CN108181836A - A kind of boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations - Google Patents
A kind of boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations Download PDFInfo
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Abstract
The invention discloses a kind of boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations, including：The dynamic characteristic of flexibility Timoshenko beam mechanical arm systems is obtained, and according to the dynamic characteristic, builds flexibility Timoshenko beam mechanical arm system models；According to the flexibility Timoshenko beam mechanical arm system models, boarder controller is built；The stability of the flexibility Timoshenko beam mechanical arm systems is verified under anti-saturation control action；Digital Simulation is carried out to the flexibility Timoshenko beams mechanical arm system using MATLAB simulation softwares, obtains simulation result；According to the simulation result, verify whether the control effect after applying control action to the flexibility Timoshenko beams mechanical arm system meets preset requirement；If the control effect does not meet the preset requirement, the gain parameter of the boarder controller is adjusted according to the simulation result, with preferable anti-saturation control and tracking performance.The present invention can realize more stable, accurate tracking and control to mechanical arm.
Description
Technical field
Resist the present invention relates to automatic control technology field more particularly to one kind for flexibility Timoshenko beam mechanical arms full
The boundary control method of sum.
Background technology
With the application of robot the fields such as ocean, building, space flight and aviation are expanded to from single industrial circle, heavy,
The Rigid Robot Manipulator that flexibility ratio is low, function is single no longer meets the job requirement of the external environment to become increasingly complex.It is and soft
Property mechanical arm have many advantages, such as that light weight, precision are high, energy consumption is small, speed is fast, many defects of Rigid Robot Manipulator can be overcome.But
It is that the influence of easy external disturbance the characteristics of due to its own material flexibility in the practical course of work, can produce violent
Vibration, the performance capabilities of the system seriously affected.
The vibration problem of flexible mechanical arm is solved using boundary control method in the prior art, is to be acknowledged as having very much
The control method of effect.But boundary control method of the prior art generally for euler beam flexible mechanical arm or for
In the case of external interference is uncertain, accurately estimation etc. to interference is realized, and for shearing deformation amount
Timoshenko beam flexible mechanical arms, in addition such as saturation of generally existing, recoil, magnetic hysteresis and dead zone are non-in industrial control device
Smooth nonlinear characteristic, there are still the defects of the reduced performance of system or even unstable system for simple boundary control method.
Invention content
Technical problem to be solved of the embodiment of the present invention is, provides a kind of for flexibility Timoshenko beam mechanical arms
The boundary control method of anti-saturation can realize more stable, accurate tracking and control to mechanical arm.
In order to solve the above-mentioned technical problem, an embodiment of the present invention provides one kind to be directed to flexibility Timoshenko beam mechanical arms
The boundary control method of anti-saturation, includes the following steps：
The dynamic characteristic of flexibility Timoshenko beam mechanical arm systems is obtained, and according to the dynamic characteristic, structure
Flexible Timoshenko beams mechanical arm system model；
According to the flexibility Timoshenko beam mechanical arm system models, boarder controller is built；
Based on the flexibility Timoshenko beam mechanical arm system models, the flexibility Timoshenko beam mechanical arms are built
The Lyapunov functions of system；
According to the Lyapunov functions, the stability of the flexibility Timoshenko beam mechanical arm systems is verified；
When judging that the flexibility Timoshenko beam mechanical arm systems meet preset stability requirement, MATLAB is utilized
Simulation software carries out Digital Simulation to the flexibility Timoshenko beams mechanical arm system, obtains simulation result；
According to the simulation result, verify after applying control action to the flexibility Timoshenko beams mechanical arm system
Whether control effect meets preset requirement；
If the control effect meets the preset requirement, the gain parameter of the boarder controller is preserved, terminates this
Operation；
If the control effect does not meet the preset requirement, the gain parameter of the boarder controller is corrected, again
Carry out Digital Simulation.
