Fractional order sliding mode synchronous control method for teleoperation system based on event trigger mechanism

Abstract

The present invention provides a fractional order sliding mode synchronous control method for a teleoperation system based on an event trigger mechanism. The method comprises: establishing a dynamics model for the teleoperation system by considering external disturbance and parameter uncertainty, selecting a master robot and a slave robot, interactively establishing the teleoperation system through a communication network, determining system parameters of the dynamics model, designing a fractional order nonsingular rapid terminal sliding mode surface equation by utilizing a position tracking error and a fractional order calculus, setting a trigger event condition of information interaction between the master robot and the slave robot, designing a self-adaptive fractional order nonsingular rapid terminal sliding mode controller based on the sliding mode, designing a Lyapunov function to carry out stability analysis, proving the boundedness of a closed-loop state signal of the system.

Claims

1. A fractional order sliding mode synchronous control method for a teleoperation system based on an event trigger mechanism, characterized in that, the method comprises steps of: S1, establishing a dynamic model for the teleoperation system by considering external disturbance and parameter uncertainty, wherein the dynamic model is:
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)+B.sub.m({dot over (q)}.sub.m)=.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)+B.sub.s({dot over (q)}.sub.s)=.sub.s+F.sub.e(1) in which, q.sub.i, {dot over (q)}.sub.i, {umlaut over (q)}.sub.1R.sup.n respectively represent a generalized displacement, a speed and an acceleration of n joints of the robot; M.sub.i(q.sub.i) represents a positive definite inertial matrix of the system; C.sub.i(q.sub.i, {dot over (q)}.sub.i) represents a matrix of Coriolis forces and centrifugal forces of the system; G.sub.i(q.sub.i) represents a gravity moment of the system; F.sub.h, F.sub.eR.sup.n respectively represent an external force applied by an operator and an external force applied by the environment; .sub.iR.sup.n represents a generalized input moment; B.sub.i({dot over (q)}.sub.i)R.sup.n represents a unknown external interference; i=m, s, m represents a master robot, s represents a slave robot; M.sub.i(q.sub.i), C.sub.i(q.sub.i,{dot over (q)}.sub.i) and G.sub.i(q.sub.i) have a relationship of: $\begin{array}{cc}\{\begin{array}{c}{M}_i\left(q_i\right)=M_{\mathrm{oi}}\left(q_i\right)+M_i\left(q_i\right)R^{nn}\\ C_i\left(q_i,{\stackrel{.}{q}}_i\right)=C_{\mathrm{oi}}\left(q_i,{\stackrel{.}{q}}_i\right)+C_i\left(q_i,{\stackrel{.}{q}}_i\right)R^{nn}\\ G_i\left(q_i\right)=G_{\mathrm{oi}}\left(q_i\right)+G_i\left(q_i\right)R^n\end{array}& \left(2\right)\end{array}$ in which, M.sub.oi(q.sub.i) represents a nominal value of the positive definite inertia matrix of the system; C.sub.oi(q.sub.i,{dot over (q)}.sub.i) represents a nominal value of the matrix of Coriolis forces and centrifugal forces of the system; G.sub.oi(q.sub.i) represents a nominal value of the gravity moment of the system; M.sub.i(q.sub.i) represents parameter change of the positive definite inertia matrix of the system; C.sub.i(q.sub.i,{dot over (q)}.sub.i) represents parameter change of the matrix of Coriolis forces and centrifugal forces of the system; .sub.Gi(q.sub.i) represents parameter change of the gravity moment of the system; at the same time, B.sub.i({dot over (q)}.sub.i) satisfies:
B.sub.i({dot over (q)}.sub.i)B.sub.i(3) in which, B.sub.i represents an unknown positive number; then, $\begin{array}{cc}\{\begin{array}{c}\begin{array}{c}{P}_m\left(q_m,{\stackrel{.}{q}}_m,{\stackrel{.Math.}{q}}_m\right)=-M_m\left(q_m\right){\stackrel{.Math.}{q}}_m-C_m\left(q_m,{\stackrel{.}{q}}_m\right){\stackrel{.}{q}}_m-\\ G_m\left(q_m\right)-B_m\left({\stackrel{.}{q}}_m\right)R^n\end{array}\\ \begin{array}{c}{P}_s\left(q_s,{\stackrel{.}{q}}_s,{\stackrel{.Math.}{q}}_s\right)=-M_s\left(q_s\right){\stackrel{.Math.}{q}}_s-C_s\left(q_s,{\stackrel{.}{q}}_s\right){\stackrel{.}{q}}_s-G_s\left(q_s\right)-\\ B_s\left({\stackrel{.}{q}}_s\right)R^n\end{array}\end{array}& \left(4\right)\end{array}$ in which, P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m) represents a parameter matrix of the master robot; P.sub.s(q.sub.s,{dot over (q)}.sub.s,{umlaut over (q)}.sub.s) represents a parameter matrix of the slave robot; P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m) and P.sub.s(q.sub.s,{dot over (q)}.sub.s,{umlaut over (q)}.sub.s) are bounded and satisfies:
P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m).sub.m
P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m).sub.s(5) in which, .sub.m represents an upper limit of P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m), .sub.m=b.sub.0m+b.sub.1mq.sub.m+b.sub.2m{dot over (q)}.sub.m.sup.2; .sub.s represents an upper limit of P.sub.s(q.sub.s,{dot over (q)}.sub.s,{umlaut over (q)}.sub.s), .sub.s=b.sub.0s+b.sub.1sq.sub.s+b.sub.2s{dot over (q)}.sub.s.sup.2; b.sub.0i, b.sub.1i, b.sub.2i(i=m, s) is a positive constant; accordingly, equation (1) for the system is re-represented as: $\begin{array}{cc}\{\begin{array}{c}\begin{array}{c}{M}_{om}\left(q_m\right){\stackrel{.Math.}{q}}_m+C_{om}\left(q_m,{\stackrel{.}{q}}_m\right){\stackrel{.}{q}}_m+G_{\mathrm{om}}\left(q_m\right)={}_m+\\ P_m\left(q_m,{\stackrel{.}{q}}_m,{\stackrel{.Math.}{q}}_m\right)+F_h\end{array}\\ \begin{array}{c}{M}_{\mathrm{os}}\left(q_s\right){\stackrel{.Math.}{q}}_s+C_{os}\left(q_s,{\stackrel{.}{q}}_s\right){\stackrel{.}{q}}_s+G_{os}\left(q_s\right)={}_s+\\ P_s\left(q_s,{\stackrel{.}{q}}_s,{\stackrel{.Math.}{q}}_s\right)-F_e\end{array}\end{array}& \left(6\right)\end{array}$ S2, selecting the master robot and the slave robot, interactively constructing the teleoperation system through a communication network, measuring respectively mass and length information of a master robot connecting rod and a slave robot connecting rod and real-time robot position information of the master robot connecting rod and the slave robot connecting rod, and determining system parameters of the dynamic model; S3, designing a sliding mode surface equation of a fractional order nonsingular rapid terminal by using position tracking errors of the master robot and the slave robot and Riemann-Liouville fractional order calculus; S4, setting a trigger event condition of information interaction between the master robot and the slave robot, designing a self-adaptive fractional order nonsingular rapid terminal sliding mode controller which can compensate measurement errors and eliminate the influence of system uncertainty based on the sliding mode, enabling the controller to avoid the Zeno phenomenon, performing stability analysis, proving the boundedness of a closed-loop state signal of the system, enabling the slave robot to track the motion of the master robot within a limited time, and realizing synchronous control of the teleoperation system; in step S4, the trigger event condition is: $\begin{array}{cc}\{\begin{array}{c}{}_i\left(t\right)=u_i\left(t_k^i\right),t[t_k^i,t_{k+1}^i)\\ t_{k+1}^j=\inf \{tR.Math.z_i\left(t\right).Math.z_i\left(t\right)b_i\},t_1^i=0,j=1,2,.Math.,n\end{array}& \left(10\right)\end{array}$ in which, z.sub.i(t) represents a measurement error, z.sub.i(t)=[z.sub.i1, z.sub.i2, . . . , z.sub.in].sup.T and z.sub.i(t)=u.sub.i(t).sub.i(t); .sub.i(t) represents a generalized input moment of the teleoperation system, .sub.i(t)=[.sub.i1, .sub.i2, . . . , .sub.in].sup.T; b.sub.i represents a positive definite matrix, b.sub.i(t)=[b.sub.i1, b.sub.i2, . . . , b.sub.in].sup.T R.sup.n; u.sub.i(t) represents actual controller under event trigger mechanism, u.sub.i(t)=[u.sub.i1, u.sub.i2, . . . , u.sub.in].sup.T; t.sub.k.sup.i, kz.sup.+ represents controller update time; in a period of time t.sup.i[t.sub.k.sup.i,t.sub.k+1.sup.t), the control signal remains constant until t.sup.i=t.sub.k+1.sup.i, minimum internal time of the mentioned event trigger mechanism is strictly greater than 0, which eliminates occurrence of the Zeno behavior; based on the designed sliding mode, the actual controller is chosen as:
.sub.i(t)=u.sub.i(t).sub.i(t)b.sub.i(11) in which, .sub.i(t) represents a continuous time-varying parameter satisfying conditions .sub.i(t.sub.k.sup.i)=0,.sub.i(t.sub.k+1.sup.i)=1 and |.sub.i(t)|1, t[t.sub.k.sup.i,t.sub.k+1.sup.i; the sliding mode controller of the self-adaptive fractional order nonsingular rapid terminal is designed as: $\begin{array}{cc}\{\begin{array}{c}{u}_m\left(t\right)={\stackrel{\_}{u}}_m-{\stackrel{\_}{b}}_m\tanh \left(s_m^T{M}_{\mathrm{om}}^{-1}{\stackrel{\_}{b}}_m/{}_m\right)\\ \begin{array}{c}{\stackrel{\_}{u}}_m=M_{\mathrm{om}}\left(q_m\right){\stackrel{.Math.}{q}}_s\left(t-d_s\right)+C_{\mathrm{om}}\left(q_m,{\stackrel{.}{q}}_m\right){\stackrel{.}{q}}_m+G_{\mathrm{om}}\left(q_m\right)+\\ {\stackrel{\_}{u}}_{m1}+{\stackrel{\_}{u}}_{m2}\end{array}\\ \begin{array}{c}{\stackrel{\_}{u}}_{m1}=-M_{\mathrm{om}}({}_{m}si{g\left(s_m\right)}^{}+{}_{m1}{D}^{{}_1}\left[\mathrm{si}{g\left(e_m\right)}^{{}_{m1}}\right]+\\ {}_{m2}{D}^{{}_2+1}\left[{}_m\left(e_m\right)\right])\end{array}\\ {\stackrel{\_}{u}}_{m2}=\{\begin{array}{cc}-\frac{s_m^T{M}_{om}^{-1}}{.Math.s_m^T{M}_{om}^{-1}.Math.}{\stackrel{}{}}_m,& \mathrm{if}{s}_m^T{M}_{om}^{-1}0\\ 0,& \mathrm{if}{s}_m^T{M}_{om}^{-1}=0\end{array}\end{array}& \left(12\right)\end{array}$ $\begin{array}{cc}\{\begin{array}{c}{u}_s\left(t\right)={\stackrel{\_}{u}}_s-{\stackrel{\_}{b}}_s\tanh \left(s_s^T{M}_{\mathrm{os}}^{-1}{\stackrel{\_}{b}}_s/{}_s\right)\\ \begin{array}{c}{\stackrel{\_}{u}}_s=M_{\mathrm{os}}\left(q_s\right){\stackrel{.Math.}{q}}_m\left(t-d_m\right)+C_{\mathrm{os}}\left(q_s,{\stackrel{.}{q}}_s\right){\stackrel{.}{q}}_s+G_{\mathrm{os}}\left(q_s\right)+\\ {\stackrel{\_}{u}}_{s1}+{\stackrel{\_}{u}}_{s2}\end{array}\\ \begin{array}{c}{\stackrel{\_}{u}}_{s1}=-M_{\mathrm{os}}({}_{s}si{g\left(s_s\right)}^{}+{}_{s1}{D}^{{}_1}\left[\mathrm{si}{g\left(e_s\right)}^{{}_{s1}}\right]+\\ {}_{s2}{D}^{{}_2+1}\left[{}_s\left(e_s\right)\right])\end{array}\\ {\stackrel{\_}{u}}_{s2}=\{\begin{array}{cc}-\frac{s_s^T{M}_{\mathrm{os}}^{-1}}{.Math.s_s^T{M}_{\mathrm{os}}^{-1}.Math.}{\stackrel{}{}}_s,& \mathrm{if}{s}_s^T{M}_{\mathrm{os}}^{-1}0\\ 0,& \mathrm{if}{s}_s^T{M}_{\mathrm{os}}^{-1}=0\end{array}\end{array}& \left(13\right)\end{array}$ in which, b.sub.m represents a positive definite matrix, b.sub.m=[b.sub.m1, b.sub.m2, . . . , b.sub.mn].sup.T and satisfies the constant b.sub.mj>b.sub.mj(j=1, 2, . . . , n); b.sub.s represents a positive definite matrix, b.sub.s=[b.sub.s1, b.sub.s2, . . . , b.sub.sn].sup.T and satisfies the constant b.sub.sj>b.sub.sj(j=1, 2, . . . , n); .sub.m, .sub.s represents a positive constant to be designed; .sub.i represents a positive definite diagonal matrix, .sub.i=diag[.sub.i1, .sub.i2, . . . , .sub.in] and satisfies the constant .sub.ij>0(j=1, 2, . . . , n); represents a positive constant to be designed and satisfies (0,1); {circumflex over ()}.sub.m represents an estimated value of an upper bound of P.sub.m(q.sub.m, {dot over (q)}.sub.n, {umlaut over (q)}.sub.m), {circumflex over ()}.sub.m={circumflex over (b)}.sub.0m+{circumflex over (b)}.sub.1m+{circumflex over (b)}.sub.2m{dot over (q)}.sub.m.sup.2; {circumflex over ()}.sub.s represents an estimated value of an upper bound of P.sub.s(q.sub.s, {dot over (q)}.sub.s, {umlaut over (q)}.sub.s), {circumflex over ()}.sub.s={circumflex over (b)}.sub.0s+{circumflex over (b)}.sub.1s+{circumflex over (b)}.sub.2s{dot over (q)}.sub.s.sup.2; variables {circumflex over (b)}.sub.0i, {circumflex over (b)}.sub.1i, {circumflex over (b)}.sub.2i representing estimated values of parameters b.sub.0i, b.sub.1i, b.sub.2i, respectively; the self-adaptation law is designed as:
{circumflex over ({dot over (b)})}.sub.0i=.sub.0is.sub.i.sup.TM.sub.oi.sup.1
{circumflex over ({dot over (b)})}.sub.1i=.sub.1is.sub.i.sup.TM.sub.oi.sup.1q.sub.i(14)
{circumflex over ({dot over (b)})}.sub.2i=.sub.2is.sub.i.sup.TM.sub.oi.sup.1q.sub.i.sup.2 in which, .sub.0i, .sub.1i, .sub.2i represents positive constants; a Lyapunov function is selected as: $$\begin{gathered} \begin{array}{cc}V=V_1+V_2 \\ {V}_1=\underset{i=m,s}{.Math.}\frac{1}{2}{s}_i^T{s}_i \\ {V}_2=\underset{i=m,s}{.Math.}\frac{1}{2}\underset{j=0}{\overset{2}{.Math.}}{}_{ji}^{-1}{\tilde{b}}_{ji}^2& \left(15\right)\end{array} \end{gathered}$$ in which, V represents a Lyapunov function; V.sub.1 represents a first Lyapunov function, V.sub.2 represents a second Lyapunov function, {tilde over (b)}.sub.ji=b.sub.ji{circumflex over (b)}.sub.ji represents a self-adaptive estimation error, j=1, 2, . . . , n; limited time stable operation of the nonlinear uncertain teleoperation system is ensured through stability analysis.

2. The method according to claim 1, characterized in that, system parameters of the dynamical model in step S2 comprise nominally determined parts of the inertia matrix, the matrix of Coriolis forces and centrifugal forces, and the gravity matrix of the master robot and the slave robot, and are all calculated from the length, mass, and joint position information of a mechanical arm.

3. The method according to claim 1, characterized in that, the position tracking errors of the master robot and the slave robot in step S3 are:
e.sub.m=q.sub.mq.sub.s(td.sub.s)
e.sub.s=q.sub.sq.sub.m(td.sub.m)(7) in which, e.sub.m represents position tracking error of the master robot; e.sub.s represents position tracking error of the slave robot; d.sub.m represents fixed-length time delay of forward information transmission from the master robot to the slave robot; d.sub.s represents fixed-length time delay of backward information transmission from the slave robot to the master robot, and d.sub.md.sub.s; t represents an operating time; the fractional order nonsingular rapid terminal sliding mode surface equation is:
s.sub.m(t)=.sub.m+.sub.m1D.sup..sup.1.sup.1[sig(e.sub.m).sup..sup.m1]+.sub.m2D.sup..sup.2[.sub.m(e.sub.m)]
s.sub.s(t)=.sub.s+.sub.s1D.sup..sup.1.sup.1[sig(e.sub.s).sup..sup.s1]+.sub.s2D.sup..sup.2[.sub.s(e.sub.s)](8) in which, s.sub.i represents designed fractional order nonsingular fast terminal sliding mode surface, s.sub.i=[s.sub.i1, s.sub.i2, . . . s.sub.in].sup.TR.sup.n; .sub.i1 is a positive constant, .sub.m1,.sub.s1>1; .sub.1, .sub.2 represent fractional orders and .sub.1(0,1],.sub.2[0,1); D.sup..sup.i.sup.1sig(e.sub.i).sup..sup.i1] represents .sub.11 ordered Riemann-Liouville fractional order derivative of sig(e.sub.i).sup.i1; D.sup..sup.2[.sub.i(e.sub.i)] represents .sub.2 ordered Riemann-Liouville fractional order derivative of .sub.i(e.sub.1); .sub.i1 and .sub.i2 both represent parameters of the sliding mode surface equation and are positive definite diagonal matrix, .sub.i1=diag(.sub.i11, .sub.i12, . . . , .sub.i1n), .sub.i2=diag(.sub.i21, .sub.i22, . . . , .sub.i2n); functions .sub.m(e.sub.m), .sub.s(e.sub.s) are described as: $\begin{array}{cc}{}_m\left(e_m\right)=\{\begin{array}{cc}\mathrm{si}{g\left(e_m\right)}^{{}_{m2}},& \mathrm{if}{\stackrel{\_}{s}}_m=0\mathrm{or}{\stackrel{\_}{s}}_{m}0,.Math.e_m.Math.>{}_m\\ k_{m1}{e}_m+k_{m2}\mathrm{sign}\left(e_m\right)e_m^2,& \mathrm{if}{\stackrel{\_}{s}}_{m}0,.Math.e_m.Math.{}_m\end{array}{}_s\left(e_s\right)=\{\begin{array}{cc}\mathrm{si}{g\left(e_s\right)}^{{}_{s2}},& \mathrm{if}{\stackrel{\_}{s}}_s=0\mathrm{or}{\stackrel{\_}{s}}_{s}0,.Math.e_s.Math.>{}_s\\ k_{s1}{e}_s+k_{s2}\mathrm{sign}\left(e_s\right)e_s^2,& \mathrm{if}{\stackrel{\_}{s}}_{s}0,.Math.e_s.Math.{}_s\end{array}& \left(9\right)\end{array}$ in which, s.sub.m represents a terminal sliding mode surface which shows that the main robot has a singularity problem, s.sub.m=.sub.m+.sub.m1D.sup..sup.1.sup.1[sig(e.sub.m).sup..sup.m1]+.sub.m2D.sup..sup.2[sig(e.sub.m).sup..sup.m2]; s.sub.s represents a terminal sliding mode surface which shows that the slave robot has a singularity problem, s.sub.s=.sub.s+.sub.s1D.sup..sup.1.sup.1[sig(e.sub.s).sup..sup.s1]+.sub.s2D.sup..sup.2[sig(e.sub.s).sup..sup.s2]; .sub.i2 is a positive constant, 0<.sub.m2,.sub.s2<1; k.sub.m1 is a positive constant, k.sub.m1=(2.sub.m2).sub.m.sup..sup.m2.sup.1; k.sub.m2 is a positive constant, k.sub.m2=(.sub.m21).sub.m.sup..sup.m2.sup.2; k.sub.s1 is a positive constant, k.sub.s1=(2.sub.s2).sub.s.sup..sup.m2.sup.1; k.sub.s2 is a positive constant, k.sub.s2=(.sub.s21).sub.s.sup..sup.s2.sup.2; .sub.m,.sub.s represents designed small constant.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Other features, objects and advantages of the present application will become more apparent upon reading of the following detailed description of non-limiting embodiments thereof with reference to the accompanying drawings.

(2) FIG. 1 is a control schematic diagram of a fractional order sliding mode synchronization control method for a teleoperation system based on an event trigger mechanism according to the present invention;

(3) FIG. 2 is a block diagram of the architecture of a bilateral teleoperation system according to the present invention;

(4) FIG. 3 is a curve graph of the trajectory of the teleoperational system according to the present invention;

(5) FIG. 4A is a graph comparing the tracking error of a first joint of the teleoperation system according to the present invention with that of the prior art integer order sliding mode technique;

(6) FIG. 4B is a graph comparing the tracking error of a second joint of the teleoperation system according to the present invention with that of the prior art integer order sliding mode technique;

(7) FIG. 4C is a graph comparing the tracking error of a third joint of the teleoperation system according to the present invention with that of the prior art integer order sliding mode technique; and

(8) FIG. 5 is a curve graph of the controller of the teleoperational system according to the present invention.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(9) The present application will be described in further detail with reference to the drawings and embodiments below. It is to be understood that the specific embodiments described herein are merely illustrative of the relevant invention and not restrictive of the present invention. It should be noted that, for convenience of description, only the portions related to the related invention are shown in the drawings.

(10) It should be noted that the embodiments and features of the embodiments in the present application may be combined with each other without conflict. The present application will be described in detail below with reference to the embodiments with reference to the attached drawings.

(11) FIG. 2 is a block diagram of the architecture of a teleoperation system according to the present invention, depicting a complete teleoperation system, where the operator operates the master robot locally, while the slave robot works on site. The master robot transmits the relevant control information to the slave robot through a communication network, the slave robot feeds back the state information and the acting force of the external environment to the master robot through the network, and the operator can truly sense the change of the remote environment. And the master robot and the slave robot utilize the feedback information to adjust the control strategy, and finally realize the effective operation on the remote controlled object.

(12) FIG. 1 shows a fractional order sliding mode synchronous control method for a teleoperation system based on an event trigger mechanism. The method comprises steps of:

(13) S1, establishing a dynamic model for the teleoperation system by considering external disturbance and parameter uncertainty, wherein the dynamic model is:
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)+B.sub.m({dot over (q)}.sub.m)=.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)+B.sub.s({dot over (q)}.sub.s)=.sub.s+F.sub.e(1)

(14) in which, q.sub.i, {dot over (q)}.sub.i, {umlaut over (q)}.sub.iR.sup.n respectively represent a generalized displacement, a speed and an acceleration of n joints of the robot; M.sub.i(q.sub.i) represents a positive definite inertial matrix of the system; C.sub.i(q.sub.i, {dot over (q)}.sub.i) represents a matrix of Coriolis forces and centrifugal forces of the system; G.sub.i(q.sub.i) represents a gravity moment of the system; F.sub.h, F.sub.eR.sup.n respectively represent an external force applied by an operator and an external force applied by the environment; .sub.iR.sup.n represents a generalized input moment; B.sub.i({dot over (q)}.sub.i)R.sup.n represents a unknown external interference; i=m, s, m represents a master robot, s represents a slave robot.

(15) M.sub.i(q.sub.i), C.sub.i(q.sub.i,{dot over (q)}.sub.i) and G.sub.i(q.sub.i) have a relationship of:

(16) $\begin{array}{cc}\{\begin{array}{c}{M}_i\left(q_i\right)=M_{\mathrm{oi}}\left(q_i\right)+M_i\left(q_i\right)R^{nn}\\ C_i\left(q_i,{\stackrel{.}{q}}_i\right)=C_{\mathrm{oi}}\left(q_i,{\stackrel{.}{q}}_i\right)+C_i\left(q_i,{\stackrel{.}{q}}_i\right)R^{nn}\\ G_i\left(q_i\right)=G_{\mathrm{oi}}\left(q_i\right)+G_i\left(q_i\right)R^n\end{array}& \left(2\right)\end{array}$

(17) in which, M.sub.oi(q.sub.i) represents a nominal value of the positive definite inertia matrix of the system; C.sub.oi(q.sub.i,{dot over (q)}.sub.i) represents a nominal value of the matrix of Coriolis forces and centrifugal forces of the system; G.sub.oi(q.sub.i) represents a nominal value of the gravity moment of the system; M.sub.i(q.sub.i) represents parameter change of the positive definite inertia matrix of the system; C.sub.i(q.sub.i,{dot over (q)}.sub.i) represents parameter change of the matrix of Coriolis forces and centrifugal forces of the system; .sub.Gi(q.sub.i) represents parameter change of the gravity moment of the system.

(18) At the same time, B.sub.i({dot over (q)}.sub.i) satisfies:
B.sub.i({dot over (q)}.sub.i)B.sub.i(3)

(19) in which, B.sub.i represents an unknown positive number.

(20) Then,

(21) $\begin{array}{cc}\{\begin{array}{c}\begin{array}{c}{P}_m\left(q_m,{\stackrel{.}{q}}_m,{\stackrel{.Math.}{q}}_m\right)=-M_m\left(q_m\right){\stackrel{.Math.}{q}}_m-C_m\left(q_m,{\stackrel{.}{q}}_m\right){\stackrel{.}{q}}_m-\\ G_m\left(q_m\right)-B_m\left({\stackrel{.}{q}}_m\right)R^n\end{array}\\ \begin{array}{c}{P}_s\left(q_s,{\stackrel{.}{q}}_s,{\stackrel{.Math.}{q}}_s\right)=-M_s\left(q_s\right){\stackrel{.Math.}{q}}_s-C_s\left(q_s,{\stackrel{.}{q}}_s\right){\stackrel{.}{q}}_s-G_s\left(q_s\right)-\\ B_s\left({\stackrel{.}{q}}_s\right)R^n\end{array}\end{array}& \left(4\right)\end{array}$

(22) in which, P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m) represents a parameter matrix of the master robot; P.sub.s(q.sub.s,{dot over (q)}.sub.s,{umlaut over (q)}.sub.s) represents a parameter matrix of the slave robot.

(23) P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m) and P.sub.s(q.sub.s,{dot over (q)}.sub.s,{umlaut over (q)}.sub.s) are bounded and satisfies:
P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m).sub.m
P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m).sub.s(5)

(24) in which, .sub.m represents an upper limit of P.sub.m(q.sub.m,{dot over (q)}.sub.m,{umlaut over (q)}.sub.m), .sub.m=b.sub.0m+b.sub.1mq.sub.m+b.sub.2m{dot over (q)}.sub.m.sup.2; .sub.s represents an upper limit of P.sub.s(q.sub.s,{dot over (q)}.sub.s,{umlaut over (q)}.sub.s), .sub.s=b.sub.0s+b.sub.1sq.sub.s+b.sub.2s{dot over (q)}.sub.s.sup.2; b.sub.0i, b.sub.1i, b.sub.2i(i=m, s) is a positive constant.

(25) Accordingly, equation (1) for the system may be re-represented as:

(26) 0$\begin{array}{cc}\{\begin{array}{c}\begin{array}{c}{M}_{om}\left(q_m\right){\stackrel{.Math.}{q}}_m+C_{om}\left(q_m,{\stackrel{.}{q}}_m\right){\stackrel{.}{q}}_m+G_{\mathrm{om}}\left(q_m\right)={}_m+\\ P_m\left(q_m,{\stackrel{.}{q}}_m,{\stackrel{.Math.}{q}}_m\right)+F_h\end{array}\\ \begin{array}{c}{M}_{\mathrm{os}}\left(q_s\right){\stackrel{.Math.}{q}}_s+C_{os}\left(q_s,{\stackrel{.}{q}}_s\right){\stackrel{.}{q}}_s+G_{os}\left(q_s\right)={}_s+\\ P_s\left(q_s,{\stackrel{.}{q}}_s,{\stackrel{.Math.}{q}}_s\right)-F_e\end{array}\end{array}.& \left(6\right)\end{array}$

(27) S2 is of selecting the master robot and the slave robot, interactively constructing the teleoperation system through a communication network, measuring respectively mass and length information of a master robot connecting rod and a slave robot connecting rod and real-time robot position information of the master robot connecting rod and the slave robot connecting rod, and determining system parameters of the dynamic model.

(28) System parameters of the dynamical model in step S2 comprise nominally determined parts of the inertia matrix, the matrix of Coriolis forces and centrifugal forces, and the gravity matrix of the master robot and the slave robot, and are all calculated from the length, mass, and joint position information of a mechanical arm.

(29) S3 is of designing a sliding mode surface equation of a fractional order nonsingular rapid terminal by using position tracking errors of the master robot and the slave robot and Riemann-Liouville fractional order calculus.

(30) The position tracking errors of the master robot and the slave robot in step S3 are:
e.sub.m=q.sub.mq.sub.s(td.sub.s)
e.sub.s=q.sub.sq.sub.m(td.sub.m)(7)

(31) in which, e.sub.m represents position tracking error of the master robot; e.sub.s represents position tracking error of the slave robot; d.sub.m represents fixed-length time delay of forward information transmission from the master robot to the slave robot; d.sub.s represents fixed-length time delay of backward information transmission from the slave robot to the master robot, and d.sub.md.sub.s; t represents an operating time.

(32) The fractional order nonsingular rapid terminal sliding mode surface equation is:
s.sub.m(t)=.sub.m+.sub.m1D.sup..sup.1.sup.1[sig(e.sub.m).sup..sup.m1]+.sub.m2D.sup..sup.2[.sub.m(e.sub.m)]
s.sub.s(t)=.sub.s+.sub.s1D.sup..sup.1.sup.1[sig(e.sub.s).sup..sup.s1]+.sub.s2D.sup..sup.2[.sub.s(e.sub.s)](8) in which, s.sub.i represents designed fractional order nonsingular fast terminal sliding mode surface, s.sub.i=[s.sub.i1, s.sub.i2, . . . s.sub.in].sup.TR.sup.n; .sub.i1 is a positive constant, .sub.m1,.sub.s1>1; .sub.1, .sub.2 represent fractional orders and .sub.1(0,1],.sub.2[0,1); D.sup..sup.i.sup.1sig(e.sub.i).sup..sup.i1] represents .sub.11 ordered Riemann-Liouville fractional order derivative of sig(e.sub.i).sup.i1; D.sup..sup.2[.sub.i(e.sub.i)] represents .sub.2 ordered Riemann-Liouville fractional order derivative of .sub.i(e.sub.1); .sub.i1 and .sub.i2 both represent parameters of the sliding mode surface equation and are positive definite diagonal matrix, .sub.i1=diag(.sub.i11, .sub.i12, . . . , .sub.i1n), .sub.i2=diag(.sub.i21, .sub.i22, . . . , .sub.i2n); functions .sub.m(e.sub.m), .sub.s(e.sub.s) are described as:

(33) $\begin{array}{cc}{}_m\left(e_m\right)=\{\begin{array}{cc}\mathrm{si}{g\left(e_m\right)}^{{}_{m2}},& \mathrm{if}{\stackrel{\_}{s}}_m=0\mathrm{or}{\stackrel{\_}{s}}_{m}0,.Math.e_m.Math.>{}_m\\ k_{m1}{e}_m+k_{m2}\mathrm{sign}\left(e_m\right)e_m^2,& \mathrm{if}{\stackrel{\_}{s}}_{m}0,.Math.e_m.Math.{}_m\end{array}{}_s\left(e_s\right)=\{\begin{array}{cc}\mathrm{si}{g\left(e_s\right)}^{{}_{s2}},& \mathrm{if}{\stackrel{\_}{s}}_s=0\mathrm{or}{\stackrel{\_}{s}}_{s}0,.Math.e_s.Math.>{}_s\\ k_{s1}{e}_s+k_{s2}\mathrm{sign}\left(e_s\right)e_s^2,& \mathrm{if}{\stackrel{\_}{s}}_{s}0,.Math.e_s.Math.{}_s\end{array}& \left(9\right)\end{array}$

(34) in which, s.sub.m represents a terminal sliding mode surface which shows that the main robot has a singularity problem, s.sub.m=.sub.m+.sub.m1D.sup..sup.1.sup.1[sig(e.sub.m).sup..sup.m1]+.sub.m2D.sup..sup.2[sig(e.sub.m).sup..sup.m2]; s.sub.s represents a terminal sliding mode surface which shows that the slave robot has a singularity problem, s.sub.s=.sub.s+.sub.s1D.sup..sup.1.sup.1[sig(e.sub.s).sup..sup.s1]+.sub.s2D.sup..sup.2[sig(e.sub.s).sup..sup.s2]; .sub.i2 is a positive constant, 0<.sub.m2,.sub.s2<1; k.sub.m1 is a positive constant, k.sub.m1=(2.sub.m2).sub.m.sup..sup.m2.sup.1; k.sub.m2 is a positive constant, k.sub.m2=(.sub.m21).sub.m.sup..sup.m2.sup.2; k.sub.s1 is a positive constant, k.sub.s1=(2.sub.s2).sub.s.sup..sup.m2.sup.1; k.sub.s2 is a positive constant, k.sub.s2=(.sub.s21).sub.s.sup..sup.s2.sup.2; .sub.m,.sub.s represents designed small constant.

(35) S4 is of setting a trigger event condition of information interaction between the master robot and the slave robot, designing a self-adaptive fractional order nonsingular rapid terminal sliding mode controller which can compensate measurement errors and eliminate the influence of system uncertainty based on the sliding mode, enabling the controller to avoid the Zeno phenomenon, performing stability analysis, proving the boundedness of a closed-loop state signal of the system, enabling the slave robot to track the motion of the master robot within a limited time, and realizing synchronous control of the teleoperation system.

(36) In step S4, the trigger event condition is:

(37) $\begin{array}{cc}\{\begin{array}{cc}{}_i\left(t\right)=u_i\left(t_k^i\right),& t[t_k^i,t_{k+1}^i)\\ t_{k+1}^j=\inf \{tR.Math.z_i\left(t\right).Math.z_i\left(t\right)b_i\},& t_1^i=0,j=1,2,.Math.,n\end{array}& \left(10\right)\end{array}$

(38) in which, z.sub.i(t) represents a measurement error, z.sub.i(t)=[z.sub.i1, z.sub.i2, . . . , z.sub.in].sup.T and z.sub.i(t)=u.sub.i(t).sub.i(t); .sub.i(t) represents a generalized input moment of the teleoperation system, .sub.i(t)=[.sub.i1, .sub.i2, . . . , .sub.in].sup.T; b.sub.i represents a positive definite matrix, b.sub.i(t)=[b.sub.i1, b.sub.i2, . . . , b.sub.in].sup.T R.sup.n; u.sub.i(t) represents actual controller under event trigger mechanism, u.sub.i(t)=[u.sub.i1, u.sub.i2, . . . , u.sub.in].sup.T; t.sub.k.sup.i, kz.sup.+ represents controller update time;

(39) in a period of time t.sup.i[t.sub.k.sup.i,t.sub.k+1.sup.t), the control signal remains constant until t.sup.i=t.sub.k+1.sup.i, minimum internal time of the mentioned event trigger mechanism is strictly greater than 0, which eliminates occurrence of the Zeno behavior.

