# Fundamental Framework to Plan 4D Robust Descent Trajectories for Uncertainties in Weather Prediction

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## Abstract

**:**

## 1. Introduction

## 2. Models for Robust Trajectory Planning

#### 2.1. Uncertainty Models of Weather Prediction

**Assumption**

**1**

**Assumption**

**2**

#### 2.2. Flight Performance Models

**Assumption**

**3**

**Assumption**

**4**

**Assumption**

**5**

**Assumption**

**6**

**Assumption**

**7**

**Assumption**

**8**

## 3. Robust Descent Trajectory Planning

#### 3.1. Deterministic Formalization as a Basis

**Constraint D1: Aircraft EoM**

**Constraint D2: Initial/final conditions**

**Constraint D3: Path constraints**

**Constraint D4: Phase-edge event conditions**

**Constraint D5: Phase-linkage conditions**

**Objective: Minimum operational costs**

#### 3.2. Robust Trajectory Planning

**Robust constraint R1: Aggregated aircraft EoM**

**Robust constraint R2: Commonality constraints**

**Robust constraint R3: Initial/final conditions**

**Robust constraint R4: Path constraints**

**Robust constraint R5: Phase-edge event conditions**

**Robust constraint R6: Phase-linkage conditions**

**Robust constraint R7: Required time of arrival**

**Robust objective: Expected value of the operational costs**

## 4. Case Studies

#### 4.1. Scenario Settings

#### 4.2. Robust vs. Scenario-Optimal Trajectories

#### 4.3. Robust vs. Inappropriately-Controlled Trajectories

#### 4.4. Effects of Cost-Index and RTA Variations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

**Family 4 Release 4.2**which has been made available by EUROCONTROL to TU DRESDEN. EUROCONTROL has all relevant rights to BADA. ©2021 The European Organisation for the Safety of Air Navigation (EUROCONTROL). All rights reserved. EUROCONTROL shall not be liable for any direct, indirect, incidental or consequential damages arising out of or in connection with this product or document, including with respect to the use of BADA.

## Appendix A. Three Dimensional B-Spline Function

## Appendix B. Derivation of Fuel Burn

## References

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**Figure 1.**GEFS consists of ${N}_{w}$ different ensemble members (${N}_{w}=21$ for this study). A different b-spline approximation surface (blue surface) is obtained for a different GEFS member, based on the corresponding GRIB2 data (red dots).

**Figure 3.**Overview of the constraints and symbols specific to phases. The constraints are categorized as phase-interior constraints and phase-edge constraints. Continuity of the trajectory is assumed for $\mathit{x}$ at phase edges.

**Figure 5.**The horizontal route for the simulation. The arrival route for the CDO operation (CDO NIGHT) of RWY08R at Leipzig/Halle Airport, Germany (original source in black: AIP Germany [51], modified for research purpose in purple). The dotted lines and the symbols $c,{d}_{1},\cdots ,{d}_{6}$ show the phase separation. The lengths of $c,{d}_{1}$ and ${d}_{2}$ phases do not necessarily reflect reality in this figure.

**Figure 6.**Lateral paths of the robust (red) and the scenario-optimal (blue) trajectories (with ${C}_{I}=30$ kg/min, RTA not imposed). The red dots correspond to the waypoints and the black dots show the ToD locations of the trajectories.

**Figure 7.**Trends of u and v along the robust trajectory for each weather scenario. The components are positive for west-to-east wind and south-to-north wind, respectively.

**Figure 8.**Vertical paths of the robust (red) and the scenario-optimal (blue) trajectories (with ${C}_{I}=30$ kg/min, RTA not imposed). The trajectories cover the last part of c-phase and the following descent phases. The vertical lines corresponds to phase edges of the robust trajectory.

**Figure 9.**Arrival time at GAMKO, fuel burn and operational costs of the robust trajectory for each weather scenario (${C}_{I}=30$ kg/min, RTA not imposed).

**Figure 10.**Total operational costs for each weather scenario (with ${C}_{I}=30$ kg/min, RTA not imposed). The box plots show the ranges of costs of the inappropriately-controlled trajectories for the corresponding weather scenarios, and their mean values are depicted with the green triangles. Each box plot is based on the costs of ${N}_{w}-1=20$ inappropriately-controlled trajectories. The red dots show the costs for the robust trajectory.

**Figure 11.**CAS profiles in the last 4 phases (with ${C}_{I}=20$ kg/min, RTA not imposed). The black profiles are inappropriately-controlled trajectories and the red are the robust trajectories. The vertical lines are the phase edges and the horizontal dashed red lines show the limitations for the corresponding phases. The red diamond at the end of ${d}_{6}$ phase depicts the final condition.

**Figure 12.**Trend of the average arrival time at GAMKO vs. the average fuel burn of the robust trajectories with ${C}_{I}=[10;80]$ kg/min. RTA is not imposed.

