Full-state control method for the master-slave robot system with flexible joints and time-varying delays

Abstract

A full-state control method for a master-slave robot system with flexible joints and time-varying delays is provided. In a teleoperation system formed by connecting a master robot and a slave robot through network, a proportional damping controller based on a position error and velocities, and a full-state feedback controller based on backstepping are designed for the master robot and the slave robot, respectively. High-dimension uniform accurate differentiators are designed to realize an exact difference to the virtual controllers. Delay-dependent stability criteria are established by constructing Lyapunov functions. Therefore, the criteria for selecting controller parameters are presented such that the global stability of the master-slave robot system with flexible joints and time-varying delays is realized. For the master-slave robot system with flexible joints, the global precise position tracking performance is realized by adopting a full-state feedback controller based on the backstepping method and the high-dimensional uniform accurate differentiators. Moreover, the global asymptotic convergence of the system is guaranteed and the robustness of the system is improved.

Claims

1. A full-state control method for a master-slave robot system with flexible joints and time-varying delays, comprising: connecting a master robot and a slave robot through a network to form a teleoperation system; measuring system parameters of the master robot and the slave robot, and measuring position and velocity information of joints and motors of the master robot and the slave robot in real-time; controlling the master robot with a proportional damping controller and controlling the slave robot with a full-state feedback controller, wherein a control torque provided by the proportional damping controller and a control torque provided by the full-state feedback controller are determined according to following formulas: ${\tau }_m=-k_m\left(X_{m1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)-{\alpha }_m{X}_{m2}$ ${\tau }_s=S_s\left(X_{s3}-X_{s1}\right)+J_s\left({\stackrel{.}{X}}_{s4}^{*}-\left(X_{s3}-X_{s3}^{*}\right)-k_2\left(X_{s4}-X_{s4}^{*}\right)\right),$ wherein, τ.sub.m is the control torque provided by the proportional damping controller and τ.sub.s is the control torque provided by the full-state feedback controller, X*.sub.s3 is a first virtual controller and X*.sub.s4 is a second virtual controller, {dot over (X)}*.sub.s4 is a first derivative of X*.sub.s4, X.sub.m1 is a vector of joint displacements of the master robot, X.sub.m2 is a vector of joint velocities of the master robot, X.sub.s1 is a vector of joint displacement of the slave robot, X.sub.s3 is a vector of motor positions of the slave robot, X.sub.s4 is a vector of motor velocities of the slave robot, T.sub.s(t) is a feedback time delay from the slave robot to the master robot, α.sub.m is a damping coefficient which is a positive constant, k.sub.m is >0 a proportional coefficient, S.sub.s is a diagonal positive-definite constant matrix which contains a joint stiffness of the slave robot, J.sub.s is a diagonal constant matrix of a moment of actuator inertia, and k.sub.2 is selected to be a positive constant.

2. The full-state control method of the master-slave robot system with flexible joints and time-varying delays according to claim 1, wherein the system parameters of the master robot and the slave robot comprises: length and mass of manipulators of the master robot and the slave robot, positions and positive-definite inertia matrices M.sub.m(X.sub.m1) and M.sub.s(X.sub.s1) matrices of centripetal and coriolis torques C.sub.m(q.sub.m, {dot over (q)}.sub.m) and C.sub.s(q.sub.s, {dot over (q)}.sub.s) gravity torques G.sub.m(X.sub.m1) and G.sub.s(X.sub.s1), diagonal constant matrix J.sub.s of moment of actuator inertia of the master robot and the slave robot, and a diagonal positive-definite constant matrix S.sub.s that contains joint stiffness of the slave robot.

3. The full-state control method of the master-slave robot system with flexible joints and time-varying delays according to claim 2, wherein the first virtual controller X*.sub.s3 and the second virtual controller X*.sub.s4 are determined according to following formulas: $X_{s3}^{*}=X_{s1}+S_s^{-1}\left(k_s\left(X_{m1}\left(t-T_m\left(t\right)\right)-X_{s1}\right)-{\alpha }_s{X}_{s2}\right),X_{s4}^{*}={\stackrel{.}{X}}_{s3}^{*}-2\frac{k_m}{k_s}{S}_s^T{X}_{s2}-k_1\left(X_{s3}-X_{s3}^{*}\right),$ wherein, subscripts m and s denotes the master robot and the slave robot, respectively; {dot over (X)}*.sub.s3 is a first derivative of X*.sub.s3, X.sub.s2 is a vector of joint velocities of the slave robot, T.sub.m(t) denotes a forward time delay from the master robot to the slave robot, α.sub.s is a damping coefficient which is a positive constant, k.sub.m, k.sub.s>0 are proportional coefficients, S.sub.s.sup.−1 and S.sub.s.sup.T are an inverse matrix and a transpose matrix of the diagonal positive-definite constant matrix S.sub.s, respectively, and k.sub.1 is selected to be a positive constant.

4. The full-state control method of the master-slave robot system with flexible joints and time-varying delays according to claim 3, wherein {dot over (X)}*.sub.s3 is determined according to following formulas:
X.sub.1=X*.sub.s3,X.sub.2={dot over (X)}*.sub.s3,σ.sub.1=X.sub.1−Y.sub.1,σ.sub.2=X.sub.2−Y.sub.2, $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{X}}_1=X_2\\ {\stackrel{.}{X}}_2={\stackrel{\mathrm{.Math.}}{X}}_{s3}^{*}\end{array}& \left(1\right)\\ \{\begin{array}{c}{\stackrel{.}{Y}}_1={\lambda }_1\frac{{\sigma }_1}{{.Math.{\sigma }_1.Math.}^{1/2}}+{\lambda }_2{\sigma }_1{.Math.{\sigma }_1.Math.}^{P-1}+Y_2\\ {\stackrel{.}{Y}}_2={\alpha }_1\frac{{\sigma }_1}{.Math.{\sigma }_1.Math.}\end{array}& \left(2\right)\\ \{\begin{array}{c}{\stackrel{.}{\sigma }}_1=-{\lambda }_1\frac{{\sigma }_1}{{.Math.{\sigma }_1.Math.}^{1/2}}-{\lambda }_2{\sigma }_1{.Math.{\sigma }_1.Math.}^{P-1}+{\sigma }_2\\ {\stackrel{.}{\sigma }}_2=-{\alpha }_1\frac{{\sigma }_1}{.Math.{\sigma }_1.Math.}+{\stackrel{\mathrm{.Math.}}{X}}_{s3}^{*}\end{array}& \left(3\right)\end{array}$ wherein, {umlaut over (X)}*.sub.s3 denotes a second derivative of X*.sub.s3, Y.sub.1 is an estimate of X*.sub.s3, Y.sub.2 is an estimate of {dot over (X)}*.sub.s3, σ.sub.1 and σ.sub.2 are estimation errors, λ.sub.1, λ.sub.2, α.sub.1>0 are system control gains, P>1 is a constant, {umlaut over (X)}*.sub.s3 is supposed to be bounded and satisfies ∥{umlaut over (X)}*.sub.s3∥≤L.sub.3 with a known constant L.sub.3>0, and conditions α.sub.1>4L.sub.3, λ.sub.1>√{square root over (2α.sub.1)} are satisfied.

5. The full-state control method of the master-slave robot system with flexible joints and time-varying delays according to claim 3, wherein {dot over (X)}*.sub.s4 is determined according to following formulas:
X.sub.3=X*.sub.s4,X.sub.4={dot over (X)}*.sub.s4,σ.sub.3=X.sub.3−Y.sub.3,σ.sub.4=X.sub.4−Y.sub.4, $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{X}}_3=X_4\\ {\stackrel{.}{X}}_4={\stackrel{\mathrm{.Math.}}{X}}_{s4}^{*}\end{array}& \left(4\right)\\ \{\begin{array}{c}{\stackrel{.}{Y}}_3={\lambda }_3\frac{{\sigma }_3}{{.Math.{\sigma }_3.Math.}^{1/2}}+{\lambda }_4{\sigma }_3{.Math.{\sigma }_3.Math.}^{P-1}+Y_4\\ {\stackrel{.}{Y}}_4={\alpha }_2\frac{{\sigma }_3}{.Math.{\sigma }_3.Math.}\end{array}& \left(5\right)\\ \{\begin{array}{c}{\stackrel{.}{\sigma }}_3=-{\lambda }_3\frac{{\sigma }_3}{{.Math.{\sigma }_3.Math.}^{1/2}}-{\lambda }_4{\sigma }_3{.Math.{\sigma }_3.Math.}^{P-1}+{\sigma }_4\\ {\stackrel{.}{\sigma }}_4=-{\alpha }_2\frac{{\sigma }_3}{.Math.{\sigma }_3.Math.}+{\stackrel{\mathrm{.Math.}}{X}}_{s4}^{*}\end{array}& \left(6\right)\end{array}$ wherein, {umlaut over (X)}*.sub.s4 is a second derivative of X*.sub.s4, Y.sub.3 is an estimate of X*.sub.s4, Y.sub.4 is an estimate of {dot over (X)}*.sub.s4, σ.sub.3 and σ.sub.4 are estimation errors, λ.sub.3, λ.sub.4, α.sub.2>0 are system control gains, {umlaut over (X)}*.sub.s4 is supposed to be bounded and satisfies ∥{umlaut over (X)}*.sub.s4∥≤L.sub.4 with a known positive constant L.sub.4>0, and conditions α.sub.2>4L.sub.4, λ.sub.3>√{square root over (2α.sub.2)} are satisfied.

6. The full-state control method of the master-slave robot system with flexible joints and time-varying delays according to claim 3, wherein parameters α.sub.m, α.sub.s, k.sub.m, k.sub.s, I, T.sub.m, T.sub.s, Z, P of the controllers are selected such that the following inequalities hold, $-2{\alpha }_{m}I+{\stackrel{\_}{T}}_{m}Z+{\stackrel{\_}{T}}_s{k}_m^2{P}^{-1}<0,-2\frac{k_m{\alpha }_s}{k_s}I+{\stackrel{\_}{T}}_{s}P+{\stackrel{\_}{T}}_m{k}_m^2{Z}^{-1}<0,$ wherein, I is an identity matrix, Z.sup.−1 and P.sup.−1 are inverse matrices of positive definite matrices Z and P, respectively, and the time delay T.sub.m(t) and T.sub.s(t) are bounded, i.e. there are positive scalars T.sub.m and T.sub.s, such that the inequalities T.sub.m(t)≤T.sub.m, T.sub.s(t)≤T.sub.s hold.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic control block diagram of the present invention.

DETAILED DESCRIPTION

(2) The invention is further explained in combination with the attached FIGURE as follows.

(3) As shown in FIG. 1, the steps of the invention method are as follows:

(4) Step 1: connecting the master robot and the slave robot through the network to form a teleoperation system, measuring the system parameters of the master robot and the slave robot, and measuring a force exerted by an operator and a force exerted by the external environment by using force sensors.

(5) The system parameters of the master robot and the slave robot includes the length and the mass of the manipulators, the positions and the positive-definite inertia matrices M.sub.m(X.sub.m1) and M.sub.s(X.sub.s1), the matrices of centripetal and coriolis torques C.sub.m(q.sub.m, {dot over (q)}.sub.m) and C.sub.s(q.sub.s, {dot over (q)}.sub.s), the gravity torques G.sub.m(X.sub.m1) and G.sub.s(X.sub.s1), the diagonal constant matrix J.sub.s of the moment of actuator inertia of the master robot and the slave robot, and a diagonal positive-definite constant matrix S.sub.s that contains the joint stiffness of the slave robot respectively according to the length and the mass of the manipulators. Moreover, a force F.sub.h exerted by the operator to the master robot and a force F.sub.e exerted by the external environment to the slave robot are measured by using force sensors.

(6) The system dynamics equation can be described as
M.sub.m(q.sub.m){umlaut over (q)}.sub.m+C.sub.m(q.sub.m,{dot over (q)}.sub.m){dot over (q)}.sub.m+G.sub.m(q.sub.m)=τ.sub.m+F.sub.h
M.sub.s(q.sub.s){umlaut over (q)}.sub.s+C.sub.s(q.sub.s,{dot over (q)}.sub.s){dot over (q)}.sub.s+G.sub.s(q.sub.s)=S.sub.s(θ.sub.s−q.sub.s)−F.sub.e
J.sub.s{umlaut over (θ)}.sub.s+S.sub.s(θ.sub.s−q.sub.s)=τ.sub.s  (1)
where the subscripts m and s denotes the master robot and the slave robot, respectively. q.sub.m, q.sub.s∈R.sup.n are the vectors of joint displacements. {dot over (q)}.sub.m, {dot over (q)}.sub.s∈R.sup.n are the vectors of joint velocities. {umlaut over (q)}.sub.m, {umlaut over (q)}.sub.s∈R.sup.n are the vectors of joint accelerations. θ.sub.s∈R.sup.n is the vector of motor displacements, {umlaut over (θ)}.sub.s∈R.sup.n is the vector of motor accelerations. M.sub.m(q.sub.m), M.sub.s(q.sub.s)∈R.sup.n×n are the positive-definite inertia matrices of the system, C.sub.m(q.sub.m, {dot over (q)}.sub.m), C.sub.s(q.sub.s, {umlaut over (q)}.sub.s)∈R.sup.n are the matrices of centripetal and coriolis torques, G.sub.m(q.sub.m), G.sub.s(q.sub.s)∈R.sup.n are the gravitational torques. J.sub.s∈R.sup.n×n is the diagonal constant matrix of the moments of actuator inertia. S.sub.s∈R.sup.n is a diagonal positive-definite constant matrix that contains the joint stiffness of the slave robot. F.sub.h, F.sub.e∈R.sup.n are the force exerted by the operator to the master robot and the force exerted by the external environment to the slave robot, respectively. τ.sub.m, τ.sub.s∈R.sup.n are control torques provided by the controllers.

(7) Step 2: measuring the position and velocity information of joints and motors in real-time, designing a proportional damping controller for the master robot based on position error and velocities, and designing a full-state feedback controller for the slave robot based on backstepping recursive technology in combination with Lyapunov equation.

(8) For the master robot with rigid joints, letting X.sub.m1=q.sub.m, X.sub.m2={dot over (q)}.sub.m, the state space expression of the system is obtained as follows:

(9) $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{X}}_{m1}=X_{m2}\\ {\stackrel{.}{X}}_{m2}=M_m^{-1}\left(X_{m1}\right)\left({\tau }_m+F_h-C_m\left(X_{m1},X_{m2}\right)X_{m2}-G_m\left(X_{m1}\right)\right)\end{array}& \left(2\right)\end{array}$
where the subscripts m denotes the master robot, q.sub.m∈R.sup.n is the vector of joint displacements, {dot over (q)}.sub.m∈R.sup.n is the vector of joint velocities, M.sub.m.sup.−1(X.sub.m1) is the inverse matrix of the positive-definite inertia matrix M.sub.m(X.sub.m1), τ.sub.m∈R.sup.n is a control torque provided by the controller, C.sub.m(q.sub.m, {dot over (q)}.sub.m)∈R.sup.n is the matrix of centripetal and coriolis torque, G.sub.m(q.sub.m)∈R.sup.n is the gravitational torque, and F.sub.h∈R.sup.n is a force exerted by an operator to the master robot.

(10) For the slave robot with flexible joints, letting X.sub.s1=q.sub.s, X.sub.s2={dot over (q)}.sub.s, X.sub.s3=θ.sub.s, X.sub.s4={dot over (θ)}.sub.s, the state space expression of the system is obtained as follows:

(11) $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{X}}_{s1}=X_{s2}\\ {\stackrel{.}{X}}_{s2}=M_s^{-1}\left(X_{s1}\right)\left(S_s\left(X_{s3}-X_{s1}\right)-F_e-C_s\left(X_{s1},X_{s2}\right)X_{s2}-G_s\left(X_{s1}\right)\right)\\ {\stackrel{.}{X}}_{s3}=X_{s4}\\ {\stackrel{.}{X}}_{s4}=J_s^{-1}\left({\tau }_s-S_s\left(X_{s3}-X_{s1}\right)\right)\end{array}& \left(3\right)\end{array}$
where the subscripts s denotes the master robot, q.sub.s∈R.sup.n is the vector of joint displacements, {dot over (q)}.sub.s∈R.sup.n is the vector of joint velocities, θ.sub.s∈R.sup.n is the vector of motor displacements, {dot over (θ)}.sub.s∈R.sup.n is the vector of motor velocities. M.sub.s.sup.−1(X.sub.s1) is the inverse matrix of the positive-definite inertia matrix M.sub.s(X.sub.s1), J.sub.s.sup.−1 is the inverse matrix of the diagonal constant matrix J.sub.s of the moments of actuator interia, τ.sub.s∈R.sup.n is the control torque of the controller, C.sub.s(q.sub.s, {dot over (q)}.sub.s)∈R.sup.n is the matrix of centripetal and coriolis torque, G.sub.s(q.sub.s)∈R.sup.n is the gravitational torque, F.sub.e is a force exerted by the external environment to the slave robot, and, S.sub.s∈R.sup.n×n is a diagonal positive-definite constant matrix that contains the joint stiffness of the slave robot.

(12) The first Lyapunov Equation is selected as follows,

(13) $\begin{array}{cc}{V}_1=V_{11}+V_{12}+V_{13}{V}_{11}=X_{m2}^T{M}_m\left(X_{m1}\right)X_{m2}+\frac{k_m}{k_s}{X}_{s2}^T{M}_S\left(X_{s1}\right)X_{s2}+2\left(U_m\left(X_{m1}\right)-{\beta }_m\right)+\frac{2k_m}{k_s}\left(U_s\left(X_{s1}\right)-{\beta }_s\right)+2{\int }_0^t\left(-X_{m2}^T\left(\sigma \right)F_h\left(\sigma \right)+\frac{k_m}{k_s}{X}_{s2}^T\left(\sigma \right)F_e\left(\sigma \right)\right)d\sigma V_{12}={k_m\left(X_{m1}-X_{s1}\right)}^T\left(X_{m1}-X_{s1}\right)V_{13}={\int }_{-{\overline{T}}_m}^0{\int }_{t+\theta }^t{X}_{m2}^T\left(\xi \right)Z{X}_{m2}\left(\xi \right)d\xi d\theta +{\int }_{-\overline{T_s}}^0{\int }_{t+\theta }^t{X}_{s2}^T\left(\xi \right)P{X}_{s2}\left(\xi \right)d\xi d\theta & \left(4\right)\end{array}$
where the integral terms satisfy ∫.sub.0.sup.t−X.sub.m2.sup.T(σ)F.sub.h(σ)dσ≥0, ∫.sub.0.sup.tX.sub.s2.sup.T(σ)F.sub.e(σ)dσ≥0. Z and P are positive definite matrices. It is supposed that the time delay T.sub.m(t) and T.sub.s(t) are bounded, i.e. there are positive scalars T.sub.m and T.sub.s, such that T.sub.m(t)≤T.sub.m, T.sub.s(t)≤T.sub.s. M.sub.m(M.sub.m1) and M.sub.s(X.sub.s1) are positive-definite inertia matrices of the master robot and the slave robot, respectively.
U.sub.m(X.sub.m1) and U.sub.s(X.sub.s1) are the potential energy of the master robot and the slave robot satisfying

(14) $G_m\left(X_{m1}\right)=\frac{\partial U_m\left(X_{m1}\right)}{\partial X_{m1}},G_s\left(X_{s1}\right)=\frac{\partial U_s\left(X_{s1}\right)}{\partial X_{s1}}.$
There are positive scalars β.sub.m and β.sub.s such that U.sub.m(X.sub.m1)≥β.sub.m, U.sub.s(X.sub.s1)≥β.sub.s, and k.sub.m, k.sub.s>0 are proportional coefficients.

(15) The time derivatives of V.sub.1, V.sub.11, V.sub.12, V.sub.13 are given by

(16) 0$\begin{array}{cc}\begin{array}{c}{\stackrel{.}{V}}_{11}=2X_{m2}^T\left(F_h-k_m\left(X_{m1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)-{\alpha }_m{X}_{m2}-C_m\left(X_{m1},X_{m2}\right)X_{m2}-G_m\left(X_{m1}\right)\right)\\ +2\frac{k_m}{k_s}{X}_{s2}^T\left(s,\left(X_{s3}-X_{s1}\right)-F_e-C_s\left(X_{s1},X_{s2}\right)X_{s2}-G_s\left(X_{s1}\right)\right)+\frac{2k_m}{k_s}{G}_s\left(X_{s1}\right)X_{s2}^T\\ +X_{m2}^T{\stackrel{.}{M}}_m\left(X_{m1}\right)X_{m2}+\frac{k_m}{k_s}{X}_{s2}^T{\stackrel{.}{M}}_s\left(X_{s1}\right)X_{s2}+2G_m\left(X_{m1}\right)X_{m2}^T-2X_{m2}^T{F}_h+2\frac{k_m}{k_s}{X}_{s2}^T{F}_e\\ =-2k_m{X}_{m2}^T\left(X_{m1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)-2{\alpha }_m{X}_{m2}^T{X}_{m2}+2\frac{k_m}{k_s}{X}_{s2}^T{S}_s\left(X_{s3}-X_{s1}\right)\\ =-2k_m{X}_{m2}^T\left(X_{m1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)-2{\alpha }_m{X}_{m2}^T{X}_{m2}+2\frac{k_m}{k_s}{X}_{s2}^T\left(S_s\left(X_{s3}-X_{s3}^{*}\right)+S_s{X}_{s3}^{*}-S_s{X}_{s1}\right)\end{array}& \left(5\right)\end{array}$

(17) With Equation (5), the first virtual controller is derived as X*.sub.s3=X.sub.s1±S.sub.s.sup.−1(k.sub.s(X.sub.m1(t−T.sub.m(t))−X.sub.s1)−α.sub.sX.sub.s2). By substituting the first virtual controller X*.sub.s3 into Equation (5),

(18) $\begin{array}{cc}\begin{array}{c}{\stackrel{.}{V}}_{11}=-2k_m{X}_{m2}^T\left(X_{m1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)-2{\alpha }_m{X}_{m2}^T{X}_{m2}\\ +2\frac{k_m}{k_s}{X}_{s2}^T\left(S_s\left(X_{s3}-X_{s3}^{*}\right)+k_s\left(X_{m1}\left(t-T_m\left(t\right)\right)-X_{s1}\right)-{\alpha }_s{X}_{s2}\right)\\ =-2{\alpha }_m{X}_{m2}^T{X}_{m2}-2\frac{k_m{\alpha }_s}{k_s}{X}_{s2}^T{X}_{\mathrm{s2}}-2k_m{X}_{m2}^T\left(X_{m1}-X_{s1}\right)-2k_m{X}_{m2}^T\left(X_{s1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)\\ -2k_m{X}_{s2}^T\left(X_{s1}-X_{m1}\right)-2k_m{X}_{s2}^T\left(X_{m1}-X_{m1}\left(t-T_m\left(t\right)\right)\right)+2\frac{k_m}{k_s}{X}_{s2}^T{S}_s\left(X_{s3}-X_{s3}^{*}\right)\\ =-2{\alpha }_m{X}_{m2}^T{X}_{m2}-2{k_m\left(X_{m1}-X_{s1}\right)}^T\left(X_{m2}-X_{s2}\right)-2\frac{k_m{\alpha }_s}{k_s}{X}_{s2}^T{X}_{s2}-2k_m{X}_{m2}^T{\int }_{t-T_s\left(t\right)}^t{X}_{s2}\left(\xi \right)d\xi \\ -2k_m{X}_{s2}^T{\int }_{t-T_m\left(t\right)}^t{X}_{m2}\left(\xi \right)d\xi +2\frac{k_m}{k_s}{X}_{s2}^T{S}_s\left(X_{s3}-X_{s3}^{*}\right)\end{array}{\stackrel{.}{V}}_{12}=2{k_m\left(X_{m2}-X_{s2}\right)}^T\left(X_{m1}-X_{s1}\right)\begin{array}{c}{\stackrel{.}{V}}_{13}={\overline{T}}_m{X}_{m2}^{T}Z{X}_{m2}-{\int }_{t-{\overline{T}}_m}^t{X}_{m2}^T\left(\xi \right)Z{X}_{m2}\left(\xi \right)d\xi +\overline{T_s}{X}_{s2}^{T}P{X}_{s2}-{\int }_{t-{\overline{T}}_s}^t{X}_{s2}^T\left(\xi \right)P{X}_{s2}\left(\xi \right)d\xi \\ \le {\overline{T}}_m{X}_{m2}^{T}Z{X}_{m2}-{\int }_{t-T_m\left(t\right)}^t{X}_{m2}^T\left(\xi \right)Z{X}_{m2}\left(\xi \right)d\xi +\overline{T_s}{X}_{s2}^{T}P{X}_{s2}-{\int }_{t-T_s\left(t\right)}^t{X}_{s2}^T\left(\xi \right)P{X}_{s2}\left(\xi \right)d\xi \end{array}& \left(6\right)\end{array}$
is obtained. Furthermore, with the inequality, it holds that

(19) $\begin{array}{cc}-2k_m{X}_{m2}^T{\int }_{t-T_s\left(t\right)}^t{X}_{s2}\left(\xi \right)d\xi -{\int }_{t-T_s\left(t\right)}^t{X}_{s2}^T\left(\xi \right)P{X}_{s2}\left(\xi \right)d\xi \le \overline{T_s}{k}_m^2{X}_{m2}^T\left(t\right)P^{-1}{X}_{m2}\left(t\right)-2k_m{X}_{s2}^T{\int }_{t-T_m\left(t\right)}^t{X}_{m2}\left(\xi \right)d\xi -{\int }_{t-T_m\left(t\right)}^t{X}_{m2}^T\left(\xi \right)Z{X}_{m2}\left(\xi \right)d\xi \le {\stackrel{\_}{T}}_m{k}_m^2{X}_{s2}^T\left(t\right)Z^{-1}{X}_{s2}\left(t\right)\mathrm{So}& \left(7\right)\\ \begin{array}{c}{\stackrel{.}{V}}_1={\stackrel{.}{V}}_{11}+{\stackrel{.}{V}}_{12}+{\stackrel{.}{V}}_{13}\le X_{m2}^T\left(-2{\alpha }_{m}I+{\stackrel{\_}{T}}_{m}Z+\overline{T_s}{k}_m^2{P}^{-1}\right)X_{m2}\\ +X_{s2}^T\left(-2\frac{k_m{\alpha }_s}{k_s}I+\overline{T_s}P+{\stackrel{\_}{T}}_m{k}_m^2{Z}^{-1}\right)X_{s2}+2\frac{k_m}{k_s}{X}_{s2}^T{S}_s\left(X_{s3}-X_{s3}^{*}\right)\end{array}& \left(8\right)\end{array}$
is derived, where I is the identity matrix, z.sup.−1 and P.sup.−1 are the inverse matrices of positive definite matrices Z and P, respectively. S.sub.s is a diagonal positive-definite constant matrix which contains the joint stiffness of the slave robot. α.sub.m and α.sub.s are damping coefficients which are positive constants. k.sub.m, k.sub.s>0 are proportional coefficients. It is supposed that the time delay T.sub.m(t) and T.sub.s(t) are bounded, i.e. there are positive scalars T.sub.m and T.sub.s, such that T.sub.m(t)≤T.sub.m, T.sub.s(t)≤T.sub.s, X*.sub.s3 is the first virtual controller.

(20) The second Lyapunov Equation is selected as follows,
V.sub.2=V.sub.1+½(X.sub.s3−X*.sub.s3).sup.T(X.sub.s3−X*.sub.s3)  (9)

(21) The time derivative of V.sub.2 is given by
{dot over (V)}.sub.2={dot over (V)}.sub.1+(X.sub.s3−X*.sub.s3).sup.T(X.sub.s4−{dot over (X)}*.sub.s3)={dot over (V)}.sub.1+(X.sub.s3−X*.sub.s3).sup.T(X.sub.s4−X*.sub.s4+X*.sub.s4−{dot over (X)}*.sub.s3)  (10)
With equation (10), the second virtual controller is derived as

(22) $X_{s4}^{*}={\stackrel{.}{X}}_{s3}^{*}-2\frac{k_m}{k_s}{S}_s^T{X}_{s2}-k_1\left(X_{s3}-X_{s3}^{*}\right).$
By substituting the second virtual controller X*.sub.s4 into Equation (10),

(23) $\begin{array}{cc}{\stackrel{.}{V}}_2\le X_{m2}^T\left(-2{\alpha }_{m}I+{\overline{T}}_{m}Z+\overline{T_s}{k}_m^2{P}^{-1}\right)X_{m2}+X_{s2}^T\left(-2\frac{k_m{\alpha }_s}{k_s}I+\overline{T_s}P+{\overline{T}}_m{k}_m^2{Z}^{-1}\right)X_{s2}+{\left(X_{s3}-X_{s3}^{*}\right)}^T\left(X_{s4}-X_{s4}^{*}\right)-{k_1\left(X_{s3}-X_{s3}^{*}\right)}^T\left(X_{s3}-X_{s3}^{*}\right)& \left(11\right)\end{array}$
is derived.

