# UAV Mission Planning Resistant to Weather Uncertainty

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. A Motivational Example

^{2}and contains 13 nodes (one base: ${N}_{1}$ and 12 customers: ${N}_{2}-{N}_{13}$; see Figure 1). The fleet consists of three homogeneous UAVs with the technical parameters presented in Table 1. The horizon time and goods delivery demand of individual customers is known in advance and the goods are transported under any weather conditions where the UAVs are capable to operate. In that context, the problem under consideration can be reduced to answering the following question: Is the available UAV fleet able to guarantee the delivery of the required quantity of goods using the given transport network within the assumed time horizon under the forecasted weather conditions?

## 4. Modeling

#### 4.1. Assumptions

- -
- The weather forecast is known in advance with sufficient accuracy to specify the so-called weather time windows ${W}_{l}$.
- -
- The weather time windows can be subdivided into flying time windows ${F}_{l}$.
- -
- The weather (which is known in advance) is specified by vector $\overrightarrow{{W}_{l}}=\left[v{w}_{l},{\theta}_{l}\right]$ where $v{w}_{l}^{}$ is the wind speed and ${\theta}_{l}$ is the direction of wind for each ${F}_{l}$. Vector $\overrightarrow{{W}_{l}}$ is constant for a given weather time window.
- -
- Every route traveled starts and terminates within a given flying time window.
- -
- All UAVs are charged to their full energy capacity before the start of a flying time window, and a UAV can only fly once during a flying time window.
- -
- The same kind of cargo is delivered to different customers in different amounts (kg).
- -
- The weight of a UAV is decreased as the cargo is successively unloaded at customers located along its route.
- -
- The network consists of customer locations (delivery points) and flying corridors.

- -
- Strategy 1—which assumes that a UAV travels at a constant ground speed. The airspeed must compensate adverse changes in wind direction and speed.
- -
- Strategy 2—which assumes that the UAV airspeed is constant throughout the mission. The ground speed is different for different segments and depends of the wind parameters specified by $\overrightarrow{{W}_{l}}$.

#### 4.2. Declarative Model

Parameters | |

Network | |

$G=\left(N,E\right)$ | graph of a transportation network: $N=\left\{1\dots n\right\}$ is a set of nodes, $E=\left\{\left\{i,j\right\}\right|i,j\in N,i\ne j\}$ is a set of edges |

$C{L}_{m,l}=\left({N}_{m,l},{E}_{m,l}\right)$ | subgraph of $G$ representing the mth cluster in the lth flying time window: ${N}_{m,l}\subseteq N$ and ${E}_{m,l}\subseteq E$ |

${z}_{i}$ | demand at node $i\in N$, ${z}_{1}=0$ |

$p{r}_{i}$ | priority of the node $i\in N$, $p{r}_{1}=0$ |

${d}_{i,j}$ | travel distance from node $i$ to node $j$ |

${t}_{i,j}$ | travel time from node $i$ to node $j$ |

$w$ | time spent on take-off and landing of a UAV |

$ts$ | time interval at which UAVs can take off from the base |

${b}_{\left\{i,j\right\};\{\alpha ,\beta \}}^{}$ | binary variable corresponding to crossed edges |

${b}_{\left\{i,j\right\};\{\alpha ,\beta \}}^{}=\{\begin{array}{cc}1& \mathrm{when}\text{}\mathrm{edges}\text{}\{i,j\}\text{}\mathrm{and}\text{}\{\alpha ,\beta \}\text{}\mathrm{are}\text{}\mathrm{crossed}\\ 0& \mathrm{otherwise}.\end{array}$ | |

UAV Technical Parameters | |

K | size of the fleet of UAVs |

Q | maximum loading capacity of a UAV |

${C}_{D}$ | aerodynamic drag coefficient of a UAV |

A | front facing area of a UAV |

ep | empty weight of a UAV |

D | air density |

b | width of a UAV |

$CAP$ | maximum energy capacity of a UAV |

Environmental Parameters | |

$H$ | time horizon $H=\left[0,{t}_{max}\right]$ |

${W}_{T}$ | weather time window $T$: ${W}_{T}=\left[W{S}_{T},W{E}_{T}\right]$, $W{S}_{T}$/$W{E}_{T}$ is a start/end time of ${W}_{T}$ |

