Safety guaranteed longitudinal motion control for connected and autonomous vehicles in a lane-changing scenario

This paper aims at utilising the ideology of controlling a constrained dynamical system to address the longitudinal motion control problem during lane-changing process subject to time-varying uncertainties. To this end, a robust controller is designed based on Udwadia–Kalaba (UK) approach and Lyapunov stability theory. Most of the studies on lane-changing considered only the equality constraint, namely the desired inter-vehicle distance. In a bilateral inequality constraint, the upper bound avoids unpredictable cut-ins and the lower bound collisions. This research applies the constraint to guarantee a safe and efﬁcient lane-changing process. The original UK approach cannot handle bilateral inequalities. Therefore, a diffeomorphism method is proposed to transform the bounded state to an unbounded one. The latter enables the UK approach to deal with both equality and bilateral inequality constraints. The proposed controller can render each vehicle to adjust its distance with the predecessor without violating the prescribed bounds. Numerical experiments have been conducted to validate the effectiveness of the proposed controller under different trafﬁc demands.


INTRODUCTION
With the rapid development of automotive industry, the highway system has been facing challenges. The explosion of automotive industry has posed challenges to highway systems. An overabundance of vehicles causes troubles such as speed breakdown, traffic flow oscillation and congestion [1]. Connected and automated vehicles (CAVs) have great potential in dealing with traffic problems [2,3]. Equipped with sensing and communication apparatuses, CAVs can share information and employ more aggressive control strategies, thus yielding better traffic throughput and road safety [4,5]. Moreover, driving vehicles in a platoon can change the aerodynamic forces on them, thus leads to better fuel economy [6][7][8]. However, a platoon can be flexible because vehicles on other lanes may enter the platoon out of economy or destination considerations. Hence, the lane-changing maneuver is common for CAVs on the highway system.
Recently, a number of researches have been explored lanechanging strategies for CAVs [9][10][11][12][13][14][15][16][17] and Bevly et al. [18] [19] introduced a dynamic adaptive algorithm to guide on-ramp vehicles to merge. The effectiveness of the proposed algorithm was simulated under different traffic demands to optimise the travel time for vehicles on both lanes. Hsu and Liu [20] proposed a platoon-lane-change strategy with different initial spacing, achieving safety and passenger comfort in Smart AHS simulations. Lu et al. [21] proposed a real-time merging algorithm to generate a smooth reference speed trajectory for the merging vehicle based on the speed of the main lane vehicle. In the experiments, the vehicle speeds varied from 20 km/h to 30 km/h. Results showed that the proposed algorithm succeeded to assist the on-ramp vehicle to merge smoothly and safely. Rathgeber et al. [22] proposed two different kinds of trajectory estimation approaches for lane-changing process of CAVs. The two approaches adopt  [27] Motion control Simplified dynamics model 75 Yes (Unilateral inequality) N/A nonlinear vehicle model and simplified vehicle model, respectively. It was concluded that trajectory planning for the driver assistance system was necessary to guarantee safety and comfort. Min et al. [23] proposed a distributed model predictive controller for lane-changing process. The controller considered both safe space and acceleration limitations. The dynamic programming method was adopted to seek for the optimal value of the complex minimisation problem. The desired vehicle speed was set at 20 km/h in the simulation. Michalek and Adouane [24] explored the platoon forming problem that can be viewed as a special version of lane-changing problem. The problem required all vehicles to find and tracking the lane center line while avoid collisions. The authors hence imposed inter-vehicle distance requirements and formulated this problem with kinematics model. Ntousakis et al. [25] presented a longitudinal trajectory planning method to assist the merging on highways.
