# An Informed-Bi-Quick RRT* Algorithm Based on Offline Sampling: Motion Planning Considering Multiple Constraints for a Dual-Arm Cooperative System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Problem Description

#### 2.2. Constraint Tolerance and Task Error

#### 2.3. Construction of Offline Sampling Set

#### 2.4. Informed Bidirectional Quick-RRT* Algorithm

#### 2.4.1. Multi-Objective Bias Strategy

- Tree 1 and tree 2 randomly pick points in the whole map. The purpose is to enhance the ability to jump out of the obstacle area faster when facing more complex obstacles, reduce the number of invalid samples, and improve the sampling efficiency.
- Tree 1 takes the starting point of tree 2 as the biased target for sampling points, and tree 2 takes the starting point of tree 1 as the biased target for sampling points. The purpose is to ensure the overall direction of random sampling and ensure the efficiency of the algorithm.
- Tree 1 uses the previous sampling point of tree 2 as a biased target for sampling points, and tree 2 uses the previous sampling point of tree 1 as a biased target for sampling points. The purpose is to speed up the connection of double trees and shorten the path process when facing simple and uncomplicated obstacles.

Algorithm 1: InformedSampleFree Function. |

$1:Fori=1\mathrm{to}\mathrm{K}do$ |

$2:\hspace{1em}\mathrm{P}=\mathrm{Rand}\left(0,1\right)$ |

$3:\hspace{1em}if0\mathrm{P}\le 0.5$ |

$4:\hspace{1em}\hspace{1em}{q}_{\mathrm{rand}1}=\mathrm{SampleFree}({q}_{\mathrm{rand}1});{q}_{\mathrm{rand}2}=\mathrm{SampleFree}({q}_{\mathrm{rand}2});$ |

$6:\hspace{1em}elseif0.5\mathrm{P}\le 0.75$ |

$7:\hspace{1em}\hspace{1em}{q}_{\mathrm{rand}1}={q}_{\mathrm{goal}};{q}_{\mathrm{rand}2}={q}_{\mathrm{start}};$ |

$8:\hspace{1em}else0.5\mathrm{P}\le 0.75$ |

$9:\hspace{1em}\hspace{1em}{q}_{\mathrm{rand}1}={q}_{\mathrm{rand}2};{q}_{\mathrm{rand}2}={q}_{\mathrm{rand}1};$ |

$10:\hspace{1em}endif$ |

$11:endfor$ |

$12:return{q}_{\mathrm{rand}1},{q}_{\mathrm{rand}2}$ |

#### 2.4.2. Informed Bidirectional Quick-RRT* algorithm

Algorithm 2: Informed-Bi-Quick-RRT*. |

$1:{\mathrm{T}}_{1}\leftarrow {\mathrm{T}}_{\mathrm{init}}({q}_{\mathrm{start}}{),\mathrm{T}}_{2}\leftarrow {\mathrm{T}}_{\mathrm{init}}\left({q}_{\mathrm{goal}}\right)$ |

$2:fori=1toKdo$ |

$3:{q}_{\mathrm{rand}1},{q}_{\mathrm{rand}2}\leftarrow \mathrm{InformedSampleFree}(i);$ |

$4:{q}_{\mathrm{nearest}1}\leftarrow \mathrm{Nearest}({q}_{\mathrm{rand}1}{,\mathrm{T}}_{1});{q}_{\mathrm{nearest}2}\leftarrow \mathrm{Nearest}({q}_{\mathrm{rand}2}{,\mathrm{T}}_{2});$ |

$5:{q}_{\mathrm{new}1}\leftarrow \mathrm{Steer}({q}_{\mathrm{nearest}1},{q}_{\mathrm{rand}1},\delta );{q}_{\mathrm{new}}\leftarrow \mathrm{Steer}({q}_{\mathrm{nearest}2},{q}_{\mathrm{rand}2},\delta );$ |

$6:if\mathrm{CollisionDetection}({q}_{\mathrm{nearest}1,2},{q}_{\mathrm{new}1,2},\mathrm{map})$ |

