# Online Deterministic 3D Trajectory Generation for Electric Vertical Take-Off and Landing Aircraft

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## Abstract

**:**

## 1. Introduction

## 2. Trajectory Construction: Overview

#### 2.1. On Plane Curves

#### 2.2. On Straight Line Segments

#### Curvature and Arc Length of the Line Segment

#### 2.3. On Circle Segment

#### 2.3.1. Curvature of the Circle Segment

#### 2.3.2. Arc Length of the Circle Segment

#### 2.4. On Clothoid Segment

#### 2.4.1. Curvature of the Clothoid Segment

#### 2.4.2. Arc Length of the Clothoid Segment

#### 2.4.3. Change in Direction

#### 2.4.4. Clothoid Approximation

## 3. Trajectory Construction: Flyby

- Fixed parameters resulting from the flight plan:course angle change between two subsequent legs $\Delta {\chi}_{T}$, kinematic speed ${V}_{T}$;
- Fixed parameters resulting from the inner loop controller design, the resulting inner loop dynamics and aircraft performance:roll time constant ${T}_{p}$, roll rate command ${p}_{cmd}$;
- Trajectory design parameter:desired turn rate ${\dot{\chi}}_{{T}_{d}}$.

#### 3.1. Horizontal Plane

- 1.
- Straight line: the aircraft is flying in a straight line with a kinematic bank angle $\mu =0$.
- 2.
- Circle: The aircraft is performing a turn with a kinematic bank angle as given by Equation (33)$$\mu =arctan\left(\frac{{V}_{T}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\dot{\chi}}_{{T}_{d}}}{{g}_{0}}\right)$$

#### 3.1.1. Turn Distance

#### 3.1.2. Implementation Considerations

- 1.
- Calculate the maneuver circle ${r}_{c}$ using Equation (34), where ${\dot{\chi}}_{{T}_{d}}$ is defined during design.
- 2.
- Calculate the corresponding bank angle ${\mu}_{cmd}$ using Equation (33).
- 3.
- Calculate the clothoid transition time ${t}_{cl}$ using Equation (35), where ${p}_{cmd}$ is the maximum achievable roll rate.
- 4.
- Calculate the clothoid shaping parameter A using Equation (40).
- 5.
- Calculate the clothoid running parameter (dimensionless time) ${\tau}_{cl}$ using Equation (37).
- 6.
- Calculate the clothoid displacement $\left[\Delta {x}_{cl}\left(\tau \right),\phantom{\rule{0.277778em}{0ex}}\Delta {y}_{cl}\left(\tau \right)\right]$ using Equation (32).
- 7.
- Calculate the turn distance ${d}_{turn}$ using Equation (52).

#### 3.2. Vertical Plane

#### 3.2.1. Altitude Polynomial Definition

#### 3.2.2. Polynomial Coefficients

- ${h}_{T}^{{P}_{n}}\to {h}_{{p}_{n}}$;
- $tan{\gamma}_{T}^{{P}_{1}}\to {h}_{{b}_{n}}$;
- ${s}_{tot,hor}\to {s}_{tot}$.

## 4. Trajectory Construction: Flyby Implications for eVTOLs

#### 4.1. Minimum Distance between Waypoints

- Aircraft speed ${V}_{T}$: The slower the aircraft, the less distance we need to turn.
- Desired turn rate ${\dot{\chi}}_{{T}_{d}}$: The higher the turn rate, the less distance we need to turn.

**Theorem**

**1.**

#### 4.2. Maximum Angle ${\alpha}_{T}$ between Legs

**Theorem**

**2.**

#### 4.2.1. Making the Turn Rate Dynamic

**Figure 11.**Typical arctan argument for eVTOL aircraft (see Equation (63)).

**Proof.**

#### 4.2.2. Integrating the Dynamic Turn Rate

## 5. Consolidation and Results

#### 5.1. Turn Distance

- Roll rate command: ${p}_{cmd}={30}^{\circ}$
- Roll time constant: ${T}_{p}=0.5$ s
- Design turn rate: ${\dot{\chi}}_{{T}_{d}}={10}^{\circ}/\mathrm{s}$

#### 5.2. Maximum Angle between Legs

#### 5.3. Simulation Results

#### 5.3.1. Testing Environment Overview

^{®}following strict guidelines, allowing us to obtain a code-compliant software according to the DO-178/DO-331 [33]. The clothoid-based lateral implementation of the trajectory generation system has been successfully flight tested on a fixed wing aircraft [13].