It was designed to minimise the acceleration to reduce engine effort and passenger discomfort. The constrained optimisation problem was solved by a discrete time quadratic programming method. The simulation was carried out with six vehicles in a predefined sequence. Xu et al. [26] studied different driving styles of human drivers during the lane-changing process and imposed constraints on driver's behaviour. Duret and Wang [27] set constraints on the admissible gap, speed and acceleration, and solved the multi-vehicle coordination problem under model predictive control framework. Although the computing power of modern vehicular equipment has made great progress in the past few years, the solving of the optimisation problem can still be tough when facing complex constraints. The summary shown in Table 1 indicates that the majority of the researches formulate the lane-changing problem as an optimisation one by kinematics model or simplified dynamics model. Yet, the safety of the merging process cannot be guaranteed. Some researches have introduced inequality constraints to further improve the reliability of their strategies, but the complex optimisation problem may place a heavy computational burden [28,29]. This paper aims to overcome the shortcomings of the previous studies. It developed a longitudinal motion control strategy for lane-changing scenarios in CAV environments with three main objectives: (1) assist the merging vehicle to reach a smooth and safe lane-changing process under low to heavy traffic conditions; (2) consider both the nonlin-ear character of the vehicle longitudinal motion and the external disturbance and formulate an uncertain dynamical system control problem to design the lane-changing strategy; (3) impose a bilateral inequality constraint on the motion control problem to guarantee safety in lane-changing process.
When it comes to the control of a dynamical system, Udwadia and Kalaba [30] developed a constraint-following approach, where constraints were integrated in system dynamics in the form of 'constraint force'. They introduced a series of explicit and legible equations to solve the control law of a dynamical system subject to equality constraints. Owing to its simplicity, the Udwadia-Kalaba (UK) approach has been applied to various dynamics control problems. Liu et al. [31] applied the UK approach to constrained under-actuated systems and validated it with the two-mass spring system. Yu et al. [32] adopted the UK approach as the basis for the energy control of the Toda Lattice. Pappalardo and Carmine [33] utilised the UK approach to model the dynamical behaviour of a rigid multi-body system, and they demonstrated the effectiveness of the control scheme. Because it originally is made to cope with equality constraints, although UK approach has been widely applied to constrained [34] motion control, how to deal with bilateral constraints is still challenging. Besides, the real-time uncertainty (possibly time-varying) will bring more difficulty to control the constraint dynamical system. In fact, there was not such a general approach that can deal with the system control problem that involves the equality constraint, the bilateral inequality constraint and the time-varying uncertainty. The current approach fits into this gap.
To summarise, this paper explores the lane-changing strategy for CAVs in the context of longitudinal motion control. Our strategy targets to guarantee safety in the merging process. To this end, we first formulate the nonlinear longitudinal dynamics model for each vehicle considering the possible time-varying uncertainty and obtain the multi-vehicle dynamical system. Then, an equality constraint and a bilateral inequality constraint are incorporated in the dynamical system (hence constrained dynamical system). On one hand, the equality constraint is set to ensure that the error between the current inter-vehicle distance and the desired distance will converge to zero. On the other hand, during the process, the bilateral inequality constraint is adopted to strictly restrict the inter-vehicle distance in a proper range, thus to guarantee safety and traffic efficiency. The UK approach is utilised to render the explicit control force for the dynamical system. Because the original UK approach cannot handle inequality constraints, the diffeomorphism method is used to ease the inequality constraint, transforming the bounded state to an unbounded one. By doing so, the UK approach does work. Finally, Lyapunov stability theory is used to verify the performance of the proposed control strategy. We prove that the proposed control force can render the uniform boundedness and uniform ultimate boundedness performance for the unbounded state. The contributions are (1) using bilateral constraints in lane-changing process to guarantee the proper intervehicle distance to avoid collisions and to maintain sufficient traffic throughput; (2) a generic framework containing diffeomorphism method is proposed to enable UK approach to handle bilateral inequality constraints; and (3) an analytical, closedform control force is proposed for controlling the constrained dynamical system in the presence of uncertainties.
This paper is outlined as follows. Section 2 derives the vehicle longitudinal dynamics model, system uncertainty and constraints. Section 3 introduces the diffeomorphism method. The analytical control force is proposed in Section 4. Section 5 evaluates the effectiveness of the proposed methodology. Section 6 concludes the study and gives an outlook on future research.

