according to the calibration results of the LC model.
where α LCV ∈(0, 1); β LCV > 0; C LCV 1 is a constant; is always a positive value, as tanh is a monotonically increasing function. The model in (12) simulates a scenario that: when the LCV gradually enters its target lane, the LV's influence on the LCV increases while the FV's influence decreases. Till the end of its inserting process, the LCV is only influenced by the LV.
It assumes that the LCV driver always pursues the desired vehicle spacing and speed difference defined in Section III.A. To reflect this desire in the LC model, the adjusting parameters, δ xj LCV (t) and δ vj LCV (t), are used in (14) to measure the safety states of current vehicle spacing and velocity difference. With the parameters, the model in (12) is reformulated as: The values of the adjusting parameters in (13) are set according to specific driving scenarios which will explained as follows.
In the scenario 1 shown in Fig. 4a, the LCV has to shorten its longitudinal spacing to the FV if it wants to thrust into the gap between the FV and the LV, but it does not completely overtake the FV ( x FV LCV (t) ≤ 0). So the LCV should have a larger acceleration or a smaller deceleration than the FV. To reflect this trend, the adjusting parameters should be set as: 1) δ xFV The LCV also needs to shorten the spacing to the LV, which does not increase its forward collision risk to the LV. So the parameters should be set as: 1) δ xLV In the scenario 2 shown in Fig. 4b, in order to insert between the FV and the LV, the LCV should shorten the distances to them, but the LV still does not overtake the LCV ( x LV LCV (t) ≤ 0). The LCV's adjustment of its spacing to the FV can be modelled by setting: 1) δ xFV To reflect the LCV's adjustment of its spacing to the LV, the parameters should be set as: 1) δ xLV In the scenario 3 shown in Fig. 4c, the LCV continuously travels between the LV and the FV ( x LCV FV (t) ≥ 0 and x LV LCV (t) ≥ 0). At this time, the LCV has the opportunity to conducting its insertion in case of the vehicle keeping safe distance to the other two vehicles. To reflect this trend, the parameters should be set as: 1) δ xFV LCV (t) = +1, and δ vFV . Such setting could also be followed if the FV has been overtaken by the LCV in the scenario 1, or the LV has overtaken the LCV in the scenario 2.

3) TRAFFIC SIGNAL
When observing the red signal of downstream intersection, the LCV driver knows that all vehicles will stop behind the stop line or at least continuously decelerate in red phase. Such anticipation could affect the driver's decision. This can be modelled by reformulating the a LCV (t + τ LCV ) in (13) as the a r LCV (t + τ LCV ) in (14): where the optimal velocity function V x stop j (t) defined in (15) is used to reflect the influence of red signal on the LCV driver's decision. Such definition means that the optimal velocity of the LCV is determined by the horizontal distance of the vehicle j to the stop line, x stop j (t), rather than x j LCV (t) in (13). The minus sign of V x stop j (t) in (14) indicates a fact that the approaching stop line at downstream reduces the LCV driver's intention to pursue his or her desired velocity V e in red phase.
In addition, since the driver can hardly anticipate other vehicles' moving trends in green phase, so the LCV's maneuver in green phase can still be simulated by (13), yet namely a g LCV (t + τ LCV ).

4) LC TARGET
The target of a LC vehicle could be 1) to obtain a faster speed, 2) or change to the lane heading to its destination. The inserting maneuver to fulfil the first target is often namely discretionary LC, while the second target can be VOLUME 8, 2020 achieved by conducting mandatory LC. Previous study had found that the mandatory inserting maneuver could be more aggressive than the discretionary one [29]. Such influence can be reflected by the adjusting parameters of LC target, ξ xFV LCV and ξ xLV LCV (or ξ vFV LCV and ξ vLV LCV ), which are respectively used to adjust the driver's sensitivity of the variance of vehicle spacing (or speed difference). The parameters are set to 0.5 in the scenario of discretionary LC, while they are subjected to the following constraint in the scenario of mandatory LC: where (ξ xFV LCV , ξ xLV LCV , ξ vFV LCV , ξ vLV LCV ) ∈ [0, 1]. If one of them is smaller than 0.5, it means that the LCV driver is less sensitive to the variance of specific vehicle spacing or velocity difference in mandatory LC scenario respect to discretionary LC scenario.
To sum up, with the parameters defined above, the basic LC model in (15) can be extended as the two models in (17), as shown at the bottom of this page, where a r LCV (t + τ LCV ) and a g LCV (t + τ LCV ) are the LCV's accelerations in red phase and green phases.

