# Path Planning for Fixed-Wing Unmanned Aerial Vehicles: An Integrated Approach with Theta* and Clothoids

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## Abstract

**:**

## 1. Introduction

## 2. Path Transition Curve

#### Flight Path Transition Curve between Two Directions

**Procedure 1. Flight path transition curve between two directions**

- STEP 1. Firstly, compute the angles ${\psi}_{j}$ and ${\psi}_{k}$ between an arbitrary axis and ${r}_{j}$ and ${r}_{k}$, respectively.
- STEP 2. Assuming $\Delta \psi =\left(\right)open="|"\; close="|">{\psi}_{k}-{\psi}_{j}$, ${\kappa}_{max}$ the maximum curvature and ${\sigma}_{max}$ the maximum sharpness, it is possible to compute $\Delta s=2{\kappa}_{max}/{\sigma}_{max}$, as the length of a virtual curve with maximum sharpness and maximum curvature. It is worth noting that it is called virtual because the heading change constraint is not considered yet.
- STEP 3. The area of the trapezium with major base $\Delta s$, minor base l as the length of the circular arc, and height ${\kappa}_{max}$ must be equal to $\Delta \psi $. The minor base can be computed as$$l=\frac{\Delta \psi}{{\kappa}_{max}}-\frac{\Delta s}{2}$$If $l>0$, the path includes a circular arc with curvature ${\kappa}_{max}$. If $l=0$, then the path includes only two clothoids and the maximum curvature ${\kappa}_{max}$ is reached at the middle point. If $l<0$, the path includes only two clothoids that do not reach the maximum curvature.
- STEP 4. Starting from the intersection point $Q=({x}_{Q},{y}_{Q})$, the path can be computed by integrating the following equations as follows:$$\left(\right)$$
- -
- For the half-circular arc, the initial conditions are $x\left(0\right)={x}_{Q},y\left(0\right)={y}_{Q}$, $\dot{\psi}\left(0\right)=V{\kappa}_{max}$, while $\sigma \left(s\right)=0$ and $s\in \left(\right)open="["\; close="]">0,\frac{l}{2}$.
- -
- For the spiral curve, in the case of $l>0$ ($l\le 0$), the initial conditions are $x\left(0\right)={x}_{Q}+x(l/2),y\left(0\right)={y}_{Q}+y(l/2)$, $\dot{\psi}\left(0\right)=V{\kappa}_{max}$ ($x\left(0\right)={x}_{Q},y\left(0\right)={y}_{Q}$, $\dot{\psi}\left(0\right)=V\sqrt{\Delta \psi \xb7{\sigma}_{max}}$), while $\sigma \left(s\right)=-{\sigma}_{max}$ and $s\in \left(\right)open="["\; close="]">\frac{l}{2},\frac{\Delta s-l}{2}$.

These segments represent the second half of the overall curve. The first part can be computed by mirroring the results with respect to the median line between the considered directions. The set of Equation (9) has been employed to mitigate non-linearities in the calculation of heading variation $\dot{\psi}$. This implies that, in the simulation of aircraft motion, the variation law of the bank angle $\dot{\varphi}$ can be computed as follows:$$\begin{array}{c}\dot{\varphi}=\sigma \frac{{V}^{2}}{g}{\mathrm{cos}}^{2}\varphi \\ \varphi =\mathrm{arctan}\left(\right)open="("\; close=")">\frac{{V}^{2}}{g}\kappa \end{array}$$ - STEP 5. The curve must be moved in order to be tangent to both the assigned directions.