MODELLING OF THE LANE-CHANGING PROBLEM
To describe a common setting for the lane-changing problem, we introduce N vehicles made up of various types (e.g. sedan, van and truck), and Figure 1 depicts a single vehicle lane-changing occasion. In Figure 1, the i th vehicle is the merging vehicle and the (i + 1) th vehicle, its follower, needs to slow down to generate a proper gap for the merging vehicle to fit in. This paper investigates the lane-changing strategy in the context of the CAV environment, which means that all vehicles are CAVs and equipped with on-board sensors that measure their state information (e.g. velocity, acceleration and location coordinates). Besides, these vehicles follows the predecessor-follower (PF) vehicle to vehicle (V2V) communication topology [35], that is, the predecessor sends its current state information to its follower. Thus, the inter-vehicle distance are presented from the differences among vehicles' positions. In this section, the lanechanging model is established on the basis of nonlinear vehicle longitudinal dynamics model, constraints on system state and time-varying uncertainties.

Nonlinear vehicle dynamics model
The nonlinear longitudinal vehicle dynamics model for the i th (i = 1, … , N ) vehicle is described bẏ where t is the time, x i is the position based on the geodetic coordinate system. v i is the velocity. u i is the control input that represents the driving or braking force. M i is a scalar constant representing the vehicle mass.c i v i (t )|v i (t )| is the nominal aerodynamic resistance andF i is the nominal road resistance including the rolling resistance and the gradient resistance. The inter-vehicle distance is defined as the interval between the front of one vehicle and the rear of its predecessor. It is denoted as where l i−1 is the length of the vehicle. Then the error of between the desired inter-vehicle distance and the actual inter-vehicle distance, namely spacing error, can be given by where d d is a scalar constant which represents the desired intervehicle distance. Substitute Equation (3) into Equation (1), we have the dynamical model of the spacing error (i.e. the error dynamics model): Rewriting (4) in the form of generalised one-dimension Newtonian and/or Lagrangian mechanics, we have According to [36],ë i (t ) play the role of actual acceleration and the impressed (or applied) force, respectively, in system (5). Furthermore, the acceleration introduced by the Nature can be denoted as Remark 1. The system states (i.e. position, velocity and acceleration) are measured by vehicular sensors. For example, the position of the vehicle can be measured by the product-level carrierphase differential GPS (CDGPS) whose measurement error is within the range of centimeters [37]. The states of a vehicle can be transmitted to its follower through V2V communication, and the communication delay is not considered in this paper.
Remark 2. As mentioned above, the core of multi-vehicle longitudinal coordination is to provide a gap for the merging vehicle once initiate the lane-changing process. Hence, we introduce the error dynamics system (4) which, obviously, is nonlinear due to the resistance forces. It is worth pointing out that a linear system can be considered as a special case of non-linear systems, hence all developed theories be applied to linear systems as well.

Constraints on system state
A large inter-vehicle distance reduces traffic input and may lead to unpredictable cut-ins, while a small one may lead to collisions.
To ensure a safe and efficient lane-changing process, the distance must be appropriate. So the bilateral inequality constraint is denoted as follows where d max and d min are scalar constants representing the upper bound and the lower bound of the inter-vehicle distance.
Apparently, d < 0 and d > 0. Then, the bilateral inequality constraint on the spacing error of i th vehicle can be written as Apart from constraint Equation (8), the spacing error e i (t ) should approach 0 as t → ∞ to make the vehicle to follow the desired inter-vehicle distance (i.e. d d i ). Therefore, the following equality constraint [38] is also imposed to the error dynamics where h i > 0 is a scalar constant. By solving the differential equation Equation (9), we get Equation (10) indicates that for any given non-zero e i (t 0 ), e i (t ) will converge to 0 as t → ∞ if constraint (9) is strictly satisfied. The convergence rate of e i (t ) can be regulated by adjusting h i .
Rewriting (8) and (9) in accordance with Lagrangian mechanics, we have the generalised equation Remark 3. The bilateral inequality constraint maintains a proper distance to avoid collisions and cut-ins. It is required that all initial state, transient state and steady-state of the spacing error satisfy these constraints. Therefore, we usually use the largest initial spacing error among all vehicles as the upper bound of the bilateral inequality constraint.

