# An Effort to Use a Solid Propellant Engine Arrangement in the Moon Soft Landing Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Powered Descend Control

#### 1.2. Research Proposal

## 2. Problem Definition

#### 2.1. Landing Module Description

#### 2.2. Dynamic Model

#### 2.3. Thrust Description

#### 2.3.1. PGCS-Regressive

#### 2.3.2. PGCS-Neutral

#### 2.3.3. PGCS-Progressive

#### 2.4. Boundary Conditions

#### 2.5. Uncertain Parameters

#### 2.5.1. Altitude and Velocity

#### 2.5.2. Specific Impulse with Bias and Noise

#### 2.5.3. Ignition Dead Time

## 3. Optimization Problem

GA are optimization methods, which operate on a population of points, designated as individuals. Each individual of the population represents a possible solution of the OP. Individuals are evaluated depending upon their fitness. The fitness indicates how well an individual of the population solves the OP.

#### 3.1. GA Functionality

#### 3.2. Design of Control Function

#### 3.3. Individual Parameters

#### 3.4. Individuals Cost Function

#### 3.4.1. Direct Cost Function

#### 3.4.2. Average Cost Function

## 4. Study Scenarios

#### 4.1. First Approach: Controller Optimization without Trajectory Uncertainties

#### 4.2. Second Approach: Controller Optimization with Trajectory Uncertainties

## 5. Simulation and Results

Parameters | Value |
---|---|

${y}_{0}$ | 2000 m |

${v}_{0}$ | 0 m/s |

${\sigma}_{alt}$ | 50 m |

${\sigma}_{vel}$ | 5 m/s |

${I}_{sp}$ | 300 s |

${\sigma}_{bias}$ | 10.83 s (10% at $3\sigma $) |

${\sigma}_{noise}$ | 3.25 s (3% at $3\sigma $) |

${m}_{0}$ | 24 kg |

g Acceleration of the moon | 1.67 [m/s^{2}] |

$\Delta t$ of simulation | 0.1 s |

Numerical method | Runge-Kutta 4th Order |

${\tau}_{d}$ | 0.2 s |

${\tau}_{l}$ | 0.5 s |

Parameters | Value |
---|---|

${N}_{e}$ | 10 |

${N}_{case}$ | 30 in trained |

${R}_{a},{R}_{b},{R}_{c}$ | $0.1,1.0,10.0$ |

${b}_{1},{b}_{2}$ | 1.0, 1.0 |

${\alpha}_{k}$ | $\in [0.0,1.0]$ |

${\gamma}_{k}$ | $\in [0.0,{\gamma}_{max}]$ |

${t}_{a,k}$ | $\in [1.0,20.0]$ s |

$\dot{m}$ | $\in [{\dot{m}}_{min},{\dot{m}}_{max}]$ |

Probability of mutation | 0.25 |

Number of individuals | 40 |

Number of generations | 300 |

$center$ | 0.7 |

#### 5.1. Optimization and Evaluation of First Approach

#### 5.2. Optimization and Evaluation of Second Approach

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations/Nomenclature

SPE | Solid propellant engine |

GA | Genetic algorithm |

PGCS | Propellant-grain cross-section |

DCF | Direct cost function |

ACF | Average cost function |

T | Total thrust |

${t}_{ig,k}$ | Ignition time of the k-th engine |

${t}_{k}^{*}$ | Current ignition time |

${\tau}_{d}$ | Delay time to reach the $10\%$ of the thrust level |

${\tau}_{l}$ | Lag time to reach the $99.99\%$ of the thrust level from the $10\%$ |

${t}_{a}$ | Action time from the initial $10\%$ of the thrust level and the drop to $10\%$ |

