机器人 2022, Vol. 44 Issue (1): 19-34, 44 0

LI Longnan, HUANG Panfeng, MA Zhiqiang. Finite-time Control Method for Robot Teleoperation Based on Time-varying Output Constraints[J]. ROBOT, 2022, 44(1): 19-34, 44.

1. 西北工业大学航天飞行动力学技术重点实验室, 陕西 西安 710072;
2. 西北工业大学智能机器人研究中心, 陕西 西安 710072

Finite-time Control Method for Robot Teleoperation Based on Time-varying Output Constraints
LI Longnan1,2 , HUANG Panfeng1,2 , MA Zhiqiang1,2
1. National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi'an 710072, China;
2. Research Center of Intelligent Robotics, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: Limited by the operation time window and working space, the space teleoperation tasks need to be completed in a finite time while ensuring that the end effector meets the physical constraints. In addition, time delay and external disturbance seriously affect the stability and control performance of uncertain teleoperation system. Therefore, a finitetime control method for robot teleoperation based on time-varying output constraints is proposed. Firstly, the integral barrier Lyapunov function is used to deal with the time-varying constraints of the operation space, and the rapid stability of the system is guaranteed by the practical finite-time Lyapunov stability theorem. Then, the neural network is utilized to estimate the environment force and resolve the impact of model uncertainty, and the robust term is used to compensate for the estimation bias of the neural network and eliminate the influence of unknown external disturbances. Finally, the proposed algorithm is compared with other algorithms in Matlab/Simulink simulation environments and verified on the ground experiment platform. The results of theoretical simulation and experiment show that the proposed method can further improve the error convergence rate and convergence accuracy, and the output of the system never violates the prescribed time-varying boundary.
Keywords: space teleoperation    integral barrier Lyapunov function (IBLF)    finite-time    time-varying output constraint    neural network

1 引言（Introduction）

 图 1 空间遥操作系统框架 Fig.1 The architecture of space teleoperation system

2 问题描述和基础理论（Problem description and basic theory） 2.1 遥操作系统的动力学

 \begin{align} &{{{\mathit{\boldsymbol{M}}}}}_{j} ({{{\mathit{\boldsymbol{q}}}}}_{j}) {{\ddot{{{\mathit{\boldsymbol{q}}}}}}}_{j} +{{{\mathit{\boldsymbol{C}}}}}_{j} ({{{\mathit{\boldsymbol{q}}}}}_{j}, {{\dot{{{\mathit{\boldsymbol{q}}}}}}}_{j}) {{ \dot{\mathit{\boldsymbol{q}}}}}_{j} +{{{\mathit{\boldsymbol{g}}}}}_{j} ({{{\mathit{\boldsymbol{q}}}}}_{j}) +{{{\mathit{\boldsymbol{f}}}}}_{j}^{{\rm f}} ({{\dot{{{\mathit{\boldsymbol{q}}}}}}}_{j})+ \\ &\; \; \; \; \; {{\delta}} _{j} ({{{\mathit{\boldsymbol{q}}}}}_{j}, {{\dot{{{\mathit{\boldsymbol{q}}}}}}}_{j}, t)={{{\mathit{\boldsymbol{u}}}}}_{j} +{{{\mathit{\boldsymbol{J}}}}}_{j}^{\rm T} ({{{\mathit{\boldsymbol{q}}}}}_{j}) {{{\mathit{\boldsymbol{F}}}}}_{j{ν}} \end{align} (1)

 $$${{{\mathit{\boldsymbol{F}}}}}_{{\rm e}} ={{{\mathit{\boldsymbol{B}}}}}_{{\rm e}} {{\dot{{{\mathit{\boldsymbol{x}}}}}}}_{{\rm s}} +{{{\mathit{\boldsymbol{K}}}}}_{{\rm e}} {{{\mathit{\boldsymbol{x}}}}}_{{\rm s}} +{{{\mathit{\boldsymbol{C}}}}}_{{\rm e}}$$$ (2)

