CN108181836B - Boundary control method for anti-saturation of flexible Timoshenko beam mechanical arm - Google Patents
Boundary control method for anti-saturation of flexible Timoshenko beam mechanical arm Download PDFInfo
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Abstract
The invention discloses a boundary control method for anti-saturation of a flexible Timoshenko beam mechanical arm, which comprises the following steps: acquiring the dynamic characteristics of a flexible Timoshenko beam mechanical arm system, and constructing a flexible Timoshenko beam mechanical arm system model according to the dynamic characteristics; constructing a boundary controller according to the flexible Timoshenko beam mechanical arm system model; verifying the stability of the flexible Timoshenko beam mechanical arm system under the action of anti-saturation control; carrying out digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software to obtain a simulation result; verifying whether the control effect of the flexible Timoshenko beam mechanical arm system after control action is applied meets the preset requirement or not according to the simulation result; and if the control effect does not meet the preset requirement, adjusting the gain parameter of the boundary controller according to the simulation result to enable the boundary controller to have better anti-saturation control and tracking performance. The invention can realize more stable and accurate tracking and control of the mechanical arm.
Description
Technical Field
The invention relates to the technical field of automatic control, in particular to a boundary control method for anti-saturation of a flexible Timoshenko beam mechanical arm.
Background
With the application of robots extending from a single industrial field to the fields of oceans, buildings, aerospace and the like, rigid mechanical arms which are heavy, low in flexibility and single in function no longer meet the working requirements of more and more complex external environments. The flexible mechanical arm has the advantages of light weight, high precision, low energy consumption, high speed and the like, and can overcome a plurality of defects of the rigid mechanical arm. However, due to the characteristic of flexibility of the material, the system is easily affected by external interference, and severe vibration is generated in the actual working process, so that the performance of the system is seriously affected.
The boundary control method adopted in the prior art to solve the vibration problem of the flexible mechanical arm is recognized as a very effective control method. However, the boundary control method in the prior art is generally directed at an euler beam flexible mechanical arm, or is used for realizing accurate estimation of interference and the like under the condition that external interference is uncertain, and for a Timoshenko beam flexible mechanical arm with a shear deformation amount, in addition to non-smooth non-linear characteristics such as saturation, backlash, hysteresis, dead zones and the like which are generally existing in an industrial control device, the simple boundary control method still has the defects that the performance of a system is reduced, and even the system is unstable.
Disclosure of Invention
The technical problem to be solved by the embodiment of the invention is to provide a boundary control method for the saturation resistance of a flexible Timoshenko beam mechanical arm, which can realize more stable and accurate tracking and control of the mechanical arm.
In order to solve the technical problem, an embodiment of the present invention provides a boundary control method for saturation resistance of a flexible Timoshenko beam mechanical arm, including the following steps:
acquiring the dynamic characteristics of a flexible Timoshenko beam mechanical arm system, and constructing a flexible Timoshenko beam mechanical arm system model according to the dynamic characteristics;
constructing a boundary controller according to the flexible Timoshenko beam mechanical arm system model;
constructing a Lyapunov function of the flexible Timoshenko beam mechanical arm system based on the flexible Timoshenko beam mechanical arm system model;
verifying the stability of the flexible Timoshenko beam mechanical arm system according to the Lyapunov function;
when the flexible Timoshenko beam mechanical arm system is judged to meet the preset stability requirement, performing digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software to obtain a simulation result;
verifying whether the control effect of the flexible Timoshenko beam mechanical arm system after control action is applied meets the preset requirement or not according to the simulation result;
if the control effect meets the preset requirement, saving the gain parameter of the boundary controller, and ending the operation;
and if the control effect does not meet the preset requirement, correcting the gain parameter of the boundary controller, and carrying out digital simulation again.