#### 5.3.2. Point Mass Model

- The prime vertical radius of curvature, ${N}_{\phi}$ is:$${N}_{\phi}=\frac{a}{1-{e}^{2}{sin}^{2}\phi}$$
- The meridian radius of curvature, ${M}_{\phi}$ is:$${M}_{\phi}={N}_{\phi}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{1-{e}^{2}}{1-{e}^{2}si{n}^{2}\phi}$$

- $\phi $ is the latitude;
- $a=6,378,137\phantom{\rule{0.277778em}{0ex}}$m is the length of the semi-major axis;
- $b=6,356,752.3142\phantom{\rule{0.277778em}{0ex}}$m is the length of the semi-minor axis;
- e is the first eccentricity, calculated as:$${e}^{2}=2f-{f}^{2}$$$$f=\frac{a-b}{a}$$

#### 5.3.3. Results

**Figure 30.**Horizontal plane commands during maneuver of Figure 29.

**Figure 31.**Vertical plane commands during maneuver of Figure 29.

## 6. Conclusions

^{®}R2021a, together with a point mass model assuming perfect tracking of the commands generated by the TrajGen module. The results of our analysis are shown in the paper, along with simulation results.

## 7. Recommendations for Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

eVTOL | Electric Vertical Take-Off and Landing |

RRT | Rapidly exploring Random Tree |

BLP | Bi-Level Programming |

UAV | Unmanned Aerial Vehicle |

TrajGen | Trajectory Generation |

Symbol | |

A | Clothoid shaping parameter |

${a}_{m}$ | Coefficient of a polynomial |

$\overrightarrow{c}$ | Curve in a 2D Euclidean space |

${b}_{0}$ | Line coefficient for approximation of atan |

${c}_{x}$, ${c}_{y}$ | x- and y- components of a curve $\overrightarrow{c}$ |

$cl{p}_{n}$ | Clothoid point n |

subscript${}_{cmd}$ | Command |

${d}_{ap}$ | Distance from the TO waypoint to the auxiliary point |

${d}_{cc}$ | Distance from the TO waypoint to the center of the maneuver circle |

${d}_{turn}$ | Turn distance |

subscript${}_{d}$ | Desired |

${g}_{0}$ | gravity acceleration |

${h}_{T}^{F}$ | Altitude of a reference point F |

${M}_{\phi}$ | Meridian radius of curvature |

${N}_{\phi}$ | Prime vertical radius of curvature |

p | Roll rate |

${r}_{c}$ | Radius of a maneuver circle |

s | Arc length of a curve $\overrightarrow{c}$ |

${s}_{hor}$ | Distance traveled in the horizontal plane |

$\overrightarrow{t}$ | tangent vector |

$\overrightarrow{T}$ | Unit tangent vector |

${T}_{p}$ | Roll time constant |

${V}_{p}$ | Planning speed |

${V}_{buffer}$ | Buffer speed used for in the planning speed |

${V}_{T}$ | Kinematic speed |

$\Delta {x}_{cl}$, $\Delta {y}_{cl}$ | Change in position of induced by a clothoid segment |