${t}_{d}$ | Dead time from the control signal to start the ignition |

${t}_{land}$ | Landing time |

y | Altitud |

v | Velocity |

m | Mass |

${\sigma}_{X}$ | Standard deviation from and X variable |

${I}_{sp}$ | Specific impulse |

${\beta}_{k}$ | Control function from the k-th engine |

$\mathcal{P}$ | Population of genetic algorithm |

$\mathcal{G}$ | Individual of genetic algorithm for tentative solution |

J | Direct cost function |

$\mathcal{J}$ | Average cost function |

${N}_{e}$ | Number of the engines in the arrangement |

## References

- Siddiqi, A.A. Beyond Earth: A Chronicle of Deep Space Exploration, 1958–2016, 2nd ed.; Number 2018-4041 in NASA SP; National Aeronautics and Space Administration, Office of Communications, NASA History Division: Washington, DC, USA, 2018.
- Kajon, D.; Masson, F.; Wagner, T.; Welberg, D. Development of an Attitude Control and Propellant Settling System for the aA5ME Upper Stage. In Proceedings of the Space Propulsion Conference, Cologne, Germany, 19–22 May 2014; p. 9. [Google Scholar]
- Morrisey, D.C. Historical perspective—Viking Mars Lander propulsion. J. Propuls. Power
**1992**, 8, 320–331. [Google Scholar] [CrossRef] - Soffen, G.A.; Snyder, C.W. The First Viking Mission to Mars. Science
**1976**, 193, 759–766. [Google Scholar] [CrossRef] [PubMed] - Maggi, F.; Bandera, A.; Galfetti, L.; De Luca, L.T.; Jackson, T.L. Efficient solid rocket propulsion for access to space. Acta Astronaut.
**2010**, 66, 1563–1573. [Google Scholar] [CrossRef] - Chavers, D.G. NASA Lander Technologies Project Status. In AIAA SPACE 2016; American Institute of Aeronautics and Astronautics: Long Beach, CA, USA, 2016. [Google Scholar] [CrossRef]
- Spear, A.J. Low cost approach to Mars Pathfinder and small landers. Acta Astronaut.
**1995**, 35, 345–354. [Google Scholar] [CrossRef] - Kalita, H.; Thangavelautham, J. Lunar CubeSat Lander to Explore Mare Tranquilitatis pit. In AIAA Scitech 2020 Forum; American Institute of Aeronautics and Astronautics: Orlando, FL, USA, 2020. [Google Scholar] [CrossRef]
- Grimm, C.D.; Witte, L.; Schröder, S.; Wickhusen, K. Size matters-The shell lander concept for exploring medium-size airless bodies. Acta Astronaut.
**2020**, 173, 91–110. [Google Scholar] [CrossRef] - Hashimoto, T.; Yamada, T.; Otsuki, M.; Yoshimitsu, T.; Tomiki, A.; Torii, W.; Toyota, H.; Kikuchi, J.; Morishita, N.; Kobayashi, Y.; et al. Nano Semihard Moon Lander: OMOTENASHI. IEEE Aerosp. Electron. Syst. Mag.
**2019**, 34, 20–30. [Google Scholar] [CrossRef] - Meditch, J. On the problem of optimal thrust programming for a lunar soft landing. IEEE Trans. Autom. Control
**1964**, 9, 477–484. [Google Scholar] [CrossRef] - Fleming, W.; Rishel, R. Deterministic and Stochastic Optimal Control; Springer: New York, NY, USA, 1975. [Google Scholar] [CrossRef]
- Acikmese, B.; Ploen, S.R. Convex Programming Approach to Powered Descent Guidance for Mars Landing. J. Guid. Control Dyn.
**2007**, 30, 1353–1366. [Google Scholar] [CrossRef] - Xu, B.; Sun, J.; Li, S.; Cao, T. Finite time sliding sector control for spacecraft atmospheric entry guidance. Acta Astronaut.
**2018**, 163, 108–113. [Google Scholar] [CrossRef] - Zhang, Y.; Guo, Y.; Ma, G.; Wie, B. Fixed-time pinpoint mars landing using two sliding-surface autonomous guidance. Acta Astronaut.
**2019**, 159, 547–563. [Google Scholar] [CrossRef] - Liu, X.L.; Duan, G.R.; Teo, K.L. Optimal soft landing control for moon lander. Automatica
**2008**, 44, 1097–1103. [Google Scholar] [CrossRef] - Zhang, H.; Li, J.; Wang, Z.; Guan, Y. Guidance Navigation and Control for Chang’E-5 Powered Descent. Space Sci. Technol.
**2021**, 2021, 1–15. [Google Scholar] [CrossRef] - Rom, H. Thrust Control of Hydrazine Rocket Motors by Means of Pulse Width Modulation. Acta Astronaut.
**1992**, 26, 313–316. [Google Scholar] [CrossRef] - Mishra, D.P. Fundamentals of Rocket Propulsion; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar] [CrossRef]
- Park, S.; Choi, S. A Study on the Characteristics of Solid Propellant Using a Defoamer. Propellants Explos. Pyrotech.
**2020**, 45, 486–492. [Google Scholar] [CrossRef] - Klumpp, A.R. Apollo lunar descent guidance. Automatica
**1974**, 10, 133–146. [Google Scholar] [CrossRef] - Marrison, C.; Stengel, R. Robust control system design using random search and genetic algorithms. IEEE Trans. Autom. Control
**1997**, 42, 835–839. [Google Scholar] [CrossRef] [Green Version] - Herreros, A.; Baeyens, E.; Perán, J.R. MRCD: A genetic algorithm for multiobjective robust control design. Eng. Appl. Artif. Intell.
**2002**, 15, 285–301. [Google Scholar] [CrossRef] - Gao, Y.; Wang, J.; Gao, S.; Cheng, Y. An Integrated Robust Design and Robust Control Strategy Using the Genetic Algorithm. IEEE Trans. Ind. Inform.
**2021**, 17, 8378–8386. [Google Scholar] [CrossRef] - Sekaj, I. Genetic Algorithm Based Controller Design. IFAC Proc. Vol.
**2003**, 36, 125–128. [Google Scholar] [CrossRef] - Jamanca Lino, G. Space Resources Engineering: Ilmenite Deposits for Oxygen Production on the Moon. Am. J. Min. Metall.
**2021**, 6, 6–11. [Google Scholar] [CrossRef] - Moore, J.; Calvert, D.; Frady, G.; Chavers, G.; Hull, P.; Lowery, E.; Farmer, J.; Trinh, H.; Rojdev, K.; Piatek, I.; et al. Resource Prospector Lander: Architecture and trade studies. In Proceedings of the 2015 IEEE Aerospace Conference, Big Sky, MT, USA, 7–14 March 2015. [Google Scholar]
- Sutton, G.P. Rocket Propulsion Elements, 9th ed.; John Wiley & Sons (US): Hoboken, NJ, USA, 2017; Google-Books-ID: GS9gswEACAAJ. [Google Scholar]
- Stein, S.D. Benefits of the Star Grain Configuration for a Sounding Rocket; United States Air Force Academy: Colorado Springs, CO, USA, 2008; p. 11. [Google Scholar]
- Rao, A. A Survey of Numerical Methods for Optimal Control. Adv. Astronaut. Sci.
**2010**, 135, 1–32. [Google Scholar] - Jamshidi, M.; Krohling, R.A.; Coelho, L.d.S.; Fleming, P.J. Robust Control Systems with Genetic Algorithms; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar] [CrossRef]

