Traditional vehicle system modelling relies on geodetic coordinate systems, body coordinate systems and polar coordinates. As the motion trajectory of the self-driving vehicle grows, the trajectory becomes more difficult to fit and the cumulative error of the calculation increases. The Frenet coordinate system has excellent decoupling ability, which can plan the vehicle’s longitudinal and transverse motions independently and separately [24]. It can drastically improve the performance of decision models.

The polynomial-based lane change trajectory computation requires only the initial and target states of the vehicle. In the lane-changing scenario, there are FV (Front Vehicle) and RV (Rear Vehicle) in front and rear of the target lane, PV (Preceding Vehicle) in front of the current lane, and LV (Lane-changing Vehicle) in the current lane. Set H_fs as the safe lane-changing distance between LV and FV, H_fs = v·1.5·7/8; set H_rs as the safe lane-changing distance between LV and RV, H_rs = v·1.5·5/8. v is the vehicle speed, and the time interval between vehicles is 1.5 s. When the actual distances between LV, FV and RV are greater than the safe lane-changing distances, lane-changing planning is performed. The vehicle state at this time is the initial state. The target state is initially selected considering the predicted trajectory of the other vehicles in the target lane. The target state is finally adjusted by the safety constraints described below. As shown in Fig. 2, the boundary conditions in a complete lane-changing process can be expressed in terms of vertical and horizontal constraints as follows:(1){S(0)=s0,S˙(0)=vs0,S¨(0)=as0,d(0)=d0,d˙(0)=vd0,d¨(0)=ad0S(τ)=sτ,S˙(τ)=vsτ,S¨(τ)=asτ,d(τ)=dτ,d˙(τ)=vdτ,d¨(τ)=adτwhere S and d are the longitudinal and lateral distances of the vehicle along the reference path in the Frenet coordinate system, respectively. τ is the lane-changing duration.

To facilitate solving for the curvature of the planned trajectory, the lateral boundary conditions are expressed by the geodetic to Frenet coordinate system transformation formula as [25]: (2)d(s0)=d0,d˙(s0)=vd0vs0,d¨(s0)=ad0−d˙(s0)⋅as0vs0,d(sτ)=dτ,d˙(sτ)=vdτvsτ,d¨(sτ)=adτ−d˙(sτ)⋅asτvsτ

The fifth-degree polynomial plan obtained by re-coupling the vertical and horizontal boundary conditions is: (3){S(t)=a0+a1⋅t+a2⋅t2+a3⋅t3+a4⋅t4+a5⋅t5d(s)=b0+b1⋅s+b2⋅s2+b3⋅s3+b4⋅s4+b5⋅s5where S(t) and d(t) are the coordinates of the vehicle mass center at time t in the Frenet coordinate system. a₀, …, a₅ and b₀, …, b₅ are the polynomial coefficients.

θ_h is the heading angle of the vehicle on the planning trajectory at any moment during the planning period. k_h is the curvature of the vehicle on the planning trajectory at any moment during the planning period, which can be represented by Fig. 2 as: (4)tan⁡(θh)=limt→0d(n+t)−d(n)S(n+t)−S(n)=d(d(S))dS (5)kh=d′′(S)(1+(d′(S))2)1.5

As shown in Fig. 3, different alternative target points can be taken before and after the target point (S_τ, d_τ) to get the alternative trajectory set in trajectory planning.

In order to clearly describe the motion law of the vehicle, as shown in Fig. 4, a two-degree-of-freedom vehicle dynamics model is established as follows [26]:(6)[y¨φ¨]=[Cαf+CαrmvxaCαf−bCαrmvx−vxaCαf−bCαrIvxa2Cαf+b2CαrIvx][y˙φ˙]+[−CαfmCαfI]δwhere v_x is the component of v in the x-direction, C_αf and C_αr are the lateral stiffnesses of the front and rear wheels, a and b are the distances of the vehicle’s center of gravity from the front and rear axles, I is the rotational moment of inertia of the whole vehicle, and m is the vehicle mass. The vehicle parameters are in Table 1. As shown in Fig. 4, α_f and α_r are the front and rear wheel lateral deflection angles, F_yf and F_yr are the front and rear axle lateral forces, δ is the turning angle of the driving wheels, φ is the transverse pendulum angle, and β is the lateral deflection angle of the center of mass. It is usually assumed that cos(δ) = 1.

