Automotive Innovation

, Volume 1, Issue 1, pp 85–94 | Cite as

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

The model used to verify the performance of the proposed method is shown in Fig. 1. A total of seven degrees of freedom (DOFs) are considered, including the longitudinal, lateral, and yaw motion and the rotation of the four wheels. The governing equations are
$$\begin{aligned} \left[ {\begin{array}{c} \dot{V}_x \\ \dot{V}_y \\ \dot{\gamma } \\ \end{array}} \right] =\left[ {\begin{array}{c} V_y \gamma \\ -V_x \gamma \\ 0 \\ \end{array}} \right] +\left[ {\begin{array}{c@{\quad }c@{\quad }c} 1/m&{} 0 &{}0 \\ 0&{}1/m&{}0 \\ 0&{} 0 &{}1/{I_z } \\ \end{array}} \right] \left[ {\begin{array}{c} F_{xd} \\ F_{yd} \\ M_{zd} \\ \end{array}} \right] \end{aligned}$$
(1)
where 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
$$\begin{aligned} \left[ {F_d } \right] ^\mathrm{T}= & {} H_x \times \left[ {F_{xi} } \right] ^\mathrm{T}+H_y \times \left[ {F_{yi} } \right] ^\mathrm{T}\\ H_x= & {} \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \cos \delta _1 &{}\hbox { cos}\delta _2 &{} 1&{} 1 \\ \sin \delta _1 &{}\sin \delta _2 &{} 0 &{}0 \\ a\sin \delta _1 -\frac{d_\mathrm{f} }{2}\hbox {cos}\delta _1 &{}a\sin \delta _2 +\frac{d_\mathrm{f} }{2}\hbox {cos}\delta _2 &{}-\frac{d_\mathrm{r} }{2}&{}\frac{d_\mathrm{r} }{2} \\ \end{array}} \right] \nonumber \\ H_y= & {} \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} -\sin \delta _1 &{}-\sin \delta _2 &{}0 &{}0 \\ \hbox { cos}\delta _1 &{}\hbox { cos}\delta _2 &{}1 &{}1 \\ a\cos \delta _1 +\frac{d_\mathrm{f} }{2}\sin \delta _1 &{} a\cos \delta _2 -\frac{d_\mathrm{f} }{2}\sin \delta _2 &{}-b&{}-b \\ \end{array}} \right] \nonumber \end{aligned}$$
(2)
where \(F_{d }= [F_{xd}, F_{yd}, M_{d}]\). \(F_{xi}\) and \(F_{yi}\) are the longitudinal and lateral tire forces, respectively. The subscript \(i = 1, 2, 3, 4\) represents the 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 \(d_\mathrm{f}=d_\mathrm{r}=d\).
The in-wheel motors of steer-by-wire technology mean that the longitudinal tire forces of FWID EVs can be controlled independently. The rotation dynamics of the four wheels are expressed as
$$\begin{aligned} F_{xi} =\frac{(T_i -J\dot{\omega }_i )}{R} (i=1,2,3,4) \end{aligned}$$
(3)
where \(T_{i}\) is the output torque of each in-wheel motor (note that \(T_{i} \ge 0\) denotes driving torque and \(T_{i} <0\) denotes braking torque). \(\omega _{i}\) is the wheel rotational speed. J and R are the wheel moment of inertia and effective tire radius, respectively.
Fig. 1

Seven-DOF vehicle model and the simplified single-track model

For the reference lateral dynamics, a single-track system is simplified as shown in the middle of Fig. 1. The state space of the vehicle can be expressed by setting \(V_{y}\) and \(\gamma \) as the state variables and the front wheel steering angle \(\delta _{\mathrm{f}}\) as the input.
$$\begin{aligned} \left[ {\begin{array}{c} \dot{V}_y \\ \dot{\gamma } \\ \end{array}} \right] =\left[ {\begin{array}{c@{\quad }c} \frac{k_\mathrm{f} +k_\mathrm{r} }{mV_x }&{}\frac{ak_\mathrm{f} -bk_\mathrm{r} }{mV_x }-V_x \\ \frac{ak_\mathrm{f} -bk_\mathrm{r} }{I_z V_x }&{}\frac{a^{2}k_\mathrm{f} +b^{2}k_\mathrm{r} }{I_z V_x } \\ \end{array}} \right] \left[ {\begin{array}{c} V_y \\ \gamma \\ \end{array}} \right] +\left[ {\begin{array}{l} -\frac{k_\mathrm{f} }{m} \\ -\frac{ak_\mathrm{f} }{I_z } \\ \end{array}} \right] \delta _\mathrm{f}\nonumber \\ \end{aligned}$$
(4)
where \(k_\mathrm{f}\) and \(k_\mathrm{r}\) are the tire cornering stiffness of the front and rear axles, respectively.
The vehicle sideslip angle \(\beta \) is
$$\begin{aligned} \beta =\arctan \left( {V_y /V_x } \right) \end{aligned}$$
(5)