Further, the dynamic characteristic includes the kinetic energy of flexibility Timoshenko beam mechanical arm systems, the flexibility
The potential energy and nonconservative force of Timoshenko beam mechanical arm systems do the flexibility Timoshenko beam mechanical arm systems
Virtual work；Wherein,
The kinetic energy is：
Wherein, x ∈ [0, L] be each position of flexibility Timoshenko beam mechanical arms, t ∈ [0, ∞) be the time, IhFor wheel
The rotary inertia of hub, L are the length of flexible machine Timoshenko beam mechanical arms, and ρ is the list of flexibility Timoshenko beam mechanical arms
Bit length homogeneous quality, IρFor the unit turn inertia of flexible Timoshenko beams mechanical arm, m is the quality of end load, and J is
The rotary inertia of end load, φ (x, t) for flexible mechanical arm under xoy coordinate systems in the cross torsion shape of position x moment t
Become, absolute displacement y (x, t) of the mechanical arm under xoy coordinates is defined as y (x, t)=w (x, t)+x θ (t), wherein w (x, t) be
Under xoy coordinate systems in time t position x flexibility Timoshenko beam mechanical arm systems elastic deformation, θ (t) is mechanical arm
Rotational angle；
The potential energy is：
Wherein, EI is the bending stiffness of flexibility Timoshenko beam mechanical arms, and K=kGA, k are for one by soft and fine
The constant that Timoshenko beam mechanical arms cross-sectional shape determines, cross-sectional areas of the A for flexibility Timoshenko beam mechanical arms, G
The coefficient of rigidity for flexible Timoshenko beams mechanical arm；
The virtual work is：
δ W=u (t) δ y (L, t)+τ1(t)δφ(L,t)+τ2(t)δθ(t)；
Wherein, δ be variation symbol, u (t), τ1(t) and τ2(t) device in order to control.
The structure flexibility Timoshenko beam mechanical arm system models, specifically, by the kinetic energy, the potential energy, institute
It states virtual work and substitutes into Hamiton's principle, obtaining flexible Timoshenko beams mechanical arm system model is：
W (0, t)=φ (0, t)=0；
Further, the boarder controller is u (t), τ1(t) and τ2(t)；Wherein,
Wherein, α1,α2,α3,α4,k1,k2,k3,k4Gain parameter for the boarder controller；α1,α2,α3,α4,k1,k2,
k3,k4Value be more than 0；E (t) is angular error, and e (t)=θ (t)-θd。
Further, it is described based on the flexibility Timoshenko beam mechanical arm system models, build the flexibility
The Lyapunov functions of Timoshenko beam mechanical arm systems, specially：
Based on the flexibility Timoshenko beam mechanical arm system models, the flexibility Timoshenko beam mechanical arms are designed
The Lyapunov functions of system,
V (t)=Va(t)+Vb(t)+Vc(t)；
Wherein,
It represents
Energy term；
Represent auxiliary item；
Vc (t)=α3ln(cosh(k3E (t))), represent addition Item.
Further, it is described according to the Lyapunov functions, verify the flexibility Timoshenko beam mechanical arm systems
Stability, specially：
It verifies the orthotropicity of Lyapunov functions, show that the flexibility Timoshenko beam mechanical arm systems meet
Stabilization under Lyapunov meanings；
It verifies the negative definiteness of Lyapunov function first derivatives, obtains the flexibility Timoshenko beams mechanical arm system symbol
Close asymptotically stability.
Further, the boarder controller includes anti-saturation controller and angle controller.
Further, the boarder controller includes movable sensor, disturbance observer, central controller and driving dress
It puts.
Further, if the control effect does not meet the preset requirement, the boarder controller is corrected
Gain parameter re-starts Digital Simulation, specially：
Correct the gain parameter of the boarder controller, according to the gain parameter verify the Lyapunov functions and
The orthotropicity and negative definiteness of Lyapunov function first derivatives, and using MATLAB simulation softwares to the flexibility Timoshenko
Beam mechanical arm system carries out Digital Simulation.
Further, the simulation result includes the vibration amplitude of flexibility Timoshenko beam mechanical arms, shearing deformation amount
And angle value.
Implement the embodiment of the present invention, have the advantages that：
A kind of boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations provided in an embodiment of the present invention,
Including：The dynamic characteristic of flexibility Timoshenko beam mechanical arm systems is obtained, and according to the dynamic characteristic, structure flexibility
Timoshenko beam mechanical arm system models；According to the flexibility Timoshenko beam mechanical arm system models, the control of structure boundary
Device processed；The stability of the flexibility Timoshenko beam mechanical arm systems is verified under anti-saturation control action；Utilize MATLAB
Simulation software carries out Digital Simulation to the flexibility Timoshenko beams mechanical arm system, obtains simulation result；According to described imitative
It is true as a result, verification whether the control effect after flexibility Timoshenko beams mechanical arm system application control action is met it is pre-
If it is required that；If the control effect does not meet the preset requirement, the boarder controller is adjusted according to the simulation result
Gain parameter, with preferable anti-saturation control and tracking performance.The present invention can realize more stable, smart to mechanical arm
True tracking and control.
