# Adaptive Nonsingular Fast Terminal Sliding Mode-Based Direct Yaw Moment Control for DDEV under Emergency Conditions

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Dynamics Modeling of DDEV

#### 2.1. 7-DOF Model

_{zx}is derived from the longitudinal tire forces, M

_{zy}is derived from the lateral tire forces, and $\sum {M}_{z}$ is described as follows:

_{x}and a

_{y}correspond to the longitudinal and lateral acceleration. v

_{x}and v

_{y}correspond to the longitudinal and lateral speeds. δ is the steering angle of the front wheels, assuming the steering angles of the front left and right wheels are equal. γ is the yaw rate. ∑M

_{z}is the resultant yaw moment. I

_{z}is the inertia moment about the z-axis. B

_{w}is the wheel track. F

_{yij}is lateral tire force. F

_{xij}is longitudinal tire force. The symbol ij (ij = fl, fr, rl, rr) denotes front left, front right, rear left, and rear right.

_{ij}is the wheel angular speed. J

_{ωij}is the wheel inertia moment. R

_{e}is the effective rolling radius. T

_{dij}and T

_{bij}are the motor driving and braking torque.

#### 2.2. Magic Formula Tire Model

_{h}and S

_{v}are the shifts of horizontal and vertical.

_{i}(i = 0, 1, …, 8) are the fitting parameters of lateral tire force, b

_{i}(i = 0, 1, …, 8) are the fitting parameters of longitudinal tire force, and they can be obtained by the tire force testing given in the following Table 1.

_{wxij}denotes the wheel center velocity, as shown in Equation (16):

_{zij}is the vertical load, h

_{g}is the height of the gravity center, l is the wheelbase, and g is the gravity acceleration.

#### 2.3. Reference Model

_{yf}and F

_{yr}are the lateral tire forces of the front and rear wheels, respectively.

_{x}is constant, δ is small, and the lateral tire force is a linear function of tire slip angle as follows:

_{f}is the cornering stiffness of the front wheel, and C

_{r}is the cornering stiffness of the rear wheel. α

_{f}and α

_{r}are the slip angles of the front and rear wheels, respectively.

## 3. Stability Boundary Function

_{1}and E

_{2}are constant coefficients.

_{x}, and front-wheel steering angle δ are the major factors that determine the boundary of the stability domain. This paper focuses on emergency conditions, so the effect of road adhesion coefficient on stability boundary is emphasized.

_{x}, μ, and δ are given, the initial vehicle state point $({\dot{\beta}}_{0},{\gamma}_{0})$ is set, and the solution of this state equation is a phase trajectory curve. Set δ = 0, v

_{x}= 80 km/h, 0.1 ≤ μ ≤ 0.8, and the simulation is carried out under different road adhesion coefficients at 0.1 intervals. The phase trajectories can be drawn in Figure 6, and, based on the division principle of whether the phase trajectory curve converges to the equilibrium point, the $\beta -\dot{\beta}$ phase plane is partitioned into two parts (stable and unstable domains). It is observed that when the road adhesion coefficient is less than 0.6, the stable region expands as the road adhesion coefficient increases, which means that the high road adhesion coefficient can provide a large stability margin. When the road adhesion coefficient is beyond 0.6, the stable region is almost invariable. Therefore, based on the two-line method, the boundary coefficients E

_{1}and E

_{2}under different road adhesion coefficients are obtained in Table 2.

## 4. Decision Controller

_{zc}, which is related to longitudinal tire forces. Based on Equation (5), M

_{zc}is expressed as follows:

_{1}, x

_{2}]

^{T}is the system state vector. f(x) and g(x) are the nonlinear functions of

**x**. u is the control input. d(x) represents the system uncertainties and external disturbances, D is the upper boundary, d(x) $\le $ D and D > 0.

#### 4.1. Side Slip Angle Tracking Controller

_{1}= β and x

_{2}= $\dot{\beta}$, and the control system model is described as follows:

_{β}(x,t) is the system uncertainties and external disturbances, satisfying |d

_{β}(x,t)| ≤ D

_{β}and D

_{β}> 0, D

_{β}is the upper boundary.

#### 4.2. Yaw Rate Tracking Controller

_{γ}(x,t) is the system uncertainties and external disturbances, and satisfying |d

_{γ}(x,t)| ≤ D

_{γ}and D

_{γ}> 0, D

_{γ}is the upper boundary.

