# The use of Rodrigues-Hamilton parameters in mathematical model of cargo landing system with dampers in tasks of its overturn

DOI: 10.34759/trd-2022-126-07

### Аuthors

Averyanov I. O.1*, Zinin A. V.2**

1. Moscow design industrial complex "MKPK "Universal", 79A, Altufevskoe shosse, Moscow, 127410, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: i.averyanov@mail.ru
**e-mail: allzin@yandex.ru

### Abstract

In the previous article [1] we described the mathematical model of dynamic landing process for cargo dropped systems with dampers (System) that we developed for statistical modelling of the process. That dynamic model consisted of spatial equations of forces and moments equilibrium, kinematic Euler’s equations and contained contact boundaries from rigid plane to dampers and to cargo. The method of forth integration was used to solve the equations and the idea was that such modeling of landing process had allowed us to detect all the failures well known for such Systems. One of the possible System failure that may occur during landing is its overturn. In this case we have a deal with large angles that may lead to problems with kinematic Euler’s equations. Researching of sources shows that the problem of System overturn usually considers separately with analytical approaches, while the method we used implies continuity of the solution from the beginning of the process to its end.

To avoid such problems in this article we consider the use of Rodrigues-Hamilton parameters (or quaternion) instead of kinematic Euler’s equations to solve the dynamic task of complex System moving. We use the fourth order of Runge-Kutta method to realize the algorithm of Rodrigues-Hamilton parameters. To convert the quaternion to spatial angles Krilov’s equations are used.

To demonstrate accuracy and stability of the developed algorithm the task of complex rigid body spatial free motion is considered. Comparison of these results with the solution that comes from commonly used CAE shows us their similarity. After that, we also consider the task of System landing with its overturn. These results are considered from qualitative analysis point of view.

Thus, the use of algorithm we realized with Rodrigues-Hamilton parameters instead of kinematic Euler’s equation in the mathematical model of System landing process allows us to avoid «special points» and to generalize the solution to the tasks of large spatial angles, including System overturn.

### Keywords:

soft landing system, air damper, dropped cargo landing, dependability model of landing process, statistics modelling tasks

### References

1. Averyanov I.O., Zinin A.V. Trudy MAI, 2022, no. 124. URL: http://trudymai.ru/eng/published.php?ID=167067. DOI: 34759/trd-2022-124-12
2. Tryamkin A.V., Emel’yanov Yu.N. Trudy MAI, 2000, no. 1. URL: http://trudymai.ru/eng/published.php?ID=34731
3. Tryamkin A.V., Skidanov S.N. Trudy MAI, 2001, no. 3. URL: https://trudymai.ru/eng/published.php?ID=34686
4. Averyanov I.O. Trudy MAI, 2020, no. 115. URL: http://trudymai.ru/eng/published.php?ID=119896. DOI: 34759/trd-2020-115-03
5. Averyanov I.O., Suleimanov T.S., Tarakanov P.V. Trudy MAI, 2017, no. 92. URL: http://trudymai.ru/eng/published.php?ID=77448
6. Ponomarev P.A., Skidanov S.N., Timokhin V.A. Trudy MAI, 2000, no. 2. URL: http://trudymai.ru/eng/published.php?ID=34708
7. Ponomarev P.A. Issledovanie i vybor ratsional’nykh parametrov pnevmaticheskogo amortizatora dlya posadki distantsionno-pilotiruemykh letatel’nykh apparatov (Analysis and choice of rational parameters of pneumatic shock absorber for the landing aircrafts), Doctor’s thesis, Moscow, MAI, 2000, 145 p.
8. Titov V.A. Proektirovanie ratsional’nykh sistem penoplastovykh amortizatorov dlya ob"ektov desantirovaniya (Improvement of design efficiency for cargo dropped systems foam plastic dampers), Doctor’s thesis, Moscow, MIREA, 1989, 170 p.
9. Qu Pu, Yang Zhen, Shi Rui. Research on Airbags Landing System for Airborne Vehicle Airdrop, Journal of Information and Computational Science, 2015, vol. 12 (5), pp. 2035–2042. DOI: 12733/jics20105798.
10. Yves de Lassat de Pressigny, Vincent Lapoujade. Numerical simulation of ground impact after airdrop, 5th European LS-Dyna Users Conference, 2005. URL: https://www.dynalook.com/conferences/european-conf-2005
11. Yves de Lassat de Pressigny, Thierry Baylot. Simulation of the impact on ground of airdrop loads to define a standard worst case test, 6th European LS-Dyna Users Conference, URL: https://www.dynalook.com/conferences/european-conf-2007
12. Masoud Alizadeh, Ahmad Sedaghat, Ebrahim Kargar. Shape and Orifice Optimization of Airbag Systems for UAV Parachute Landing, International Journal of Aeronautical and Space Sciences, DOI 10.5139/IJASS 2014.15.3.335
13. Lebedev A.A., Chernobrovkin L.S. Dinamika poleta (Flight dynamics), Moscow, Mashinostroenie, 1973, 616 p.
14. Godunov S.K., Ryaben’kii V.S. Raznostnye skhemy (Finite- difference schemes), Moscow, Nauka, 1977, 440 p.
15. Amosov A.A., Dubinskii Yu.A., Kopchenova N.V. Vychislitel’nye metody dlya inzhenerov (Numerical methods for engineers), Moscow, Vysshaya shkola, 1994, 544 p.
16. Branec V.N., Shmiglevskiy I.P. Primenenie kvaternionov v zadachah orientacii tverdogo tela (The use of quaternions in the tasks of rigid body orientation), Moscow, Nauka, 1973, 320 p.
17. Biryukov V.G. Zadachi opredeleniya orientacii i upravleniya uglovim dvizheniem tverdogo tela (kosmicheskogo apparata) (Tasks of orientation determination and angular motion control for rigid body (spacecraft)), Doctor’s thesis, Saratov, IPTMU RAS, 2005, 151 p.
18. Zhidkova N.V., Volkov V.L. Sovremennye problemy nauki i obrazovaniya, 2015, no. 1. URL: https://www.elibrary.ru/item.asp?id=25323023
19. Volkov V.L., Zhidkova N.V. Nauchnoe obozrenie. Tekhnicheskie nauki, 2016, no. 4, pp. 5-19.
20. Dmitrochenko L.A., Sachkov G.P. Trudy MAI, 2015, no. 80. URL: https://trudymai.ru/eng/published.php?ID=56986
21. Chelnokov U.N. Kvaternionnie i bikvaternionnie modeli i metody mechaniki tverdogo tela i ih prilozheniya (Quaternion and biquaternion models and methods of rigid body mechanics and their applications), Moscow, FIZMATLIT, 2006, 512 p.
22. Golubev Yu.F. Preprinty IPM im. M.V. Keldysha, 2013, no. 39. pp. 1-23.