# Local Path Planning of the Autonomous Vehicle Based on Adaptive Improved RRT Algorithm in Certain Lane Environments

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## Abstract

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## 1. Introduction

- This method uses the sampling node method based on heuristic information to make the random tree grow more directionally and utilizes the adaptive sampling space and the adaptive step size to speed up its convergence.
- The road environment constraints and the vehicle’s constraint method gained by a magnifying mechanism considering the motion characteristics of the vehicle are considered to make the vehicle achieve an excellent behavior of avoiding obstacles.
- A heuristic node selection mechanism is introduced to select the nearest tree node so that the vehicle moves smoothly at less cost.
- The post-processing, including the pruning method and the cubic B-spline, is used to optimize the generated path to make the path satisfy the requirement of autonomous vehicle driving.

## 2. Preliminaries

#### 2.1. Road Environment Method

#### 2.1.1. Road Environment Geometries

#### 2.1.2. Constraints of Road Environment

#### 2.2. The Vehicle Model with Non-Integrality Constraint

## 3. Adaptive Improved RRT Path Planning

#### 3.1. Basic RRT

Algorithm 1:$\mathrm{Build}\_\mathrm{RRT}({P}_{init}$) |

$1.\mathrm{T}\_\mathrm{init}({P}_{init}$) |

$2.\mathrm{While}\mathrm{DISTANCE}({P}_{new}$$,{P}_{goal}$) do |

$3.{P}_{rand}$←RANDOM_STATE ( ) |

$4.{P}_{near}$←$\mathrm{NEAREST}\_\mathrm{NEIGHBOR}(\mathrm{T},{P}_{rand}$) |

$5.{P}_{new}$←$\mathrm{EXTEND}(\mathrm{T},{P}_{rand}$$,{P}_{near}$) |

$6.\mathrm{if}\mathrm{DISTANCE}({P}_{new}$$,{P}_{goal}$$)\le {D}_{\mathrm{l}\mathrm{i}\mathrm{m}it}$ then |

7. Return T |

8. endif |

9. endWhile |

10. path←GET_PATH (T) |

Algorithm 2: Function GET_PATH (T) |

1. Var path_set; $2.\mathrm{path}\_\mathrm{set}.\mathrm{Add}\_\mathrm{Node}(\mathrm{T}{.\mathrm{node}}_{\mathrm{n}}$); 3. while 4. i←$\mathrm{Pre}\_{\mathrm{Node}\_\mathrm{Index}}_{\mathrm{n}}$; $5.\mathrm{path}\_\mathrm{set}.\mathrm{Back}\_\mathrm{Add}\_\mathrm{Node}(\mathrm{T}{.\mathrm{node}}_{\mathrm{i}}$); 6. if i = 1 7. break; 8. endif 9. i←i + 1; 10. endwhile 11. path←path_set |

#### 3.2. Adaptive Improved RRT Algorithm

Algorithm 3:$\mathrm{Build}\mathrm{Adaptive}\mathrm{Improved}\_\mathrm{RRT}({P}_{init}$) |

$1.\mathrm{T}.\mathrm{init}({P}_{init}$) |

$2.\mathrm{While}\mathrm{DISTANCE}({P}_{new}$$,{P}_{goal}$$)>{D}_{\mathrm{l}\mathrm{i}\mathrm{m}it}$ do |

$3.{P}_{rand}$←EFFECTIVE_RANDOM_STATE ( ) |

$4.{P}_{near}$←EFFECTIVE_NEAREST_NEIGHBOR ( ) |

$5.{P}_{new}$←EFFECTIVE_EXTEND ( ) |

$6.\mathrm{if}\mathrm{COLLISION}\_\mathrm{DISTANCE}({P}_{new}$$,{P}_{goal}$$)\le {D}_{\mathrm{l}\mathrm{i}\mathrm{m}it}$ then |

7. Return T |

8. endif |

9. endWhile |

10. path←GET_PATH (T) |

11. path←PRUNING (path) |

12. trajectory←SMOOTHING (path) |

#### 3.2.1. Collision Detection

#### 3.2.2. Adaptive Directed Sampling Strategy

#### Adaptive Sampling Space

#### Dynamic Sampling

Algorithm 4: Adaptive Improved RRT EFFECTIVE_RANDOM_STATE ( ) |

1. Sample_zone←$\mathrm{ADAPTIVE}\_\mathrm{ZONE}({P}_{goal}$,T) |

$2.\rho $←RANDOM_VALUE (0,1) |

$3.\mathrm{if}\rho $$\ge {\rho}_{goal}$ |

$4.{P}_{rand}$←Double_Rand ( ) |

5. else |

$6.{P}_{rand}$←${P}_{goal}$; |

7. endif |

#### 3.2.3. Reasonable Node Selection Strategy

#### 3.2.4. Adaptive Node Extension Strategy

#### 3.2.5. Post-Processing Strategy

Algorithm 5: Adaptive Improved RRT POST_PROCESSING ( ) |

$1.\mathrm{Var}{\mathrm{Q}}_{0}$$,{Q}_{1}$$,{Q}_{2}$$,{Q}_{3}$$,{Q}_{4}$: path |

$2.{\mathrm{Q}}_{0}$$({\mathrm{q}}_{\mathrm{n}}$$\dots ,{\mathrm{q}}_{2}$$,{\mathrm{q}}_{1}$$,{\mathrm{q}}_{0}$)←GET_PATH (T) |

