Speed Control of Inflow Vehicles for Merging Support on Motorways with Limited I2V Communication

Abstract

Since merging is one of the most difficult situations for automated driving, merging support provided by infrastructure is extremely helpful for automated vehicles on motorways. However, in the early stages of automated vehicle penetration, infrastructure-side equipment to support merging is limited. In the first half of this chapter, a method to support speed control for inflow vehicles is proposed for the case where there is one detection device on the main lanes and one transmission/detection device for the vehicles in the inflow lane. In the second half, the case equipped with an additional detector and transmitter downstream on each lane is dealt with. New accurate information about the merging vehicles concerned could be given to the speed controller in the infrastructure. The control should be recalculated and sent to the inflow vehicle that has already started. Each control problem is formulated in the optimal control theory and calculated numerically by a nonlinear programming algorithm. The computational experiment is performed in each realistic situation.

1. Introduction

Automated driving is entering a new stage. In the United Nations WP29 (World Forum for the Harmonization of Regulations), a proposal was adopted in 2022 to extend automated driving in certain traffic environments (that is Level 3 defined by SAE (Society of Automotive Engineers)) from the limit of 60 km/h to up to 130 km/h (in Ref. [1]). Nevertheless, V2I (Vehicle to Infrastructure)/I2V (Infrastructure to Vehicle)/V2V (Vehicle to Vehicle) communication has not evolved explosively.

Merging is one of the most difficult situations for automated driving on motorways. Previous studies on vehicle speed control in merging areas have mainly focused on the control in situations where V2I/I2V/V2V communication is fully equipped. However, in the early stages of automated vehicle penetration, the infrastructure-side equipment associated with V2I/I2V is limited.

The Strategic Innovation Promotion Program (SIP) of the Japanese national project includes Automated Driving for Universal Services (ADUS, 2014–2023). SIP-ADUS classifies the development of merging support systems into the following four stages:

Day 1: Support for vehicles on the inflow lane (basic).

Detection on main lanes is done at one spot, transmission on inflow lane is made at one spot.

Day 2: Support for vehicles on the inflow lane (advanced).

Detection on main lanes is done in a consecutive section, and transmission on inflow lane is made consecutively.

Day 3: Support for vehicles on both inflow lane and main lanes via I2V.

Day 4: Support for vehicles on both inflow lane and main lanes with full V2I/I2V and V2V.

The situations on Days 1 and 2 are illustrated precisely in Ref. [2]. In the Days 1 and 2 stages, the infrastructure only provides information about vehicles on main lanes to inflow vehicles. It is assumed that vehicle speed in inflow lane is determined by its in-vehicle function.

In this chapter, the situation where the speed control of inflow vehicles is provided by the infrastructure is considered for advanced merging support. In order to determine the control by the infrastructure, a detector has to be added at the same position as the transmitter on inflow lane. This situation is called Day 1⁺ by the authors in this chapter. The Day 2⁺ is also used in the same manner.

In Section 2, a practical method is proposed that provides speed control to an inflow vehicle for merging under the limited conditions in the Day 1⁺ situation.

Next, in Section 3, a modified method that provides speed control to the inflow vehicles under intermediate conditions between Days 1⁺ and 2⁺, which is called Day 1.5⁺ provisionally, is also proposed for near future use. If equipped with an additional detector and transmitter downstream, new accurate information about the merging vehicles concerned could be given to the speed controller in the infrastructure. The control should be recalculated and sent to the inflow vehicle that has already started.

The control problem is formulated in the optimal control theory and calculated numerically by a nonlinear programming algorithm. The computational experiment is performed in each realistic situation associated with Days 1⁺ and 1.5⁺ respectively.

Finally, in Section 4, the conclusion and perspective are presented. The studies which should succeed in the proposed contents in this chapter are described.