Further, the dynamic characteristics comprise kinetic energy of the flexible Timoshenko beam mechanical arm system, potential energy of the flexible Timoshenko beam mechanical arm system, and virtual work done by non-conservative force on the flexible Timoshenko beam mechanical arm system; wherein,
the kinetic energy is:
wherein x ∈ [0, L ]]For each position of the flexible Timoshenko beam mechanical arm, t epsilon [0, infinity) is time, IhIs the rotational inertia of the hub, L is the length of the Timoshenko beam mechanical arm of the flexible machine, rho is the uniform mass per unit length of the Timoshenko beam mechanical arm, and IρThe unit moment of inertia of the flexible Timoshenko beam mechanical arm is shown, m is the mass of the tail end load, J is the moment of inertia of the tail end load, phi (x, t) is the transverse torsional deformation of the flexible mechanical arm at a position x and a moment t under an xoy coordinate system, the absolute displacement y (x, t) of the mechanical arm under the xoy coordinate system is defined as y (x, t) w (x, t) + x theta (t), wherein w (x, t) is the elastic deformation of the flexible Timoshenko beam mechanical arm system at the time t and the position x under the xoy coordinate system, and theta (t) is the rotation angle of the mechanical arm;
the potential energy is as follows:
the bending stiffness of the flexible Timoshenko beam mechanical arm is represented by EI (intrinsic elastic modulus), K is kGA, K is a constant determined by the cross section shape of the flexible Timoshenko beam mechanical arm, A is the cross section area of the flexible Timoshenko beam mechanical arm, and G is the shear elastic modulus of the flexible Timoshenko beam mechanical arm;
the deficiency work is as follows:
δW＝u(t)δy(L,t)+τ1(t)δφ(L,t)+τ2(t)δθ(t)；
wherein, delta is a variation symbol, u (t), tau1(t) and τ2And (t) is a controller.
The method for constructing the flexible Timoshenko beam mechanical arm system model comprises the following steps of:
wherein,
w(0,t)＝φ(0,t)＝0；
wherein,
further, the boundaryThe controller is u (t), tau1(t) and τ2(t); wherein,
wherein alpha is1,α2,α3,α4,k1,k2,k3,k4Is a gain parameter of the border controller; alpha is alpha1,α2,α3,α4,k1,k2,k3,k4All values of (a) are greater than 0; e (t) is an angle error, and e (t) is θ (t) - θd。
Further, constructing a Lyapunov function of the flexible Timoshenko beam mechanical arm system based on the flexible Timoshenko beam mechanical arm system model, specifically:
designing a Lyapunov function of the flexible Timoshenko beam mechanical arm system based on the flexible Timoshenko beam mechanical arm system model,
V(t)＝Va(t)+Vb(t)+Vc(t)；
wherein,
Vc(t)＝α3ln(cosh(k3e (t))) representing additional items.
Further, verifying the stability of the flexible Timoshenko beam mechanical arm system according to the Lyapunov function, specifically:
verifying the positive nature of the Lyapunov function to obtain the stability of the flexible Timoshenko beam mechanical arm system in accordance with the Lyapunov meaning;
and verifying the negative nature of the first-order derivative of the Lyapunov function to obtain that the flexible Timoshenko beam mechanical arm system conforms to gradual stability.
Further, the boundary controller includes an anti-saturation controller and an angle controller.
Further, the boundary controller includes a movement sensor, a disturbance observer, a central controller, and a driving device.
Further, if the control effect does not meet the preset requirement, modifying the gain parameter of the boundary controller, and performing digital simulation again, specifically:
and correcting the gain parameters of the boundary controller, verifying the positive and negative characteristics of the Lyapunov function and the first-order derivative of the Lyapunov function according to the gain parameters, and performing digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software.
Further, the simulation result comprises the vibration amplitude, the shearing deformation amount and the angle value of the flexible Timoshenko beam mechanical arm.