$7:\hspace{1em}{\mathrm{Q}}_{\mathrm{near}1}\leftarrow {\mathrm{Near}(\mathrm{T}}_{1},{q}_{\mathrm{new}1},{r}_{\mathrm{near}1});{\mathrm{Q}}_{\mathrm{near}2}\leftarrow {\mathrm{Near}(\mathrm{T}}_{2},{q}_{\mathrm{new}2},{r}_{\mathrm{near}2});$ |

$8:\hspace{1em}{\mathrm{Q}}_{\mathrm{P}1-\mathrm{near}}\leftarrow {\mathrm{Ancestry}(\mathrm{T}}_{1},{\mathrm{Q}}_{\mathrm{near}1},{d}_{\mathrm{near}1}{);\mathrm{Q}}_{\mathrm{P}2-\mathrm{near}}\leftarrow {\mathrm{Ancestry}(\mathrm{T}}_{2},{\mathrm{Q}}_{\mathrm{near}2},{d}_{\mathrm{near}2});$ |

$9:\hspace{1em}{q}_{\mathrm{parent}1}\leftarrow {\mathrm{ChooseParent}(\mathrm{Q}}_{\mathrm{near}1}\cup {\mathrm{Q}}_{\mathrm{P}1-\mathrm{near}},{q}_{\mathrm{new}1});$ $\hspace{1em}\hspace{1em}{q}_{\mathrm{parent}2}\leftarrow {\mathrm{ChooseParent}(\mathrm{Q}}_{\mathrm{near}2}\cup {\mathrm{Q}}_{\mathrm{P}2-\mathrm{near}},{q}_{\mathrm{new}2});$ |

$10:\hspace{1em}{\mathrm{T}}_{1}\leftarrow {\mathrm{Link}(q}_{\mathrm{parent}},{q}_{\mathrm{new}}{);\mathrm{T}}_{2}\leftarrow {\mathrm{Link}(q}_{\mathrm{parent}2},{q}_{\mathrm{new}2});$ |

$11:\hspace{1em}{\mathrm{Q}}_{\mathrm{P}1-\mathrm{near}}\leftarrow {\mathrm{Ancestry}(\mathrm{T}}_{1},{q}_{\mathrm{new}1},{d}_{\mathrm{new}1}{);\mathrm{Q}}_{\mathrm{P}2-\mathrm{near}}\leftarrow {\mathrm{Ancestry}(\mathrm{T}}_{2},{q}_{\mathrm{new}2},{d}_{\mathrm{new}2});$ |

$12:\hspace{1em}{\mathrm{T}}_{1}\leftarrow {\mathrm{Rewire}(\mathrm{Q}}_{\mathrm{near}1},{q}_{\mathrm{new}1}\cup {\mathrm{Q}}_{\mathrm{P}1-\mathrm{near}}{);\mathrm{T}}_{2}\leftarrow {\mathrm{Rewire}(\mathrm{Q}}_{\mathrm{near}2},{q}_{\mathrm{new}2}\cup {\mathrm{Q}}_{\mathrm{P}2-\mathrm{near}});$ |

$13:\hspace{1em}{\mathrm{T}}_{1}\leftarrow {\mathrm{Connnect}(\mathrm{T}}_{2},{q}_{\mathrm{new}1}{);\mathrm{T}}_{2}\leftarrow {\mathrm{Connnect}(\mathrm{T}}_{1},{q}_{\mathrm{new}2});$ |

$14:\hspace{1em}\mathrm{if}{\mathrm{IsConnected}(\mathrm{T}}_{1}{,\mathrm{T}}_{2})$ |

$15:\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{Path}=\mathrm{FillPath}(\mathrm{T}}_{1}{,\mathrm{T}}_{2});$ |