2.2 Nonlinear Tire Model

The Pacejka tire model is adopted to determine the coupling of longitudinal and lateral tire forces. This model uses combinations of trigonometric functions to describe the relationship between the tire slips and the tire forces. The longitudinal and lateral tire forces are described as complex nonlinear functions of the vertical load, longitudinal slip ratio, and sideslip angle of tires:
$$\begin{aligned} F_{xi,yi} (F_{zi} ,\lambda _i ,\alpha _i )= & {} D\sin (C\arctan (B\lambda _i (\alpha _i )(1-E) \nonumber \\&+\,E\arctan (B\lambda _i (\alpha _i )))) \end{aligned}$$
(6)
where 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).
Fig. 2

Tire forces versus \(\alpha \) and \(\lambda \) a normalized longitudinal tire force. b Normalized lateral tire force

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.

The proposed steering FTC strategy for FWID EVs is shown in Fig. 3. In the steering modes, the wheel torques are dynamically allocated based on the desired yaw moment when the steering fails, which is determined according to a driver model. A multi-loop architecture is used to clarify the proposed method. In the outer loop, a revised adaptive driver model is regulated to obtain the desired dynamic response and an EKF is adopted to estimate the sideslip angle requested as feedback by the controller. In the inner loop, a motion controller based on the sliding mode control method is designed, and a new allocation strategy is realized with two coordinate distribution modes.
Fig. 3

Structure of the FTC strategy for steering failure

3.2 Modified Driver Model

To ensure acceptable performance after the steering system has failed, the desired motion must first be determined. In this study, a revised adaptive driver model based on optimal preview lateral acceleration is employed [17]. Unlike the general driver model, the desired vehicle states are derived as the output instead of the corrective steering wheel by considering the inherent delay of the human and inertia system. As shown in Fig. 4, a single-point preview model combined with a fuzzy proportional–integral–derivative (PID) controller is formed as a closed-loop adaptive driver model. In this figure, 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
$$\begin{aligned} \varepsilon =f(t+T_\mathrm{p} )-y(t)-T_\mathrm{p} \dot{y}(t) \end{aligned}$$
(7)
Fig. 4

Adaptive preview driver model

For the fuzzy PID controller in the driver model, the lateral acceleration error 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
$$\begin{aligned} u\left( t \right) =K_p \left[ e(t)+\frac{1}{T_i }\int _0^t {e(t)\mathrm{d}t+T_d \frac{\mathrm{d}e(t)}{\mathrm{d}t}} \right] \end{aligned}$$
(8)
where \(T_{i }=K_{p}\)/\(K_{i}\), \(T_{d }=K_{d}\)/\(K_{p}\) are the integral and differential time parameters, respectively. Assuming \( T_{i }= { nT}_{d}\), we have that \(K_{i }=K_{p}^{2}/{ nK}_{d}\). In dynamic situations, \(K_{p}\), \(K_{d}\), and 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.