The embodiment of the invention has the following beneficial effects:
the embodiment of the invention provides a boundary control method for anti-saturation of a flexible Timoshenko beam mechanical arm, which comprises the following steps: acquiring the dynamic characteristics of a flexible Timoshenko beam mechanical arm system, and constructing a flexible Timoshenko beam mechanical arm system model according to the dynamic characteristics; constructing a boundary controller according to the flexible Timoshenko beam mechanical arm system model; verifying the stability of the flexible Timoshenko beam mechanical arm system under the action of anti-saturation control; carrying out digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software to obtain a simulation result; verifying whether the control effect of the flexible Timoshenko beam mechanical arm system after control action is applied meets the preset requirement or not according to the simulation result; and if the control effect does not meet the preset requirement, adjusting the gain parameter of the boundary controller according to the simulation result to enable the boundary controller to have better anti-saturation control and tracking performance. The invention can realize more stable and accurate tracking and control of the mechanical arm.
Drawings
Fig. 1 is a flowchart of a boundary control method for anti-saturation of a flexible Timoshenko beam mechanical arm according to a first embodiment of the present invention;
FIG. 2 is another flow chart diagram according to the first embodiment of the present invention;
fig. 3 is a schematic structural diagram of the operation of the flexible Timoshenko beam mechanical arm in the first embodiment of the invention;
fig. 4 is a diagram illustrating the simulation result of the elastic deformation w (x, t) of the flexible Timoshenko beam mechanical arm without control applied in the first embodiment of the invention;
fig. 5 is a diagram showing simulation results of the shear deformation amount phi (x, t) of the flexible Timoshenko beam manipulator without control applied in the first embodiment of the present invention;
fig. 6 is a schematic diagram showing the simulation result of the angular position θ (t) of the mechanical arm hub of the flexible Timoshenko beam without control in the first embodiment of the invention;
fig. 7 is a diagram illustrating the simulation result of the elastic deformation amount w (x, t) of the controlled flexible Timoshenko beam mechanical arm in the first embodiment of the present invention;
fig. 8 is a diagram showing simulation results of the shear deformation amount phi (x, t) of the controlled flexible Timoshenko beam manipulator in the first embodiment of the present invention;
fig. 9 is a schematic diagram showing simulation results of the angular position θ (t) of the hub of the mechanical arm of the flexible Timoshenko beam after control is applied in the first embodiment of the invention;
fig. 10 is a schematic diagram of simulation results of an angle position error e (t) of a mechanical arm hub of a flexible Timoshenko beam after control is applied according to the first embodiment of the present invention;
FIG. 11 is a diagram illustrating simulation results of the boundary control force u (t) designed according to the first embodiment of the present invention;
FIG. 12 shows a boundary torque τ designed in the first embodiment of the present invention1(t) a simulation result schematic diagram;
FIG. 13 shows the control torque τ according to the first embodiment of the present invention2(t) a simulation result diagram.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The first embodiment of the present invention:
referring to fig. 1, fig. 1 is a flowchart illustrating a boundary control method for anti-saturation of a flexible Timoshenko beam manipulator according to a first embodiment of the present invention. The boundary control method for the anti-saturation of the flexible Timoshenko beam mechanical arm comprises the following steps:
s101, obtaining dynamic characteristics of the flexible Timoshenko beam mechanical arm system, and constructing a flexible Timoshenko beam mechanical arm system model according to the dynamic characteristics.
In this embodiment, the dynamic characteristics include kinetic energy of the flexible Timoshenko beam-arm system, potential energy of the flexible Timoshenko beam-arm system, and virtual work done by non-conservative forces on the flexible Timoshenko beam-arm system. Wherein,
the kinetic energy is:
wherein x ∈ [0, L ]]For each position of the flexible Timoshenko beam mechanical arm, t epsilon [0, infinity) is time, IhThe moment of inertia of the hub, L is the length of the Timoshenko beam mechanical arm of the flexible machine, rho is the uniform mass per unit length of the Timoshenko beam mechanical arm, and T isρIs a flexible Timoshenko beamThe unit moment of inertia of the mechanical arm, m is the mass of the end load, J is the moment of inertia of the end load, phi (x, t) is the transverse torsional deformation of the flexible mechanical arm at the position x and the moment t in the xoy coordinate system, the absolute displacement y (x, t) of the mechanical arm in the xoy coordinate system is defined as y (x, t) w (x, t) + x theta (t), w (x, t) is the elastic deformation of the flexible Timoshenko beam mechanical arm system at the time t and the position x in the xoy coordinate system, and theta (t) is the rotation angle of the mechanical arm.