## 3. Path-Planning Algorithm

**Procedure 2. Flyable edge between two points ${P}_{i}$, with direction ${r}_{i}$, and ${P}_{j}$.**

- STEP 0—Set an initial guess for the building distance ${d}_{c}$.
- STEP 1—Consider a point ${Q}^{0}=({x}_{{Q}^{0}},{y}_{{Q}^{0}})$ to be used as the intersection point between the direction ${r}_{i}$ and the candidate direction ${r}_{j}^{0}$ such that$$\left(\right)$$
- STEP 2—Compute ${r}_{j}^{0}$ as the direction ${P}_{j}-{Q}^{0}$.
- STEP 3—Build the transition curve using Procedure 1.
- STEP 4—Add two segments: the former connecting ${P}_{i}$ with the initial point of the transition curve and the latter connecting the ending point of the transition curve with ${P}_{j}$.
- STEP 5—Compute $\Delta L$ the length of the first segment of the just-built flyable path between ${P}_{i}$ and ${P}_{j}$.
- STEP 6—If $\Delta L>0$, then ${d}_{c}={d}_{c}-\Delta L$ and return to STEP 1 (see Figure 2b, otherwise return the obtained flyable path $\Gamma ({P}_{i},{P}_{j})$ and the final direction ${r}_{j}={r}_{j}^{0}$.

**Procedure 3. Theta* exploration process to find the flyable path between two points A and B, with assigned directions ${r}_{A}$ and ${r}_{B}$, respectively.**

- STEP 0—Consider the set $\mathcal{C}$ of explored nodes and the set $\mathcal{O}$ of unexplored nodes consisting of tuples $<{P}_{i},{r}_{i},{P}_{i}^{h},{\mathfrak{F}}_{\mathfrak{i}}>$, where ${P}_{i}$ is the considered node, ${r}_{i}$ is the departing direction, ${P}_{i}^{h}$ is the parent node, and ${\mathfrak{F}}_{\mathfrak{i}}$ is the global cost:$${\mathfrak{F}}_{i}=\mathfrak{H}(A,{P}_{i}^{h})+L({P}_{i}^{h},{P}_{i})+{\parallel B-{P}_{i}\parallel}_{2}$$Initialize $\mathcal{C}=\varnothing $ and $\mathcal{O}=<A,{r}_{A},A,{\mathfrak{F}}_{A}>$.
- STEP 1—Select and remove from $\mathcal{O}$ the tuple $<{P}_{i},{r}_{i},{P}_{i}^{h},{\mathfrak{F}}_{i}>$ with minimum global cost. Add the tuple to $\mathcal{C}$. If ${P}_{i}=B$, then terminate the procedure.
- STEP 2—Compute the vision cone ${\mathbb{V}}_{i}$ and build the set ${\mathcal{V}}_{i}$. If B is in ${\mathbb{V}}_{i}$, add B to the set of neighbors ${\mathcal{V}}_{i}$.
- STEP 3—For each node ${P}_{j}\in {\mathcal{V}}_{i}$, build the flyable paths $\Gamma ({P}_{i},{P}_{j})$ and $\Gamma ({P}_{i}^{h},{P}_{j})$.If ${P}_{j}=B$, then two flyable edges are needed to find a trajectory between two points with assigned initial and final directions. In particular, said $\tilde{B}=\frac{{P}_{i}+B}{2}$ an intermediate point, two flyable edges $\Gamma ({P}_{i},\tilde{B})$ and $\Gamma (\tilde{B},B)$ must be computed.$$\Gamma ({P}_{i},B)=\Gamma ({P}_{i},\tilde{B})\cup \Gamma (\tilde{B},B)$$
- STEP 4—For each node ${P}_{j}$, compute the two possible global costs associated to ${P}_{j}$:$${F}_{j}^{1}=\left(\right)open="\{"\; close>\begin{array}{cc}\mathfrak{H}(A,{P}_{i}^{h})+L({P}_{i}^{h},{P}_{j})+{\parallel B-{P}_{j}\parallel}_{2}& \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\Gamma ({P}_{i}^{h},{P}_{j})\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{feasible}\\ +\infty & \mathrm{otherwise}\end{array}$$$${F}_{j}^{2}=\left(\right)open="\{"\; close>\begin{array}{cc}\mathfrak{H}(A,{P}_{i})+L({P}_{i},{P}_{j})+{\parallel B-{P}_{j}\parallel}_{2}& \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\Gamma ({P}_{i},{P}_{j})\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{feasible}\\ +\infty & \mathrm{otherwise}\end{array}$$
- STEP 5—For each node ${P}_{j}$ assign to ${\mathfrak{F}}_{j}$ the minimum global cost associated to ${P}_{j}$ and the relative parent node$${P}_{j}^{h}=\left(\right)open="\{"\; close>\begin{array}{cc}{P}_{i}& \mathrm{if}\phantom{\rule{4.pt}{0ex}}{F}_{j}^{2}{F}_{j}^{1}\\ {P}_{i}^{h}& \mathrm{otherwise}\end{array}$$
- STEP 6—Consider the tuple $<{P}_{j},{r}_{j},{P}_{j}^{h},{\mathfrak{F}}_{j}>$. If $\mathcal{C}$ contains a tuple $<{P}_{j},{r}_{j}^{0},{P}_{j}^{{h}^{0}},{\mathfrak{F}}_{j}^{0}>$, if ${\mathfrak{F}}_{j}^{0}>{\mathfrak{F}}_{j}$ then remove $<{P}_{j},{r}_{j}^{0},{P}_{j}^{{h}^{0}},{\mathfrak{F}}_{j}^{0}>$ and add $<{P}_{j},{r}_{j},{P}_{j}^{h},{\mathfrak{F}}_{j}>$ to $\mathcal{C}$. If $\mathcal{C}$ does not contain any tuple with ${P}_{j}$, then add $<{P}_{j},{r}_{j},{P}_{j}^{h},{\mathfrak{F}}_{j}$ to $\mathcal{C}$.
- STEP 7—Go to STEP 1.

## 4. Results

#### 4.1. Test Case #1

#### 4.2. Test Case #2

#### 4.3. Test Case #3

#### 4.4. Test Case #4: Urban Scenario

#### 4.5. Test Case #5: Simulation of a Short Cruise Path

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Curvature variation along the path between two straight segments. (

**a**) Trapezoid construction: the oblique edges represent the spirals, and the minor base is the circular arc. (

**b**) Path transition curve: dot lines represent the spiral-only trajectory without curvature constraint, solid lines are for the curve made of three pieces: two spirals and an arc of circumference.

**Figure 2.**The optimal path between two graph nodes is found with an iterative method. (

**a**) First step of the method. (

**b**) Final result.

**Figure 3.**Image-based solution adopted to speed up the process. The distance between two nodes of the grid depends on their distances from the obstacles. (

**a**) Example of binary representation of the scenario. (

**b**) Example of filtered scenario; the blue areas indicate the low-resolution zones because they are far from the obstacles while the green ones stand for the high-resolution regions because they are close to the obstacles.

**Figure 4.**Test case #0: Scenario with one obstacle, represented in red, starting point at $A=(0,0)\mathrm{m}$ and target point at $B=(3000,3000)\mathrm{m}$. (

**a**) Paths (blue lines) explored by the algorithm during the optimization process. (

**b**) Final path (black line).

**Figure 5.**Test case #0: Details of the optimal flyable path. (

**a**) Initial turn maneuver. (

**b**) Turn around the obstacle.

**Figure 6.**Test case #1. Comparison between ${\theta}_{C}^{\ast}$ and ${\theta}_{V}^{\ast}$ algorithms. Obstacles are represented in red, explored paths with solid blue lines, and final paths with solid black lines. (

**a**,

**b**) ${\theta}_{C}^{\ast}$ algorithm. (

**c**,

**d**) ${\theta}_{V}^{\ast}$ algorithm.

**Figure 7.**Test case #2. Comparison between ${\theta}_{C}^{\ast}$ and ${\theta}_{G}^{\ast}$ algorithms. Obstacles are represented in red, explored nodes with blue dots, and final paths with solid black lines. (

**a**,

**b**) ${\theta}_{C}^{\ast}$ algorithm with fixed search distance ${d}_{search}=50$ m. (

**c**,

**d**) ${\theta}_{C}^{\ast}$ algorithm with fixed search distance ${d}_{search}=100$ m. (

**e**,

**f**) ${\theta}_{G}^{\ast}$ algorithm.

**Figure 8.**Test case #3. Comparison between ${\theta}_{C}^{\ast}$, ${\theta}_{G}^{\ast}$, ${\theta}_{V}^{\ast}$, and ${\theta}_{F}^{\ast}$ algorithms. Obstacles are represented in red, and explored paths with solid blue lines. (

**a**) Explored paths with ${\theta}_{C}^{\ast}$ algorithm. (

**b**) Explored paths with ${\theta}_{G}^{\ast}$ algorithm. (

**c**) Explored paths with ${\theta}_{V}^{\ast}$ algorithm. (

**d**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm.

**Figure 9.**Test case #3. Final paths. Obstacles are represented in red, and final paths with solid black line. (

**a**) Final path with ${\theta}_{C}^{\ast}$ and ${\theta}_{V}^{\ast}$ algorithms. (

**b**) Final path with ${\theta}_{G}^{\ast}$ algorithm. (

**c**) Final path with ${\theta}_{F}^{\ast}$ algorithm.

**Figure 10.**Test case #4. Simulation results for the search of the path between the starting and the target point in a realistic urban scenario in Naples. Obstacles are represented in red, explored paths with solid blue lines, and final paths with solid black lines. (

**a**) Exploration phase. (

**b**) Final path.

**Figure 11.**Test-case #5. Simulation results. (

**a**) Planned path in black solid line and tracked trajectory in dashed cyan. (

**b**) Reference and tracked heading during flight.

**Figure 12.**Comparison ${\theta}_{F}^{\ast}$—EVG. Final paths obtained with ${\theta}_{F}^{\ast}$ algorithm for the scenario proposed in [61]. Obstacles are represented in red and final paths with solid black lines. (

**a**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm and ending point ${\mathit{B}}_{\left(a\right)}$. (

**b**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm and ending point ${\mathit{B}}_{\left(b\right)}$. (

**c**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm and ending point ${\mathit{B}}_{\left(c\right)}$. (

**d**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm and ending point ${\mathit{B}}_{\left(d\right)}$.

**Figure 13.**Comparison ${\theta}_{F}^{\ast}$—clothoid-based solution. Final path obtained with ${\theta}_{F}^{\ast}$ (dotted line) and clothoid-based algorithm (solid line) for the scenario proposed in Section 4.3. Obstacles are represented in red and final paths with solid black lines.

**Figure 14.**Comparison ${\theta}_{F}^{\ast}$ - Stack-RRT*. Final paths obtained with ${\theta}_{F}^{\ast}$ algorithm for the scenario proposed in [62]. Obstacles are represented in red and final paths with solid black lines. (

**a**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm for the scenario presented in Figure 5 of Ref. [62]. (

**b**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm for the scenario presented in Figure 6 of Ref. [62]. (

**c**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm for the scenario presented in Figure 7 of Ref. [62]. (

**d**) Explored paths with ${\theta}_{F}^{\ast}$ algorithm for the scenario presented in Figure 8 of Ref. [62].

Value | |
---|---|

${d}_{search}$ [m] | 100 |

Time [s] | 16.61 |

Path length [m] | 4527.4 |

Explored nodes | 190 |

Total generated nodes | 286 |

Smooth Transition | Vision Cone | Variable Grid | |
---|---|---|---|

${\theta}^{\ast}$ | x | x | x |

${\theta}_{C}^{\ast}$ | + | x | x |

${\theta}_{V}^{\ast}$ | + | + | x |

${\theta}_{G}^{\ast}$ | + | x | + |

${\theta}_{F}^{\ast}$ | + | + | + |

Value | |
---|---|

$\mathit{A}\phantom{\rule{0.166667em}{0ex}}$ [m] | (1300,100) |

${\mathit{\psi}}_{\mathit{start}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 270 |

