A Novel Integrated Stability Control Based on Differential Braking and Active Steering for Four-axle Trucks

Rollover and lateral instability are two common issues of heavy-duty trucks, which can result in fatal accidents and massive losses. Differential braking (DB) and active steering (AS) are two of the most effective methods to improve the vehicle yaw stability and untripped rollover accidents [1]. DB and AS systems have their own disadvantages, and researchers have proposed several integrated controllers to overcome them. However, existing research mainly focuses on two-axle vehicles or multi-axle vehicles with two-axle equivalent models [2]. Heavy-duty trucks usually have more than two axles for increased transportation efficiency [3]. DB or AS individually might not control the lateral instability and rollover accident of multi-axle trucks, because multi-axle trucks usually have higher center of gravity and heavier mass and they work under difficult conditions. In a multi-axle truck, distributing the reasonable braking torques and steering angles to suitable wheels is complex. The integrated controller design in a two-axle vehicle cannot be used directly in a multi-axle vehicle. The integrated control system for a multi-axle truck has not been fully studied. Therefore, a practical and effective integrated control system is needed urgently, especially for multi-axle trucks.

DB systems can slow down the vehicle and make the driver feel safe under high-speed conditions. In addition, the trajectory deviation caused by DB is smaller than that caused by AS. DB systems can also be easily implemented in conventional braking systems. For instance, electronic stability programs, electronic stability control (ESC), and direct yaw moment control (DYC) are widely used in heavy-duty vehicles, which apply DB or driving torques on wheels such that the vehicle active safety can be greatly improved [4‐6]. By contrast, AS systems can generate an external yaw movement by actively turning the vehicle wheels in a small range [7]. Three types of AS, namely active front steering (AFS) [8, 9], active rear steering (ARS) [10], and four-wheel steering [11], are defined in existing literature. The AS system cannot generate the required yaw moment under certain circumstances [12]. The control effects and limits of AS and DB have been extensively discussed in literature. Koibuchi et al. [13] discussed the pros and cons of the DB system for a passenger car and concluded that when an inward moment is needed, the braking force is applied to the rear-inner wheel and vice versa. Yang et al. [14] showed the effects of all-wheel AS for a three-axle truck. Balázs et al. [15] discussed the steering angle limit and steering rate limit and proposed different actuation-level steering control methods for articulated vehicles. They presented the effects and limitations of each control method without comparing them and without providing selection criteria for specific conditions.

Most integrated control methods work in two steps. First, the total control inputs are calculated. Then, they are distributed using a coordination logic, rule, or optimization process. The controller in Ref. [16] is developed on the basis of a two-level control structure. In the upper level, the required yaw moment and rear steering angle are calculated, while in the lower level, the braking torque is distributed. The most important part of the integrated control is the coordination of DB and AS. A popular method is the phase plane \(\left( {\beta ,\dot{\beta }} \right)\) analysis. In Ref. [17], AFS and DB are activated together or separately based on regions of stability index derived from the phase plane analysis. In Ref. [18], AFS and DYC have been integrated via fuzzy logic based on stability index obtained by the phase plane analysis. The phase plane analysis is widely discussed and applied for vehicle stability analysis and control [19‐21]. Empirical methods such as setting gains [22, 23] and fuzzy logic [18, 24] are also employed in integrated control methods. They are practical, but they require significant simulation or experimental data. Other researchers use optimization methods and model predictive control (MPC) to distribute AS and DB [25‐27]. They cannot completely replace the functions of rules or logic. The coordination still mainly depends on the rule and logic design. The MPC or optimization may also make the control system complex. In other research, DB and AS are used to compensate for the other when one of them has reached its saturation limit. In Ref. [28], ARS was used to extend tire limitations, and four-wheel drive/ESC was used to optimally distribute the longitudinal forces. In Ref. [29], researchers controlled the vehicle lateral stability by DB and AS sequentially. Once the vehicle stability cannot be maintained only by using DB, AS starts to work. In Ref. [30], the braking system is used to generate large lateral forces when the steering tire is saturated. In Ref. [31], the priority of DB is set lower than that of AFS, and DB is functional only when activated. Existing studies have not compared DB and AS, and the characteristics of multi-axle truck are also not considered. Moreover, most studies have not activated different control methods at the same time [28‐33]. The advantages of DB and AS can only be found when they are activated. If they are activated simultaneously, the integrated control system can genuinely overcome the disadvantages of DB and AS. If the integrated controller has an optimization system in the control process, DB and AS can be activated simultaneously. However, for a multi-axle truck, more axles mean more variables, making the optimization more complex and slower.

