Collision-free control strategy for on-ramp merging: A spatial-dependent constraint following approach

This paper focuses on the problem of on-ramp merging control under the cooperation of intelligent and connected vehicles. A decentralized collision-free control strategy is proposed for on-ramp merging control. Each vehicle in the virtual platoon constructed by all vehicles on the arterial road and the on-ramp is equipped with a spatial-dependent constraint following controller. Under nonlinear vehicle dynamics, the proposed controller is proved to be uniformly bounded, thus assuring that each vehicle can satisfy the safety requirements to avoid collision at any specific spatial location, especially at the most dangerous merging point. Compared with time-dependence, this spatial-dependence means much more stability because spatial conditions during the on-ramp merging process are more static and invariant. Finally, a simulation containing six vehicles with relatively extreme testing conditions is conduct to validate the effectiveness of the proposed approach. The results demonstrate that the spacing errors can converge to 0 with respect to varying spatial-dependent desired spacings. The spacing errors of the six vehicles are kept at a relatively low level with a maximum value of 3.0778m. The maximal acceleration is 0.6060 m/s 2 and the maximal deceleration is -1.4042 m/s 2. All vehicles can achieve collision-free safety for on-ramp merging with a smooth and non-saturated control input generated by the proposed controller.


Introduction
The transportation systems and the automotive industry have been experiencing a fast development around the whole world. With the estimation that the total ownership of motor vehicles all around the world is about to increase to exceed one billion, the experts predict that the number will double during the following ten or twenty years [1]. The dense transportation activities provide people convenience, however, together with resulting in many problems, including speed breakdown, extra fuel consumption, increase of wasted hours, traffic flow vibration and more traffic congestions in converging areas [2][3], where on-ramp merging scenarios have become main triggers [4].
With various kinds of sensing and communication equipment, intelligent and connected vehicles (ICVs) have achieved considerable momentum in dealing with aforementioned issues. Compared with traditional manually driven vehicles, ICVs can provide shorter spacings and faster responses for intervehicle communication [5][6]. The vehicle-to-vehicle (V2V) communication and vehicle-toinfrastructure (V2I) communication enable the vehicles to communicate with others to achieve cooperative driving by adopting new-generation techniques.
There is a large number of literature on cooperative on-ramp merging control of ICVs [7][8][9][10][11][12]. Rios-Torres et al. [6], Scarinci et al. [13] and Bevly et al. [14] have reviewed the research efforts on the onramp merging but with different focuses on control techniques, benefit evaluation and vehicular IOP Publishing doi: 10.1088/1742-6596/2234/1/012011 2 technologies, respectively. Research efforts on coordinating ICVs' merging can be roughly divided into two categories, i.e. centralized and decentralized approaches. Centralized approaches are used when there is at least one task in the system that is globally decided for all vehicles by a single central controller [15]. Meanwhile, in decentralized approaches, all vehicles are treated as autonomous agents that attempt to achieve safety through strategic interaction and maximize their own efficiency [15].
There have been many studies focusing on designing controllers to achieve merging safety and high efficiency, but most of them do not focus on binding the controller performance with static spatial locations in the environment. Our team's previous work [16][17][18][19][20][21][22][23] mainly concentrates on the robust controller design and doesn't deal with this spatial-dependence, neither. Spatial-dependence means that the controller can assure that each vehicle satisfies the safety requirements to avoid collision at specific spatial locations, especially at the most dangerous merging point. Compared with the time-dependent controller, which plans a velocity or acceleration profile based on specific time, the spatial-dependence means much more stability and reliability since spatial environment of the on-ramp merging scenario is not likely to change abruptly (the position of the merging point is just with no accident).By binding the controller performance with static spatial locations in the merging scenario, merging safety at specific spatial locations is more straightforward and foreseeable. This spatial-dependence is exactly the main goal and emphasis of this paper based on our previous work. The kernal contribution of this paper is proposing such a decentralized spatial-dependent constraint following controller with an analytical, closed-form solution to the on-ramp merging control problem.
In this paper, we propose a decentralized spatial-dependent collision-free control strategy. The vehicles on the arterial road and the on-ramp together construct a virtual platoon [24] based on the initial conditions. A spatial-dependent constraint following controller is equipped on each vehicle to follow the preceding vehicle in the virtual platoon, with the target of decreasing the spacing error with respect to the desired safe spacing. By constraint following, we mean that the controller is designed based on the equlity constraint with expected performance. Through the performance of spatial-dependent safety assurance, we guarantee a safe and stable on-ramp merging process with the proposed method.
The rest of the structure of the paper is as follows. Section 2 formulates the problem, including nonlinear vehicle dynamics, spatial-dependent desired spacing design and state transformation. Section 3 designs the spatial-dependent controller with the uniform boundedness performance guaranteed. Section 4 shows the simulation results. Finally, concluding remarks are provided in Section 5.

