Stability control of single-wheel failure in four-wheel independent drive electric vehicles

The integration of the motor and transmission mechanism into the wheels of a four-wheel independent drive electric vehicle with hub motors increases the frequency of failures, especially torque output failures, which seriously affect vehicle stability and jeopardize driving safety. This paper focuses on the perspective of single-wheel torque output failure and studies the vehicle stability control strategy. Firstly, based on the constructed simulation dynamics model, a slip controller is designed to obtain the direct lateral moment. Then, based on the control allocation principle, upper-level lateral moment allocation rules are designed, and torque allocation rules under single-wheel torque output failure are formulated. Finally, simulation experiments are conducted under different operating conditions. The simulation results show that this control strategy can effectively control vehicle stability under single-wheel failure conditions and ensure safe driving.


Introduction
To ensure the safety of vehicle operation, a stable control system is required to address torque output failures.Hub motor electric vehicles have significant advantages in terms of stability control.Therefore, the development of stability control systems for fault conditions has great research potential and practical significance [1] .
There have been many related studies on fault-tolerant control for four-wheel independent drive vehicles.Ghaedi et al. proposed a robust two-layer control scheme to improve the lateral motion performance in the existence of parameter uncertainties and exterior disturbances of Four-Wheel Independent Drive (4-WID) electric vehicles [2] .Hu et al. proposed a model-free adaptive fault-tolerant control strategy to address the drawbacks of relying excessively on fault diagnosis and system models in line control steering systems [3] .Zhang et al. proposed a fault-tolerant control method based on coordinated control of trajectory tracking and lateral stability, which utilizes differential steering and direct yawing moment cooperation, for a 4-wheel independent drive intelligent electric vehicle.This approach is developed to address the line control steering system's actuator failure issues [4] .Chen et al. proposed a passive fault-tolerant path following the control method of autonomous distributed drive electric vehicles considering the vehicle steering system fault [5] .
The main research objective of this paper is to address the faults in the torque output of individual tires in wheel-motor and power transmission electric vehicles.A highly stable drive control system is established to effectively ensure the vehicle's driving and safety on the road.The main research content includes the following: building a simulation dynamic model and establishing a slip control algorithm; designing a rational stability control strategy for single-wheel faults and proposing a torque allocation strategy for such faults; conducting simulation verification of the control strategy and validating its effectiveness.

Design of direct yaw moment controller
To study the stability control of a hub motor electric vehicle, it is necessary to establish an accurate vehicle dynamics model.In this paper, a joint simulation modeling method using Carsim/Simulink is employed, where the hub motor and driver speed control model are built in Simulink, and the four-wheel independent drive form is achieved through the interface settings in Carsim [6] .
When designing the controller, it is crucial to consider ways to eliminate the effects of non-linear systems and external disturbances such as crosswinds and inclines on the controller's accuracy.Therefore, a sliding mode controller is chosen as the upper-level controller [7] .
By selecting a two-degree-of-freedom vehicle model as the reference model, the desired vehicle parameters, such as the desired yaw rate and desired lateral displacement of the center of mass, are obtained.These desired states can serve as the sliding surface for sliding mode control.To meet the requirements of stable control, the sliding surface should be chosen in such a way that the yaw rate and lateral displacement of the center of mass can both track the desired parameters calculated by the reference model [8] .This paper uses the following sliding surface for the sliding mode controller: ( ) where β represents the synovial coefficient, which is usually a positive constant.The damper can control both the yaw rate and the lateral velocity of the center of mass at the desired values under any working conditions.To minimize the vibration generated when the state variable reaches the synovial surface, this paper chooses power approaching rate to design the lateral moment controller: sgn( ) Due to the discontinuity of the sign function in the approaching rate, the control system exhibits highfrequency oscillations around the synovial surface.To eliminate vibrations without affecting stability, we approximate the sign function with a continuous function: sgn( ) On this basis, we can calculate the required direct lateral torque z M Δ input for the reference state parameters obtained from the tracking reference model of a four-wheel independent drive electric vehicle.
Taking the derivative of the sliding surface yields: ( ) Vehicle lateral motion can be expressed as: Since the front wheel steering angle f δ is typically very small during high-speed driving and can be neglected, we take cos 1 The sum of the last two terms in Equation ( 6) represents the direct yaw moment Based on Equations ( 5)-( 7), we can conclude that: Direct yaw moment can be represented as follows: The final direct lateral moment z M Δ can be expressed as:

Stability control strategy for single-round fault state
Many factors can cause a failure in the hub motor driving the car tire, but for the stability control studied in this paper, the focus is on analyzing torque output failures caused by the motor's actual output torque not reaching the target torque.
When a driving wheel fails to provide the required output torque, to improve the utilization of the drive system, the fault rate of the faulty motor can be determined while diagnosing the fault.The lost drive torque will be compensated by another normal motor on the same side, using the remaining output torque capacity of the motor.Regardless of the failure condition, it is important to always maintain the same desired torque for the steering wheels.Additionally, when detecting the fault condition, the distribution of lateral forces needs to be considered.Therefore, the fault coefficient of the fault torque is determined only by the desired torque of the speed controller, while the rules for distributing additional lateral forces are the same as in normal conditions [9] .
The specific allocation strategy is as follows: where ( , , , ) At the same time, in control allocation, we also need to consider situations where the wheel driving force exceeds the traction with the ground, leading to wheel slippage and instability.Therefore, constraints need to be applied to the output torque, requiring that the torque on each wheel does not exceed the maximum traction between the tire and the ground and that the torque remains below the peak torque of the motor.Hence, the constraint conditions are derived as follows [10] : max min( , ) where μ represents the traction coefficient between the driving wheel and the ground; ( , , , )

zi F i fl fr rl rr =
represents the normal force acting on each wheel; max T represents the peak torque of the hub motor.