The lateral force in Pacejka’s tire model [16] is described as
$$\begin{aligned} F_y= & {} D\sin \{C\arctan [B(\alpha +S_\mathrm{H} )-E(B(\alpha +S_\mathrm{H} ) \nonumber \\&-\,\arctan (B(\alpha +S_\mathrm{H} )))]\}+S_\mathrm{V} \end{aligned}$$
(9)
where \(\alpha \) is the tire sideslip angle. \(S_\mathrm{H}\) and \(S_\mathrm{V}\) are the horizontal and vertical drift, respectively. Ignoring the drift factor, Eq. (9) can be simplified as
$$\begin{aligned} \bar{{F}}_y (\alpha )= & {} D\sin \{C\arctan [B\alpha -BE\alpha \nonumber \\&+\,E\arctan (B\alpha )]\} \end{aligned}$$
(10)
Equation (10) can be used to describe the steady state lateral tire force. To illustrate the transient behavior of tires, a relaxation length \(\sigma \) is introduced. Based on the relaxation model, the dynamic lateral force is described as
$$\begin{aligned} \dot{F}_y =[\bar{{F}}_y (\alpha )-F_y ]\times V_x /\sigma \end{aligned}$$
(11)
where \(\sigma \) is the relaxation length.
Taking the vehicle system with the state of the sideslip angle as a nonlinear stochastic state space, we obtain
$$\begin{aligned}&\left\{ {\begin{array}{l} \dot{X}=f(X,U)+W \\ Y=GX+V \\ \end{array}} \right. \\&f(X,U)=\left[ {\begin{array}{c} \frac{(F_{y1} +F_{y2} )\cos (\delta -\beta )+(F_{y3} +F_{y4} )\cos \beta }{mV_x }-\gamma \\ \frac{a(F_{y1} +F_{y2} )\cos \delta -b(F_{y3} +F_{y4} )}{I_z } \\ \frac{V_x }{\sigma }(\overline{F_{y1} } -F_{y1} ) \\ \frac{V_x }{\sigma }(\overline{F_{y2} } -F_{y2} ) \\ \frac{V_x }{\sigma }(\overline{F_{y3} } -F_{y3} ) \\ \frac{V_x }{\sigma }(\overline{F_{y4} } -F_{y4} ) \\ \end{array}} \right] \nonumber \\&G=\left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{}0&{}\frac{1}{m}&{}\frac{1}{m}&{}\frac{1}{m}&{}\frac{1}{m} \\ 0&{}1&{}0&{}0&{}0 &{}0 \\ \end{array}} \right] \nonumber \end{aligned}$$
(12)
where \(X = [\begin{array}{c@{\quad }c@{\quad }c} \beta&\gamma&F_{yi}\end{array}]^{\mathrm{T}}\), \(\hbox {U} = [\begin{array}{c@{\quad }c@{\quad }c} \delta&V_{x}&F_{zi}\end{array}]^{\mathrm{T}}\) with \(i = 1, 2, 3, 4\). \(f(\cdot )\) is a nonlinear function of the state equations, \(Y = [\begin{array}{c@{\quad }c}a_{y}&\gamma \end{array}]^{\mathrm{T}}\), and \(G\in \mathbf{R}^{2\times 6}\) is the measurement matrix. W and V are the process and measurement noise vectors, respectively; both are assumed to be zero mean white noise.
The nonlinear stochastic system described by Eq. (12) can be discretized as
$$\begin{aligned} \left\{ {\begin{array}{l} x(k)=f(x(k-1),u(k-1))+w(k) \\ y(k)=g(x(k))+v(k) \\ \end{array}} \right. \end{aligned}$$
(13)
According to the idea of EKF, the nonlinear functions \(f(\cdot )\) and \(g(\cdot )\) can be approximated from their Taylor series around the estimated states at step k. Ignoring second-order terms and above, the system can be linearized as
$$\begin{aligned} A(k)=\frac{\partial f}{\partial x}, \quad H(k)=\frac{\partial g}{\partial x} \end{aligned}$$
With the approximate linear system, the system state with the sideslip angle can be recursively estimated by the EKF as follows:
$$\begin{aligned} x(k\left| k \right. )= & {} x(k\left| {k-1} \right. )\nonumber \\&+\, K_g (k)\left[ {y(k)-g(x(k\left| {k-1} \right. ))} \right] \end{aligned}$$
(14)
where \(K_{g}(k)\) is the EKF gain matrix.

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.