Problem formulation
As demonstrated in Figure 1, the virtual platoon containing a variety of heterogenous vehicle models (e.g. sedans, vans, trucks) is constructed by vehicles on the two roads under the on-ramp merging scenario. The virtual platoon contains + 1 vehicles, where the index 0 represents the leading vehicle and the indexes from 1 to represent the following vehicles from front to back. The position of the th vehicle is expresses as , = 0,1, … , . The th vehicle's length is denoted as . The spacing between the th vehicle and its preceding vehicle is denoted as . In this problem, we presume that every vehicle owns specific sensors to detect its real-time spacing between itself and the preceding and other necessary state data. Besides, the follower can acquire information of its preceding vehicle with communication apparatus.
This section mainly describes the platoon's dynamics modeling, the design of the spatial-dependent desired spacing and the state transformation from the bounded spacing error state to the unbounded.

System dynamics
In this on-ramp merging problem, the longitudinal dynamics model for the th ( = 0,1, … , ) vehicle is formulated as where is time, is the velocity, is the vehicle mass, is the driving or braking force input. Besides, − · ( ) · | ( )| denotes the aerodynamic resistance and − denotes the sum of the rolling resistance and gradient resistance.
The real-time spacing of the th vehicle ( = 1, … , ) in real time is

Spatial-dependent desired spacing
The target of this on-ramp merging control system is to control all inter-vehicle spacings in the virtual platoon to approach their respective desired spacing. In other words, the spacing error should be as small as possible for safety. Here the spacing error of the th vehicle, i.e. ( ), is defined as where des ( ) is the spatial-dependent desired spacing. While small inter-vehicle spacing is critically dangerous, too large spacing will degrade the performances of fuel economy and traffic efficiency. Therefore, the spacing error need to be properly bounded. Let the expected boundedness of the spacing error be where the constants min and max denote the minimal and the maximal allowable spacing error(s) respectively. This system aims to control the spacing error to converge to 0 as time going while strictly satisfying this bidirectional constraint. Next, it should be noted that the desired spacing is designed as spatial-dependent, which means the desired spacing is the function of the position .
where (·) is a function that maps the position to desired spacing. What's more, the desired spacing satisfies the following boundary conditions where 0 is the initial position, 0 is the initial spacing, m is the position of the merging point and constant safe is the expected safe spacing at the merging point. The first boundary constraint indicates that the initial spacing error ( 0 ) is 0, which guarantees that the bidirectional constraint is satisfied at the initial time. The second boundary constraint indicates that the final desired spacing at the merging point ( m ) is an expected proper safe value, which, at the same time, guarantees the safety at the merging point for each vehicle since the spacing error is bidirectionally constrained around safe .
After that, we design the specific formula of the function (·). The trigonometric form is chosen as a candidate, that is, where the derivative of des with respect to position , equals 0. It means that the desired spacing has smooth changing at both the beginning and the ending of the merging control process. What's more, it is known that the derivative of a trigonometric function also owns the trigonometric form. Therefore, the multiorder derivatives of des with respect to position are bounded and relatively not large.

State transformation
To employ the Lyapunov stability analysis, this subsection discusses how to perform a transformation to convert the domain of the spacing error from bounded to unbounded.

Controller design
Physically speaking, the square of can be interpreted as a potential energy function [25] between the th vehicle and its preceding vehicle, which approximates to a hazard coefficient for vehicles [20]. In this section, a controller is proposed to maintain the vehicle at the state with a low potential energy. What's more, the boundedness of under the controller means that the spacing error will satisfy the bidirectional constraint in equation (4) so that a hign traffic efficiency and the merging safety are both achieved. This inspires us to construct a controller rendering the uniformly bounded (UB) performance.

