Planning and Tracking Control of Full Drive-by-Wire Electric Vehicles in Unstructured Scenario

The artificial potential field (APF) method is widely used for the path generator. However, it currently suffers from the local optimal solution problem and target unreachability. Based on it, some scholars have proposed some improved algorithms[34, 35]. In this paper, modifying the repulsive force function and repulsive direction have been presented.

(1) Repulsive force function optimization

When the moving vehicle, the target point, and the obstacle are all in the same line and the target point is close to the obstacle, the attractive force of the target point to the moving vehicle decreases while the repulsive force of the obstacle from the moving vehicle increases. As a result, the moving vehicle cannot reach the target point since the repulsive force does not vary as the driving vehicle gets closer to the target point in this scenario. Therefore, modifying the repulsive force function decreases the repulsive force as the moving vehicle gets closer to the target point. In this way, the repulsive and the attractive force are equal to zero at the target point, and the vehicle can maintain the target point. The improved repulsive field function is described as follows:

where $U_{rep,j}$ denotes the improved repulsive force function, $\eta_{j}$ is the coefficient constant of the repulsive force function. $d_{j}(x)$ is the shortest distance between the moving vehicle and the $j_{th}$ obstacle. $Q^{*}_{j}$ is the maximum range of the influence for $j_{th}$ obstacle on the moving vehicle. When the shortest distance between the moving vehicle and the obstacle is less than $Q^{*}_{j}$, the repulsive force function will affect the moving vehicle. $d(x,x_{goal})$ denotes the shortest distance between the moving vehicle and the target point.

The corresponding repulsive force function is:

where:

$F_{rep1}$ and $F_{rep2}$ are the two splitting forces of $F_{rep}$, where the direction of $F_{rep1}$ is from the center of the obstacle pointing to the moving vehicle, the direction of $F_{rep2}$ and the attractive force are the same.

This improved method incorporates an adjustment factor $\mathrm{d}(x,x_{goal})$ so that the repulsive force decreases while the moving vehicle is approaching the target point. The repulsive force is zero, and the attractive force is also zero when reaching the target point.

(2) Repulsive force direction reformation

Altering the repulsive force described above is insufficient to solve either of the difficulties. As a result, based on the repulsive force function, modifying the direction of repulsive force $F_{rep1}$ has been conducted. $F_{rep1}$ and $F_{rep2}$ are the two splitting forces of the repulsive force, and the direction of $F_{rep1}$ is from the obstacle to the moving vehicle, and the direction of $F_{rep2}$ is from the moving vehicle to the target point. When the angle between $F_{rep2}$ and the attractive force is larger than $90^{\circ}$, this may nevertheless result in a locally optimal solution. Therefore, modifying the direction of the repulsive force is to define $F_{rep1}$ as a tangent direction along the influencing range of the obstacle and the angle between it and the attractive force, which is less than or equal to $90^{\circ}$.

(1) Vehicle dynamics constraints

According to the vehicle dynamics theory analysis, the under-steering gain can be defined as $K=\frac{m}{(l_{f}+l_{r})^{2}}(\frac{l_{f}}{C_{\alpha r}}-\frac{l_{r}}{C_{\alpha f}})$. Then, steering gain is the ratio of the yaw rate to the front wheel angle, which is $\frac{\varphi}{\delta_{f}}=\frac{V_{x}/(l_{f}+l_{r})}{1+KV^{2}_{x}}$, ensuring the vehicle’s stability and safety.[36].

In the vehicle coordinate system, the vehicle lateral acceleration can be expressed as:

To avoid obstacles, the vehicle is steering in such a way as to satisfy steady-state steering and to make certain that the lateral side slip angle of the center of mass is as small as possible. Assuming the velocity of the vehicle along the longitudinal axis remains constant, the lateral acceleration can be represented as follows:

To make the tires in the linear area, the lateral acceleration should be no more than 0.4 g, and the front wheel needs to be satisfied as follows:

Therefore, the vehicle’s steering would be simultaneously impacted by the velocity and steering radius. And the front wheel steering angle should meet smoothly and securely:

During steering, the longitudinal speed is far too high to have any impact on the safety of the vehicle, and the vehicle speed at any given curve can be satisfied:

Eq. 28 the limited speed under ideal conditions, it is required to adjust the vehicle speed by inserting an adjustment factor $\lambda_{d}$ by considering the real instantaneous scenario:

It can be seen that the only factors that influence the limited speed are the radius of the road, denoted by $\lambda_{d}$, and the maximum adhesion coefficient, denoted by $\mu$. Then the longitudinal reference speed should also consider the maximum legal speed restriction. Therefore, the following can be obtained[37] :

where $V_{set}$ is the desired target speed.

