Safe trajectory analysis and validation for multi-axle vehicles considering the stable constraints

Abstract

Vehicle safety and stability in various driving conditions is the key issue for multi-axle heavy vehicles on long haul routes. Due to the characteristics of high center-of-gravity height and high inertia of multi-axle vehicles, these vehicles are more prone to instability and rollover during high-speed maneuvers, which affects the driving safety. Aiming at the problem of safe trajectory planning in the obstacle avoidance process under the adaptive cruise driving situation, this paper considers the obstacle avoidance stability of the vehicle while driving and creates a safe driving trajectory for the five-axle vehicle with automatic driving function. First, the vehicle dynamics model is established in this paper. Then, considering the current and future motion state of the multi-axle vehicle and its surrounding vehicles, the vehicle collision risk model and the obstacle avoidance stability model are established. Finally, a local path-speed collaborative planning method based on the model predictive control (MPC) method is developed, to generate a safe and stable driving trajectory of the multi-axle vehicle. The simulations and experiments show that, compared to the traditional planning method, the yaw and roll motion of the vehicle can be reduced about 28.5% and 25.7% when driving along the trajectory planned by the proposed method, and the risk of driving collision can be reduced effectively.

Introduction

Multi-axle vehicles are widely used in transportation due to their heavy load bearing characteristics and wide area mobility. Compared with two axle vehicles, multi-axle vehicles have a higher center of gravity height and larger mass, which makes them more prone to instability and rollover when driving at high speeds. Traditionally, the safety and stability control of multi-axle vehicles mainly depends on the experience of the driver and the vehicle motion controllers. By designing a vehicle motion controller and tracking the desired state of the vehicle, it is possible to achieve yaw stability and prevent rollover to a certain extent.^1–3 However, due to the complex driving environment, the actuators of the vehicle with large inertia often can hardly follow the expected targets, which can make the vehicle unstable.

To overcome the limitations described above, autonomous vehicles achieve safe driving by adopting the perception, planning and control methods and technologies. ⁴ Through the perception of the external environment, the safe driving trajectory can be output in real time by the trajectory planner and tracked by the motion controller. This method makes up for the shortcomings of traditional driving methods. However, as the key link of perception, planning and control, the trajectory planning directly determines the stability and driving safety of the vehicle.

In general, there are three types of methods to achieve trajectory planning, namely, sampling-based, search-based, and optimization-based methods. Sampling-based planning methods generate a feasible trajectory by randomly sampling around the surrounding environment considering obstacle constraints. Representative algorithms mainly include the rapidly exploring random trees (RRT) and the probabilistic road map (PRM).^5–7 The search-based planning method is mainly based on graph search, which generates feasible driving trajectories by rasterizing the surrounding environment information and generating feasible grid paths. Representative algorithms mainly include the A* and D* algorithms.^8–10 However, to obtain the feasible trajectory, the above methods need to discretize the search space. For this reason, the curvature of the driving trajectory will become discontinuous, and it will be difficult to obtain the optimized driving path using these approaches. Therefore, trajectory optimization should be done separately, which reduces the efficiency of the algorithm. To overcome this shortcoming, the optimization-based trajectory generation method could be used, which describes the surrounding environment through the parametric model and then constructs the vehicle evaluation index to generate the trajectory with the optimal comprehensive performance. For this reason, this method has attracted widespread attention.

In this regard, Huang et al. ¹¹ constructed a resistance network and then generated a feasible trajectory without changing the speed. Then Huang et al. ¹² designed a lower motion planner based on longitudinal MPC and lateral MPC to realize adaptive cruise control and lane changing functions. To improve the smoothness of the planned trajectory, Huang et al. ¹³ in another work used the sinusoidal resistance network to mesh the road and reduced the jerkiness. These trajectory generation methods are all based on the artificial potential field, which reduces the real-time performance of the algorithm in the complex road environment. To improve the efficiency of the algorithm, Wang et al. ¹⁴ designed an optimal solution objective based on convex quadratic programing to achieve real-time collision-free vehicle motion planning. Zhang et al. ¹⁵ presented a new collision avoidance method for cases where obstacles and controlled objects could be represented as convex sets. Pek and Althoff ¹⁶ proposed a fail-safe trajectory generation method to ensure the driving safety of autonomous vehicles in complex environments. Eiras et al. ¹⁷ used a two-stage optimization method in motion planning to solve non-convex problems. The mixed-integer linear programing was used in the first stage to obtain an approximate solution, and nonlinear programing was used in the second stage to obtain the exact solution. In addition, Cichella et al. ¹⁸ generated safe and feasible motion trajectories based on Bernstein polynomials for multi-vehicle motion planning problems. However, the reliability of these methods depends on accurate sensor detection and prediction results of the surrounding environment, which is often unrealistic in practical applications. To cope this problem, Artunedo et al. ¹⁹ considered localization uncertainty, and proposed a motion planning strategy based on the probabilistic occupancy grid. Feng et al. ²⁰ considered the uncertain pedestrian movement as well as spatial and temporal sequences, then proposed a candidate trajectory planning method.

The above optimal trajectories were always generated based on the vehicle kinematics model. In practice, trajectory often contains both path and speed, so the dynamic characteristics of the vehicle should also be considered. For example, Hu et al. ²¹ considered the safety, comfort, and dynamic performance of the vehicle during planning, and generated a trajectory with the best local comprehensive performance in real-time. Besides, considering that the optimal vehicle trajectory planning problem can be transformed into a continuous-time optimal control problem, Zhang et al. ²² designed a three-layer trajectory planner that generated the trajectory cluster based on quadratic programing. Further, MPC-based trajectory generation methods could effectively predict vehicle states and achieve rolling optimization for trajectories that considering dynamic characteristics. Ji et al. ²³ considered the constraints of the vehicle dynamics and constructed a three-dimensional potential field based on road obstacle information to generate a collision-free trajectory for the tracking controller in real time. Rasekhipour et al. ²⁴ considered the dynamic characteristics of the vehicle in an artificial potential field to generate a safe vehicle trajectory. Tang et al. ²⁵ considered the risk of position uncertainty of surrounding vehicles and the risk of rollover, and proposed a driving environment uncertainty-aware motion planning method. For the prediction and optimization problem in MPC-based method, Tian et al. ²⁶ used the MPC-based method to energy management problem and the estimate distribution and particle swarm optimization (ED-PSO) algorithm was introduced to solve optimal control sequences over a predicting horizon. Yang et al. ²⁷ proposed a stochastic predictive energy management strategy based on fast rolling optimization for plug-in hybrid electric vehicles.

Nevertheless, the aforementioned approach encounters challenges in addressing multi-dimensional constraints within complex scenarios, particularly in the context of multi-vehicle cooperation faced by intelligent connected vehicles, thereby constraining its applicability. Therefore, Reinforcement Learning (RL) based methods have been widely concerned in recent years. In this regard, Zhu et al. ²⁸ proposed a safe off-policy model-based reinforcement learning algorithm for eco-driving. By using the online constrained optimization formulation and the approximate safe set, the trajectory planned by the learned strategy can consume approximately 22% less fuel. Liu et al. ²⁹ used the RL algorithm to train the connected and automated vehicle agent to drive on a single-lane road with multiple signalized intersections. By using the multi-light training to enhance the visual distance of the agent to make a better overall decision in speed planning, the algorithm had better performance on fuel economy and computational efficiency. Li et al. ³⁰ employed a deep Reinforcement Learning algorithm with a safe action set technique to enable flexible, robust, and safe lane-changing maneuvers. Nan et al. ³¹ presented an interaction-aware planning method for merge scenarios based on deep inverse reinforcement learning. The planned trajectories with the presented method was highly similar to those of human drivers, and can be applied to more complex scenarios, such as intersections with stronger interaction.