 \begin{align} &{{{\mathit{\boldsymbol{M}}}}}_{{\rm x}j} ({{{\mathit{\boldsymbol{q}}}}}_{j}) \ddot{{{\mathit{\boldsymbol{x}}}}}_{j} + {{{\mathit{\boldsymbol{C}}}}}_{{\rm x}j} ({{{\mathit{\boldsymbol{q}}}}}_{j}, {{\dot{{{\mathit{\boldsymbol{q}}}}}}}_{j}) {{\mathit{\boldsymbol{x}}}}_{j} +{ {{\mathit{\boldsymbol{g}}}}}_{{\rm x}j} ({{{\mathit{\boldsymbol{q}}}}}_{j}) +{{\mathit{\boldsymbol{f}}}}_{{\rm x}j}^{{\rm f}} ({{{\mathit{\boldsymbol{q}}}}}_{j}, {{\dot{{{\mathit{\boldsymbol{q}}}}}}}_{j})+ \\ &{{\delta}} _{{\rm x}j} ({{{\mathit{\boldsymbol{q}}}}}_{j}, {{ \dot{\mathit{\boldsymbol{q}}}}}_{j}, t)={{{\mathit{\boldsymbol{u}}}}}_{{\rm x}j} +{{{\mathit{\boldsymbol{F}}}}}_{jν} \end{align} (5)

 $$$s_{k} ({{\mathit{\boldsymbol{X}}}})=\exp \left(-\frac{\left\| {{{{\mathit{\boldsymbol{x}}}}}-{{{\mathit{\boldsymbol{c}}}}}_{k} } \right\|^{2}}{b_{k}^{2}}\right)$$$ (8)

 \begin{align*} \left\{\!\lim\limits_{t\to T_{{\rm reach}}} {{\varsigma }}\Big|V({{\varsigma}}) \leqslant \min \left\{{\frac{\varXi }{(1\!-\!{\rlap{--} \lambda}) \iota_{1}}, \left({\frac{\varXi }{(1\!-\!{\rlap{--} \lambda}) \iota_{2}}} \right)^{\frac{1}{\xi}}} \right\} \!\right\} \end{align*}

 \begin{align*} T_{{\rm reach}} &\leqslant \max \left\{t_{0} +\frac{1}{{\rlap{--} \lambda} \iota_{1} (1-\xi) }\ln \frac{{\rlap{--} \lambda} \iota_{1} V^{1-\xi} ({{\varsigma}} _{0}) +\iota_{2}} {\iota_{2}}, \right. \\ &\qquad\quad\; \; \left.t_{0} +\frac{1}{\iota_{1} (1-\xi) }\ln \frac{\iota_{1} V^{1-\xi }({{\varsigma}} _{0}) +{\rlap{--} \lambda} \iota_{2} }{{\rlap{--} \lambda} \iota_{2}} \right\} \end{align*}

3 控制器设计和稳定性分析（Controller design and stability analysis） 3.1 遥操作系统的控制框架

 图 2 空间遥操作系统控制框架 Fig.2 Control framework of the space teleoperation system
3.2 参考轨迹生成器 3.2.1 主端参考轨迹生成器

 $$${{{\mathit{\boldsymbol{F}}}}}_{{\rm e}} ={{{\mathit{\boldsymbol{W}}}}}_{{\rm e}}^{\rm T} {{{\mathit{\boldsymbol{S}}}}}_{{\rm e}} ({{\mathit{\boldsymbol{X}}}}_{{\rm e}})$$$ (10)

 $$${{{\mathit{\boldsymbol{F}}}}}_{{\rm e}} ={{{\mathit{\boldsymbol{W}}}}}_{{\rm e}}^{\rm T} (t-T_{{\rm b}} (t)){{{\mathit{\boldsymbol{S}}}}}_{{\rm m}} ({{\mathit{\boldsymbol{X}}}}_{{\rm m}})$$$ (14)

 $$${{{\mathit{\boldsymbol{M}}}}}_{{\rm r}} \ddot{{{\mathit{\boldsymbol{x}}}}}_{{\rm m}}^{{\rm r}} +{{{\mathit{\boldsymbol{C}}}}}_{{\rm r}} \dot{{{\mathit{\boldsymbol{x}}}}}_{{\rm m}}^{{\rm r}} +{{{\mathit{\boldsymbol{g}}}}}_{{\rm r}} =\Im_{1} {{{\mathit{\boldsymbol{F}}}}}_{{\rm h}} -\Im_{2} {{\hat{\mathit{\boldsymbol{F}}}}}_{{\rm e}}$$$ (15)