2. Speed control of inflow vehicles for merging support equipped with one detector/transmitter on each lane

The traffic situation in the Day 1⁺ stage is illustrated in Figure 1. In the following sections, results of computational experiments are provided based on the situations in actual Japanese expressways, and diagrams associated with traffic on the left side are consistently shown in this chapter. In Japan, as in the UK, Australia, etc., traffic keeps to the left side, thus the inflow lane is located on the left side of the main lanes. The lane on the outside of the main lanes is called the outer main lane in this chapter.

Figure 1.

Schematic illustration of merging support area in Day1⁺ stage.

The vehicle A passing the transmitter/detector spot in inflow lane receives the speed control computed by the traffic management center or an edge computer on roadside considering the information about the vehicles P, Q, and R already passed through the detector spot in the outer main lane. The position and the speed of each of these vehicles in the merging zone can be estimated from the time and the speed when passing through the detector spot.

Based on all the information, the traffic management center or the edge computer searches for the possible headway between two successive vehicles in the outer main lane where vehicle A can merge into. If the headway between P and Q would be impossible for the merging of A, the next headway between Q and R should be tried. If the appropriate headway is found, the center or the edge computer computes the speed control of vehicle A and sends it to vehicle A. The control support finishes at the start of the merging zone. The merging action itself should be conducted by the in-vehicle automated driving function or the human driver in the Day 1⁺ situation. The method for speed control is described precisely in the next subsection.

2.1 Method of speed control of inflow vehicle

Various methods of control of vehicles have been proposed for merging. Most of them are based on optimization techniques such as dynamic programming [3], game theory [4], and so forth. Recently, deep Reinforcement Learning (deep RL) has been attracting attention in various application fields. A vehicle control method for merging based on the deep RL is also proposed in [5]. Although the deep RL is extremely attractive in the theoretical aspect, each iterative procedure of the RL tries to choose the best action based on consideration of all possible estimated states up to the distant future. Hence, computational efforts expand exponentially and the procedures for implementation are quite complicated.

Computational time should not be ignored in practical applications because the speed control must be applied to the inflow vehicle as soon as possible after receiving the control from the transmitter. Therefore, a simple method of computing speed control is proposed in this section. The method is based on a numerical approach to the traditional optimal control theory.

2.1.1 Scheme of numerical approach to optimal speed control of the inflow vehicle A

A scheme of the numerical approach to optimal speed control of the inflow vehicle A is illustrated in Figure 2. The position and the speed of vehicle A at the transmitter/detector and the position and the speed of vehicles P and Q detected in the outer main lane are used to calculate the position and the velocity of vehicle A after passing the start of the merging zone. The velocity of vehicle A on the inflow lane is usually lower than the running speed of vehicles on main lanes. It is assumed that the preceding vehicle B in the inflow lane does not prevent vehicle A.

Figure 2.

Scheme of numerical approach to optimal speed control of the inflow vehicle A in Day 1⁺ stage.

2.1.2 Motion equation of vehicle A

Let

$x\left(k\right),v_x\left(k\right)$: position (m) and velocity(m/s) of the vehicle A at time k: state variables;

$u_x\left(k\right)$: acceleration (m/s²) of vehicle the A at time k: control variable;

$t_s$: sampling time.

$$\begin{array}{lll}\mathit{x}\left(k\right)& =& {\left[\begin{array}{cc}x\left(k\right)& v_x\left(k\right)\end{array}\right]}^T\\ A& =& \left[\begin{array}{cc}1& t_s\\ 0& 1\end{array}\right],B={\left[\begin{array}{cc}0& t_s\end{array}\right]}^T\end{array}$$

Then, discrete the motion equation of vehicle A is described in (1).

$$\left\{\begin{array}{c}x\left(k+1\right)=x\left(k\right)+t_s{v}_x\left(k\right)\\ v_x\left(k+1\right)=v_x\left(k\right)+t_s{u}_x\left(k\right)\end{array}\right.$$E1

The motion equation Eq. (1) can be expressed as discrete-time state-space representation of a linear system Eq. (2).