Description of the drawings
Fig. 1 is the boundary Control for flexibility Timoshenko beam mechanical arm anti-saturations that first embodiment of the invention provides
The flow diagram of method；
Fig. 2 is another flow diagram based on first embodiment of the invention；
Fig. 3 is the structure diagram of the flexible Timoshenko beams mechanical arm operation in first embodiment of the invention；
Fig. 4 is the elastic deformation of the flexible Timoshenko beams mechanical arm for not applying control in first embodiment of the invention
W (x, t) simulation result schematic diagram；
Fig. 5 is the shearing deformation of the flexible Timoshenko beams mechanical arm for not applying control in first embodiment of the invention
Measure φ (x, t) simulation result schematic diagram；
Fig. 6 is the angle of the flexible Timoshenko beams mechanical arm wheel hub for not applying control in first embodiment of the invention
Position θ (t) simulation result schematic diagrams；
Fig. 7 is the elastic deformation of the flexible Timoshenko beams mechanical arm after the application control in first embodiment of the invention
Measure w (x, t) simulation result schematic diagram；
Fig. 8 is the shearing deformation of the flexible Timoshenko beams mechanical arm after the application control in first embodiment of the invention
Measure φ (x, t) simulation result schematic diagram；
Fig. 9 is the angle of the flexible Timoshenko beams mechanical arm wheel hub after the application control in first embodiment of the invention
Position θ (t) simulation result schematic diagrams；
Figure 10 is the angle of the flexible Timoshenko beams mechanical arm wheel hub after the application control in first embodiment of the invention
Spend site error e (t) simulation result schematic diagram；
Figure 11 is boundary Control power u (t) simulation result schematic diagrams designed in first embodiment of the invention；
Figure 12 is the boundary torque tau designed in first embodiment of the invention1(t) simulation result schematic diagram；
Figure 13 is the control torque tau designed in first embodiment of the invention2(t) simulation result schematic diagram.
Specific embodiment
Below in conjunction with the attached drawing in the embodiment of the present invention, the technical solution in the embodiment of the present invention is carried out clear, complete
Site preparation describes, it is clear that described embodiment is only part of the embodiment of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained every other without making creative work
Embodiment shall fall within the protection scope of the present invention.
First embodiment of the invention：
Referring to Fig. 1, Fig. 1 is first embodiment of the invention provide for flexibility Timoshenko beam mechanical arm anti-saturations
Boundary control method flow diagram.The boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations,
Include the following steps：
S101, the dynamic characteristic for obtaining flexibility Timoshenko beam mechanical arm systems, and according to the dynamic characteristic,
Build flexibility Timoshenko beam mechanical arm system models.
In the present embodiment, the kinetic energy of the dynamic characteristic including flexibility Timoshenko beam mechanical arm systems, described
The potential energy and nonconservative force of flexible Timoshenko beams mechanical arm system are to the flexibility Timoshenko beam mechanical arm systems
The virtual work done.Wherein,
The kinetic energy is：
Wherein, x ∈ [0, L] be each position of flexibility Timoshenko beam mechanical arms, t ∈ [0, ∞) be the time, IhFor wheel
The rotary inertia of hub, L are the length of flexible machine Timoshenko beam mechanical arms, and ρ is the list of flexibility Timoshenko beam mechanical arms
Bit length homogeneous quality, TρFor the unit turn inertia of flexible Timoshenko beams mechanical arm, m is the quality of end load, and J is
The rotary inertia of end load, φ (x, t) for flexible mechanical arm under xoy coordinate systems in the cross torsion shape of position x moment t
Become, absolute displacement y (x, t) of the mechanical arm under xoy coordinates is defined as y (x, t)=w (x, t)+x θ (t), wherein w (x, t) be
Under xoy coordinate systems in time t position x flexibility Timoshenko beam mechanical arm systems elastic deformation, θ (t) is mechanical arm
Rotational angle.