The sliding surface is defined as a weighted combination of yaw rate error and sideslip angle [21, 22], written as
$$\begin{aligned} S=r-r_d +\xi (\beta -\beta _d ) \end{aligned}$$
(15)
where \(\xi \) is a positive constant used to tune the trade-off between the yaw rate error and the sideslip angle error.
In the proposed system, the following equal control law is adopted:
$$\begin{aligned} \dot{S}=-\eta \hbox {sgn}(S) (\eta >0) \end{aligned}$$
(16)
To validate the stability of the proposed control law, the Lyapunov method is used. The Lyapunov function is chosen as
$$\begin{aligned} V=\frac{1}{2}S^{2} \end{aligned}$$
(17)
The derivative of the above Lyapunov function is always negative, which proves that the system is stable.
When the steering system fails, the front steering angles are assumed to be zero. Hence, the yaw motion in Eq. (1) can be represented as
$$\begin{aligned} I_z \dot{r}=a(F_{y1} +F_{y2} )-(F_{y3} +F_{y4} )b+M \end{aligned}$$
(18)
where \(M = (F_{x2}-F_{x1}+F_{x4}-F_{x3})\times d/2\) is the target control yaw moment realized by the longitudinal tire forces alone.
Combining Eqs. (15)–(18), M can be derived as
$$\begin{aligned} M= & {} b(F_{y3} +F_{y4} )-a(F_{y1} +F_{y2} )+I_z \dot{r}_d\nonumber \\&-\, I_z \xi (\dot{\beta }-\dot{\beta }_d )-I_z \eta \hbox {sgn}(S) \end{aligned}$$
(19)
However, the switching function sgn(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
$$\begin{aligned} M= & {} b(F_{y3} +F_{y4} )-a(F_{y1} +F_{y2} )+I_z \dot{r}_d \nonumber \\&-\,I_z \xi (\dot{\beta }-\dot{\beta }_d )-I_z \eta \mathrm{sat}(S/\varphi ) \end{aligned}$$
(20)
where \(\varphi \) is the thickness of the boundary layer. The saturation function is defined as
$$\begin{aligned} \mathrm{sat}(S/\varphi )=\left\{ {\begin{array}{l@{\quad }l} S/\varphi ,&{}\quad \left| S \right| \le \varphi \\ \hbox {sgn}(S),&{}\quad \left| S \right| >\varphi \\ \end{array}} \right. \end{aligned}$$
(21)
When the steering system fails, the corresponding yaw moment and sideslip angle deviate from the desired values. Because of the function redundancy of the actuator system, both driving and braking can be used to compensate the steering failures. Hence, the vehicle handling and stability can be guaranteed by achieving the virtual control of the yaw moment M in SMC.

3.5 Torque Control Allocator

In FWID EVs, the driving/braking is controlled independently at each wheel. This makes the torque allocation control more flexible compared with that of traditional vehicles. The wheel torque distribution should not only meet the needs of the generalized force, but also satisfy actuator constraints such as the peak torque of the motor and adhesive boundary of the tire force. In this study, the torque distribution is considered under multiple constraints and a new allocation strategy including two coordination modes is designed.
Fig. 5

Process of 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.