Parametric uncertainties
Parametric uncertainties are inevitable in real world and they may have great impact on vehicle dynamics behavior especially in high speed conditions. In fact, parameters such asc i and F i cannot be stable. Therefore, this study considers parametric uncertainties, thus the multi-vehicle system is subject to (possible) time-varying parametric uncertainties. The actual parameters c i , F i are divided into the nominal part and the time-varying part. Then where Δc i and ΔF i are the uncertain portions; i ∈ Σ i is the uncertain parameter. Here Σ i ⊂ R p is compact, which stands for the bound of i .c i andF i denote the predefined nominal parameters.We assume that the functionsc i (⋅),F i (⋅), ΔM i (⋅), Δc i (⋅) and ΔF i (⋅) are continuous.
Substituting Equation (12) into Equations (4) and (5), the error dynamics model under parametric uncertainties can be rewritten as:

THE DIFFEOMORPHISM METHOD
This section introduces the motivation of using the diffeomorphism method, and the diffeomorphism method is applied to handle the bilateral inequality constraint.

Motivation
We have obtained the error dynamical model (13), the inequality constraint (8) and the inequality constraint (9) in Section 2.
As mentioned above, the UK approach cannot solve the constrained motion control problem, because the control force brought by the UK approach may cause the spacing error to violate the inequality constraint, that is, violating the safety requirement. To strictly guarantee the range of the spacing error, we intend to ease the bilateral bound on spacing error by transforming the bounded state to an unbounded one. The transformation should follow several basic rules: (1) the variable after transformation should be unbounded so that UK approach can be leveraged; (2) the correspondence between the initial state and the transformed value should be exclusive; (3) the mapping function is two times continuously differentiable.
Considering the above requirements, we apply the diffeomorphism method, through which a continuously differentiable map and its inverse are denoted [39]. The approach involves an invertible mapping function to transform the spacing error e i (t ) and the bilateral inequality constraints (8) into an unbounded variable. By using the inverse of the mapping function, the new variable can replace e i . Hence, an error dynamical system with the unbounded variable is reconstructed. As a result, the inequality constraint is included in the unbounded variable. In this way, the new dynamical system contains only the equality constraint in mathematical form, and the UK approach can be used. In a word, with the diffeomorphism method, the usage of the UK approach is expanded to handle inequality constraints. Accordingly, the control inputs guide vehicles to keep suitable distances without violating predefined boundaries.

