A Hybrid PID-LQR Control Applied in Positioning Control of Robotic Manipulators Subject to Excitation from Non-ideal Sources

Abstract

This paper proposes the use of a hybrid controller that combines concepts of the Proportional-Integral-Derivative (PID) controller with the Linear-Quadratic-Regulator (LQR) and a Feedforward gain to control the positioning of a 2 DOF robotic arm with flexible joints. As the joints are flexible, there is in this system a non-ideal coupling between the links of the robotic arm, where the angular movement of one link can generate oscillations that spread and impact the response of the system. The non-ideal excitation source originates from the coupling between the electric motor used to move the link of the robotic arm with the flexible element of the structure. These mechanical oscillations interfere in the positioning of the motor, and thus in the electrical power consumed by the system. In the results section of this work, numerical simulations are used to show the functionality and performance of the proposed controller in the studied system.

Appendix

${\mathbf{Q}} = \left[ {\begin{array}{*{20}c} {10^{7} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {10^{7} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {10^{7} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {10^{7} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]$, ${\mathbf{B}} = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ {\frac{{k_{t} }}{J}} & 0 \\ 0 & 0 \\ 0 & {\frac{{k_{t} }}{J}} \\ \end{array} } \right]$,

$$ {\mathbf{A}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - kp_{11} } & \alpha & { - kp_{12} } & \beta & {kp_{11} } & {b_{s} p_{11} } & {kp_{12} } & {b_{s} p_{12} } \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ { - kp_{21} } & \gamma & { - kp_{22} } & \delta & {kp_{21} } & {b_{s} p_{21} } & {kp_{22} } & {b_{s} p_{22} } \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ {\frac{ - k}{J}} & {\frac{{b_{s} }}{J}} & 0 & 0 & {\frac{ - k}{J}} & {\frac{{ - \left( {b_{s} + b_{v} } \right)}}{J}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & {\frac{ - k}{J}} & {\frac{{b_{s} }}{J}} & 0 & 0 & {\frac{ - k}{J}} & {\frac{{ - \left( {b_{s} + b_{v} } \right)}}{J}} \\ \end{array} } \right] $$

${\mathbf{R}} = \left[ {\begin{array}{*{20}c} {10^{ - 2} } & 0 \\ 0 & {10^{ - 2} } \\ \end{array} } \right]$,

and

$$ {\mathbf{K}} = \left[ {\begin{array}{*{20}c} {21250.11} & {3008.71} & { - 21760.09} & { - 993.00} & {222386.01} & {1.50} & { - 2111.67} & {0.31} \\ {15721.43} & { - 925.88} & {289671.93} & {1925.28} & {16458.92} & {0.31} & {30071.57} & {1.70} \\ \end{array} } \right] $$