The potential energy is as follows:
the bending stiffness of the flexible Timoshenko beam mechanical arm is represented by EI (intrinsic elastic modulus), K is kGA, K is a constant determined by the cross section shape of the flexible Timoshenko beam mechanical arm, A is the cross section area of the flexible Timoshenko beam mechanical arm, and G is the shear elastic modulus of the flexible Timoshenko beam mechanical arm;
the deficiency work is as follows:
δW＝u(t)δy(L,t)+τ1(t)δφ(L,t)+τ2(t)δθ(t)；
wherein, delta is a variation symbol, u (t), tau1(t) and τ2And (t) is a controller.
The method for constructing the flexible Timoshenko beam mechanical arm system model comprises the following steps of:
wherein,
w(0,t)＝φ(0,t)＝0 (3)
wherein,
and S102, constructing a boundary controller according to the flexible Timoshenko beam mechanical arm system model.
In the embodiment, a boundary controller with a reasonable hyperbolic tangent function design is utilized, so that the flexible Timoshenko beam mechanical arm system achieves a stable bounded state and tracking performance, the problem of input tremble generated by using a symbolic function is avoided, and the mechanical arm is tracked and controlled more stably and accurately.
In this embodiment, the boundary controllers are u (t), τ1(t) and τ2(t); wherein,
wherein alpha is1,α2,α3,α4,k1,k2,k3,k4Is a gain parameter of the border controller; alpha is alpha1,α2,α3,α4,k1,k2,k3,k4All values of (a) are greater than 0; e (t) is an angle error, and
e(t)＝θ(t)-θd (10)
in the present embodiment, please refer to fig. 11, fig. 12 and fig. 13, wherein fig. 11 is a schematic diagram of a simulation result of the boundary control force u (t) designed in the first embodiment of the present invention. FIG. 12 shows a boundary torque τ designed in the first embodiment of the present invention1(t) a simulation result diagram. FIG. 13 shows the control torque τ according to the first embodiment of the present invention2(t) a simulation result diagram.
S103, constructing a Lyapunov function of the flexible Timoshenko beam mechanical arm system based on the flexible Timoshenko beam mechanical arm system model; and verifying the stability of the flexible Timoshenko beam mechanical arm system according to the Lyapunov function.
In this embodiment, the constructing a Lyapunov function of the flexible Timoshenko beam mechanical arm system based on the flexible Timoshenko beam mechanical arm system model specifically includes:
designing a Lyapunov function of the flexible Timoshenko beam mechanical arm system based on the flexible Timoshenko beam mechanical arm system model,
V(t)＝Va(t)+Vb(t)+Vc(t)；
wherein,
Vc(t)＝α3ln(cosh(k3e (t))) representing additional items.
And S104, when the flexible Timoshenko beam mechanical arm system is judged to meet the preset stability requirement, performing digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software to obtain a simulation result.
In this embodiment, the flexible Timoshenko beam mechanical arm system is judged to meet a preset stability requirement, that is, the positive nature of the Lyapunov function is verified, and the stability of the flexible Timoshenko beam mechanical arm system in accordance with the Lyapunov meaning is obtained; and verifying the negative nature of the first-order derivative of the Lyapunov function to obtain that the flexible Timoshenko beam mechanical arm system conforms to gradual stability.
In the present embodiment, the positive nature of the Lyapunov function v (t) is verified by the following method:
the value range of the cosh function is [1, ∞), i.e. cosh (k)1e (t) > 1, so Vc(t)＝α3ln(cosh(k3e(t)))＞0；
At the same time, Va(t)＞0,Vb(t) > 0 to give V (t) ═ Va(t)+Vb(t)+Vc(t) > 0, i.e., the positive nature of the Lyapunov function V (t) was verified.