${F}_{l}$ | flying time window $l$: ${F}_{l}=\left[F{S}_{l},F{E}_{l}\right]$, $F{S}_{l}$/$F{E}_{l}$ is a start time of ${F}_{l}$ |

$v{w}_{l}$ | wind speed in the lth flying time window |

${\theta}_{l}$ | wind direction in the lth flying time window |

$v{a}_{i,j}^{l}$ | airspeed of a UAV traveling from node $i$ to node $j$ in the lth flying time window |

${\phi}_{i,j}$ | heading angle, angle of the airspeed vector when the UAV travels from node $i$ to node $j$ |

$v{g}_{i,j}^{l}$ | ground speed of a UAV travelling from node $i$ to node $j$ in the lth flying time window |

${\vartheta}_{i,j}$ | course angle, angle of the ground speed vector when the UAV travels from node $i$ to node $j$ |

Decision Variables | |

${x}_{i,j}^{k}$ | binary variable used to indicate if the kth UAV travels from node $i$ to node $j$ |

${x}_{i,j}^{k}=\{\begin{array}{cc}1& \mathrm{if}\text{}k\mathrm{th}\text{}\mathrm{UAV}\text{}\mathrm{travels}\text{}\mathrm{from}\text{}\mathrm{node}\text{}i\text{}\mathrm{to}\text{}\mathrm{node}\text{}j\\ 0& \mathrm{otherwise}.\end{array}$ | |

${y}_{i}^{k}$ | time at which the kth UAV arrives at node $i$ |

${c}_{i}^{k}$ | weight of freight delivered to node $i$ by the kth UAV |

${f}_{i,j}^{k}$ | weight of freight carried from node $i$ to node $j$ by the kth UAV |

${P}_{i,j}^{k}$ | energy per unit of time, consumed by kth UAV during a flight from node $i$ to node $j$ |

${s}^{k}$ | take-off time of the kth UAV |

$c{p}_{i}$ | total weight of freight delivered to node $i$ |

${\pi}_{m,l}^{k}$ | route of the kth UAV in the mth cluster in the lth flying time window ${\pi}_{m,l}^{k}=\left({v}_{1},\dots ,{v}_{i},{v}_{i+1},\dots ,{v}_{\mu}\right)$, ${v}_{i}\in {N}_{m,l}$, ${x}_{{v}_{i},{v}_{i+1}}^{k}=1$ |

Sets | |

${Y}_{}^{k}$ | set of times ${y}_{i}^{k}$—schedule of the kth UAV |

$Y$ | family of ${Y}_{}^{k}$—schedule of UAV fleet |

${C}_{}^{k}$ | set of ${c}_{i}^{k}$—payload weight delivered by the kth UAV |

$C$ | family of ${C}_{}^{k}$ |

$\Pi $ | set of UAV routes ${\pi}_{m,l}^{k}$ |

${S}_{m,l}$ | sub-mission in the mth cluster in the lth flying time window ${S}_{m,l}=\left(\Pi ,Y,C\right)$ |

**Constraints**

- -
- Strategy 1—ground speed $v{g}_{i,j}^{l}$ is constant and airs peed $v{a}_{i,j}^{l}$ is calculated from:$$v{a}_{i,j}^{l}=\sqrt{{\left(v{g}_{i,j}^{l}\times cos{\vartheta}_{i,j}-v{w}_{l}\times cos{\theta}_{l}\right)}^{2}+{\left(v{g}_{i,j}^{l}\times sin{\vartheta}_{i,j}-v{w}_{l}\times sin{\theta}_{l}\right)}^{2}}$$$${t}_{i,j}=\frac{{d}_{i,j}}{v{g}_{i,j}^{l}}$$
- -
- Strategy 2—air speed $v{a}_{i,j}^{l}$ is a constant and time ${t}_{i,j}$ is calculated due to Formula (29) where ground speed $v{g}_{i,j}^{l}$ is [45,47].$$v{g}_{i,j}^{l}=\sqrt{{\left(v{a}_{i,j}^{l}\times cos{\phi}_{i,j}+v{w}_{l}\times cos{\theta}_{l}\right)}^{2}+{\left(v{a}_{i,j}^{l}\times sin{\phi}_{i,j}+v{w}_{l}\times sin{\theta}_{l}\right)}^{2}}$$$${\phi}_{i,j}={\vartheta}_{i,j}-arcsin\left(\frac{v{w}_{l}}{v{a}_{i,j}^{l}}sin\left({\theta}_{l}-{\vartheta}_{i,j}\right)\right)$$