C. LC-ADAPTIVE CF MODEL
Similar to the LCV driver, the FV driver also has the multi-directional and multianticipative driving ability, so the FV's LC-adaptive CF maneuver during the LCV's inserting process is modelled by following the way described in Section III.B.
Three forms of lateral influential parameter of the CF model, θ FV (t), is defined in (18), where k FV is a constant and p LCV (t) refers to (10). According to the definitions, the value of θ FV (t) decreases when the LCV gradually inserts into its target lane, as the LCV's influence on the FV increases while the FV's influence on the LV decreases (see the Fig. 1).
Then θ FV (t) is used to simulate the FV's LC-adaptive CF maneuver during the LCV's inserting process by reforming the multianticipative CF model in (10) as: where V FV x j FV is the FV's optimal velocity, which is defined in (20); b FV is the driver's sensitivity of the difference between V e and V FV x j FV to v FV (t). According to the definition of p LCV (t) and θ FV (t), the value of θ FV (t) decreases when the LCV gradually approaches to its inserting ending. It means that the influence of the LV on the FV become less and less. Proper form of θ FV (t) and the value of τ FV will be confirmed based on calibration results of the CF model. For the V FV x j FV shown in (22), the definitions of its parameter are β FV > 0, C FV 2 = 1, and C FV 1 is a constant.
Similar to the definition of the LC model in (13), a group of adjusting parameters of vehicle relative position, δ xj FV (t) or δ vj FV (t) (j = LCV and LV), are applied in (19) to reflect the FV driver's desired varying trends of vehicle spacing and velocity difference. The desires include: 1) shortening the FV's spacing to the LCV or LV in the scenario 1; and 2) increasing its spacing to the LCV and decreasing its spacing to the LV in the scenario 2 and scenario 3. The settings of the adjusting parameters are listed in (21).
77672 VOLUME 8, 2020 Moreover, to reflect the influences of traffic signal and LC target change on the FV's maneuver, the model in (19) can be extended to the one in (22), as shown at the bottom of this page. The adjusting parameters of LC target in (22) measure the influence of LC target difference on the FV driver's sensitivity of the variance of vehicle spacing (or speed difference). They are subjected to the constraint shown in (23), and they are assumed to equal 0.5 in the case of discretionary LC.  Fig. 5 shows the study site. It is an urban arterial link with signalized intersections at upstream and downstream. The link length is about 220m, and its speed limit is 60km/h. Due to its limited length, all drivers can observe the traffic signal at downstream intersection in driving. Traffic flow video was collected from a roadside tall building. A total of 250 groups of vehicles in the video were selected as study objects, and their trajectories were extracted from the video in 10hz. Each group consists of one LCV, one FV and one LV. The trajectory collecting point is the head bumper or rear bumper of a vehicle (see Fig. 4). Applying the method proposed in [30], several control points (see Fig. 5) were measured in field and used to calibrate the video-extracted trajectories to the ones in real world coordinates. The noise of the trajectories was filtered referring to the way designed in [31]. The speed of some vehicles were metered by radar guns in field. By comparing the measurements with the speeds calculated from the trajectories, it is found that the error of the vehicle speed can be controlled in 0.1m/s. The information of LC target and traffic signal light was recorded manually from the pedestrian overpass located at upstream intersection.
Only the part of a trajectory which covers the study period defined in Section II.A was used in this study. According to the calculated length of each LCV's inserting process, the longest process lasts 7.4s, while the shortest one last 1.1s. The average length is 3.42s. Fig. 6 illustrates the trajectories of No. 126 and No. 134 vehicle groups. The LCV in Fig.  6a failed its insertion at the first attempt, while it finished the insertion at the second try. Obviously, the LCV perceived collision risk at its first attempt and given up it, and gives it a second chance. In contrast, the inserting process of the