**Figure 1.**Example of 12U landers with engine arrangements. From left to right, 4 examples are shown: For a configuration of one, two, nine and sixteen engines.

**Figure 2.**Example of a lander with 10 thrusters in the engine array in a one-dimensional dynamic model. The inertial reference frame is fixed on the lunar surface, and is perpendicular and positive to it. This configuration frees up space equivalent to 4U for electronic components.

**Figure 7.**Representation of the evolutionary process of the GA, where the landing simulation is used to create a new generation of population $\mathcal{P}$.

**Figure 8.**Example of activation of each engine. When the control function $\beta $ is less than zero, crossing the red line, the engine turns on at the instant ${t}_{ig,k}$ of the simulation. The ignition delay seen in the figure is due to the dead time of the propellant.

**Figure 11.**Behavior of altitude and velocity in the evaluation of the best individual $\mathcal{G}$ obtained from the optimization without uncertainties in training. The SPEs array has ${N}_{e}=9$ and Regressive PGCS. Color map is used to differentiate different random trajectories.

**Figure 12.**Distribution of altitude and velocity landing in the evaluation of the best result obtained from the optimization without uncertainties in training. The SPEs array has ${N}_{e}=9$ and Regressive PGCS. The vertical axis shows the frequency of the values.

**Figure 13.**First approach: performance comparison for an array between 1 and 10 configuration engines, and for the three types of PGCS: Regressive, Neutral and Progressive.

**Figure 14.**Behavior of altitude and velocity in the evaluation of the best $\mathcal{G}$ obtained from the optimization with uncertainties and ${N}_{e}=10$. Color map is used to differentiate different random trajectories.

**Figure 15.**Distribution of altitude and velocity landing in the evaluation of the best result obtained from the optimization with uncertainties and ${N}_{e}=10$. The vertical axis shows the frequency of the values.

**Figure 16.**Second approach: Performance comparison for an array between 1 and 10 configuration engines, and for the three types of PGCS: Regressive, Neutral and Progressive.

**Figure 17.**Behavior of altitude and velocity in the evaluation of the best $\mathcal{G}$ obtained from the optimization with uncertainties and ${N}_{e}=16$. Color map is used to differentiate different random trajectories.

**Figure 18.**Distribution of altitude and velocity landing in the evaluation of the best result obtained from the optimization with uncertainties and ${N}_{e}=16$. The vertical axis shows the frequency of the values.

