Observer-based continuous adaptive sliding mode control for soft actuators

Abstract

Fabricated by high elastic materials, soft actuators provide a prominent solution for soft rehabilitation gloves, soft graspers and locomotion robots. However, the control of soft actuators is a grant challenge due to dynamic modeling error and unavailable system states. This paper proposes an observer-based continuous adaptive sliding mode controller for soft actuators in the presence of system uncertainties without knowledge of its upper bound in prior. By exploiting a novel nonsingular fast terminal sliding mode (NFTSM) surface and a high-order sliding mode (HOSM) observer, the proposed control scheme features adaptive-tuning gains, continuity, singularity-free, stronger robustness and higher tracking accuracy. The stability of the proposed controller is analyzed by the Lyapunov method. Corresponding comparative simulations and experiments of a soft pneumatic network actuator verify the effectiveness and related features of the proposed controller.

Appendices

Appendix A

According to the update law (17) and (20), some auxiliary variances are defined as

$$\begin{aligned} \rho _1=\sqrt{z} s^{[\frac{1}{2}]},\rho _2=s,\rho _3=\xi . \end{aligned}$$

(38)

Therefore,

$$\begin{aligned}&\dfrac{z}{\left| \rho _1\right| }\rho _1\rho _2=\zeta \rho _1\rho _1, \dfrac{z }{\left| \rho _1\right| }\rho _2\rho _2=z \rho _1\rho _2, \nonumber \\&\quad \dfrac{z }{\left| \rho _1\right| }\rho _2\rho _3=z \rho _1\rho _3, \end{aligned}$$

(39)

and

$$\begin{aligned} \begin{array}{l} {\dot{\rho }}\triangleq \begin{bmatrix}{{\dot{\rho }}}_1\\ {{\dot{\rho }}}_2\\ {{\dot{\rho }}}_3\\ \end{bmatrix} =-\dfrac{z }{2\left| \rho _1\right| }{\underbrace{\begin{bmatrix}k_{z1}&{}k_{z2}&{}-1\\ 0&{}0&{}0\\ 2k_{z3}&{}0&{}0\\ \end{bmatrix}}_{A_1}}\begin{bmatrix}\rho _1\\ \rho _2\\ \rho _3\\ \end{bmatrix} \\ \qquad -{\underbrace{\begin{bmatrix}0&{}0&{}0\\ z k_{z1}&{}z k_{z2}&{}-z \\ 0&{} k_{z4}z &{}0\\ \end{bmatrix}}_{A_2}}\begin{bmatrix}\rho _1\\ \rho _2\\ \rho _3\\ \end{bmatrix}+{\underbrace{\begin{bmatrix}\dfrac{{\dot{z}}}{2z }\rho _1\\ \dfrac{{\dot{z}}}{z }\rho _2\\ d_2\\ \end{bmatrix}}_{A_3}} \\ =-\dfrac{z }{2\left| \rho _1\right| }A_1\rho -A_2\rho +A_3. \end{array}\nonumber \\ \end{aligned}$$

(40)

The candidate Lyapunov function is chosen as,

$$\begin{aligned} V_1=\dfrac{1}{2}\rho ^TP\rho , \end{aligned}$$

(41)

where $P=\begin{bmatrix}4k_{z3}+k_{z1}^2&{}k_{z1}k_{z2}&{}-k_{z1}\\ k_{z1}k_{z2}&{}k_{z2}^2+4k_{z4}&{}-k_{z2}\\ -k_{z1}&{}-k_{z2}&{}2\\ \end{bmatrix}$ is positive definite.

