Compensation control of hydraulic manipulator under pressure shock disturbance

Abstract

This study aims to develop an advanced controller for high-accuracy tracking control ofhydraulic manipulators. The primary technical challenges identified in previous research are friction, leakage, external disturbance, and modelling uncertainties. However, this study for the first time discovers that pressure shock disturbance generated by the supply pump significantly impairs the tracking performance of the hydraulic robotic arm. To address these issues, a shock disturbance compensation controller (SDCC) based on backstepping is proposed in this research. A newly developed adaptive controller and compensation controller are used to handling uncertainties and pressure shock disturbance in the hydraulic system, respectively. The controller theoretically guarantees the asymptotic tracking performance of the hydraulic manipulator under uncertainties and mixed disturbances. Extensive comparative experimental results show that the addition of SDCC reduces the maximum tracking error and variance of PID by an average of 68.7$\%$ and 68.55$\%$, respectively.

Appendices

Appendix A Proof of Theorem 1

Given (23) (24) and (25), we have

$$\begin{aligned} \begin{aligned} e_{2}^T {m} \dot{e_{2}}=\,&e_{2}^T\left( e_3{-}k_pe_1{-}k_v{{\dot{e}}}_1{+}{\varvec{\Phi }}\widetilde{{{\varvec{P}}}}{+}u{-}\varvec{{c}}e_{2}{-}d\right) \\&\quad {-}\varvec{{c}}\,e_{2}^Te_{2} \end{aligned} \end{aligned}$$

(45)

Then, we can transform (33) as follows

$$\begin{aligned} \begin{aligned} {{\dot{V}}}_1\left( t\right) =&\frac{1}{2}e_{2}^T\dot{\varvec{{m}}}e_{2}{-}e_{2}^T\varvec{{c}}e_{2}{+}e_1^T\left( k_{p1}{+}\gamma _1k_{v1}\right) \dot{e_1}{+}e_{2}^Te_3\\&{+}e_{2}^T\left( {-}k_pe_1{-}k_v{{\dot{e}}}_1\right) \\&+e_{2}^T{\varvec{\Phi }}\widetilde{{{\varvec{P}}}}+{\widetilde{{{\varvec{P}}}}}^T{\varvec{\Gamma }}^{-1}\dot{\widetilde{{{\varvec{P}}}}}+e_{3}^T(u-d)\\&+e_3^T{{\dot{e}}}_3+\lambda _1^{-1}{\widetilde{d}}{\dot{{\widetilde{d}}}}+\lambda _2^{-1}\epsilon {\dot{\epsilon }} \end{aligned} \end{aligned}$$

(46)

According to (29), we have

$$\begin{aligned} \begin{aligned} e_{1}^{T}\left( k_{p 1}+\gamma _{1} k_{v 1}\right) {\dot{e}}_{1}=e_{1}^{T} k_{p} {\dot{e}}_{1}-e_{1}^{T} {\frac{k_{p 2}}{\alpha +\left| e_{1 }\right| }} {\dot{e}}_{1}\\ +e_{1}^{T} \gamma _{1}(k_{v}-{\frac{k_{v 2}}{\beta +\left| {\dot{e}}_{1}\right| }}) \dot{e_{1}} \end{aligned} \end{aligned}$$

(47)

Using Property 1 in [33], the following formula can be obtained

$$\begin{aligned} \frac{1}{2}e_{3}^T\dot{\varvec{{m}}}e_{3}-e_{3}^T\varvec{{c}}e_{3}=0 \end{aligned}$$

(48)

Then, combining (18) and (31), the following equation can be obtained

$$\begin{aligned} \begin{aligned} e_{3}^T\left( {-}k_pe_1-k_v{{\dot{e}}}_1\right) {=}&{-}e_1^Tk_p\dot{e_1}{-}{{\dot{e}}}_1^Tk_v\dot{e_1}-\gamma _1e_1^Tk_pe_1\\ -\gamma _1e_1^Tk_v{{\dot{e}}}_1 \end{aligned} \end{aligned}$$

(49)