Popular Types of PGCS | Advantages | Disadvantages |
---|---|---|

EndBurn (Neutral) | - Long action time | - Low thrust |

- Stable and neutral thrust | - The chamber wall must be thick (greater weight) due to direct and long exposure of high-pressure and high-temperature combustion gases | |

Star (Neutral, Progressive, and Regressive) | - The shape of the star can be modified depending on the thrust requirements | - Short action time |

- The shape is more difficult to manufacture compared to the others | ||

BATES (Progressive) | - High final thrust | - Very low thrust at the start of ignition |

- Low weight | - Short action time |

gf | Confidence Interval | Percentage within the Distribution |
---|---|---|

1 | +1 s | 68.27% |

2 | +2 s | 95.45% |

3 | +3 s | 99.73% |

Genes | Random Method |
---|---|

$\dot{m}$ | $\mathrm{Unif}({\dot{m}}_{min},\phantom{\rule{4pt}{0ex}}{\dot{m}}_{max})$ |

${t}_{a}$ | $\mathrm{Unif}(0.0,\phantom{\rule{4pt}{0ex}}{\left({t}_{a}\right)}_{max})$ |

${\alpha}_{k}$ | $\mathrm{Triang}(left={\left({\alpha}_{k}\right)}_{min},right={\left({\alpha}_{k}\right)}_{max},\phantom{\rule{4pt}{0ex}}center)$ |

${\gamma}_{k}$ | $\mathrm{Triang}(left={\left({\gamma}_{k}\right)}_{min},right={\left({\gamma}_{k}\right)}_{max},\phantom{\rule{4pt}{0ex}}center)$ |

**Table 6.**Design parameters of the best result obtained from the optimization scenario without uncertainties.

Parameters | Value |
---|---|

PGCS | Regressive |

${N}_{e}$ | 9 |

$\dot{m}$ | 14.15 × 10${}^{-3}$ kg/s |

${t}_{a}$ | $15.75$ s |

${v}_{land}$ | $-5.87$ m/s |

Std Dev. of ${v}_{land}$ | $2.49$ m/s |

Mass used | $1.51$ kg |

**Table 7.**Control parameters from the best result obtained in the scenario without optimization uncertainties. The SPEs array has ${N}_{e}=9$ and Regressive PGCS.

k | ${\mathsf{\alpha}}_{\mathit{k}}$ | γ_{k} |
---|---|---|

1 | 0.511 | 3.632 |

2 | 0.992 | 4.393 |

3 | 0.300 | 10.071 |

4 | 0.755 | 19.199 |

5 | 0.531 | 20.431 |

6 | 0.795 | 11.292 |

7 | 0.560 | 17.099 |

8 | 0.323 | 8.667 |

9 | 0.729 | 16.382 |

**Table 8.**Design parameters of the best result obtained from the optimization scenario with uncertainties.

Parameters | Value |
---|---|

PGCS | Regressive |

${N}_{e}$ | 10 |

$\dot{m}$ | 10.16 × 10${}^{-3}$ kg/s |

${t}_{a}$ | $20.0$ s |

${v}_{land}$ | $-2.97$ m/s |

Std Dev. of ${v}_{land}$ | $0.99$ m/s |

Mass used | $1.35$ kg |

**Table 9.**Control parameters from the best result obtained in the scenario with optimization uncertainties.

k | ${\mathsf{\alpha}}_{\mathit{k}}$ | γ_{k} |
---|---|---|

1 | 0.639 | 12.111 |

2 | 0.138 | 11.165 |

3 | 0.862 | 4.742 |

4 | 0.551 | 12.844 |

5 | 0.999 | 11.465 |

6 | 0.091 | 6.036 |

7 | 0.768 | 22.285 |

8 | 0.608 | 5.219 |

9 | 0.624 | 17.851 |

10 | 0.867 | 24.439 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Obreque, E.; Díaz, M.
An Effort to Use a Solid Propellant Engine Arrangement in the Moon Soft Landing Problem. *Aerospace* **2022**, *9*, 540.
https://doi.org/10.3390/aerospace9100540

**AMA Style**

Obreque E, Díaz M.
An Effort to Use a Solid Propellant Engine Arrangement in the Moon Soft Landing Problem. *Aerospace*. 2022; 9(10):540.
https://doi.org/10.3390/aerospace9100540

**Chicago/Turabian Style**

Obreque, Elías, and Marcos Díaz.
2022. "An Effort to Use a Solid Propellant Engine Arrangement in the Moon Soft Landing Problem" *Aerospace* 9, no. 10: 540.
https://doi.org/10.3390/aerospace9100540