The derivative of $V_1$ with respect to time along with equations (40) is,

$$\begin{aligned} \begin{array}{l} {\dot{V}}_1=-\dfrac{\zeta }{2\left| \rho _1\right| }\rho ^TPA_1\rho -\rho ^TPA_2\rho +\rho ^TPA_3 \\ \quad = -\zeta \rho ^T\begin{bmatrix}k_{z1}^2k_{z2}&{}k_{z1}k_{z2}^2+2k_{z1}k_{z4}&{}-k_{z1}k_{z2}\\ *&{}k_{z2}\left( k_{z2}^2+4k_{z4}\right) &{}-k_{z2}^2-k_{z4}\\ *&{}*&{}k_{z2}\\ \end{bmatrix}\rho \\ \qquad -\dfrac{\zeta }{2\left| \rho _1\right| }\rho ^T\begin{bmatrix}k_{z1}^3+2k_{z3}k_{z1}&{}(k_{z1}^2+k_{z3})k_{z2}&{}-k_{z1}^2\\ *&{}k_{z1}k_{z2}^2&{}-k_{z1}k_{z2}\\ *&{}*&{}k_{z1}\\ \end{bmatrix}\rho \\ \\ \qquad +\dfrac{{\dot{z}}}{2z }\rho ^T{\underbrace{\begin{bmatrix}4k_{z2}+k_{z1}^2&{}\dfrac{3}{2}k_{z1}k_{z3}&{}-\dfrac{1}{2}k_{z1} \\ *&{} 2k_{z3}^2+8k_{z4} &{}-k_{z3}\\ *&{}*&{}0\\ \end{bmatrix}}_{A_3'}}\rho \\ \qquad {\underbrace{-k_{z1}\rho _1d_2-k_{z2}\rho _2d_2+2\rho _3d_2}_\varDelta }.\nonumber \end{array}\!\!\!\!\!\!\!\\ \end{aligned}$$

(42)

The time derivative of the $V_1$ is further equal to

$$\begin{aligned} \begin{array}{l}{\dot{V}}_1 =\dfrac{{\dot{\zeta }}}{2\zeta }\rho ^TA_3'\rho +\varDelta \\ -\dfrac{\zeta \rho ^T}{2\left| \rho _1\right| }{\underbrace{\begin{bmatrix}k_{z1}^3+2k_{z3}k_{z1}&{}0&{}-k_{z1}^2\\ *&{}5k_{z1}k_{z2}^2+8k_{z1}k_{z4}&{}-3k_{z1}k_{z2}\\ *&{}*&{}k_{z1}\\ \end{bmatrix}}_{A_4}}\rho \\ -\zeta \rho ^T{\underbrace{\begin{bmatrix}2k_{z1}^2k_{z2}+k_{z3}k_{z2}&{}0&{}0\\ *&{}k_{z2}\left( k_{z2}^2+4k_{z4}\right) &{}-k_{z2}^2-k_{z4}\\ *&{}*&{}k_{z2}\\ \end{bmatrix}}_{A_5}}\rho \\ \qquad =-\dfrac{\zeta }{2\left| \rho _1\right| }\rho ^TA_4\rho -\zeta \rho ^TA_5\rho +\dfrac{{\dot{\zeta }}}{2\zeta }\rho ^TA_3'\rho +\varDelta .\nonumber \end{array}\!\!\!\!\!\!\!\\ \end{aligned}$$

(43)

The * in the matrices $A_4$ and $A_5$ denotes the symmetrical term of the matrix. The matrices $A_4$ and $A_5$ are positive definite if the inequality (18) holds. Further, noticing that the following inequalities are satisfied,

$$\begin{aligned}&\lambda _{min}\{A_3'\}||\rho ||^2\le \rho ^TA_3'\rho \le \lambda _{max}\{A_3'\}||\rho ||^2, \nonumber \\&\lambda _{min}\{A_4\}||\rho ||^2\le \rho ^TA_4\rho \le \lambda _{max}\{A_4\}||\rho ||^2, \nonumber \\&\lambda _{min}\{A_5\}||\rho ||^2\le \rho ^TA_5\rho \le \lambda _{max}\{A_5\}||\rho ||^2, \nonumber \\&\lambda _{min}\{P_a\}||\rho ||^2\le V_1\le \lambda _{max}\{P_a\}||\rho ||^2, \end{aligned}$$

(44)