$\mu $ | Kinematic bank angle command |

${\dot{\chi}}_{T}$ | Turn rate |

$\Delta {\chi}_{T}$ | Change in course |

${\alpha}_{T}$ | Angle between two legs in the flight plan |

${\gamma}_{T}^{F}$ | Climb angle of a reference point F |

$\omega $ | Bandwidth |

$\tau $ | Running parameter/dimensionless time of a curve $\overrightarrow{c}$ |

${\chi}_{T}^{R}$ | Kinematic course angle of a reference point R |

$\lambda $ | Longitude |

$\kappa $ | Curvature |

$\phi $ | Angle representing change in direction over an arc length s |

## References

- Goerzen, C.; Kong, Z.; Mettler, B. A survey of motion planning algorithms from the perspective of autonomous UAV guidance. J. Intell. Robot. Syst.
**2010**, 57, 65–100. [Google Scholar] [CrossRef] - Karaman, S.; Frazzoli, E. Sampling-based algorithms for optimal motion planning. Int. J. Robot. Res.
**2011**, 30, 846–894. [Google Scholar] [CrossRef] - Devaurs, D.; Siméon, T.; Cortés, J. Optimal path planning in complex cost spaces with sampling-based algorithms. IEEE Trans. Autom. Sci. Eng.
**2015**, 13, 415–424. [Google Scholar] [CrossRef] - LaValle, S.M.; James J Kuffner, J. Randomized Kinodynamic Planning. SageJournals
**2001**, 20, 378–400. [Google Scholar] [CrossRef] - Zhao, Y.; Zheng, Y.L. Survey on computational-intelligence-based UAV path planning. Knowl.-Based Syst.
**2018**, 158, 54–64. [Google Scholar] [CrossRef] - Qu, C.; Gai, W.; Zhong, M.; Zhang, J. A novel reinforcement learning based grey wolf optimizer algorithm for unmanned aerial vehicles (UAVs) path planning. Appl. Soft Comput.
**2020**, 89, 106099. [Google Scholar] [CrossRef] - Satai, H.A.; Zahra, M.M.A.; Rasool, Z.I.; Abd-Ali, R.S.; Pruncu, C.I. Bézier Curves-Based Optimal Trajectory Design for Multirotor UAVs with Any-Angle Pathfinding Algorithms. Sensors
**2021**, 21, 2460. [Google Scholar] [CrossRef] [PubMed] - Liu, W.; Zheng, Z.; Cai, K.Y. Bi-level programming based real-time path planning for unmanned aerial vehicles. Knowl.-Based Syst.
**2013**, 44, 34–47. [Google Scholar] [CrossRef] - Bousson, K.; Machado, P.F. 4D trajectory generation and tracking for waypoint-based aerial navigation. WSEAS Trans. Syst. Control
**2013**, 3, 105–119. [Google Scholar] - Hentschel, M.; Lecking, D.; Wagner, B. Deterministic path planning and navigation for an autonomous fork lift truck. IFAC Proc. Vol.
**2007**, 40, 102–107. [Google Scholar] [CrossRef] - Schneider, V.; Piprek, P.; Schatz, S.P.; Baier, T.; Dörhöfer, C.; Hochstrasser, M.; Gabrys, A.; Karlsson, E.; Krause, C.; Lauffs, P.J.; et al. Online trajectory generation using clothoid segments. In Proceedings of the 2016 14th International Conference on Control, Automation, Robotics and Vision (ICARCV), Phuket, Thailand, 13–15 November 2016; pp. 1–6. [Google Scholar]
- Schneider, V.; Mumm, N.C.; Holzapfel, F. Trajectory generation for an integrated mission management system. In Proceedings of the 2015 IEEE International Conference on Aerospace Electronics and Remote Sensing Technology (ICARES), Bali, Indonesia, 3–5 December 2015; pp. 1–7. [Google Scholar]
- Schatz, S.P.; Schneider, V.; Karlsson, E.; Holzapfel, F.; Baier, T.; Dörhöfer, C.; Hochstrasser, M.; Gabrys, A.; Krause, C.; Lauffs, P.J.; et al. Flightplan flight tests of an experimental DA42 general aviation aircraft. In Proceedings of the 2016 14th International Conference on Control, Automation, Robotics and Vision (ICARCV), Phuket, Thailand, 13–15 November 2016; pp. 1–6. [Google Scholar]
- Dubins, L.E. On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math.
**1957**, 79, 497–516. [Google Scholar] [CrossRef] - Schneider, V. Trajectory Generation for Integrated Flight Guidance. Ph.D. Thesis, Technische Universität München, Munich, Germany, 2018. [Google Scholar]
- Casey, J. Exploring Curvature; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Walton, D.J.; Meek, D.S. A controlled clothoid spline. Comput. Graph.
**2005**, 29, 353–363. [Google Scholar] [CrossRef] - Levien, R. The Euler Spiral: A Mathematical History; Technical Report No. UCB/EECS-2008-111; University of California: Berkeley, CA, USA, 2008. [Google Scholar]
- Sendra, J.R.; Winkler, F. Symbolic parametrization of curves. J. Symb. Comput.
