Adaptive sliding mode control of $n$ flexible-joint robot manipulators in the presence of structured and unstructured uncertainties

Abstract

This study investigates a voltage-based adaptive sliding mode control (VB-ASMC) to tracking the position of an $n$ rigid-link flexible-joint (RLFJ) robot manipulator under the presence of uncertainties and external disturbances. First, the dynamic equations of the $n$-RLFJ robot manipulator have been divided into $n$ subsystems, and for each of them a voltage-based sliding mode control (VB-SMC) is designed simultaneously. The mathematical proof shows that the closed-loop system under VB-SMC has global asymptotic stability. Second, due to the use of the sign function in the VB-SMC structure, the occurrence of chattering is inevitable. Therefore, to overcome this problem, an adaptive estimator is designed to estimate the boundary of uncertainties. Since the adaptive estimator part in the VB-ASMC has only one law, the proposed control has a very low computational volume. The Lyapunov stability theorem shows that the controlled closed-loop system under the VB-ASMC has global asymptotic stability. Finally, extensive simulations on the single and 2-RLFJ robot manipulator and practical implementation on the single-RLFJ robot manipulator are presented to demonstrate the effectiveness and improved performance of the proposed control scheme.

Appendix

Dynamic equations of Quanser 2-RLFJ serial robot manipulator according to (14) can be defined as [51]:

where the vector of the state variables is considered as

$$ x= [ \theta _{m_{1}}, \theta _{m_{2}},\alpha _{1}, \alpha _{2}, \dot{\theta }_{m_{1}}, \dot{\theta }_{m_{2}}, \dot{\alpha }_{1}, \dot{\alpha }_{2} ]^{T}. $$

(74)

The other coefficients expressed in (72) and (73) are as bellow:

$$\begin{aligned} f_{1} &= \left . \frac{R_{m_{1}} K_{s_{1}} x_{3} - ( K_{g_{1}} ^{2} K_{m_{1}} K_{t_{1}} \eta _{g_{1}} \eta _{m_{1}} + B_{r_{1}} R_{m _{1}} ) x_{5}}{R_{m_{1}} J_{eq_{1}}} \right \vert _{x=e+ x_{d}}, \end{aligned}$$

(75)

$$\begin{aligned} f_{2} &= \left .\frac{R_{m_{2}} K_{s_{2}} x_{4} - ( K_{g_{2}} ^{2} K_{m_{2}} K_{t_{2}} \eta _{g_{2}} \eta _{m_{2}} + B_{r_{2}} R_{m _{2}} ) x_{6}}{R_{m_{2}} J_{eq_{2}}} \right \vert _{x=e+ x_{d}}, \end{aligned}$$

(76)

$$\begin{aligned} f_{3} &= \frac{1}{R_{m_{1}} J_{eq_{1}} J_{l_{1}}} \bigl( g J _{eq_{1}} R_{m_{1}} L_{2} m_{1} \sin ( x_{1} + x_{3} ) - R_{m_{1}} K_{s_{1}} ( J_{eq_{1}} + J_{l_{1}} ) x_{3} \\ &\quad {}+ \bigl( ( - B_{l_{1}} J_{eq_{1}} + B_{r_{1}} J_{l_{1}} ) R_{m_{1}} + K_{g_{1}}^{2} K_{m_{1}} K_{t_{1}} \eta _{g_{1}} \eta _{m _{1}} J_{l_{1}} \bigr) x_{5} - B_{l_{1}} R_{m_{1}} J_{eq_{1}} x_{7} \bigr) \Biggr\vert _{x=e+ x_{d}}, \end{aligned}$$

(77)

$$\begin{aligned} f_{4} &= \frac{1}{R_{m_{2}} J_{eq_{2}} J_{l_{2}}} \bigl( g J _{eq_{2}} R_{m_{2}} L_{2} m_{2} \sin ( x_{2} + x_{4} ) - R_{m_{2}} K_{s_{2}} ( J_{eq_{2}} + J_{l_{2}} ) x_{4} \\ &\quad {}+ \bigl( ( - B_{l_{2}} J_{eq_{2}} + B_{r_{2}} J_{l_{2}} ) R_{m_{2}} + K_{g_{2}}^{2} K_{m_{2}} K_{t_{2}} \eta _{g_{2}} \eta _{m _{2}} J_{l_{2}} \bigr) x_{6} - B_{l_{2}} R_{m_{2}} J_{eq_{2}} x_{8} \bigr) \Biggr\vert _{x=e+ x_{d}}, \end{aligned}$$

(78)

$$\begin{aligned} &\left \{ \textstyle\begin{array}{l} b_{1} = \frac{\eta _{g_{1}} K_{g_{1}} \eta _{m_{1}} K_{t_{1}}}{R_{m_{1}} J_{eq_{1}}}, \\ b_{2} = \frac{\eta _{g_{2}} K_{g_{2}} \eta _{m_{2}} K_{t_{2}}}{R_{m_{2}} J_{eq_{2}}}, \end{array}\displaystyle \right . \end{aligned}$$

(79)

$$\begin{aligned} &\left \{ \textstyle\begin{array}{l} b_{3} =- \frac{\eta _{g_{1}} K_{g_{1}} \eta _{m_{1}} K_{t_{1}}}{R_{m _{1}} J_{eq_{1}}}, \\ b_{4} =- \frac{\eta _{g_{2}} K_{g_{2}} \eta _{m_{2}} K_{t_{2}}}{R_{m _{2}} J_{eq_{2}}}. \end{array}\displaystyle \right . \end{aligned}$$

(80)

The parameters, as well as their values, for both subsystems are shown in Table 4.

Table 4 Motor and manipulator parameters of Quanser 2-RLFJ robot manipulator [51]