The derivative of the Lyapunov function $V_1$ is

$$\begin{aligned} {\dot{V}}_1= & {} -\dfrac{z }{2\left| \rho _1\right| }\rho ^TA_4\rho -\rho ^TA_5\rho +\dfrac{{\dot{z}}}{2z }\rho ^TA_3'\rho +\varDelta \nonumber \\\le & {} -\dfrac{z }{2\left| \rho _1\right| }\rho ^TA_4\rho -\rho ^TA_5\rho +\dfrac{{\dot{z}}}{2z }\lambda _{max}\{A_3'\}\rho ^T\rho \nonumber \\&+\sqrt{k_{z1}^2+k_{z2}^2+4} D ||\rho || \nonumber \\\le & {} -\dfrac{z }{2\left| \rho _1\right| }\lambda _{min}\{A_4\}\rho ^T\rho -\lambda _{min}\{A_5\}\rho ^T\rho \nonumber \\&+\dfrac{{\dot{z}}}{2z }\lambda _{max}\{A_3'\}\rho ^T\rho +\sqrt{k_{z1}^2+k_{z2}^2+4}D||\rho ||.\nonumber \\ \end{aligned}$$

(45)

The time derivative of the $V_1$ is further equal to

$$\begin{aligned} {\dot{V}}_1\le & {} -\dfrac{z }{2|\rho _1|}\lambda _{min}\{A_4\}||\rho ||^2 +\sqrt{k_{z1}^2+k_{z2}^2+4}D||\rho || \nonumber \\&-\lambda _{min}\{A_5\}||\rho ||^2+\dfrac{{\dot{z}}}{2z }\lambda _{max}\{A_3'\}||\rho ||^2.\nonumber \\ \end{aligned}$$

(46)

In virtue of the well-known inequality $(\rho _1^2+\rho _2^2+\rho _3^2)^{\frac{1}{2}}\le (|\rho _1|+|\rho _2|+|\rho _3|)$, $||\rho ||^2\ge ||\rho _1||^2$, $\rho ^T\rho \le \dfrac{2}{\lambda _{min}\left( P_a\right) }V_1$, $-\rho ^T\rho \le -\dfrac{2}{\lambda _{max}\left( P_a\right) }V_1$, then the derivative satisfies

$$\begin{aligned} {\dot{V}}_1\le & {} -\left[ \dfrac{2\lambda _{min}\{A_5\}}{\lambda _{max}\left( P_a\right) }-\dfrac{{\dot{z}}\left| \lambda _{max}\{A_3'\}\right| }{z \lambda _{min}\left( P_a\right) }\right] V_1 \nonumber \\&-\left[ \dfrac{z }{2}\lambda _{min}\{A_4\}-\sqrt{k_{z1}^2+k_{z2}^2+4}D\right] \nonumber \\&\times \sqrt{\dfrac{2}{\lambda _{max}\left( P_a\right) }}V_1^{1/2}. \end{aligned}$$

(47)

Based on the update law ${\dot{z}}\ge 0$, then z is increased all the time except when ${\tilde{q}}_1=0$, so $\varphi _1=\dfrac{2\lambda _{min}\{A_5\}}{\lambda _{max}\left( P_a\right) }-\dfrac{{\dot{z}}\left| \lambda _{max}\{A_3'\}\right| }{z \lambda _{min}\left( P_a\right) }$ and $\varphi _2=\dfrac{z }{2}\lambda _{min}\{A_4\}-\sqrt{k_{z1}^2+k_{z2}^2+4}D$ will be positive in a short adjusting time. Then, ${\dot{V}}_{1}\le -\varphi _1V_{1}-\varphi _2{V}_{1}^{1/2}$. According to Lemma.1, the estimated errors converge to zero in finite time,

$$\begin{aligned} t_s\le \dfrac{2}{\varphi _1}\ln \dfrac{\varphi _1 V_1^{1/2}(x_0)+\varphi _2}{\varphi _2}. \end{aligned}$$

(48)

This completes the proof. $\square$

Appendix B

As discussed before, soft actuators are widely used as soft gloves. To mimic a human middle finger, the details of the simulated actuator are $W=1.5\times 10^{-2}\text {m},L_1=8\times 10^{-3}\text {m},W_1=3\times 10^{-3}\text {m},W_2=2\times 10^{-3}\text {m},H_1=7\times 10^{-3}\text {m}, H_3=1.1\times 10^{-2}\text {m},L_{20}=2\times 10^{-3}\text {m},N=14, k_p=6.2\times 10^{-7}, G=2.387\times 10^{6}\text {Pa},m=4\times 10^{-2}\text {kg}$.

