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

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.

TECHNICAL FIELD

This present invention relates to the control field of the flexible master-slave robot system, in particular to a full-state control method for a master-slave robot system with flexible joints and time-varying delays.

BACKGROUND

The master-slave robot system with flexible joints is a very complex dynamic system with high-nonlinearity, strong-coupling, and time-varying delays. To complete some complex and difficult work, flexible joint manipulator has the advantages of small volume, high flexibility, high deadweight ratio of load, low energy consumption, and wider working space compared with rigid joint manipulators. In most cases, the teleoperation system is a kind of remote operating system to complete more complex operations. It mainly consists of an operator, a master robot system, a network information transmission channel, a slave robot system, and external working environment. Its workflow can be roughly described as follows: the operator manipulates the master robot directly, so the control instructions will be sent to the slave robot through the network information transmission channel, then the slave robot will act on the remote environment according to the instructions. Meanwhile, the relevant information will also be feedback to the operator timely. Therefore, a closed-loop teleoperation system can be constituted and the operating tasks can be completed effectively. For slave robot systems, n-DOF (Degree Of Freedom) robots with flexible joints are usually used to execute commands from the master side.

With the development and progress of science and technology, the exploration of human beings has been constantly expanding. Since flexible joints are characterised by small size and high flexibility, a flexible master-slave robot system is more widely used. However, vibration of the flexible joint may affect the control accuracy and system stability. Therefore, the proposed control method can not only enable the robot to complete the work objectives but also eliminate the vibration of the flexible joint to ensure the stability of the system.

For the master-slave robot system with flexible joints, a full-state feedback control method based on the backstepping technique is proposed. The position error between the master and slave robots with time delays are used to analyze the global tracking performance. The globally accurate position tracking is realized and the robustness of the closed-loop system is improved. The high-dimension uniform accurate differentiators are employed to realize the precise difference to the virtual controllers and the convergence of the system is improved.

SUMMARY

The purpose of the invention is to provide a full-state control method for the master-slave robot system with flexible joints and time-varying delays to solve the problem of position tracking and stability existing in the master-slave robot system with flexible joints. To achieve these above objectives, the following technical schemes are adopted.

A full-state control method for the master-slave robot system with flexible joints and time-varying delays includes the following steps:

Step 1: connecting a master robot and a slave robot through the network to form a teleoperation system.

Step 2: measuring system parameters of the master robot and the slave robot, among which the position and velocity information of joints and motors are measured in real-time.

Step 3: designing the first virtual controller and the second virtual controller, respectively.

The first virtual controller is X*s3=Xs1+Ss−1(ks(Xm1(t−Tm(t))−Xs1)−αsXs2), the second virtual controller is

Xs⁢4*=X.s⁢⁢3*-2⁢kmks⁢SsT⁢Xs⁢2-k1⁡(Xs⁢3-Xs⁢3*),
the subscripts m and s denotes the master robot and the slave robot, respectively, X*s3and X*s4are virtual controllers, {dot over (X)}*s3, is the first derivative of the virtual controller X*s3, Tm(t) and Ts(t) are forward time delay (from the master robot to the slave robot) and feedback time delay (from the slave robot to the master robot), respectively, αsis a damping coefficient which is a positive constant, km, ks>0 are proportional coefficients, Ss−1and SsTare the inverse matrix and the transpose matrix of a diagonal positive-definite constant matrix Sswhich contains the joint stiffness of the slave robot, respectively, and k1is selected to be a positive constant.

Step 4: designing high-dimension uniform accurate differentiators to carry out a precise difference to the first virtual controller and the second virtual controller.

{X.1=X2X.2=X¨s⁢3*(1){Y.1=λ1⁢σ1σ11/2+λ2⁢σ1⁢σ1P-1+Y2Y.2=α1⁢σ1σ1(2)
with the Equations (1) and (2),

{σ.1=-λ1⁢σ1σ11/2-λ2⁢σ1⁢σ1P-1+σ2σ.2=-α1⁢σ1σ1+X¨s⁢3*(3)
is derived, where X*s3denotes the virtual controller, {dot over (X)}*s3denotes the first derivative of the virtual controller X*s3, {umlaut over (X)}s3denotes the second derivative of the virtual controller X*s3, Y1is an estimate of the virtual controller X*s3, Y2is an estimate of {dot over (X)}*s3, σ1and σ2are estimation errors, λ1, λ2, α1>0 are system control gains, P>1 is a constant, {umlaut over (X)}*s3supposed to be bounded and satisfies ∥{umlaut over (X)}*s3∥≤L3with a known constant L3>0, and if the parameters are selected to satisfy the conditions α1>4L3, λ1>√{square root over (2α1)}, the estimation errors σ1, σ2will globally converge to the origin quickly, thus a precise difference value {dot over (X)}*s3of X*s3is obtained.

{X.3=X4X.3=X¨s⁢⁢4*(4){Y.3=λ3⁢σ3σ31/2+λ4⁢σ3⁢σ3P-1+Y4Y.4=α2⁢σ3σ3(5)
with the Equations (4) and (5),

Step 5: designing the controllers of the master-slave robot system with flexible joints by using the backstepping method as follows

Step 6: 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.

When the controller parameters are selected such that the following inequalities hold,

-2⁢αm⁢I+T_m⁢Z+T_s⁢km2⁢P-1<0,-2⁢km⁢αsks⁢I+T_s⁢P+T_m⁢km2⁢Z-1<0
the joint and motor velocities {dot over (q)}m, {dot over (q)}s, {dot over (θ)}sand position error qm−qsof the master-slave robot system with flexible joints are all bounded.

Furthermore, when a force Fhexerted by an operator to the master robot and a force Feexerted by the external environment to the slave robot are both zero, the controllers are designed as follows:

When the controller parameters are selected such that the following inequalities hold,

-2⁢αm⁢I+T_m⁢Z+T_s⁢km2⁢P-1<0,-2⁢km⁢αsks⁢I+T_s⁢P+T_m⁢km2⁢Z-1<0
it can be guaranteed that the joint and motor velocities {dot over (q)}m, {dot over (q)}s, {dot over (θ)}sand the position error qm−qsof 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, where I is the identity matrix, Z−1and P−1are the inverse matrices of positive definite matrices Z and P, respectively, αmand αsare damping coefficients which are positive constants, km, ks>0 are proportional coefficients, and it is supposed that the time delay Tm(t) and Ts(t) are bounded, i.e. there are positive scalarsTmandTs, such that the inequalities Tm(t)≤Tm, Ts(t)≤Tshold.

Preferably, the system parameters of the master robot and the slave robot include the length and the mass of the robot manipulators of the master robot and the slave robot, the positions and the velocities of the robot joints and motor of the master robot and the slave robot, and the force exerted by the operator and the force exerted by the external environment measured by using force sensors.

Preferably, the design process of the first virtual controller and the second virtual controller is as follows:

As for the first virtual controller, the first Lyapunov Equation is selected as follows,

Gm(Xm⁢1)=∂Um(Xm⁢1)∂Xm⁢1,Gs(Xs⁢1)=∂Us(Xs⁢1)∂Xs⁢1,
there are positive scalars βmand βssuch that Um(Xm1)≥βm, Us(Xs1)≥βs, and km, ks>0 are the proportional coefficients.

The time derivative of V11is given by

With Equation (9), the first virtual controller
X*s3=Xs1+Ss−1(ks(Xm1(t−Tm(t))−Xs1)−αsXs2) is derived.

As for the second virtual controller, the second Lyapunov equation is selected as follows,
V2=V1+½(Xs3−X*s3)T(Xs3−X*s3)  (10)

The time derivative of V2is given by
{dot over (V)}2={dot over (V)}1+(Xs3−X*s3)T(Xs4−{dot over (X)}*s3)={dot over (V)}1+(Xs3−X*s3)T(Xs4−X*s4+X*s4−{dot over (X)}*s3)  (11)

With Equation (11), the second virtual controller

Preferably, a high-dimensional uniform accurate differentiators is designed as follows.

With Equations (12) and (13),

{σ.1=-λ1⁢σ1σ11/2-λ2⁢σ1⁢σ1P-1+σ2σ.2=-α1⁢σ1σ1+X¨s⁢3*(14)
is derived, where X*s3denotes the virtual controller, {dot over (X)}*s3denotes the first derivative of the virtual controller X*s3, {umlaut over (X)}*s3denotes the second derivative of the virtual controller X*s3, Y1is an estimate of the virtual controller X*s3, Y2is an estimate of {dot over (X)}*3, σ1and σ2are estimation errors, λ1, λ2, α1>0 are system control gains, P>1 is a constant, {umlaut over (X)}*s3is supposed to be bounded and satisfies ∥{umlaut over (X)}*s3∥≤L3with a known constant L3>0, and if the parameters are selected to satisfy the conditions α1>4L3, λ1>√{square root over (2α1)}, the estimation errors σ1, σ2will globally converge to the origin quickly, thus a precise difference value {dot over (X)}*s3of X*s3is obtained,

is derived, where X*s4denotes a virtual controller, {dot over (X)}*s4is the first derivative of the virtual controller X*s4, {umlaut over (X)}*s4is the second derivative of the virtual controller X*s4, Y3is an estimate of the virtual controller X*s4, Y4is an estimate of {dot over (X)}*s4, σ3and σ4are estimation errors, λ3, λ4, α2>0 are system control gains, {umlaut over (X)}*s4supposed to be bounded and satisfies ∥{umlaut over (X)}*s4∥≤L4with a known positive is constant L4>0, and, if the parameters are selected to satisfy the conditions α2>4L4, λ3>√{square root over (2α2)}, the estimation errors σ3, σ4will globally converge to the origin quickly, thus a precise difference value {dot over (X)}*s4of X*s4is obtained.

Preferably, a delay-dependent system stability criteria is established by constructing the Lyapunov Equations, thus a criteria for selecting controller parameters are provided. The detailed steps are as follows.

S1 selecting the first Lyapunov Equation as follows,

The time derivative of V1is given by

S2 selecting the second Lyapunov Equation as follows:
V2=V1+½(Xs3−X*s3)T(Xs3−X*s3)  (20)

The time derivative of V2is given by

S3 selecting the third Lyapunov Equation as follows:
V3=V2+½(Xs4−X*s4)T(Xs4−X*s4)  (22)

The time derivative of V3is given by

When the controller parameters are selected such that the following inequalities hold,

-2⁢αm⁢I+T¯m⁢Z+Ts¯⁢km2⁢P-1<0,-2⁢km⁢αsks⁢I+Ts¯⁢P+T¯m⁢km2⁢Z-1<0
the joint and motor velocities {dot over (q)}m, {dot over (q)}s, {dot over (θ)}sand the position error qm−qsof the master-slave robot system with flexible joints are all bounded.

When a force Fhexerted by an operator to the master robot and a force Feexerted by the external environment to the slave robot are both zero, the controllers are designed as follows:

When the controller parameters αm, αs, km, ks, I,Tm,Ts, Z, P are selected such that the following conditions hold,

-2⁢αm⁢I+T¯m⁢Z+Ts¯⁢km2⁢P-1<0,-2⁢km⁢αsks⁢I+Ts¯⁢P+T¯m⁢km2⁢Z-1<0
it is guaranteed that the joint and motor velocities {dot over (q)}m, {dot over (q)}s, {dot over (θ)}sand the position error qm−qsof 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.

Compared with the existing methods, the present invention has the following advantages:

1. Compared with the local state control method, the proposed full-state control method can realize the global asymptotic stability of the master-slave system better.

2. The exact position tracking accuracy can be guaranteed by applying the full-state feedback controller based on the backstepping method globally and the robustness of closed-loop system also can be improved.

3. The precise difference to the virtual controllers can be realized by designing high-dimension uniform accurate differentiators. Additionally, this method is more applicable to the high-order control systems.

DETAILED DESCRIPTION

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

As shown inFIG. 1, the steps of the invention method are as follows:

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.

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 Mm(Xm1) and Ms(Xs1), the matrices of centripetal and coriolis torques Cm(qm, {dot over (q)}m) and Cs(qs, {dot over (q)}s), the gravity torques Gm(Xm1) and Gs(Xs1), the diagonal constant matrix Jsof the moment of actuator inertia of the master robot and the slave robot, and a diagonal positive-definite constant matrix Ssthat contains the joint stiffness of the slave robot respectively according to the length and the mass of the manipulators. Moreover, a force Fhexerted by the operator to the master robot and a force Feexerted by the external environment to the slave robot are measured by using force sensors.

The system dynamics equation can be described as
Mm(qm){umlaut over (q)}m+Cm(qm,{dot over (q)}m){dot over (q)}m+Gm(qm)=τm+Fh
Ms(qs){umlaut over (q)}s+Cs(qs,{dot over (q)}s){dot over (q)}s+Gs(qs)=Ss(θs−qs)−Fe
Js{umlaut over (θ)}s+Ss(θs−qs)=τs(1)
where the subscripts m and s denotes the master robot and the slave robot, respectively. qm, qs∈Rnare the vectors of joint displacements. {dot over (q)}m, {dot over (q)}s∈Rnare the vectors of joint velocities. {umlaut over (q)}m, {umlaut over (q)}s∈Rnare the vectors of joint accelerations. θs∈Rnis the vector of motor displacements, {umlaut over (θ)}s∈Rnis the vector of motor accelerations. Mm(qm), Ms(qs)∈Rn×nare the positive-definite inertia matrices of the system, Cm(qm, {dot over (q)}m), Cs(qs, {umlaut over (q)}s)∈Rnare the matrices of centripetal and coriolis torques, Gm(qm), Gs(qs)∈Rnare the gravitational torques. Js∈Rn×nis the diagonal constant matrix of the moments of actuator inertia. Ss∈Rnis a diagonal positive-definite constant matrix that contains the joint stiffness of the slave robot. Fh, Fe∈Rnare the force exerted by the operator to the master robot and the force exerted by the external environment to the slave robot, respectively. τm, τs∈Rnare control torques provided by the controllers.

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.

For the master robot with rigid joints, letting Xm1=qm, Xm2={dot over (q)}m, the state space expression of the system is obtained as follows:

{X.m⁢1=Xm⁢2X.m⁢2=Mm-1⁡(Xm⁢1)⁢(τm+Fh-Cm⁡(Xm⁢1,Xm⁢2)⁢Xm⁢2-Gm⁡(Xm⁢1))(2)
where the subscripts m denotes the master robot, qm∈Rnis the vector of joint displacements, {dot over (q)}m∈Rnis the vector of joint velocities, Mm−1(Xm1) is the inverse matrix of the positive-definite inertia matrix Mm(Xm1), τm∈Rnis a control torque provided by the controller, Cm(qm, {dot over (q)}m)∈Rnis the matrix of centripetal and coriolis torque, Gm(qm)∈Rnis the gravitational torque, and Fh∈Rnis a force exerted by an operator to the master robot.

For the slave robot with flexible joints, letting Xs1=qs, Xs2={dot over (q)}s, Xs3=θs, Xs4={dot over (θ)}s, the state space expression of the system is obtained as follows:

The first Lyapunov Equation is selected as follows,

⁢V1=V1⁢1+V1⁢2+V1⁢3⁢⁢V1⁢1=Xm⁢2T⁢Mm⁡(Xm⁢1)⁢Xm⁢2+kmks⁢Xs⁢2T⁢MS⁡(Xs⁢1)⁢Xs⁢2+2⁢(Um⁡(Xm⁢1)-βm)+2⁢kmks⁢(Us⁡(Xs⁢1)-βs)+2⁢∫0t⁢(-Xm⁢2T⁡(σ)⁢Fh⁡(σ)+kmks⁢Xs⁢2T⁡(σ)⁢Fe⁡(σ))⁢d⁢⁢σ⁢⁢⁢V1⁢2=km⁡(Xm⁢1-Xs⁢1)T⁢(Xm⁢1-Xs⁢1)⁢⁢V1⁢3=∫-T¯m0⁢∫t+θt⁢Xm⁢2T⁡(ξ)⁢Z⁢⁢Xm⁢2⁡(ξ)⁢d⁢ξ⁢d⁢θ+∫-Ts¯0⁢∫t+θt⁢Xs⁢2T⁡(ξ)⁢P⁢Xs⁢2⁡(ξ)⁢d⁢⁢ξ⁢⁢d⁢⁢θ(4)
where the integral terms satisfy ∫0t−Xm2T(σ)Fh(σ)dσ≥0, ∫0tXs2T(σ)Fe(σ)dσ≥0. Z and P are positive definite matrices. It is supposed that the time delay Tm(t) and Ts(t) are bounded, i.e. there are positive scalarsTmandTs, such that Tm(t)≤Tm, Ts(t)≤Ts. Mm(Mm1) and Ms(Xs1) are positive-definite inertia matrices of the master robot and the slave robot, respectively.
Um(Xm1) and Us(Xs1) are the potential energy of the master robot and the slave robot satisfying

The time derivatives of V1, V11, V12, V13are given by

With Equation (5), the first virtual controller is derived as X*s3=Xs1±Ss−1(ks(Xm1(t−Tm(t))−Xs1)−αsXs2). By substituting the first virtual controller X*s3into Equation (5),

-2⁢km⁢Xm⁢2T⁢∫t-Ts⁡(t)t⁢Xs⁢2⁡(ξ)⁢d⁢⁢ξ-∫t-Ts⁡(t)t⁢Xs⁢2T⁡(ξ)⁢P⁢Xs⁢2⁡(ξ)⁢d⁢ξ≤Ts¯⁢km2⁢Xm⁢2T⁡(t)⁢P-1⁢Xm⁢2⁡(t)⁢-2⁢km⁢Xs⁢2T⁢∫t-Tm⁡(t)t⁢Xm⁢2⁡(ξ)⁢d⁢⁢ξ-∫t-Tm⁡(t)t⁢Xm⁢2T⁡(ξ)⁢Z⁢⁢Xm⁢2⁡(ξ)⁢d⁢⁢ξ≤T_m⁢km2⁢Xs⁢2T⁡(t)⁢Z-1⁢Xs⁢2⁡(t)⁢⁢So(7)V.1=⁢V.1⁢1+V.1⁢2+V.1⁢3≤Xm⁢2T⁡(-2⁢αm⁢I+T_m⁢Z+Ts¯⁢km2⁢P-1)⁢Xm⁢2⁢+Xs⁢2T⁡(-2⁢km⁢αsks⁢I+Ts¯⁢P+T_m⁢km2⁢Z-1)⁢Xs⁢2+2⁢kmks⁢Xs⁢2T⁢Ss⁡(Xs⁢3-Xs⁢⁢3*)(8)
is derived, where I is the identity matrix, z−1and P−1are the inverse matrices of positive definite matrices Z and P, respectively. Ssis a diagonal positive-definite constant matrix which contains the joint stiffness of the slave robot. αmand αsare damping coefficients which are positive constants. km, ks>0 are proportional coefficients. It is supposed that the time delay Tm(t) and Ts(t) are bounded, i.e. there are positive scalarsTmandTs, such that Tm(t)≤Tm, Ts(t)≤Ts, X*s3is the first virtual controller.

The second Lyapunov Equation is selected as follows,
V2=V1+½(Xs3−X*s3)T(Xs3−X*s3)  (9)

The time derivative of V2is given by
{dot over (V)}2={dot over (V)}1+(Xs3−X*s3)T(Xs4−{dot over (X)}*s3)={dot over (V)}1+(Xs3−X*s3)T(Xs4−X*s4+X*s4−{dot over (X)}*s3)  (10)
With equation (10), the second virtual controller is derived as

Xs⁢⁢4*=X.s⁢⁢3*-2⁢kmks⁢SsT⁢Xs⁢2-k1⁡(Xs⁢3-Xs⁢3*).
By substituting the second virtual controller X*s4into Equation (10),

The third Lyapunov Equation is selected as follows,
V3=V2+½(Xs4−X*s4)T(Xs4−X*s4)  (12)

The time derivative of V3is given by

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

τs=Ss(Xs3−Xs1)+Js({dot over (X)}*s4−(Xs3−X*s3)−k2(Xs4−X*s4)). By substituting the obtained all-state feedback controller τsinto Equation (13),

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

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

{X.1=X2X.2=X¨s⁢⁢3*(16){Y.1=λ1⁢σ1σ11/2+λ2⁢σ1⁢σ1P-1+Y2Y.2=α1⁢σ1σ1(17)
with Equations (16) and (17),

{σ.1=-λ1⁢σ1σ11/2-λ2⁢σ1⁢σ1P-1+σ2σ.2=-α1⁢σ1σ1+X¨s⁢⁢3*(18)
is derived, where X*s3denotes the first virtual controller, {dot over (X)}*s3denotes the first derivative of the first virtual controller X*s3, {umlaut over (X)}s3denotes the second derivative of the first virtual controller X*s3, Y1is an estimate of the first virtual controller X*s3, Y2is an estimate of {dot over (X)}*s3, σ1and σ2are estimation errors, λ1, λ2, α1>0 are system control gains, P>1 is a constant, {umlaut over (X)}*s3is supposed to be bounded and satisfies ∥{umlaut over (X)}*s3∥≤L3with a known constant L3>0. If the parameters are selected to satisfy the conditions α1>4L3, λ1>√{square root over (2α1)}, the estimation errors σ1, σ2will converge to the origin quickly, thus a precise difference value {dot over (X)}*s3of X*s3is obtained.

{X.3=X4X.4=X¨s⁢⁢4*(19){Y.3=λ3⁢σ3σ31/2+λ4⁢σ3⁢σ3P-1+Y4Y.4=α2⁢σ3σ3(20)
with Equations (19) and (20),

{σ.3=-λ3⁢σ3σ31/2-λ4⁢σ3⁢σ3P-1+σ4σ.4=-α2⁢σ3σ3+X¨s⁢⁢4*(21)
is derived, where X*s4denotes the second virtual controller, {dot over (X)}*s4denotes the first derivative of the second virtual controller X*s4, {umlaut over (X)}*s4denotes the second derivative of the second virtual controller X*s4, Y3is an estimate of the second virtual controller X*s4, Y4is an estimate of {dot over (X)}*s4, σ3and σ4are estimation errors, λ3, λ4, α2>0 are system control gains, {umlaut over (X)}*s4is supposed to be bounded and satisfies ∥{umlaut over (X)}*s4∥≤L4with a known positive constant L4>0. If the parameters are selected to satisfy the conditions α2>4L4, λ3>√{square root over (2α2)}, the estimation errors σ3, σ4will converge to the origin quickly, thus a precise difference value {dot over (X)}*s4of X*s4is obtained.

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.

The Lyapunov Equation is selected as

The time derivative of V3is given by

When the controller parameters αm, αs, km, ks, I,Tm,Ts, Z, P are selected such that the following conditions hold,

-2⁢αm⁢I+T_m⁢Z+T_s⁢km2⁢P-1<0,-2⁢km⁢αsks⁢I+T_s⁢P+T_m⁢km2⁢Z-1<0
the joint and motor velocities {dot over (q)}m, {dot over (q)}s, {dot over (θ)}sand position error qm−qsof the master-slave robot system with flexible joints are all bounded.

If the force Fhexerted by an operator to the master robot and a force Feexerted by the external environment to the slave robot are both zero, the controllers are designed as follows:

When the controller parameters αm, αs, km, ks, I,Tm,Ts, Z, P are selected such that the following conditions hold,

-2⁢αm⁢I+T_m⁢Z+T_s⁢km2⁢P-1<0,-2⁢km⁢αsks⁢I+T_s⁢P+T_m⁢km2⁢Z-1<0
it can be guaranteed that the joint and motor velocities {dot over (q)}m, {dot over (q)}s, {dot over (θ)}sand the position error qm−qsof 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.

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 τmand 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 τsexecutes 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.

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.

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.