A gravitational search algorithm-based control of an underactuated system with experimental verifications

Abstract

Underactuated systems, such as the double inverted pendulum (DIP) on a cart, are widely used in various applications such as anthropomorphic robot design, assistive robotics, slosh modeling etc. However, the design of a control strategy for such systems is challenging due to their underactuated, nonlinear and unstable nature. This paper addresses the aforementioned challenges by introducing a linear–quadratic-regulator (LQR) controller assisted by the gravitational search algorithm (GSA) to stabilize the pendulums on the cart, while also allowing it to follow the desired movements and overcoming the unmodeled dynamics of the system. The GSA is used to improve the system performances while maintaining stability. The suitability of the proposed method is demonstrated through simulation and experimental studies, including cases with additional weights on the bob and on the cart and external disturbances. The improvements in system performances are shown through a comparative study with the performances of particle swarm optimization (PSO) and quantum PSO-based LQR controllers. The findings corroborate the efficacy of the proposed method in stabilizing the underactuated double inverted pendulum system on a cart in the nominal operating conditions and situations with additional weights and external disturbances.

Appendix A

System parameters of the considered system

$m_\textrm{p1}=0.072$ kg, $m_\textrm{p2}=0.127$ kg, $L_\textrm{p1}=0.2095$ m, $L_\textrm{p2}=0.3365$ m, $l_\textrm{p1}=0.1143$ m, $l_\textrm{p2}=0.1778$ m, $m_\textrm{c}=0.70313$ kg, $m_\textrm{h}=0.17$ kg, $\beta _\textrm{p1}=0.0024$ N m s/rad, $\beta _\textrm{p2}=0.0024$ N m s/rad, $\beta _\textrm{eq}=4.3$ N m s/rad, $K_g=3.71$, $K_t=0.00768$ N m/A, $K_\textrm{m}=0.00767$ V s/rad, $R_\textrm{m}=2.6$ $\Omega$, $r_\textrm{mp}=0.00635$ m, $\eta _g$=1, $\eta _\textrm{m}=1$, $g=9.8$ m/s$^2$.