However, the above research was majority used the path-speed decoupling planning method to complete the planning task of the vehicle, without considering the longitudinal speed or assuming the longitudinal speed constant. Thus, the time-varying properties of dynamic constraints of the vehicle was hardly noticed by the above trajectory planner, the stability and accuracy of the trajectory tracking could not be simultaneously ensured by the vehicle controller in high-speed obstacle avoidance. Although velocity and acceleration are considered as dynamic constraints in some reference,^32,33 these variables could not characterize the instability of the vehicle sufficiently. Meantime, due to large mass bearing, multi-axle driven, and multi-axle steering characteristics, the multi-axle vehicles have larger roll angle and tire forces are more likely saturated during high-speed maneuvers, if the above planning method is directly introduced into the multi-axle vehicles, it can easily cause the vehicle to exceed the dynamic limit during trajectory tracking, resulting instability and rollover.

In response to the above issues, considering the movement state of surrounding vehicles, high-speed cruising mobility, and obstacle avoidance stability of the multi-axle vehicle, a local path-speed collaborative planning method is proposed in this paper based on a dynamic model to achieve safe and stable driving. First, a dynamic model is established by considering four degrees of freedom (4-DOF) for a five-axle vehicle. Then, a vehicle driving risk field model is established according to the relative motion state of the surrounding vehicles and the five-axle vehicle. Also, the vehicle driving stable constraint index is established considering the dynamic stability and rollover stability of the vehicle. After that, an MPC trajectory planner is designed that uses the risk field model as soft constraints and vehicle stable constraint index as hard constraints, while considering the smoothness of the trajectory to generate a feasible driving trajectory. Finally, the correctness of the proposed method is verified by simulation and experiment. The execution of the algorithm is shown in Figure 1. During the vehicle driving, the MPC planner, designed in accordance with the dynamics model outlined in Section 4.1, continuously receives sensor signals and solves the optimization problem described in Section 4.2 while considering constraints detailed in Sections 3.1 and 3.2, ultimately generating the optimal trajectory in a recurring cycle.

Figure 1.

The framework of the proposed method.

The main contributions of this paper are as follows:

• (1) A more reasonable risk field model is built, in which the current and future relative motion states between vehicles (including their position, speed, acceleration, etc.) are introduced to assess vehicle collision risk more accurately.

• (2) The stability envelope is constructed considering the multi-axle driven characteristics of the vehicle, and the tire vertical force model of multi-axle vehicle is developed to generate instability boundary based on the geometric relationship of vehicle suspension deformation.

• (3) By using the risk field model and stable constraint index, a local path-speed collaborative planning method is proposed based on the verified dynamic model to achieve safe and stable driving on cruise driving situation.

Compared with the existing method, the stability and accuracy of the trajectory tracking could be simultaneously ensured in high-speed driving by employing the trajectories generated through this paper’s approach. Thus, the method especially suitable for large mass and high center of gravity vehicles which are more prone to instability and rollover during high-speed maneuvers.

Dynamic model for five-axle vehicle

A single-track dynamic model is developed in this section to describe the stability characteristics of the vehicle. Additionally, both the tire model and the steering model are also considered to provide a comprehensive description of the dynamic model.

The single-track dynamic model

As shown in Figure 2, the dynamic model for the five-axle vehicle is a 4-DOF system, including its longitudinal, lateral, yaw, and roll movements.

Figure 2.

4-DOF dynamic model.

The dynamic model of the vehicle can be expressed as:

$$m({\stackrel{·}{V}}_x-V_y\stackrel{·}{\phi })=2\sum _{i=1}^5{F}_{\mathrm{xi}}$$

(1)

$$m({\stackrel{·}{V}}_y+V_x\stackrel{·}{\phi }-h\stackrel{··}{\theta })=2\sum _{i=1}^5{F}_{\mathrm{yi}}$$

(2)

$$I_z\stackrel{··}{\phi }=2(\sum _{i=1}^2{L}_i{F}_{\mathrm{yi}}-\sum _{i=3}^5{L}_i{F}_{\mathrm{yi}})$$

(3)

$$I_x\stackrel{··}{\theta }=\mathrm{mh}{a}_y\cos \theta +\mathrm{mgh}\sin \theta -M_R$$

(4)

where $m$ is the mass of vehicle, $V_x$ and $V_y$ are the longitudinal and lateral velocity of the vehicle, respectively (see Figure 1), $\stackrel{·}{\phi }$ is the yaw rate of the vehicle, $I_x$ and $I_z$ represent the rolling moment of inertia and the yawing moment of inertia, respectively, $F_{\mathrm{xi}}$ and $F_{\mathrm{yi}}$ represent the longitudinal and lateral forces exerted on the vehicle by the ground through the i-th (i = 1, 2,…, 5) tire, $L_i$ represents the distance between the i-th axis and the center of gravity of the vehicle (CG), $\theta$ represents the rolling angle of the vehicle, $h$ is the vertical distance between CG and ground, and $a_y$ is the lateral acceleration at CG, expressed as $a_y={\stackrel{·}{V}}_y+V_x\stackrel{·}{\phi }-h\stackrel{··}{\theta }$ . Due to the suspension effect, the vehicle is also subjected to an anti-roll moment $M_R$ . This moment can be expressed as $M_R=K\theta +D\stackrel{·}{\theta }$ , where K and D represent the stiffness coefficient and damping coefficient of the vehicle suspension system, respectively.

Besides, the kinematic model of the five-axle vehicle can also be determined as follows:

$$\{\begin{array}{c}\stackrel{·}{Y}=V_x\sin \phi +V_y\cos \phi \hfill \\ \stackrel{·}{X}=V_x\cos \phi -V_y\sin \phi \hfill \end{array}$$

(5)

where $\phi$ is the yaw angle of the vehicle, while $X$ and $Y$ are the longitudinal and lateral positions of the center of gravity of the vehicle.

Tire model

The longitudinal and lateral forces exerted on the vehicle by the ground (through the tires) can be expressed as:

$$\{\begin{array}{c}{F}_{\mathrm{xi}}=F_{li}\cos {\delta }_i-F_{ci}\sin {\delta }_i\hfill \\ F_{\mathrm{yi}}=F_{li}\sin {\delta }_i+F_{ci}\cos {\delta }_i\hfill \end{array}$$

(6)

where ${\delta }_i$ represents the steering angle of the i-th axle, while $F_{\mathrm{li}}$ and $F_{\mathrm{ci}}$ represents the longitudinal and lateral forces of the tire on the i-th axle.

To establish a dynamic relationship between the tire and the vehicle body, thereby accurately characterize the vehicle’s motion state accurately, the magic formula is employed to develop the tire model:

$$\{\begin{array}{c}y(x)=D\sin (C\arctan (\mathrm{Bx}-E(\mathrm{Bx}-\arctan (\mathrm{Bx}))))\hfill \\ Y(x)=y(x)+S_v\hfill \\ x=X+S_h\hfill \end{array}$$

(7)

where $Y$ represents the longitudinal or lateral force of the tire, $x$ represents the sideslip angle or slip rate, $S_V$ represents the vertical offset, and $S_H$ represents the horizontal offset. The coefficients B, C, D, and E are the stiffness factor, shape factor, peak value factor, and curvature factor, respectively. The parameters mentioned above can be obtained through parameter identification for actual tires.