$16:\hspace{1em}endif$ |

$17:\mathrm{Swap}({\mathrm{T}}_{1}{,\mathrm{T}}_{2});$ |

$18:endif$ |

$19:endfor$ |

$20:returnPath$ |

- 1.
- The InformedSampleFree($i$) function is used to generate random sampling points for both trees. Algorithm 2 presents the pseudocode for the InformedSampleFree($i$) function. In the exploration process, the two trees are consistent in steps 2 through 4, so the related steps use the same expression.
- 2.
- Two new nodes are used: ${q}_{\mathit{nearest}}$ and ${q}_{\mathit{new}}$. The Nearest (${q}_{\mathit{rand}}$, ${T}_{1}$) function should be used to produce ${q}_{\mathit{nearest}}$, and the Steer (${q}_{\mathit{nearest}}$, ${q}_{\mathit{rand}}$, δ) function should be used to create ${q}_{\mathit{new}}$.
- 3.
- Under the premise that there is no collision between ${q}_{\mathit{nearest}}$ and ${q}_{\mathit{new}}$, the path from ${q}_{\mathrm{start}}$ to ${q}_{\mathrm{goal}}$ is optimized. The Near (${T}_{1}$, ${q}_{\mathit{new}}$, ${r}_{\mathit{near}}$) and Ancestry (${T}_{1}$, ${Q}_{\mathit{near}}$, ${d}_{\mathit{near}}$) functions are used to identify the potential parent node ${Q}_{\mathit{\text{P-near}}}$ of ${q}_{\mathit{new}}$. Then, the ChooseParent (‘${Q}_{\mathit{near}}$ ∪ ${Q}_{\mathit{\text{P-near}}}$, ${q}_{\mathit{new}}$) function is used to select the parent node ${q}_{\mathit{parent}}$ in the potential parent node, so that the path distance from ${q}_{\mathrm{start}}$ to ${q}_{\mathit{new}}$ is the smallest; finally, the Link (${q}_{\mathit{parent}}$, ${q}_{\mathit{new}}$) function is used to form ${T}_{1}$.
- 4.
- The path from ${q}_{\mathrm{start}}$ to the midpoint of ${Q}_{\mathit{near}}$ is optimized. The Ancestry (${T}_{1}$, ${q}_{\mathit{new}}$, ${d}_{\mathit{new}}$) function is used to find the parent node ${Q}_{\mathit{\text{P-near}}}$ of ${q}_{\mathit{new}}$; the ${q}_{\mathit{new}}$ and ${q}_{\mathit{parent}}$ are used as the potential parent nodes of the midpoint of ${Q}_{\mathit{\text{P-near}}}$, and the Rewire (${Q}_{\mathit{near}}$, ${q}_{\mathit{new}}$ ∪ ${Q}_{\mathit{\text{P-near}}}$) function is used to find the parent node at the potential parent node, so that the ${T}_{1}$ path distance is minimized.
- 5.
- The Connect (${T}_{2}$, ${q}_{\mathit{new}1}$) function is used to connect ${T}_{2}$ and ${q}_{\mathit{new}1}$. Firstly, the nodes closest to ${q}_{\mathit{new}1}$ in ${T}_{2}$: ${q}_{\mathit{newest}1}^{\prime}$ are determined, and then ${q}_{\mathit{newest}}^{\prime}$ continuously advances δ with ${q}_{\mathit{new}1}$ as the goal until they encounter obstacles or link to ${q}_{\mathit{new}1}$.
- 6.
- The FillPath (${T}_{1}$, ${T}_{2}$) function is used to stitch paths. In the case of a link between ${T}_{2}$ and ${q}_{\mathit{new}1}$, ${T}_{1}$ traces the parent node from ${q}_{\mathit{new}1}$ to ${q}_{\mathit{start}1}$, thus forming the final ${T}_{1}$. ${T}_{1}$ begins to trace the parent node to ${q}_{\mathrm{goal}}$ from the node connected to ${q}_{\mathit{new}1}$, thus forming the final ${T}_{2}$.
- 7.
- The Swap (${T}_{1}$, ${T}_{2}$) function is used to exchange the contents of two trees, so that the number of nodes of the two trees remains balanced after several iterations.