$\mathit{B}\phantom{\rule{0.166667em}{0ex}}$ [m] | (3500,2500) |

${\mathit{\psi}}_{\mathit{end}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 50 |

$\eta \phantom{\rule{0.166667em}{0ex}}$ [deg] | 80 |

**Table 4.**Test-case #1. Comparison of time, length, and grid nodes between the algorithm with and without the vision cone.

${\mathit{\theta}}_{\mathit{C}}^{\ast}$ | ${\mathit{\theta}}_{\mathit{V}}^{\ast}$ | |
---|---|---|

Time [s] | 42.40 | 13.21 |

Path length [m] | 4389.2 | 4606.9 |

Explored nodes | 305 | 245 |

Total nodes generated | 368 | 294 |

Value | |
---|---|

$\mathit{A}\phantom{\rule{0.166667em}{0ex}}$ [m] | (1800, 500) |

${\mathit{\psi}}_{\mathit{start}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 0 |

$\mathit{B}\phantom{\rule{0.166667em}{0ex}}$ [m] | (5000, 1500) |

${\mathit{\psi}}_{\mathit{end}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 90 |

**Table 6.**Test-case #2. Comparison of time, length, and grid nodes between the algorithm with fixed grid and search distance 50 m; fixed grid and search distance 100 m; variable grid.

${\mathit{\theta}}_{\mathit{C}}^{\ast}$ | ${\mathit{\theta}}_{\mathit{C}}^{\ast}$ | ${\mathit{\theta}}_{\mathit{G}}^{\ast}$ | |
---|---|---|---|

${d}_{search}$ [m] | 50 | 100 | variable |

Time [s] | 41.59 | 32.88 | 22.92 |

Path length [m] | 3761.2 | 4348.1 | 3845.5 |

Explored nodes | 336 | 299 | 285 |

Total generated nodes | 465 | 418 | 450 |

Value | |
---|---|

$\mathit{A}\phantom{\rule{0.166667em}{0ex}}$ [m] | (1800, 500) |

${\mathit{\psi}}_{\mathit{start}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 0 |

$\mathit{B}\phantom{\rule{0.166667em}{0ex}}$ [m] | (3500, 3000) |

${\mathit{\psi}}_{\mathit{end}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 0 |

**Table 8.**Test-case #3. Comparison of time, length, and grid nodes between the algorithm with Theta* clothoid; Theta* clothoid with variable grid; Theta* clothoid with vision cone; Theta∗ clothoid with both variable grid and vision cone.

${\mathit{\theta}}_{\mathit{C}}^{\ast}$ | ${\mathit{\theta}}_{\mathit{V}}^{\ast}$ | ${\mathit{\theta}}_{\mathit{G}}^{\ast}$ | ${\mathit{\theta}}_{\mathit{F}}^{\ast}$ | |
---|---|---|---|---|

${d}_{search}\phantom{\rule{0.166667em}{0ex}}$ [m] | 50 | 50 | variable | variable |

Time [s] | 35.39 | 15.36 | 33.94 | 11.80 |

Path length [m] | 3168.1 | 3168.1 | 3194.0 | 3183.5 |

Explored nodes | 329 | 316 | 343 | 258 |

Total generated nodes | 447 | 386 | 624 | 384 |

Value | |
---|---|

$\mathit{A}$ (lon,lat) [deg, deg] | (14.277, 40.8566) |

${\mathit{\psi}}_{\mathit{start}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 45 |

$\mathit{B}$ (lon,lat) [deg, deg] | (14.284, 40.859) |

${\mathit{\psi}}_{\mathit{end}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 45 |