FIGURE 7. Relationship between p LCV (t)−θ LCV (t) and p LCV (t)−θ FV (t).
LCV in Fig. 6b lasted for a shorter time and finished it at the first try. From the comparison, it is easy to find the necessity of setting lateral influential parameters, θ LCV (t) and θ FV (t), in the LC and LC-adaptive CF models. Fig. 7c and

V. MODEL CALIBRATION AND VALIDATION A. CALIBRATION
The LC and LC-adaptive CF models respectively developed in Section III.B and III.C are calibrated in two steps. In the first step, the values of reaction time of the LCV and FV drivers, τ LCV and τ FV , are identified together with proper forms of lateral influential parameters, θ LCV (t) or θ FV (t).
Since the trajectory is logged every 0.1s, so the value range of τ LCV and τ FV are set from 0.2s to 1s in the interval of 0.1s. All combinations of the two parameters are calibrated by the whole 250 groups of vehicle trajectories. The proper form of θ i (t) and value of τ FV are identified by comparing the model fitness to the trajectories at each parameter combination.
In the second step, other parameters of the LC and CF models are calibrated and validated in the cross-validation way adopted by [14]. Table 2 lists value ranges of the parameters, some of which refer to [29], [32], according to the physical boundaries of the parameters. The vehicle trajectories are divided into 5 groups. Four groups are applied in model calibration while the left one is used in model validation. Such process is repeated 5 times and each data group is used only once as the validation data. Average results of the 5 times processes are the calibrated parameters and final performances of the models. The calibration work of a microscopic traffic flow model can be abstracted as a nonlinear optimization problem. In this study, the objective function is set to minimize the error of the model prediction with respect to the trajectory data in terms of a r LCV (t + τ LCV ), a g LCV (t + τ LCV ), a r FV (t + τ FV ) or a g FV (t + τ FV ). Theil's U function is formulated in (24) as the objective function as previous studies did [14], [33]. The function is to minimize the error of the predicted acceleration, a m sim (m = 1,. . . , M ), to the one calculated from the trajectory data, a m field . f obj measures the fitness of the developed model to the trajectory data. M is the number of the trajectory samples. The genetic algorithm package in MATLAB software is applied in model calibration, and the algorithm parameters are set referring to [34]: population size is 50; maximum number of generations is 200; fraction of population created by the crossover function is 0.8; migration interval is 20; migration fraction is 0.2; the initial value of penalty parameter is 10; termination tolerance on function value is 10 −6 . Five indicators are used to evaluate the prediction accuracy of the LC and CF models to the collected vehicle trajectories. The indicators include the mean error (ME), the mean absolute error (MAE), the root mean square error (RMSE), the coefficient of determiniation (R 2 ) and the adjusted R 2 . Their formations are shown in (25), in which a m r is the real acceleration of the LCV in the mth vehicle group calculated from its vehicle trajectory, a m p is its predicted acceleration of the LC or LC-adaptive CF model, a ave r is the average value of a m r , M is the number of vehicle groups, and K is the number of the parameters.

1) TEST MODEL
Four pairs of test models, each of which consists of a LC model and a LC-adaptive CF model, are formulated to observe how some critical components of the LC and CF models in (17) and (22) works. The LC model in each pair is shown in Table 3. In the 1 st pair of test models, the FV and the LCV are assumed to be only affect the movements of LV, while their interactive influences are not taken into account. In the 2 nd pair of test models, it assumes that the LCV or the FV is be affected by the counterpart's lateral movements.
In the 3 rd pair of test models, the drivers' decisions are assumed to not be influenced by the difference of LC target. The 4 th pair of test models do not consider the influence of traffic signal on the two drivers' decisions. The test models are calibrated with the same data and method mentioned in Section V.A.