## 5. Problem Formulation

- $\mathcal{V}=\left\{\Pi ,Y,C\right\}$—a set of decision variables determining sub-mission,
- ${S}_{m,l}:\Pi $—a set of UAV routes,
- $Y$—a schedule of a UAV fleet,
- $C$—a set of payload weights delivered by the UAVs,
- $\mathcal{D}$—a finite set of decision variable domain descriptions,
- $\mathcal{C}$—a set of constraints specifying the relationships between UAV routes, UAV schedules, and transported materials Formulae (3)–(32).

## 6. Computational Experiments

#### 6.1. Cluster #1

^{2}, three UAVs deliver goods to six customers. Node ${N}_{1}$ represents the location of the company (i.e., the base from which the UAVs take off from/land) and nodes ${N}_{2}-{N}_{7}$ representing the locations of individual customers. Known is the demand of the individual customers for the goods transported by the UAVs, which is the same for each customer and equals 30 kg: ${z}_{1}=0$, ${z}_{2}=\dots ={z}_{7}=30$. It is assumed that the UAVs must deliver to each customer the exact quantity of goods they demand.

#### 6.2. Cluster #2

^{2}. Nodes ${N}_{8}-{N}_{13}$ represent the locations of the individual customers. The demand of the individual customers is equal to: ${z}_{1}=0$, ${z}_{8}=\dots ={z}_{13}=30$. The flying time window is equal to ${F}_{2}=\left[2500,5000\right]$ [s]. The weather conditions are changed, whereby the wind speed is higher, i.e., $v{w}_{2}=12$ m/s and the direction of wind is equal to ${\theta}_{2}=150\xb0$.

#### 6.3. Mission Planning

#### 6.4. Quantitative Results

_{)}. Experiments were conducted in the environment IBM ILOG (Intel Core i7-M4800MQ 2.7 GHz, 32 GB RAM).

- -
- Interactive (i.e., online: $t<300$ s) support can be provided for networks consisting of no more than eight nodes. In practice, this means limiting decision making supported by DSSs to the distribution networks not exceeding eight nodes.
- -
- An increase in the number of UAVs increases the route resistance (i.e., increasing of ${v}_{MIN}$ and ${v}_{MAX}$) to changes in weather conditions. For example, in a network of four nodes, the change from two to four UAVs increases value ${v}_{MIN}$ from 24.8 to 29.4 and ${v}_{MAX}$ from 28.1 to 33.9 for Strategy 1 (see yellow cells), as well as changing ${v}_{MIN}$ from 18.1 to 18.5 and ${v}_{MAX}$ from 19.0 to 19.2 for Strategy 2 (see green cells).
- -
- The ${v}_{MIN}$ and ${v}_{MAX}$ values for route resistance in Strategy 2 are limited by the value of the airspeed ($v{a}_{i,j}^{l}=20\mathrm{m}/\mathrm{s}$). This type of restriction does not exist in Strategy 1. This means that in situations where the wind speed exceeds the value vw > 20 m/s, it is recommended to use Strategy 1 (for this strategy, it is possible to get v
_{MIN}and ${v}_{MAX}$ above 20 m/s).

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Strategies for delivering goods following a constant ground speed (