**Table 10.**Test-case #4. Results in terms of time, length and grid nodes of the algorithm for an urban scenario.

Value | |
---|---|

${d}_{search}\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{m}\right]$ | 40 |

Time [s] | 148.91 |

Path length [m] | 734.04 |

Explored nodes | 37 |

Total generated nodes | 63 |

Number of obstacles | 336 |

Value | |
---|---|

$\mathit{A}\phantom{\rule{0.166667em}{0ex}}$ [m] | (20,000, 20,000) |

${\mathit{\psi}}_{\mathit{start}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 225 |

${\mathit{B}}_{\left(a\right)}\phantom{\rule{0.166667em}{0ex}}$ [m] | (−15,000, −15,000) |

${\mathit{B}}_{\left(b\right)}\phantom{\rule{0.166667em}{0ex}}$ [m] | (−10,000, 0) |

${\mathit{B}}_{\left(c\right)}\phantom{\rule{0.166667em}{0ex}}$ [m] | (0, 0) |

${\mathit{B}}_{\left(d\right)}\phantom{\rule{0.166667em}{0ex}}$ [m] | (0, −15,000) |

${\mathit{\psi}}_{\mathit{end}}\phantom{\rule{0.166667em}{0ex}}$ [deg] | 180 |

**Table 12.**Comparison ${\theta}_{F}^{\ast}$—EVG. Comparison of the algorithms in terms of time and path length.

(a) | (b) | (c) | (d) | |
---|---|---|---|---|

Time ${\mathit{\theta}}_{\mathit{F}}^{\ast}$ [s] | 85.98 | 122.81 | 45.40 | 71.21 |

Time EVG [s] | 1.26 | 1.52 | 0.967 | 1.48 |

Path length ${\mathit{\theta}}_{\mathit{F}}^{\ast}$ [m] | 51,595.9 | 38,782.1 | 29,616.0 | 41,846.7 |

Path length EVG [m] | 51,402.1 | 38,169.0 | 29,298.4 | 41,597.1 |

**Table 13.**Comparison ${\theta}_{F}^{\ast}$—clothoid-based model. Comparison of the algorithms in terms of time and path length.

${\mathit{\theta}}_{\mathit{F}}^{\ast}$ | Clothoid-Based Model | |
---|---|---|

Time [s] | 11.50 | 16.83 |

Path length [m] | 5273.0 | 5754.1 |

**Table 14.**Results of ${\theta}_{F}^{\ast}$ for the scenarios presented in Figure 14. Results in terms of time and path length.

(a) | (b) | (c) | (d) | |
---|---|---|---|---|

Time [s] | 7.82 | 4.68 | 6.90 | 21.35 |

Path length [m] | 81.73 | 74.73 | 46.30 | 123.99 |

Explored nodes | 93 | 49 | 19 | 136 |

Total generated nodes | 143 | 86 | 49 | 206 |

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## Share and Cite

**MDPI and ACS Style**

Bassolillo, S.R.; Raspaolo, G.; Blasi, L.; D’Amato, E.; Notaro, I.
Path Planning for Fixed-Wing Unmanned Aerial Vehicles: An Integrated Approach with Theta* and Clothoids. *Drones* **2024**, *8*, 62.
https://doi.org/10.3390/drones8020062

**AMA Style**

Bassolillo SR, Raspaolo G, Blasi L, D’Amato E, Notaro I.
Path Planning for Fixed-Wing Unmanned Aerial Vehicles: An Integrated Approach with Theta* and Clothoids. *Drones*. 2024; 8(2):62.
https://doi.org/10.3390/drones8020062

**Chicago/Turabian Style**

Bassolillo, Salvatore Rosario, Gennaro Raspaolo, Luciano Blasi, Egidio D’Amato, and Immacolata Notaro.
2024. "Path Planning for Fixed-Wing Unmanned Aerial Vehicles: An Integrated Approach with Theta* and Clothoids" *Drones* 8, no. 2: 62.
https://doi.org/10.3390/drones8020062