# Extension Coordinated Multi-Objective Adaptive Cruise Control Integrated with Direct Yaw Moment Control

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Review and Analysis

#### 1.3. Contribution and Organization

## 2. Vehicle Models

#### 2.1. Longitudinal Dynamics Model

_{d}represents the net traction force, m is the vehicle mass, a

_{x}is vehicle longitudinal acceleration, g is the gravitational acceleration, f denotes the rolling coefficient, θ is the grade of road, and F

_{w}is the aerodynamic drag as shown in Equation (2).

_{D}is the drag coefficient, A is the windward area of the vehicle, and v

_{x}represents vehicle longitudinal speed.

#### 2.2. Four-Wheel Vehicle Dynamics Model

_{xi}and F

_{yi}are the longitudinal and lateral forces of the four wheels respectively, and the subscript i is 1, 2, 3, and 4, representing the front-left, front-right, rear-left and rear-right wheel respectively; δ

_{f}is the front wheel steering angle, l

_{f}and l

_{r}are the distance from vehicle gravity center to the front axle and rear axle, respectively; l is the wheelbase, and T is the track width. The longitudinal, lateral and yaw motion are presented as follows:

_{y}represents vehicle lateral velocity, β and ω represent vehicle sideslip angle and yaw rate respectively, and I

_{z}represents the inertia moment.

#### 2.3. Tire Model

_{x}, lateral force F

_{y}or aligning torques M

_{z}, X is wheel slip ratio or wheel sideslip angle, B is stiffness coefficient, C is shape coefficient, D is peak value, E is curvature coefficient, S

_{H}is horizontal offset, and S

_{V}is vertical offset.

## 3. Control System Design

- On the premise of ensuring vehicle lateral stability, the additional yaw moment should be as small as possible to reduce the impact on longitudinal car-following performance and improve the fuel economy.
- On the premise of ensuring the longitudinal car-following performance, the longitudinal acceleration and its change rate should be as small as possible to improve the longitudinal ride comfort.

#### 3.1. Predictive Model

#### 3.1.1. Longitudinal Car-Following Model

_{des}is the desired car-following distance, T

_{h}is the time headway, v

_{x}is the longitudinal speed of host vehicle, and d

_{0}is the static inter-vehicle distance. Here, T

_{h}= 2, d

_{0}= 10.

_{p}is the longitudinal speed of the preceding vehicle. The derivative of Equation (8) can be derived as follows:

_{p}is the longitudinal acceleration of preceding vehicle.

_{x}and a

_{des}are the actual longitudinal acceleration and desired longitudinal acceleration of host vehicle respectively, T

_{ax}is time-constant and T

_{ax}= 0.45.

#### 3.1.2. Lateral Dynamic Model

_{f}and C

_{r}are the cornering stiffness of the front wheel and rear wheel, respectively, and M

_{des}is the desired additional yaw moment.

_{d}and side slip angle β

_{d}are defined according to vehicle parameters, longitudinal speed, and front steering angle δ

_{f}directly manipulated by driver’s steering action [26], as shown in Equation (12).

#### 3.1.3. Model Discretization

_{d}, B

_{d}, D

_{d}can be calculated by Taylor expansion method, as shown in Equation (20).

_{s}is the sampling time and I is the unit matrix.

#### 3.2. Performance Index

#### 3.2.1. Longitudinal Car-Following Performance

_{ref}, Δv

_{ref}, a

_{x,ref}are set as zero.

#### 3.2.2. Lateral Dynamics Stability

_{des}is used to form the cost function for lateral stability, as shown in Equation (22).

#### 3.2.3. Longitudinal Ride Comfort

#### 3.2.4. Cost Function Design

_{p}and N

_{c}denote the predictive horizon and control horizon, respectively. Q(k) and R(k) are non-negative weight matrices, as shown in Equation (26). x

_{ref}is the reference value of MPC, and x

_{ref}= [β

_{d}ω

_{d}0 0 0]

^{T}.

#### 3.3. Extension Control Design

#### 3.3.1. Extracting Character Variable

_{Δd}of the distance error, and sets the weight w

_{Δv}of the relative speed as a constant, then the distance error is selected to form the longitudinal car-following feature status S(Δd).

_{1}and B

_{2}are the parameters related to the road friction coefficient μ, here B

_{1}= 0.064 and B

_{2}= 0.214 [27].

_{region}and the desired yaw rate ω

_{d}are selected as the character variables of lateral stability to form the feature status S(X

_{region}, ω

_{d}).