$$\begin{array}{c}\mathit{x}\left(k+1\right)=A\mathit{x}\left(k\right)+B{u}_x\left(k\right)\mathit{.}\end{array}$$E2

2.1.3 Precise formulation of speed control based on optimal control theory and numerical approach

From Eq. (1), when the final time is set as n, the entire motion equation of vehicle A is expressed as the following Eq. [3].

$$\begin{array}{c}\left\{\begin{array}{c}x\left(1\right)-x\left(0\right)-t_s{v}_x\left(0\right)=0\\ x\left(2\right)-x\left(1\right)-t_s{v}_x\left(1\right)=0\\ ⋮\\ x\left(n\right)-x\left(n-1\right)-t_s{v}_x\left(n-1\right)=0\\ v_x\left(1\right)-v_x\left(0\right)-t_s{u}_x\left(0\right)=0\\ v_x\left(2\right)-v_x\left(1\right)-t_s{u}_x\left(1\right)=0\\ ⋮\\ v_x\left(n\right)-v_x\left(n-1\right)-t_s{u}_x\left(n-1\right)=0\end{array}\right.\end{array}$$E3

The initial condition of Eq. (3) is set as follows:

$$\begin{array}{c}x\left(0\right)=x_0,v\left(0\right)=v_0\left(k=0,\cdots ,h\right)\end{array}$$E4

where $x_0$,$v_0$ are the initial position and velocity of vehicle A at passing the transmitter/detector, and h is the time when the in-vehicle controller in the vehicle A can start operation after receiving the control information.

The limitation of velocity of vehicle A is

$$\begin{array}{c}0\le v_x\left(k\right)\le v_{\max }\left(k=0,\cdots ,n\right)\end{array}$$E5

where $v_{\max }$ is the maximal speed of vehicles A.

Moderate acceleration and deacceleration are imposed as follows (2 m/s²$\approx$0.2G):

$$\begin{array}{c}-2\le u_x\left(k\right)\le 2\left(k=0,\cdots ,n\right)\end{array}$$E6

Let

$p\left(k\right),q\left(k\right)$: the estimated position of vehicle P, Q at time k.

$v_p\left(k\right),v_q\left(k\right)$: the estimated velocity of vehicle P, Q at time k,

then additional constraints are set as follows:

$$\begin{array}{c}q\left(k\right)+f\le x\left(k\right)\le p\left(k\right)-e\left(k=m+1,\cdots ,n\right)\end{array}$$E7

where e is safe headway to the preceding vehicle P and f is safe headway to the succeeding vehicle Q, while m is the time when the vehicle A arrives the start of the merging zone.

$$\begin{array}{c}v\left(k\right)=v_p\left(k\right)\left(k=m+1,\cdots ,n\right)\end{array}$$E8

The velocity of the vehicle A must be equal to the velocity of vehicle P after m.

Subject to the constraints Eqs. (3)–(8), the following evaluation function Eq. (9) is to be minimized.

$$\begin{array}{c}f=-\sum _{k=0}^{n}x\left(k\right)+a\sum _{k=0}^{n-1}{u}_x{\left(k\right)}^2+b\sum _{k=0}^{n-2}{\left\{u_x\left(k+1\right)-u_x\left(k\right)\right\}}^2\end{array}$$E9

The first term in Eq. (9) is to move the vehicle A forward as much as possible and the second and the third terms are to make the motion of the vehicle A moderate.

This problem belongs to nonlinear programming problems. Because it is difficult to solve it analytically in general, numerical approaches are usually adopted.

2.2 Computational experiment

In this subsection, a computational experiment in the situation in the Day 1⁺ stage is presented.

