# Real-Time Collision-Free Navigation of Multiple UAVs Based on Bounding Boxes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art

## 3. Bounding Box Collision Avoidance

#### 3.1. Conflict Detection Mechanism

#### 3.2. Computation of the Valid Velocity Space

#### 3.3. Selection of the Final Velocity

#### 3.4. Robustness and Safety Margins

## 4. Simulation Results

#### 4.1. Two-UAV Study

**w**). On the other hand, a UAV ${a}_{2}$ has 18 different starting points (${p}_{1}\dots {p}_{18}$) ranging from 0° to 170° with a 10° interval, associated to the respective target points (${w}_{1}\dots {w}_{18}$). Both nodes start moving simultaneously with a constant velocity of 13.9 m/s (50 km/h), so they will collide at the center of the circumference if the navigation algorithm does not prevent it. A safety radius of 50 m is applied to both nodes, which will cause a pre-crash conflict, at a time that depends on the relative starting angle.

#### 4.2. Multi-UAV Study

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Example scenario. Circles represent the position of 6 UAVs. The green box represents the initial valid velocity space for node 1.

**Figure 7.**Direct velocity to the destination. The black radius (southeast bound) corresponds to the current velocity. The blue radius (south bound) is the direct velocity, which in this case becomes final since it is found inside the green box.

**Figure 8.**Twelve candidate points: eight ${v}_{\left\{\mathrm{N}\mathrm{S}\mathrm{E}\mathrm{W}\right\}\left\{\pm \right\}}$ cutting points (left) and four ${\mathrm{v}}_{\left\{\mathrm{NE}\mathrm{SE}\mathrm{SW}\mathrm{NW}\right\}}$ corner points (right).

**Figure 9.**Choosing the best velocity among the candidate points. In both cases, the red circle delimits the maximum velocity; the green box defines the set of valid velocities; the black radius indicates the current velocity; the blue radius indicates the direct velocity to the destination; the green radius indicates the velocity finally selected for the next time slot.

**Figure 14.**Random scenarios with 30 (left) and 50 (right) UAVs applying BBCA. Numbers in the plots represent initial UAV positions; circles represent current UAV positions and the established safety radius; each line represents a UAV trajectory (from the initial to the final position).

**Figure 15.**Multi-UAV study: conflicts produced; average and standard deviation values (left), and relative decrement of BBCA related to “direct” (right).

**Figure 16.**Multi-UAV study: distance covered; average and standard deviation values (left) and relative increment of BBCA related to “direct” (right).

**Figure 17.**Multi-UAV study: flying time; average and standard deviation values (left) and relative increment (right).

$\mathit{c}=\mathbf{C}\mathbf{o}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{i}\mathbf{c}\mathbf{t}\left({\mathbf{a}}_{1},{\mathbf{a}}_{2}\right)$ | |
---|---|

01 | $\left[{p}_{1}{r}_{1}~~\right]={a}_{1}$ |

02 | $\left[{p}_{2}{r}_{2}~~\right]={a}_{2}$ |

03 | if $\left|\overline{{p}_{1}{p}_{2}}\right|<{r}_{1}+{r}_{2}$ then |

04 | $c=true$ |

05 | else |

06 | $c=false$ |

07 | end if |

$\mathbf{v}=\mathbf{B}\mathbf{B}\mathbf{C}\mathbf{A}\left({\mathbf{a}}_{1},{\mathit{v}}_{\mathbf{m}\mathbf{a}\mathbf{x}},\mathbf{\tau}\right)$ | |
---|---|

01 | $\left[{p}_{1}{r}_{1}{v}_{1}{w}_{1}\right]={a}_{1}$ |

02 | $\left[{G}_{\mathrm{N}}{G}_{\mathrm{S}}{G}_{\mathrm{E}}{G}_{\mathrm{W}}\right]=\left[{v}_{\mathrm{max}}-{v}_{\mathrm{max}}{v}_{\mathrm{max}}-{v}_{\mathrm{max}}\right]$ |