Once again, through the coordinate transformation, the vehicle system dynamics in the Frenet coordinate system can be obtained as follows: (7)e˙r=A⋅er+B⋅δ+C⋅θ˙rwhere er=[ede˙deφe˙φ]′,u=δ,ed=d,eφ=φ−θr, A=[01000Cαf+Cαrmvx−Cαf+CαrmaCαf−bCαrmvx00010aCαf−bCαrIvx−aCαf−bCαrIa2Cαf+b2CαrIvx],B=[0−Cαfm0−a⋅CαfI],C=[0aCαf−bCαrmvx−vx0a2Cαf+b2CαrIvx].e_r is the new state variable, u is the control vector, e_d is the lateral error, e_φ the heading error, and θ_r is the angle between the reference heading angle at the vehicle projection point and the x-axis of the Cartesian coordinate system.

Game theory can be classified into non-cooperative and cooperative games based on the ability to form binding agreements between participants. Because vehicles travelling on non-smart networked roads cannot share the driving strategies of other vehicles, the lane-changing game discussed in this paper is a non-cooperative game. During the game, each participant prioritizes their payoffs. In such Prisoner’s Dilemma situations, the Nash equilibrium is a suboptimal solution acceptable to all participants [27]. During lane-changing, vehicles are concerned with metrics such as safety, comfort, and speed payoffs [28]. The payoffs from adopting different strategies vary, so a payoff function needs to be designed to include the set of strategies among all participants [29].

The LV’s lane change strategy is divided into two types, lane change and keep moving, which are set to C and N, respectively. The corresponding probabilities are l₁ and l₂ (l₁ + l₂ = 1), respectively. FV and RV have both incentive and disincentive strategies for lane-changing. Their corresponding probabilities are f₁, f₂ (f₁ + f₂ = 1), r₁, and r₂ (r₁ + r₂ = 1), respectively. When LV makes different strategies, the payoff matrix is shown in Table 2.

When there is only FV or RV in the target lane, it is a two-party game. When both cars exist, to improve driving safety and traffic efficiency, this model assumes that the two vehicles maintain goal consistency. It can still be regarded as a two-party game, which simplifies the complexity of the game and also considers the impact of FV and RV on lane changing. Table 2 shows the two-party game payoff matrix. The following equation gives the payoffs for the three vehicles with different strategies: (8)[LijFijRij]=α[EsLijEsFijEsRij]+β[EvLijEvFijEvRij]+γ[EaLijEaFijEaRij]where L_ij, F_ij, and R_ij denote their payoffs. EsL_ij, EvL_ij, and EaL_ij denote LV’s safety, speed and comfort payoff, respectively. FV and RV are similar to LV, α, β, and γ are regulation factors.

The positions of the vehicles participating in the game before and after the lane-changing are shown in Fig. 5. Taking LV as an example, L₀ and L_τ denote the position of LV before and after the change of channel.

Es is the safety payoff and is understood as the safety space expansion factor for the corresponding vehicle before and after lane-changing. Similarly, Ev is the speed payoff and is embodied as the spatial potential for acceleration of the corresponding vehicle before and after the adoption of the corresponding strategy. Es and Ev are defined as: (9){EsL=min(|Lτ−Rτ|,|Lτ−Fτ|,|Lτ−Pτ|)min(|L0−R0|,|L0−F0|,|L0−P0|)EsF=min(|Fτ−Rτ|,|Fτ−Lτ|)min(|F0−R0|,|F0−L0|)EsR=min(|Rτ−Fτ|,|Rτ−Lτ|)min(|R0−F0|,|R0−L0|) (10){EvL=|Fτ−Lτ||P0−L0|EvF=EvL+EvR2EvR=|Lτ−Rτ||F0−R0|where v_L and v_R are the LV and RV vehicle speeds, respectively. Because the speed payoff for FV is not probable, the average speed payoff for LV and RV is assigned to EvF. Ea is the comfort payoff, which expresses the change in acceleration of the vehicle when the strategy is taken and is defined as follows: (11){EaL=1∫0Td⃛(t)2dt+∫0TS⃛(t)2dtEaF=0EaR=0

The overall payoffs before and after changing lanes are focused on LV, the size of which is determined by EsL and EvL. When EsL = 1 and EvL = 1, there are no payoffs to safety space and acceleration space potential, regardless of whether the vehicle changes lanes. Thus, the lane-changing strategy probability is obtained by ab normalization. (12){l1={0,(EsL⋅EvL≤lD)EsL⋅EvL−lDlU−lD,(lD<EsL⋅EvL<lU)1,(lU≤EsL⋅EvL)l2=1−l1where l_U > 1, 0 < l_D < 1. l_U and l_D are the maximum and minimum values set during the normalization process. LV’s lane-changing aggressiveness is determined by its values; the smaller all values are, the more this vehicle tends to change lanes in the game.

The respective strategy probability for FV and RV is calculated similarly for LV. The relative position change between vehicles measures the strategies adopted by competing vehicles. At the initial stage of lane change the changes in the position, velocity, and acceleration of the LV are small and complex. The PV will not actively respond to the behavior of the LV, and the stability of the PV can be obtained through the onboard sensors. Using PV instead of LV as the calculation anchor can also express relative distance changes. And enlarge the calculation target and improve the calculation efficiency.

As shown in Fig. 6, the position of each vehicle change in a short period after the LV gives the intention to change lane is used to calculate the strategy probabilities of FV.(13){f1={0,(Ff≤fD)Ff−fDfU−fD,(fD<Ff<fU)1,(fU≤Ff)f2=1−f1Ff=EsF⋅EvF⋅|Pt−Rt||P0−R0|where f_U and f_D are the extremes set during the normalization process. Strategy probability for RV is calculated in the same way. Their value determines the tolerance of competing vehicles for lane-changing aggressiveness; the closer the upper and lower limits are, the more sensitive the aggressiveness judgements of competing vehicles becomes.

The mixed expected payoffs under each strategy combination are: (14){EL(l,f,r)=∑i=12li[(∑j=i2fj⋅Lij)+(∑j=i2rj⋅Lij)]EF(l,f,r)=∑i=12li(∑j=i2fj⋅Lij)ER(l,f,r)=∑i=12li(∑j=i2rj⋅Rij)where E_L, E_F, and E_R denote their mixed expected payoffs. When the set of strategies is finite, there must exist an optimal solution for the game model, where all participants can achieve maximum returns when adopting these strategies. The maximum value of each participant’s expected payoff can be computed by Eq. (15). By letting the mixed probability of maximum payoff be (l*, f*, r*), the following equation can be obtained: (15){EL(l∗,f∗,r∗)≥EL(l,f∗,r∗)EF(l∗,f∗,r∗)≥EF(l∗,f,r∗)ER(l∗,f∗,r∗)≥ER(l∗,f∗,r)

The resulting l*, f*, and r* are the Nash equilibrium solutions for the three vehicles in the game’s current state. Focusing only on LV’s payoff during lane-changing, the maximum payoff table for all optimal strategies in the strategy set of LV can be obtained. The maximum payoff determines the lane-changing strategy and optimal trajectory.