**a**) and a constant airspeed (

**b**).

**Figure 3.**Energy consumption under a constant Unmanned Aerial Vehicles (UAV) weight (

**a**,

**b**), and under a variable UAV weight (

**c**,

**d**).

**Figure 6.**Obtained solution: (

**a**) routes for Strategy 1, a constant ground speed; (

**b**) routes for Strategy 2, a constant airspeed.

**Figure 7.**Obtained solution: (

**a**) flight schedule for Strategy 1; (

**b**) flight schedule for Strategy 2.

**Figure 11.**Obtained solution: (

**a**) flight schedule for Strategy 1; (

**b**) flight schedule for Strategy 2.

**Figure 13.**Example of the flying mission for the network from Figure 1.

Technical Parameters of UAVs | Value | Unit |
---|---|---|

Payload capacity ($\mathrm{Q}$) | 90 | kg |

Battery capacity ($\mathrm{CAP}$) | 8000 | kJ |

Airspeed ($va$) | 20 | m/s |

Drag coefficient (${\mathrm{C}}_{\mathrm{D}}$) | 0.54 | - |

Front surface of UAV ($\mathrm{A}$) | 1.2 | m |

UAV width ($\mathrm{b}$) | 8.7 | m |

$\mathit{n}$^{1)} | $\mathit{K}$ | Assumptions | ${\mathit{v}}_{\mathit{w}}=10\text{}\mathbf{m}/\mathbf{s}$ | ${\mathit{v}}_{\mathit{w}}=11\text{}\mathbf{m}/\mathbf{s}$ | ${\mathit{v}}_{\mathit{w}}=12\text{}\mathbf{m}/\mathbf{s}$ | NDV | NC | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\Theta}=30\xb0$ | $\mathit{\Theta}=130\xb0$ | $\mathit{\Theta}=230\xb0$ | ||||||||||||||

$\mathit{E}$ | $\mathit{T}\mathit{C}$ | ${\mathit{v}}_{\mathit{M}\mathit{I}\mathit{N}}$ | ${\mathit{v}}_{\mathit{M}\mathit{A}\mathit{X}}$ | $\mathit{E}$ | $\mathit{T}\mathit{C}$ | ${\mathit{v}}_{\mathit{M}\mathit{I}\mathit{N}}$ | ${\mathit{v}}_{\mathit{M}\mathit{A}\mathit{X}}$ | $\mathit{E}$ | $\mathit{T}\mathit{C}$ | ${\mathit{v}}_{\mathit{M}\mathit{I}\mathit{N}}$ | ${\mathit{v}}_{\mathit{M}\mathit{A}\mathit{X}}$ | |||||

4 | 2 | Strategy 1 | 29.80 | 3.73 | 24.8 | 28.1 | 19.17 | 3.71 | 24.6 | 28.2 | 29.8 | 3.74 | 24.8 | 28.1 | 828 | 356 |

Strategy 2 | 19.1 | 3.82 | 18.1 | 19.0 | 19.18 | 3.78 | 18.1 | 19.1 | 19.1 | 3.76 | 18.1 | 19.0 | 828 | 356 | ||

3 | Strategy 1 | 13.59 | 3.89 | 29.4 | 33.9 | 13.59 | 3.97 | 29.4 | 33.9 | 13.59 | 3.95 | 29.4 | 33.9 | 1774 | 654 | |

Strategy 2 | 13.57 | 3.75 | 18.5 | 19.2 | 13.57 | 3.96 | 18.5 | 19.2 | 13.57 | 3.81 | 18.5 | 19.2 | 1774 | 654 | ||

4 | Strategy 1 | 13.59 | 4.24 | 29.4 | 33.9 | 13.59 | 4.17 | 29.4 | 33.9 | 13.59 | 4.35 | 29.4 | 33.9 | 3076 | 1036 | |

Strategy 2 | 13.57 | 4.31 | 18.5 | 19.2 | 13.59 | 4.28 | 18.5 | 19.2 | 13.57 | 4.36 | 18.5 | 19.2 | 3076 | 1036 | ||

6 | 2 | Strategy 1 | 35.67 | 4.44 | 23.6 | 25.6 | 22.62 | 4.23 | 23.6 | 25.6 | 22.62 | 4.60 | 23.6 | 25.6 | 3014 | 910 |

Strategy 2 | 22.59 | 4.31 | 17.9 | 18.6 | 22.59 | 4.38 | 17.9 | 18.6 | 22.81 | 4.12 | 17.9 | 18.6 | 3014 | 910 | ||

3 | Strategy 1 | 19.4 | 7.12 | 25.8 | 27.7 | 19.40 | 5.25 | 25.8 | 27.7 | 19.4 | 8.08 | 25.8 | 27.7 | 7476 | 1902 | |

Strategy 2 | 19.37 | 6.34 | 18.2 | 18.7 | 19.38 | 5.24 | 18.2 | 18.7 | 19.37 | 9.32 | 18.2 | 18.7 | 7476 | 1902 | ||

4 | Strategy 1 | 19.4 | 9.98 | 25.8 | 27.7 | 19.40 | 6.44 | 25.8 | 27.7 | 19.4 | 13.67 | 25.8 | 27.7 | 13,910 | 3528 | |