3.2.2 从端参考轨迹生成器

 $$$\begin{cases} {{\dot{{\zeta}}}} _{1} ={{\zeta}} _{2} +c_{1}^{\prime} c_{2}^{\prime} ({{{\mathit{\boldsymbol{x}}}}}_{{\rm md\ast y}} (t)-{{\zeta}} _{1}) \\ {{\dot{{\zeta}}}} _{2} =c_{1}^{\prime2} c_{3}^{\prime} ({{{\mathit{\boldsymbol{x}}}}}_{{\rm md\ast y}} (t)-{{\zeta}} _{1}) \\ \end{cases}$$$ (16)

 $$$\begin{cases} {{\dot{{{\mathit{\boldsymbol{x}}}}}}}_{\eta 1} ={{{\mathit{\boldsymbol{x}}}}}_{\eta 2} \\ {{\dot{{{\mathit{\boldsymbol{x}}}}}}}_{\eta 2} ={{{\mathit{\boldsymbol{M}}}}}_{{\rm x}j}^{-1} ({{{\mathit{\boldsymbol{u}}}}}_{{\rm x}j} +{{{\mathit{\boldsymbol{F}}}}}_{jν} -{{{\mathit{\boldsymbol{C}}}}}_{{\rm x}j} {{\mathit{\boldsymbol{x}}}}_{\eta 2} -\\ \quad {{{\mathit{\boldsymbol{g}}}}}_{{\rm x}j} - {{\mathit{\boldsymbol{f}}}}_{{\rm x}j}^{{\rm f}} -{{\delta}}_{{\rm x}j}) \\ \end{cases}$$$ (17)

 \begin{align} {{\alpha}} _{j1} &=-\frac{k_{j1} {{\mathit{\boldsymbol{e}}}}_{j1}^{2\xi_{j} -1} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2\xi_{j} -2} (t)}{ ({{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1} }^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} )^{\xi_{j} -1}}+ \frac{{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} }{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{\dot{{{\mathit{\boldsymbol{x}}}}}}}_{j}^{{\rm r}} \varpi_{j1} +\\ &\quad\; \frac{\dot{{{{\mathit{\boldsymbol{k}}}}}}_{{\rm c}_{1}} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} (t)}{{\mathit{\boldsymbol{e}}}}_{j1} -\frac{ {{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} }{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}\dot{{{{\mathit{\boldsymbol{k}}}}}}_{{\rm c}_{1}} (t)\gamma_{j1} -{{{\mathit{\boldsymbol{K}}}}}_{j1} {{\mathit{\boldsymbol{e}}}}_{j1} \end{align} (19)
 \begin{align} {{{\mathit{\boldsymbol{u}}}}}_{{\rm x}j} &=-({{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T}) ^{+}\frac{{ {{\mathit{\boldsymbol{e}}}}}_{j1} {{\mathit{\boldsymbol{e}}}}_{j2} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} - ({{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T}) ^{+}k_{j2} ({{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} { {{\mathit{\boldsymbol{e}}}}}_{j2}} )^{\xi_{j}} -\\ &\quad\; {{{\mathit{\boldsymbol{K}}}}}_{j2} {{\mathit{\boldsymbol{e}}}}_{j2} - {{\hat{\mathit{\boldsymbol{W}}}}}_{j}^{\rm T} {{{\mathit{\boldsymbol{S}}}}}({{\mathit{\boldsymbol{X}}}}_{j}) -{{\hat{\mathit{\boldsymbol{\beta}}}}} _{j} \tanh \frac{{{\mathit{\boldsymbol{e}}}}_{j2}} {b_{j}} -{{{\mathit{\boldsymbol{F}}}}}_{jν} \end{align} (20)

 \begin{align} \lim\limits_{{{\mathit{\boldsymbol{e}}}}_{j1} \to 0} \varpi_{j1} &=\frac{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-({{{\mathit{\boldsymbol{x}}}}}_{j}^{{\rm r}}) ^{2}} \end{align} (27)
 \begin{align} \lim\limits_{{{\mathit{\boldsymbol{e}}}}_{j1} \to 0} \gamma_{j1} &=\frac{({{{\mathit{\boldsymbol{x}}}}}_{j}^{{\rm r}}) ^{2}-3{{{\mathit{\boldsymbol{x}}}}}_{j}^{{\rm r}} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-({{{\mathit{\boldsymbol{x}}}}}_{j}^{{\rm r}}) ^{2}} \end{align} (28)