#### 3.3.2. Dividing the Extension Set

_{1}and Δd

_{2}are the boundaries of the classic domain and the extension domain, respectively. The distance error should be in driver’s permissible longitudinal car-following range to reduce the driver intervention. The boundary of extension domain reflects the boundary of permissible region and impermissible region. Therefore, Δd

_{2}is set to the driver’s maximum permissible value. The driver’s permissible longitudinal car-following range [13] is shown in Equation (28).

_{2}= Δd

_{max}·SDE

^{−1}. The SDE

^{−1}is calculated as follows:

_{max}= 7.2 m, k

_{SDE}= 0.06, and d

_{SDE}= 0.12. The boundary of classic domain Δd

_{1}is set to a relatively small value and Δd

_{1}= 0.1 × Δd

_{2}.

_{region}, as shown in the Figure 7, where ω

_{1}and ω

_{2}are the boundaries of the classic domain and the extension domain in the x-axis direction, X

_{region1}and X

_{region2}are the boundaries of the classic domain and the extension domain in the y-axis direction, respectively. Here, X

_{region1}and X

_{region2}are set to 0.1 and 1 respectively. The extension boundary ω

_{2}in the x-axis direction reflects the boundary under large steering condition. Based on the experience and previous works [25], 0.2 μrad/s is set as the threshold of large steering condition. Therefore, the boundary ω

_{2}is set as 0.2 μrad/s. The classic boundary ω

_{1}is set as 0.1 × ω

_{2}.

#### 3.3.3. Calculating Dependent Degree

_{region}and ω

_{d}are zero. The point Q is supposed as a point in the extension domain. Connecting the point O with the point Q, the intersection points of the line OQ and the domains’ boundaries are the points Q

_{1}and Q

_{2}, respectively. Obviously, in 1-D extension set of car-following distance error, the points Q

_{1}and Q

_{2}correspond to Δd

_{1}and Δd

_{2}respectively. As shown in the Figure 7, the line segment OQ is the shortest distance for the point Q to approach the ideal point O. In the extension sets, the extension distance is defined as the distance from a point to a set, which is defined in a 1-D coordinate system. Therefore, it is required to convert the extension distance of 2-D extension set of lateral stability to a 1-D extension form, as shown in Figure 8.

_{1}> = X

_{c}, the extension domain <Q

_{1}, Q

_{2}> = X

_{e}. The extension distance from the point Q to classic domain is represented as ρ(Q, X

_{c}), and the extension distance from point Q to extension domain is represented as ρ(Q, X

_{e}). The extension distance can be calculated as follows:

#### 3.3.4. Identifying Measure Pattern

_{1}, M

_{2}and M

_{3}, respectively.

#### 3.3.5. Weight Matrix Design

_{Δβ}, w

_{Δω}and w

_{Δd}are set as the real-time weights which are adjusted by the corresponding values of the dependent degree K(S), and the other weights w

_{Δv}, w

_{ae}, w

_{Mdes}, w

_{ades}are set as constants.

_{1}, it means that the distance error is in a small range, and it is not necessary to increase the corresponding weight. When the car-following distance error belongs to the measure pattern M

_{2}, the distance error is in a relatively large range, and it is possible to exceed the driver’s sensitivity limit of the distance error if the corresponding weight is not adjusted timely. When the car-following distance error belongs to the measure pattern M

_{3}, the distance error exceeds the driver’s sensitivity limit, and the corresponding weight should be maximized to reduce the distance error by control. The real-time weight for longitudinal car-following distance is designed as follows:

_{ACC}and K

_{ACC}(S) are defined as the adjustment factor and dependent degree for vehicle longitudinal control.

_{1}, it indicates that the vehicle lateral stability status is in a stability region, and it is not necessary to adjust the corresponding weight. When the lateral stability status belongs to the measure pattern M

_{2}, the lateral stability status is in the area between stability region and instability region and the vehicle may lose stability if the corresponding weight is not adjusted timely. When the lateral stability status belongs to the measure pattern M

_{3}, the lateral stability status is in the instability region, the corresponding weight should be maximized to maintain vehicle lateral stability by control. The real-time weights for lateral stability are designed as follows:

_{VLS}and K

_{VLS}(S) are defined as the adjustment factor and dependent degree for vehicle stability control.

#### 3.4. Lower Layer Design

_{b}and braking torque T

_{b}at the wheels [28], as shown in Equation (40).

