Two Ways to Control a Pendulum-Type Spherical Robot on a Moving Platform in a Pursuit Problem

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

We consider the problem of controlling a spherical robot with a pendulum actuator rolling on a platform that is capable of moving translationally in the horizontal plane of absolute space. The spherical robot is subject to holonomic and nonholonomic constraints. Some point target moves at the level of the geometric center of the spherical robot and does not touch the moving platform itself. The motion program that allows the spherical robot to pursue a target is specified through two servo-constraints. The robot can follow a target from any position and with any initial conditions. Two ways to control this system in absolute space are proposed: by controlling the forced motion of the platform (the pendulum oscillates freely) and by controlling the torque of the pendulum (the platform is stationary or oscillates inconsistently with the spherical robot). The equations of motion of the system are constructed. In the case of free oscillations of the pendulum, the system of equations of motion has first integrals and, if necessary, can be reduced to a fixed level of these integrals. When a spherical robot moves in a straight line, for a system reduced to the level of integrals, phase curves, graphs of the distance from the geometric center of the spherical robot to the target, the trajectory of the selected platform point when controlling the platform, and the square of the control torque when controlling the pendulum drive are constructed. When the robot moves along a curved path, integration is carried out in the original variables. Graphs of the squares of the angular velocity of the pendulum and the spherical robot itself are constructed, as well as the trajectory of the robot’s motion in absolute space and on a moving platform. Numerical experiments were performed in the Maple software package.

Full Text

Restricted Access

About the authors

E. A. Mikishanina

I.N. Ulianov Chuvash State University

Author for correspondence.
Email: evaeva_84@mail.ru
Russian Federation, Cheboksary, 428015