There are three types of game participants in this article: (1) only FV (2) only RV, (3) both FV and RV. Trajectory planning from initial to target state, with the subject vehicle speed of 60 km/h and the curvature of 1/300 m⁻¹ on the bend. Difine the individual payoff weights and aggressiveness prediction parameters, l_U = 1.2, l_D = 0.8, f_U = 1.3, f_D = 0.8, r_U = 1.3, r_D = 0.7. (1)

The first scenario is ‘on FV’. In this scenario, there is FV ahead of the target lane, indicating that the presence of acceleration space is crucial for lane-changing behaviour. The payoff of safety, speed, and comfort are respectively: α = 0.3, β = 0.5, γ = 0.2. The payoff matrix for this vehicle can be obtained by performing a blended expected payoff calculation from the payoff matrices of all strategies, as shown in Tables 3 and 4. The optimal trajectory is the alternative trajectory #4, with the largest payoff during the lane-changing game. LV has the best safety, comfort, and acceleration potential in this condition. Fig. 7 shows the performance of the optimal trajectory in different coordinate systems. In curved road scenarios, the Frenet coordinate system can simplify complex path problems. In the initial stage, the Cartesian coordinate system is converted to the Frenet coordinate system, and the conversion error is taken into account. It is particularly suitable for the set highway scene and can achieve higher trajectory tracking accuracy. The Cartesian coordinate system more clearly shows the actual planned trajectory and driving trajectory. It is more intuitive and easier to determine whether a collision occurs.

To evaluate whether the optimal trajectory is applicable to real traffic, the NGSIM data is used for validation. NGSIM is a general tool that collects video to capture real traffic information and verify the effectiveness of trajectory planning. The data used mainly includes road data and status data of related vehicles. A series of vehicles with lane-changing behaviors are selected as target vehicles in NGSIM. The evaluation is performed by calculating the overlap rate and similarity between the actual trajectory and the planned trajectory.

Fig. 10 shows a representative trajectory planning diagram. The red curve is an actual left lane change trajectory in the NGSIM dataset. Select the target vehicle, surrounding vehicles, and road status information when the lane change starts. The blue curve represents the trajectory planned by substituting this information into the model. Find overlapping points by computing the closest point match between two trajectory point sets. The distance threshold is set to 0.3 m. Points less than this threshold are considered to be overlapping points. The overlap rate of the two trajectories shown in Fig. 10 is 88.7%. The Euclidean distance between two trajectory points is calculated, and the root mean square error (RMSE) is used to quantify the difference between the trajectories. The smaller the RMSE, the more similar the trajectories are. The RMSE of the planned trajectory shown in Fig. 10 is 0.1371. Trajectories with overlap greater than 80% and RMSE less than 0.2 were considered usable plans. Table 7 shows that there are 110 trajectories available for this model, and the lane-changing accuracy is 82.7%.

Since the actual system is discrete, Eq. (7) needs to be converted to a discrete-time model. Defining the sampling time as T_sa and the time step as k, the discrete-time model is expressed as: (16)e(k+1)=A¯⋅e(k)+B¯⋅u(k)where A¯=A⋅Tsa+I,B¯=B⋅Tsa, I is the unit matrix. Define the cost function: (17)JL=∑k=0+∞(e(k)T⋅Q⋅e(k)+u(k)T⋅R⋅u(k))where Q and R are the state weighting matrix and control weighting matrix, which influence the rate at which deviations converge to zero and the magnitude of the control inputs.

For closed-loop control, construct the Hamiltonian function. Multiplying the front of the constraint equation by the covariance matrix λ yields the Hamilton function as follows: (18)H(k)=e(k)T⋅Q⋅e(k)+u(k)T⋅R⋅u(k)+λ(k+1)T⋅(A¯⋅e(k)+B¯⋅u(k))

Performance metrics rewritten according to system constraints and design control laws. (19)JL=∑k=0n−1(H(k)−λ(k+1)T⋅e(k+1))+e(k)T⋅Q⋅e(k) (20)u(k)=−R−1⋅B¯Tsa⋅λ(k+1)

Construct the Riccati differential equation, where P(k) is a semi-positive definite symmetric matrix, λ(k)=P(k)⋅e(k). (21)PA+ATP−PBR−1BPT+Q=0

The control law and final state feedback control are calculated as follows: (22)u(k)=−R−1⋅BT⋅P(k)⋅e(k) (23)UL=−K⋅erwhere K is a 1 × 4 matrix, denoting the gain of the LQR controller.

Solving the Riccati differential equation offline improves the speed of solving the system controller. Create a look-up table of feedback gain coefficients for different vehicle speeds. This offline look-up table approach improves the system computation speed and enhances the real-time performance of the control system.

