Neural Network-Based Hybrid Position/Force Tracking Control for Robotic Systems Without Velocity Measurement

Abstract

In this paper, a hybrid position/force tracking control scheme based on neural network observer is proposed for robotic systems with uncertain parameters and external disturbances. First, an observer based on neural network is designed to estimate joint velocities. Then, a neural network-based adaptive hybrid position/force controller is proposed based on the observed joint velocities. By using strict positive real method and Lyapunov stability theory, it is proved that all the signals of the closed-loop system are ultimately uniformly bounded. Finally, the simulation tests on a two-link manipulator are conducted. The simulation results show the feasibility and effectiveness of the control scheme.

Appendices

Appendix A: Proof of Theorem 1

Considering the following Lyapunov function,

$$\begin{aligned} V_o=\frac{1}{2}{\tilde{x}}^{{\mathrm {T}}}P{\tilde{x}}+\frac{1}{2} {\tilde{W}}_o^{{\mathrm {T}}}{\varGamma }_{W_o}\dot{{\tilde{W}}}_o \end{aligned}$$

(63)

Taking the differentiation of Eq. (63) and substituting Eqs. (21) and (26), yields

$$\begin{aligned} {\dot{V}}_o&= \frac{1}{2}\dot{{\tilde{x}}}^{{\mathrm {T}}}P{\tilde{x}}+ \frac{1}{2}{\tilde{x}}^{{\mathrm {T}}}P\dot{{\tilde{x}}}+{\tilde{W}}_o^{{\mathrm {T}}} {\varGamma }_{W_o}\dot{{\tilde{W}}}_o\nonumber \\&= \frac{1}{2}{\tilde{x}}^{{\mathrm {T}}}(A_o^{{\mathrm {T}}}P+PA_o){\tilde{x}} +\frac{1}{2}U_o^{{\mathrm {T}}}B_o^{{\mathrm {T}}}P{\tilde{x}}+\frac{1}{2} {\tilde{x}}^{{\mathrm {T}}}PB_oU_o+{\tilde{W}}_o^{{\mathrm {T}}}{\varGamma }_{W_o} \dot{{\tilde{W}}}_o\nonumber \\&= -\frac{1}{2}{\tilde{x}}^{{\mathrm {T}}}Q{\tilde{x}}+U_o^{{\mathrm {T}}} B_o^{{\mathrm {T}}}P{\tilde{x}}+{\tilde{W}}_o^{{\mathrm {T}}} {\varGamma }_{W_o}\dot{{\tilde{W}}}_o \end{aligned}$$

(64)

According to Eq. (20), we can obtain,

$$\begin{aligned}&U_o^{{\mathrm {T}}}B_o^{{\mathrm {T}}}P{\tilde{x}}+{\tilde{W}}_o^{{\mathrm {T}}} {\varGamma }_{W_o}\dot{{\tilde{W}}}_o\nonumber \\&\quad = [{\tilde{W}}^{{\mathrm {T}}}_o\xi _o(x_1,{\hat{x}}_2)-M^{-1}(x_1) \tau _d-M^{-1}(x_1){\varDelta }M(x_1) M^{-1}_0(x_1)(\tau -\tau _e)\nonumber \\&\qquad +\varepsilon _o(x_1,x_2)-v_o-W^{{\mathrm {T}}}_o\tilde{\xi _o}(x_1,x_2, {\hat{x}}_2)]^{{\mathrm {T}}}B_o^{{\mathrm {T}}}P{\tilde{x}}+ {\tilde{W}}_o^{{\mathrm {T}}}{\varGamma }_{W_o}\dot{{\tilde{W}}}_o\nonumber \\&\quad = [{\tilde{W}}^{{\mathrm {T}}}_o\xi _o(x_1,{\hat{x}}_2)]^{{\mathrm {T}}} B_o^{{\mathrm {T}}}P{\tilde{x}}+{\tilde{W}}_o^{{\mathrm {T}}}{\varGamma }_{W_o} \dot{{\tilde{W}}}_o+[-M^{-1}(x_1){\varDelta }M(x_1) M^{-1}_0(x_1)(\tau -\tau _e)\nonumber \\&\qquad -M^{-1}(x_1)\tau _d-W^{{\mathrm {T}}}_o\tilde{\xi _o}(x_1,x_2,{\hat{x}}_2) +\varepsilon _o(x_1,x_2)-v_o]^{{\mathrm {T}}}B_o^{{\mathrm {T}}}P{\tilde{x}} \end{aligned}$$

(65)

Now, using the updating laws Eq. (27), the facts $\dot{{\tilde{W}}}=-\dot{{\hat{W}}}$ and $B_oP^{{\mathrm {T}}}{\tilde{x}}=C_o{\tilde{x}}={\tilde{x}}_1$, we can obtain,

$$\begin{aligned}{}[{\tilde{W}}^{{\mathrm {T}}}_o\xi _o(x_1,{\hat{x}}_2)]^{{\mathrm {T}}} B_o^{{\mathrm {T}}}P{\tilde{x}}+{\tilde{W}}_o^{{\mathrm {T}}} {\varGamma }_{W_o}\dot{{\tilde{W}}}_o \le 0 \end{aligned}$$

(66)

According to inequality (25) and Eq. (28), we can obtain,

$$\begin{aligned}&[-M^{-1}(x_1){\varDelta }M(x_1) M^{-1}_0(x_1)(\tau -\tau _e)-M^{-1}(x_1) \tau _d-W^{{\mathrm {T}}}_o\tilde{\xi _o}(x_1,x_2,{\hat{x}}_2)\nonumber \\&\quad +\varepsilon _o(x_1,x_2)-v_o]^{{\mathrm {T}}}B_o^{{\mathrm {T}}}P {\tilde{x}} \le 0 \end{aligned}$$