## 3. Different Task Constraints in Dual-Arm Cooperative Systems

#### 3.1. Collaborative Movement with Loose Constraints

#### 3.2. Collaborative Movement with Tightly Constraints

## 4. Experiment

#### 4.1. Algorithm Comparison

#### 4.1.1. Simple Environment

#### 4.1.2. Complex Environment

#### 4.2. Three-Dimensional Simulation Considering Horizontal Constraints

#### 4.2.1. Loose Collaborative Motion: Assembly Task

#### 4.2.2. Tightly Collaborative Motion: Handling Task

#### 4.3. Three-Dimensional Simulation Considering Multiple Constraints

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Algorithm | Average Path Length (mm) | Average Running Time (s) | Average Iterations |
---|---|---|---|

RRT* | 245.90 | 11.09 | 877 |

Quick-RRT* | 244.74 | 3.97 | 624 |

Bi-Quick-RRT* | 240.35 | 2.85 | 605 |

Informed-Bi-Quick-RRT* | 229.25 | 0.17 | 480 |

Algorithm | Average Path Length (mm) | Average Running Time (s) | Average Iterations |
---|---|---|---|

RRT* | 234.157 | 37.34 | 986 |

Quick-RRT* | 225.80 | 10.31 | 733 |

Bi-Quick-RRT* | 222.47 | 8.70 | 715 |

Informed-Bi-Quick-RRT* | 214.41 | 5.52 | 590 |

$\mathbf{Joint}/\mathit{i}$ | ${\mathit{\alpha}}_{\mathit{i}}$/(°) | ${\mathit{a}}_{\mathit{i}}$$/\mathbf{m}\mathbf{m}$ | ${\mathit{d}}_{\mathit{i}}$$/\mathbf{m}\mathbf{m}$ | ${\mathit{\theta}}_{\mathit{i}}$/(°) |
---|---|---|---|---|

1 | 90 | 0 | 89.2 | ${\theta}_{1}$ |

2 | 0 | −425 | 0 | ${\theta}_{2}$ |

3 | 0 | −392 | 0 | ${\theta}_{3}$ |

4 | −90 | 0 | 109.3 | ${\theta}_{4}$ |

5 | −90 | 0 | 94.75 | ${\theta}_{5}$ |

6 | 0 | 0 | 82.5 | ${\theta}_{6}$ |

$\mathbf{Joint}/\mathit{i}$ | ${\mathit{\alpha}}_{\mathit{i}}$/(°) | ${\mathit{a}}_{\mathit{i}}$$/\mathbf{m}\mathbf{m}$ | ${\mathit{d}}_{\mathit{i}}/\mathbf{m}\mathbf{m}$ | ${\mathit{\theta}}_{\mathit{i}}$/(°) |
---|---|---|---|---|

1 | 90 | 0 | 128 | ${\theta}_{1}$ |

2 | 0 | −612.7 | 0 | ${\theta}_{2}$ |

3 | 0 | −571.6 | 0 | ${\theta}_{3}$ |

4 | 90 | 0 | 163.9 | ${\theta}_{4}$ |

5 | −90 | 0 | 115.7 | ${\theta}_{5}$ |

6 | 0 | 0 | 92.2 | ${\theta}_{6}$ |

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**MDPI and ACS Style**

Zhang, Q.; Liu, Y.; Qin, J.; Duan, J.
An Informed-Bi-Quick RRT* Algorithm Based on Offline Sampling: Motion Planning Considering Multiple Constraints for a Dual-Arm Cooperative System. *Actuators* **2024**, *13*, 75.
https://doi.org/10.3390/act13020075

**AMA Style**

Zhang Q, Liu Y, Qin J, Duan J.
An Informed-Bi-Quick RRT* Algorithm Based on Offline Sampling: Motion Planning Considering Multiple Constraints for a Dual-Arm Cooperative System. *Actuators*. 2024; 13(2):75.
https://doi.org/10.3390/act13020075

**Chicago/Turabian Style**

Zhang, Qinglei, Yunfeng Liu, Jiyun Qin, and Jianguo Duan.
2024. "An Informed-Bi-Quick RRT* Algorithm Based on Offline Sampling: Motion Planning Considering Multiple Constraints for a Dual-Arm Cooperative System" *Actuators* 13, no. 2: 75.
https://doi.org/10.3390/act13020075