This paper proposes a novel integrated control system designed for a four-axle truck with fewer experiments. AS and DB can work simultaneously to improve the roll and yaw stability. To improve the robustness of the control system without many experiments, a novel comparison method for DB and AS is also proposed. This research is organized as follows. In Section 2, a novel method to accurately predict vertical tire forces in a four-axle truck is proposed. In Section 3, the effects of DB and AS are analyzed and compared and the limits of AS and DB are presented. In Section 4, a sliding mode controller is designed to determine the total yaw moment and lateral force. They are distributed on the basis of the DB and AS analysis results and transferred by optimization systems to steering angles and braking forces to each wheel. Because DB and AS are integrated on the basis of a real-time analysis of their ability and potential, the system has better robustness. In Section 5, the simulation results show that the proposed control system can obviously enhance the stability of multi-axle trucks. Finally, the conclusions are drawn and future work is discussed. Based on the results, this paper has proposed an integrated control system that can reasonably solve the problems discussed above and utilize the ability of two stability control systems further. Moreover, this paper proposes a comparison method of DB and AS, which can help for better coordination between them.

When analyzing DB and AS, the vertical forces of tires for the four-axle truck need to be accurately estimated beforehand. In existing methods, a dynamic load transfer coefficient is used, making the calculation more complex. Therefore, a novel method is proposed to identify these vertical forces and avoid using this coefficient. Assume that the variations of tire vertical forces are only caused by the lateral acceleration, longitudinal acceleration, roll angle, and roll angle rate. The four-axle truck body is separated into three parts and the center of gravity (CG) of the truck is not necessary. As shown in Figure 1, \(cg_{1}\), \(cg_{2}\), and \(cg_{3}\) are the local CG points of each part. Part 1 consists of the unloaded truck and first axle; the height of \(cg_{1}\) is equal to that of the unloaded truck. Part 2 consists of half the load and second axle; \(cg_{2}\) refers to the CG of the front part of the cargo, which is identical to the height of the CG of the entire cargo. Part 3 consists of the other half part of load, third axle, and fourth axle; \(cg_{3}\) is the CG of the rear part of the cargo. Several hypothetical internal forces (\(F_{zali,zari}\), i = 1, 2, 3) are introduced to each part. The symbols used in figures and equations are introduced in the Nomenclature (in Appendix). The tire vertical forces can be defined as Eq. (1):where \(F_{{zri_{0} ,zli_{0} }}\) represents the static vertical forces of tires and can be easily obtained from sensors or calculation. ΔF_zri,zli is the vertical tire force variation. Equations (2)–(10) are derived from the conservation of moment. As an example of the modeling of different parts, Figure 2 shows a simple model of Part 1, and Eqs. (2)–(4) can be derived easily.

Part 1

Part 2

Part 3

Figure 3 shows the comparison of results from calculation and TruckSim. The loaded truck is 5000 kg, and the unloaded truck is 4457 kg. The truck in TruckSim only has one steering axle. The maximum percentage deviation of vertical forces is 9.19%.

Both AS and DB controllers are designed as yaw rate feedback controllers. Therefore, the yaw moment or yaw rate generated by AS or DB systems can act as an index to evaluate their control ranges and potentials. In this section, these two indexes of both DB and AS are analyzed and estimated. A novel comparison method of DB and AS is also proposed and used in the integrated control system.

Using one optimization process to determine the braking torques and steering angle of a multi-axle truck is complex. Moreover, introducing too many variables slows down the optimization process. A novel control strategy, as shown in Figure 7, is proposed with several simple optimal calculations based on the analysis method discussed in Section 3. In this study, assume the roll angle, roll angle rate, yaw rate, and slip angle are acquired by sensors or observations such as those in Ref. [34]. The corresponding estimation algorithms for unmeasurable variables will be designed in future work for real applications. Based on the lateral load transfer ration (LTR) and yaw rate judgment, the strategy determines whether to activate the control system. When a control action for preventing the accident is required, the analysis module and sliding mode controller are activated. Based on the analysis and outputs of the sliding mode controller, seven optimization subsystems are designed to determine the final control outputs and to send to the local controllers of the braking system and the active steering system. Each optimization subsystem will be determined whether it needs to be activated. The DB system controls all wheels of the truck, and the AS system only controls the fourth axle.

LTR and ideal yaw rate in the control strategy are used as indexes to indicate the rollover and yaw stability accidents. LTR is a widely used index for rollover warning or rollover prevention, based on Eqs. (1)–(10). LTR in this research can be defined as Eq. (31). The LTR threshold is set as 0.55. The details of gains are in Appendix (Eqs. 51–53).

The ideal yaw rate is calculated by Eq. (32). Once the deviation between the ideal and real yaw rate goes beyond a designed range (\(\pm 0.02\) rad), indicating the vehicle is in the oversteer or understeer situation, the controller starts to work.

A sliding mode control algorithm is designed because of its robustness [35], usually designed in two steps. The first step is to design a sliding surface, and the second step is to design a control law, which drives or keeps the states to the designed sliding manifold [36]. The proposed controller is designed based on a discrete sliding mode control. A vehicle model including roll angle rate, roll angle, and yaw rate is described as Eq. (33):

The discrete model is described as

The details in Eq. (34) are shown in Table 3.

The sliding surface is defined aswhere \(E = \left[ {\begin{array}{*{20}c} {100} & 0 \\ 0 & 1 \\ \end{array} } \right]\).

The target d is different under different conditions and is shown in Table 4.

The reaching law can be described as follows:where\(k_{s} = \left[ {\begin{array}{*{20}c} {0.9} & 0 \\ 0 & {0.5} \\ \end{array} } \right]\), \(k_{sm1} = 0.001\), \(k_{sm2} = 0.001, \quad a = \frac{S\left( k \right)}{\Theta }\), \(\Theta_{1} = 0.025\), \(\Theta_{2} = 0.05.\)

The sliding mode control law is derived as \(u_{s} \left( k \right)\):

The outputs of the sliding mode control are the desired yaw moment and lateral force of the truck. Eq. (39) presents the deviations of the desired yaw moment and lateral force between real yaw moment and lateral force. From the analysis in Section 3, the maximum yaw moment and maximum braking force of each tire can be calculated. The results can be used for evaluating the effects of each stability control method. In general, the wheel with more potentials should take more responsibilities. Therefore, the entire lateral force and yaw moment can be separated by reasonable gains, which are described by the maximum yaw moment changes of different wheels. The gains of yaw moment and force distribution are defined as indicated in Eq. (40). When the extra yaw moment is positive, the gains of the right side are zero. Otherwise, the gains of the left side are zero. If the gain of the wheel is zero, its optimization will not be activated.

\(K_{Mzri,Mzli}\) is introduced as the weight of DB and AS. The smaller \(K_{Mzri,Mzli}\) is, the lesser is the involvement of integrated control. Zero \(K_{Mzri,Mzli}\) means the wheel cannot improve the stability by braking and steering. To compare the differences between AS, DB, and proposed integrated control system, a single AS system and a single DB system are also given based on the desired yaw moment and lateral force distribution. In the AS system \(\Delta M_{{zbr4_{\text{max} } ,zbl4_{\text{max} } }} = 0\) and \(k_{ms} = 1\). This means only steering system is active during control in \(K_{Mzr4,Mzl4}\). In the DB system, \(\Delta M_{{zst4_{ \text{max} } }} = 0\), which means only braking system works in Eq. (40). Based on the allocation gains, the lateral force and yaw moment change caused by each wheel can be described by Eq. (41). After the deviations of desired lateral force and yaw moment of each wheel are known, an optimal problem occurs. To each controlled wheel, the lateral forces and yaw moment should be close to ideal. Therefore, the deviations represented as Eqs. (43) and (44) should be as close as zero. The objective function of the optimization is presented as Eq. (42). In the equation,\(K_{J1}\) and \(K_{J2}\) are weights of deviations:where \(i = 1,2,3,4\), \(K_{J1} = 1\), \(K_{J2} = 1\).

In Eq. (45), \(F_{xri,xli}\) and \(F_{yri,yli}\) are the longitudinal forces and lateral forces at the moment, respectively, based on the vertical forces of tires, slip angles, and slip ratios searched from maps (shown in Appendix Figure 17, maps are from TruckSim). \(F_{xdri,xdli}\) is the desired braking force.