(24) The third Lyapunov Equation is selected as follows,
V.sub.3=V.sub.2+½(X.sub.s4−X*.sub.s4).sup.T(X.sub.s4−X*.sub.s4)  (12)

(25) The time derivative of V.sub.3 is given by

(26) $\begin{array}{cc}\begin{array}{c}{\stackrel{.}{V}}_3={\stackrel{.}{V}}_2+{\left(X_{s4}-X_{s4}^{*}\right)}^T\left({\stackrel{.}{X}}_{s4}-{\stackrel{.}{X}}_{s4}\right)\\ ={\stackrel{.}{V}}_2+{\left(X_{s4}-X_{s4}^{*}\right)}^T\left(J_s^{-1}\left({\tau }_s-S_s\left(X_{s3}-X_{s1}\right)\right)-X_{s4}^{*}\right)\end{array}& \left(13\right)\end{array}$

(27) The full-state feedback controller is derived from Equation (13) as

(28) τ.sub.s=S.sub.s(X.sub.s3−X.sub.s1)+J.sub.s({dot over (X)}*.sub.s4−(X.sub.s3−X*.sub.s3)−k.sub.2(X.sub.s4−X*.sub.s4)). By substituting the obtained all-state feedback controller τ.sub.s into Equation (13),

(29) $\begin{array}{cc}{\stackrel{.}{V}}_3\le X_{m2}^T\left(-2{\alpha }_{m}I+{\overline{T}}_{m}Z+\overline{T_s}{k}_m^2{P}^{-1}\right)X_{m2}+X_{s2}^T\left(-2\frac{k_m{\alpha }_s}{k_s}I+\overline{T_s}P+{\overline{T}}_m{k}_m^2{Z}^{-1}\right)X_{s2}-{k_1\left(X_{s3}-X_{s3}^{*}\right)}^T\left(X_{s3}-X_{s3}^{*}\right)-{k_2\left(X_{s4}-X_{s4}^{*}\right)}^T\left(X_{s4}-X_{s4}^{*}\right)& \left(14\right)\end{array}$
is derived.

(30) The controllers of the master-slave robot system with flexible joints are obtained by using the backstepping method as

(31) $\begin{array}{cc}{\tau }_m=-k_m\left(X_{m1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)-{\alpha }_m{X}_{m2}{\tau }_s=S_s\left(X_{s3}-X_{s1}\right)+J_s\left({\stackrel{.}{X}}_{s4}^{*}-\left(X_{s3}-X_{s3}^{*}\right)-k_2\left(X_{s4}-X_{s4}^{*}\right)\right)X_{s3}^{*}=X_{s1}+S_s^{-1}\left(k_s\left(X_{m1}\left(t-T_m\left(t\right)\right)-X_{s1}\right)-{\alpha }_s{X}_{s2}\right)X_{s4}^{*}={\stackrel{.}{X}}_{s3}^{*}-2\frac{k_m}{k_s}{S}_s^T{X}_{s2}-k_1\left(X_{s3}-X_{s3}^{*}\right)& \left(15\right)\end{array}$
where τ.sub.m and τ.sub.s are control torques provided by the controllers, X*.sub.s3 and X*.sub.s4 are the first virtual controller and the second virtual controller, respectively, {dot over (X)}*.sub.s3 and {dot over (X)}*.sub.s4 are the first derivatives of the virtual controllers X*.sub.s3 and X*.sub.s4, respectively, T.sub.m(t) and T.sub.s(t) are forward time delay (from the master robot to the slave robot) and backward time delay (from the slave robot to the master robot), respectively. α.sub.m and α.sub.s are damping coefficients which are positive constants. k.sub.m, k.sub.s>0 are proportional coefficients. S.sub.s.sup.−1 and S.sub.s.sup.T are the inverse matrix and the transpose matrix of a diagonal positive-definite constant matrix S.sub.s which contains the joint stiffness of the slave robot, respectively. J.sub.s is the diagonal constant matrix of the moments of actuator inertia. k.sub.1 and k.sub.2 are selected to be positive constants.

(32) Step 3: designing high-dimensional uniform accurate differentiators to carry out a precise difference to the first virtual controller and the second virtual controller.

(33) Letting X.sub.1=X*.sub.s3, X.sub.2={dot over (X)}*.sub.s3, σ.sub.1=X.sub.1−Y.sub.1, σ.sub.2=X.sub.2−Y.sub.2,

(34) $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{X}}_1=X_2\\ {\stackrel{.}{X}}_2={\stackrel{\mathrm{.Math.}}{X}}_{s3}^{*}\end{array}& \left(16\right)\\ \{\begin{array}{c}{\stackrel{.}{Y}}_1={\lambda }_1\frac{{\sigma }_1}{{.Math.{\sigma }_1.Math.}^{1/2}}+{\lambda }_2{\sigma }_1{.Math.{\sigma }_1.Math.}^{P-1}+Y_2\\ {\stackrel{.}{Y}}_2={\alpha }_1\frac{{\sigma }_1}{.Math.{\sigma }_1.Math.}\end{array}& \left(17\right)\end{array}$
with Equations (16) and (17),