Verifying first-order derivative of Lyapunov function
The negative qualitative method of (1) is as follows:
Va(t) taking the first derivative of time as,
substituting formulae (1) and (2) into formula (Ka) to yield:
partial integration of equation (Kb) yields:
substituting formula (3) into formula (Kc), and combining the same terms to obtain:
Vb(t) taking the first derivative of time to obtain:
substituting formulae (7) to (9) for formula (Ke) to obtain:
Vc(t) taking the first derivative of time to obtain:
substituting formula (Kc) to (Kg) into V (t) or Va(t)+Vb(t)+Vc(t), obtaining:
namely, it is
The negative nature of (A) was verified.
From the formula (Kh):
this can be obtained from the formula (10):
thus, there are:
combining formulas (4) and (5) can yield phi '(L, t) equal to 0 and phi (L, t) equal to w' (L, t), indicating that
By combining the analysis in formulas (6) and (9) and the above analysis, e (t) ═ 0 can be finally obtained, which indicates that the flexible Timoshenko beam manipulator system has better angle tracking performance.
S105, verifying whether the control effect of the flexible Timoshenko beam mechanical arm system after control action is applied meets the preset requirement or not according to the simulation result; if the control effect meets the preset requirement, saving the gain parameter of the boundary controller, and ending the operation; and if the control effect does not meet the preset requirement, correcting the gain parameter of the boundary controller, and carrying out digital simulation again.
Please refer to fig. 2 and fig. 3, wherein fig. 2 is another flowchart according to the first embodiment of the present invention. Fig. 3 is a schematic structural diagram of the operation of the flexible Timoshenko beam robot in the first embodiment of the present invention. As shown in fig. 2, if the control effect does not meet the preset requirement, the gain parameter of the boundary controller is modified, and digital simulation is performed again, specifically:
and correcting the gain parameters of the boundary controller, verifying the positive and negative characteristics of the Lyapunov function and the first-order derivative of the Lyapunov function according to the gain parameters, and performing digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software. It can be understood that whether the vibration, the shearing deformation amount and the angle of the flexible Timoshenko beam mechanical arm meet the requirements or not is judged according to the simulation result, and if the vibration, the shearing deformation amount and the angle of the flexible Timoshenko beam mechanical arm do not meet the requirements, the gain parameter alpha of the boundary controller is readjusted1,α2,α3,α4,k1,k2,k3,k4. And if the requirements are met, ending.
In this embodiment, the simulation result includes the vibration amplitude, the shear deformation amount and the angle value of the flexible Timoshenko beam mechanical arm. The boundary controller includes an anti-saturation controller and an angle controller. The boundary controller includes a motion sensor, a disturbance observer, a central controller, and a driving device.
In this embodiment, please refer to fig. 4 and fig. 5, wherein fig. 4 is a diagram illustrating a simulation result w (x, t) of the flexible Timoshenko beam robot without control according to the first embodiment of the present invention. Fig. 5 is a diagram showing simulation results of the shear deformation amount phi (x, t) of the flexible Timoshenko beam manipulator without control applied in the first embodiment of the present invention. As shown in fig. 4 and 5, when not controlled, there are vibration (lateral displacement) and shear deformation quantities throughout the robot arm.
In this embodiment, please refer to fig. 7 and 8, wherein fig. 7 is a diagram illustrating a simulation result of an elastic deformation amount w (x, t) of the controlled flexible Timoshenko beam robot according to the first embodiment of the present invention. Fig. 8 is a diagram showing simulation results of the shear deformation amount phi (x, t) of the controlled flexible Timoshenko beam manipulator in the first embodiment of the present invention. As shown in fig. 7 and 8, the flexible Timoshenko beam mechanical arm is used for vibration suppression, and when t is 2s, the amplitude of the flexible Timoshenko beam mechanical arm tends to be relatively stable and is near an equilibrium position.