2.2.1 Environment and conditions in the computational experiment

In 2019, the field operational tests on merging support in the Day 1 stage were conducted by SIP-ADUS at the Airport West on-ramp in Tokyo Metropolitan Expressway. In the test and its preliminary investigation, the allowable positions of the detector and transmitter were specified. Within the permissible limits, the distance between the detector on the main lanes and the start of the merging zone is set to 160 m, while the distance between the transmitter/detector on inflow lane and the start of the merging zone is set to 95 m in this computational experiment. The situation is illustrated in Figure 3. In the area, the number of main lanes is two.

Figure 3.

Positions of detectors and transmitters equipped in Day 1⁺ stage.

Because in the Day 1 situation, vehicles running on main lanes are detected only once at the detector on main lanes, velocity of each vehicle is assumed to be constant throughout the control period. Parameters and initial values in the computational experiment in the Day 1⁺ are listed in Table 1. The $x$-axis is set along the road and the positive direction is due to the vehicle running. Its origin point is the start of the merging zone.

Item	Attribute value
Time increment	0.1 s
Initial position and velocity of vehicle A	$x_0=-95\mathrm{m},v_0=40\mathrm{km}/\mathrm{h}$
Maximal speed of vehicles	$v_{\max }=60\mathrm{km}/\mathrm{h}$
Safe headway to the preceding vehicle P and the succeeding vehicle Q	$e=f=16.7\mathrm{m}$
Initial positions of vehicle P, Q, and R	$p\left(0\right)=-85\mathrm{m},q\left(0\right)=-120\mathrm{m}$; $r\left(0\right)=-155\mathrm{m}$,
Velocity of vehicle P, Q, and R	$v_p\left(k\right)=v_q\left(k\right)=v_r\left(k\right)\equiv 60\mathrm{km}/\mathrm{h}$†
Delay of transmission‡	1.3 s, i.e.,$h=13$
Period of control	12 s, i.e., $n=120$

Table 1.

Parameters and initial value set in the computational experiment on Day 1⁺.

^†

Because vehicles on main lanes are detected only once, the velocity must be dealt with constant.

^‡

Delay between receiving control information through the in-vehicle antenna and actuation of the in-vehicle control unit.

In the computational experiments in this chapter, a programming and numeric computing platform MATLAB is used to find the solution to the problem. The discrete optimal control problem Eqs. (3)–(9) belongs to the constrained nonlinear programming problems. Therefore, the optimization tool ‘fmincon’ in ‘Optimization Toolbox’ is used. Moreover, because the constraints Eqs. (3)–(8) are linear and the evaluation function Eq. (9) is quadratic, the sequential quadratic programming algorithm ‘sqp’ is adopted to find the solution.

An initial solution has to be given to the iterative optimization procedure. In this computational experiment,

$$v_x\left(k\right)=40\mathrm{km}/\mathrm{h},u_x\left(k\right)=0\left(k=0,\cdots ,n\right)$$E10

these values were adopted for the initial solution. Of course, it is infeasible. The procedure starts at the infeasible initial solution and proceeds to the feasible region and then searches for the optimal solution.

2.2.2 Result of the computational experiment

The result of the computational experiment under the conditions specified above is as follows. In this case, because the initial position of the vehicle P is too far ahead, the merging between P and Q is impossible. Thus, another merging between Q and R is tried and computed, and then it is successfully achieved. The initial positions and the final positions of the vehicles concerned are shown in Figure 4.

Figure 4.

Initial positions and final positions of vehicles concerned.

The trajectory of positions, velocity, and acceleration of vehicles concerned are shown in Figure 5. In (b) and (c), graphs associated with vehicle R are omitted because the behavior of vehicle R is assumed to be quite equal to it of vehicle Q.

Figure 5.

Positions, velocity, and acceleration of vehicles concerned in the result in Day1⁺ stage.

From (a), the accomplishment of the merging of vehicle A into the headway between vehicles Q and R in the merging zone is observed.

From (b) and (c), it is observed that vehicle A starts at 40 km/h and then deaccelerates for a while, and finally accelerates to 60 km/h before the start of the merging zone. The acceleration in the final phase is to match the speed of vehicles on outer main lane. Due to the acceleration, the initial deceleration is required to adjust the final position of vehicle A.