(67)

Therefore,

$$\begin{aligned} U_o^{{\mathrm {T}}}B_o^{{\mathrm {T}}}P{\tilde{x}}+\tilde{W_o}^{{\mathrm {T}}} {\varGamma }_{W_o}\dot{\tilde{W_o}}\le 0 \end{aligned}$$

(68)

Substituting Eq. (68) into Eq. (64) yields,

$$\begin{aligned} {\dot{V}}_o \le -\frac{1}{2}{\tilde{x}}^{{\mathrm {T}}}Q{\tilde{x}} \end{aligned}$$

(69)

This implies $V_o>0$ and ${\dot{V}}_o \le 0$, the stability of the observer can be then ensured so that ${\tilde{x}}$ and ${\tilde{W}}_o$ are bounded.

Appendix B: Proof of Theorem 2

Considering the following Lyapunov function,

$$\begin{aligned} V_c=\frac{1}{2}{\hat{s}}^{{\mathrm {T}}}M_0(x_1){\hat{s}}+ \frac{1}{2}{\tilde{W}}_c^{{\mathrm {T}}}{\varGamma }_{W_c}\dot{{\tilde{W}}}_c \end{aligned}$$

(70)

Taking the differentiation of Eq.(70) yields

$$\begin{aligned} {\dot{V}}_c&= \frac{1}{2}{\hat{s}}^{{\mathrm {T}}}{\dot{M}}_0(x_1){\hat{s}}+ {\hat{s}}^{{\mathrm {T}}}M_0(x_1)\dot{{\hat{s}}}+{\tilde{W}}_c^{{\mathrm {T}}} {\varGamma }_{W_c}\dot{{\tilde{W}}}_c \end{aligned}$$

(71)

Substituting Eqs. (54) and (56) into Eq. (71), and considering the fact $\dot{{\tilde{W}}}_c=-\dot{{\hat{W}}}_c$, we can obtain,

$$\begin{aligned} {\dot{V}}_c&= \frac{1}{2}{\hat{s}}^{{\mathrm {T}}}{\dot{M}}_0(x_1){\hat{s}} +{\hat{s}}^{{\mathrm {T}}}\{-[K_d+C_0(x_1,x_2)]{\hat{s}}- {\tilde{W}}^{{\mathrm {T}}}_c\xi _c(x_1,{\hat{x}}_2)+v_c\nonumber \\&\quad -W^{{\mathrm {T}}}_c{\tilde{\xi }}_c(x_1,x_2,{\hat{x}}_2)- \varepsilon _c(x_1,x_2)-\tau _d\}+{\tilde{W}}_c^{{\mathrm {T}}}\xi _c(x_1, {\hat{x}}_2){\hat{s}}^{{\mathrm {T}}}\nonumber \\&=\frac{1}{2}{\hat{s}}^{{\mathrm {T}}}[{\dot{M}}_0(x_1)-2C_0(x_1,x_2)] {\hat{s}}-{\hat{s}}^{{\mathrm {T}}}K_d{\hat{s}}+{\hat{s}}^{{\mathrm {T}}} \{v_c-\varepsilon _c(x_1,x_2)\nonumber \\&\quad -W^{{\mathrm {T}}}_c{\tilde{\xi }}_c(x_1,x_2,{\hat{x}}_2)-\tau _d\} \end{aligned}$$

(72)

According to property 2, we have,

$$\begin{aligned} {\dot{V}}_c = -{\hat{s}}^{{\mathrm {T}}}K_d{\hat{s}}+{\hat{s}}^{{\mathrm {T}}} [v_c-\varepsilon _c(x_1,x_2)-W^{{\mathrm {T}}}_c{\tilde{\xi }}_c (x_1,x_2,{\hat{x}}_2)-\tau _d] \end{aligned}$$

(73)

Substituting Eqs. (55) and (57) into Eq. (73), we can obtain,

$$\begin{aligned} {\dot{V}}_c \le -{\hat{s}}^{{\mathrm {T}}}K_d{\hat{s}} \end{aligned}$$

(74)

It can be concluded that the closed-loop system is asymptotically stable. Integrating Eq. (74) from time $t=0$ to $t=T$ yields,

$$\begin{aligned} \int ^{{\mathrm {T}}}_0\parallel {\hat{s}}\parallel ^2dt \le \frac{V_c(0)-V_c(T)}{\lambda _{{\mathrm {min}}}(K_d)} \end{aligned}$$

(75)

Using Eq. (70), we can obtain,

$$\begin{aligned} V_c(0)=\frac{1}{2}{\hat{s}}^{{\mathrm {T}}}(0)M_0(x_1){\hat{s}}(0) +\frac{1}{2}{\tilde{W}}_c^{{\mathrm {T}}}(0){\varGamma }_{W_c}\dot{{\tilde{W}}}_c(0) \end{aligned}$$

(76)

According to inequality (75), we have,

$$\begin{aligned} \parallel {\hat{s}}\parallel \le \sqrt{\frac{{\hat{s}}^{{\mathrm {T}}}(0) M_0(x_1) {\hat{s}}(0)+ {\tilde{W}}_c^{{\mathrm {T}}}(0) {\varGamma }_{W_c}\dot{{\tilde{W}}}_c(0)}{2\lambda _{{\mathrm {min}}}(K_d)}} \end{aligned}$$

(77)

This implies that ${\hat{s}}$ and ${\tilde{W}}_c$ are bounded. Since $W_c$ is bounded, ${\hat{W}}_c$ is hence bounded.