In this embodiment, please refer to fig. 6 and fig. 9, wherein fig. 6 is a schematic diagram of simulation results of the angular position θ (t) of the mechanical arm hub of the flexible Timoshenko beam without control according to the first embodiment of the present invention. Fig. 9 is a schematic diagram of simulation results of the angular position θ (t) of the flexible Timoshenko beam mechanical arm hub after control is applied in the first embodiment of the present invention. Referring to fig. 10, fig. 10 is a schematic diagram of an angle position error e (t) of a mechanical arm hub of a controlled flexible Timoshenko beam in a first embodiment of the present invention. As shown in fig. 10, by performing angle tracking suppression on the flexible Timoshenko beam mechanical arm, the angle of the flexible T-beam mechanical arm has better tracking performance.
The embodiment provides a boundary control method for anti-saturation of a flexible Timoshenko beam mechanical arm, which comprises the following steps: acquiring the dynamic characteristics of a flexible Timoshenko beam mechanical arm system, and constructing a flexible Timoshenko beam mechanical arm system model according to the dynamic characteristics; constructing a boundary controller according to the flexible Timoshenko beam mechanical arm system model; verifying the stability of the flexible Timoshenko beam mechanical arm system under the action of anti-saturation control; carrying out digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software to obtain a simulation result; verifying whether the control effect of the flexible Timoshenko beam mechanical arm system after control action is applied meets the preset requirement or not according to the simulation result; and if the control effect does not meet the preset requirement, adjusting the gain parameter of the boundary controller according to the simulation result to enable the boundary controller to have better anti-saturation control and tracking performance. The invention can realize more stable and accurate tracking and control of the mechanical arm.
The foregoing is directed to the preferred embodiment of the present invention, and it is understood that various changes and modifications may be made by one skilled in the art without departing from the spirit of the invention, and it is intended that such changes and modifications be considered as within the scope of the invention.
It will be understood by those skilled in the art that all or part of the processes of the methods of the embodiments described above can be implemented by a computer program, which can be stored in a computer-readable storage medium, and when executed, can include the processes of the embodiments of the methods described above. The storage medium may be a magnetic disk, an optical disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), or the like.
Claims (7)
1. A boundary control method for anti-saturation of a flexible Timoshenko beam mechanical arm is characterized by comprising the following steps:
acquiring the dynamic characteristics of a flexible Timoshenko beam mechanical arm system, and constructing a flexible Timoshenko beam mechanical arm system model according to the dynamic characteristics;
constructing a boundary controller according to the flexible Timoshenko beam mechanical arm system model; the boundary controller is u (t), tau1(t) and τ2(t); wherein,
wherein alpha is1，α2，α3，α4，k1，k2，k3，k4Is a gain parameter of the border controller; alpha is alpha1，α2，α3，α4，k1，k2，k3，k4All values of (a) are greater than 0; e (t) is an angle error, and e (t) is θ (t) - θd(ii) a Constructing a Lyapunov function of the flexible Timoshenko beam mechanical arm system based on the flexible Timoshenko beam mechanical arm system model; specifically, a Lyapunov function of the flexible Timoshenko beam mechanical arm system is designed based on the flexible Timoshenko beam mechanical arm system model,
V(t)＝Va(t)+Vb(t)+Vc(t)；
wherein,
representing an energy term;
Vc(t)＝α3ln(cosh(k3e (t))) representing additional items;
verifying the stability of the flexible Timoshenko beam mechanical arm system according to the Lyapunov function;
when the flexible Timoshenko beam mechanical arm system is judged to meet the preset stability requirement, performing digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software to obtain a simulation result;
verifying whether the control effect of the flexible Timoshenko beam mechanical arm system after control action is applied meets the preset requirement or not according to the simulation result;
if the control effect meets the preset requirement, saving the gain parameter of the boundary controller, and ending the operation;
and if the control effect does not meet the preset requirement, correcting the gain parameter of the boundary controller, and carrying out digital simulation again.