Incorporating the bilateral inequality constraints
By a transformation easing the two-side-bound on e i (t ), the diffeomorphism approach is used to relax the inequality constraints (8). That is to say, we should choose a proper function T (⋅) that satisfies the diffeomorphism method, which means T (⋅) should meet the following properties T −1 (⋅)is two times continuously. (14) In this paper, the follow mapping function is chosen for e i (t ) where z i are variables after transformation; q 0,i , q 1,i , q 2,i are coefficients, which can be determined through conditions Equation (14).
Obviously, both T (⋅) and T −1 (⋅) are monotonic. Combining T (0) = 0, it can be concluded that z i → 0 is a necessary and sufficient condition for e i → 0. Figure 2 shows the mapping relationship of Equation (15) between the state e i (t ) and the transformed state z i (t ). Therefore, the one-to-one state transformation can guarantee the spacing error in the desired interval. According to Equation (7), if e i (t ) is bounded, d i (t ) is bounded.
According to Equation (14), e i can be represented by z i as Substituting Equation (16) into Equation (3), we have Differentiating (17) with respect to t yields To simplify the following equations, the partial differential part is denoted as Then, according to Equations (3), (18), (19), the derivative of e i anḋe i can be written aṡ As discussed above, the constrained dynamical system with regard to z i contains only the equality constrain Equation (9). Combining Equations (5), (11) and (15), the dynamical system without uncertainty after transformation can be described as Remark 4. The unbounded state z i is determined by the initial condition z i (t 0 ) anḋz i (t 0 ), the control force u i and the predecessor's velocity and acceleration, that is, v i−1 anḋv i−1 . Equation (22) only provides the expression of system (5) after transformation. The expression of system (13) after transformation can be obtained in a similar way and will be presented in Section 4.2.
Remark 5. Our innovation is twofold when using the diffeomorphism method and the UK method. First, we formulate the merging problem as a constrained control problem. The control force is consequently viewed as the constraint force. Therefore, we adopt the UK method to produce the constraint force and thus exhibits a modest control magnitude in practice. Second, we adopt the diffeomorphism method to ease the inequality bound, so that the UK method can be used. Our method can be called the constraint-based diffeomorphism approach. It is different from the system-based diffeomorphism method used in [40,41] which transforms the system equation and the state variable.

VEHICLE LONGITUDINAL MOTION CONTROL
In this section, control design upon the dynamical system of z i (t ) will be carried out either the parametric uncertainties are present or not, considering both the presence and absence of parametric uncertainties. Below proves the asymptotically stable performance of z i = 0 in the absence of uncertainties, and the uniform boundedness (UB) and uniform ultimate bounded (UUB) performance of z i → 0 in the presence of uncertainties.

Control design without parametric uncertainties
In system (22), the control force will render its acceleration to follow the equality constraint. In Newtonian and Lagrangian mechanics, force (including torque) is the only agent that can change the system's acceleration. As there is no non-ideal constraint involved in the multi-vehicle system in this paper, the ideal constraint force vector is the only consideration. The explicit expression of the constraint force u c i can be rewritten as i ;â i =M −1 iÊ i ; the superscript "+" stands for the Moore-Penrose generalised inverse [38]. Theorem 1. Given system (22), the control force in Equation (24) will render the system state to follow the equality constraint in (22), that is, i is one-dimensional and it is consistent in this particular problem, we have the following property for the Moore-Penrose generalised inverse ofB î Combining Equations (22), (24) and (25), we havê Here, we use a property of MP inverse:B iB According to the properties of the diffeomorphism function T (⋅), the corresponding equality constraint of z i has a similar form of Equation (9) and can be written as where h i is a scalar constant related to the converging rate. We define i = h i z i (t ) +ż i (t ) to represent the deviation from the equality constraint Equation (27). According to Equation (11), we obtain the equality constraint of z i (t ) in the form of Lagrangian mechanics as where P i = 1; Q i = −h iżi .

Lemma 1. System (22) is asymptotically stable under the following control force
where u f i = − i i M i 1 is a proportional feedback control force to steer the system back to the constraint in the presence of initial offset. Here, i is a scalar constant.
Proof. According to Equations (24) and (28), the expansion equation of u c i after transformation can be defined as follows Substituting u c i and u f i into Equation (22), the motion equation of the system can be written as We choose the following Lyapunov function candidate: Using the differential form of V i along with Equation (30), we haveV In view of i > 0, then we havė Therefore, system (22) is asymptotically stable. □