#### 4.3. Proof of the Stability and Finite-Time Convergence

_{1}can be obtained as follows:

_{1}, s, ${\tilde{D}}_{\beta}$ are bounded and set at $\left|{\tilde{D}}_{\beta}\right|\le {\eta}_{\beta}$, ${\eta}_{\beta}>0$.

**Lemma**

**1.**

_{o}time is $x({t}_{0})={x}_{0}$. Then, it is considered that the system can converge to the equilibrium pointwithin a finite time T:

_{r}in Equation (73), the tracking error can converge to the equilibrium point:

_{s}given as follows [35]:

#### 4.4. Adaptive Weight between the Handling and Stability

_{1}is the width of the stability region, d

_{2}is the distance between point P and the stability boundary, and |PO| is the distance between point P and the centerline of the stability region.

_{th}is the threshold value, commonly set as 10 deg [37]. The vehicle is severely unstable when β > β

_{th}, and the boundary of the severely unstable region is formulated as follows:

_{β}is the weight coefficient, and 1 ≤ C

_{β}≤ 0.

_{β}, and inside the severely unstable region, the control weight is 0. Finally, the extra yaw moment is deduced as follows:

## 5. Torque Allocation Controller

#### 5.1. Allocation Objective

- $V={[{F}_{xreq},{M}_{zc}]}^{\mathrm{T}}$,
- $u={[{F}_{xfl},{F}_{xfr},{F}_{xrl},{F}_{xrr}]}^{\mathrm{T}}$,
- $B=\left[\begin{array}{cccc}\mathrm{cos}\delta & \mathrm{cos}\delta & 1& 1\\ a\mathrm{sin}\delta -\frac{{B}_{w}}{2}\mathrm{cos}\delta & a\mathrm{sin}\delta +\frac{{B}_{w}}{2}\mathrm{cos}\delta & -\frac{{B}_{w}}{2}& \frac{{B}_{w}}{2}\end{array}\right]$.

_{1}is constructed to minimize the torque allocation error:

_{F}and W

_{M}are the weight matrix of longitudinal and lateral force, and, here,${W}_{v}=diag(1,2/{B}_{w})$ [38].

_{ij}is the tire utilization ratio, and 0 ≤ η

_{ij}≤ 1. The higher tire utilization ratio means that more road adhesion force is consumed, and less road adhesion force is used to keep stable; at the same time, the tire forces tend to saturate, and the vehicle stability margin is reduced.

_{2}is constructed. The longitudinal tire force can be adjusted by the motor controller, but the lateral tire force is difficult to control directly. With the assumption that the road adhesion coefficient is fixed, J

_{2}can be expressed as follows:

- ${W}_{u}=diag\left(\frac{1}{\mu {F}_{zfl}},\frac{1}{\mu {F}_{zfr}},\frac{1}{\mu {F}_{zrl}},\frac{1}{\mu {F}_{zrr}}\right)$.

#### 5.2. Constraints

_{d}

_{max}is the maximum driving torque, and T

_{b}

_{max}is the maximum braking torque.

#### 5.3. Solution of the Optimization Problem

## 6. Simulation and Analyses

#### 6.1. Double Lane Change Maneuver

_{x}= 80 km/h, μ = 0.3). The scheme of double lane change is shown in Figure 9. The total length of the test road is 200 m, and the length and width of each road section are indicated.

#### 6.2. Serpentine Maneuver

#### 6.3. Sine Steering Angle Input Maneuver

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The characteristic curve of tire forces: (