2) CONTRAST MODEL
To testify the merit of the LC model developed in this study, its performance is compared with the LC model proposed in [14]. The previous model also derives from a Gipps-based where θ LCV (t) adopts the formation used in (17); Although the LC-adaptive CF model is not formulated in [14], it actually can be developed in the similar way as (26). The CF model is presented in (28), where v LCV (t + τ FV ) and v LV (t + τ FV ) are defined in (29).

VI. RESULTS AND DISCUSSIONS A. MODEL PARAMETERS
The value of the two driver's reaction time, τ LCV and τ FV , and the proper forms of the lateral influential parameters, θ LCV (t) or θ FV (t), of the LC model in (17) and the LC-adaptive CF model in (22) are identified by comparing the model fitness to vehicle trajectories at each combination of the two parameters. The calculated fitness is listed in Table 4. The parameter combination, which lets the model fitness (f obj ) achieves the highest value, is bolded in the table. The bolded parameters are taken as the inputs of the calibration work of the rest parameters of the LC and CF models. The results of model calibration are listed in Table 5. The estimated parameters will be interpreted as follows.

1) THE PARAMETER OF REACTION TIME
The best fitted parameter of the driver's reaction time τ i is τ LCV = 0.3 for the LC model or τ FV = 0.9 for the CF model, respectively. It indicates that the LCV driver has the shorter reaction time than the FV driver at the average level. In other words, the LCV driver could be more sensitive to the external stimuli than the FV driver on average.

2) THE PARAMETER OF LATERAL INFLUENCE
The best fitted lateral influential parameter θ i (t) is θ 2 LCV (t) for the LC model or θ 1 FV (t) for the CF model. Their relationships with the determinant variable p LCV (t) are illustrated in Fig. 7. According to the definitions of θ 2 LCV (t) in (8), θ 1 FV (t) in (18) and p LCV (t) in (9), it is known that the values of θ 2 LCV (t) and θ 1 FV (t) decrease if the value of p LCV (t) decreases. In other words, when the LCV approaches to its inserting ending gradually, the values of θ 2 LCV (t) and θ 1 FV (t) decrease. From the formations of the LC model in (17) and the CF model in (22), the decrease of θ LCV (t) means that the LCV driver is less affected by the FV, while the decrease of θ FV (t) indicates that the LCV's impact increases for the FV driver. However, the increasing rate of the LCV's impact on the FV driver is larger than the decreasing rate of the FV's impact on the LCV driver. This can be inferred from the smaller gradient of blue straight solid line than the red dot line in the high value range of p LCV (t) shown in Fig. 7. This result could be caused by the different varying trends of the two drivers' visual fields. The FV driver is always able to observe the movements of the LCV and the LV ahead, so the driver can adjust maneuvers steadily. But this work becomes harder and harder for the LCV driver when the LCV gradually inserts into target lane.

3) THE OTHER PARAMETERS
The LC model in (17) and the LC-adaptive CF model in (22) share a united model framework. The feature makes us easily observe the differences of the two vehicles' responses to the same stimulus by comparing the estimates of the variable parameters of the two models that are related to the stimulus. This way is used to infer the characteristics of the vehicle maneuvers during the LCV's inserting process as follows.
Firstly, the parameter b LCV or b FV reflects the sensitivity of the LCV or FV driver to the difference of his or her desired speed and current speed. From the results mentioned above, it can find that comparing to the discretionary LC scenario, both of the LCV and FV drivers put more attentions on their maneuvering interactions in the mandatory LC scenario than that to the LV. Such tendency could be caused by the higher collision risk of the mandatory LC than that of the discretionary LC which has been found by previous study [18]. The risk mainly comes from the interactive maneuvers of the LCV and FV, as the LV driver is rarely disturbed by their interactions occurred in his or her blind side.

B. MODEL PERFORMANCE
The values of the model performance indicators of the proposed LC model in (17) and LC-adaptive CF model in (22) are calculated and listed in Table 6. The indicator values of the contrast model and the test models are also reported in the table. The performances of different models are compared and interpreted as follows.

