3.1. Design of a Steering Anti-Rollover Controller Based on Sliding Mode Control
In order to ensure that the vehicle still has good lateral stability when steering under different road conditions, the design of the steering anti-rollover controller must consider the influence of the yaw velocity and the center-of-mass lateral deflection angle at the same time. This paper assumes that, in the ideal state of vehicle motion, based on the two-degree-of-freedom vehicle dynamics model established in
Section 2.1, we can obtain the ideal yaw velocity and center-of-mass lateral deflection angle. At this time, the yaw velocity and lateral acceleration of the vehicle are constant values and can be obtained as follows:
,
. Therefore, the ideal yaw rate can be obtained [
27] as follows:
In that, .
Simultaneously, due to the limitations imposed by the ground adhesion limit, the desired yaw rate needs to satisfy the following condition:
Wherein
f represents the safety factor, set at 0.85;
denotes the road adhesion coefficient; and
g stands for the acceleration due to gravity. Consequently, the final desired value for the yaw rate is obtained as follows:
In that, .
Incorporating the additional yaw moment
into the two-degree-of-freedom model yields the following result:
The yaw rate tracking error and its derivative are defined as follows:
Simultaneously, from several sliding mode control laws, the uniform velocity approximation law is selected for its lower computational complexity, superior real-time performance, and enhanced robustness, specifically:
In the formula,
represents the approach velocity to the sliding mode surface, which is a positive constant. By converting Equation (23) to Equation (26), the additional yaw moment can be obtained as follows:
3.2. Design of an Active Suspension Controller Based on Fuzzy PID
The basic principle of the fuzzy PID controller is to regulate the increment of PID parameters based on the error e between the ideal rollover evaluation index and the rate of change of this error as inputs to the controller. In this article, the ideal rollover evaluation index RI is set to 0. The actuating forces of the vehicle’s left and right suspension systems are considered output parameters of the controller. In this paper, the values of the three parameters
Kp,
Ki, and
Kd are obtained through the empirical trial-and-error method, being 1000, 10, and 20, respectively. The input variables of error and error rate of change, as well as the output variable for the fuzzy controller, are all described using the following seven fuzzy subsets:
The fuzzy domains of the input variables and output variables are set as {−3, −2, −1, 0, 1, 2, 3}. The physical domains of error e and error change rate ec are [−1, 1] and [−25, 25], respectively, and the physical domains of the three output variables are [−1800, 1800], [−12, 12], and [−150, 150]. In this paper, we achieve the purpose of order of magnitude unification by adding the scaling factor; thus, according to the above physical domains, the quantization factor can be obtained as 3 and 0.2, respectively, and the scaling factors are 600, 4, and 50, respectively. At the same time, Gaussian and triangular affiliation functions are selected to describe the input and output quantities and the corresponding fuzzy subset affiliation. The fuzzy control rules are formulated according to the relevant literature [
28], as shown in
Table 1,
Table 2 and
Table 3. Finally, the three output variables can be obtained as a function of the input variables, as shown in
Figure 5.
3.3. Design of a Path Tracking Controller Based on a Linear Quadratic Regulator (LQR)
The LQR control method is adopted for the design of the coupled controller in this paper due to its ability to accommodate multiple performance indices and meet the engineering practice requirements of unmanned mining trucks in complex mine area environments. LQR optimal design theory refers to the concept in which the designed state feedback controller
K minimizes the quadratic objective function
J. The quadratic objective function
J(
u) can be represented as follows:
In the expression, Q and R represent the weighting matrices of the controller, with R being a positive definite matrix.
The essence of the LQR controller is to design a control law that minimizes the quadratic objective function
J(
u). By employing the principle of least action to solve the optimal control problem, one ultimately obtains the optimal control input
u*(
t):
In the formula, K(t) = [k1, k2, k3, k4] represents the feedback gain matrix of the LQR controller.
Let the front wheel steering angle outputted by the feedforward control be denoted as
, then the feedback control law, upon incorporation of feedforward control, is expressed as follows:
Substituting the above equation into Equation (2), when the system stabilizes, thus
becoming zero, it follows that:
By substituting
A1,
B1,
K, and
C1 into the previous expression, we obtain:
When
ye is zero, and assuming that the vehicle is traveling at a given speed, the desired path can be approximated as a curve with a constant curvature
. Thus, the quantity of the feedforward controller
can be determined as follows:
3.4. Design of a Path following and Anti-Roll Coordinated Control System
When unmanned mining trucks operate under unstructured road conditions, their stability and tracking performance exhibit certain variations. Based on anti-roll control and path tracking control strategies, this study utilizes the theory of extension to switch between different control strategies according to vehicle state and tracking accuracy, ensuring stable and precise vehicle operation. The yaw rate deviation and the lateral deviation angle of the center of mass are selected as characteristic quantities in this paper.
Taking the lateral deviation angle of the center of mass as the abscissa of the extensible set and the yaw rate error as the ordinate of the extensible set, a two-dimensional set based on the characteristic states of the vehicle is established as
. Furthermore, the range of variation for the extensible set is defined as
, and three distinct zones are generated that can map the vehicle tracking control state quantities in real-time: the stable zone, the extensible zone, and the unstable zone. For the lateral deviation angle of the center of mass, serving as the abscissa of the extensible set as
, the boundary of the extensible zone can be determined based on the formula for vehicle instability conditions as follows:
The determination of the extendable region boundary
varies with changes in the road surface adhesion coefficient. On the other hand, the boundary of the stable region
is ascertained based on the linear region of the yaw rate gain. By employing multiple simulations and fitting a curve to the resulting data of vehicle speed
vx, the road surface adhesion coefficient
, and the maximum front wheel steering angle
, a relationship amongst the three parameters can be established as follows:
Substituting into Equation (12), the boundary of the stable region can be obtained. Meanwhile, for the extendable set, the longitudinal coordinate is the yaw rate error, which is determined using the tolerance band division method.
When occurs, the vehicle is in an unstable state and is at risk of rollover.
When , the state of the vehicle is within the extendable region, corresponding to the boundary of the extendable region .
When , the vehicle is in a stable condition, corresponding to the stable boundary .
As can be obtained from Equation (8), parameters
and
, according to related references and empirical methods, are known to be 0.05 and 0.15, respectively. Hence, the boundary of the controllable extension set region can be determined, which varies with changes in the road surface adhesion coefficient and the tripping-type rollover index. The range of each region within this extension set is illustrated in
Figure 6.
Based on the one-dimensional extension distance theory, the nearest extension distances from point S in the above figure to the stable region and the extendable region can be represented as
and
, respectively. Their values are as follows:
The formula for the correlation degree function is as follows:
Concurrently, different control strategies are selected based on the varying numerical values of the correlation degree function. When , the vehicle resides within the stable region, indicating that the current state of vehicle motion is steady, necessitating an increase in the weight of the front wheel steering angle to ensure higher tracking precision. When , the vehicle is situated in the expandable region, signifying a tendency towards instability where solely controlling the front wheel steering angle may lead to vehicle rollover instability. This necessitates the involvement of additional yaw torque, and a redistribution of weights to ensure an improvement in vehicle body posture at bends and enhance vehicle stability. When , the vehicle is in the unstable region, indicating that the vehicle has reached the conditions for instability. It is necessary to increase the weight of the additional yaw torque and adopt anti-rollover control to prevent vehicle rollover.
To prevent the weight distribution from being biased towards a single aspect and to ensure greater stability in the intervention and withdrawal of the two control modes, a sigmoid function [
29] is employed to redistribute the two control strategies. Consequently, the control weight for the front wheel steering angle can be represented as follows:
The control weight for the additional yaw moment is denoted as .