 \begin{align} \dot{{V}}_{j1}&=-\frac{k_{j1} {{\mathit{\boldsymbol{e}}}}_{j1}^{2\xi_{j}} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2\xi_{j}} (t)}{ ({{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{ {{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} )^{\xi_{j}}} -\frac{{{{\mathit{\boldsymbol{K}}}}}_{j1} { {{\mathit{\boldsymbol{e}}}}}_{j1}^{2} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} +\\ &\quad\; \frac{{{\mathit{\boldsymbol{e}}}}_{j1} {{\mathit{\boldsymbol{e}}}}_{j2} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1} }^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} \end{align} (29)

 $$$V_{j2} =\frac{1}{2}{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} {{{\mathit{\boldsymbol{M}}}}}_{{\rm x}j} {{\mathit{\boldsymbol{e}}}}_{j2}$$$ (30)

$V_{j2}$求导可得

 \begin{align} \dot{{V}}_{j2}&=\frac{1}{2}{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} {{\dot{{{\mathit{\boldsymbol{M}}}}}}}_{{\rm x}j} {{\mathit{\boldsymbol{e}}}}_{j2} +{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} {{{\mathit{\boldsymbol{M}}}}}_{{\rm x}j} \dot{{{\mathit{\boldsymbol{e}}}}}_{j2} \\ & = {{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} ({{{\mathit{\boldsymbol{u}}}}}_{{\rm x}j} +{{{\mathit{\boldsymbol{F}}}}}_{jν} -{{{\mathit{\boldsymbol{C}}}}}_{{\rm x}j} {{\alpha}} _{j1} -\\ &\quad\; {{{\mathit{\boldsymbol{g}}}}}_{{\rm x}j} -{{\mathit{\boldsymbol{f}}}}_{{\rm x}j}^{{\rm f}} -{{\dot{\mathit{\boldsymbol{M}}}}}_{{\rm x}j} {{\alpha}} _{j1} -{{\delta}} _{{\rm x}j}) \end{align} (31)

 $$$V_{j3} =\frac{1}{2}{\rm tr}({{\tilde{\mathit{\boldsymbol{W}}}}}_{j}^{\rm T} {{\varLambda}} _{j}^{-1} {{\tilde{\mathit{\boldsymbol{W}}}}}_{j}) +\frac{1}{2}{\rm tr}({ { \tilde{\mathit{\boldsymbol{\beta}}}}} _{j}^{\rm T} {{\varLambda}} _{\beta, j}^{-1} {{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j})$$$ (33)

$V_{j3}$求导，并将式(22) 代入可得：

 \begin{align} \dot{{V}}_{j3} &={\rm tr}\:({{\tilde{\mathit{\boldsymbol{W}}}}}_{j}^{\rm T} {{\varLambda}} _{j}^{-1} {{\dot{{\tilde{\mathit{\boldsymbol{W}}}}}}}_{j}) +{\rm tr}\:({{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j}^{\rm T} {{\varLambda}} _{\beta, j}^{-1} { { \dot{{\tilde{\mathit{\boldsymbol{\beta}}}}}}} _{j}) \\ &={\rm tr}\: ({{{\tilde{\mathit{\boldsymbol{W}}}}}_{j}^{\rm T} ({{{\mathit{\boldsymbol{S}}}}}({{\mathit{\boldsymbol{X}}}}_{j}) {{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} -\chi_{j} {{\hat{\mathit{\boldsymbol{W}}}}}_{j}) })+ \\ &\quad\; {\rm tr}\: \left({{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j}^{\rm T} \left((\tanh \frac{{{\mathit{\boldsymbol{e}}}}_{j2}} {b_{j}}){{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} -\chi _{\beta, j} {{\hat{\mathit{\boldsymbol{\beta}}}}} _{j}\right)\right) \end{align} (34)

 $$$V_{j} =V_{j1} +V_{j2} +V_{j3}$$$ (35)