However, the above model is applicable solely under conditions of pure sideslip or pure slip. In practice, it is essential to account for the interaction between longitudinal and lateral tire forces. Therefore, the final expression for the tire force can be written as follows:

$$\{\begin{array}{c}{F}_{li}=\frac{\tan ({\alpha }_i)}{\sqrt{\tan {({\alpha }_i)}^2+{S_i}^2}}{F}_{li0}\hfill \\ F_{ci}=\frac{S_i}{\sqrt{\tan {({\alpha }_i)}^2+{S_i}^2}}{F}_{ci0}\hfill \end{array}$$

(8)

where ${\alpha }_i$ and $S_i$ are the sideslip angle and the slip rate of the i-th axle of the vehicle, respectively. Furthermore, $F_{\mathrm{li}0}$ and $F_{\mathrm{ci}0}$ are the longitudinal and lateral tire forces calculated by equation (7), respectively.

The sideslip angle ${\alpha }_i$ and slip rate $S_i$ is calculated by the following equation:

$${\alpha }_i={\delta }_i-a\tan \left(\frac{v_y\pm L_i\stackrel{·}{\phi }}{v_x}\right)$$

(9)

$$S_i=\frac{{\omega }_i{r}_w-v_{\mathrm{l}i}}{max({\omega }_i{r}_w,v_{\mathrm{l}i})}$$

(10)

In equation (9), the sign convention is considered positive for the first two axles of the vehicle and negative for the last three axles. In addition, in equation (10), ${\omega }_i$ is the wheel speed, $r_w$ is the rolling radius of the tire, and $v_{\mathrm{li}}$ is the longitudinal speed of the wheel center. $v_{\mathrm{li}}$ can be expressed as follows:

$$v_{li}=V_x\cos {\delta }_i+(V_y\pm L_i\stackrel{·}{\phi })\sin {\delta }_i$$

(11)

Steering model

The steering angle model of the five-axle vehicle is based on the Ackerman steering model, that is, all wheels share a common instantaneous steering center during maneuvering. This configuration ensures that the wheels experience nearly pure rolling motion. In this context, the first, second, fourth, and fifth axes of the five-axle vehicle are the steering axes, and the instantaneous center is located on the extension line of the third axis (see Figure 2). Hence, the following relationship holds:

$$\frac{\tan {\delta }_i}{\tan {\delta }_1}=\frac{L_3\pm L_i}{L_1+L_3}$$

(12)

Then, the steering angle of each vehicle axis can be obtained as:

$${\delta }_i=a\tan \left(\frac{L_3\pm L_i}{L_1+L_3}\tan {\delta }_1\right)$$

(13)

Model validation

To verify the correctness of the simplified 4-DOF dynamic model, the MATLAB/Simulink software is used to establish its simulation model in this section.

To ensure precise verification, the TruckSim model is first established according to the parameters of a real five-axle vehicle. By comparing the dynamic response of the real vehicle and the TruckSim model, the accuracy of the TruckSim model can be verified. Subsequently, a comparison between the dynamic response of the 4-DOF model and that of the TruckSim model is then presented to further confirm the validity of the established 4-DOF model. Some parameters of the vehicle model are shown in Table 1.

Table 1.

Parameters of the vehicle dynamic model.

Parameter (unit)	Parameter meaning	Value
m (kg)	Mass	56,244
$L_1$ (m)	Distance from axle 1 to CG	5.7
$L_2$ (m)	Distance from axle 2 to CG	3.2
$L_3$ (m)	Distance from axle 3 to CG	1.1
$L_4$ (m)	Distance from axle 4 to CG	3.4
$L_5$ (m)	Distance from axle 5 to CG	5.9
h (m)	CG height	1.4
$I_x$ /(kg·m2)	Rolling moment of inertia	242,300
$I_z$ /(kg·m2)	Yawing moment of inertia	734,251
$R_w$ (m)	Tire radius	0.55

The SpeedBox-mini is used to measure the lateral acceleration, yaw rate and roll rate of the five-axle vehicle. A comparison of the dynamic response between the real vehicle and the TruckSim model is shown in Figure 3. From Figure 3, the TruckSim model response and the real vehicle test response show a certain degree of deviation due to uncertainties such as test site, driver operation and sensor noise. However, the response trend of these two models is clearly consistent. Therefore, the TruckSim model can be used for real verification.

Figure 3.

Comparison between real vehicle and TruckSim model: (a) test procedure for the dynamic response of the real vehicle and TruckSim model, (b) trajectory comparison between real vehicle and TruckSim model, (c) yaw rate comparison between real vehicle and TruckSim model, (d) lateral acceleration comparison between real vehicle and TruckSim model, and (e) roll rate comparison between real vehicle and TruckSim model.

According to Figure 4, under the same given input stimulus, the simplified 4-DOF dynamic model has certain errors in terms of sideslip angle, yaw rate, lateral velocity, and roll angle compared to the TruckSim model. This is due to the neglect of the impact of chassis components such as suspension when establishing the 4-DOF model. This will lead to the deviation between the output predicted by the dynamic model and the real vehicle state, so the optimization problem solved by the MPC planner is not exactly corresponding to the actual scene, and the accuracy of generated trajectory is reduced. However, some necessary nonlinear dynamics effects (such as the stiffness and damping characteristics of the suspension, the interaction between longitudinal and lateral tire forces) are partially taken into consideration to limit the dynamic error to a certain range. The maximum response error between these two models is within 10%, and the dynamic response is consistent, meeting the dynamic requirements of vehicle trajectory generation. In the next part of this paper, considering the dynamic state of the vehicle itself and surrounding vehicles while driving, a safe and stable driving trajectory is generated for a five-axle vehicle based on its 4-DOF dynamic model.

Figure 4.

Simulation results of 4-DOF model validation. Note that, “Sin Input” represents the sinusoidal input stimulus to the model and “Step Input” represents the step input stimulus: (a) two kinds of steering angle stimulus for 4-DOF model and TruckSim model, (b) lateral acceleration comparison between 4-DOF model and TruckSim model, (c) sideslip comparison between 4-DOF model and TruckSim model, (d) yaw rate comparison between 4-DOF model and TruckSim model, and (e) roll angle comparison between 4-DOF model and TruckSim model.

Driving safety and stable constraints

Five-axle vehicle is significantly influenced by the behavior of surrounding vehicles when cruising on structured roads. To avoid collisions, it is essential for the trajectory planner of the five-axle vehicle to perform local trajectory planning based on real-time estimation of other vehicle’s states. However, due to the characteristics of large mass and high gravity center of the five-axle vehicle, tracking these locally planned trajectories can lead to instability and potential rollover incidents. To address this issue, first, the obstacle risk field model for other vehicles is established in this section. Then, the instability and rollover model is considered in the second step to avoid the obstacle of the five-axle vehicle. In the next section, it is used as a constraint in designing a MPC trajectory planner aimed at achieving effective local vehicle trajectory planning local vehicle trajectory planning.

Risk field model for surrounding vehicles

The risk field model must be adjustable according to the vehicle state. Considering that the Gaussian function can be adjusted by its mean value and covariance matrices, and can also be normalized, the vehicle risk field is established based on the bivariate Gaussian distribution function in this paper:

$$P(x,y)=Z·\exp (-\frac{1}{2}(\frac{X^2}{{{\sigma }_x}^2}+\frac{Y^2}{{{\sigma }_y}^2}))$$

(14)

where $X$ and $Y$ are the relative positions between the five-axle vehicle and the surrounding vehicles, while the ${\sigma }_x$ and ${\sigma }_y$ coefficients are the risk field adjustment factor. Moreover, the relative driving direction between the five-axle vehicle and other vehicles is considered in this paper. The position of the i-th surrounding vehicle relative to the five-axle vehicle is expressed as follows:

$$\left[\begin{array}{c}X\hfill \\ \multicolumn{1}{c}{Y}\end{array}\right]=\left[\begin{array}{cc}\cos {\phi }_i\hfill & -\sin {\phi }_i\hfill \\ \multicolumn{1}{c}{\sin {\phi }_i}& \cos {\phi }_i\hfill \end{array}\right]\left[\begin{array}{c}{x}_i-x_0\hfill \\ \multicolumn{1}{c}{y_i-y_0}\end{array}\right]$$

(15)

where ${\phi }_i$ represents the heading angle between the five-axle vehicle and the i-th surrounding vehicle. Furthermore, $(x_i,y_i)$ represents the position of the i-th surrounding vehicle relative to the five-axle vehicle.

Generally, the differences in the relative motion states of the surrounding vehicles and the five-axle vehicle leads to different collision risks. Assume that the motion state of the five-axle vehicle at a certain moment is $(P_0,V_0,a_0,{\phi }_0)$ (representing position, velocity, acceleration, and heading angle, respectively), while the motion state of other vehicles is $(P_i,V_i,a_i,{\phi }_i)$ . The positive or negative relative velocity or acceleration between these vehicles often indicates whether a collision will occur in the future (see Table 2).

Table 2.

The relationship between vehicle collision risk and vehicle motion state.

Relative position	Relative velocity	Relative acceleration	Collision risk level
Rear vehicles ( $P_0>P_i$ )	$V_i\le V_0$	$a_i\le a_0$	Low
		$a_i>a_0$	Middle
	$V_i>V_0$	$a_i\le a_0$	Middle
		$a_i>a_0$	High
Front vehicles ( $P_0<P_i$ )	$V_i\le V_0$	$a_i\le a_0$	High
		$a_i>a_0$	Middle
	$V_i>V_0$	$a_i\le a_0$	Middle
		$a_i>a_0$	Low

Considering the relationship between the relative motion state and collision risk in Table 2, the coefficient ${\sigma }_x$ and ${\sigma }_y$ are defined as:

$$\{\begin{array}{c}{\sigma }_x=\mathrm{sgn}(\mathrm{\Delta }x)(k_1^x{V}_{\mathrm{xi}}-k_2^x{V}_{x0})(k_3^x{a}_{\mathrm{xi}}-k_4^x{a}_{x0}))+C_x\hfill \\ {\sigma }_y=\mathrm{sgn}(\mathrm{\Delta }y)(k_1^y{V}_{\mathrm{yi}}-k_2^y{V}_{y0})(k_3^y{a}_{\mathrm{yi}}-k_4^y{a}_{y0}))+C_y\hfill \end{array}$$

(16)

where $k_i^x\mathrm{and}{k}_i^y(i=1,2,3,4)$ are constant values that can be selected based on actual conditions, while $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ represent the difference between the vehicle and the surrounding vehicles in the $X$ and $Y$ directions, respectively. Furthermore, $C_x$ and $C_y$ are constant values determined by the vehicle speed.

For structured roads, apart from moving obstacles, static obstacles such as road boundaries and falling objects on the road should also be considered. The risk field adjustment factors ${\sigma }_x$ and ${\sigma }_y$ for static obstacles are defined in the same way as the equation (16), where both velocity and acceleration values are set to 0.

The road boundary is expressed as follows:

$$P(y)=Z·(\exp (-\frac{1}{2}\frac{{Y_1}^2}{{{\delta }_y}^2})+\exp (-\frac{1}{2}\frac{{Y_2}^2}{{{\delta }_y}^2}))$$

(17)

where $Z$ is a constant coefficient, selected according to the actual situation, while $Y_1$ and $Y_2$ are the closest distance from the vehicle to both sides of the road boundary. The definition of ${\sigma }_y$ is the same as equation (16). The generated risk field is shown in Figure 5(a) and (b).

Figure 5.

Obstacle risk field: (a) risk field of the vehicle, (b) risk field of the road boundary, and (c) risk field of surrounding vehicles with different motion states.

Considering the real driving situation of the five-axle vehicle applied in this paper, it is assumed that the five-axle vehicle drives at a constant speed of 20 m/s, that is, 72 km/h. The risk field of surrounding vehicles with different motion states are shown in Figure 5(c). It is easy to see that different speed and acceleration of surrounding vehicles lead to different collision risk field distributions. For vehicle A, while its speed is the same as the five-axle vehicle, it is in acceleration mode. Therefore, there is little risk of collision when the five-axle vehicle is driving behind. However, when the five-axle vehicle driving ahead, the risk of collision increases as the distance between the five-axle vehicle and vehicle A decreases. For vehicle B, the driving speed of the vehicle is greater than the cruising speed of the five-axle vehicle, so there is little risk of collision, while for vehicle C, the opposite is true.

The obstacle risk field at future moments should also be considered. However, due to the short duration of trajectory planning cycle, this paper assumes that the acceleration of the surrounding vehicles remains constant over brief periods. Assuming that the motion state of the surrounding vehicles has been estimated at time $t$ , their motion state at the subsequent moment $t+1$ can be expressed as:

$$\{\begin{array}{c}{X}_i(t+1)=X_i(t)+{\stackrel{·}{X}}_i(t)T+0.5{\stackrel{··}{X}}_i(t)T^2\hfill \\ {\stackrel{·}{X}}_i(t+1)={\stackrel{·}{X}}_i(t)+{\stackrel{··}{X}}_i(t)T\hfill \\ Y_i(t+1)=Y_i(t)+{\stackrel{·}{Y}}_i(t)T+0.5{\stackrel{··}{Y}}_i(t)T^2\hfill \\ {\stackrel{·}{Y}}_i(t+1)={\stackrel{·}{Y}}_i(t)+{\stackrel{··}{Y}}_i(t)T\hfill \end{array}$$

(18)

To establish the risk field model for the next $t+n$ moment, the surrounding vehicles motion state at the next $t+n$ moment should also be known. If the acceleration components are constant values of $a_x$ and $a_y$ in a short time (at moment $t$ ) the predicted motion state for the $t+n$ moment is simply:

$$\{\begin{array}{c}{X}_i(t+n)=X_i(t)+\frac{n{T}^2}{2}{a}_x{\stackrel{·}{X}}_i(t)+\frac{n(n-1)T}{2}{a}_x\hfill \\ {\stackrel{·}{X}}_i(t+n)={\stackrel{·}{X}}_i(t)+\mathrm{nT}{a}_x\hfill \\ Y_i(t+n)=Y_i(t)+\frac{n{T}^2}{2}{a}_y{\stackrel{·}{Y}}_i(t)+\frac{n(n-1)T}{2}{a}_y\hfill \\ {\stackrel{·}{Y}}_i(t+n)={\stackrel{·}{Y}}_i(t)+\mathrm{nT}{a}_y\hfill \end{array}$$

(19)

The above vehicle motion states can be obtained based on the on-board sensors of the vehicle (such as LiDAR) and the motion state estimation algorithms of the surrounding vehicles in Liu et al. ³⁴ In dynamically changing environments, the above speed and position states at future moments can also be derived through the integration of predicted acceleration. To address the uncertainties inherent in dynamic environments, methods such as incorporating the uncertainty of predicted motion covariance, as discussed in Na et al., ³⁵ can be employed.

