# MPC-Based Obstacle Avoidance Path Tracking Control for Distributed Drive Electric Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

_{f}and additional yaw moment ∆M

_{z}are output by the MPC controller in Section 3.2. By processing the information of the path planning layer, the torque distribution controller distributes the torque through the additional yaw moment and the wheel vertical force ratio in Section 3.3. The MPC controller and the torque distribution controller together form a path-tracking controller. During the obstacle avoidance process, the vehicle avoids obstacles through the front wheel angle and four-wheel torque output by the controller. In Section 4, the simulation platform is established, and the effect of the controllers is verified.

## 2. Path Planning

_{ob}, Y

_{ob}), and the radius of the circumscribed circle of the vehicle and the obstacle as R

_{car}and R

_{ob}, this is shown in Figure 2,the constraint condition of anti-collision can be expressed as:

_{y}is the lateral acceleration, ${m}_{s}$ is the sprung mass, $h$ is the distance from the center of mass to the ground, ${F}_{zl}$ represents the sum of forces on the left front and rear wheels in the vertical direction, ${F}_{zr}$ represents the sum of forces on the right front and rear wheels, $s$ represents the track width, g is the gravitational acceleration, $\Delta y=h\cdot \mathrm{sin}\varphi $.

_{y}should be fully taken into account in path planning. Given the maximum roll risk LTR

_{max},

_{0}and t

_{d}are the initial time and the end time, respectively, a

_{y}is the lateral acceleration, X

_{ob}and Y

_{ob}are the coordinate centers of the obstacles, the coordinate center of the obstacle:(X

_{ob}, Y

_{ob}) = (100, 1), R

_{car}and R

_{ob}are the radiuses of the circumscribed circles between the vehicle and the obstacle, ∅ is the body roll angle, and w is the wheelbase, h is the distance from the center of mass to the ground, g is the acceleration of gravity, H is the self-conjugate matrix, J

_{anti-r}is the anti-rollover evaluation index function, and LTR is the roll risk coefficient.

_{3}, b

_{4}, b

_{5}and b

_{6}can be obtained by the optimization algorithm.

## 3. Path Tracking Control

#### 3.1. Establishment of Vehicle Dynamics Model

_{xf}and F

_{yf}are the force exerted on a single tire, it needs to be multiplied by 2. According to Newton’s second law:

_{f}and l

_{r}are the distances from the center of mass to the front and rear axles, respectively. m is the overall mass of the vehicle, and I

_{z}is the central moment of inertia of the vehicle around the z axis.

_{xf}and F

_{yf}are the resultant forces on the front wheels in the x and y directions, respectively. F

_{xr}and F

_{yr}are the resultant forces on the front wheels in the x and y directions, respectively, and F

_{lf}and F

_{lr}are the longitudinal forces on the front and rear wheels, respectively. F

_{cf}and F

_{cr}are the lateral forces on the front and rear wheels, respectively. δ

_{f}is the front wheel angle, δ

_{r}is the rear wheel angle.

_{z}is the vertical load on the tire, μ is the road friction coefficient, s is the slip ratio, and α is the wheel side angle.

_{l}is the longitudinal stiffness of the vehicle tire, and C

_{c}is the cornering stiffness of the vehicle tire.

_{c}and v

_{l}represent the lateral and longitudinal speeds of the tire, respectively, which can be represented by the speeds v

_{x}and v

_{y}in the direction of the coordinate system:

_{x}, $\dot{y}$ = v

_{y}, Equation (22) can be simplified as:

**A**,

**B**, and

**C**are coefficient matrices, respectively:

#### 3.2. MPC Controller Design

_{0}= 18 m/s; The actual measured value is v

_{y}, r, φ, y; The control input value is α

_{f}, ∆M

_{z}.

**A**= I + A

_{k}_{T}, B

_{k}= BT, C

_{k}= C.

_{ref}is obtained by the path planning layer;

**Q**and

**R**are the trajectory tracking error weight matrix and the control amount increment weight matrix, respectively; N

_{p}and N

_{c}are the prediction time domain and the control time domain, respectively. Some parameter settings in model predictive control are shown in Table 1.

#### 3.3. Torque Distribution Controller Design

_{z}output by the MPC controller is used to calculate the total demand torque T

_{all}. Then using the proportion of the vertical load, calculates the torque of the four wheels. On the premise of ensuring lateral stability, the flexible torque distribution is realized.

_{z}and the four-wheel torque should satisfy the following relationship:

_{Fl}, T

_{Fr}, T

_{Rl}and T

_{Rr}are the torques of the left front wheel, the right front wheel, the left rear wheel and the right rear wheel, respectively.

_{z,Fl}, F

_{z,Fr}, F

_{z,Rl}, F

_{z,Rr}are the vertical loads of the left front wheel, the right front wheel, the left rear wheel and the right rear wheel, respectively. Force, k

_{1}, k

_{2}, k

_{3}, k

_{4}are the proportional coefficients of the four-wheel vertical force and the total vertical force.

