# Driver Model-Based Fault-Tolerant Control of Independent Driving Electric Vehicle Suffering Steering Failure

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## Abstract

This paper presents a fault-tolerant control (FTC) strategy for a four-wheel independent driving electric vehicle suffering steering failure. The method is based on the functional redundancy of driving and braking actuators to recover the vehicle’s steering ability. A dynamic vehicle model is derived with the function of four-wheel driving. A sliding mode controller with a combined sliding surface is employed as a motion controller, allowing the desired vehicle motion to be tracked by the adaptive driver model. An extended Kalman filter-based state estimator is adopted to virtually measure the sideslip angle while considering the nonlinear tire force. A new allocation strategy, involving two distribution modes of coordination, is designed. In addition, a weight coefficient adjustment strategy is implemented in optimal mode based on the lateral load transfer, thus improving the steering performance. Simulations are conducted to verify the proposed FTC algorithm. The results demonstrate that steering failure can be effectively covered by the functional redundancy of the driving/braking actuators.

## Keywords

Fault tolerance control Independent driving Steering failure Driver model Functional redundancy## 1 Introduction

Electric vehicles (EVs) offer remarkable potential in terms of reducing emissions and fuel consumption and are thus regarded as the most promising vehicle architecture of the future. In particular, EVs that implement four-wheel independent driving (FWID) with steer-by-wire technology have attracted considerable attention from both academia and industry [1, 2, 3]. The propulsion power of FWID EVs is generated from four motors positioned in each wheel, and the driving/braking of each wheel can be controlled independently. Thus, for a FWID EV with front wheel steering, there are a total of six controllable actuators that can be used to enhance the traction control and direct yaw moment control, as well as other advanced strategies such as energy-efficient control [4, 5, 6, 7]. The steer-by-wire system of FWID EVs eliminates the mechanical linkage to offer greater flexibility in locating and designing novel control technologies that improve vehicle handling and stability [3, 8]. The current manner of electronic driving means that this kind of EV is regarded as a safety-critical system—regardless of which key chassis actuator fails, a serious accident may result. Thus, the reliability of safety-critical systems, including fault detection and/or appropriate fault-tolerant strategies, has received increasing attention and become the most critical factor in process monitoring.

One possible solution for ensuring safe and reliable system performance is model-based fault detection and isolation (FDI) and fault-tolerant control (FTC). FDI systems are designed to enhance the sensitivity of faults, whereas FTC makes the system asymptotically stable and satisfies a prescribed level of performance. A number of studies have examined FTC techniques, and their failures can now be effectively controlled. Mutoh et al. [9] compared the dynamic performance of single-wheel failure between the front/rear wheels of an independent driving EV and an FWID EV. The results show that the latter deviates from the lane of travel in less than 2 s, which does not allow sufficient time for an ordinary driver to steer the vehicle to safety after noting the failure. Wang et al. [10] designed a fault-tolerant control system to accommodate in-wheel-motor driver faults by allocating the control effort to the other healthy wheels. Tian et al. [11] studied wheel hub motor failure modes and proposed an integrated coordination control strategy based on stability performance. Zong et al. [12] proposed a fault-tolerant control approach based on reconfigurable control allocation for four-wheel independent drive and steered (FWID/FWIS) EVs against driving motor failures. In their system, a control allocator reconfigures the control assigned to the healthy motors. Kim et al. [13] proposed an FTC strategy for four-wheel distributed braking systems at the vehicle dynamics level using a sliding mode algorithm and verified the effectiveness of this approach with a series of hard-in-loop simulations. Ki et al. [14] proposed a fault-tolerant logic to detect sensor faults during driving or braking. To maintain performance in the case of faults, a bumpless transfer technique was used. Lu et al. [15] quantitatively analyzed the transient behavior of different braking failure cases and proposed three FTC steering strategies. The above-mentioned studies mainly focus on the FTC of braking or driving failures, which are critical for vehicle safety. However, relatively little attention has been paid to the failure and FTC of steering actuators in FWID EVs. As a key actuator in ensuring the safety and reliability of vehicles, the FTC of steering is especially important.