Control design with parametric uncertainties
In this subsection, a robust control part is integrated in the overall control law u i (t ) to render the UB and UUB performance of z i in the presence of time-varying uncertainties.
According to Equations (13) and (21), the expression of the uncertain dynamical system (13) with respect to z i (t ) can be written as Substituting Equation (12) into Equation (35), we obtain the expansion equation of z ï As mentioned above, the overall robust control force contains three parts: the constraint force as in Equation (24), the feedback control to eliminate initial offset as in Equation (29), and the control force to compensate for the uncertainties.
The control law for the nominal part can be denoted as: whereâ i =M −1 iÊ i is the acceleration that corresponds to Equation (6) in the uncertain system. Substituting Equation (13) and (28) into Equation (37), we have the expansion equation of control law u i,1 Similarly, the feedback control force is defined as The control action u i,1 + u i,2 is enough to stabilise the nominal system.
The following assumption is provided to handle the timevarying parametric uncertainties.
where Π i (⋅) is the parameterisation of the worst case effect of the uncertainties. The concrete parametric uncertainties are unknown, while their boundary is restricted by the known function Π(⋅).
We denote the robust control part as follows where i > 0 is a scalar constant.
Therefore, the overall control input for system (35) can be written as (ii) Uniform ultimate boundedness (UUB) : For any r i > 0, with Proof. We chose the following Lyapunov function candidate: Combining (36), the derivative of V i is given by Equation (46). Then we further investigate J k , k = 1, 2, 3, 4 in Equation (46).
Expanding J 1 , we have Substituting Equation (38) into Equation (47), we have According to Assumption 1, we have Using Equation (39), we get Similarly, according to Equations (41), (42) and (43), we have Combining J k (k = 1, 2, 3, 4) with Equations (48)-(51), for Since i ∕4 > 0, then the following equation is satisfied for all Therefore, referring to the standard arguments as in [35], we get the uniform boundedness of the system: where The uniform ultimate boundedness also follows with: where □ Remark 6. Due to the fact that both i and i are scalar constants in Equation (54),V i is negative for sufficiently large | i |. That means the deviation i will reduce once it gets large enough. The boundary of i is obviously decided by i and i and will be further investigated in the following part of this section.
Proof. e i (t ) should be proved to be bounded after z i (t ) being verified bounded. On the basis of the UB performance of i , we have where i (r i ) is a scalar constant under the given initial condition We rearrange Equation (59) as . (60) The corresponding differential equations of these two differential inequalities arė Solving Equations (61) and (62), we have Therefore, solutions of the differential inequality Equation (60) are denoted as Equation (65) manifests that z i (t ) is bounded for all t > t 0 . Furthermore, we can obtain the supremum and the infimum for z i (t ): Since e i (t ) = T −1 (z i (t )) and T (⋅) follows the properties in (14), it is concluded that e i (t ) is bounded for all t > t o . Let Substituting Equations (66)-(68) into Equation (59), we have According to the diffeomorphism mapping function T (⋅) in Equation (15), we havė z i = q 0,i̇ei cos 2 (q 0,i e i + q 1,i ) .
Combining Equations (69) and (70), we have Note that i (r i ), h i , 1 , 2 , q 0,i and q 1,i are constants and e i is bounded. Therefore, e i (t ) = v i (t ) − v i−1 (t ) is bounded, which means that the velocity erroṙe i (t ) is bounded. □ Remark 7. The fact that e i (t ) is bounded for all t ≥ t 0 means that if d i < e i (t 0 ) <d i , then d i < e i (t ) <d i for all t ≥ t 0 . Put another way, the spacing error will be maintained in a proper range during the whole lane-changing process, which is enable to avoid collisions and increase traffic throughput. Furthermore, we prove that the proposed control algorithm renders consensus in velocity through the boundedness of velocity error, as is shown in Equation (71).