(35) $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{\sigma }}_1=-{\lambda }_1\frac{{\sigma }_1}{{.Math.{\sigma }_1.Math.}^{1/2}}-{\lambda }_2{\sigma }_1{.Math.{\sigma }_1.Math.}^{P-1}+{\sigma }_2\\ {\stackrel{.}{\sigma }}_2=-{\alpha }_1\frac{{\sigma }_1}{.Math.{\sigma }_1.Math.}+{\stackrel{\mathrm{.Math.}}{X}}_{s3}^{*}\end{array}& \left(18\right)\end{array}$
is derived, where X*.sub.s3 denotes the first virtual controller, {dot over (X)}*.sub.s3 denotes the first derivative of the first virtual controller X*.sub.s3, {umlaut over (X)}.sub.s3 denotes the second derivative of the first virtual controller X*.sub.s3, Y.sub.1 is an estimate of the first virtual controller X*.sub.s3, Y.sub.2 is an estimate of {dot over (X)}*.sub.s3, σ.sub.1 and σ.sub.2 are estimation errors, λ.sub.1, λ.sub.2, α.sub.1>0 are system control gains, P>1 is a constant, {umlaut over (X)}*.sub.s3 is supposed to be bounded and satisfies ∥{umlaut over (X)}*.sub.s3∥≤L.sub.3 with a known constant L.sub.3>0. If the parameters are selected to satisfy the conditions α.sub.1>4L.sub.3, λ.sub.1>√{square root over (2α.sub.1)}, the estimation errors σ.sub.1, σ.sub.2 will converge to the origin quickly, thus a precise difference value {dot over (X)}*.sub.s3 of X*.sub.s3 is obtained.

(36) Letting X.sub.3=X*.sub.s4, X.sub.4={dot over (X)}*.sub.s4, σ.sub.3=X.sub.3−Y.sub.3, σ.sub.4=X.sub.4−Y.sub.4,

(37) 0$\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{X}}_3=X_4\\ {\stackrel{.}{X}}_4={\stackrel{\mathrm{.Math.}}{X}}_{s4}^{*}\end{array}& \left(19\right)\\ \{\begin{array}{c}{\stackrel{.}{Y}}_3={\lambda }_3\frac{{\sigma }_3}{{.Math.{\sigma }_3.Math.}^{1/2}}+{\lambda }_4{\sigma }_3{.Math.{\sigma }_3.Math.}^{P-1}+Y_4\\ {\stackrel{.}{Y}}_4={\alpha }_2\frac{{\sigma }_3}{.Math.{\sigma }_3.Math.}\end{array}& \left(20\right)\end{array}$
with Equations (19) and (20),

(38) $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{\sigma }}_3=-{\lambda }_3\frac{{\sigma }_3}{{.Math.{\sigma }_3.Math.}^{1/2}}-{\lambda }_4{\sigma }_3{.Math.{\sigma }_3.Math.}^{P-1}+{\sigma }_4\\ {\stackrel{.}{\sigma }}_4=-{\alpha }_2\frac{{\sigma }_3}{.Math.{\sigma }_3.Math.}+{\stackrel{\mathrm{.Math.}}{X}}_{s4}^{*}\end{array}& \left(21\right)\end{array}$
is derived, where X*.sub.s4 denotes the second virtual controller, {dot over (X)}*.sub.s4 denotes the first derivative of the second virtual controller X*.sub.s4, {umlaut over (X)}*.sub.s4 denotes the second derivative of the second virtual controller X*.sub.s4, Y.sub.3 is an estimate of the second virtual controller X*.sub.s4, Y.sub.4 is an estimate of {dot over (X)}*.sub.s4, σ.sub.3 and σ.sub.4 are estimation errors, λ.sub.3, λ.sub.4, α.sub.2>0 are system control gains, {umlaut over (X)}*.sub.s4 is supposed to be bounded and satisfies ∥{umlaut over (X)}*.sub.s4∥≤L.sub.4 with a known positive constant L.sub.4>0. If the parameters are selected to satisfy the conditions α.sub.2>4L.sub.4, λ.sub.3>√{square root over (2α.sub.2)}, the estimation errors σ.sub.3, σ.sub.4 will converge to the origin quickly, thus a precise difference value {dot over (X)}*.sub.s4 of X*.sub.s4 is obtained.

(39) Step 4: Establishing the delay-dependent system stability criteria by constructing Lyapunov Equations, providing the criteria for selecting controller parameters, and realizing the global stability of the master-slave robot system with flexible joints and time-varying delays.

(40) The Lyapunov Equation is selected as

(41) $\begin{array}{cc}{V}_{11}=X_{m2}^T{M}_m\left(X_{m1}\right)X_{m2}+\frac{k_m}{k_s}{X}_{s2}^T{M}_s\left(X_{s1}\right)X_{s2}+2\left(U_m\left(X_{m1}\right)-{\beta }_m\right)+\frac{2k_m}{k_s}\left(U_s\left(X_{s1}\right)-{\beta }_s\right)+2{\int }_0^t\left(-X_{m2}^T\left(\sigma \right)F_h\left(\sigma \right)+\frac{k_m}{k_s}{X}_{s2}^T\left(\sigma \right)F_e\left(\sigma \right)\right)d\sigma V_{12}={k_m\left(X_{m1}-X_{s1}\right)}^T\left(X_{m1}-X_{s1}\right)V_{13}={\int }_{-{\stackrel{\_}{T}}_m}^0{\int }_{t+\theta }^t{X}_{m2}^T\left(\xi \right)Z{X}_{m2}\left(\xi \right)d\xi d\theta +{\int }_{-{\stackrel{\_}{T}}_s}^0{\int }_{t+\theta }^t{X}_{s2}^T\left(\xi \right)P{X}_{s2}\left(\xi \right)d\mathrm{\xi d\theta }{V}_1=V_{11}+V_{12}+V_{13}{V}_2=V_1+\frac{1}{2}{\left(X_{s3}-X_{s3}^{*}\right)}^T\left(X_{s3}-X_{s3}^{*}\right)V_3=V_2+\frac{1}{2}{\left(X_{s4}-X_{s4}^{*}\right)}^T\left(X_{s4}-X_{s4}^{*}\right)& \left(22\right)\end{array}$