The speed control of vehicle A that tries merging between the vehicles Q and R are computed in 1.72 s on a note PC (Windows11 with AMD Ryzen 7 5800H 3.2GHz). In the computation shown above, iterations of solution update are limited to 40. When iterations are limited to 10, the computational time decreases to 0.66 s, while the difference in calculation accuracy is only 0.0024%.

2.2.3 Discussions

The result of the computational experiment presented in the previous sub-subsection shows the effectiveness of the method of speed control of inflow vehicles in merging areas in the Day 1⁺ stage.

Nevertheless, if the speed of the vehicle on the main line changes significantly due to the slow down around the merging zone after being sensed at the detector, the merging using the computed speed control will not be successful. Generally speaking, the effect of the merging support system in the Day 1⁺ stage is limited into the stationary traffic flow.

In order to enhance the effect of merging support, more detectors on main lanes and more detectors/transmitters on inflow lanes are required. The control method of inflow vehicles in the Day 1.5⁺ situation is proposed in the following section.

3. Speed control of inflow vehicles equipped with two detectors/transmitters

As explained in Section 1, it is assumed that multiple detectors on main lanes and multiple transmitters/detectors on inflow lanes are equipped in the Day 1.5⁺ stage. In Figure 6, two detectors on main lanes and two transmitters/detectors on inflow lane are equipped. Based on the additional information provided by the additional facilities, the speed control of the inflow vehicle could be improved.

Figure 6.

Merging support area associated with the computational experiment in Day 1.5⁺ stage.

3.1 Introduction of ‘event-driven sequential optimal control’

In the 1980s, the Model Predictive Control (MPC) was proposed and has been applied to various fields. A survey paper including recent applications is presented [6]. Studies on the application of MPC to vehicle control for merging support have been presented, which are based on complete V2I/I2V and V2V communication [7, 8]. The key concept of MPC is that optimal control problems are solved repeatedly at certain time intervals under the latest conditions, which are the sum of state transitions due to past control already applied and unexpected changes due to other factors. Although MPC is an effective and powerful control method, computational efforts are considerably large due to repeated optimization.

In the merging situation illustrated in Figure 6, an additional control could be computed when new information is detected. Therefore, a two-phase optimization scheme could be applied to the situation. As in this example, the method of recalculating optimal control each time an event occurs is called ‘event-driven sequential optimal control’ by the authors. The scheme is illustrated in Figure 7.

Figure 7.

Scheme of ‘event-driven sequential optimal speed control’ of the inflow vehicle A in Day 1.5⁺ stage.

3.2 Computational experiment

In the computational experiment presented in this subsection, most of the environment and conditions are inherited from the experiment shown in Subsection 2.2, in order to focus on the substantial difference between situations/results in Days 1⁺ and 1.5⁺. The only exception is the exit of the second detectors.

3.2.1 Environment and conditions in the computational experiment

The positions of transmitters and detectors used in the following computational experiment are illustrated in Figure 8. Please note that positions of the second transmitter/detector on main lanes and the second detector in inflow lane are determined virtually by the authors, while the first transmitter/detector on main lanes and the first detector in inflow lane are inherited from the previous experiment illustrated in Figure 3.

Figure 8.

Positions of detectors and transmitters equipped in Day 1.5⁺ stage.

It is assumed that while running the inflow vehicle A under the speed control computed in Sub-subsection 3.2.2, the current position and velocity of the target vehicle Q passing at the second detector on the outer main lane is informed.

In the following computational experiment, the velocity of the vehicles P, Q, and R passing at the second detector is set at 47.8 km/h. By using the new information, a new speed control for vehicle A is computed and transmitted to it by the second transmitter/detector in inflow lane.

3.2.2 Result of the computational experiment

The result of the computational experiment under the conditions specified above is illustrated in Figure 9.

