# Trajectory Planner CDT-RRT* for Car-Like Mobile Robots toward Narrow and Cluttered Environments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Kinematic Model and Dual-Tree RRT

#### 2.1. Kinematic Model of a Car-Like Robot

#### 2.2. Dual-Tree RRT

## 3. Car-like DT-RRT* (CDT-RRT*)

#### 3.1. Motion Controller for Trajectory Generations

#### 3.2. Dual-Tree Structure of CDT-RRT*

Algorithm 1 BuildTree*$({q}_{0},{x}_{0},goal)$ |

$\mathbf{TW}.\mathrm{INIT}({q}_{0},goal);$ $\mathbf{TX}.\mathrm{INIT}({x}_{0},goal);$ while ${t}_{limit}$ do${q}_{rand}\leftarrow $ RandomSample(); $[{Q}_{new},{Q}_{near},type]\leftarrow $ ExtWorkSpace$(\mathbf{TW},{q}_{rand});$ ${\mathbb{Q}}_{nearlist}\leftarrow $ NearNodeList$({Q}_{new},{B}_{r})$ $[{X}_{p},{X}_{min}]\leftarrow $ FindParentState*$({\mathbb{Q}}_{nearlist},{q}_{new},type);$ if ${X}_{p}=nil$ then$\mathbf{continue}$; end if$\mathbf{TW}.\mathrm{Add}({Q}_{near},{Q}_{new});$ $\mathbf{TX}.\mathrm{Add}({X}_{p},{X}_{min});$ ReConnectTrees$({Q}_{new},{X}_{min});$ end while$\mathbf{return};$ |

Algorithm 2 FindingParentState*$\{{\mathbb{Q}}_{nearlist},{q}_{new},type\}$ |

${X}_{p},{X}_{min}\leftarrow nil;$ $cos{t}_{min}\leftarrow inf;$ for all ${Q}_{near}$ ∈ ${\mathbb{Q}}_{nearlist}$ do${X}_{cur}\leftarrow $ GetStateNode$({Q}_{near});$ $[{X}_{new},{e}_{r},cost]\leftarrow $ ExtendState$({\mathrm{q}}_{\mathrm{new}},{\mathrm{X}}_{\mathrm{cur}},\mathrm{type})$ if ${e}_{r}\le {B}_{r}\phantom{\rule{3.33333pt}{0ex}}\mathbf{and}\phantom{\rule{3.33333pt}{0ex}}cost\le cos{t}_{min}$ then$[{X}_{p},{X}_{min},cos{t}_{min}]\leftarrow [{X}_{cur},{X}_{new},cost];$ end ifend for$\mathbf{return}\phantom{\rule{1.33333pt}{0ex}}{X}_{p},{X}_{min};$ |

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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RRT [10] | CL-RRT [25] | High Speed RRT* [23] | DT-RRT [29] | CDT-RRT* | |
---|---|---|---|---|---|

Car-like model | None | Good | Good | None | Good |

Piecewise motion | None | None | None | Good | Good |

Goal-biased sampling | Good | Fair | Fair | Good | Good |

Cluttered Environments | Good | Fair | Fair | Fair | Good |

Replanning | None | Fair | Fair | Good | Good |

Saving orphan nodes | None | None | None | Good | Good |

Parameters | Values | Specifications | |
---|---|---|---|

${v}_{max}$ | 2.7 m/s | CPU | Intel i7-2620M 2.7 Ghz |

${a}_{max}$ | 1.8 m/s${}^{2}$ | RAM | 4 GB (1333 MHz) |

$\mathbf{L}$ | 2.8 m | OS | Ubuntu 11.04 |

$[{\delta}_{min},{\delta}_{max}]$ | $[-{30}^{\circ},{30}^{\circ}]$ | Middleware | Robot Operating System |

${\dot{\delta}}_{max}$ | ${20}^{\circ}/$s${}^{2}$ | Language | C++ |

Mean (s) | $\mathit{\sigma}$ | Min (s) | Max (s) | Failure Count | ||
---|---|---|---|---|---|---|

RRT | Computing time | 8.62 | 7.93 | 0.02 | 27.59 | 66 |

Travel time | 35.01 | 7.15 | 19.20 | 49.80 | ||

DT-RRT | Computing time | 0.36 | 0.20 | 0.06 | 1.04 | 0 |

Travel time | 18.67 | 0.62 | 17.30 | 19.30 | ||

CDT-RRT* | Computing time | 0.36 | 0.52 | 0.07 | 5.00 | 0 |

Travel time | 18.87 | 0.82 | 17.30 | 21.70 |

Mean (s) | $\mathit{\sigma}$ | Min (s) | Max (s) | Failure Count | ||
---|---|---|---|---|---|---|

