# A Path Planning Method for Autonomous Vehicles Based on Risk Assessment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Lane Change Curve Models

#### 2.1. Analysis of Potential Collision Points of Vehicles

_{1}and S

_{2}are the critical points of collision between autonomous vehicles and vehicles in other lanes, respectively. X

_{a}and X

_{f}represent the location of autonomous vehicles when collision occurs. At the same time, the vehicle model is simplified to an elliptical model, taking the vehicle center as the origin of the ellipse, assuming that the minor axis is equal to the width of the lane, and the major axis a is:

_{a}is the rear vehicle speed, v

_{f}is the speed of the vehicle ahead, T

_{d}is the driver’s style factor, and the aggressive style is 0.8 while the normal style is 0.2, w is the vehicle width, and l is the vehicle length.

_{ego}> v

_{f}, the relative distance between the two vehicles will be reduced, and a collision will occur when it is less than the minimum safe distance. At this time, the collisional point is recorded as S

_{2}, and the collisional time is t

_{S2}(t

_{1}< t

_{S2}< t

_{2}). The coordinates of S

_{2}are:

_{ego}< v

_{a}, the collisional point is recorded as S

_{1}, and the collisional time is t

_{S1}(t

_{1}< t

_{S1}< t

_{2}). The coordinates of S

_{1}are:

_{a}and X

_{f}, the collision can be avoided. The vehicle center coordinate x is obtained according to the following formula.

_{1}and f

_{2}are the focus points of ellipse, c is the focal length, θ is the heading angle of the vehicle, and |f

_{1}S

_{1}| is the distance from f

_{1}to S

_{1}.

#### 2.2. Mathematical Model of Trajectory

_{0}= [0 0]′ is the start point and P

_{5}is the end point. The first control point is the position of the vehicle itself, so it is known, and the rest of the control points are unknown during the obstacle-avoidance process.

_{1}and P

_{4}are midpoints. The heading angle of the vehicle is always parallel to the centerline of the lane at the start and end.

_{0}, P

_{2}and P

_{3}, P

_{5}is a straight line and the value is L

_{1}. Therefore, we can obtain a series of Bézier curve clusters by constraining the positions of the points P

_{2}and P

_{3}. For straight line L

_{1}, the vehicle is just on the center line of the lane and the heading angle θ is 0. At this time, the obstacle vehicle is directly in front of the vehicle. The constraint case is denoted as

_{obs}refers to the longitudinal distance between the vehicle and the front vehicle, and L

_{b}is the distance from the center of mass of the vehicle to the front of the vehicle. Therefore, under the constraint of L

_{1}, it can be obtained that P

_{2}= [0, L

_{1}]′. The coordinates of point O can be expressed as (L1 + L2, y

_{0}), X

_{a}< L

_{1}+ L

_{2}< X

_{f}. The curvature at point O is obtained as

_{1}+ L

_{2}, 4k

_{o}L

^{2}

_{1}/3]′. The operation between L

_{1}and L

_{2}is

_{o}≤ 0.5gt

^{2}. Then, we can get all the control points of the Bézier curve, such as P

_{3}= [L

_{1}+ L

_{2}, 8k

_{o}L

^{2}

_{1}/3]′, P

_{5}= [2L

_{1}+ 2L

_{2}, 8k

_{o}L

^{2}

_{1}/3]′, P

_{1}= [0, L

_{1}/2], and P4 = [3L

_{1}/2 + 2L

_{2}, 8k

_{o}L

^{2}

_{1}/3]′.

## 3. Risk Assessment Process

#### 3.1. Establishing Road Risk Assessment Model

_{obs}= (x

_{obs},y

_{obs}), and U

_{obs}and U

_{road}represent the potential field of the obstacle and the road, respectively. The attractive potential field is

_{r}is the current lane centerline, and D

_{s}is the search distance.

_{x}is the variation function of potential energy in the width direction, A

_{y}is the potential energy variation function in the length direction of the road, and A represents the amplitude of A

_{y}.