This paper focuses on fault accommodation control for actuator failures in a steering system. A new control algorithm based on the functional redundancy of driving/braking systems is proposed to realize emergency steering when steering failure occurs. The proposed algorithm employs two torque distribution modes for the four in-wheel motors and a weight coefficient adjustment mechanism based on the normal load transfer. An inner–outer loop structure is used to clarify the control strategy. In the outer loop, an adaptive driver model and an extended Kalman filter (EKF)-based state estimator are implemented. In the lower layer, a weight coefficient for adjusting the FTC torque allocation is realized. The aim is to prevent the faulty vehicle from becoming out of control and possibly drive it back to the nominal condition.

## 2 System Dynamic Modeling

### 2.1 Vehicle Dynamic Model

*m*is the vehicle mass, \(V_{x}\), \(V_{y}\) are the longitudinal and lateral velocity, respectively, \(\gamma \) is the yaw rate of the center-of-gravity (c.g.), \(I_{z}\) is the inertia around the

*z*-axis of the vehicle coordinate system, \(F_{xd}\), \(F_{yd}\) are the total longitudinal and lateral tire forces, respectively, and \(M_{zd}\) is the desired yaw moment. The virtual control variables can be described as

*front left*,

*front right*,

*rear left*, and

*rear right*tires, respectively. \(\delta _{1}\) and \(\delta _{2}\) are the steering angles of the left front and right front wheels, respectively.

*a*and

*b*are the distances from

*c*.

*g*. to the front and rear axles, respectively. \(d_\mathrm{f}\) and \(d_{\mathrm{r}}\) are the track width of the front and rear axles, respectively, assuming \(d_\mathrm{f}=d_\mathrm{r}=d\).

*J*and

*R*are the wheel moment of inertia and effective tire radius, respectively.

### 2.2 Nonlinear Tire Model

*B*,

*C*,

*D*, and

*E*are factors related to the stiffness, shape, peak, and curvature, respectively. These could be expressed as a function of vertical load \(F_{zi}\), which can be estimated. The tire slip ratio \(\lambda _{i}\) is defined as the difference between the wheel velocity and the vehicle velocity. The tire slip angle \(\alpha _{i}\) is defined as the angle between the wheel orientation and its velocity vector (see Fig. 1).

With the parameters provided by Bakker et al. [16], the normalized longitudinal and lateral tire forces are generated as shown in Fig. 2. It is clear that both the longitudinal tire force \(F_{x}\) and lateral tire force \( F_{y}\) demonstrate strong nonlinearity with respect to \(\lambda \) and \(\alpha \). For small tire slip angles, the lateral forces could be approximately linearized.

## 3 Fault-Tolerant Strategy for Steering Failure

### 3.1 Global Control Scheme

When a steering system fails, the ultimate aim should be to stop the vehicle in a safe way. In particular, emphasis should be placed on ensuring that the vehicle avoids traveling away from the road. For FWID EVs, both driving/braking and steering actuators could provide an appropriate corrective yaw moment to improve the lateral dynamics. Even if the steering fails, the function redundancy of the driving/braking subsystem means there are four controllable actuators that could still be used to compensate for the lost steering function.