**a**) the longitudinal tire force under different vertical forces; (

**b**) the lateral tire force under different vertical forces; (

**c**) the combined condition under different tire slip angles.

**Figure 10.**The simulation results of double lane change. (

**a**) Trajectory; (

**b**) longitudinal velocity; (

**c**) lateral acceleration; (

**d**) yaw rate; (

**e**) side slip angle; (

**f**) $\beta -\dot{\beta}$ phase plane; (

**g**) yaw moment; (

**h**) longitudinal torque of ANFSMC.

**Figure 12.**The simulation results of the serpentine. (

**a**) Lateral acceleration; (

**b**) yaw rate; (

**c**) side slip angle; (

**d**) β−$\dot{\beta}$phase plane; (

**e**) yaw moment; (

**f**) longitudinal torque of ANFSMC.

**Figure 13.**The simulation results of the sine steering angle input. (

**a**) Steering wheel angle; (

**b**) yaw rate; (

**c**) side slip angle; (

**d**) β−$\dot{\beta}$phase plane; (

**e**) yaw moment; (

**f**) longitudinal torque of ANFSMC.

Parameters | Value | Parameters | Value |
---|---|---|---|

a_{0} | 1.30 | b_{0} | 1.65 |

a_{1} | −22.1 | b_{1} | −21.3 |

a_{2} | 1011 | b_{2} | 1144 |

a_{3} | 1078 | b_{3} | 49.6 |

a_{4} | 1.82 | b_{4} | 226 |

a_{5} | 0.208 | b_{5} | 0.069 |

a_{6} | 0 | b_{6} | −0.006 |

a_{7} | −0.354 | b_{7} | 0.056 |

a_{8} | 0.707 | b_{8} | 0.486 |

μ | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|

E_{1}/(s) | 0.5424 | 0.3934 | 0.3311 | 0.2923 | 0.2295 | 0.1736 | 0.1398 | 0.1684 |

E_{2}/(rad) | 0.0163 | 0.0690 | 0.0761 | 0.0877 | 0.0918 | 0.0955 | 0.0964 | 0.0973 |

Description | Symbol | Value |
---|---|---|

Vehicle mass | m | 1350 kg |

Vehicle rotational inertia about z-axis | I_{z} | 1343 kg∙m^{2} |

Inertia of wheel | ${J}_{\omega}$ | 0.6 kg∙m^{2} |

Height of the center of gravity (CoG) | h_{g} | 0.54 m |

Distance from COG to front axle | a | 1.04 m |

Distance from COG to rear axle | b | 1.56 m |

Wheel track | B_{w} | 1.481 m |

Effective wheel radius | r_{e} | 0.298 m |

Front cornering stiffness | C_{f} | 58,070 N/rad |

Rear cornering stiffness | C_{r} | 58,070 N/rad |

Parameter | Value | Parameter | Value |
---|---|---|---|

r_{1γ} | 5/3 | r_{1β} | 5/3 |

r_{2γ} | 7/5 | r_{2β} | 7/5 |

ζ_{1γ} | 0.5 | ζ_{1β} | 0.5 |

ζ_{2γ} | 0.5 | ζ_{2β} | 0.5 |

ε_{γ} | 1000 | ε_{β} | 1000 |

k_{γ} | 100 | k_{β} | 100 |

∆ | 0.5 | D_{β} | 0.3 |

D_{γ} | 0.3 | - | - |

Control Strategy | Yaw Rate/(deg/s) | Side Slip Angle/(deg) |
---|---|---|

SMC | 0.7294 | 0.1480 |

ANFTSMC | 0.3363 | 0.0581 |

Control Strategy | Yaw Rate/(deg/s) | Side Slip Angle/(deg) |
---|---|---|

SMC | 1.3272 | 0.2805 |

ANFTSMC | 0.4939 | 0.0943 |

Control Strategy | Yaw Rate/(deg/s) | Side Slip Angle/(deg) |
---|---|---|

SMC | 0.8159 | 0.2593 |

ANFTSMC | 0.5699 | 0.0943 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Ma, J.; Zhou, Y.
Adaptive Nonsingular Fast Terminal Sliding Mode-Based Direct Yaw Moment Control for DDEV under Emergency Conditions. *Actuators* **2024**, *13*, 170.
https://doi.org/10.3390/act13050170

**AMA Style**

Zhang Y, Ma J, Zhou Y.
Adaptive Nonsingular Fast Terminal Sliding Mode-Based Direct Yaw Moment Control for DDEV under Emergency Conditions. *Actuators*. 2024; 13(5):170.
https://doi.org/10.3390/act13050170

**Chicago/Turabian Style**

Zhang, Yixi, Jian Ma, and Yang Zhou.
2024. "Adaptive Nonsingular Fast Terminal Sliding Mode-Based Direct Yaw Moment Control for DDEV under Emergency Conditions" *Actuators* 13, no. 5: 170.
https://doi.org/10.3390/act13050170