_{l}is the centerline of the left lane, and d

_{ac}is the best distance that the vehicle should keep from the obstacle vehicle. When the distance between the vehicle and the obstacle is less than d

_{ac}, the influence of the road length on the potential field function is 0. Then, the driver’s reaction time t

_{r}, the maximum braking force of the wheel F

_{b}, the body length L

_{f}, and the speed of the preceding vehicle should be considered, and the distance formula from the vehicle to the obstacle can be expressed as:

#### 3.2. Obstacle Risk Assessment Model

_{1}and q

_{2}are the variable coefficients of the potential field in the lateral and longitudinal directions of the road. U

_{t}is the threshold of the obstacle repulsion potential field. The longitudinal coefficient q

_{2}is

_{f}is the length of obstacle vehicle. The longitudinal and variable coefficient prevents the vehicle from colliding with an obstacle; the horizontal and variable coefficient q

_{1}is related to the width of the road, and mainly it prevents collision with the road boundary. That is to say, when the vehicle and the obstacle are not in the same lane, or the distance is more than the d

_{ac}, the q

_{1}is 0.

_{i}is the i-th path point. The candidate paths are determined by the lowest risk assessment.

## 4. Optimal Path Selection

_{c}and w

_{e}are the weight value for comfort and efficiency of the lane change, f

_{c}(i) is the comfort evaluation function for the i-th path, and f

_{e}(i) is the efficiency of lane change evaluation function for the i-th path.

_{0}to the end point P

_{5}. S(i) is the length of the path i.

_{c}(i) = k(i), in [29] k(i) is

_{a}is introduced separately for the cost functions.

## 5. Results and Analysis

#### 5.1. The Scene of Static Obstacles

_{0}and P

_{5}of the fifth order Bézier curve. The comparison of the yaw angles between aggressive-style and normal-style vehicles is shown in Figure 8d. The aggressive-style angle is turned earlier, which means the steering is earlier than the normal style, and the maximum yaw angle is 6.696, which is larger than the 6.01 of the normal style.

#### 5.2. The Scene of Dynamic Obstacles

#### 5.3. Experimental Results

## 6. Conclusions

- (1)
- In this paper, the vehicle is simplified as an ellipse considering the length, width, and speed information, which makes the model more accurate in collision solution. Then, the control points of the fifth order Bézier curve are constrained to generate a series of trajectories in a safe range.
- (2)
- The APF model, which takes the driver’s reaction time into account, conducts risk assessment on each path and selects the path most suitable for the driver’s habits under the aggressive or the normal style. The results of both simulation and experiment show that the algorithm proposed in this paper has a good effect on driverless vehicles’ lane changing and obstacle avoidance. In the future, continuous lane changing and obstacle avoidance under the condition of multiple obstacles will be considered, and the trajectory prediction of lane changing will be introduced.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**(

**a**) Static cost function values of candidate paths, (

**b**) the total cost values of different styles in static scene, (

**c**) comparison of different style paths, and (

**d**) comparison of different styles’ yaw angles.

**Figure 11.**(

**a**) Dynamic cost function values of candidate paths, (

**b**) the total cost values of different styles in the dynamic scene, (

**c**) comparison of different style paths, (

**d**) comparison of different styles’ yaw angles, and (

**e**) comparison of the lateral velocity.

**Figure 12.**(

**a**) Scene before lane change in aggressive style, (

**b**) Begin lane change, (

**c**) End of lane change, (

**d**) Scene after lane change in aggressive style.

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

Sprung mass | 1370 kg |

Speed | 60 km/h |

Wheelbase | 2866 mm |

Single lane width | 4000 mm |

Obstacle vehicle length | 4600 mm |

Potential energy threshold | 0.01 |

Driver reaction time | 0.35 s |

Preview time | 0.6 s |

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**MDPI and ACS Style**

Yang, W.; Li, C.; Zhou, Y.
A Path Planning Method for Autonomous Vehicles Based on Risk Assessment. *World Electr. Veh. J.* **2022**, *13*, 234.
https://doi.org/10.3390/wevj13120234

**AMA Style**

Yang W, Li C, Zhou Y.
A Path Planning Method for Autonomous Vehicles Based on Risk Assessment. *World Electric Vehicle Journal*. 2022; 13(12):234.
https://doi.org/10.3390/wevj13120234

**Chicago/Turabian Style**

Yang, Wei, Cong Li, and Yipeng Zhou.
2022. "A Path Planning Method for Autonomous Vehicles Based on Risk Assessment" *World Electric Vehicle Journal* 13, no. 12: 234.
https://doi.org/10.3390/wevj13120234