Figure 9.

Positions, velocity, and acceleration of vehicles concerned in the result in the Day 1.5⁺ stage.

After starting with the initial positions of the vehicles shown in Figure 4, new information about vehicle Q is provided at 2.6 s, which is the passing instant of vehicle Q at the second detector. A new computation is executed immediately. In Figure 9, data concerned with previous control is expressed as phase 1, while new data concerned with new control computation is expressed as phase 2. (At 2.6 s, vehicle R is not detected yet, and then assumed to follow exactly the move of vehicle Q in phase 2.)

Please note the following points in Figure 9:

1. All data in phase 1 are absolutely the same as in Figure 5.

2. Trajectories of position, velocity, and acceleration of vehicle Q are discontinuous at 2.6 s due to new detection.

3. The velocity of the vehicle Q has to be constant from 2.6 s because additional detection could not be provided in the Day 1.5 situation.

4. Trajectories of position and velocity of vehicle A are continuous at 2.6 s due to the exact inheritance of data of vehicle A.

5. Velocity of vehicle A are constant for 1.3 s from 2.6 s due to the delay of transmission set in Table 1.

6. The timing of acceleration of vehicle A is delayed due to the slower velocity of vehicle Q in the phase 2. As a result, the arrival of the vehicle A at the start of the merging zone is 9.9 s. It delayed 1.6 s compared to the result of the previous result in Figure 5.

The computational time is 1.76 s on the note PC (Windows 11 with AMD Ryzen 7 5800H 3.2 GHz). In the computation shown above, iterations of solution update are limited to 40. When iterations are limited to 10, computational time decreases to 0.64 s, while the difference in calculation accuracy is only 0.0033%.

3.2.3 Discussion

In the result of the computational experiment presented in the previous sub-subsection, the method of speed control could appropriately cope with the second detection of the vehicle information in outer main lane. Moreover, the computational effort is also suitable for practical use.

The speed control of the inflow vehicle in the Day 1.5⁺ stage is expected to be improved in comparison with it in the Day1⁺ stage.

4. Conclusion and perspectives

Provision of the control to inflow vehicles for merging in situations with limited detection and transmission is required during the low penetration of automated driving. In this chapter, speed control methods of the inflow vehicle were proposed for the early stages of penetration in merging support of infrastructure to cope with automated driving development.

In Section 2, the method of speed control of the inflow vehicle was introduced in the situation where one detector for each lane was equipped (Day 1⁺). The method is formulated as an optimal control formulation and solved by a numerical approach. The computational experiment brings the expected result.

In Section 3, the method of speed control was developed for the application to the situation where an additional detector for each lane was equipped (Day 1.5⁺). The method belongs to the sequential approach which is composed of two phases of optimization coping with two detectors. The computational experiment also brings the expected result.

Although the performance of speed control with two detectors is of course better than with one, the merging cannot be successful when there are large changes in traffic flow. In order to analyze quantitatively this problem, further computational experiments in various types of traffic flow should be executed. It is expected to conduct verification in a more realistic experiment environment which combines a traffic simulator that reproduces the actual traffic situation with this control method.

In the situation where consecutive detection and transmission are equipped (Day 2⁺), speed control of inflow vehicles could be solved by Model Predictive Control (MPC). Because MPC needs a large computational effort, some contrivance should be included in the solution procedure.

The main method for merging support is the speed control of inflow vehicles as dealt with in this chapter. In order to proceed to the next stage of merging support (Day 3), collaborative speed control of both inflow vehicles and main lane vehicles must be investigated. To prepare for that, the study on merging support for main lane vehicles is essential, which includes lane change to the inner lane. The control methods proposed in this chapter can be applied also to this problem.

Since communication equipment for automated driving will be expanded by several stages, it is necessary to develop the practical merging support technology in accordance with each stage. The practical methodology should be developed to bridge the gap between theoretical studies assuming that all information is available and the actual communication situation.