_{d}and engine output torque T

_{e}is as follows [14].

_{t}/ω

_{e}) is the torque characteristic function of torque converter, and i

_{g}and i

_{o}denote the transmission ratio of the gearbox and main reducer, respectively.

## 4. DIL Test Results and Analysis

^{2}and drives at a low constant speed 54 km/h in the curve, as shown in Figure 12a. Finally, the preceding vehicle speeds up with an acceleration of 1 m/s

^{2}to drive away from the curve. During the driver in the loop test, in order to reduce the influence of driver’s subjective factors on the results, the driver is not told what kind the controller is, and the steering wheel angle from driver is shown in Figure 12b. It can be seen that the driver’s steering angle under the three controllers is almost the same as a whole, and the driver’s steering wheel angle is little different with the different three controllers.

_{ACC}(S) which can reflect the control effect of the longitudinal control. The proposed control determines the weights of sideslip angle error and yaw rate error by the dependent degree K

_{VLS}(S) which can reflect the risk of losing vehicle lateral stability. As can be seen from Figure 16b, when the longitudinal distance error increases, the KACC will be increased to adjust the weights and ensure the longitudinal car-following capability; when the value of driver steering wheel angle and X

_{region}increase, the K

_{VLS}will be increased to adjust the weights and ensure the lateral stability. The maximum errors and X

_{region}with three controllers are shown in Table 2. Obviously, the overall performance of the control system is improved. The proposed control can intelligently determine the weight matrices by the control effect of the longitudinal distance error and the risk of losing lateral stability. Thus, on the premise of ensuring the car-following performance and lateral stability, the fuel economy and longitudinal ride comfort are improved as much as possible.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**Fuel consumption rate and map of engine: (

**a**) fuel consumption rate of engine; (

**b**) engine map.

**Figure 12.**Longitudinal speed and steering wheel angle: (

**a**) longitudinal speed; (

**b**) steering wheel angle.

**Figure 13.**Longitudinal car-following errors, (

**a**) Longitudinal car-following distance error, (

**b**) Relative speed.

**Figure 15.**Phase plane of errors: (

**a**) phase plane of longitudinal car-following errors; (

**b**) phase plane of lateral errors.

**Figure 17.**Fuel consumption and longitudinal acceleration: (

**a**) fuel consumption; (

**b**) longitudinal acceleration.

**Figure 19.**Brake pressure on wheels: (

**a**) brake pressure with ACC; (

**b**) brake pressure with ACC&DYC; (

**c**) brake pressure with proposed control.

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

Vehicle mass | m | 1301 kg |

Gravitational acceleration | g | 9.8 m/s^{2} |

Inertial of z axis | I_{z} | 1600 kg·m^{2} |

Track width | T | 1.544 m |

Distance from vehicle gravity center to front axle | l_{f} | 0.97 m |

Distance from vehicle gravity center to rear axle | l_{r} | 1.567 m |

Road adhesion coefficient | μ | 0.6 |

Maximum acceleration | a_{max} | 2.5 m/s^{2} |

Maximum jerk | j_{max} | 0.5 m/s^{3} |

ACC | ACC&DYC | Proposed Control | |
---|---|---|---|

Δd (m) | 12.539 | 20.836 | 9.311 |

Δv (m/s) | 3.232 | 3.425 | 3.092 |

Δβ (rad) | 0.021 | 0.021 | 0.020 |

Δω (rad/s) | 0.090 | 0.045 | 0.067 |

X_{region} | 0.321 | 0.289 | 0.273 |

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## Share and Cite

**MDPI and ACS Style**

Wang, H.; Sun, Y.; Gao, Z.; Chen, L.
Extension Coordinated Multi-Objective Adaptive Cruise Control Integrated with Direct Yaw Moment Control. *Actuators* **2021**, *10*, 295.
https://doi.org/10.3390/act10110295

**AMA Style**

Wang H, Sun Y, Gao Z, Chen L.
Extension Coordinated Multi-Objective Adaptive Cruise Control Integrated with Direct Yaw Moment Control. *Actuators*. 2021; 10(11):295.
https://doi.org/10.3390/act10110295

**Chicago/Turabian Style**

Wang, Hongbo, Youding Sun, Zhengang Gao, and Li Chen.
2021. "Extension Coordinated Multi-Objective Adaptive Cruise Control Integrated with Direct Yaw Moment Control" *Actuators* 10, no. 11: 295.
https://doi.org/10.3390/act10110295