Geodesic
Control of an Inverted Wheeled Pendulum
Russian journal of nonlinear dynamics, Tome 16 (2020) no. 3, pp. 421-436.

Voir la notice de l'article provenant de la source Math-Net.Ru

The dynamics of an inverted wheeled pendulum controlled by a proportional plus integral plus derivative action controller in various cases is investigated. The properties of trajectories are studied for a pendulum stabilized on a horizontal line, an inclined straight line and on a soft horizontal line. Oscillation regions on phase portraits of dynamical systems are shown. In particular, an analysis is made of the stabilization of the pendulum on a soft surface, modeled by a differential inclusion. It is shown that there exist trajectories tending to a semistable equilibrium position in the adopted mathematical model. However, in numerical simulations, as well as in the case of real robotic devices, such trajectories turn into a limit cycle due to round-off errors and perturbations not taken into account in the model.
Keywords: pendulum, control, stability, differential inclusion. 
@article{ND_2020_16_3_a1,
     author = {O. M. Kiselev},
     title = {Control of an {Inverted} {Wheeled} {Pendulum}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {421--436},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2020_16_3_a1/}
}
TY  - JOUR
AU  - O. M. Kiselev
TI  - Control of an Inverted Wheeled Pendulum
JO  - Russian journal of nonlinear dynamics
PY  - 2020
SP  - 421
EP  - 436
VL  - 16
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2020_16_3_a1/
LA  - en
ID  - ND_2020_16_3_a1
ER  - 
%0 Journal Article
%A O. M. Kiselev
%T Control of an Inverted Wheeled Pendulum
%J Russian journal of nonlinear dynamics
%D 2020
%P 421-436
%V 16
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2020_16_3_a1/
%G en
%F ND_2020_16_3_a1
O. M. Kiselev. Control of an Inverted Wheeled Pendulum. Russian journal of nonlinear dynamics, Tome 16 (2020) no. 3, pp. 421-436. http://geodesic.mathdoc.fr/item/ND_2020_16_3_a1/

[1] Andrievsky, B. R. and Fradkov, A. L., Selected Chapters of Control Theory with Examples in MATLAB, Nauka, St. Petersburg, 1999, 475 pp. (Russian)

[2] Åström, K. J., Block D. J., and Spong M. W., The Reaction Wheel Pendulum, Morgan Claypool, San Rafael, Calif., 2007, 105 pp.

[3] Halkyard, C. R., Chan, R. P. M., and Stol, K. A., “Review of Modelling and Control of Two-Wheeled Robots”, Annu. Rev. Control, 37:1 (2013), 89–103 | DOI

[4] Formalskii, A. M., Stabilization and Motion Control of Unstable Objects, Stud. Math. Phys., 33, de Gruyter, Berlin, 2015, 239 pp. | MR

[5] Izv. Akad. Nauk. Mekh. Tverd. Tela, 2013, no. 1, 9–23 | DOI | MR

[6] Colombi, S., Grasser, F., D'Arrigo, A., and Rufer, A. C., “Joe: A Mobile, Inverted Pendulum”, IEEE Trans. Ind. Electron., 49:1 (2002), 107–114 | DOI | MR

[7] Pathak, K., Franch, J., and Agrawal, S. K., “Velocity and Position Control of a Wheeled Inverted Pendulum by Partial Feedback Linearization”, IEEE Trans. Robot., 21:3 (2005), 505–513 | DOI

[8] Francovský, P., Dominik, L., Gmiterko, A., Virgala, I., Kurylo, P., and Perminova, O., “Modeling of Two-Wheeled Self-Balancing Robot Driven by DC Gearmotors”, Int. J. Appl. Mech. Eng., 22:3 (2017), 739–747 | DOI

[9] Nasrallah, D. S., Michalska, L., and Angeles, J., “Controllability and Posture Control of a Wheeled Pendulum Moving on an Inclined Plane”, IEEE Trans. Robot., 23:3 (2007), 564-577 | DOI | MR

[10] Kunz, T., Teeyapan, K., Wang, J., and Stilman M., “Robot Limbo: Optimized Planning and Control for Dynamically Stable Robots under Vertical Obstacles”, IEEE Int. Conf. Robot. Autom. (Anchorage, Alaska, May 3–8, 2010), 4519–4524

[11] Stol, K., Kausar, Z., and Patel, N., “Performance Enhancement of a Statically Unstable Two Wheeled Mobile Robot Traversing on an Uneven Surface”, 2010 IEEE Conference on Robotics Automation and Mechatronics (RAM), 156–162

[12] Stol, K., Kausar, Z., and Patel, N., “The Effect of Terrain Inclination on Performance and the Stability Region of Two-Wheeled Mobile Robots”, Int. J. Adv. Robot. Syst., 9:5 (2012), 11 pp.

[13] Åström, K. J. and Hägglund, T., PID Controllers: Theory, Design, and Tuning, 2nd ed., 1994, 343 pp.

[14] Kiselev, O. M., Mathematical Foundations of Robotics, Kartush, Oryol, 2019, 228 pp.

[15] Shimada, A. and Hatakeyama, N., “Movement Control Using Zero Dynamics of Two-Wheeled Inverted Pendulum Robot”, 10th IEEE International Workshop on Advanced Motion Control (Trento, Italy, March 26–28, 2008), 38–43 | MR

[16] Ahmad, M. A., Nasir, A. N. K., and Raja Ismail, R. M.,T., “Performance Comparison between Sliding Mode Control (SMC) and PD-PID Controllers for a Nonlinear Inverted Pendulum System”, 11th Internat. Conf. on Control, Automation, Robotics and Vision (ICARCV 2010), 122–127

[17] Maddahi, A., Shamekhi, A. H., and Ghaffari, A., “A Lyapunov Controller for Self-Balancing Two-Wheeled Vehicles”, Robotica, 33:1 (2015), 225–239 | DOI

[18] Filippov, A. F., Differential Equations with Discontinuous Righthand Sides, Math. Appl., 18, Springer, Dordrecht, 1988, X, 304 pp. | MR