NUMERICAL SIMULATIONS
In this section, simulations are carried out to validate the effectiveness of the proposed control algorithm in MATLAB. The progress consists of three cases, covering both low traffic demand condition and heavy demand condition, and two discussions.
In the simulation, the initial speed for highway vehicles is approximately 20 m/s, since vehicles on the main lane tend to running in a platoon. It starts when a vehicle activates the lane-changing process. The types of the vehicles and their relevant parameters are listed in Table 2. The 3 rd vehicle serves as the merging vehicle and the 4 th vehicle the facilitating vehicle. It is assumed that all the vehicles are CAVs and applied with the same distributed robust controller. Situations with different penetration rates of CAVs under mixed traffic conditions will be studied in future research. The uncertainties of the dynamical system are unknown, while their bounds meet Assumption 1. In the following simulations, we assume that Assumption 1 is met with The time-varying uncertainties occur in tire friction coefficient and resistance force. Their values are randomly chosen, and the detailed information is shown in Table 3. Uncertainties of each vehicle is implemented with random initial phase in the following simulations.
The desired inter-vehicle distance is set as 15 m. The inequality constraint for inter-vehicle distance is chosen as [7 m, 18 m], and hence the constraint for spacing error is [−3 m, 8 m] accordingly. The initial inter-vehicle distance for the merging vehicle and the facilitating vehicle are 9 m and 7 m, respectively, while that of the other vehicles are randomly chosen between [−2 m, 2 m]. The control force for the leading vehicle is given byū for Case 1, 2 u 1 − 3000 sin(0.2 (t − 13)), 13 < t ≤ 23 for Case 3 . (74)

Case study 1: Coordination of 10 vehicles without uncertainties
In this case, the proposed control law is enforced upon the system without parametric uncertainties. The purpose of the case is to validate the feasibility of both safety and traffic throughput during the lane-changing process. As is shown in Figure 3, the lane-changing maneuver has caused dense traffic flow at the beginning. As a result, vehicles, especially the facilitating vehicles, on the main lane have to adjust their speed to avoid collisions. The proposed control strategy can successfully convert the original densely packed vehicles to loosely packed vehicles.
To further investigate the adjustment process, Figure 4 presents the spacing error. The relative large initial spacing error of both merging vehicle and facilitating vehicle can be eliminated through the proposed control. Moreover, the spacing error of all vehicles shrinks within 2 m after approximately 8 s, which verifies the safety and efficiency of the merging process. The velocity error is defined as the speed difference between one vehicle and its predecessor. The velocity error for each vehicle is illustrated in Figure 5. It is evident that the speed error converges to zero after 20 s. The control force is shown in Figure 6. Except for the initial adjustment stage, the control force is smooth and satisfies the power limitation of vehicles.

Case study 2: Coordination of 10 vehicles with uncertainties under low traffic flow
Same as Case 1, the simulation is carried out under low traffic demand. That is to say, the speed of the main lane traffic, represented by the speed of vehicle 1, is stable in the process.  Different from that of Case 1, fluctuations appears in the curves of both spacing error, velocity error and control force as shown in Figures 8, 9 and 10. It is more obvious in Fig-FIGURE 9 The velocity error history of 10 vehicles with uncertainty under low traffic flow The speed profile of the heavy traffic flow ure 9 (see the partial enlarged figures). These fluctuations are reasonable because they are triggered by parametric uncertainties. Our greatest concern is that these fluctuations will not amplify through the propagation of traffic flow. According to Figure 8 and Figure 9, the spacing error and velocity error do not amplify though uncertainties exist. It proves that the proposed method renders the uniform boundedness (UB) and uniform ultimate bounded (UUB) performance of the constrained uncertain dynamical system.

Case study 3: Coordination of 10 vehicles with uncertainties under heavy traffic flow
The simulations in Case 3 contains uncertainties. The background is switched to the heavy traffic flow condition, which is usually accompanied by severe speed variation situations. Hence we simulate this kind of situation by imposing a different speed profile on the 1 st vehicle, which is given in Equation (74). Figure 11 illustrates the speed variation vividly. The emulation is rational, because a slight change in vehicle speed form upstream is likely to be amplified when propagating downstream under heavy traffic demand.
As shown in Figure 12, a bend occurs in the position profile due to speed variation. Meanwhile, all vehicles can still modify their clearance under the robust control law. Figure 13 depicts  Figure 14. Corresponding control force is shown in Figure 15. Combining these facts, verdict can be made that the merging process remains safe under high traffic flow. To summarise, the proposed control method fulfills a safe, steady and efficient lane-changing process.

Discussion 1: Comparisons of the performance in guaranteeing safety
To further illustrate the capability that the proposed control algorithm (namely UK-Diff) can strictly guarantee the merging safety, we adopt the LQR control algorithm and the UK approach without inequality constraints (namely UK) as contrast. The initial inter-vehicle distance of the merging vehicle is 9m and that of the facilitating vehicle is 7 m. As mentioned above, the lower bound is set at the minimum initial value of the inter-vehicle distance, that is, 7 m. The results are shown in Figure 16 and the merging vehicle and the facilitating vehicle are named, respectively, as MV and FV for short. As is shown in the partial enlarged figure in Figure 16, the inter-vehicle distance of FV under LQR and UK method will shrink in the beginning and exceed the lower bound, which violates the inequality constraint. The inter-vehicle distances under UK-Diff method, however, range strictly within the prescribed bound, that is, 7-18 m. Besides, the UK-Diff method achieves a gentler control process compared with the LQR method. Besides, the performance under LQR method is greatly affected by the timevarying uncertainties and has an obvious overshoot. To summarise, the proposed control algorithm guarantees the merging safety and achieves better performance than the LQR and the UK method.

Discussion 2: Control performance when multiple vehicles merging simultaneously
In this subsection, we explore the effectiveness of the proposed lane-changing strategy when multiple vehicles are merging simultaneously. A total of 16 vehicles are used, of which the 3 rd , 4 th , 9 th and 13 th vehicles are merging vehicles. We can have a global perspective through Figure 17, which validates that the proposed strategy can successfully guide all vehicle to reach a smooth and safe merging process. From Figure 18, it is concluded that all merging vehicles and facilitating vehicles can eliminate their initial spacing error. Additionally, the control force can limit the spacing error in a small region, which verifies the UB and UUB performance of the proposed control

FIGURE 17
The position history when multiple vehicles merge simultaneously method. Moreover, the velocity error shown in Figure 19 indicates the consensus with respect to velocity as we have proved in Lemma 2. Finally, the overall control force history is presented in Figure 20. Therefore, the proposed lane-changing strategy is capable of handling multiple vehicles merging situation. Also, its potential is worth exploited in other vehicle coordination situations, such as on-ramp driving.

FIGURE 18
The spacing error history when multiple vehicles merge simultaneously

FIGURE 19
The velocity error history when multiple vehicles merge simultaneously

FIGURE 20
The control force history when multiple vehicles merge simultaneously

CONCLUSION
This paper has proposed a safety guaranteed control method for lane-changing maneuver aiming at avoiding accidents while improving system-wide traffic throughput. In order to maintain the reasonable inter-vehicle distances, we introduce the bilateral inequality constraint, whose the upper bound ensures sufficient traffic throughput while the lower bound guarantees the merging safety. Since the Udwadia-Kalaba approach cannot handle inequality constraints directly, the diffeomorphism method is applied to transfer the bounded spacing error to an unbounded variable. The UK approach can thus be used to generate the control force. What is more, parametric uncertainties are considered in the dynamic system. An analytical, closed-form robust control law is formulated for the uncertain dynamic system. Simulations have been conducted, including comparisons on control performance with and without uncertainties. The results indicate that the proposed control law renders asymptotically stable performance for the certain system and the UB and UUB performance for the uncertain system. Furthermore, the intervehicle distances can be confined within the specified range, which guarantees a safety lane-changing process under both the high traffic demand and the low traffic demand. Future works will consider a more sophisticated vehicle dynamics model, including constraints on spacing error against current position to strictly guarantee the safety of the lane-changing process. In addition, the implementation of such approach is worth exploring.