$V_{j}$求导，利用式(29)(32)(34) 及如下的等式和不等式：

 \begin{align} &{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} ({{\varepsilon}} ({{\mathit{\boldsymbol{X}}}}_{j} )-{{\delta}} _{{\rm x}j}) =\sum\limits_{i=1}^n {{{\mathit{\boldsymbol{e}}}}_{j2i}} \beta _{j, i} \leqslant \lambda_{\beta j} \sum\limits_{i=1}^n {\|{{\mathit{\boldsymbol{e}}}}_{j2i}} \| \end{align} (36)
 \begin{align} &0\leqslant |x|-x\tanh \frac{x}{b}\leqslant 0.2785b, \quad \forall b>0, \; \; x\in {\rm R} \end{align} (37)
 \begin{align} &\!\!\!\!\begin{cases} {-\chi_{j} {\rm tr}({{{\tilde{\mathit{\boldsymbol{W}}}}}_{j}^{\rm T} {{\hat{\mathit{\boldsymbol{W}}}}}_{j}} )\leqslant -\frac{\chi_{j}} {2}\|{{\tilde{\mathit{\boldsymbol{W}}}}}\|_{F}^{2} +\frac{\chi_{j}} {2}\|{{{\mathit{\boldsymbol{W}}}}}\|_{F}^{2}} \\[6pt] {-\chi_{\beta, j} {\rm tr}({{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j}^{{\rm T}} {{\hat{\mathit{\boldsymbol{\beta}}}}} _{j}} )\leqslant -\frac{\chi_{\beta, j} }{2}\|{{\tilde{\mathit{\boldsymbol{\beta}}}}} \|_{F}^{2} +\frac{\chi_{\beta, j} }{2}\|{{\beta}} \|_{F}^{2}} \\ \end{cases}\!\!\!\! \end{align} (38)

 \begin{align} \dot{{V}}_{j} &\leqslant -\frac{{{{\mathit{\boldsymbol{K}}}}}_{j1} {{\mathit{\boldsymbol{e}}}}_{j1}^{2} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} -\frac{k_{j1} {{\mathit{\boldsymbol{e}}}}_{j1}^{2\xi_{j}} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2\xi_{j}} (t)}{ ({{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-{ {{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} )^{\xi_{j}}}-\\ &\quad\; {{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} {{{\mathit{\boldsymbol{K}}}}}_{j2} {{\mathit{\boldsymbol{e}}}}_{j2} -k_{j2} ({{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} {{\mathit{\boldsymbol{e}}}}_{j2}} )^{\xi_{j}} +0.2785n\lambda_{\rm\beta m} b_{\rm m} + \\ &\quad\; \chi_{j} \bigg(-\frac{1}{4}\|{{\tilde{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}^{2} -\frac{1}{4} ({\|{{{\mathit{\boldsymbol{W}}}}}_{j} \|_{F} -\sqrt{\|{{{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}}} )^{2} +\\ &\quad\; \frac{1}{4}\|{{\tilde{\mathit{\boldsymbol{W}}}}}_{j} \|_{F} -\frac{1}{2}\|{ { \tilde{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}^{\frac{3}{2}} +\frac{1}{2}\|{{{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}^{2}\bigg)+\\ &\quad\; \chi_{\beta, j} \Bigg(-\frac{1}{4}\|{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|_{F}^{2} -\frac{1}{4} ({\|{{\beta}} _{j} \|_{F} -\sqrt{\|{ {\beta}} _{j} \|_{F}}} )^{2}+\\ &\quad\; \frac{1}{4}\|{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|_{F} -\frac{1}{2}\|{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|_{F}^{\frac{3}{2}} +\frac{1}{2}\|{{\beta}} _{j} \|_{F}^{2}\Bigg) \end{align} (39)

 $$$-\dfrac{k_{j1} {{\mathit{\boldsymbol{e}}}}_{j1}^{2} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}}^{2} (t)-{{{\mathit{\boldsymbol{x}}}}}_{\eta 1}^{2}} \leqslant -k_{j1} \int_0^{{ {{\mathit{\boldsymbol{e}}}}}_{j1}} \dfrac{{{\sigma}} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)- ({{{\sigma}} +{{{\mathit{\boldsymbol{x}}}}}_{j}^{{\rm r}}} )^{2}} {\rm d}{{\sigma}}$$$ (40)