Driving stability model for five-axle vehicles

Under heavy load conditions, the tire force of the five-axle vehicle tends to be saturated, which can lead to a loss of stability. Additionally, the structural characteristics of the high center of gravity cause the vehicle body to roll significantly, increasing the load transfer between the tires and increasing the risk of the vehicle rollover. To ensure the driving safety of the five-axle vehicle, a vehicle dynamic constraint model is established in this section, considering vehicle yaw stability and rollover safety.

Constraints on vehicle stability

Tire force saturation is considered as a constraint on vehicle stability. As shown in Figure 6, a vehicle stability envelope based on the yaw rate-sideslip angle phase plane is generated and used in this paper. ¹

Figure 6.

Vehicle stability constraints.

Due to the minor fluctuations of the vehicle speed during cruising, the assumption of constant speed is adopted for driving in this paper. Then, the vehicle body dynamics can be simplified to a 2-DOF model as:

$$\begin{array}{c}\stackrel{·}{\beta }=\frac{2}{m{V}_x}\sum _{i=1}^5{F}_{\mathrm{yi}}-\stackrel{·}{\phi }\hfill \\ \stackrel{··}{\phi }=\frac{2}{I_z}(\sum _{i=1}^2{L}_i{F}_{\mathrm{yi}}-\sum _{i=3}^5{L}_i{F}_{\mathrm{yi}})\hfill \end{array}$$

(20)

where $\beta$ is the sideslip angle. According to equation (20), under the steady-state motion of the vehicle, the sideslip angle rate is 0. Considering the maximum force provided by the ground, the maximum rate of stable yaw is expressed as:

$${\stackrel{·}{\phi }}_{\mathrm{ss},\max }=\frac{g\mu }{v_x}$$

(21)

As shown in Figure 6, when the vehicle is unstable, the tire force is first saturated at the first or fifth axle of the vehicle. Therefore, in this paper, the saturation of tire force on axles 1 and 5 of the vehicle is used as a basis for establishing stability envelope. Then, the maximum sideslip angle can be obtained from equation (9) as:

$$\{\begin{array}{c}{\beta }_{1,max}={\alpha }_{1,\mathrm{sat}}+\frac{L_1\stackrel{·}{\phi }}{V_x}-{\delta }_1\hfill \\ {\beta }_{5,max}={\alpha }_{5,\mathrm{sat}}+\frac{L_5\stackrel{·}{\phi }}{V_x}-{\delta }_5\hfill \end{array}$$

(22)

The saturated tire sideslip angle can be approximately calculated by the following formula:

$${\alpha }_{3,\mathrm{sat}}=\mathrm{a}\tan \left(\frac{3\mu F_z}{C_{\alpha }}\right)$$

(23)

where $C_{\alpha }$ is the lateral stiffness of the tire, and $F_{\mathrm{zi}}$ is the vertical force of the i-th tire.

Since the stress of the five-axle vehicle body is statically indeterminate, the force and moment balance equation can be expressed as:

$$\{\begin{array}{c}{F}_{z1}+F_{z2}+F_{z3}+F_{z4}+F_{z5}=m_{b}g\hfill \\ F_{z1}{L}_1+F_{z2}{L}_2=F_{z3}{L}_3+F_{z4}{L}_4+F_{z5}{L}_5\hfill \end{array}$$

(24)

According to the geometric relationship of deformation in Figure 7, $f_i$ can be expressed as:

$$\{\begin{array}{c}\frac{f_1-f_2}{f_1-f_3}=\frac{L_1-L_2}{L_1-L_3}\hfill \\ \frac{f_1-f_2}{f_1-f_4}=\frac{L_1-L_2}{L_1-L_4}\hfill \\ \frac{f_1-f_2}{f_1-f_5}=\frac{L_1-L_2}{L_1-L_5}\hfill \end{array}$$

(25)

where $F_{\mathrm{zi}}=K_i{f}_i$ , in which $K_i$ is the stiffness coefficient of the vehicle suspension. Therefore, $F_{\mathrm{zj}}$ can be obtained from equations (24) and (25) as:

$$F_{\mathrm{zj}}=\frac{m_{\mathrm{b}}g(\sum _{i=1}^5{L_i}^2-\sum _{i=1}^5(\pm L_i)(\pm L_3))}{2(2\sum _{i=1}^5{L_i}^2-\sum _{i=1}^5\sum _{n>i}^5((\pm L_i)(\pm L_{i+n})))}+m_{w}g$$

(26)

where $m_b$ is the vehicle sprung mass, and $m_w$ is the tire mass. The sign convention is the same as before.

Figure 7.

Vehicle stress state.

Constraints on vehicle roll

The vehicle roll constraint is based on the zero moment point method. ³⁶ Then, the following relationship can be obtained:

$$\mathrm{mg}\cos (\theta )·y_{\mathrm{ZM}}=(m{a}_y+\mathrm{mg}\sin (\theta ))h-I_x\stackrel{··}{\theta }$$

(27)

where $y_{\mathrm{zm}}$ is the lateral offset of zero moment point (see Figure 1). Based on the small angle assumption of roll angle, one has:

$$y_{\mathrm{ZM}}=(\frac{{\stackrel{·}{V}}_y+V_x\stackrel{·}{\phi }}{g}+\theta )h-\frac{I_x\stackrel{··}{\theta }}{\mathrm{mg}}$$

(28)

The roll angle is constrained by the $y_{\mathrm{ZM}}$ range:

$$-y_{\mathrm{ZM},max}\le y_{\mathrm{ZM}}\le y_{\mathrm{ZM},max}$$

(29)

Ultimately, the driving stability constraint on the vehicle can be expressed as:

$$\{\begin{array}{c}-{\stackrel{·}{\phi }}_{\mathrm{ss},\max }\le \stackrel{·}{\phi }\le {\stackrel{·}{\phi }}_{\mathrm{ss},\max }\hfill \\ -{\beta }_{i,max}\le \beta \le {\beta }_{i,max};i=[1,5]\hfill \\ -y_{\mathrm{ZM},\max }\le y_{\mathrm{ZM}}\le y_{\mathrm{ZM},\max }\hfill \end{array}$$

(30)

Safe trajectory generation for five-axle vehicle

A 4-DOF model has been previously established. Building upon this foundation, a driving safety model that considers surrounding vehicles as well as the five-axle vehicle has also been developed. In this section, an MPC trajectory planner will be designed based on the aforementioned model to generate a safe driving trajectory, in which constrained by Section 3.2.

Design of MPC trajectory planner

The vehicle dynamics model established according to equations (1)–(13) can be described as:

$$\stackrel{·}{\xi }=F({\delta }_1,V_x,V_y,\stackrel{·}{\phi },\phi ,\stackrel{·}{\theta },\theta ,{\omega }_i)$$

(31)

where $\xi$ is the state variables, specified as $[V_x,V_y,\stackrel{·}{\phi },\phi ,\stackrel{·}{\theta },\theta ,X,Y]$ . The control variable is $u=[{\delta }_1,{\omega }_i]$ . Therefore, the following state equation can be established:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\xi }(t)=\left[\begin{array}{c}{f}_{V_x}(\xi (t),u(t))\hfill \\ \multicolumn{1}{c}{f_{V_y}(\xi (t),u(t))}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{f_Y(\xi (t),u(t))}\end{array}\right]}\end{array}$$

(32)

Since the above system is nonlinear, to reduce the computational complexity, it is linearized in the current execution time, and the output of the system is chosen as $\eta =[V_x,V_y,\stackrel{·}{\phi },X,Y]$ . Then, the system is expressed as:

$$\begin{array}{c}\stackrel{·}{\xi }(t)=A_t\xi (t)+B_{t}u(t)+d_t(t)\hfill \\ \eta (t)=C_t\xi (t)\hfill \end{array}$$

(33)

where the matrices $A_t$ and $B_t$ are the Jacobian matrix. In addition, $d_t$ is the linearization error, which is calculated in real-time based on the motion state. The detailed representation is as follows:

$$\begin{array}{c}{A}_t=\underset{8\times 8}{\underbrace{\left[\begin{array}{cccc}\frac{\partial f_{V_x}}{\partial V_x}\hfill & \frac{\partial f_{V_x}}{\partial V_y}\hfill & \cdots \hfill & \frac{\partial f_{V_x}}{\partial Y}\hfill \\ \multicolumn{1}{c}{\frac{\partial f_{V_y}}{\partial V_x}}& \frac{\partial f_{V_y}}{\partial V_y}\hfill & ⋮\hfill & \ddots \hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{\frac{\partial f_Y}{\partial V_x}}& \frac{\partial f_Y}{\partial V_y}\hfill & \cdots \hfill & \frac{\partial f_Y}{\partial Y}\hfill \end{array}\right]}};B_t=\underset{8\times 2}{\underbrace{\left[\begin{array}{cc}\begin{array}{c}\frac{\partial f_{V_x}}{\partial {\delta }_1}\hfill \\ \multicolumn{1}{c}{\frac{\partial f_{V_y}}{\partial {\delta }_1}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\frac{\partial f_Y}{\partial {\delta }_1}}\end{array}\hfill & \begin{array}{c}\frac{\partial f_{V_x}}{\partial {\omega }_i}\hfill \\ \multicolumn{1}{c}{\frac{\partial f_{V_y}}{\partial {\omega }_i}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\frac{\partial f_Y}{\partial {\omega }_i}}\end{array}\hfill \end{array}\right]}}\hfill \\ d_t=f(\xi (t),u(t-1))-[A_t\xi (t)+B_{t}u(t-1)]\hfill \\ C_t=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & \begin{array}{cc}\begin{array}{cc}\cdots \hfill & 0\hfill \end{array}\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & \begin{array}{cc}\begin{array}{cc}\cdots \hfill & 0\hfill \end{array}\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & \begin{array}{ccc}\cdots \hfill & 0\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}}& \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{ccc}\begin{array}{c}\cdots \hfill \\ \multicolumn{1}{c}{\cdots }\end{array}\hfill & \begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\hfill & \begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\hfill \end{array}\hfill \end{array}\right]\}5\times 8;\hfill \end{array}$$

(34)

To generate the driving trajectory in $n$ seconds based on the current moment $t$ , the above system is discretized by the forward Euler method as:

$$\{\begin{array}{c}\xi (t+1)=A_t^d\xi (t)+B_t^{d}u(t)+d_t(t)\hfill \\ \eta (t)=C_t^d\xi (t)\hfill \end{array}$$

(35)

where $A_t^d=I+T{A}_t$ , $B_t^d=T{B}_t$ , and $C_t^d=C_t$ .

The control increment input $\mathrm{\Delta }u\left(t\right)$ is used instead of the control input $u\left(t\right)$ , hence the vehicle smooth trajectory planning can be achieved by directly constraining $\mathrm{\Delta }u\left(t\right)$ . If the control increment is written as $\mathrm{\Delta }u\left(t\right)=u\left(t\right)-u(t-1)$ , then equation (33) can be expressed as:

$$\begin{array}{c}\left[\begin{array}{c}\xi (t+1)\hfill \\ \multicolumn{1}{c}{u(t)}\end{array}\right]=\left[\begin{array}{cc}{A}_k^d\hfill & B_k^d\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right]\left[\begin{array}{c}\xi (t)\hfill \\ \multicolumn{1}{c}{u(t-1)}\end{array}\right]+\left[\begin{array}{c}{B}_k^d\hfill \\ \multicolumn{1}{c}{I}\end{array}\right]△u(t)+\left[\begin{array}{c}{d}_k\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \\ \eta (t)=\left[\begin{array}{cc}{C}_k^d\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}\xi (t)\hfill \\ \multicolumn{1}{c}{u(t-1)}\end{array}\right]\hfill \end{array}$$

(36)

Further, the above equation can be rewritten in the following compact form:

$$\{\begin{array}{c}\tilde{\xi }(t+1)={\tilde{A}}_t\tilde{\xi }(t)+{\tilde{B}}_t\mathrm{\Delta }u(t)+\tilde{d}(t)\hfill \\ \eta (t)={\tilde{C}}_t\tilde{\xi }(t)\hfill \end{array}$$

(37)

Suppose the prediction domain is $H_p$ , the control domain is $H_c$ , and the relationship between these two domains is $H_p>H_c$ . Then at the current moment t, given a series of control inputs $[\mathrm{\Delta }{u}_1,\mathrm{\Delta }{u}_2,\dots ,\mathrm{\Delta }{u}_M]$ in time domain $[t,t+H_c-1]$ , and [0, 0, …,0] in time domain $[t+H_c,t+H_p-1]$ . Finally, the output can be expressed as follows:

$$Y_t={\mathrm{\Psi }}_t{\tilde{\xi }}_t+{\mathrm{\Theta }}_t\mathrm{\Delta }{U}_t+{\mathrm{\Gamma }}_t{\mathrm{\Phi }}_t$$

(38)

where:

$$\begin{array}{c}Y=\left[\begin{array}{c}Y(t+1|t)\hfill \\ \multicolumn{1}{c}{Y(t+2|t)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{Y(t+H_p|t)}\end{array}\right];\psi =\left[\begin{array}{c}\tilde{C}\tilde{A}\hfill \\ \multicolumn{1}{c}{\tilde{C}{\tilde{A}}^2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\tilde{C}{\tilde{A}}^{H_p}}\end{array}\right];\hfill \\ \mathrm{\Theta }=\left[\begin{array}{cccc}\tilde{C}\tilde{B}\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{\tilde{C}\tilde{A}\tilde{B}}& \tilde{C}\tilde{B}\hfill & 0\hfill & \ddots \hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{\tilde{C}{\tilde{A}}^{H_p-1}\tilde{B}}& \tilde{C}{\tilde{A}}^{H_p-2}\tilde{B}\hfill & \cdots \hfill & \tilde{C}{\tilde{A}}^{H_p-H_c-1}\tilde{B}\hfill \end{array}\right]\hfill \\ \mathrm{\Delta }U(t)=\left[\begin{array}{c}\mathrm{\Delta }U(t|t)\hfill \\ \multicolumn{1}{c}{\mathrm{\Delta }U(t+1|t)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\mathrm{\Delta }U(t+H_c|t)}\end{array}\right];\mathrm{\Gamma }=\left[\begin{array}{cccc}\tilde{C}\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{\tilde{C}\tilde{A}}& \tilde{C}\hfill & ⋮\hfill & \ddots \hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{\tilde{C}{\tilde{A}}^{H_p-1}}& \tilde{C}{\tilde{A}}^{H_p-2}\hfill & \cdots \hfill & \tilde{C}\hfill \end{array}\right]\hfill \\ \mathrm{\Phi }(t)=\left[\begin{array}{c}\tilde{d}(t|t)\hfill \\ \multicolumn{1}{c}{\tilde{d}(t+1|t)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\tilde{d}(t+H_p-1|t)}\end{array}\right];\hfill \end{array}$$

(39)

From above equation, the optimal output trajectory can be obtained by calculating the optimal input.