Strategy 2 | 19.37 | 8.19 | 18.2 | 18.7 | 19.38 | 9.46 | 18.2 | 18.7 | 19.37 | 8.08 | 18.2 | 18.7 | 13,910 | 3528 | ||

8 | 2 | Strategy 1 | 22.62 | 46.04 | 20.9 | 25.6 | 24.21 | 7.93 | 18.8 | 25.1 | 22.62 | 8.63 | 23.6 | 25.6 | 9552 | 2248 |

Strategy 2 | 22.59 | 102.93 | 17.9 | 18.6 | 24.18 | 281.67 | 17.7 | 18.6 | 22.59 | 9.43 | 17.9 | 18.6 | 9552 | 2248 | ||

3 | Strategy 1 | 22.62 | t > 300 | 23.6 | 25.6 | 20.62 | 19.94 | 25.3 | 26.9 | 25.15 | 231.63 | 18.8 | 24.2 | 25,898 | 5358 | |

Strategy 2 | 25.13 | 59.49 | 17.8 | 18.2 | 23.79 | 71.73 | 17.8 | 18.6 | 25.13 | 126.89 | 17.8 | 18.2 | 25,898 | 5358 | ||

4 | Strategy 1 | 22.55 | t > 300 | 23.6 | 25.6 | 20.62 | 128.00 | 25.3 | 26.9 | 25.15 | 105.18 | 18.8 | 24.2 | 49,960 | 9800 | |

Strategy 2 | 25.13 | 110.94 | 17.8 | 18.2 | 21.33 | 95.87 | 18.0 | 18.7 | 25.49 | 115.29 | 17.7 | 18.2 | 49,960 | 9800 | ||

10 | 2 | Strategy 1 | 27.54 | t > 300 | 18.9 | 22.9 | 28.59 | 183.53 | 18.2 | 22.8 | 29.46 | t > 300 | 17.8 | 21.7 | 21,402 | 4530 |

Strategy 2 | 24.18 | t > 300 | 17.9 | 18.4 | 25.65 | t > 300 | 17.5 | 18.0 | 29.35 | t > 300 | 19.5 | 20.6 | 21,402 | 4530 | ||

3 | Strategy 1 | 28.38 | t > 300 | 18.7 | 22.5 | 23.83 | t > 300 | 20.0 | 25.1 | 24.29 | t > 300 | 18.9 | 25.3 | 59,920 | 11,502 | |

Strategy 2 | 24.23 | t > 300 | 17.5 | 18.5 | 24.23 | t > 300 | 17.5 | 18.5 | 24.23 | t > 300 | 17.5 | 18.5 | 59,920 | 11,502 | ||

4 | Strategy 1 | 26.21 | t > 300 | 19.5 | 23.2 | 24.29 | t > 300 | 18.9 | 25.3 | 26.2 | t > 300 | 18.4 | 24.3 | 116,986 | 21,622 | |

Strategy 2 | 26.18 | t > 300 | 17.6 | 18.3 | 26.19 | t > 300 | 17.3 | 18.4 | 26.19 | t > 300 | 17.3 | 18.4 | 116,986 | 21,622 |

^{1)}—number of nodes; $K$—size of the UAV fleet; $E$—maximum consumed energy (%); $TC$—time of computation (s); $NC$—number of constraints; $NDV$—number of decision variables

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## Share and Cite

**MDPI and ACS Style**

Thibbotuwawa, A.; Bocewicz, G.; Radzki, G.; Nielsen, P.; Banaszak, Z.
UAV Mission Planning Resistant to Weather Uncertainty. *Sensors* **2020**, *20*, 515.
https://doi.org/10.3390/s20020515

**AMA Style**

Thibbotuwawa A, Bocewicz G, Radzki G, Nielsen P, Banaszak Z.
UAV Mission Planning Resistant to Weather Uncertainty. *Sensors*. 2020; 20(2):515.
https://doi.org/10.3390/s20020515

**Chicago/Turabian Style**

Thibbotuwawa, Amila, Grzegorz Bocewicz, Grzegorz Radzki, Peter Nielsen, and Zbigniew Banaszak.
2020. "UAV Mission Planning Resistant to Weather Uncertainty" *Sensors* 20, no. 2: 515.
https://doi.org/10.3390/s20020515