On one problem of quadcopter control with given intermediate values of different parts of coordinates

V. R. Barseghyan ; T. A. Simonyan ; A. G. Matevosyan

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 145, pp. 29-42

Résumé

The work is devoted to the issues of mathematical modeling of the spatial motion of a quadcopter and the construction of program control laws that ensure flight with the values of part of the coordinates of the phase vector specified at intermediate times. A structural diagram of a quadcopter with four propeller engines is used, which allows for movement in space, vertical takeoff and landing. Based on the laws of theoretical mechanics, a system of differential equations is obtained that describes the spatial motion of such a quadcopter. For a linearized mathematical model of quadcopter motion, the problem of constructing program control laws with given initial and final values of the phase vector, as well as the values of part of the coordinates of the phase vector at two intermediate moments of time, has been solved. A necessary and sufficient condition for the existence of program control is obtained and the corresponding movement of the quadcopter is described. Control functions and corresponding phase trajectories of motion are constructed. To illustrate the results obtained, for specific initial, final and intermediate values, explicit expressions of the program control function, program motion are obtained and the corresponding graphs are constructed.

Détail

Keywords: mathematical model of quadcopter motion, flight control, multi-point intermediate conditions, phase trajectories

Bibliographie

[1] D. V. Sitnikov, Y. A. Burian, G. S. Russkih, “Motion control system of multicopter”, Proceedings Of The Tula States University. Technical sciences, 2012, no. 7, 213–221 (In Russian)

[2] D. T. Rubin, V. N. Konev, A. V. Starikovsky, A. A. Sheptunov, A. S. Smirnov, A. M. Tolstaya, “Development of special quadcopters for survey works”, Special Machinery and Communications, 2012, no. 1, 28–30 (In Russian)

[3] M. I. Epov, I. N. Zlygostev, “Application of unmanned aerial vehicles in airborne geophysical reconnaissance”, Interekspo GEO-Sibir, 2:3 (2012), 22–27 (In Russian)

[4] S. V. Telukhin, V. V. Matveev, “Unmanned aerial vehicle as an object management”, Mekhatronika, Avtomatizatsiya, Upravlenie, 2008, no. 10, 7–10 (In Russian)

[5] T. Chettibi, M. Haddad, “Dynamic modelling of a quadrotor aerial robot”, Journees D’etudes Nationales de Mecanique, 2007, 22–27

[6] A. Mokhtari, A. Benallegue, “Dynamic feedback controller of Euler angles and wind parameters estimation for a quadrotor unmanned aerial vehicle”, IEEE International Conference on Robotics and Automation, Proceedings of the International Conference ICRA '04 (New Orleans, LA USA, 26 April 2004 – 01 May 2004), v. 3, IEEE, 2004, 2359–2366

[7] L. Derafa, T. Madani, A. Benallegue, “Dynamic modelling and experimental identification of four rotors helicopter parameters”, IEEE International Conference on Industrial Technology, Proceedings of the International Conference (Mumbai, India, 15–17 December 2006), v. Art. 4237837, IEEE, 2006, 1834–1839

[8] A. A. Margun, K. A. Zimenko, D. N. Bazylev, A. A. Bobtsov, A. S. Kremlev, D. D. Ibraev, “Control system for unmanned aircraft equipped with robotics arm”, Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2014, no. 6 (94), 54–62 (In Russian)

[9] T. Luukkonen, Modelling and control of quadcopter, 2011 https://sal.aalto.fi/publications/pdf-files/eluu11_public.pdf

[10] T. Puls, A. Hein, “3D trajectory control for quadrocopter”, IEEE International Conference on Intelligent Robots and Systems, Proceedings of the International Conference (Taipei, Taiwan, 18–22 October 2010), IEEE, 2010, 640–645

[11] Z. Benić, P. Piljek and D. Kotarski, “Mathematical modelling of unmanned aerial vehicles with four rotors”, Interdisciplinary Description of Complex Systems, 14:1 (2016), 88–100 | DOI

[12] L. T. Aschepkov, “Optimalnoe upravlenie sistemoi s promezhutochnymi usloviyami”, Prikladnaya matematika i mekhanika, 45:2 (1981), 215–222 | MR | Zbl

[13] V. A. Dykhta, O. N. Samsonyuk, “Printsip maksimuma dlya gladkikh zadach optimalnogo impulsnogo upravleniya s mnogotochechnymi fazoogranicheniyami”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki \year 2009, 49:6, 981–997 ; V. A. Dykhta, O. N. Samsonyuk, “A maximum principle for smooth optimal impulsive control problems with multipoint state constraints”, Computational Mathematics and Mathematical Physics, 49:6 (2009), 942–957 | MR | Zbl | DOI

[14] V. R. Barsegyan, “Controlling of linear dynamic systems with restrictions on values of parts of coordinates of a phase vector in the intermediate moments of time”, Reports NAS RA, 110:3 (2010), 251–260 (In Russian) | MR

[15] V. R. Barseghyan, T. V. Barseghyan, “On an approach to the problems of control of dynamic system with nonseparated multipoint intermediate conditions”, Automation and Remote Control, 76:4 (2015), 549–559 | DOI | MR | Zbl

[16] V. R. Barsegyan, Control of Composite Dynamic Systems and Systems with Multipoint Intermediate Conditions, Nauka Publ., Moscow, 2016, 230 pp. (In Russian)

[17] V. Barseghyan, S. Solodusha, “On one problem in optimal boundary control for string vibrations with a given velocity of points at an intermediate moment of time”, Proceedings of the International Conference, IEEE International Russian Automation Conference (RusAutoCon) (Sochi, Russian Federation, 05–11 September 2021), IEEE, 2021, 343–349 | MR

[18] N. N. Krasovskii, Theory of Control of Motion, Nauka Publ., Moscow, 1968, 476 pp. (In Russian)

[19] V. I. Zubov, Lectures on Control Theory, Nauka Publ., Moscow, 1975, 496 pp. (In Russian)