Assuming that the front and rear wheel tracks are equal to d, the longitudinal tire forces at each wheel are constrained by
$$\begin{aligned} \left\{ {\begin{array}{l} F_{x1} +F_{x2} +F_{x3} +F_{x4} =\Sigma F \\ \frac{d}{2}(F_{x2} -F_{x1} +F_{x4} -F_{x3} )=M \\ \end{array}} \right. \end{aligned}$$
(22)
To simplify and reduce the number of variables, we assume that \(P =F_{x1}+F_{x3}\) and \(Q =F_{x2}+F_{x4}\). Thus,
$$\begin{aligned} \left\{ {\begin{array}{l} P={(\Sigma F-{2M}/d)}/2 \\ Q={(\Sigma F+{2M}/d)}/2 \\ \end{array}} \right. \end{aligned}$$
(23)
In addition, the maximum driving torque of the individual in-wheel motors is limited. Hence, the actuator saturation of each tire should also be considered. The actual longitudinal force generated by a specific tire is limited by the vertical load and tire–road friction. Thus, an additional constraint on the control input can be written as
$$\begin{aligned} \begin{array}{l} -A_i \le F_{xi} \le A_i \\ A_i =\min (\mu _i F_{zi} ,F_m )\ (i=1,2,3,4) \\ \end{array} \end{aligned}$$
(24)
where \(F_{\mathrm{m}}\) represents the maximum driving force converted by the in-wheel motor.
Substituting Eqs. (22) into (24), we obtain
$$\begin{aligned} \left\{ {\begin{array}{l} -A_1 \le F_{x1} \le A_1 \\ -A_3 +P\le F_{x1} \le A_3 +P \\ -A_2 \le F_{x2} \le A_2 \\ -A_4 +Q\le F_{x2} \le A_4 +Q \\ \end{array}} \right. \end{aligned}$$
(25)
For convenience, Eq. (19) is written as
$$\begin{aligned} \left\{ {\begin{array}{l} -2(A_1 +A{ }_3)\le (\Sigma F-{2M}/d)\le 2(A_1 +A_3 ) \\ -2(A_2 +A{ }_4)\le (\Sigma F+{2M}/d)\le 2(A_2 +A_4 ) \\ \end{array}} \right. \end{aligned}$$
(26)
The equations above are the optimal judgment conditions, and two allocation modes are implemented depending on whether these are satisfied or not.
In the optimal mode, the multivariable constrained optimal allocation algorithm is applied to solve the problem of wheel torque control allocation. The objective of the control allocator is to minimize the total tire workload and maximize the vehicle–road grip margin. The objective function to be minimized is defined as
$$\begin{aligned} \min J=\sum _{i=1}^4 {c_i \frac{F_{xi}^2 +F_{yi}^2 }{\mu _i^2 F_{zi}^2 }} \end{aligned}$$
(27)
where \(F_{xi}\), \( F_{yi}\), and \(F_{zi}\) represent the longitudinal, lateral, and vertical tire forces, respectively. \(\mu _{i}\) is the tire–road friction coefficient and \(c_{i}\) is the weight coefficient. In this study, the FTC of steering failure is mainly applied through the distribution of the longitudinal force; hence, the lateral tire forces are neglected.
Assuming that the four wheels of the vehicle are driving with the same adhesion coefficient, we can replace the tire forces \(F_{x1}\), \(F_{x2}\) with \(F_{x3}, F_{x4}\), respectively. Therefore, Eq. (21) can be represented as
$$\begin{aligned} \min J=J_\mathrm{H}= & {} \underbrace{\frac{c_1 (P^{2}-2PF_{x3} +F_{x3}^2 )}{F_{z1}^2 }+\frac{c_3 F_{x3}^2 }{F_{z3}^2 }}_{J_a } \nonumber \\&+\,\underbrace{\frac{c_2 (Q^{2}-2QF_{x4} +F_{x4}^2 )}{F_{z2}^2 }+\frac{c_4 F_{x4}^2 }{F_{z4}^2 }}_{J_b } \end{aligned}$$
(28)
To obtain the minimum value, \(J_{a}\) and \(J_{b}\) should be minimized. We differentiate the two parts of Eq. (28) and calculate the minimum points as
$$\begin{aligned} \begin{array}{l} \hat{{F}}_{x3} =\frac{2c_1 PF_{z3}^2 }{2c_1 F_{z3}^2 +2c_3 F_{z1}^2 } \\ \hat{{F}}_{x4} =\frac{2c_2 QF_{z4}^2 }{2c_2 F_{z4}^2 +2c_4 F_{z2}^2 } \\ \end{array} \end{aligned}$$
(29)
The tire forces \(F_{x1}\) and \(F_{x2}\) are calculated according to Eq. (22). Under the multiple constraints, the optimal result should be determined so as to avoid a minimum point beyond the domain of the constraints. The results are summarized in Table 1.
Table 1

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.