In view of [40], Property 3 in [33] and (27), we have

$$\begin{aligned} \begin{aligned} {\dot{V}}_{1}(t)= {\dot{e}}_{1}^{T} e_{3}-e_{1}^{T} {\frac{k_{p 2}}{\alpha +\left| e_{1}\right| }} {\dot{e}}_{1} -e_{1}^{T} {\frac{\gamma _{1} k_{v 2}}{\beta +\left| {\dot{e}}_{1}\right| }} {\dot{e}}_{1}\\ -{\dot{e}}_{1}^{T} k_{v} {\dot{e}}_{1} -\gamma _{1} e_{1}^{T} k_{p} {\dot{e}}_{1}+\gamma _{1} e_{1}^{T} e_{3} \end{aligned} \end{aligned}$$

(50)

We can obtain the following inequalities by using Theorem 1.

$$\begin{aligned}{} & {} {\gamma _{1}}+0.5\le 0,\ \ \ {\gamma _{1}}\left( 0.5-{k_{p}}\right) \le 0\\\\{} & {} 0.5-{k_{v}} \le 0, \quad -\frac{0.5 {k_{p 2 }}}{\alpha +\left| {e_{1}}\right| } \le 0, \quad -\frac{0.5 {\gamma _{1} k_{v 2}}}{\beta +\left| {{\dot{e}}_{1}}\right| } \le 0 \end{aligned}$$

Then, by using Young’s inequality, (50) can be deduced that ${{\dot{V}}}_1\left( t\right) \le 0$. Appendix 1 is proven.

Appendix B Proof of Theorem 2

Differentiating $\mathrm {\Xi }$ yields

$$\begin{aligned} {\dot{\Xi }}=-r^{T}\left[ \dot{\Delta _i}(t)-\lambda {\text {sgn}}\left( e_{3}\right) \right] \end{aligned}$$

(51)

Combining (34) and (51), we have

$$\begin{aligned} \begin{aligned} {\dot{V}}_{2}&={\dot{V}}_{1}(t)+e_{3}^{T} {\dot{e}}_{2}+r^{T} {\dot{r}}+{\dot{\Xi }} \\&=e_{1}^{T}\left[ \gamma _{1}\left( 0.5-k_{p}\right) -\frac{0.5 k_{p 2}}{\alpha +\left| e_{1}\right| }-\frac{0.5 \gamma _{1} k_{v 2}}{\beta +\left| {\dot{e}}_{1}\right| }\right] e_{1}\\&\quad +e_{3}^{T}\left( \gamma _{1}-\gamma _{2}+1\right) e_{3}\\&\quad +{\dot{e}}_{1}^{T}\left( 0.5-k_{v}-\frac{0.5 k_{p 2}}{\alpha +\left| e_{1}\right| }-\frac{0.5 \gamma _{1} k_{v 2}}{\beta +\left| {\dot{e}}_{1}\right| }\right) {\dot{e}}_{1}\\&\quad +\left( 0.5-k_{s}\right) r^{T} r \\ \end{aligned}\nonumber \\ \end{aligned}$$

(52)

By reusing Theorem 1, The following inequalities can be obtained.

$$\begin{aligned} \begin{aligned} 0.5-{k_{s}} \le 0, \quad {\gamma _{1}}-{\gamma _{2}}+1 \le 0 \\ {\gamma _{1}}\left( 0.5-{k_{p}}\right) -\frac{0.5 {k_{p 2 }}}{\alpha +\left| {e_{1}}\right| }-\frac{0.5 {\gamma _{1} k_{v 2}}}{\beta +\left| {{\dot{e}}_{1}}\right| } \le 0 \\ 0.5-{k_{v}}-\frac{0.5 {k_{p 2 }}}{\alpha +\left| {e_{1}}\right| }-\frac{0.5 {\gamma _{1} k_{v 2}}}{\beta +\left| {{\dot{e}}_{1}}\right| } \le 0 \end{aligned} \end{aligned}$$

Finally, simplify (52) yields ${{\dot{V}}}_2\le 0$.