In Eq. (42) when the brake is on the first, second, or third axle, except for the fourth axle (i = 1, 2, 3), or under the condition that only DB system is working on the truck (i = 1, 2, 3, 4), the optimization settings are

In the integrated control system, when both DB and AS are activated on the fourth axle in Eq. (42),

When only the AS control system is activated on the fourth axle, the optimization settings in Eq. (42) are

The limitation of braking forces on each wheel should be within the braking force limits from Table 2. \(\Delta M_{zJri,zJli}\) and \(\Delta F_{yJri,yJli}\) are also limited for the optimization to be more accurate and faster. The limitation of these wheels controlled only by the DB system are

For the fourth axle, besides the limitations in limit 1, the steering angle should also be limited by Eq. (23). To keep the tires in the linear area, the absolute slip angles of tires are limited under 10°. The limitation of the fourth axle wheels is given in limit 2.

If there is only AS system on the truck, the limitation will be

With the vertical force, the desired braking force can be transformed to the desired slip ratio based on the maps (in Appendix Figure 17). The slip ratio of the tire is limited under 0.15, which makes the tire stay in the linear area. The longitudinal slip ratio is realized by the braking pressure calculated by a simple sliding mode control. Eq. (47) shows the wheel model for sliding mode controller. The simple sliding mode control is shown in Eq. (48) for calculating the desired braking torque. Based on the braking torque, the air pressure in braking system is obtained finally from Eq. (49). The sliding mode control reach law is \(S_{bsri,bsli} = C_{b} \left( {\kappa_{ri,li} - \kappa_{dri,dli} } \right)\), where \(\kappa_{ri,li}\) is the slip ratio and \(\kappa_{dri,dli}\) is the desired slip ratio of the ith left or right wheel. When \(\left| {\kappa_{ri,li} - \kappa_{dri,dli} } \right| \le 0.001\), the sliding mode control stops. The existences of these sliding mode controls are certified in Appendix (Eqs. 54–60).

In Eq. (48), \(C_{b}^{{}} = 1\), \(k_{bs} = 0.8\), \(k_{bsm} = 0.001\), \(A_{b}^{{}} = \left| {\frac{r}{{v_{x} J_{ytire} }}} \right|\), i = 1, 2, 3, 4.

To test the effectiveness of the proposed control system, TruckSim–Matlab Simulink cosimulation is conducted. TruckSim develops the truck model, while Matlab Simulink develops the control system, as shown in Figure 8. The truck runs under different velocities with a designed steering input, as shown in Figure 9. The coefficient of friction is 0.85 and the cargo weight is 5000 kg.

To find the best steering speed, 5°/s, 10°/s, 20°/s, 40°/s, and 80°/s are used and their control performances are compared. The driver inputs a 180° step steer under 90 km/h (LTR is larger than 0.55 without control), the mean and mean variances of deviation (LTR to 0.55) with the AS control system under different steering speeds are shown in Table 5. As the steering speed increases, the LTR value oscillation becomes larger. However, if the steering variation is too small, the vehicle cannot be controlled back to steady. The AS system with 5°/s and 10°/s steering variation cannot prevent the rollover accident. Therefore, the best steering variation of AS is 20°/s.

Figures 10 and 11 show the LTR and yaw rate of the four-axle truck with integrated control system under different velocities. It is clear from Figure 11 that without the control system, the truck will rollover completely at 3.278 s, whereas with the integrated control system the vehicle goes through without rollover. The real yaw rate with integrated controller follows the ideal yaw rate very well, and the LTR is controlled within or back to 0.55 quickly.