03 | for all ${a}_{2}\in \mathbb{A},{a}_{1}\ne {a}_{2}$ do |

04 | $\left[{p}_{2}{r}_{2}{v}_{2}~\right]={a}_{2}$ |

05 | $O=\frac{\left[\begin{array}{cc}{p}_{2}-{p}_{1}& {r}_{1}+{r}_{2}\end{array}\right]}{\tau}$ |

06 | $R=\mathrm{QPlane}\left(O\right)$ |

07 | $R+={v}_{2}\left[\begin{array}{c}0011\\ 1100\end{array}\right]$ |

08 | $L=\left({v}_{1}\left[\begin{array}{c}0011\\ 1100\end{array}\right]-R\right)\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]$ |

09 | $R=\mathrm{HPlane}\left(L,R\right)$ |

10 | $R=\frac{R+{v}_{1}\left[\begin{array}{c}0011\\ 1100\end{array}\right]}{2}$ |

11 | $G=\mathrm{Truncate}\left(L,G,R\right)$ |

12 | end do |

13 | $v=\mathrm{VSelect}\left(G\right)$ |

$\mathbf{R}=\mathbf{Q}\mathbf{P}\mathbf{l}\mathbf{a}\mathbf{n}\mathbf{e}\left(\mathbf{O}\right)$ | |
---|---|

14 | $\left[{O}_{x}{O}_{y}{O}_{r}\right]=O$ |

15 | $\left[{R}_{\mathrm{N}}{R}_{\mathrm{S}}{R}_{\mathrm{E}}{R}_{\mathrm{W}}\right]=\left[{O}_{x}{O}_{y}\right]\left[\begin{array}{c}0011\\ 1100\end{array}\right]+{O}_{r}\left[1-11-1\right]$ |

16 | if ${O}_{y}<0$ then |

17 | ${R}_{\mathrm{S}}=-\infty $ |

18 | else |

19 | ${R}_{\mathrm{N}}=+\infty $ |

20 | end if |

21 | if ${O}_{x}<0$ then |

22 | ${R}_{\mathrm{W}}=-\infty $ |

23 | else |

24 | ${R}_{\mathrm{E}}=+\infty $ |

25 | end if |

26 | $R=\left[{R}_{\mathrm{N}}{R}_{\mathrm{S}}{R}_{\mathrm{E}}{R}_{\mathrm{W}}\right]$ |

${\mathbf{R}}^{\prime}=\mathbf{H}\mathbf{P}\mathbf{l}\mathbf{a}\mathbf{n}\mathbf{e}\left(\mathbf{L},\mathbf{R}\right)$ | |
---|---|

27 | $\left[{L}_{\mathrm{N}}{L}_{\mathrm{S}}{L}_{\mathrm{E}}{L}_{\mathrm{W}}\right]=L$ |

28 | case $\mathrm{max}\left(L\right)$ of |

29 | ${L}_{\mathrm{N}}$: |

30 | ${R}^{\prime}=R+\left[\begin{array}{cccc}0& -\infty & +\infty & -\infty \end{array}\right]$ |

31 | ${L}_{\mathrm{S}}$: |

32 | ${R}^{\prime}=R+\left[\begin{array}{cccc}+\infty & 0& +\infty & -\infty \end{array}\right]$ |

33 | ${L}_{\mathrm{E}}$: |

34 | ${R}^{\prime}=R+\left[\begin{array}{cccc}+\infty & -\infty & 0& -\infty \end{array}\right]$ |

35 | ${L}_{\mathrm{W}}$: |

36 | ${R}^{\prime}=R+\left[\begin{array}{cccc}+\infty & -\infty & +\infty & 0\end{array}\right]$ |

37 | end case |

${\mathbf{G}}^{\prime}=\mathbf{T}\mathbf{r}\mathbf{u}\mathbf{n}\mathbf{c}\mathbf{a}\mathbf{t}\mathbf{e}\left(\mathbf{L},\mathbf{G},\mathbf{R}\right)$ | |
---|---|

38 | $\left[{L}_{\mathrm{N}}{L}_{\mathrm{S}}{L}_{\mathrm{E}}{L}_{\mathrm{W}}\right]=L$ |