 $$$\begin{cases} {\dfrac{1}{4}\|{{\tilde{\mathit{\boldsymbol{W}}}}}_{j} \|_{F} \leqslant \frac{1}{8}\|{{\tilde{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}^{2} +\frac{1}{8}} \\[6pt] {\dfrac{1}{4}\|{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|_{F} \leqslant \frac{1}{8}\|{ { \tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|_{F}^{2} +\frac{1}{8}} \\ \end{cases}$$$ (41)

 \begin{align} \dot{{V}}_{j}&\leqslant -{{{\mathit{\boldsymbol{K}}}}}_{j1} \int_0^{{{\mathit{\boldsymbol{e}}}}_{j1}} {\frac{{{\sigma}} {{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{{\rm 1}}} ^{2} (t)\!-\!({{{\sigma}}\!+\!{{{\mathit{\boldsymbol{x}}}}}_{j}^{{\rm r}}} )^{2}}} {\rm d}{{\sigma}}\! +\!0.2785n\lambda_{\beta j} b_{j}- \\ &\quad\; k_{j1} \left({\int_0^{{{\mathit{\boldsymbol{e}}}}_{j1}} {\frac{{{\sigma }}{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)}{{{\mathit{\boldsymbol{k}}}}_{{\rm c}_{1}} ^{2} (t)-( {{{\sigma}} +{{{\mathit{\boldsymbol{x}}}}}_{j}^{{\rm r}}} )^{2}}} {\rm d}{ {\sigma}}} \right)^{\xi_{j}} -\\ &\quad\; \frac{\chi_{\beta, j}} {8}\|{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|_{F}^{2} -{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} {{{\mathit{\boldsymbol{K}}}}}_{j2} {{\mathit{\boldsymbol{e}}}}_{j2} -k_{j2} \left({{{\mathit{\boldsymbol{e}}}}_{j2}^{\rm T} {{\mathit{\boldsymbol{e}}}}_{j2}} \right)^{\xi_{j}} - \\ &\quad\; \frac{\chi_{j}} {8}\|{{\tilde{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}^{2}-\frac{\chi_{j}} {2} ({\|{{\tilde{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}^{2} } )^{\frac{3}{4}}-\frac{\chi_{\beta, j}} {2} ({\|{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|_{F}^{2}} )^{\frac{3}{4}}+ \\ &\quad\; \chi_{j} \left({\frac{1}{8}+\frac{1}{2}\left\|{{{\mathit{\boldsymbol{W}}}}}_{j} \right\|_{F}^{2}} \right)+\chi_{\beta, j} \left( {\frac{1}{8}+\frac{1}{2}\|{{\beta}} _{j} \|_{F}^{2}} \right) \\ &\leqslant -\iota_{j3}^{1} V_{j3} -\iota_{j3}^{2} V_{j3}^{\xi_{j}} +\varXi _{j3} \end{align} (42)

 \begin{align*} &\iota_{j3}^{1} =\min \Bigg\{\lambda_{\min} ({{{\mathit{\boldsymbol{K}}}}}_{j1} ), \frac{2\lambda_{\min} ({{{\mathit{\boldsymbol{K}}}}}_{j2}) }{\lambda_{\max} ({{{\mathit{\boldsymbol{M}}}}}_{{\rm x}j}) }, \\ & \quad \frac{\chi_{j}} {4\lambda_{\max} ({{{\varLambda}} _{j}^{-1}} )}, \frac{\chi_{\beta, j}} {4\lambda_{\max} ({{{\varLambda}} _{\beta, j}^{-1}} )} \Bigg\}\\ &\iota_{j3}^{2}=\min \left\{{k_{j1}, \frac{(2)^{\xi_{j}} k_{j2} }{(\lambda^{\ast}) ^{\xi_{j}}}, \frac{2^{^{{-1}/4}}\chi_{j}} {\lambda _{\max} ^{3/4} ({{\varLambda}} _{j}^{-1}) }, \frac{2^{{-1}/4}\chi _{\beta, j}} {\lambda_{\max}^{3/4} ({{\varLambda}} _{\beta, j}^{-1} )}}\right\} \\ &\varXi_{j3} =\chi_{j} \left(\frac{1}{8}+\frac{1}{2}\|{{{\mathit{\boldsymbol{W}}}}}_{j} \|_{F}^{2} \right)+\chi_{\beta, j} \left(\frac{1}{8}+\frac{1}{2}\|{{\beta}} _{j} \|_{F}^{2}\right) +\\ & \quad 0.2785n\lambda_{\beta j} b_{j} \end{align*}