Optimal driving trajectory generation

When the five-axle vehicle cruises on the structured road, the designed MPC planner observes the vehicle motion state and solves the optimal trajectory conforming to the dynamic constraints according to the predetermined time interval.

The optimal trajectory is defined as:

$$\begin{array}{c}\multicolumn{1}{c}{J=\sum _{i=t+1}^{t+H_{\mathrm{p}}}(\Vert \mathrm{Tra}-\mathrm{Tr}{a}^{\mathrm{ref}}{\Vert }_D^2+\Vert V_x-{V_x}^{\mathrm{ref}}{\Vert }_Q^2+\Vert {\kappa }_i{\Vert }_R^2)}\\ +\sum _{i=t+1}^{t+H_{\mathrm{p}}}\left(\sum _{j=1}^n{\Vert {P_{\mathrm{obs}}}^j\Vert }_K\right)+\sum _{i=t+1}^{t+H_{\mathrm{c}}}\Vert \mathrm{\Delta }U{\Vert }_G^2\hfill \end{array}$$

(40)

In equation (40), the optimal function $J$ considers the driving trajectory $\mathrm{Tra}$ , vehicle speed, driving smoothness, and the motion state of other vehicles in the prediction time domain. $P_{\mathrm{obs}}$ is the motion risk field that is generated in the prediction domain according to the observation results of the motion state of other vehicles and road boundary, which is calculated by equations (14)–(19). In addition, $\kappa$ is the curvature of the generated trajectory at a certain moment, which is calculated as follows:

$$\kappa =\frac{X\prime (t)Y″(y)-X″(t)Y\prime (t)}{\sqrt{{(X\prime (t)+Y\prime (t))}^3}}$$

(41)

where $X$ and $Y$ are the cubic spline curves fitted by the output trajectory points of the planner. If the maximum allowable lateral acceleration of the vehicle is ${\left|a_{\mathrm{lat}}\right|}_{\max }$ , then the vehicle speed constraint is expressed as:

$$V_{xlim}=min(v_k=\sqrt{\frac{{\left|a_{\mathrm{lat}}\right|}_{max}}{{\kappa }_{max}}},v_{\mathrm{sign}})$$

(42)

where $v_{\mathrm{sign}}$ is the road speed limit. Ultimately, the optimization problem is expressed as:

$$\begin{array}{c}minJ(\mathrm{Tra},V_x,\kappa ,P_{\mathrm{obs}},\mathrm{\Delta }U)\hfill \\ \mathrm{s}.\mathrm{t}.V_x\le V_{xlim},\left|y_{\mathrm{ZM}}\right|\le y_{\mathrm{ZM},\max }\hfill \\ \left|U\right|\le U_{max},\left|\mathrm{\Delta }U\right|\le \mathrm{\Delta }{U}_{max}\hfill \\ \left|\stackrel{·}{\phi }\right|\le {\stackrel{·}{\phi }}_{\mathrm{ss},\max },\left|\beta \right|\le {\beta }_{max}\hfill \end{array}$$

(43)

Equation (43) is the nonlinear programing problem with multi-constrains, and can be solved by conventional optimization algorithms, such as interior-point method. The planner generates continuous vehicle driving trajectories by solving the optimization problem (43) cyclically. The lower-level controller enables automatic cruising on structured roads by tracking the driving trajectory in real time.

Simulation and experiment

To verify the proposed method, the software simulation and hardware in the loop (HIL) testing are both used in adaptive cruising scenarios.

The straight and curved roads are used in simulation and experiment, respectively. The scenario includes road boundary lines, surrounding vehicles, and static obstacles in the center of the road. The trajectory planner is implemented using the proposed method, and the PID controller is used for trajectory tracking. The relevant parameters of the trajectory planner are shown in Table 3, while the parameters of the vehicle model are the same as Table 1.

Table 3.

Parameters of the trajectory planner.

Parameters	Value	Parameters	Value
Control horizon $H_c$	3	Prediction horizon $H_p$	20
Max control increment $[\mathrm{\Delta }{\delta }_1,\mathrm{\Delta }\omega ]$	[0.01 rad, 0.15 rad/s]	Max control input $[{\delta }_{1\max },{\omega }_{\max }]$	[0.3 rad, 50 rad/s]
Planning cycle	0.2 s	Discrete time interval T	0.1 s

In Table 3, with reference to the actual driver reaction time, the planning cycle is set to 0.2 s, that is, the planner calculates the trajectory within 0.2 s. With reference to the parameters in Ji et al. ²³ and Rasekhipour et al. ²⁴ the MPC planner’s control horizon and prediction horizon are set to 3 and 20, respectively. The discrete time interval is set to 0.1 s, so that the planner can generate the trajectory control inputs in 0.3 s by predicting the vehicle state within 2 s. Considering the actual driving conditions of multi-axle vehicles, the max control increment is established to signify the vehicle’s maximum steering velocity and acceleration capabilities, and the max control input is set to indicate the maximum steering angle and driving velocity.

Simulation for straight road driving scenarios

Vehicle simulation tests at two reference speeds, namely 18 and 20 m/s, are used to validate the proposed method. The motion status of each vehicle is shown in Table 4. Among them, the data format in the table is $[P_i,V_i,a_i],i=A,B,C$ .

Table 4.

Driving state of the vehicle on the straight road.

Case 1		Case 2
Simulation speed	18 m/s	Simulation speed	25 m/s
Vehicle A	[130,10,0]	Vehicle A	[150,20,1]
Vehicle B	[80,10,0]	Vehicle B	[100,15,0]
Vehicle C	[50,20,0]	Vehicle C	[50,30,0]

The planned trajectories under different speeds are shown in Figures 8 and 9, where the upper figures are the planned trajectory using the proposed method (shown by the thick blue line), and the lower figures are the planned cruising speed of the vehicle (shown by the thick red line). The thin solid lines with various colors are the predicted trajectory and speed in real time. Moreover, $A_{\mathrm{obs}}$ is the static obstacle. The initial relative position and final relative position between vehicles can also be seen in the figure.

Figure 8.

Driving trajectory at speed of 18 m/s.

Figure 9.

Driving trajectory at speed of 25 m/s.

In Figure 8, when the vehicle is driving at 18 m/s, due to the presence of obstacles $A_{\mathrm{obs}}$ ahead, the five-axle vehicle decelerates to reduce the risk of collision. Then, the vehicle bypasses obstacle $A_{\mathrm{obs}}$ at 100 m. However, there is vehicle B driving at 10 m/s in front of the five-axle vehicle, meanwhile, vehicle A is further away at the same speed. To avoid collision with two vehicles A and B, the five-axle vehicle decelerates and moves laterally away from vehicle B while ensuring its own driving stability (100–150 m). After approaching vehicle A on the front left, the five-axle vehicle accelerates beyond vehicle B (150–220 m). Throughout the entire process, the five-axle vehicle and vehicle A always maintain a safe longitudinal distance. Afterward, the five-axle vehicle accelerates at 270 m, overtakes vehicle A, and moves away from vehicles A and B to avoid collision. Since the vehicle C always moves forward at 20 m/s, there is no effect on the five-axle vehicle.

In Figure 9, the five-axle vehicle cruising at speed of 25 m/s. In this case, vehicle A is far away from the five-axle vehicle and in acceleration mode, meantime, vehicle C is always faster than the five-axle vehicle. Therefore, these two vehicles have no influence on the five-axle vehicle. After bypassing the obstacle $A_{\mathrm{obs}}$ by decelerating to reduce the risk of collision, the five-axle vehicle transcends vehicle B at 240 m and accelerates until it reaches the reference speed. The spatio-temporal relationship of the vehicle at different speeds is illustrated in Figure 10, and the variation of the sideslip angle-yaw rate and roll angle during driving are all illustrated in Figure 11.