39 | $\left[{G}_{\mathrm{N}}{G}_{\mathrm{S}}{G}_{\mathrm{E}}{G}_{\mathrm{W}}\right]=G$ |

40 | $\left[{R}_{\mathrm{N}}{R}_{\mathrm{S}}{R}_{\mathrm{E}}{R}_{\mathrm{W}}\right]=R$ |

41 | case $\mathrm{max}\left(L\right)$ of |

42 | ${L}_{\mathrm{N}}$: |

43 | ${G}^{\prime}=\left[{G}_{\mathrm{N}},\mathrm{max}\left({G}_{\mathrm{S}},{R}_{\mathrm{N}}\right),{G}_{\mathrm{E}},{G}_{\mathrm{W}}\right]$ |

44 | ${L}_{\mathrm{S}}$: |

45 | ${G}^{\prime}=\left[\mathrm{min}\left({G}_{\mathrm{N}},{R}_{\mathrm{S}}\right),{G}_{\mathrm{S}},{G}_{\mathrm{E}},{G}_{\mathrm{W}}\right]$ |

46 | ${L}_{\mathrm{E}}$: |

47 | ${G}^{\prime}=\left[{G}_{\mathrm{N}},{G}_{\mathrm{S}},{G}_{\mathrm{E}},\mathrm{max}\left({G}_{\mathrm{W}},{R}_{\mathrm{E}}\right)\right]$ |

48 | ${L}_{\mathrm{W}}$: |

49 | ${G}^{\prime}=\left[{G}_{\mathrm{N}},{G}_{\mathrm{S}},\mathrm{min}\left({G}_{\mathrm{E}},{R}_{\mathrm{W}}\right),{G}_{\mathrm{W}}\right]$ |

50 | end case |

$\mathbf{v}=\mathbf{V}\mathbf{S}\mathbf{e}\mathbf{l}\mathbf{e}\mathbf{c}\mathbf{t}\left(\mathbf{p},\mathbf{w},\mathbf{G},\mathbf{R},{\mathit{v}}_{\mathbf{m}\mathbf{a}\mathbf{x}},\mathbf{\tau}\right)$ | |
---|---|

51 | $v=0$ |

52 | if $p=w$ then |

53 | return |

54 | end if |

55 | $\left[{G}_{\mathrm{N}}{G}_{\mathrm{S}}{G}_{\mathrm{E}}{G}_{\mathrm{W}}\right]=G$ |

56 | if ${G}_{\mathrm{N}}<{G}_{\mathrm{S}}$ or ${G}_{\mathrm{E}}{G}_{\mathrm{W}}$ then |

57 | $v=\left[\begin{array}{cc}\frac{{G}_{\mathrm{W}}+{G}_{\mathrm{E}}}{2}& \frac{{G}_{\mathrm{S}}+{G}_{\mathrm{N}}}{2}\end{array}\right]$ |

58 | return |

59 | end if |

60 | $m=w-p$ |

61 | ${v}_{\mathrm{d}}=\frac{m}{\left|m\right|}\cdot \mathrm{min}\left(\frac{\left|m\right|}{\mathsf{\tau}},{v}_{\mathrm{max}}\right)$ |

62 | if $\mathrm{VInside}\left({v}_{\mathrm{d}},G\right)$ then |

63 | $v={v}_{\mathrm{d}}$ |

64 | return |

65 | end if |

66 | ${v}_{\left\{\mathrm{N}\mathrm{S}\right\}\left\{\pm \right\}}=\left[\pm \sqrt{{v}_{\mathrm{max}}^{2}+{G}_{\left\{\mathrm{N}\mathrm{S}\right\}}^{2}},{G}_{\left\{\mathrm{N}\mathrm{S}\right\}}\right]$ |

67 | ${v}_{\left\{\mathrm{E}\mathrm{W}\right\}\left\{\pm \right\}}=\left[{G}_{\left\{\mathrm{E}\mathrm{W}\right\}},\pm \sqrt{{v}_{\mathrm{max}}^{2}+{G}_{\left\{\mathrm{E}\mathrm{W}\right\}}^{2}}\right]$ |