RRT | Computing time | 29.89 | 1.11 | 18.90 | 30.00 | 99 |

Travel time | 85.50 | 0.00 | 85.50 | 85.50 | ||

T-RRT | Computing time | 12.08 | 12.19 | 0.60 | 30.00 | 26 |

Travel time | 58.79 | 2.22 | 55.70 | 73.30 | ||

CDT-RRT* | Computing time | 2.79 | 1.47 | 0.70 | 8.60 | 0 |

Travel time | 57.91 | 1.07 | 55.70 | 61.30 |

Mean (s) | $\mathit{\sigma}$ | Min (s) | Max (s) | Failure Count | ||
---|---|---|---|---|---|---|

RRT | Computing time | 29.81 | 1.93 | 10.71 | 30.00 | 99 |

Travel time | 62.70 | 0.00 | 62.70 | 62.70 | ||

DT-RRT | Computing time | 10.20 | 12.30 | 0.20 | 30.00 | 23 |

Travel time | 40.56 | 1.91 | 35.70 | 45.20 | ||

CDT-RRT* | Computing time | 6.78 | 7.03 | 0.32 | 30.00 | 3 |

Travel time | 40.71 | 4.71 | 35.20 | 61.30 |

Mean (s) | $\mathit{\sigma}$ | Min (s) | Max (s) | Failure Count | ||
---|---|---|---|---|---|---|

RRT | Computing time | 30.00 | - | 30.00 | 30.00 | 100 |

Travel time | - | - | - | - | ||

DT-RRT | Computing time | 14.50 | 12.39 | 0.98 | 30.00 | 36 |

Travel time | 72.83 | 11.60 | 58.30 | 102.70 | ||

CDT-RRT* | Computing time | 6.46 | 3.44 | 1.60 | 17.34 | 0 |

Travel time | 66.54 | 8.52 | 57.60 | 86.00 |

Mean (s) | $\mathit{\sigma}$ | Min (s) | Max (s) | Failure Count | ||
---|---|---|---|---|---|---|

RRT | Computing time | 30.00 | - | 30.00 | 30.00 | 100 |

Travel time | - | - | - | - | ||

DT-RRT | Computing time | 7.61 | 10.03 | 0.90 | 30.00 | 14 |

Travel time | 62.40 | 3.37 | 57.50 | 74.60 | ||

CDT-RRT* | Computing time | 5.25 | 2.06 | 2.20 | 11.10 | 0 |

Travel time | 61.33 | 4.22 | 56.00 | 75.70 |

Mean (s) | $\mathit{\sigma}$ | Min (s) | Max (s) | Failure Count | ||
---|---|---|---|---|---|---|

RRT | Computing time | 30.00 | - | 30.00 | 30.00 | 100 |

Travel time | - | - | - | - | ||

DT-RRT | Computing time | 3.53 | 3.78 | 1.23 | 30.00 | 1 |

Travel time | 47.10 | 5.19 | 42.60 | 66.90 | ||

CDT-RRT* | Computing time | 1.74 | 0.69 | 0.60 | 4.45 | 0 |

Travel time | 46.43 | 4.29 | 42.60 | 61.80 |

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**MDPI and ACS Style**

Kwon, H.; Cha, D.; Seong, J.; Lee, J.; Chung, W.
Trajectory Planner CDT-RRT* for Car-Like Mobile Robots toward Narrow and Cluttered Environments. *Sensors* **2021**, *21*, 4828.
https://doi.org/10.3390/s21144828

**AMA Style**

Kwon H, Cha D, Seong J, Lee J, Chung W.
Trajectory Planner CDT-RRT* for Car-Like Mobile Robots toward Narrow and Cluttered Environments. *Sensors*. 2021; 21(14):4828.
https://doi.org/10.3390/s21144828

**Chicago/Turabian Style**

Kwon, Hyunki, Donggeun Cha, Jihoon Seong, Jinwon Lee, and Woojin Chung.
2021. "Trajectory Planner CDT-RRT* for Car-Like Mobile Robots toward Narrow and Cluttered Environments" *Sensors* 21, no. 14: 4828.
https://doi.org/10.3390/s21144828