(2) Vehicle kinematic constraints

The front wheels need to be in accordance with the ideal Ackermann geometry steering, and the vehicle kinematic model can be described as follows.

The incomplete constraint can be expressed as:

where $\theta$ is the angle between the vehicle and the X-axis direction. Based on the above constraints, it is clear that the kinematic constraints include a speed limit $V_{\lim}$ without side slip, with different curvatures:

The reference speed $V_{ref}$ will be selected as the smallest speed value.

In the lateral dynamics model, the defined preview deviation $y_{L2}$ contains three parts, $\Delta y_{L2}$ indicating the future path change, $\Delta y$ representing the current lateral position deviation, and $\Delta\varphi$ indicating the current heading angle deviation.

(1) Curvature calculation based on current lateral deviation $\Delta y$

To calculate and correct the deviation between the current path and the reference path, PID controllers, as the most common feedback controllers used in industrial fields, were first employed due to no representation of the internal system and the straightforward application to determine the ideal curvature $\kappa_{\Delta y}$ based on the present deviance. The following is an equation representing the PID controller.

The $K_{p}$ is the proportional coefficient, the $K_{i}$ is the integral coefficient, and the $K_{d}$ is the differential coefficient. The I controller provides the most effective results by modifying the three parameters in the repeated simulations. As a result, the I controller is used to determine the ideal curvature. Then, the sensor measurement has a certain delay in the real-world case, which results in a delay in the acquired current lateral deviation $\Delta y$. So a delayed module $e^{sT_{Delay}}$ needs to be introduced to provide a more accurate control in Fig. 3.

(2) Curvature calculation based on preview deviation $y_{L2}$

For the control loop of the $y_{L2}$, the data delay is first considered to get the delayed preview deviation $y_{L2delayed}$, and then the derivation from the lateral dynamics model is given:

The preview deviation distance can therefore be converted into the ideal curvature as shown below:

The conversion of the preview deviation into the corresponding curvature value by gain $2/{L^{2}_{e}ff}$ is conducted. Then a parameter $G_{out}$ can be added between the two distances, the range of which is set to $0<G_{out}<1$ to make the steering less aggressive. Since the vehicle is subject to side slip, a compensation amount is considered to calculate the preview distance deviation to obtain $y_{L2_{slipcomp}}$, which is:

where

The compensated preview distance deviation is brought into the curvature calculation module as the actual preview deviation, which is $y_{L2_{slipcomp}}=y_{L2}$. The control block diagram can be obtained as shown in Fig. 4.

Finally, the ideal curvature $\kappa_{\Delta y}$ based on the current deviation and the ideal curvature $\kappa_{y_{L2}}$ based on the preview distance are transformed into the corresponding steering angle in different cases, the conversion between the steering angle and the ideal curvature is derived as follows:

The effective preview time can be obtained from the fuzzy controller and the speed limit obtained from the vehicle dynamics and kinematics. An effective preview distance can be obtained by multiplying the speed with the effective preview time. The CC-based tracking controller framework can therefore be obtained as shown in Fig. 5. $G_{out}$ is an adaptive parameter, and $LocalOffsetLimit$ is set to 0.2m, and the logic pseudo-code for the judgment is:

(1) Optimization objectives

The aim is not only to require the vehicle to produce the smallest lateral deviation but also to achieve the smallest steering wheel angle variation based on the consideration of the stability and safety [38, 39]. Therefore, the optimization objective function is defined as follows:

The reference path information is defined as:

Normalization is required due to different magnitudes of three terms for the objective function; then the objective function can be expressed as:

The vehicle lateral tracking deviation is defined as:

Bring the lateral tracking deviation into the objective function as follows:

The above equation is simplified as follows:

It is easy to see that M and N are irrelevant to $\Delta U(k)$. Then the dimensions of the input, output, and state variables in the system are assumed to be l, m, and n; therefore, the dimensions of the matrix can be obtained as shown in Table 1.