 \begin{align*} &{ \varDelta}_{{{\mathit{\boldsymbol{e}}}}_{j1}} =\{{{\mathit{\boldsymbol{e}}}}_{j1} \in {\rm R}^{3\times 1}\Big| \|{{\mathit{\boldsymbol{e}}}}_{j1} \|\leqslant \sqrt{2\varXi_{j}} \}\\ &{ \varDelta}_{{{\mathit{\boldsymbol{e}}}}_{j2}} =\Big\{{{\mathit{\boldsymbol{e}}}}_{j2} \in {\rm R}^{3\times 1}\Big| \|{{\mathit{\boldsymbol{e}}}}_{j2} \|\leqslant \sqrt{{2\varXi_{j} }/{\lambda_{\min} ({{{\mathit{\boldsymbol{M}}}}}_{{\rm x}j}) }} \Big\}\\ &{ \varDelta}_{{{\tilde{\mathit{\boldsymbol{W}}}}}_{j}} =\Big\{{\tilde{\mathit{\boldsymbol{W}}}}_{j} \in {\rm R}^{l\times n}\Big| \|{{\tilde{\mathit{\boldsymbol{W}}}}}_{j} \|\leq \sqrt{{2\varXi_{j}}/{\lambda_{\min} ({{\varLambda}} _{j}^{-1}) }} \Big\}\\ &{ \varDelta}_{{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j}} =\Big\{{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \in {\rm R}^{l\times n}\Big| \|{{\tilde{\mathit{\boldsymbol{\beta}}}}} _{j} \|\leqslant \sqrt{{2\varXi_{j}}/{\lambda_{\min} ({ {\varLambda}} _{\beta, j}^{-1}) }} \Big\} \end{align*}

 \begin{align*} \frac{1}{2}{{\mathit{\boldsymbol{e}}}}_{j1}^{\rm T}{{\mathit{\boldsymbol{e}}}}_{j1} \leqslant V_{j1} \end{align*}

 \begin{align*} V_{j} \leqslant \min \left\{{\frac{\varXi_{j3}} {(1-{\rlap{--} \lambda} )\iota_{j3}^{1}}, \left({\frac{\varXi_{j3} }{(1-{\rlap{--} \lambda}) \iota_{j3}^{2}}} \right)^{\frac{1}{\xi_{j}}}} \right\} \end{align*}

 \begin{align*} \frac{1}{2}{{\mathit{\boldsymbol{e}}}}_{j1}^{\rm T}{{\mathit{\boldsymbol{e}}}}_{j1} \leqslant V_{jh} \leqslant \min \left\{{\frac{\varXi _{j3}} {(1-{\rlap{--} \lambda}) \iota_{j3}^{1}}, \left( {\frac{\varXi_{j3}} {(1-{\rlap{--} \lambda}) \iota_{j3}^{2}}} \right)^{\frac{1}{\xi_{j}}}} \right\} \end{align*}

 \begin{align*} \varXi_{j}=\min \{{{\varXi_{j3}}/({(1-{\rlap{--} \lambda} )\iota_{j3}^{1}}), ({{\varXi_{j3}}/({(1-\varXi_{j3}) \iota_{j3}^{2}})} )^{1/{\xi_{j}}}}\} \end{align*}

4 仿真和实验验证（Simulation and experimental verification） 4.1 遥操作平台理论模型

 \begin{align*} {{{\mathit{\boldsymbol{M}}}}}_{j} &= \begin{bmatrix} {m_{11}} & 0 & 0 \\ 0 & {m_{22}} & {m_{23}} \\ 0 & {m_{32}} & {m_{33}} \\ \end{bmatrix}, \; \; {{{\mathit{\boldsymbol{g}}}}}_{j} = \begin{bmatrix} 0 \\ {\theta_{5} gc_{2} +\theta_{6} gc_{23}} \\ {\theta_{6} gc_{23}} \\ \end{bmatrix}\\ {{{\mathit{\boldsymbol{C}}}}}_{j} &= \begin{bmatrix} {-a_{1} \dot{{q}}_{2}} & {-a_{1} \dot{{q}}_{1}} & {-a_{2} \dot{{q}}_{1}} \\ {a_{1} \dot{{q}}_{1}} & {-a_{3} \dot{{q}}_{3}} & {-a_{3} (\dot{{q}}_{2} +\dot{{q}}_{3}) } \\ {-a_{2} \dot{{q}}_{1}} & {a_{3} \dot{{q}}_{2}} & 0 \\ \end{bmatrix} \end{align*}