(42) The time derivative of V.sub.3 is given by

(43) $\begin{array}{cc}{\stackrel{.}{V}}_3\le X_{m2}^T\left(-2{\alpha }_{m}I+{\stackrel{\_}{T}}_{m}Z+{\stackrel{\_}{T}}_s{k}_m^2{P}^{-1}\right)X_{m2}+X_{s2}^T\left(-2\frac{k_m{\alpha }_s}{k_s}I+{\stackrel{\_}{T}}_{s}P+{\stackrel{\_}{T}}_m{k}_m^2{Z}^{-1}\right)X_{s2}-{k_1\left(X_{s3}-X_{s3}^{*}\right)}^T\left(X_{s3}-X_{s3}^{*}\right)-{k_2\left(X_{s4}-X_{s4}^{*}\right)}^T\left(X_{s4}-X_{s4}^{*}\right)& \left(23\right)\end{array}$

(44) When the controller parameters α.sub.m, α.sub.s, k.sub.m, k.sub.s, I, T.sub.m, T.sub.s, Z, P are selected such that the following conditions hold,

(45) $-2{\alpha }_{m}I+{\stackrel{\_}{T}}_{m}Z+{\stackrel{\_}{T}}_s{k}_m^2{P}^{-1}<0,-2\frac{k_m{\alpha }_s}{k_s}I+{\stackrel{\_}{T}}_{s}P+{\stackrel{\_}{T}}_m{k}_m^2{Z}^{-1}<0$
the joint and motor velocities {dot over (q)}.sub.m, {dot over (q)}.sub.s, {dot over (θ)}.sub.s and position error q.sub.m−q.sub.s of the master-slave robot system with flexible joints are all bounded.

(46) If the force F.sub.h exerted by an operator to the master robot and a force F.sub.e exerted by the external environment to the slave robot are both zero, the controllers are designed as follows:

(47) $\begin{array}{cc}{\tau }_m=-k_m\left(X_{m1}-X_{s1}\left(t-T_s\left(t\right)\right)\right)-{\alpha }_m{X}_{m2}+G_m\left(X_{m1}\right){\tau }_s=S_s\left(X_{s3}-X_{s1}\right)+J_s\left({\stackrel{.}{X}}_{s4}^{*}-\left(X_{s3}-X_{s3}^{*}\right)-k_2\left(X_{s4}-X_{s4}^{*}\right)\right)X_{s3}^{*}=X_{s1}+S_s^{-1}\left(k_s\left(X_{m1}\left(t-T_m\left(t\right)\right)-X_{s1}\right)-{\alpha }_s{X}_{s2}+G_s\left(X_{s1}\right)\right)X_{s4}^{*}={\stackrel{.}{X}}_{s3}^{*}-2\frac{k_m}{k_s}{S}_s^T{X}_{s2}-k_1\left(X_{s3}-X_{s3}^{*}\right)& \left(24\right)\end{array}$
where X*.sub.s3 and X*.sub.s4 are the first controller and the second virtual controller, respectively, {dot over (X)}*.sub.s3 and {dot over (X)}*.sub.s4 are the first derivatives of the virtual controllers X*.sub.s3 and X*.sub.s4, respectively, T.sub.m(t) and T.sub.s(t) are forward time delay (from the master robot to the slave robot) and backward time delay (from the slave robot to the master robot), respectively, α.sub.m and α.sub.s are damping coefficients which are positive constants, k.sub.m, k.sub.s>0 are proportional coefficients, S.sub.s.sup.−1 and S.sub.s.sup.T are the inverse matrix and the transpose matrix of a diagonal positive-definite constant matrix S.sub.s which contains the joint stiffness of the slave robot, respectively, G.sub.m(X.sub.m1), G.sub.s(X.sub.s1) are the gravity torques of the master robot and the slave robot, J.sub.s is a diagonal constant matrix of the moment of actuator inertia, and k.sub.1 and k.sub.2 are selected to be positive constants.

(48) When the controller parameters α.sub.m, α.sub.s, k.sub.m, k.sub.s, I, T.sub.m, T.sub.s, Z, P are selected such that the following conditions hold,

(49) $-2{\alpha }_{m}I+{\stackrel{\_}{T}}_{m}Z+{\stackrel{\_}{T}}_s{k}_m^2{P}^{-1}<0,-2\frac{k_m{\alpha }_s}{k_s}I+{\stackrel{\_}{T}}_{s}P+{\stackrel{\_}{T}}_m{k}_m^2{Z}^{-1}<0$
it can be guaranteed that the joint and motor velocities {dot over (q)}.sub.m, {dot over (q)}.sub.s, {dot over (θ)}.sub.s and the position error q.sub.m−q.sub.s of the master-slave robot system with flexible joints will converge to zero asymptotically and that the global master-slave robot system with flexible joints will asymptotically become stable.

(50) For a master-slave robot system with flexible joints, first a control command is sent by the operator to the master robot, and the master robot is controlled by the proportional damping controller τ.sub.m and transmits the received control command to the slave robot through the network information transmission channel. Then the slave robot controlled by the full-state feedback controller τ.sub.s executes the control command on the external environment, feedbacks the measured position and force information to the operator in time, thus a closed-loop system is formed and tasks can be completed effectively.

(51) Based on the backstepping technique and the high-dimension uniform accurate differentiators, a full-state feedback controller is proposed for a flexible master-slave robot system. With this flexible master-slave robot system, accurate position tracking performance is achieved in the global scope. Additionally, the global asymptotic convergence is guaranteed and the robustness of the closed-loop master-slave system is improved.

(52) The above-mentioned embodiment merely describes the preferred embodiment of the present invention. The scope of the invention should not be limited by the described embodiment. It is intended that various changes and modifications to the technical solutions of the present invention made by those skilled in the art without departing from the spirit of the present invention shall fall within the protection scope determined by the claims of the present invention.