Control input
This subsection designs the control input .
With the aim of improving ride comfort, the converging of the transformed state should be gently regulated. An equation constraint for this expected performance is defined, that is, where ℎ > 0 is scalar constant to be prescribed. Equation (18) implies that Equation (19) implies that for any initial ( 0 ), ( ) will converge to 0 as → ∞. The time of ( ) approaching 0 can be modulated by ℎ . Define = ℎ ( ) +̇ (20) to represent the deviation from the equation constraint. By differentiating equation (18) with respect to , we obtain ℎ̇( ) +̈= 0 By using the constraint-following control method [26], is assured to follow this equation constraint. Through Udwadia-Kalaba approach [26], the following control force 1 is introduced.
This control force is model-based. It is interpreted as a feedforward part. Considering the initial condition offset, we design a feedback control part to modify the system to approach the expected equation constraint (18), that is, where is a scalar constant.
Therefore, the overall control force is designed as

Performance analysis
In this subsection, we prove uniformly bounded (UB) performance of the designed controller.
The UB performance is defined as follows.
The proofs are as follows.
First, the Lyapunov function candidate is chosen as Then, the derivative of with respect to is Substitute ̈ into equation (26), we can get The first term containing 1 is 0. Thus, we get where ̇= 0 when and only when = 0. Therefore, we derive the uniform boundedness of the system: The UB performance of is satisfied under any initial condition ( 0 ). That is, if ‖ ( 0 )‖ ≤ , then ‖ ( )‖ ≤ ( ) for all ≥ 0 . The boundedness of ( ) indicates the boundedness of ( ). The proofs are as follows.
Since ‖ ( )‖ ≤ ( ), we have where ( ) is a constant depending on the initial conditions. The differential inequalities are solved, and we obtain which shows that ( ) is also bounded.
According to the state transformation, the boundedness of ( ) implies that if the initial spacing error ( 0 ) satisfies the bidirectional constraint, the subsequent spacing error ( ) will also satisfy the bidirectional constraint. Therefore, compact stable formation and collision avoidance of the virtual platoon are realized, which also means that the safety of on-ramp merging control is realized.

Simulation results
In this section, a numerical simulation is conducted to verify the effectiveness of the proposed method.
Considering six vehicles ( =5), with three on the arterial road and the rest on the on-ramp, after forming a virtual platoon, the 1 st , 3 rd and 5 th vehicles in the virtual platoon are originally from the arterial road, while the 2 nd , 4 th and 6 th are originally from the on-ramp. The initial conditions are listed in Table  1 (The units of position, velocity, mass and vehicle length are m, m/s, kg and m, respectively). The initial positions of adjacent vehicles ( 0 0 and 1 0 , 2 0 and 3 0 , 4 0 and 5 0 ) are the same, which forms a relatively extreme testing conditions. Some parameters are listed in Table 2. Table 1. Initial conditions of 6 vehicles.     Figure 4 demonstrates the velocities. It's evident that the velocity regulation is very smooth, that is, all the velocities maintain at a range within 15~20 m/s. In Figure 5, the accelerations have a relatively sharp change in the beginning and then keep at a stable and low level within -0.5~0.5 m/s 2 . Even though the acceleration increases sharply in the beginning, the maximum acceleration and deceleration are 0.6060 m/s 2 and -1.4042 m/s 2 respectively.
As for the spacing and spacing errors, we can refer to Figures 6~7. The spacings are successfully regulated from initial dangerous values to the final desired spacing. The spacing errors with respect to the spatial-dependent desired spacing are kept at a relatively low level, with a maximum value of 3.0778m. The spacing errors associated with spatial positions are illustrated in Figure 8. It's clear that all spacing errors remains within a safe range during the on-ramp merging process and converges to 0 finally.
The simulation results validate the effectiveness of the proposed method in the on-ramp merging control problem. The spacing errors corresponded with the spatial-dependent desired spacing can be controlled at a relatively low level, thus ensuring the safety of the on-ramp merging process.

Conclusion
This paper proposes a decentralized spatial-dependent collision-free control strategy for on-ramp merging. The designed spatial-dependent controller is proved to be uniformly bounded and assure that each vehicle satisfies the safety requirements to avoid collision at specific spatial locations. The spatialdependence is one kernel point in the proposed strategy. It brings much more stability and reliability