68 | if not $\mathrm{VInside}\left({v}_{\left\{\mathrm{N}\mathrm{S}\mathrm{E}\mathrm{W}\right\}\left\{\pm \right\}},G\right)$ then |

69 | ${v}_{\left\{\mathrm{N}\mathrm{S}\mathrm{E}\mathrm{W}\right\}\left\{\pm \right\}}=0$ |

70 | end if |

71 | ${v}_{\left\{\mathrm{NE}\mathrm{SE}\mathrm{SW}\mathrm{NW}\right\}}=\left[\begin{array}{cc}{G}_{\left\{\mathrm{E}\mathrm{E}\mathrm{W}\mathrm{W}\right\}}& {G}_{\left\{\mathrm{N}\mathrm{S}\mathrm{S}\mathrm{N}\right\}}\end{array}\right]$ |

72 | if $\left|{v}_{\left\{\mathrm{NE}\mathrm{SE}\mathrm{SW}\mathrm{NW}\right\}}\right|{v}_{\mathrm{max}}$ then |

73 | ${v}_{\left\{\mathrm{NE}\mathrm{SE}\mathrm{SW}\mathrm{NW}\right\}}=0$ |

74 | end if |

75 | for all $\mathrm{L}\in \left\{\mathrm{N}\mathrm{S}\mathrm{E}\mathrm{W}\right\}\left\{\pm \right\}\cup \left\{\mathrm{NE}\mathrm{SE}\mathrm{SW}\mathrm{NW}\right\}$ do |

76 | ${\alpha}_{\mathrm{L}}=\mathrm{acos}\left(\frac{{v}_{\mathrm{d}}\cdot {v}_{\mathrm{L}}}{\left|{v}_{\mathrm{d}}\right|\left|{v}_{\mathrm{L}}\right|}\right)$ |

77 | if $\left|{v}_{\mathrm{L}}\right|>\left|v\right|$ or ($\left|{v}_{\mathrm{L}}\right|=\left|v\right|$ and ${\alpha}_{\mathrm{L}}<\alpha $) then |

78 | $v={v}_{\mathrm{L}}$ |

79 | $\alpha ={\alpha}_{\mathrm{L}}$ |

80 | end if |

81 | end do |

$\mathbf{b}=\mathbf{V}\mathbf{I}\mathbf{n}\mathbf{s}\mathbf{i}\mathbf{d}\mathbf{e}\left(\mathbf{v},\mathbf{G}\right)$ | |
---|---|

82 | $\left[{v}_{\mathrm{x}}{v}_{\mathrm{y}}\right]=v$ |

83 | $\left[{G}_{\mathrm{N}}{G}_{\mathrm{S}}{G}_{\mathrm{E}}{G}_{\mathrm{W}}\right]=G$ |

84 | if ${G}_{\mathrm{S}}\le {v}_{\mathrm{y}}\le {G}_{\mathrm{N}}$ and ${G}_{\mathrm{W}}\le {v}_{\mathrm{x}}\le {G}_{\mathrm{E}}$ then |

85 | $b=\mathrm{true}$ |

86 | else |

87 | $b=\mathrm{false}$ |

88 | end if |

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**MDPI and ACS Style**

Sánchez, P.; Casado, R.; Bermúdez, A.
Real-Time Collision-Free Navigation of Multiple UAVs Based on Bounding Boxes. *Electronics* **2020**, *9*, 1632.
https://doi.org/10.3390/electronics9101632

**AMA Style**

Sánchez P, Casado R, Bermúdez A.
Real-Time Collision-Free Navigation of Multiple UAVs Based on Bounding Boxes. *Electronics*. 2020; 9(10):1632.
https://doi.org/10.3390/electronics9101632

**Chicago/Turabian Style**

Sánchez, Paloma, Rafael Casado, and Aurelio Bermúdez.
2020. "Real-Time Collision-Free Navigation of Multiple UAVs Based on Bounding Boxes" *Electronics* 9, no. 10: 1632.
https://doi.org/10.3390/electronics9101632