_{all}and the distributed four-wheel torques T

_{Fl}, T

_{Fr}, T

_{Rl}, T

_{Rr}can be obtained:

## 4. Simulation Analysis and Verification

#### 4.1. Verification of Co-Simulation Platform

#### 4.2. Simulation of Tracking Control Performance

## 5. Conclusions

- (1)
- Sixth-order polynomial for obstacle avoidance path planning is presented. Through the simulation results, it is verified that the planned path can be accurately tracked, and the LTR values are always within the safe range. The vehicle has no risk of rollover. The obstacle avoidance path and tracking controller in this paper can effectively meet the requirements of safe obstacle avoidance.
- (2)
- In this paper, path planning, path tracking and torque distribution are combined to achieve safe obstacle avoidance through the tracking control of the obstacle avoidance path. The path-tracking controller not only realizes the intelligent obstacle avoidance process of unmanned vehicles but also combines it with distributed vehicles. The safety and stability are improved by the torque distribution strategy in the obstacle avoidance process compared with traditional vehicles. Simultaneously, simulations are carried out under different road adhesion coefficient conditions, and the simulation results show that the vehicle can still perform safe and stable automatic obstacle avoidance under the conditions of road adhesion coefficient μ = 0.2, indicating that the controller has good robustness.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Vehicle control flow chart. Where, X

_{c}

_{0}and Y

_{c}

_{0}are the lateral distance and longitudinal distance from the obstacle; v

_{c}is the vehicle speed of the obstacle; X

_{d}and Y

_{d}are the expected lateral and longitudinal distance of the vehicle; φ

_{d}is the expected yaw angle of the vehicle; ∆M

_{z}is the additional yaw moment; T

_{fl}, T

_{rl}, T

_{rr}and T

_{fr}are the torques of the left front wheel, the left rear wheel, the right rear wheel and the right front wheel, respectively; X and Y are the transverse and vertical coordinates under the inertial coordinate system; φ is the yaw angle under the body coordinate system; r is the turning radius of the vehicle; v

_{x}and v

_{y}are the lateral and speed longitudinal of the vehicle; a

_{x}is the lateral acceleration of the vehicle.

**Figure 6.**Co-simulation results. (

**a**) Path tracking. (

**b**) Yaw angle tracking. (

**c**) Lateral acceleration. (

**d**) Front-wheel turning angle.

**Figure 8.**Simulation comparison under different road adhesion coefficient conditions. (

**a**) Path tracking. (

**b**) Yaw tracking. (

**c**) Front-wheel angle. (

**d**) Lateral acceleration.

**Figure 9.**Comparison of vertical forces on wheels. (

**a**) vertical force of left wheel; (

**b**) vertical force of right wheel.

**Figure 10.**Comparison of torque distribution among wheels. (

**a**) Torque comparison of left front wheel. (

**b**) Torque comparison of left rear wheel. (

**c**) Torque comparison of right front wheel. (

**d**) Torque comparison of right rear wheel.

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

Nc | 8 |

Np | 25 |

$\Delta {u}_{min}$ | −0.02 |

$\Delta {u}_{max}$ | 0.2 |

${a}_{ymin}$ | −0.25 g |

${a}_{ymax}$ | 0.25 g |

Q | [2000 0; 0 2000] |

R | [3000 0; 0 1] |

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

Vehicle sprung mass ${m}_{s}/\mathrm{kg}$ | 1743 |

Vehicle mass m/kg | 1907 |

Moment of inertia ${I}_{z}/\left(\mathrm{kg}\cdot {\mathrm{m}}^{2}\right)$ | 3246.9 |

front wheelbase ${l}_{f}$/m | 1.33 |

rear wheelbase ${l}_{r}$/m | 1.81 |

The height of the center of mass above the ground h/m | 0.781 |

Wheeltrack b/m | 2.029 |

Lateral stiffness of front wheels $\text{}{C}_{cf}/\left(\mathrm{N}\cdot {\mathrm{rad}}^{-1}\right)$ | 116,050 |

Lateral stiffness of rear wheels ${C}_{cr}/\left(\mathrm{N}\cdot {\mathrm{rad}}^{-1}\right)$ | 104,590 |

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**MDPI and ACS Style**

Wu, H.; Zhang, H.; Feng, Y.
MPC-Based Obstacle Avoidance Path Tracking Control for Distributed Drive Electric Vehicles. *World Electr. Veh. J.* **2022**, *13*, 221.
https://doi.org/10.3390/wevj13120221

**AMA Style**

Wu H, Zhang H, Feng Y.
MPC-Based Obstacle Avoidance Path Tracking Control for Distributed Drive Electric Vehicles. *World Electric Vehicle Journal*. 2022; 13(12):221.
https://doi.org/10.3390/wevj13120221

**Chicago/Turabian Style**

Wu, Hongchao, Huanhuan Zhang, and Yixuan Feng.
2022. "MPC-Based Obstacle Avoidance Path Tracking Control for Distributed Drive Electric Vehicles" *World Electric Vehicle Journal* 13, no. 12: 221.
https://doi.org/10.3390/wevj13120221