**1991**, 12, 607–631. [Google Scholar] [CrossRef] - Peterson, J.W. Arc Length Parameterization of Spline Curves. J. Comput. Aided Des. 2006. Available online: http://www.saccade.com/writing/graphics/RE-PARAM.PDF (accessed on 10 December 2023).
- Morvan, J.M.; Morvan, J. Generalized Curvatures; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Alexander, D.C.; Koeberlein, G.M. Elementary Geometry for College Students; Cengage Learning: Singapore, 2014. [Google Scholar]
- Herman, E.; Strang, G. Calculus Volume 2; OpenStax: Houston, TX, USA, 2016. [Google Scholar]
- Farouki, R.T.; Pelosi, F.; Sampoli, M.L. Approximation of monotone clothoid segments by degree 7 Pythagorean–hodograph curves. J. Comput. Appl. Math.
**2021**, 382, 113110. [Google Scholar] [CrossRef] - Sandoval-Hernandez, M.; Vazquez-Leal, H.; Hernandez-Martinez, L.; Filobello-Nino, U.; Jimenez-Fernandez, V.; Herrera-May, A.; Castaneda-Sheissa, R.; Ambrosio-Lazaro, R.; Diaz-Arango, G. Approximation of Fresnel integrals with applications to diffraction problems. Math. Probl. Eng.
**2018**, 2018, 4031793. [Google Scholar] [CrossRef] - Abramson, J. Precalculus 2e; OpenStax: Houston, TX, USA, 2021. [Google Scholar]
- Euclid. Euclid’s Elements; 300 BC; Printed by Erhard Ratdolt; Green Lion Press: Santa Fe, Mexico, 1482. [Google Scholar]
- Heimsch, D.; Söpper, M.; Speckmaier, M.; Mbikayi, Z.; Kellringer, S.; Holzapfel, F. Development and Implementation of a Safety Gateway for a Medical Evacuation eVTOL Aircraft. In Proceedings of the 2024 AIAA AVIATION Forum, Las Vegas, NV, USA, 29 July–3 August 2024. [Google Scholar]
- McLain, T.; Beard, R.W.; Owen, M. Implementing dubins airplane paths on fixed-wing uavs. In Handbook of Unmanned Aerial Vehicles; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Lugo-Cárdenas, I.; Flores, G.; Salazar, S.; Lozano, R. Dubins path generation for a fixed wing UAV. In Proceedings of the 2014 International Conference on Unmanned Aircraft Systems (ICUAS), Orlando, FL, USA, 27–30 May 2014; pp. 339–346. [Google Scholar]
- Sabetghadam, B.; Cunha, R.; Pascoal, A. Real-time trajectory generation for multiple drones using bézier curves. IFAC-PapersOnLine
**2020**, 53, 9276–9281. [Google Scholar] [CrossRef] - Petit, P.J.; Wartmann, J.; Fragnière, B.; Greiser, S. Waypoint based online trajectory generation and following control for the ACT/FHS. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7–11 January 2019; p. 0918. [Google Scholar]
- Dmitriev, K.; Zafar, S.A.; Schmiechen, K.; Lai, Y.; Saleab, M.; Nagarajan, P.; Dollinger, D.; Hochstrasser, M.; Holzapfel, F.; Myschik, S. A lean and highly-automated model-based software development process based on do-178c/do-331. In Proceedings of the 2020 AIAA/IEEE 39th Digital Avionics Systems Conference (DASC), San Antonio, TX, USA, 11–15 October 2020; pp. 1–10. [Google Scholar]
- Piprek, P.; Schneider, V.; Fafard, V.; Schatz, S.P.; Dörhöfer, C.; Lauffs, P.J.; Peter, L.; Holzapfel, F. Enhanced kinematics calculation for an online trajectory generation module. Transp. Res. Procedia
**2018**, 29, 312–322. [Google Scholar] [CrossRef] - Vanicek, P.; Krakiwsky, E.J. Geodesy: The Concepts; Elsevier: Amsterdam, The Netherlands, 2015. [Google Scholar]
- Scherer, S.; Speckmaier, M.; Gierszewski, D.; Mishra, C.; Steinert, A.C.; Steffensen, R.; Wulf, S.; Holzapfel, F. AUTOMATIC TAKE-OFF AND LANDING OF A VERY LIGHT ALL ELECTRIC OPTIONALLY PILOTED AIRCRAFT. In Proceedings of the 33rd Congress of the International Council of the Aeronautical Sciences (ICAS), Stockholm, Sweden, 4–9 September 2022. [Google Scholar]
- Scherer, S.; Mishra, C.; Holzapfel, F. Extension of the capabilities of an automatic landing system with procedures motivated by visual-flight-rules. In Proceedings of the 33rd Congress of the International Council of the Aeronautical Sciences (ICAS), Stockholm, Sweden, 4–9 September 2022. [Google Scholar]
- Labakhua, L.; Nunes, U.; Rodrigues, R.; Leite, F.S. Smooth trajectory planning for fully automated passengers vehicles: Spline and clothoid based methods and its simulation. In Proceedings of the Informatics in Control Automation and Robotics: Selected Papers from the International Conference on Informatics in Control Automation and Robotics 2006; Springer: Berlin/Heidelberg, Germany, 2008; pp. 169–182. [Google Scholar]
- Aeronautical Radio, Inc. ARINC 424-20 Navigation System Database; Aeronautical Radio, Inc.: Annapolis, ML, USA, 2011. [Google Scholar]