Simulation validation and analysis
To verify the effectiveness of the above control strategy, three types of simulation scenarios were set up for normal vehicle operation, controlled state in case of failure, and uncontrolled state in case of failure.

Straight-line driving condition
The simulated vehicle had an initial speed of zero and accelerated to 60 km/h within 12 seconds.Then, at the 15th second, the left front wheel malfunctioned, losing 50% of its driving force.Simulation comparisons were made for the fault-free state, controlled fault state, and uncontrolled fault state, and the simulation results are shown in Figure 1.

Turning driving condition
The simulation vehicle starts from zero velocity and accelerates to 60 km/h within 12 s.At the 13th second, the steering wheel turns to the left, with a steering angle of 180 degrees, and maintains this position.At the 15th second, the left front wheel experiences a fault and loses 50% of its driving force.A simulation comparison was conducted for the normal state, controlled fault state, and uncontrolled fault state.The simulation results are shown in Figure 2.  2(a) and 2(b), it can be observed that the torque on the left front and rear wheels decreases, while the torque on the right front and rear wheels increases.At 15 s, when a torque output failure occurred on the left front wheel, the output torque on the left front wheel instantly decreased by 50% without control, while the torque output on other wheels remained unchanged.Under controlled conditions, the output torque on the right front wheel decreases along with the left front wheel, while the output torque on the left rear and right rear wheels increases.It can be seen that the torque on each wheel can be accurately output according to the rules.
Based on Figures 2(c) and 2(d), it can be observed that both the yaw angular velocity and the lateral deviation angle of the vehicle have significant errors under fault conditions, while both parameters can be stably tracked to the reference values under controlled conditions.
According to Figure 2(e), without control, the vehicle deviates from the reference trajectory by approximately 1.4 m during the steering motion.This poses a great danger for a vehicle during cornering.Conversely, under controlled conditions, although the vehicle trajectory still deviates from the normal state, the deviation within a distance of 200 m is less than 0.05 m, which can be neglected.It can be considered that the vehicle trajectory stably tracks the reference trajectory.

Single lane change condition
The simulated vehicle starts with an initial velocity of zero and accelerates to 60 km/h within 12 s.At 13 s, it enters the single lane change driving condition.At 15 s, the left front wheel experiences a failure, resulting in a loss of 60% of the driving force.Simulation results for the no-fault state, fault state with control, and fault state without control are shown in Figure 3. should increase when the current wheel rotation angle increases.According to Figure 3(b), it can be observed that the torques of each wheel can be accurately output according to the rules, and the difference in driving torque between the left and right wheels does change with the steering wheel angle.
Combined with Figures 3(c) to 3(f), it can be seen that the stability control strategy proposed in this paper can effectively control the vehicle's yaw angular velocity and lateral deviation of the center of mass.
From the wheel diameter comparison in Figure 3(g), it can be seen that the influence of single-wheel torque output failure on the vehicle's path is effectively suppressed under the action of the stability control strategy.

Conclusion
This paper focuses on the stability control strategy of vehicles under single-wheel failure by studying electric vehicles driven by hub motors.Based on the construction of the vehicle dynamics model, suitable sliding surfaces, convergence rates, and approximation functions are selected based on the reference model.A sliding mode controller is designed to obtain the desired direct yaw moment input during vehicle steering.A torque distribution strategy is formulated for single-wheel failure conditions, and simulation experiments are conducted under straight-line driving, steering driving, and lateral motion conditions.The simulation results demonstrate that the proposed control strategy can ensure stable vehicle operation in the presence of a single-wheel failure, thereby improving vehicle driving safety.
is the fault coefficient, indicating the fault status of torque output of the tires; a and b are the allocation coefficients, which are determined by the rules of the fault coefficient, as shown in the following equation:

Figure 1 .
Figure 1.The schematic diagram for straight-line driving simulation.Analyzing the simulation results for straight-line driving conditions, according to Figure1(a)and Figure1(b), it can be observed that when a torque output failure occurs in the left front wheel at 15 s, the output torque of the left front wheel instantly decreases by 50% in the uncontrolled state, while the torque output of the other wheels remains unchanged.However, in the controlled state, the output torque

Figure 2 .
Figure 2. The schematic diagram for steering driving simulation.When the steering angle changes at 13 s, the additional yaw moment z M Δ is allocated.According to Figures 2(a) and 2(b), it can be observed that the torque on the left front and rear wheels decreases, while the torque on the right front and rear wheels increases.At 15 s, when a torque output failure occurred on the left front wheel, the output torque on the left front wheel instantly decreased by 50% without control, while the torque output on other wheels remained unchanged.Under controlled conditions, the output torque on the right front wheel decreases along with the left front wheel, while the output torque on the left rear and right rear wheels increases.It can be seen that the torque on each wheel can be accurately output according to the rules.

Figure 3 .7
Figure 3.The schematic diagram for single lane change simulation.Figure 3(a) shows a schematic diagram of the steering wheel angle during lateral motion.The singlewheel mobility simulation condition starts at 13 s, and the torque distribution strategy responds quickly.