$3.{\mathrm{q}}_{\mathrm{root}}$←${\mathrm{q}}_{0}$$;{\mathrm{q}}_{\mathrm{temp}}$←${\mathrm{q}}_{1}$; |

$4.{\mathrm{Q}}_{1}$←$({\mathrm{q}}_{\mathrm{root}}$$,{\mathrm{q}}_{\mathrm{temp}}$) |

$5.{Q}_{2}$←$({\mathrm{q}}_{\mathrm{n}}$$\dots ,{\mathrm{q}}_{2}$) |

$6.\mathrm{While}{\mathrm{q}}_{\mathrm{temp}}$$!={\mathrm{q}}_{\mathrm{n}-1}$ do |

$7.{\mathrm{Q}}_{3}$←$({\mathrm{q}}_{\mathrm{temp}}$$,{\mathrm{q}}_{\mathrm{root}}$$);{Q}_{4}$←$({\mathrm{q}}_{\mathrm{root}}$$,{\mathrm{q}}_{\mathrm{temp}}$) |

$8.\mathrm{for}\mathrm{each}\mathrm{node}{\mathrm{q}}_{\mathrm{i}}$∈${\mathrm{Q}}_{2}$ |

$9.\mathrm{if}\mathrm{Collision}\_\mathrm{Free}({\mathrm{q}}_{\mathrm{temp}}$$,{\mathrm{q}}_{\mathrm{i}}$) |

$10.{\mathrm{Q}}_{3}$$.\mathrm{Forward}\_\mathrm{Add}\_\mathrm{Node}({\mathrm{q}}_{\mathrm{i}}$); |

11. else |

$12.{\mathrm{Q}}_{3}$$.\mathrm{Forward}\_\mathrm{Add}\_\mathrm{Node}({\mathrm{q}}_{\mathrm{i}}$); break |

13. end if |

14. end for |

$15.\mathrm{for}\mathrm{each}\mathrm{node}{\mathrm{q}}_{\mathrm{k}}$∈${\mathrm{Q}}_{3}$$\mathrm{and}{\mathrm{q}}_{\mathrm{k}-1}$∈${\mathrm{Q}}_{3}$ |

$16.{\mathrm{q}}_{\mathrm{dou}\_\mathrm{head}}$←${\mathrm{q}}_{\mathrm{k}}$$;{\mathrm{q}}_{\mathrm{head}}$←${\mathrm{q}}_{\mathrm{k}-1}$; |

$17.\mathrm{if}(\pi -\mathrm{Angle}(\overrightarrow{{\mathrm{q}}_{\mathrm{temp}}{\mathrm{q}}_{\mathrm{root}}},\overrightarrow{{\mathrm{q}}_{\mathrm{temp}}{\mathrm{q}}_{\mathrm{head}}}))\beta $ |

$\mathrm{and}(\pi -\mathrm{Angle}(\overrightarrow{{\mathrm{q}}_{\mathrm{head}}{\mathrm{q}}_{\mathrm{temp}}},\overrightarrow{{\mathrm{q}}_{\mathrm{head}}{\mathrm{q}}_{\mathrm{dou}\_\mathrm{head}}}\left)\right)\beta $ |

$18.{\mathrm{Q}}_{4}$$.\mathrm{Backward}\_\mathrm{Add}\_\mathrm{Node}({\mathrm{q}}_{\mathrm{k}-1}$); |

$19.{\mathrm{q}}_{\mathrm{root}}$←${\mathrm{q}}_{\mathrm{temp}}$$;{\mathrm{q}}_{\mathrm{temp}}$←${\mathrm{q}}_{\mathrm{head}}$; a←k; break |

20. end if |

21. end for |

$22.{\mathrm{Q}}_{1}$←${\mathrm{Q}}_{1}\cup {\mathrm{Q}}_{4}$ |

$23.{\mathrm{Q}}_{2}$←$({\mathrm{q}}_{\mathrm{n}}$$\dots ,{\mathrm{q}}_{\mathrm{a}}$) |

24. endWhile |

$25.{\mathrm{Q}}_{1}$$.\mathrm{Backward}\_\mathrm{Add}\_\mathrm{Node}({\mathrm{q}}_{\mathrm{n}}$) |

26. trajectory←$\mathrm{Cubic}\_\mathrm{Bspline}({\mathrm{Q}}_{1}$) |

#### Path Pruning

_{1}, P

_{2}, P

_{3}, P

_{4}, P

_{5}, and P

_{6}do not intersect with the expanded safe ellipse, these nodes can be connected directly by line segments to delete the redundant nodes between them and have no sharp angles between those line segments, thus obtaining a relatively gentle path sequence. In addition, the included angles of the connecting lines between these nodes are less than $\beta $; that is, α

_{1}, α

_{2}, α

_{3}, and α

_{4}are less than $\beta $.