Figures 12 and 13 show the comparisons between single control methods, integrated control strategy, and system without control. The control system outputs such as braking pressures and fourth axle steering angle under 80 km/h are given as Figure 18 (Appendix). Figure 12 shows the LTR under different conditions, where the truck without control rollovers at 3.6 s under 100 km/h. The DB system controls the rollover more smoothly, but the action speed is slower than AS. On the other hand, the AS system responds more quickly, but the LTR is still fluctuating. Since the DB slows the vehicle down, the more the DB system is working and the smaller is the LTR. Under a high velocity (100 km/h in Figure 12) with a large steering input, as indicated in Figure 9, the AS system cannot prevent rollover and the truck controlled by AS will rollover at 13.13 s, making the DB system more effective. From the figures, the proposed integrated controller combines both DB and AS; thus, the LTR can be controlled as quickly as AS and as smoothly as DB. Figure 13 shows the yaw rate under different conditions with different control system. Similarly, the truck with the DB system controls the truck smoothly, and the AS system responds more quickly than the DB system. Therefore, the performance of integrated control system is much better than that of DB or AS. The truck (cargo weight 5000 kg) with proposed control system can keep away from rollover under 150 km/h when the steering angle input is as per Figure 9, whereas the maximum velocity of the truck without control is just 89 km/h, the integrated control system can improve 68% of the safe velocity (DB 16.9%, AS 12.4%).

To verify the robustness of the proposed control system, the friction coefficients of the road are changed to 0.3 and 0.6. The steering input is a 200° step input and the steering operation lasts for 0.66 s. On a low friction road, the truck does not rollover, but the yaw stability becomes the main problem. Figure 14 shows the control performances under 70 km/h on roads with 0.3 and 0.6 friction coefficients. The proposed integrated control system also works well under low friction road. Figure 15 shows the performances of integrated control system under 70 km/h and 80 km/h with 20000 kg cargo weight on high friction road (road friction coefficient 0.85); the steering input is as shown in Figure 9. The proposed integrated control system can also keep the truck from rollover. In Figure 15(b), the truck without control will rollover at 13.64 s under 80 km/h, but the proposed control system can keep the truck from rollover.

The simulation results demonstrate that the proposed integrated controller outperforms the DB or AS in terms of vehicle stability. The integrated control system combines the advantages of both DB and AS, thus improving the disadvantages of DB and AS. The proposed control system presents a good performance even under high velocities with a large steering angle input. In addition, the proposed integrated control system can work well on low friction or different loads. From Figures 12 and 13, the integrated control system can unfold the advantages of both AS and DB simultaneously.

This paper proposed a novel integrated control system to improve the yaw and lateral stability for a four-axle truck. The proposed control system has a good performance in the multi-axle truck stability control. It has better control performance than DB or AS individually. First, a novel method was proposed to calculate the vertical forces of tires on four axles. Then, the analysis was presented based on the extra yaw moments generated by these two control systems. The effects and potentials of DB and AS were analyzed and compared according to how much extra yaw moment they can generate. The comparison results are transformed into gains for DB and AS coordination. In this way, the control system is more robust. Moreover, DB and AS can work simultaneously and efficiently. From the control results, the control system can unfold the characteristics of both DB and AS. The response is faster because distinct optimization is much more efficient and easier for calculation, and there is no expert system in the controller. From the analysis, the following conclusion can also be drawn.

For differential braking control: (1) In general, to prevent rollover or oversteer accidents, braking on outside wheels on axles in front of the CG point can generate a larger extra yaw moment than on rear axles. (2) To prevent understeering, braking on the inside wheels on axles after the CG point can generate better effects than braking on the front axles. (3) The largest yaw moment changes generated by braking on each wheel are limited not only by initial forces of the tires, ability of air braking system, and steering angle but also by distances between the axles and CG position. (4) In terms of braking wheel, when the root of the quadratic sum of braking force and lateral force is equal to the radius of Kamm’s Circle, braking wheels on the rear axle need a complex control logic. When the truck is in the danger of drifting, braking on front axles is a better choice.

For active steering control: (1) To generate an extra outward yaw moment, if active steering acts on the axles before the CG point, the steering angle needs to be outward and vice versa. (2) The extra yaw moment depends on the vertical forces of tires and distance of the active steering axle to the CG. (3) During the steering period, an extra detrimental lateral acceleration is generated, affecting vehicle stability. (4) Regarding the truck in this paper, the best AS control speed is 20°/s.

In the future research, a multi-axle active steering will be studied. The trajectory tracking based on DB and AS will also be considered. Moreover, the states of the truck such as roll angle, roll angle rate, yaw rate, and accelerations are assumed partially known from sensors. Thus, the estimation method will be considered as well. The ultimate objective is to develop an integrated control system that is suitable for trucks with different numbers of axles and tractor semitrailers with a minimum number of sensors.