**Table 1.**Conditions for altitude polynomial [15].

Condition Parameter | Condition for $\mathit{s}=0$ | Condition for $\mathit{s}={\mathit{s}}_{\mathit{tot},\mathit{hor}}$ |
---|---|---|

${h}_{T}^{F}$ | ${h}_{T}^{{P}_{1}}$ | ${h}_{T}^{{P}_{2}}$ |

${\left({h}_{T}^{F}\right)}^{\prime}$ | $tan{\gamma}_{T}^{{P}_{1}}$ | $tan{\gamma}_{T}^{{P}_{2}}$ |

${\left({h}_{T}^{F}\right)}^{\prime \prime}$ | 0 | 0 |

${\left({h}_{T}^{F}\right)}^{\prime \prime \prime}$ | 0 | 0 |

${\left({h}_{T}^{F}\right)}^{\prime \prime \prime \prime}$ | 0 | 0 |

**Table 2.**Coefficients of the altitude polynomial of Equation (56).

Coefficient | Expression |
---|---|

${a}_{0}$ | ${h}_{\mathrm{p}1}$ |

${a}_{1}$ | ${h}_{\mathrm{p}2}$ |

${a}_{2}$ | 0 |

${a}_{3}$ | 0 |

${a}_{4}$ | 0 |

${a}_{5}$ | $-\frac{14\phantom{\rule{0.166667em}{0ex}}\left(9\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}1}-9\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}2}+5\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}1}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}+4\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}2}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}\right)}{{{s}_{\mathrm{tot}}}^{5}}$ |

${a}_{6}$ | $\frac{28\phantom{\rule{0.166667em}{0ex}}\left(15\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}1}-15\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}2}+8\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}1}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}+7\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}2}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}\right)}{{{s}_{\mathrm{tot}}}^{6}}$ |

${a}_{7}$ | $-\frac{20\phantom{\rule{0.166667em}{0ex}}\left(27\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}1}-27\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}2}+14\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}1}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}+13\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}2}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}\right)}{{{s}_{\mathrm{tot}}}^{7}}$ |

${a}_{8}$ | $\frac{5\phantom{\rule{0.166667em}{0ex}}\left(63\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}1}-63\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}2}+32\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}1}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}+31\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{b}2}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}\right)}{{{s}_{\mathrm{tot}}}^{8}}$ |

${a}_{9}$ | $-\frac{35\phantom{\rule{0.166667em}{0ex}}\left(2\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}1}-2\phantom{\rule{0.166667em}{0ex}}{h}_{\mathrm{p}2}+{h}_{\mathrm{b}1}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}+{h}_{\mathrm{b}2}\phantom{\rule{0.166667em}{0ex}}{s}_{\mathrm{tot}}\right)}{{{s}_{\mathrm{tot}}}^{9}}$ |

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## Share and Cite

**MDPI and ACS Style**

Mbikayi, Z.; Steinert, A.; Heimsch, D.; Speckmaier, M.; Rudolph, P.; Liu, H.; Holzapfel, F.
Online Deterministic 3D Trajectory Generation for Electric Vertical Take-Off and Landing Aircraft. *Aerospace* **2024**, *11*, 157.
https://doi.org/10.3390/aerospace11020157

**AMA Style**

Mbikayi Z, Steinert A, Heimsch D, Speckmaier M, Rudolph P, Liu H, Holzapfel F.
Online Deterministic 3D Trajectory Generation for Electric Vertical Take-Off and Landing Aircraft. *Aerospace*. 2024; 11(2):157.
https://doi.org/10.3390/aerospace11020157

**Chicago/Turabian Style**

Mbikayi, Zoe, Agnes Steinert, Dominik Heimsch, Moritz Speckmaier, Philippe Rudolph, Hugh Liu, and Florian Holzapfel.
2024. "Online Deterministic 3D Trajectory Generation for Electric Vertical Take-Off and Landing Aircraft" *Aerospace* 11, no. 2: 157.
https://doi.org/10.3390/aerospace11020157