#### Cubic B-Spline Smoothing Method

## 4. Simulation Experiments

#### 4.1. Performance Comparison of Several RRT Algorithms

#### 4.2. Path Following Validation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 11.**Path tracking results in the straight road environment. (

**a**) Planned path on straight road. (

**b**) Path following result on straight road. (

**c**) Path following error on straight road. (

**d**) Yaw velocity on straight road. (

**e**) Lateral acceleration on straight road.

**Figure 12.**Path tracking results in curved road environment. (

**a**) Planned path on curved road. (

**b**) Path following result on curved road. (

**c**) Path following error on curved road. (

**d**) Yaw velocity on curved road. (

**e**) Lateral acceleration on curved road.

Road Type | Road Length (m) | Single Lane Width (m) | Initial Point | Target Point | Obstacle Point |
---|---|---|---|---|---|

Straight | 120 | 3.75 | 0, −1.875 | 120, −1.875 | 60, −1.875 |

Curve | 200 | 3.75 | 20, −1.865 | 180, 4.160 | 100, −0.813 |

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

Obstacle vehicle width $\mathrm{W}$ (m) | 1.8 | Terminal step size δ (m) | 20 |

Obstacle vehicle length $\mathrm{L}$ (m) | 4.8 | Constraint angle $\phi $ (°) | 30 |

Expansion coefficient $s$ | $\sqrt{3}$ | Weighted coefficient ${\omega}_{1}$ | 0.5 |

Friction coefficient $\mu $ | 0.8 | Weighted coefficient ${m}_{1}$ | 0.5 |

Acceleration of gravity $\mathrm{g}$ (m/s^{2}) | 9.8 | Weighted coefficient ${m}_{2}$ | 0.5 |

Host vehicle speed $\mathrm{V}$ (km/h) | 60 | Weighted coefficient ${n}_{1}$ | 0.7 |

Biased probability ${\rho}_{goal}$ | 0.1 | Weighted coefficient ${n}_{2}$ | 0.3 |

Maximum step size ${L}_{max}$ (m) | 20 |

**Table 3.**Comparison of several RRT algorithms in the straight road environment: mean values in 30 runs.

Algorithm | Node | Length | Segment | Time |
---|---|---|---|---|

Basic | 37.47 | 124.435 | 15.7 | 0.026 |

Biased | 25.77 | 122.746 | 11.57 | 0.017 |

Bi | 14.90 | 122.105 | 10.03 | 0.016 |

Connect | 10.50 | 124.378 | 4.43 | 0.011 |

Adaptive-Improved | 22.50 | 120.290 | 5.23 | 0.024 |

**Table 4.**Comparison of several RRT algorithms in the curved road environment: mean values in 30 runs.

Algorithm | Node | Length | Segment | Time |
---|---|---|---|---|

Basic | 26.93 | 162.634 | 16.57 | 1.755 |

Biased | 21.13 | 161.884 | 14.10 | 1.411 |

Bi | 14.00 | 161.032 | 12.13 | 0.386 |

Connect | 4.87 | 162.243 | 3.30 | 0.299 |

Adaptive-Improved | 26.37 | 160.141 | 4.23 | 3.257 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Zhu, T.; Xu, Y.; Liu, H.; Liu, F.
Local Path Planning of the Autonomous Vehicle Based on Adaptive Improved RRT Algorithm in Certain Lane Environments. *Actuators* **2022**, *11*, 109.
https://doi.org/10.3390/act11040109

**AMA Style**

Zhang X, Zhu T, Xu Y, Liu H, Liu F.
Local Path Planning of the Autonomous Vehicle Based on Adaptive Improved RRT Algorithm in Certain Lane Environments. *Actuators*. 2022; 11(4):109.
https://doi.org/10.3390/act11040109

**Chicago/Turabian Style**

Zhang, Xiao, Tong Zhu, Yu Xu, Haoxue Liu, and Fei Liu.
2022. "Local Path Planning of the Autonomous Vehicle Based on Adaptive Improved RRT Algorithm in Certain Lane Environments" *Actuators* 11, no. 4: 109.
https://doi.org/10.3390/act11040109