### 3.2 Modified Driver Model

*f*(

*t*) is the desired path, \(T_\mathrm{p}\) is the preview time, and \(f(t+T_\mathrm{p})\) is the previewed path. The single-point preview model is used to reduce the lateral position error \(\varepsilon \) by minimizing the error of lateral acceleration. The lateral position error is defined as

*e*and the rate of change

*ec*between the desired and actual acceleration of the reference model are taken as the input, and the equivalent front wheel angle \({\delta _{sw}}^{*}\) is given as the output. The delay effect incorporates the driver action delay \( T_{d}\) and the system delay \(T_{h}\) caused by inertia. The real action angle \(\delta _{sw}\) is then transferred to the desired vehicle states \(\gamma _{d}\) and \(\beta _{d}\) to control the motion in the inner loop. For the conventional PID, the adaptive parameters based on fuzzy tuning are defined as

*n*are tuned online by fuzzy inference based on a special principle: For a large error

*e*, \(K_{p}\) should be large, \(K_{d}\) should be small, and the integral term should be limited to prevent a large overshoot. For a smaller error rate

*ec*, \(K_{p}\) should be larger; otherwise, it should be smaller. Seven language variables are defined for the three parameters (NB, NM, NS, ZO, PS, PM, and PB) representing negative big, medium, small, zero, and positive small, medium, big, respectively. The detailed fuzzy rules are determined based on the above principle and are not presented here.

### 3.3 Sideslip Angle Estimator

The yaw rate and sideslip angle are two important states for the lateral dynamics of vehicle control. The yaw rate can be measured directly using a cost-effective gyro. However, the sideslip angle cannot be measured at low cost using standard sensors, and so estimation methods are usually employed. The “\(\beta \)-estimation” has been widely discussed in the literature. The problem of an EKF-based observer in [18, 19] is synthesized and summarized as follows.

*W*and

*V*are the process and measurement noise vectors, respectively; both are assumed to be zero mean white noise.

*k*. Ignoring second-order terms and above, the system can be linearized as

### 3.4 Sliding Mode-Based Motion Controller

The motion controller determines the target control yaw moment for tracking the desired states derived from the driver model in the outer loop. Considering the nonlinear and model uncertainties of the vehicle system [20], sliding mode control (SMC) is employed to ensure appropriate performance in the control system.

*M*can be derived as

*S*) means that the control law is discontinuous during the sliding motion. This discontinuity is highly undesirable because it could result in chattering in the controller output, which may excite high-frequency dynamics. Hence, to avoid these chattering effects, the sign function in Eq. (19) is replaced by a saturation function with a boundary layer. Thus, the control law becomes

*M*in SMC.

### 3.5 Torque Control Allocator

The objective of the control allocator is to distribute the generalized moment calculated by the motion controller to each wheel. If the generalized moment and force meet the optimal conditions, the optimal mode will be implemented with multiple constraints; otherwise, the proportional load allocation mode will be implemented. The whole process is synthesized in Fig. 5, where numbers in parentheses represent the corresponding equations. In the optimal mode, an adjustment method for the weight coefficients is employed based on the variation of the vehicle normal load between the left and right sides.

*d*, the longitudinal tire forces at each wheel are constrained by

Logic for determining the tire force allocation

Conditions | Value |
---|---|

\(A_{3}\le {F_{x3}}^\prime \) | \(F_{x3}=A_{3}\) |

\(-A_{3}\le {F_{x3}}^\prime \le A_{3}\) | \(F_{x3}={F_{x3}}^\prime \) |

\({F_{x3}}^\prime \le -A_{3}\) | \(F_{x3}=-A_{3}\) |

\(A_{4}\le {F_{x4}}^\prime \) | \(F_{x4}=A_{4}\) |

\(-A_{4}\le {F_{x4}}^\prime \le A_{4}\) | \(F_{x4}={F_{x4}}^\prime \) |

\({F_{x4}}^\prime \le -A_{4}\) | \(F_{x4}=-A_{4}\) |

The longitudinal tire force of FWID EVs can satisfy the driving force requirements, and simultaneously, provide an additional yaw moment for emergency steering. However, some degree of vertical load transfer is inevitable when vehicle turning affects the steering characteristics. To further improve the steering performance, a weight coefficient adjustment strategy is proposed based on the lateral load transfer. When turning to the left, the vehicle load is transferred from left to right, and the available tire adhesion margin of the right-side wheels increases. At this point, more control should be allocated to the right-side wheels according to the turning degree to make full use of the available tire force, and vice versa when turning to the right.