Figure 10.

The spatio-temporal map in different speed: (a) the spatio-temporal map at 18 m/s and (b) the spatio-temporal map at 25 m/s.

Figure 11.

Five-axle vehicle state variation in different speeds: (a) sideslip-yaw rate and roll angle variation at 18 m/s and (b) sideslip-yaw rate and roll angle variation at 25 m/s.

The three-dimensional relationship in Figure 10 indicates the relative position between the five-axle vehicle and the surrounding vehicles at different times. According to Figure 10, the dynamic obstacle avoidance can be effectively achieved by using the path speed collaborative planning method proposed in this paper. Meanwhile, Figure 11 shows that the driving stability and safety can be ensured by limiting the wheel angle and vehicle speed.

The computational time of the algorithm is shown in Figure 12. It can be seen that most of the computational time is controlled within 0.05 s, and the maximum is no more than 0.2 s. However, the max computational time still meets the requirement of planning cycle.

Figure 12.

The computational time of the algorithm: (a) computational time at 18 m/s and (b) computational time at 25 m/s.

Experiment for curved road driving scenarios

To further verify the correctness of the proposed method, the HIL experiment is used in the curve driving situation (Figure 13). In the practical application of the methodology, a simplified vehicle dynamics model can be established according to the real vehicle parameters, and the MPC-based trajectory planner can be built accordingly. The trajectory can be calculated in real time according to the input of external sensors, and the generated trajectory can be sent to the vehicle controller.

Figure 13.

HIL verification for curved road driving scenarios.

The testbed consists of driving motors and loading motors which are driven by motor drives. To maximize the feasibility verification of the proposed method, the driving motor is used to simulate the driving ability of the vehicle, while the vehicle driving condition is simulated by the loading motor. By obtaining the speed and torque signals offered by speed and torque sensors on driving and loading motors, the vehicle performance is simulated by high-precision TruckSim dynamic model mentioned in Section 2.4 in real-time simulator. Thus, the influence of vehicle dynamics characteristics (such as tires, etc.) are also considered in the accurate dynamic model of real-time simulator. The vehicle controller in the simulator regulates the steering angle and wheel torque of the TruckSim model based on the sensor signals received from the hardware platform and the driving trajectory generated in accordance with the predefined scenario, outputs the vehicle state to verify the advantages of the trajectory generation algorithm in terms of stability.

In this case, there are three driving vehicles around the five-axle vehicle. The speed of the five-axle vehicle is set to 20 m/s, while the initial speed of the surrounding vehicles is set to 15 m/s for vehicle A, 10 m/s for vehicle B, and 13 m/s for vehicle C. These vehicles drive at variable speed, and the acceleration variation range is $0-2m/s^2$ . In this section, the vehicle trajectory is expressed in the same way as in Figures 8 and 9.

Along with Figure 14, when the vehicle is driving on the curved road, it starts to turn at 100 m. Meantime, the vehicle decelerates to ensure its stability and safety. After that, the five-axle vehicle realizes the obstacle avoidance of vehicle B with a lower speed firstly. Since vehicle B is currently accelerating, the five-axle vehicle first accelerates to the reference speed at 120 m, approaches vehicle B and continues to accelerate until it transcends vehicle B at 170 m. Due to the long lateral distance between vehicle C and the five-axle vehicle, the five-axle vehicle slightly avoids the vehicle C and transcends after approaching vehicle C. Then, the five-axle vehicle gradually slows down to the reference speed and begins to follow vehicle A around 380 m. Figure 15 shows the entire motion process of the vehicle through the spatio-temporal map.

Figure 14.

Vehicle driving trajectory on curved road.

Figure 15.

The spatio-temporal map on curved road.

Figure 16 shows that in this driving situation, the five-axle vehicle is always in the stable region when driving according to the planned trajectory. In extreme case, the vehicle can actively reduce its speed to ensure the stability of obstacle avoidance and effectively control the roll angle (right-side of Figure 16). Figure 17 illustrates the computational time of the algorithm, revealing that the average execution time has increased in comparison to the straight road, it remains within acceptable limits.

Figure 16.

Five-axle vehicle state variation at 20 m/s.

Figure 17.

The computational time of the algorithm at curved road.

To further clarify the benefits of the proposed method, the driving trajectory is generated without considering the stability of obstacle avoidance under the same conditions (blue trajectory in Figure 18). Moreover, the trajectory in Figure 14 is also presented here for comparison (red trajectory in Figure 18). Meantime, the motion states and roll angle variation are generated when the vehicle drives under unconstrained conditions, as shown in Figure 19.

Figure 18.

Vehicle driving trajectory without constraints.

Figure 19.

Five-axle vehicle state variation without constraints.

As can be seen from Figure 18, under the same obstacle avoidance situation, the maximum curvature of the trajectory generated by these two methods is 0.011 and 0.006 respectively, and the curvature of the driving trajectory planned by the proposed method is reduced by 45%. However, a larger curvature area locally (at 140 m) means a higher steering angle change rate when the vehicle controller performs trajectory tracking, which means instability.

The comparative analysis of Figures 16 and 19 shows that the maximum yaw rate of the driving trajectory generated by the proposed method is about 0.25 rad/s and the maximum roll angle is about 0.0026 rad. When the obstacle avoidance stability is not considered, the maximum yaw rate of the vehicle is 0.35 rad/s and the maximum roll angle is 0.035 rad (see Table 5). Compared to the latter, the maximum yaw rate and roll angle are reduced by about 28.5% and 25.7%, respectively, when the vehicle is driven on the trajectory generated by the proposed method. Therefore, when tracking this trajectory, the controller easily causes the vehicle to become unstable. Furthermore, compared with the latter, the average speed of the former in obstacle avoidance is increased by 5.2% under the premise of maintaining the vehicle stability. This means that the former planning trajectory has certain advantages in terms of driving efficiency.

Table 5.

Comparison of parameters between the two methods.

Items	Proposed method	No const. method	Performance improvement (%)
Yaw rate (rad/s)	[–0.2, 0.25]	[–0.25, 0.35]	28.5
Roll angle range (Rad)	[–0.026, 0.026]	[–0.026, 0.035]	25.7
Avg. speed in obstacle avoidance (100–200 m)	19.46 m/s	18.49 m/s	5.2
Maximum curvature	0.006	0.011	45

Conclusions

This paper takes a five-axis vehicle as an example, aiming at the stability and safety problem of the multi-axis vehicle during high-speed maneuvering, proposes a safe trajectory planning method for multi-axis vehicle considering the obstacle avoidance stability.

• (1) Taking a five-axle vehicle as an example, the dynamic model is established with four degrees of freedom, including longitudinal, lateral, yaw, and roll motion. The correctness of the model is verified by comparing the dynamic responses of the TruckSim model and the established model.

• (2) Aiming at the obstacle avoidance stability problem of multi-axis vehicles, the stability constraints of multi-axis vehicles are established based on the phase plane of sideslip angle and yaw rate. Furthermore, based on the vehicle motion state, the multi-axis vehicle obstacle risk field is established.

• (3) In order to generate a safe driving trajectory of multi-axis vehicles, a trajectory planner based on model predictive control is designed. Simulation experiments show that the designed planner can meet the stability requirements of obstacle avoidance and driving safety of multi-axle vehicles.