**Figure 13.**(

**Left**) Trend of the average arrival time at GAMKO vs. fuel burn: ±60 s from the average arrival time without RTA ${t}_{f}^{*}$, i.e., ${t}_{RTA}=[{t}_{f}^{*}-60;{t}_{f}^{*}+60]$ for ${C}_{I}=30$ kg/min. (

**Right**) Trend of the average arrival time at GAMKO vs. the robust operational costs ${J}_{robust}$ (Equation (37)) with the same RTA range.

Name | Coordinates | Altitude | Speed |
---|---|---|---|

MAXEB | N51 12.4 E012 13.9 | FL80 or above | IAS 250 kt |

DP808 | N51 19.3 E011 48.9 | 5500 ft or above | IAS 230 kt |

DP807 | N51 23.5 E011 48.4 | 5000 ft or above | IAS 210 kt |

DP442 | N51 23.8 E011 53.5 | - | - |

GAMKO (FAF) | N51 24.1 E011 59.7 | 3000 ft or above | IAS 180 kt |

States | Symbols | Conditions |
---|---|---|

Along-track distance | ${s}_{0}$ | 0 m |

Coordinates | $({\varphi}_{0},{\lambda}_{0})$ | (N49${}^{\xb0}$55${}^{\prime}$48.00${}^{\u2033}$, E021${}^{\xb0}$10${}^{\prime}$31.00${}^{\u2033}$) |

Altitude | ${h}_{0}$ | 35,000 ft |

Mass | ${m}_{0}$ | 63,700 kg |

Time | ${t}_{0}$ | 0 s |

Phase | Description | Path Constraints | Final Conditions |
---|---|---|---|

c | Cruise | ${M}_{min}\le M\le {M}_{MO}$ | |

$h={h}_{0}$, $\gamma =0$ ${}^{\xb0}$, ${\delta}_{SB}=0$ | |||

${d}_{1}$ | ToD to 10,000 ft | ${M}_{min}\le M\le {M}_{MO}$ | h = 10,000 ft |

${V}_{CAS}\le 350\phantom{\rule{3.33333pt}{0ex}}\mathrm{kt}$ | 230 kt $\le {V}_{CAS}\le $ 250 kt | ||

${d}_{2}$ | 10,000 ft to MAXEB | 230 kt $\le {V}_{CAS}\le $ 250 kt | $(\varphi ,\lambda )={(\varphi ,\lambda )}_{MAXEB}$ |

$h\ge $FL80, ${V}_{CAS}=$250 kt | |||

${d}_{3}$ | MAXEB to DP808 | 230 kt $\le {V}_{CAS}\le $ 250 kt | $(\varphi ,\lambda )={(\varphi ,\lambda )}_{DP808}$ |

$h\ge $5500 ft, ${V}_{CAS}=$230 kt | |||

${d}_{4}$ | DP808 to DP807 | 210 kt $\le {V}_{CAS}\le $ 230 kt | $(\varphi ,\lambda )={(\varphi ,\lambda )}_{DP807}$ |

$h\ge $ 5000 ft, ${V}_{CAS}=$ 210 kt | |||

${d}_{5}$ | DP807 to DP442 | 180 kt $\le {V}_{CAS}\le $ 210 kt | $(\varphi ,\lambda )={(\varphi ,\lambda )}_{DP442}$ |

$h\ge $ 3000 ft | |||

${d}_{6}$ | DP442 to GAMKO | 180 kt $\le {V}_{CAS}\le $ 210 kt | $(\varphi ,\lambda )={(\varphi ,\lambda )}_{GAMKO}$ |

$h=$ 3000 ft, ${V}_{CAS}=$ 180 kt |

Controls | Symbols | Lower Bounds | Upper Bounds |
---|---|---|---|

Thrust coefficient | ${C}_{T}$ | ${C}_{{T}_{idle}}$ | ${C}_{{T}_{MCRZ}}$ |

Flight path angle | $\gamma $ | −4${}^{\xb0}$ | 0${}^{\xb0}$ |

Heading | $\psi $ | −180 deg | 180 deg |

Speed brake | ${\delta}_{SB}$ | 0 | 1 |

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**MDPI and ACS Style**

Kamo, S.; Rosenow, J.; Fricke, H.; Soler, M.
Fundamental Framework to Plan 4D Robust Descent Trajectories for Uncertainties in Weather Prediction. *Aerospace* **2022**, *9*, 109.
https://doi.org/10.3390/aerospace9020109

**AMA Style**

Kamo S, Rosenow J, Fricke H, Soler M.
Fundamental Framework to Plan 4D Robust Descent Trajectories for Uncertainties in Weather Prediction. *Aerospace*. 2022; 9(2):109.
https://doi.org/10.3390/aerospace9020109

**Chicago/Turabian Style**

Kamo, Shumpei, Judith Rosenow, Hartmut Fricke, and Manuel Soler.
2022. "Fundamental Framework to Plan 4D Robust Descent Trajectories for Uncertainties in Weather Prediction" *Aerospace* 9, no. 2: 109.
https://doi.org/10.3390/aerospace9020109