In numerical simulations, gains are introduced as follows. In the control scheme of STO+LSM, the gains of the STO (22) are $k_{1a} = 7, k_{1b}=5, k_{1c}=4, k_{1d}=10$, the gain of the LSM (23) is $\lambda _1 = 50$, and the gains of the STC (25) $k_{1e} = k_{1ez}\sqrt{z}, k_{1f} = k_{11f}z, k_{1g} = k_{11g}z, k_{1h} = k_{11h}z^2$ are updated by $k_{1ez}=1.8, k_{11f} = 3, k_{11g} = 0.05, k_{11h} = 2, k_{zu} = 0.5.$

In the control scheme of HOSM+LSM, the gains of the HOSM observer (8) are $k_{11}=7, k_{12} = 20, k_{21} = 10, k_{22} = 50, k_{31} = 20$, and the gains of LSM (23) and the updated law of controller are same to the STO+LSM.

In the control scheme of HOSM+NFTSM, parameters of NFTSM (31) are $\lambda _{31} = 42,\lambda _{32} = 0.1$, and feedback gains of the STC (33) are $a_3 = 2, b_3 = 1.5, k_{3g} = 6, k_{3h} = 2.$

In the control scheme of HOSM+FTSM, the parameters of FTSM (27) are $\lambda _{21}=14, \lambda _{22}=2$, and the gains of STC (29) are updated by $k_{2e} = k_{02z}\sqrt{z}, k_{2f}= k_{02f}z, k_{2g}= k_{02g}z, k_{2h} = k_{02h}z^2$, where $k_{02z}=1.8, k_{02f} = 3, k_{2g} = 0.05, k_{02h} = 2, k_{zu} = 0.5.$

In the proposed control scheme, the observer (8), parameters of the NFTSM (10) are $\lambda _{11} = 14,\lambda _{12}=2,\mu _1=4,\mu _2=2, a=0.75, b=1.5, b_2=0.5$, and the gains of proposed controller (15) are $k_{s1} = k_{s01}\sqrt{z}, k_{s2} = k_{s02}z, k_{p1} = k_{p01}z, k_{p2} = k_{p02}z$, where $k_{s01} = 2, k_{s02} = 3, k_{p01} = 0.05, k_{p02} = 2, k_{zu} = 0.5$.

In experiments, gains are selected after trial-in-errors. In the control scheme of STO+LSM, the gains of the STO (22), the LSM (23) and the STC (25) are $k_{1a} = 4, k_{1b}=7, k_{1c}=4, k_{1d}=6, \lambda _1 = 12, k_{1ez}=0.6, k_{11f} = 2.1, k_{11g} = 0.03, k_{11h} = 1.2, k_{zu} = 0.3$. In the control scheme of HOSM+LSM, the gains of the HOSM observer (8) and the LSM (23) are $k_{11}=3, k_{12} = 12, k_{21} = 5, k_{22} = 13, k_{31} = 0.5, \lambda _1 = 12$. In the control scheme of HOSM+NFTSM, parameters of NFTSM (31) are $\lambda _{31} = 9,\lambda _{32} = 0.08$, and feedback gains of the STC (33) are $a_3 = 1.8, b_3 = 1.5, k_{3g} = 4.5, k_{3h} = 2.3$. In the control scheme of HOSM+FTSM, parameters of FTSM (27) and STC (29) are $\lambda _{21}=9, \lambda _{22}=2, k_{02z}=0.6, k_{02f} = 2.1, k_{2g} = 0.03, k_{02h} = 1.5, k_{zu} = 0.3.$ In the proposed control scheme, the observer (8), parameters of the NFTSM (10) and the controller (15) are $\lambda _{11} = 11,\lambda _{12}=2,\mu _1=3.5,\mu _2=2, a=0.75, b=1.5, b_2=0.5, k_{s01} = 0.8, k_{s02} = 2.5, k_{p01} = 0.05, k_{p02} = 3, k_{zu} = 0.3$.