On this basis of Eq. (17), the equations for predicting the future system output can be derived. In order to unify e(k) and u(k − 1), let ξ(k) be the new state variable and η(k) be the system output η(k), which leads to the state space expression: (24){ξ(k+1)=A~⋅ξ(k)+B~⋅Δu(k)η(k)=C~⋅ξ(k)where ξ(k)=[e(k)u(k−1)],Δu(k)=u(k)−u(k−1),A~=[A¯B¯0I],B~=[B¯I],C~=[I0]

H_p is the system prediction time domain, and H_c is the control time domain, where k = H_c, H_c + 1, …, H_p − 1. The prediction model is obtained by iteration. (25)Y(k+1|k)=Φ⋅ξ(k)+Θ⋅ΔU(k)

Similar to Eq. (20), the cost function in the prediction process is defined as: (26)JM=Y(k)T⋅QQ⋅Y(k)+ΔU(k)T⋅RR⋅ΔU(k)

Q_Q and R_R are the state weighting matrix and control weighting matrix. The following are essential regarding vehicle driving effects: steering angle constraints, steering angle increment constraints, and acceleration constraints. Considering the need to improve the smoothness of the vehicle travelling and to fit the desired path, the amount of control and the control increments are also constrained. (27){umin(k+i)≤u(k+i)≤umax(k+i),i=0,…,Hc−1Δumin(k+i)≤Δu(k+i)≤umax(k+i),i=0,…,Hc−1

On this basis, the quadratic objective function can be obtained as follows [30]:(28)JM=ΔU(k)T⋅Ht⋅ΔU(k)+Gt⋅ΔU(k)where Ht=ΘT⋅QQ⋅Θ+RR, Gt=2(Φ⋅ξk)T⋅QQ⋅Θ.

The optimal sequence of ΔU(k) is obtained by solving for the minimum value of J_M under constraints Eq. (27). The control input for time step k is obtained by iteration:(29)u(k|k)=u(k−1|k)+[10⋯0]δU∗(k)where ΔU*(k) denotes the optimal sequence of ΔU(k).

Experimental validation was carried out under straight road single shift line conditions, as shown in Fig. 11. In straight-line scenarios, the Cartesian coordinate system is used directly for planning and control. The coordinate conversion step can be omitted, which reduces errors and optimizes calculation speed. Where the speed of the subject vehicle is 100 km/h, the road friction coefficient is 0.75, the lane width is 3.6 m, and the sampling time is 0.01 s. Fig. 11a shows the lateral displacement of the subject vehicle relative to the road edge line Y. Fig. 11b–d illustrates the yaw angle, front wheel angle of rotation, and lateral acceleration of the subject vehicle, respectively.

In detail, the black line represents the planning trajectory, the blue line represents the simulation of LQR tracking, and the red line represents the simulation of MPC tracking. The lateral displacement shows that the LQR and MPC controllers do well in tracking the planned trajectory. Compared to the LQR control, the vehicle lateral error under MPC control is greater, up to a maximum of 0.2 m. The vehicle sway angle shows that the vehicle controlled by the MPC controller has a smaller peak sway angle and better stability during lane-changing. The front wheel angle of the vehicle shows that the vehicle starts to change lanes at 20 m and completes the change at around 85 m. In this case, the front wheel angle of the vehicle controlled by the MPC controller is smoother, with a sudden change in the initial phase of lane-changing and a maximum value of no more than 1° during the process. In contrast, the control of the front wheel angle by the LQR controller suffers from the jitter phenomenon. The lateral acceleration curve shows that both the LQR and MPC controllers have lateral acceleration within 0.3 g during vehicle lane-changing. Moreover, the lateral acceleration profile of the vehicle controlled by the MPC controller is smoother, indicating good comfort.

Similar to the straight road, the vehicle state response curve under single shift condition of constant curvature curved road can be obtained in Fig. 12. The radius of curvature of the road is 300 m.