2. The boundary control method for the saturation resistance of the flexible Timoshenko beam mechanical arm according to claim 1, wherein the dynamic characteristics comprise kinetic energy of the flexible Timoshenko beam mechanical arm system, potential energy of the flexible Timoshenko beam mechanical arm system, and virtual work done by non-conservative forces on the flexible Timoshenko beam mechanical arm system; wherein,
the kinetic energy is:
wherein x ∈ [0, L ]]For each position of the flexible Timoshenko beam mechanical arm, t epsilon [0, infinity) is time, IhIs the rotational inertia of the hub, L is the length of the Timoshenko beam mechanical arm of the flexible machine, rho is the uniform mass per unit length of the Timoshenko beam mechanical arm, and IρThe unit moment of inertia of the flexible Timoshenko beam mechanical arm is shown, m is the mass of the end load, J is the moment of inertia of the end load, phi (x, t) is the transverse torsional deformation of the flexible mechanical arm at a position x and a moment t under an xoy coordinate system, and the absolute displacement y (x, t) of the mechanical arm under the xoy coordinate system is defined as y (x, t) w (x, t) + x theta (t), wherein w (x, t) is the time t at the position x under the xoy coordinate systemElastic deformation of a flexible Timoshenko beam mechanical arm system, wherein theta (t) is the rotation angle of the mechanical arm;
the potential energy is as follows:
the bending stiffness of the flexible Timoshenko beam mechanical arm is represented by EI (intrinsic elastic modulus), K is kGA, K is a constant determined by the cross section shape of the flexible Timoshenko beam mechanical arm, A is the cross section area of the flexible Timoshenko beam mechanical arm, and G is the shear elastic modulus of the flexible Timoshenko beam mechanical arm;
the deficiency work is as follows:
δW＝u(t)δy(L，t)+τ1(t)δφ(L，t)+τ2(t)δθ(t)；
wherein, delta is a variation symbol, u (t), tau1(t) and τ2(t) is a controller;
the method for constructing the flexible Timoshenko beam mechanical arm system model comprises the following steps of:
wherein,
w(0,t)＝φ(0,t)＝0；
wherein,
3. the boundary control method for the saturation resistance of the flexible Timoshenko beam mechanical arm according to claim 1, wherein the verifying the stability of the flexible Timoshenko beam mechanical arm system according to the Lyapunov function specifically comprises:
verifying the positive nature of the Lyapunov function to obtain the stability of the flexible Timoshenko beam mechanical arm system in accordance with the Lyapunov meaning;
and verifying the negative nature of the first-order derivative of the Lyapunov function to obtain that the flexible Timoshenko beam mechanical arm system conforms to gradual stability.
4. The boundary control method for the anti-saturation of the flexible Timoshenko beam mechanical arm according to claim 1, wherein the boundary controller comprises an anti-saturation controller and an angle controller.
5. The boundary control method for the anti-saturation of the flexible Timoshenko beam mechanical arm according to claim 1, wherein the boundary controller comprises a movement sensor, a disturbance observer, a central controller and a driving device.
6. The boundary control method for the anti-saturation of the flexible Timoshenko beam mechanical arm according to claim 1, wherein if the control effect does not meet the preset requirement, the gain parameter of the boundary controller is modified, and digital simulation is performed again, specifically:
and correcting the gain parameters of the boundary controller, verifying the positive and negative characteristics of the Lyapunov function and the first-order derivative of the Lyapunov function according to the gain parameters, and performing digital simulation on the flexible Timoshenko beam mechanical arm system by using MATLAB simulation software.
7. The boundary control method for the anti-saturation of the flexible Timoshenko beam mechanical arm according to claim 1, wherein the simulation result comprises vibration amplitude, shearing deformation amount and angle value of the flexible Timoshenko beam mechanical arm.
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