1) CONTRAST MODELS
The LC and CF models proposed in this study are found to perform better than the contrast models shown in (26) and (28). This difference could be caused by the basic CF models which the two pairs of the models are developed from. The full velocity difference CF model shown in (3) is selected as the basic model of the proposed models shown in (17) and (22), while the contrast models derive from Gipps CF model. Compared with the Gipps model, the full velocity difference model is more suitable to simulate the nonlinear accelerating or decelerating maneuver of the vehicles that are commonly seen in urban traffic. The vehicle trajectories used in model calibration and validation of this study were collected in urban arterial, so the result that the proposed models outperform the contrast models in trajectory fitting are not quite surprising. This result gives us a hint that it is important to select the proper basic model according to studied driving scenario when a more complex microscopic traffic flow model is developed from the basic model.

2) TEST MODELS
By comparing the values of model performance indicators in Table 6, it can find that the maneuvers of the LCV and the FV are modelled in a reasonable way in this study. This is presented from the following points.
To figure out if the drivers' multianticipative ability work during the LCV's inserting process, the 1 st pair of test models assume both of the LCV and or the FV are only influenced by the movements LV. Result show that the models perform the worst among all test models, which means that this pair of the test models miss the most important kinematic variables, the interactive maneuvers of the LCV and the FV. It reveals the necessity of considering the driver's multianticipative ability into the LC model and LC-adaptive CF model. Previous study had pointed out the defect of some CF models that omit the LCV's movements when simulating the FV's maneuver in LC scenario [3]. The bad performance of the CF model in the 1 st pair of test model verifies his point.
The 2 nd pair of test models assumes the LCV driver and the FV driver are not affected by lateral moving of the adjacent vehicles, but this assumption is not supported by the results in Table 4. These models only perform better than the 1 st pair of test models. It means that the lateral disturbance is indeed perceived by the drivers and heavily influence their decisions. The variable that reflects the behavioral characteristics should not be neglected in the LC and LC-adaptive CF models.
The 3 rd and 4 th pairs of test models exclude parameters that reflect the impact of LC target and traffic signal on the maneuvers of the LCV and the FV. The performances of the test models are worse than the LC and CF models proposed in (17) and (22). Excluding the parameter of traffic signal leads to the heavier impairment on the model performance than that of excluding the parameter of LC target. This result indicates the importance of modelling the driver's response to traffic signal change in microscopic traffic flow model, yet which is still not done well in existing simulation software.

VII. CONCLUSIONS
This study develops a pair of LC and LC-adaptive CF models to simulate mutually adaptive maneuvers of the LCV and its adjacent CFV during the LCV's inserting process. The models are formulated in a united framework with a group of parameters to reflect the influences of longitudinal and lateral moving vehicles, change of traffic signal, LC target on the two vehicles' maneuvers. The models are calibrated and validated by vehicle trajectories collected in urban traffic. The vehicle trajectory predictive results, which is the output of the LC and CF models, indicate that the models proposed in this study can simulate LC and CF maneuvers during the LCV's inserting process in a higher accuracy than the ones proposed by previous study. By inferring the estimated model parameters, it is found that LC driver and CF driver perceives the external stimuli and response them in different ways during the inserting process.
There exists some work for future improvement. Firstly, the study period of this paper only covers the LCV's inserting period, but previous study has found that its impact on the CFVs on original lane and target lane still exists even the insertion has finished [8], [17]. The CF maneuver under this continuous LC impact is still not been modelled well now, so the existing CF model has room for further improvements. VOLUME 8, 2020 Secondly, the LC and LC-adaptive CF models derived from the full velocity difference CF model are found to perform better than the previous ones derived from Gipps CF model. It is necessary to apply the modelling method applied in this study to extend more classical CF models and compare their performances. This is helpful to improve our understandings of their simulation features in the LC scenario. Thirdly, the knowledge obtained in this study can be referred when developing the trajectory planning algorithm of the automated vehicle, as a major challenge in this field is to predict the moving trend of adjacent LCV or CFV that are manipulated manually [17], [35]. Finally, the models proposed in this study can be used in motion planning of the LCV and CFV during LC process. The generated trajectory can be further integrated with power transmission model or tyre model in the vehicle control research [36]- [38].