References

  1. S. A. Chaplygin, “On a ball’s rolling on a horizontal plane,” Mat. Sborn. 24 139–168 (1903); Reg. Chaotic Dyn. 7 (2), 131-148 (2002). https://doi.org/10.1070/RD2002v007n02ABEH000200
  2. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “The problem of drift and recurrence for the rolling Chaplygin ball,” Regul. Chaotic Dyn. 18 (6), 832–859 (2013). https://doi.org/10.1134/S1560354713060166
  3. A.V. Borisov and E. A. Mikishanina, “Dynamics of the Chaplygin ball with variable parameters,” Rus. J. Nonlin. Dyn. 16 (3), 453–465 (2020). https://doi.org/10.20537/nd200304
  4. A. A. Kilin, “The dynamics of Chaplygin ball: The qualitative and computer analysis,” Regul. Chaotic Dyn. 6 (3), 291–306 (2002). https://doi.org/10.1070/RD2001v006n03ABEH000178
  5. D. A. Schneider, “Nonholonomic Euler–Poincarè equations and stability in Chaplygin’s sphere,” Dyn. Syst. 17 (2), 87–130 (2002). https://doi.org/10.1080/02681110110112852
  6. S. Bolotin, “The problem of optimal control of a Chaplygin ball by internal rotors,” Regul. Chaotic Dyn. 17 (6), 559–570 (2012). https://doi.org/10.1134/S156035471206007X
  7. A. V. Borisov, A. A. Kilin, and I. S. Mamaev, “How to control Chaplygin’s sphere using rotors,” Regul. Chaotic Dyn. 17 (3), 258–272 (2012). https://doi.org/10.1134/S1560354712030045
  8. T. B. Ivanova and E. N. Pivovarova, “Dynamics and control of a spherical robot with an axisymmetric pendulum actuator,” Rus. J. Nonlin. Dyn. 9 (3), 507–520 (2013). https://doi.org/10.20537/nd1303008
  9. S. Gajbhiye and R. N. Banavar, “Geometric modeling and local controllability of a spherical mobile robot actuated by an internal pendulum,” Int. J. Robust Nonlin. Control 26, 2436–2454 (2015). https://doi.org/10.1002/rnc.3457
  10. T. B. Ivanova, A. A. Kilin, and E. N. Pivovarova, “Controlled motion of a spherical robot with feedback. I,” J. Dyn. Control Syst. 24 (3), 497–510 (2018). https://doi.org/10.1007/s10883-017-9387-2
  11. V. A. Joshi, R. N. Banavar, and R. Hippalgaonkar, “Design and analysis of a spherical mobile robot,” Mech. Mach. Theory. 45 (2), 130–136 (2010). https://doi.org/10.1016/j.mechmachtheory.2009.04.003
  12. E. A. Mikishanina, “Motion control of a spherical robot with a pendulum actuator for pursuing a target,” Rus. J. Nonlin. Dyn. 18 (5), 899–913 (2022). https://doi.org/10.20537/nd221223
  13. T. Ylikorpi and J. Suomela, “Ball-shaped robots,” in Climbing and Walking Robots: Towards New Applications, Ed. by H. Zhang (InTechOpen, Vienna, 2007), pp. 235–256. https://doi.org/10.5772/5083
  14. A. G. Azizov, “Motion of controllable mechanical systems with servo-constraints,” J. Appl. Math. Mech. 54 (3), 302-308 (1990). https://doi.org/10.1016/0021-8928(90)90129-X
  15. V. I. Kirgetov, “The motion of controlled mechanical systems with prescribed constraints (servoconstraints),” J. Appl. Math. Mech. 31 (3), 465-477 (1967). https://doi.org/10.1016/0021-8928(67)90029-9
  16. R. Altmann and Heiland J. Simulation of multibody systems with servo constraints through optimal control,” Multibody Syst. Dyn. 40, 75–98 (2017). https://doi.org/10.1007/s11044-016-9558-z
  17. A. H. Bajodah, D. H. Hodges, and Y. H. Chen, “Inverse dynamics of servo-constraints based on the generalized inverse,” Nonlin. Dyn. 39 (1), 179–196 (2005). https://doi.org/10.1007/s11071-005-1925-x
  18. M. H. Bèghin, Ètude Thèorique des Compas Gyrostatiques Anschütz et Sperry (Impr. Nationale, Paris, 1931).
  19. Ya. V. Tatarinov, Equations of Classical Mechanics in Concise Forms (MGU, Moscow, 2005) [in Russian].
  20. V. V. Kozlov, “The dynamics of systems with servoconstraints. I,” Regul. Chaotic Dyn. 20 (3), 205–224 (2015). https://doi.org/10.1134/S1560354715030016
  21. P. Appell, Traité de Mécanique Rationnelle, Vol. 2: Dynamique des Systèmes. Mécanique Analytique (Gauthier-Villars, Paris, 1932; Gos. Izd. Fiz.-Mat. Lit., Moscow, 1960).
  22. E. A. Mikishanina, “Rolling motion dynamics of a spherical robot with a pendulum actuator controlled by the Bilimovich servo-constraint,” Theor. Math. Phys. 211, 679–691 (2022). https://doi.org/10.1134/S0040577922050087
  23. A.V. Borisov and I. S. Mamaev, “Two nonholonomic integrable problems traicing back to Chaplygin,” Regul. Chaotic Dyn. 17 (2), 191–198 (2012). https://doi.org/10.1134/S1560354712020074

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1. System design.

Download (61KB)
Indexing metadata
3. Fig. 2. A typical phase curve on the plane of the graph of functions and when controlling the pursuit of a target moving in a straight line at a constant speed. (a) Movement of a spherical robot with a non-zero initial velocity. (b) Movement of a spherical robot from a resting state.

Download (76KB)
Indexing metadata
4. Fig. 3. A typical phase curve on the plane of the graph of functions and when controlling the pursuit of a target moving in a straight line with periodic velocity. (a) Movement of a spherical robot with a non-zero initial velocity. (b) Movement of a spherical robot from a resting state.

Download (101KB)
Indexing metadata
5. Fig. 4. A typical phase curve on the plane of the graph of functions and when controlling the pursuit of a target moving in a straight line at a constant speed. (a) Movement of a spherical robot with a non-zero initial velocity. (b) Movement of a spherical robot from a resting state.

Download (139KB)
Indexing metadata
6. Fig. 5. Movement of a spherical robot from a resting state behind the target: (a) on a stationary platform when controlling the torque of the pendulum; (b) on an oscillating platform when controlling the torque of the pendulum; (c) when controlling the movement of the platform.

Download (152KB)
Indexing metadata
7. Fig. 6. Movement of a spherical robot with non-zero initial conditions behind the target: (a) on a stationary platform when controlling the torque of the pendulum; (b) on an oscillating platform when controlling the torque of the pendulum; (c) when controlling the movement of the platform.

Download (232KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences