Method for controlling vibration of flexible mechanical arm based on cooperative tracking

Abstract

A method for controlling vibration of flexible mechanical arms based on cooperative tracking is disclosed, including: building a dynamic model of the flexible mechanical arm, according to a dynamic characteristic, constructing a flexible mechanical arm group made up of a plurality of flexible mechanical arms, assigning one of the plurality of flexible mechanical arms as a leader and the rest ones as followers which are required to track the leader's motion trajectory so as to realize cooperative work; designing cooperative control-based boundary controllers in combination with a Lyapunov method to realize cooperative work and suppress vibration of the flexible mechanical arms; and constructing a Lyapunov function using Lyapunov direct method to validate stability of the flexible mechanical arms under the control.

Claims

1. A method for controlling vibration of a plurality of flexible mechanical arms based on cooperative tracking, the method comprising: kinematic each of establishing a dynamic model equation for each of the plurality of flexible mechanical arms, according to characteristics of the flexible mechanical arms including kinetic energy, potential energy, and virtual work done by non-conservative force acting on the flexible mechanical arm; assigning one of the plurality of flexible mechanical arms as a leader and the rest ones as followers which need to track the leader's motion trajectory; designing cooperative tracking-based boundary controllers based on the plurality of flexible mechanical arms; constructing a Lyapunov function for each of the plurality of flexible mechanical arms based on the dynamic model equation and the boundary controllers; and validating stability of each of the plurality of flexible mechanical arms, according to the Lyapunov function, wherein the dynamic model equation is obtained by substituting the kinetic energy, the potential energy, and the virtual work into Hamilton's principle, and the kinematic model equation is as follows: $?{\stackrel{.Math.}{w}}_i+{\mathrm{Elw}}_i^{}-{\mathrm{Tw}}_i^{}+?{\stackrel{.}{w}}_i=-\left(r+x\right)\left(?{\stackrel{.Math.}{?}}_i+?{\stackrel{.}{?}}_i\right),\mathrm{and}{I}_h\stackrel{.Math.}{?}=-?{\mathrm{Elw}}_i^{}\left(0,t\right)+{\mathrm{Elw}}_i^{}\left(0,t\right)+{\mathrm{Tw}}_i\left(l,t\right)+u_{2l},$ where w.sub.i(x,t) represents a vibration offset of an ith flexible mechanical arm in xoy coordinate system, {dot over (w)}.sub.i(x,t) and {umlaut over (w)}.sub.i(x,t) represent the first and second derivative of time and are abbreviated as {dot over (w)}.sub.i and {umlaut over (w)}.sub.i respectively, w.sub.i(x,t), w.sub.i(x,t), w.sub.i(x,t) and w.sub.i(x,t) represent the first, second, third and fourth derivatives of w.sub.i(x,t) with respect to x and are abbreviated as w.sub.i, w.sub.i, w.sub.i and w.sub.i respectively, ? represents a uniform mass per unit length of the flexible mechanical arm, m represents a tip mass of the flexible mechanical arm, l represents a length of the mechanical arm, r represents a radius of a rigid hub, lh represents a hub inertia, ?.sub.i represents an attitude angle of the ith flexible mechanical arm, {dot over (?)}.sub.i and {umlaut over (?)}.sub.i represent the first and second derivative of ?.sub.i with respect to time respectively, T represents a tension, EI represents a bending stiffness, y represents a viscous damping coefficient, w.sub.i(0,t) represents a value of w.sub.i(x,t) at x=0, w.sub.i(x,t) represents a value of w.sub.i(x,t) at x=0, w.sub.i(l,t) represents a value of w.sub.i(x,t) at x=l, and u.sub.2l represents a controller at a fixed end position of the flexible mechanical arm, wherein a boundary condition is as follows: $m{\stackrel{.Math.}{w}}_i\left(l,t\right)={\mathrm{EIw}}_i^{}\left(0,t\right)-m\left(r+l\right){\stackrel{.Math.}{?}}_i-{\mathrm{Tw}}_i^{}\left(l,t\right)+u_{1i},\mathrm{and}$ $w_i\left(0,t\right)=w_i^{}\left(0,t\right)=w_i^{}\left(l,t\right)=0,$ where w.sub.i(l,t) and w.sub.i(l,t) represent values of w.sub.i(x,t) and w.sub.i(x,t) at x=l respectively, w.sub.i(l,t) and w.sub.i(l,t) represent values of w.sub.i(x,t) and w.sub.i(x,t) at x=l respectively, w.sub.i(0,t) and w.sub.i(0,t) represent values of w.sub.i(x,t) and w.sub.i(x,t) at x=0 respectively, and u.sub.1i represents a controller at a tip position of the flexible mechanical arm, and wherein the boundary controllers are configured to suppress vibration of the plurality of flexible mechanical arms, and to perform cooperative control in such a way that the flexible mechanical arm as one of the followers tracks the motion trajectory of the flexible mechanical arm as the leader.

2. The method of claim 1, wherein assigning one of the plurality of flexible mechanical arms as the leader and the rest ones as the followers, and designing the boundary controller includes: defining an auxiliary variable as follows: ${?}_{ri}=-v{.Math.}_{j=1}^N{a}_{ij}\left({?}_i-{?}_j\right)-b_{\mathrm{io}}\left({?}_i-{?}_o\right),$ where ?.sub.ij is an element denoted by (i, j) in an adjacency matrix A, the adjacency matrix A represents relationships between respective flexible mechanical arms as the follower, and A=[?.sub.ij]?R.sup.k?k is a non-negative matrix and defined that if there is information communication between the followers, then ?.sub.ij>0, otherwise, ?.sub.ij=0; b.sub.i0 is an element denoted by (i, j) in a diagonal matrix B which represents relationships between the leader and the flexible mechanical arms as the follower, and B=diag(b.sub.10, b.sub.20, . . . , b.sub.k0) is a non-negative diagonal matrix and defined that if there exists information communication between the leader and the followers, then b.sub.i0>0, otherwise, b.sub.i0=0; and v is a positive constant, ?.sub.0 represents an attitude angle of the flexible mechanical arm as the leader, ?.sub.i represents an attitude angle of the i.sup.th flexible mechanical arm, and ?.sub.j represent an attitude angle of a j.sup.th flexible mechanical arm; defining a generalized tracking error, a second tracking error, and a virtual control amount respectfully as follows: $e_{1i}={?}_i-{?}_{\mathrm{ri}},e_{2i}={\stackrel{.}{?}}_i-u_{\mathrm{ei}}\mathrm{and}{u}_{\mathrm{ei}}={\stackrel{.}{?}}_{\mathrm{ri}}-\frac{?}{?}{e}_{1i},$ where ?.sub.ri represents an auxiliary angle, and {dot over (?)}.sub.ri represents the first derivative of ?.sub.ri with respect to time; defining variables as follows: y.sub.ei(x,t)=(r+x)e.sub.1i+w.sub.i, and y.sub.ei(x,t) being abbreviated as y.sub.ei; and designing the boundary controllers as follows: $u_{1i}=-\frac{?m}{?}{\stackrel{.}{y}}_{\mathrm{ei}}\left(l,t\right)-k_m{S}_{1i},\mathrm{and}$ $u_{2i}=-k_{p1}{e}_{1i}-\frac{?I_h}{?}{e}_{2i}-k_{p3}{u}_{\mathrm{ei}}-k_d{S}_{2i},$ where {dot over (y)}.sub.ei(x,t) represents the first derivative of y.sub.ei(x,t) with respect to time, {dot over (y)}.sub.ei(l,t) represents a value of {dot over (y)}.sub.ei(x,t) at x=l, and S.sub.1i and S.sub.2i are as follows respectfully: $S_{1i}=a{\stackrel{.Math.}{y}}_i\left(l,t\right)+?y_{ei}\left(l,t\right)\mathrm{and}{S}_{2i}=\frac{1}{2}a{\stackrel{.Math.}{?}}_i+?e_{1i},$ where {dot over (y)}.sub.i(l,t) represents a value of {dot over (y)}.sub.i(x,t) at x=l, y.sub.ei(l,t) represents a value of y.sub.ei(x,t) at x=l, and ?, ?, k.sub.m, k.sub.p1, k.sub.p3 and k.sub.d represent control parameters and are non-negative constants.

3. The method of claim 2, wherein the Lyapunov function for each of the flexible mechanical arms is constructed as follows: V.sub.i=V.sub.1i+V.sub.2i+V.sub.3i, where $V_i=V_{1i}+V_{2i}+V_{3i},\mathrm{where}$ $V_{1i}=\frac{\mathrm{??}}{2}{?}_0^l{y}_{\mathrm{ei}}^2\mathrm{dx}+\frac{\mathrm{??}}{2}{?}_0^l{\stackrel{.}{y}}_i^2\mathrm{dx}+\frac{?T}{2}{?}_0^l{\left(w_i^{}\right)}^2\mathrm{dx}+\frac{?\mathrm{EI}}{2}{\left(w_i^{}\right)}^2\mathrm{dx},$ $V_{2i}=\left(\frac{?k_{p1}}{2}+\frac{\mathrm{??}{k}_d}{4}\right)e_{1i}^2+\frac{?I_h}{4}{e}_{2i}^2+\frac{m}{2?}{S}_{1i}^2+\frac{I_h}{?}{S}_{2i}^2+\frac{?I_h}{4}{u}_{\mathrm{ei}}^2,\mathrm{and}$ $V_{3i}=\frac{?I_h}{2}{e}_{2i}{u}_{\mathrm{ei}}+\mathrm{??}{?}_0^l{y}_{\mathrm{ei}}{\stackrel{.}{y}}_i\mathrm{dx}.$

4. The method of claim 1, wherein validating stability of the flexible mechanical arm, according to the Lyapunov function, includes: proving, by validating a positive definiteness of the Lyapunov function, the flexible mechanical arm is stable in a Lyapunov theory; and proving, by validating a negative definiteness of the first derivative of the Lyapunov function, the flexible mechanical arm is asymptotically stable.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a flow schematic diagram illustrating a method for controlling vibration of a flexible mechanical arm based on collaborative tracking according to an embodiment of the present disclosure;

(2) FIG. 2 is a structural schematic diagram illustrating a flexible mechanical arm according to an embodiment of the present disclosure; and

(3) FIG. 3 is an example diagram illustrating a network topology of a flexible mechanical arm group according to an embodiment of the present disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(4) In order to make the object, technical solutions, and advantages of embodiments in the present disclosure clearer, the technical solutions of the embodiments will be described more clearly and completely according to accompanying drawings in the embodiments in the present disclosure. Apparently, the described embodiments are part of the embodiments of the present disclosure, rather than all of the embodiments. Based on the embodiments of the present disclosure, all other embodiments obtained by those ordinary technicians without creative work shall be within the scope of the present disclosure.

Embodiments

(5) Referring to FIG. 1, FIG. 1 is a flow schematic diagram illustrating a method for controlling vibration of a flexible mechanical arm based on cooperative tracking according to an embodiment of the present disclosure. The method may include the following steps.

(6) At S101, a dynamic model of the flexible mechanical arm is built according to a dynamic characteristic of the flexible mechanical arm.

(7) As shown in FIG. 2, a typical flexible mechanical arm is with a left side boundary fixed to an origin of a coordinate system, which is called as a fixed end, and with a right side boundary loadable with a load, which is called as a tip. Boundary controllers u.sub.1i and u.sub.2i act on the tip position and the left side position of the flexible mechanical arm respectively. The flexible mechanical arm is with a length of l, a vibration offset of w.sub.i(x,t) in xoy coordinate system, and a total vibration offset of y.sub.i(x,t) in XOY coordinate system.

(8) Kinetic energy of the flexible mechanical arm may be represented as:

(9) $\begin{array}{cc}{E}_{ki}=\frac{1}{2}?{?}_0^l{\stackrel{.Math.}{y}}_i^{2}dx+\frac{1}{2}m{\stackrel{.Math.}{y}}_i^2\left(l,t\right)+\frac{1}{2}{I}_h{\stackrel{.Math.}{?}}_i^2,& \left(1\right)\end{array}$

(10) where E.sub.ki represents kinetic energy of an i.sup.th flexible mechanical arm, ? represents a uniform mass per unit length of the flexible mechanical arm, y.sub.i(x,t) represents an elastic deformation of the i.sup.th flexible mechanical arm at time t and location x in the XOY coordinate system and is abbreviated as y.sub.i, and {dot over (y)}.sub.i(x,t) is the first derivative of y.sub.i(x,t) with respect to time and is abbreviated as {dot over (y)}.sub.i, {dot over (y)}.sub.i(l,t) represents a value of y.sub.i(x,t) at x=l, m represents a tip mass of the flexible mechanical arm, l represents a length of the flexible mechanical arm, r represents a radius of a rigid hub, I.sub.h represents a hub inertia, ?.sub.i represents an attitude angle, and {dot over (?)}.sub.i is the first derivative of ?.sub.i with respect to time.

(11) Potential energy of the flexible mechanical arm is represented as:

(12) $\begin{array}{cc}{E}_{pi}=\frac{1}{2}T{?}_0^l{\left(w_i^{}\right)}^{2}dx+\frac{1}{2}\mathrm{EI}{?}_0^l{\left(w_i^{}\right)}^2\mathrm{dx},& \left(2\right)\end{array}$

(13) where w.sub.i(x,t) represents a vibration offset of the i.sup.th flexible mechanical arm in the xoy coordinate system and is abbreviated as w.sub.i, w.sub.i(x,t) and w.sub.i(x,t) represent the first and second derivative of w.sub.i(x,t) with respect to x and are abbreviated as w.sub.i and w.sub.i respectively, T represents a tension, and EI represents a bending stiffness.

(14) And virtual work done by non-conservative forces acting on the flexible mechanical arm is represented as:

(15) 0$\begin{array}{cc}?W_i=-?{?}_0^l{y}_i?y_{i}dx+u_{1i}?y_i\left(l,t\right)+u_{2i}?{?}_i,& \left(3\right)\end{array}$

(16) where ? represents a variational symbol, ? represents a viscous damping coefficient, u.sub.1i and u.sub.2i represent controllers located at the tip and the fixed end of the flexible mechanical arm respectively, and y.sub.i(l,t) represents a value of y.sub.i(l,t) at x=l.

(17) By substituting the kinetic energy, the potential energy, and the virtual work into Hamilton's principle, a dynamic model equation for the flexible mechanical arm is obtained as follows:

(18) $\begin{array}{cc}?{\stackrel{.Math.}{?}}_i+{\mathrm{EIw}}_i^{}-{\mathrm{Tw}}_i^{}+?{\stackrel{.Math.}{w}}_i=-\left(r+x\right)\left(?{\stackrel{.Math.}{?}}_i+?{\stackrel{.}{?}}_i\right),\mathrm{and}& \left(4\right)\end{array}$$\begin{array}{cc}{I}_h\stackrel{.Math.}{?}=-?{\mathrm{EIw}}_i^{}\left(0,t\right)+{\mathrm{EIw}}_i^{}\left(0,t\right)+T{w}_i\left(l,t\right)+u_{2i},,& \left(5\right)\end{array}$

(19) where {dot over (w)}.sub.i(x,t) and {umlaut over (w)}.sub.i(x,t) represent the first and second derivative of w.sub.i(x,t) with respect to time and are abbreviated as {dot over (w)}.sub.i and {umlaut over (w)}.sub.i respectively, w.sub.i(x,t) and w.sub.i(x,t) represent the third and fourth derivative of w.sub.i(x,t) with respect to x and are abbreviated as w.sub.i and w.sub.i respectively, {umlaut over (?)}.sub.i represents the second derivative of the attitude angle ?.sub.i with respect to time, w.sub.i(0,t) represents a value of w.sub.i(x,t) at x=0, w.sub.i(0,t) represents a value of w.sub.i(x,t) at x=0, w.sub.i(l,t) represents a value of w.sub.i(x,t) at x=l, and ?t|[0,?).

(20) A boundary condition is presented as follows:

(21) $\begin{array}{cc}m{\stackrel{.Math.}{w}}_i\left(l,t\right)={\mathrm{EIw}}_i^{}\left(0,t\right)-m\left(r+l\right){\stackrel{.Math.}{?}}_i-{\mathrm{Tw}}_i^{}\left(l,t\right)+u_{1i},\mathrm{and}& \left(6\right)\end{array}$$\begin{array}{cc}{w}_i\left(0,t\right)=w_i^{}\left(0,t\right)=w_i^{}\left(l,t\right)=0,& \left(7\right)\end{array}$

(22) where {dot over (w)}.sub.i(l,t) represents a value of {dot over (w)}.sub.i(x,t) at x=l, {umlaut over (w)}.sub.i(l,t) represents a value of {umlaut over (w)}.sub.i(x,t) at x=l, w(l,t) represents a value of w.sub.i(x,t) at x=l, w.sub.i(l,t) represents a value of w.sub.i(x,t) at x=l, u.sub.1i represents a controller at the tip position of the flexible mechanical arm, w.sub.i(0,t) represents a value of w.sub.i(x,t) at x=0, and w.sub.i(0,t) represents a value of w.sub.i(x,t) at x=0.

(23) At S102, based on the flexible mechanical arm, a flexible mechanical arm group is made up of a plurality of flexible mechanical arms, one of which is assigned as a leader, and the rest ones are followers. Then the boundary controllers are constructed based on cooperative tracking.

(24) As shown in FIG. 3, the flexible mechanical arm numbered 0 is assigned as a leader, and the others are followers. The follower needs to track the leader's motion trajectory to realize the cooperative control of multiple flexible arms. Arrows represent information communication between the flexible mechanical arms, and the adjacency matrix A represents information communication relationship between these followers, and the diagonal matrix B represents information communication relationship between the leader and the followers.

(25) In order to reduce or eliminate vibration of the flexible mechanical arm and to achieve cooperative tracking of the plurality of flexible mechanical arms, a kind of cooperative tracking-based boundary controller is constructed. Details are as follows

(26) An auxiliary variable is defined as:

(27) $\begin{array}{cc}{?}_{ri}=-v\underset{j=1}{\overset{N}{.Math.}}{a}_{ij}\left({?}_i-{?}_j\right)-b_{i0}\left({?}_i-{?}_0\right),& \left(8\right)\end{array}$

(28) where A=[?.sub.ij]?R.sup.k?k is a non-negative matrix and is defined that if there is information communication between two flexible mechanical arms, then ?.sub.ij>0, otherwise ?.sub.ij=0; B=diag(b.sub.10, b.sub.20, . . . , b.sub.k0) is a non-negative diagonal matrix and is defined that if there is information communication between the leader and the follower, then b.sub.i0>0, otherwise, b.sub.i0=0; and v is a positive constant, ?.sub.0 represents an attitude angle of the flexible mechanical arm as the leader, and ?.sub.i and ?.sub.j represent attitude angles of the i.sup.th and j.sup.th flexible mechanical arms respectively.

(29) A generalized tracking error, a second tracking error, and a virtual control amount are respectively defined as:

(30) $\begin{array}{cc}{e}_{1i}={?}_i-{?}_{ri},& \left(9\right)\end{array}$$\begin{array}{cc}{e}_{2i}={\stackrel{.}{?}}_i-u_{ei},\mathrm{and}& \left(10\right)\end{array}$$\begin{array}{cc}{u}_{ei}={\stackrel{.}{?}}_{ri}-\frac{?}{?}{e}_{1i},& \left(11\right)\end{array}$

(31) where {dot over (?)}.sub.ri i is the first derivative of ?.sub.ri with respect to time.

(32) A variables is defined as follows:

(33) $\begin{array}{cc}{y}_{ei}\left(x,t\right)=\left(r+x\right)e_{1i}+w_i,& \left(12\right)\end{array}$

(34) y.sub.ei(x,t) is abbreviated as y.sub.ei.

(35) Boundary controllers are constructed as follows:

(36) $\begin{array}{cc}{u}_{1i}=-\frac{?m}{?}{\stackrel{.}{y}}_{ei}\left(l,t\right)-k_m{S}_{1i},\mathrm{and}& \left(13\right)\end{array}$$\begin{array}{cc}{u}_{2i}=-k_{p1}{e}_{1i}-\frac{?I_h}{?}{e}_{2i}-k_{p3}{u}_{ei}-k_d{S}_{2i},& \left(14\right)\end{array}$

(37) where {dot over (y)}.sub.ei(x,t) represents the first derivative of y.sub.ei (x,t) with respect to time, and {dot over (y)}.sub.ei(l,t) represents a value of {dot over (y)}.sub.ei(x,t) at x=l.

(38) S.sub.1i, and S.sub.2i are proposed as follows:

(39) $\begin{array}{cc}{S}_{1i}=a{\stackrel{.Math.}{y}}_i\left(l,t\right)+?y_{ei}\left(l,t\right)\mathrm{and}{S}_{2i}=\frac{1}{2}a{\stackrel{.Math.}{?}}_i+?e_{1i},& \left(15\right)\end{array}$

(40) where {dot over (y)}.sub.i(l,t) represents a value of {dot over (y)}.sub.i(x,t) at x=l, y.sub.ei(l,t) represents a value of y.sub.ei(x,t) at x=l, and ?, ?, k.sub.m, k.sub.p1, k.sub.p3 and k.sub.d are gain parameters of the boundary controller and all are greater than 0.

(41) Most of the existing researches on vibration control of the flexible mechanical arm focus on a single flexible mechanical arm system, and many of them adopt PID control, robust control, and so on. In this embodiment, the auxiliary variable represents information communication relationship among the flexible mechanical arms, and then two boundary controllers located at the fixed end and the tip respectively are constructed, so that not only vibration suppression effect can be achieved, but also the effect of cooperative tracking of these multiple flexible mechanical arms are achieved. All of the above signals can be obtained by sensors or calculations.

(42) At S103, A Lyapunov function for the flexible mechanical arm is constructed based on the flexible mechanical arm and the boundary controllers.

(43) The Lyapunov function is constructed as:

(44) $\begin{array}{cc}{V}_i=V_{1i}+V_{2i}+V_{3i}.& \left(16\right)\end{array}$$\begin{array}{cc}{V}_{1i}=\frac{??}{2}{?}_0^l{y}_{ei}^{2}dx+\frac{??}{2}{?}_0^l{\stackrel{.}{y}}_i^{2}dx+\frac{?T}{2}{?}_0^l{\left(w_i^{}\right)}^{2}dx+\frac{?\mathrm{EI}}{2}{?}_0^l{\left(w_i^{}\right)}^2\mathrm{dx},& \left(17\right)\end{array}$$\begin{array}{cc}{V}_{2i}=\left(\frac{?k_{p1}}{2}+\frac{??k_d}{4}\right)e_{1i}^2+\frac{?I_h}{4}{e}_{2i}^2+\frac{m}{2?}{S}_{1i}^2+\frac{I_h}{?}{S}_{2i}^2+\frac{?I_h}{4}{u}_{ei}^2,\mathrm{and}& \left(18\right)\end{array}$$\begin{array}{cc}{V}_{3i}=\frac{?I_h}{2}{e}_{2i}{u}_{ei}+??{?}_0^l{y}_{ei}{\stackrel{.}{y}}_i\mathrm{dx}.& \left(19\right)\end{array}$
where V.sub.1i, V.sub.2i and V.sub.3i are as follows respectively:

(45) $\begin{array}{cc}{V}_{1i}=\frac{??}{2}{?}_0^l{y}_{ei}^{2}dx+\frac{??}{2}{?}_0^l{\stackrel{.}{y}}_i^{2}dx+\frac{?T}{2}{?}_0^l{\left(w_i^{}\right)}^{2}dx+\frac{?EI}{2}{?}_0^l{\left(w_i^{}\right)}^{2}dx,& \left(17\right)\end{array}$$\begin{array}{cc}{V}_{2i}=\left(\frac{?k_{p1}}{2}+\frac{??k_d}{4}\right)e_{1i}^2+\frac{?I_h}{4}{e}_{2i}^2+\frac{m}{2?}{S}_{1i}^2+\frac{I_h}{?}{S}_{2i}^2+\frac{?I_h}{4}{u}_{ei}^2,\mathrm{and}& \left(18\right)\end{array}$$\begin{array}{cc}{V}_{3i}=\frac{?I_h}{2}{e}_{2i}{u}_{ei}+??{?}_0^l{y}_{ei}{\stackrel{.}{y}}_{i}dx.& \left(19\right)\end{array}$

(46) At S104, stability of the flexible mechanical arm is validated according to the Lyapunov function. In this step, A Lyapunov direct method is used to validate the stability of the flexible mechanical arm.

(47) In this embodiment, if the flexible mechanical arm meets a preset requirement, that is, the stability of the flexible arms in Lyapunov theory can be drawn from the fact that the Lyapunov function is validated positive definite.

(48) By validating the Lyapunov function with the first derivative negative definite, it is obtained that the flexible mechanical arm is asymptotically stable.

(49) In this embodiment, the positive definiteness of the Lyapunov function is validated as follows.

(50) According to an inequality ab??(a.sup.2+b.sup.2), it may be obtained that:

(51) 0$\begin{array}{cc}.Math.V_{3i}.Math.?\frac{?I_h}{4}{e}_{2i}^2+\frac{?I_h}{4}{u}_{ei}^2+??{?}_1{?}_0^l{y}_{ei}^{2}dx+\frac{??}{{?}_1}{?}_0^l{\stackrel{.}{y}}_i^{2}dx.& \left(20\right)\end{array}$

(52) According to the equation (16), it may be determined that the Lyapunov function is positive definite, i.e.

(53) $\begin{array}{cc}{V}_i?\left(\frac{??}{2}-??{?}_1\right){?}_0^l{y}_{ei}^{2}dx+\left(\frac{\mathrm{??}}{2}-\frac{??}{{?}_1}\right){?}_0^l{\stackrel{.}{y}}_i^{2}dx+\frac{?T}{2}{?}_0^l{\left(w_i^{}\right)}^2\mathrm{dx}+\frac{?\mathrm{EI}}{2}{?}_0^l{\left(w_i^{}\right)}^{2}dx+\left(\frac{?k_{p1}}{2}+\frac{\mathrm{??}{k}_d}{4}\right)e_{1i}^2+\frac{m}{2?}{S}_{1i}^2+\frac{I_h}{?}{S}_{2i}^2>0,& \left(21\right)\end{array}$$\mathrm{where}\frac{2?}{?}<{?}_1<\frac{?}{2?}.$
The positive definiteness of the Lyapunov function is validated.

(54) The negative definiteness of the first derivative of the Lyapunov function is validated as follows.

(55) The derivative of V.sub.i(t) with respect to time is taken as:

(56) $\begin{array}{cc}{\stackrel{.}{V}}_i={\stackrel{.}{V}}_{1i}+{\stackrel{.}{V}}_{2i}+{\stackrel{.}{V}}_{3i}.& \left(22\right)\end{array}$

(57) Calculating the derivative of V.sub.1i, V.sub.2i and V.sub.3i in (16) with respect to time and then adding them together, it can be obtained that:

(58) $\begin{array}{cc}{\stackrel{.}{V}}_i?-\left(?T-\frac{??l^2}{?}\right){?}_0^l{\left(w_i^{}\right)}^{2}dx-\left(?\mathrm{EI}-16?l^4\right){?}_0^l{\left(w_i^{}\right)}^2\mathrm{dx}-?\left[2?k_{p1}+\frac{2{?}^3{I}_h}{{?}^2}+\frac{{?}^2{k}_d}{2}+?\left(2rl+8l\right)-\frac{?\left(??+?\right){\left(r+l\right)}^3}{6}-\frac{??l^3}{?}\right]e_{1i}^2-?{?}_0^l{y}_{ei}^{2}dx-\left(\mathrm{??}-\frac{3??}{2}\right){?}_0^l{\stackrel{.}{y}}_i^{2}dx-\left(\frac{{?}^2{k}_d}{4}-\frac{?k_{p3}}{2}\right)e_{2i}^2-k_m{S}_{1i}^2-k_d{S}_{2i}^2-\left[\frac{?k_{p3}}{2}+\frac{{?}^2{k}_d}{4}-\frac{?\left(??+?\right){\left(r+l\right)}^3}{6}\right]u_{ei}^2,& \left(23\right)\end{array}$

(59) where ? and ? are positive constants.

(60) Appropriate parameters should be selected as follows:

(61) $\begin{array}{cc}\{\begin{array}{c}?,?,k_m>0,?>1\\ k_d>\frac{?\left(??+?\right){\left(r+l\right)}^3}{3{?}^2},k_{p3}<\frac{?k_d}{2}\\ k_{p1}>\frac{?\left(??+?\right){\left(r+l\right)}^3}{12}+\frac{?l^3}{2?}-\frac{{?}^2{I}_h}{{?}^2}-\frac{?k_d}{4}-?\left(2rl+8l\right)\end{array},& \left(24\right)\end{array}$

(62) It is obtained that {dot over (V)}.sub.i?0, i.e., {dot over (V)}.sub.i is semi-negative definiteness.

(63) From (21), it can be obtained that:

(64) $\begin{array}{cc}{?}_1\left(V_{1i}+V_{2i}+V_{3i}\right)?V_i?{?}_2\left(V_{1i}+V_{2i}+V_{3i}\right),\mathrm{where}& \left(25\right)\end{array}$${?}_1=\min \left\{\frac{?-2?{?}_1}{?},\frac{?{?}_1-2?}{?{?}_1},1\right\},$${?}_2=\max \left\{\frac{?+2?{?}_1}{?},\frac{?{?}_1+2?}{?{?}_1},1\right\}.$

(65) When ? meets the following conditions:

(66) $?=\frac{1}{{?}_2}\min \left\{\frac{2?}{??},\frac{2??-3??}{??},\frac{2\left(??T-?r{1}^2\right)}{\mathrm{??T}},\frac{2\left(?\mathrm{EI}-16?l^4\right)}{?\mathrm{EI}},\frac{2?k_{p1}+\frac{2{?}^3{I}_h}{{?}^2}+\frac{?k_d}{2}+?\left(2\mathrm{rl}+8l\right)-\frac{?\left(??+?\right){\left(r+l\right)}^3}{6}-\frac{\mathrm{??}{l}^3}{?}}{2?k_{p1}+??k_d}\frac{?k_d-2k_{p3}}{I_h},\frac{2?k_m}{m},\frac{?k_d}{I_h},\frac{6?k_{p3}+3{?}^2{k}_d-2?\left(??+?\right){\left(r+l\right)}^3}{3?I_h}\right\}>0.$

(67) Multiplying both sides of (25) by e.sub.?t, it can be obtained that:

(68) $\begin{array}{cc}.Math.w_i.Math.=\sqrt{\frac{2l}{?T{?}_1}{V}_i\left(0\right)e^{-?t}}\mathrm{and}& \left(27\right)\end{array}$$.Math.e_{1i}.Math.?\sqrt{\frac{4}{\left(2?k_{p1}+??k_d\right)}{V}_i\left(0\right)e^{-?t}}.$

(69) Thus, it holds that

(70) $?\left(x,t\right)?\left[0,l\right]?\left[0,t_n\right],\underset{t.fwdarw.0}{\lim }.Math.w_i.Math.=0\mathrm{and}\underset{t.fwdarw.0}{\lim }.Math.e_{1i}.Math.=0.$

(71) According to the above analysis, the stability of the flexible mechanical arm based on cooperative tracking is validated.

(72) It should be noted that, referring to FIGS. 2 and 3, FIG. 2 is a schematic diagram illustrating a flexible mechanical arm according to an embodiment of the present disclosure. FIG. 3 is an example diagram illustrating a network topology of a flexible mechanical arm group, mainly showing information communication relationship among the flexible mechanical arms. As shown in FIG. 3, a flexible mechanical arm group made up of six flexible robotic arms is consider, in which the flexible mechanical arm numbered 0 is the leader, the rest numbered flexible mechanical arms are the followers. the followers numbered 1 and 2 have information communication with the leader numbered 0, and the followers numbered 3, 4 and 5 have information communication with the followers numbered 1, 2 and 3 respectively. Information communication relationship among the flexible mechanical arms is represented by an adjacency matrix A and a diagonal matrix B. The adjacency matrix A represents information communication relationship between the flexible mechanical arms as followers. The diagonal matrix B represents information communication relationship between the leader and the flexible mechanical arms as followers. A=[?.sub.ij]?R.sup.k?k is a non-negative matrix and is defined that if there is information communication between two flexible mechanical arms, then ?.sub.ij>0, otherwise ?.sub.ij=0. B=diag(b.sub.10, b.sub.20, . . . , b.sub.k0) is a non-negative diagonal matrix and is defined that if there is information communication between the leader and the flexible mechanical arm as the follower, then b.sub.i0>0, otherwise, b.sub.i0=0.

(73) Appropriate gain parameters are selected to validate the positive definiteness of the Lyapunov function and the negative definiteness of the first derivative of the Lyapunov function.

(74) In this embodiment, the numerical simulations of the flexible arms can be conducted on MATLAB and then the corresponding simulation results can be obtained. According to the simulation results, it can be judged whether the control effect of the flexible mechanical arm under control can meet expectations. If it is, the operation can be ended. If not, the gain parameters of the boundary controller should be corrected and the numerical simulation should be performed again.

(75) In summary, the present embodiment provides a method for controlling vibration of a flexible mechanical arm based on cooperative tracking, including: building a dynamic model of the flexible mechanical arm; constructing a flexible mechanical arm group made up of a plurality of flexible mechanical arms, assigning one of the plurality of flexible mechanical arms as a leader and the rest ones as followers; determining information communication relationship, and designing cooperative tracking-based boundary controllers located at a fixed end and a tip position of the flexible mechanical arm respectively; and validating stability of the flexible mechanical arm under control. The disclosure can realize the control of the flexible mechanical arms more stably and accurately, and can also realize the cooperative tracking of the flexible mechanical arm.

(76) The above-mentioned embodiments are preferred embodiments of the present disclosure, but embodiments of the present disclosure are not limited to the above-mentioned embodiments. Any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principles of the present disclosure are intended to be equivalent permutations and are included within the scope of the present disclosure.