(40) Based on the designed sliding mode, the actual controller is chosen as:
.sub.i(t)=u.sub.i(t).sub.i(t)b.sub.i(11)

(41) in which, .sub.i(t) represents a continuous time-varying parameter satisfying conditions .sub.i(t.sub.k.sup.i)=0,.sub.i(t.sub.k+1.sup.i)=1 and |.sub.i(t)|1, t[t.sub.k.sup.i,t.sub.k+1.sup.i).

(42) The sliding mode controller of the self-adaptive fractional order nonsingular rapid terminal is designed as:

(43) $\begin{array}{cc}\{\begin{array}{c}{u}_m\left(t\right)={\stackrel{\_}{u}}_m-{\stackrel{\_}{b}}_m\tanh \left(s_m^T{M}_{\mathrm{om}}^{-1}{\stackrel{\_}{b}}_m/{}_m\right)\\ \begin{array}{c}{\stackrel{\_}{u}}_m=M_{\mathrm{om}}\left(q_m\right){\stackrel{.Math.}{q}}_s\left(t-d_s\right)+C_{\mathrm{om}}\left(q_m,{\stackrel{.}{q}}_m\right){\stackrel{.}{q}}_m+\\ G_{\mathrm{om}}\left(q_m\right)+{\stackrel{\_}{u}}_{m1}+{\stackrel{\_}{u}}_{m2}\end{array}\\ \begin{array}{c}{\stackrel{\_}{u}}_{m1}=-M_{\mathrm{om}}({}_{m}si{g\left(s_m\right)}^{}+{}_{m1}{D}^{{}_1}\left[\mathrm{si}{g\left(e_m\right)}^{{}_{m1}}\right]+\\ {}_{m2}{D}^{{}_2+1}\left[{}_m\left(e_m\right)\right])\end{array}\\ {\stackrel{\_}{u}}_{m2}=\{\begin{array}{cc}-\frac{s_m^T{M}_{om}^{-1}}{.Math.s_m^T{M}_{om}^{-1}.Math.}{\stackrel{}{}}_m,& \mathrm{if}{s}_m^T{M}_{om}^{-1}0\\ 0,& \mathrm{if}{s}_m^T{M}_{om}^{-1}=0\end{array}\end{array}& \left(12\right)\end{array}$$\begin{array}{cc}\{\begin{array}{c}{u}_s\left(t\right)={\stackrel{\_}{u}}_s-{\stackrel{\_}{b}}_s\tanh \left(s_s^T{M}_{\mathrm{os}}^{-1}{\stackrel{\_}{b}}_s/{}_s\right)\\ \begin{array}{c}{\stackrel{\_}{u}}_s=M_{\mathrm{os}}\left(q_s\right){\stackrel{.Math.}{q}}_m\left(t-d_m\right)+C_{\mathrm{os}}\left(q_s,{\stackrel{.}{q}}_s\right){\stackrel{.}{q}}_s+G_{\mathrm{os}}\left(q_s\right)+\\ {\stackrel{\_}{u}}_{s1}+{\stackrel{\_}{u}}_{s2}\end{array}\\ \begin{array}{c}{\stackrel{\_}{u}}_{s1}=-M_{\mathrm{os}}({}_{s}si{g\left(s_s\right)}^{}+{}_{s1}{D}^{{}_1}\left[\mathrm{si}{g\left(e_s\right)}^{{}_{s1}}\right]+\\ {}_{s2}{D}^{{}_2+1}\left[{}_s\left(e_s\right)\right])\end{array}\\ {\stackrel{\_}{u}}_{s2}=\{\begin{array}{cc}-\frac{s_s^T{M}_{\mathrm{os}}^{-1}}{.Math.s_s^T{M}_{\mathrm{os}}^{-1}.Math.}{\stackrel{}{}}_s,& \mathrm{if}{s}_s^T{M}_{\mathrm{os}}^{-1}0\\ 0,& \mathrm{if}{s}_s^T{M}_{\mathrm{os}}^{-1}=0\end{array}\end{array}& \left(13\right)\end{array}$

(44) in which, b.sub.m represents a positive definite matrix, b.sub.m=[b.sub.m1, b.sub.m2, . . . , b.sub.mn].sup.T and satisfies the constant b.sub.mj>b.sub.mj(j=1, 2, . . . , n); b.sub.s represents a positive definite matrix, b.sub.s=[b.sub.s1, b.sub.s2, . . . , b.sub.sn].sup.T and satisfies the constant b.sub.sj>b.sub.sj(j=1, 2, . . . , n); .sub.m, .sub.s represents a positive constant to be designed; .sub.i represents a positive definite diagonal matrix, .sub.i=diag[.sub.i1, .sub.i2, . . . , .sub.in] and satisfies the constant .sub.ij>0(j=1, 2, . . . , n); represents a positive constant to be designed and satisfies (0,1); {circumflex over ()} represents an estimated value of an upper bound of P.sub.m(q.sub.m, {dot over (q)}.sub.n, {umlaut over (q)}.sub.m), {circumflex over ()}.sub.m={circumflex over (b)}.sub.0m+{circumflex over (b)}.sub.1m+{circumflex over (b)}.sub.2m{dot over (q)}.sub.m.sup.2; {circumflex over ()}.sub.s represents an estimated value of an upper bound of P.sub.s(q.sub.s, {dot over (q)}.sub.s, {umlaut over (q)}.sub.s), {circumflex over ()}.sub.s={circumflex over (b)}.sub.0s+{circumflex over (b)}.sub.1s+{circumflex over (b)}.sub.2s{dot over (q)}.sub.s.sup.2; variables {circumflex over (b)}.sub.0i, {circumflex over (b)}.sub.1i, {circumflex over (b)}.sub.2i representing estimated values of parameters b.sub.0i, b.sub.1i, b.sub.2i, respectively;

(45) the self-adaptation law is designed as:
{circumflex over ({dot over (b)})}.sub.0i=.sub.0is.sub.i.sup.TM.sub.oi.sup.1
{circumflex over ({dot over (b)})}.sub.1i=.sub.1is.sub.i.sup.TM.sub.oi.sup.1q.sub.i(14)
{circumflex over ({dot over (b)})}.sub.2i=.sub.2is.sub.i.sup.TM.sub.oi.sup.1q.sub.i.sup.2

(46) in which, .sub.0i, .sub.1i, .sub.2i represents positive constants;

(47) a Lyapunov function is selected as:

(48) $$\begin{gathered} \begin{array}{cc}V=V_1+V_2 \\ {V}_1=\underset{i=m,s}{.Math.}\frac{1}{2}{s}_i^T{s}_i \\ {V}_2=\underset{i=m,s}{.Math.}\frac{1}{2}\underset{j=0}{\overset{2}{.Math.}}{}_{ji}^{-1}{\tilde{b}}_{ji}^2& \left(15\right)\end{array} \end{gathered}$$

(49) in which, V represents a Lyapunov function; V.sub.1 represents a first Lyapunov function, V.sub.2 represents a second Lyapunov function, {tilde over (b)}.sub.ji=b.sub.ji{circumflex over (b)}.sub.ji represents a self-adaptive estimation error, j=1, 2, . . . , n;

(50) Its first derivative with time is:

(51) $\begin{array}{cc}\begin{array}{c}{\stackrel{.}{V}}_1=\underset{i=m,s}{.Math.}{s}_i^T{\stackrel{.}{s}}_i\\ =\underset{i=m,s}{.Math.}{s}_i^T\left[{\stackrel{.Math.}{e}}_i+{}_{i1}{D}^{{}_1}\left[{\mathrm{sig}\left(e_i\right)}^{{}_{i1}}\right]+{}_{i2}{D}^{{}_2+1}\left[{}_i\left(e_i\right)\right]\right]\\ \underset{i=m,s}{.Math.}-{\stackrel{\_}{}}_i{.Math.s_i.Math.}^{+1}+s_i^T{M}_{\mathrm{oi}}^{-1}{P}_i-.Math.s_i^T{M}_{\mathrm{oi}}^{-1}.Math.{\stackrel{}{}}_i+s_i^T{M}_{\mathrm{oi}}^{-1}\\ \left(-{\stackrel{\_}{b}}_i\tanh \left(\frac{s_i^T{M}_{\mathrm{oi}}^{-1}{\stackrel{\_}{b}}_i}{{}_i}\right)-{}_i\left(t\right)b_i\right)\end{array}\begin{array}{c}{\stackrel{.}{V}}_2=-\underset{i=m,s}{.Math.}\underset{j=0}{\overset{2}{.Math.}}{\tilde{b}}_{ji}{}_{ji}^{-1}{\stackrel{.}{\hat{b}}}_{ji}\\ =\underset{i=m,s}{.Math.}-{\tilde{b}}_{0i}.Math.s_i{M}_{\mathrm{oi}}^{-1}.Math.-{\tilde{b}}_{1i}.Math.s_i{M}_{\mathrm{oi}}^{-1}.Math..Math.q_m.Math.-\\ {\tilde{b}}_{2i}.Math.s_i{M}_{\mathrm{oi}}^{-1}.Math.{.Math.{\stackrel{.}{q}}_m.Math.}^2\\ =\underset{i=m,s}{.Math.}-.Math.s_i{M}_{\mathrm{oi}}^{-1}.Math.{\stackrel{~}{}}_i\end{array}& \left(16\right)\end{array}$

(52) in which, .sub.i represents a minimum characteristic value of a matrix .sub.i, .sub.i=.sub.min(.sub.i); {tilde over ()}.sub.i represents an estimation error, that is, {tilde over ()}.sub.i=.sub.i{circumflex over ()}.sub.i.

(53) As shown in FIG. 3, it is a curve graph of the trajectory of the self-adaptive fractional order nonsingular rapid terminal sliding mode controller based on the event trigger mechanism according to the present invention. As shown in FIGS. 4A to 4C, they are graphs comparing the tracking error of the fractional order nonsingular rapid terminal sliding mode control for the teleoperation system according to the present invention with that of the prior art integer order sliding mode technique for the teleoperation system. As shown in FIG. 5, it is a curve graph showing a comparison between the responses of the controller for the teleoperation system under the event trigger mechanism and under the time trigger mechanism in the present invention.

(54) Effectiveness of the designed controller can be proved through experimental results, a closed loop state signal of the system is bounded, and the slave robot can track the motion of the master robot within a limited time, so that the rapid high-precision synchronous control of the teleoperation system is realized.

(55) According to the present invention, the dynamic model for the teleoperation system under the conditions of unknown external interference and parameter uncertainty is established, and the fractional order nonsingular rapid terminal sliding mode surface is adopted, so that the singularity problem can be avoided, the degree of freedom of the controller is expanded, the convergence speed is accelerated, the control precision is improved, the buffeting problem existing in an integer order sliding mode surface is reduced, and the applicability is stronger. Compared with the prior art in which the synchronous control mode of the traditional periodic sampling of the teleoperation is adopted, the method provided by the present invention has the advantages that on the premise of ensuring the synchronous control performance of the system, the master robot and the slave robot only need to carry out intermittent communication and controller transmission and updating when the event trigger condition is met, the occupation of network bandwidth and communication resources is greatly reduced, and the utilization rate of the resources is improved.

(56) The present invention can avoid the singularity problem, simultaneously expand the degree of freedom of the controller, accelerate the convergence speed, improve the control precision, reduce the buffeting problem existing in the integer order sliding mode surface, have stronger applicability, greatly reduce the occupation of network bandwidth and communication resources and improve the utilization rate of the resources.

(57) Finally, it should be noted that, the above embodiments are merely illustrative of the technical solution of the present invention and not restrictive of technical solution of the present invention. Although the present invention has been described in detail with reference to the above embodiments, it should be understood by those skilled in the art that modifications and equivalents may be made thereto without departing from the spirit and scope of the present invention and it is intended to fall into the scope of the invention as defined in the appended claims.