The degree of vertical load transfer is represented by the lateral transfer ratio, which can be calculated by
$$\begin{aligned} R_t =\frac{d}{2}\times \frac{( F_{z2} -F_{z1} + F_{z4} - F_{z3} )}{ F_{z1} + F_{z2} + F_{z3} + F_{z4} } \end{aligned}$$
(30)
where \(R_{t}\) is the lateral transfer ratio, \(-d/2< R_{t }< d\)/2. The greater the value of \(R_{t}\), the greater the load transferred from left to right, and vice versa.
To simplify the system, the weight coefficients are assumed to be equal for wheels on the same side of the vehicle, and we only consider the difference between the two sides. Assuming that \(c_{1 }=c_{3 }=c_{l}\) and \(c_{2 }=c_{4 }=c_\mathrm{r}\), the weight constraints are
$$\begin{aligned} \left\{ {\begin{array}{l} c_l +c_\mathrm{r} =1 \\ 0.1\le c_l \le 0.9 \\ \end{array}} \right. \end{aligned}$$
(31)
where \(c_{l}\) and \(c_{r}\) are the weight coefficients of wheels on the left and right sides, respectively. The approximate linear relation between \(c_{l}\) and \(R_{t}\) is shown in Fig. 6, where the slope of the line depends on the vehicle track.
Fig. 6

Weight coefficient adjustment

If there is no solution to Eq. (26), a load-based proportional allocation mode will be executed. The generalized force and moment are distributed in accordance with the ratio of vertical load between the front and rear axles. To equalize the load utilization for both the total longitudinal forces and longitudinal forces on the front/rear axles, the allocated tire force can be derived as
$$\begin{aligned} \begin{aligned} F_{x1} =\frac{gb-a_x h}{2gL}\left( \Sigma F-\frac{2M}{d}\right) \\ F_{x2} =\frac{gb-a_x h}{2gL}\left( \Sigma F+\frac{2M}{d}\right) \\ F_{x3} =\frac{ga+a_x h}{2gL}\left( \Sigma F-\frac{2M}{d}\right) \\ F_{x4} =\frac{ga+a_x h}{2gL}\left( \Sigma F+\frac{2M}{d}\right) \\ \end{aligned} \end{aligned}$$
(32)
Based on the resulting longitudinal tire forces, the driving/braking torque of each wheel can easily be determined according to the effective wheel radius. Both the driving and braking torques of the in-wheel motors are limited to 500 Nm in this study.

4 Simulation Results and Discussion

The vehicle model and control strategy described in the previous sections were implemented in MATLAB/Simulink to conduct a simulation study. A typical single-lane-change maneuver is used to verify the effectiveness of the proposed FTC method. The vehicle parameters used in the simulations are listed in Table 2. The driver model is used to generate the reference signals of normal cases and compared with the two sets of simulation results.
Table 2

Key parameters for simulations

Variables

Parameter name

Units

Value

m

Vehicle mass

kg

1820

a

Distance from c.g. to front axle

m

1.46

b

Distance from c.g. to rear axle

m

1.58

h

Height of the c.g.

m

0.50

d

Front/rear track width

m

1.55

\(I_{z}\)

Yaw moment of inertia

kg m\(^{2}\)

3800

J

Wheel moment of inertia

kg m\(^{2}\)

1.28

R

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.

Figure 7 shows the equivalent steering angle of single-lane-change (SLC). The corresponding reference trajectory will be shown later.
Fig. 7

Equivalent steering angle of SLC

Fig. 8

Variation of WC

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 yaw rate and sideslip angle responses are shown in Figs. 9 and 10, respectively, for different conditions. The results show that the vehicle states follow the reference values well in the case of steering failure with the proposed weight-adjusting FTC strategy. Without WC adjustment, both the yaw rate and sideslip angle follow the desired responses at the start of the anti-fault action, but large steady state errors occur because of the tire force saturation; this will be explained later.
Fig. 9

Yaw rate of SLC

Fig. 10

Sideslip angle of SLC

Fig. 11

Wheel torques with WC

Fig. 12

Wheel torques without WC

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.

The vehicle trajectory for different conditions is shown in Fig. 14. This demonstrates that the proposed FTC strategy with WC adjustment ensures the vehicle trajectory remains much closer to the desired one than in the case of control without WC adjustment. Additionally, it is clear that the vehicle would run straightforward without any turning action when steering failure occurs if no control is applied. This is very dangerous in real scenarios.
Fig. 13

Longitudinal velocity

Fig. 14

Vehicle trajectory

5 Conclusions

  1. (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. (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. (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. (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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Copyright information

© Society of Automotive Engineers of China (SAE-China) 2018

Authors and Affiliations

  1. 1.School of Automotive Engineering, The State Key Laboratory of Mechanical TransmissionChongqing UniversityChongqingChina

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