Like state response curve for a curved road would result in proportional compression if the identical X is taken as the transverse axis, so the longitudinal displacement S in the Frenet coordinate system is taken as the transverse axis in describing the vehicle’s transverse swing angle, front wheel angle of rotation, and transverse acceleration.

The tracking effect of curved roads is similar to that of straight roads. Fig. 12b–d shows that the vehicle starts to change lanes at 40 m and completes the change at around 140 m, in which the front wheel angle of the vehicle controlled by the MPC controller is stable in the interval of 1°–2° change during the initial phase of the lane-changing. The LQR controller’s front wheel angle judder is still present and severe. The lateral acceleration change is slightly larger than the straight line but still within control. Furthermore, MPC control is still better than LQR control. Therefore, a dual controller trajectory tracking approach is used, with the MPC controller for the lane-changing process and the LQR controller for the road-holding process. It can effectively increase the speed of calculation.

In this subsection, the effectiveness of the lane-changing strategy is further validated through hardware-in-the-loop driving simulator experiments. The hardware-in-the-loop driving simulator shown in Fig. 13 consists of a model Lenovo ThinkCentre M910t-D158 host computer, a model HP Compaq DC7700p CMT implementation target machine, five joint flat-screen displays, sensors, motors, and acquisition cards. The real-time system is built with LabVIEW Real-Time module as the software core.

In the experiment, the driving route was set up as an S-shaped road with straight and left and right curves. In order to validate the effectiveness of the driving scheme, game lane-changing, adaptive cruise, lane-changing abort, and direct lane-changing are implemented continuously for four working conditions.

Fig. 14 shows that the black line is the expected steering wheel angle, and the red line is the actual steering wheel angle. The steering wheel angle during lane-changing can achieve a better tracking effect. In conjunction with the in-software timer, at 6.392 s, the following threshold is calculated to be zero, the lane-changing speed threshold to be one, and the lane-changing distance threshold to be zero. The game lane change is performed, as shown in Fig. 14a, and the lane change is performed at 6.392~9.557 s.

At 9.557 to 23.712 s, the calculation of the following and lane-changing speed threshold is zero. In the process, the subject vehicle enters the curve and gets closer to the vehicle in front of it, but there is no opportunity to change lanes. Adaptive cruise is maintained to follow the vehicle in front based on driving behaviour decisions. The actual and desired steering wheel angle is within 1°, as shown in Fig. 14b.

At 23.712 s, by calculating the thresholds, this vehicle again generates the intention to change lanes and starts a two-sided game with the vehicle behind the target lane. 24.6–25.8 s, the speed of the vehicle behind the target lane increases from 90 to 110 km/h. At 25.843 s, based on the calculation of the abort threshold of 1, the lane-changing abort command is triggered, and the vehicle immediately returns to the original lane. The steering wheel angle deviation is not more than 1.5°, as shown in Fig. 14c. The actual tracking effect is satisfactory for sudden changes in steering wheel angle within 0.5 s.

At 29.829 s, by calculating the lane-changing distance threshold, which is zero, indicating that there are no gaming vehicles, this vehicle adopts a direct lane-changing strategy. As shown in Fig. 14d, the two curves coincide, and the steering wheel angle changes smoothly during the lane-changing.

The simulations were run on a desktop computer with an Intel Core i7-12700KF (3.6 GHz), 32.0 GB RAM, Microsoft Windows 10 (64-bit), and MATLAB 2016b (64-bit). The key indicators are shown in Table 8. Lane-changing conditions are divided into the straight road and the curved road. The maximum safety distance meets the safe driving conditions. The maximum acceleration on curved roads is less than 0.52°, ensuring driving stability. The average error is within the controllable range of 0.5 m. The total computation time for the two cases including planning and control is 0.642 and 0.813 s, respectively.

The LQP controller has a small amount of calculation while the MPC controller has a higher calculation complexity. In most cases, the LQR controller provides good performance in real-time systems. The MPC controllers require stronger computing resources. Adaptive optimization algorithms and machine learning-assisted optimization controller design and parameter selection are future research directions.