(2) Constraints

To ensure the safety and stability of the vehicle, limitations on the steering wheel angle, steering wheel angle rate, lateral acceleration, lateral deviation, and actuator capacity should be considered when designing the control algorithms. Firstly, the constraint of the front wheel angle as the model input is set as $-25^{\circ}-25^{\circ}$, which is the extreme value of the steering angle. Thus, the constraints on the front wheel angle and the corresponding variation are designed as follows:

Then the lateral acceleration is $\alpha_{y}\leq\mu g$ in the theoretical case, and this paper sets the limit as $\alpha_{y}\leq 0.85\mu g$ for safety. The yaw rate can also be obtained as follows:

Finally, the side slip angle cannot be exceeded to $5^{\circ}$, within which there is a linear relationship between the lateral force and side slip angle according to the tyre characteristics. Therefore, the constraint of the side slip angle of the front wheel is:

(3) Optimization with constraints

The above constraints can first be expressed as linear inequalities as:

where the matrix $F$ is denoted as $f=[F_{1},F_{2},\cdots,F_{H_{u}},f]$, $F_{i}$ is the matrix of $q\times m$, $f$ is the matrix of $q\times 1$, it can be described as:

Because

Bring this equation into Eq. (53):

where $F_{i}=\sum_{j=1}^{H_{u}}F_{j}$

Thus Eq. (51) can be transformed into a linear constraint on $\Delta U(k)$. Similarly, Eq. (52) can be transformed into a constraint on $\Delta U(k)$.

where $G=[T,g]$, $g$ is the last column of $G$, since it is obtained:

Furthermore, it can be described as:

Finally, it can be summarized as:

Combining with three equations, it can be described as:

Thus, the optimization problem can be expressed as follows:

The optimization solution is obtained by solving the gradient function for the objective function and making it equal to 0.

So the optimal solution is obtained as follows:

A practicable path within a specific range is designed for the FDWEV, with various obstacles, the start point, and a target point to offer correct position information for the tracking control module. After obtaining the path point information, the generated points will be interpolated with a three-order spline. Then the points are processed using a five-order polynomial fitting for every 10 path points to make them meet the vehicle driving requirements and generate a reasonable path[40]. Then simulation settings with vehicle speeds of 30 km/h and 50 km/h are created in a specific scenario with considerable curvatures to validate the effectiveness of the tracking control algorithm.

(1) Vehicle speed is 30 km/h

According to the various deviations and curvature inputs, the output corresponding to the preview time $T_{p}$ can be generated using the fuzzy controller, as illustrated in Fig. 6 (a). Based on the speed limitations for the curvature of the road, in this paper, we use the safety factor, which is defined as the adjustment factor $\lambda_{d}$ = 0.65, and the adhesion coefficient $\mu=0.85$. As a result, the reference speed curve with the restriction can be obtained, and the speed tracking results show good performance by the chassis integrated control strategy, as shown in Fig. 6 (b).

Fig. 6 (c) depicts the path tracking results at a speed of 30 km/h working condition, and Fig. 6 (d) depicts the corresponding lateral deviation. To determine the present curvature, a delay is introduced with $T_{Delay}=0.065s$, and the PID controller computes a curvature value. By repeated simulations, $K_{i}=1.5$ is eventually determined, implying that the I controller is employed to regulate the deviation directly. A delay with $T_{Delay}=0.065s$ is also added to determine the preview deviation, the preview time determined by the fuzzy controller can then be multiplied by the reference speed to obtain the variable preview distance and the gain is set as $G_{out}$ = 0.8, which would produce a curvature value with $LocalOffsetLimit$ = 0.2m. As a result, as illustrated in Fig. 6 (e), the steering wheel angle can be calculated, where the transmission ratio is chosen as 16. Further, as illustrated in Fig. 6 (f) with the lateral acceleration, it shows that the vehicle is mostly in the stable area and the steering curvature is significant, and near to 0.1, the lateral acceleration reaches its maximum, increasing the lateral force and allowing it to negotiate the curve successfully.

(2) Vehicle speed is 50 km/h

Fig. 7 (a) depicts the obtained preview time $T_{p}$ when the vehicle speed increases to 50 km/h, where the road curvature is large, the preview time and distance are both reduced, and the majority of the preview time is one second. Because the maximum adhesion given by the tires is regarded as the major speed restriction and the vehicle’s limit in speed planning, when the speed is raised, however, the vehicle speed cannot be instantly lowered if the curvature is larger than 0.1. Therefore, it is necessary to slow down the vehicle earlier by establishing a lower speed threshold. The vehicle’s speed planning and tracking results are illustrated in Fig. 7 (b). The tracking path and accompanying lateral deviation are determined by lowering the vehicle speed immediately before approaching the curve and allowing the vehicle to finish the steering operation for varied curvatures, as shown in Figs. 7 (c) and 7 (d). It creates a greater deviation than 30 km/h since increasing the speed makes the vehicle more prone to side slip during the steering, and even if the vehicle is in a stable zone in advance, by reducing the speed, the actuators will have a delay in taking action. As a result, although the curvature calculation (CC-based) method is straightforward to be calculated, it does not take into account different restrictions and real-time feedback correction according to the vehicle’s condition. Therefore, it will generate larger lateral deviations at higher vehicle speeds when the road curvature is rather considerable.

Comparing the tracking results under different speeds, the CC-based tracking controller performs better when the vehicle speed is relatively lower in such a complicated environment. However, when the vehicle speed increases, it is not enough to reduce the speed to a relatively low level in advance where the curvature is significant. This is mostly because this method does not have real-time feedback correction of the vehicle’s state and considers different limits. As a result, it is not ideally suitable for an environment with greater operational difficulty. On the other hand, this approach is straightforward and does not require significant computation, so it is more applicable under typical working conditions.

The simulation settings of vehicle speed 30 km/h and 50 km/h have been specified separately to validate the effectiveness of the tracking controller based on the MPC, with the control goal and the process of continuous feedback optimization.

(1) Vehicle speed is 30 km/h

Based on the established vehicle model with both lateral and yaw directions in MPC, the road surface adhesion coefficient is $\mu=0.85$, and the MPC controller is written using the S-function in Simulink with parameters: sample period $T=0.05s$, prediction step $N_{p}=25$, control step $N_{c}=10$, and the objective function weights are set as follows:

The slack factor is taken as 1000, the output deviation constraint is [-0.5, 0.5], and the front wheel steering angle constraint is [$-20^{\circ}$, $20^{\circ}$]. Therefore, the results can be obtained and presented in Fig. 8. In Fig. 8 (a), the vehicle can maintain the longitudinal speed; the tracking control precision is within 0.3 m, which has a strong tracking capability in extremely complicated environments. From the lateral acceleration, the vehicle is in a rather stable area; only when the curvature is at its greatest extent will the lateral acceleration reach its greatest value. This is because there will be a greater requirement to increase the lateral force.

(2) Vehicle speed is 50 km/h

The simulation results are depicted in Fig. 9 following an increase in the vehicle speed. The vehicle is capable of maintaining the desired speed. The path control deviation in this complicated environment can be controlled to within about 0.5 m, and it can be seen from the lateral acceleration and yaw rate that an increase in vehicle speed does not result in a decrease in the performance of the vehicle’s stability. Similarly, except for the abrupt bends with the greatest curvature, the lateral acceleration will reach its maximum to give an adequate amount of lateral force, but the longitudinal speed will decrease. Because the combined effect is required to help the vehicle safely in a more complicated environment, in which decoupling the longitudinal and lateral motion is tough.

MPC tracking controller can produce satisfactory tracking results as well, when the speeds are relatively modest, there is no significant difference in the tracking precision provided by the two methods. However, in a complicated environment, when the vehicle speed rises, the curvature is still rather significant in some locations. MPC with the vehicle dynamics model can adjust the vehicle’s state and predict the vehicle’s output for a future period by continuous feedback optimization correction. Consequently, it also has an improved tracking capacity at faster speeds.