 \begin{align*} &m_{11} =\theta_{1} +\theta_{2} c_{2}^{2} +\theta_{3} c_{23}^{2} +2\theta _{4} c_{2} c_{23}\\ &m_{22} =\theta_{2} +\theta_{3} +2\theta_{4} c_{3}, \; \; m_{23} =m_{32} =\theta_{3} +\theta_{4} c_{3}\\ &m_{33} =\theta_{3}, \; \; c_{2\times 23} =\cos (2q_{2} +q_{3}) \\ &a_{1} =\theta_{2} c_{2} s_{2} +\theta_{3} c_{23} s_{23} +\theta_{4} c_{2\times 23}\\ &a_{2} =\theta_{3} c_{23} s_{23} +\theta_{3} c_{2} s_{23} \\ &a_{3} =\theta_{4} s_{3}, \; \; s_{2} =\sin q_{2}, \; \; c_{2} =\cos q_{2}, \; \; s_{3} =\sin q_{3}\\ &c_{3} =\cos q_{3}, \; \; s_{23} =\sin (q_{2} +q_{3}), \; \; c_{23} =\cos (q_{2} +q_{3}) \end{align*}

 \begin{align*} k_{j1} &=0.45, \; k_{j2} =2.85, \; b_{j}=20, \; \; \chi_{j} =\chi_{\beta, j} =0.001\\ {{{\mathit{\boldsymbol{K}}}}}_{j1} &={\rm diag\:}(5, 5, 5), \; {{{\mathit{\boldsymbol{K}}}}}_{j2} ={\rm diag\:}(5, 5, 5), \; \xi_{j} =0.595\\ {{\varLambda}} _{\beta, j} &={\rm diag\:}(350, 400, 450), \; \; {{\varGamma }}_{{\rm e}} ={\rm diag\:}(50, 50, 50) \end{align*}

${{\varLambda}} _{j} \in {\rm R}^{153\times 153}$为对角矩阵，其维数由神经网络的节点数决定，神经网络${{{\mathit{\boldsymbol{W}}}}}_{i}^{\rm T} {{{\mathit{\boldsymbol{S}}}}}_{i} ({{\mathit{\boldsymbol{X}}}}_{j})$中的${{{\mathit{\boldsymbol{X}}}}}_{j} \in {\rm R}^{4}$包含节点数为$51$，其中心均位于$[-5, 5]\times [-5, 5]\times [-5, 5]\times [-5, 5]$中，宽度为$0.2$。用于估计环境参数的神经网络${{{\mathit{\boldsymbol{W}}}}}_{{\rm e}}^{\rm T} {{{\mathit{\boldsymbol{S}}}}}_{{\rm e}} ({{\mathit{\boldsymbol{X}}}}_{{\rm e}})$中的${{\mathit{\boldsymbol{X}}}}_{{\rm e}} \in {\rm R}^{2}$同样包含51个节点，其中心也位于$[-5, 5]\times [-5, 5]$中，宽度也为0.2。

 图 3 环境力重建和时变时延 Fig.3 Environment force reconstruction and time-varying delay

 图 4 仿真过程中x-y轴上的追踪效果 Fig.4 Tracking effect in the x-y axes during the simulation
 图 5 仿真过程中z轴和3D空间的追踪效果 Fig.5 Tracking effect in the z axis and 3D space during the simulation
 图 6 仿真过程中x-y-z轴上的追踪误差 Fig.6 Tracking deviation in the x-y-z axes during the simulation
 图 7 仿真过程中本文方法的控制输入 Fig.7 Control input of the proposed method during the simulation

 图 14 文[31] 方法的控制输入 Fig.14 Control input of the method in literature [31]
 图 15 文[32] 方法的控制输入 Fig.15 Control input of the method in literature [32]

4.3 实验结果

5 结论（Conclusion）