## 4 Simulation Results and Discussion

Key parameters for simulations

Variables | Parameter name | Units | Value |
---|---|---|---|

| Vehicle mass | kg | 1820 |

| Distance from c.g. to front axle | m | 1.46 |

| Distance from c.g. to rear axle | m | 1.58 |

| Height of the c.g. | m | 0.50 |

| Front/rear track width | m | 1.55 |

\(I_{z}\) | Yaw moment of inertia | kg m\(^{2}\) | 3800 |

| Wheel moment of inertia | kg m\(^{2}\) | 1.28 |

| Effective radius of the wheel | m | 0.28 |

\(k_\mathrm{f}\) | Cornering stiffness of front tire | N/rad | 38,000 |

\(k_\mathrm{r}\) | Cornering stiffness of rear tire | N/rad | 46,000 |

\(T_{\mathrm{max}}\) | Peak torque of motor | N\(\cdot \)m | 500 |

To better show the effectiveness of the proposed FTC strategy, the performance of an uncontrolled vehicle with the same faults was also studied on a low-friction road. In this maneuver, a fault is added to the steering system at 1.0 s, causing the vehicle to travel straightforward if no control is applied. The initial velocity is set to 20 m/s, and the friction coefficient of the road is assumed to be 0.5. The driving/braking torques determined by the fault-tolerant controller are intended to follow the reference trajectory. To demonstrate the effect of the weight coefficient adjustment strategy, the vehicle responses with and without weight coefficient (WC) adjustment are also presented.

As shown in Fig. 8, it is clear that the left and right wheels have the same weight during the straight trajectory at the beginning. When the failure occurs after 1.0 s, the braking and driving torques are applied to the left and right wheels, respectively, replacing the steering system to turn the vehicle. To make better use of the available adhesion margin, more weight is allocated to the outside wheels because of the large load transfer. Thus, the weight of the right-hand driving side is greater than that of the left-hand braking side during the first turn, and vice versa for the second turn.

The torques allocated to each wheel with and without WC adjustment are shown in Figs. 11 and 12, respectively. Here, the first subscript \(i =f\), *r* denotes front and right, the second subscript \( j = l, r\) means left and right, respectively. The results indicate that the peak value of the outside driving torque is greater than that of the inside braking torque, with the net result being an increase in longitudinal velocity, as shown in Fig. 13. Figure 12 shows that the driving torque on the front outside is close to the motor limitation, whereas for control cases with WC adjustment, nearly all the torques remain within the range of the motor limitation.

## 5 Conclusions

- (1)
A steering FTC strategy with an actuator WC adjustment mechanism has been proposed for FWID EVs. The main aim of this system is to improve vehicle safety and stability. The proposed method realizes emergency steering by coordinating the torques of four in-wheel motors in the case of steering system failure.

- (2)
A revised adaptive driver model was formulated so that the desired dynamic response could be obtained. A motion controller based on the sliding mode method was designed and an EKF was adopted to estimate the sideslip angle requested as feedback by the controller. In addition, a WC adjustment method based on the load transfer was implemented to improve the steering characteristics.

- (3)
Simulations using the proposed method were evaluated by means of a single-lane-change maneuver and compared to the reference model in order to verify the post-fault vehicle safety and directional stability. The results demonstrate that the proposed FTC strategy can effectively cope with steering system failure.

- (4)
In future research, an actual experimental vehicle test platform will be built to test the control performance of the proposed FTC method.

## Notes

### Acknowledgements

The work was supported by the National Science Foundation of China (51675066), Chongqing Research Program of Basic Research and Frontier Technology (cstc2017jcyjAX0323), and Shanghai Aerospace Science and Technology Innovation Foundation (SAST201016).

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