The potential energy is：
Wherein, EI is the bending stiffness of flexibility Timoshenko beam mechanical arms, and K=kGA, k are for one by soft and fine
The constant that Timoshenko beam mechanical arms cross-sectional shape determines, cross-sectional areas of the A for flexibility Timoshenko beam mechanical arms, G
The coefficient of rigidity for flexible Timoshenko beams mechanical arm；
The virtual work is：
δ W=u (t) δ y (L, t)+τ1(t)δφ(L,t)+τ2(t)δθ(t)；
Wherein, δ be variation symbol, u (t), τ1(t) and τ2(t) device in order to control.
The structure flexibility Timoshenko beam mechanical arm system models, specifically, by the kinetic energy, the potential energy, institute
It states virtual work and substitutes into Hamiton's principle, obtaining flexible Timoshenko beams mechanical arm system model is：
Wherein,
W (0, t)=φ (0, t)=0 (3)
Wherein,
S102, according to the flexibility Timoshenko beam mechanical arm system models, build boarder controller.
In the present embodiment, using the boarder controller of hyperbolic tangent function reasonable design, make the flexibility
Timoshenko beam mechanical arm systems reach stable Bounded states and realize tracking performance, avoid what is generated using sign function
The problem of input is trembled, so as to fulfill tracking more stable, accurate to mechanical arm and control.
In the present embodiment, the boarder controller is u (t), τ1(t) and τ2(t)；Wherein,
Wherein, α1,α2,α3,α4,k1,k2,k3,k4Gain parameter for the boarder controller；α1,α2,α3,α4,k1,k2,
k3,k4Value be more than 0；E (t) is angular error, and
E (t)=θ (t)-θd (10)
In the present embodiment, 1, Figure 12 and Figure 13 is please referred to Fig.1, wherein, Figure 11 is designed in first embodiment of the invention
Boundary Control power u (t) simulation result schematic diagrams.Figure 12 is the boundary torque tau designed in first embodiment of the invention1(t) it imitates
True result schematic diagram.Figure 13 is the control torque tau designed in first embodiment of the invention2(t) simulation result schematic diagram.
S103, based on the flexibility Timoshenko beam mechanical arm system models, build the flexibility Timoshenko beams
The Lyapunov functions of mechanical arm system；According to the Lyapunov functions, the flexibility Timoshenko beam mechanical arms are verified
The stability of system.
In the present embodiment, it is described based on the flexibility Timoshenko beam mechanical arm system models, build the flexibility
The Lyapunov functions of Timoshenko beam mechanical arm systems, specially：
Based on the flexibility Timoshenko beam mechanical arm system models, the flexibility Timoshenko beam mechanical arms are designed
The Lyapunov functions of system,
V (t)=Va(t)+Vb(t)+Vc(t)；
Wherein,
It represents
Energy term；
Represent auxiliary item；
Vc (t)=α3ln(cosh(k3E (t))), represent addition Item.
S104, when judging that the flexibility Timoshenko beam mechanical arm systems meet preset stability requirement, utilize
MATLAB simulation softwares carry out Digital Simulation to the flexibility Timoshenko beams mechanical arm system, obtain simulation result.
In the present embodiment, it is described to judge that the flexibility Timoshenko beam mechanical arm systems meet preset stability and want
It asks, that is, verifies the orthotropicity of Lyapunov functions, show that the flexibility Timoshenko beam mechanical arm systems meet Lyapunov
Stabilization under meaning；It verifies the negative definiteness of Lyapunov function first derivatives, obtains the flexibility Timoshenko beam mechanical arms
System meets asymptotically stability.
In the present embodiment, the orthotropicity of verification Lyapunov function V (t), method are as follows：
The codomain of cosh functions for [1, ∞), i.e. cosh (k1E (t)) ＞ 1, so Vc(t)=α3ln(cosh(k3e(t)))
＞ 0；
Meanwhile Va(t) ＞ 0, Vb(t) ＞ 0 obtains V (t)=Va(t)+Vb(t)+Vc(t) ＞ 0, i.e. Lyapunov functions V
(t) orthotropicity is verified.
Verify Lyapunov function first derivativesNegative definiteness method it is as follows：
Va(t) first derivative is asked the time to be,
Formula (1) and (2) generation into formula (Ka), are obtained：
Formula (Kb) is subjected to partial integration, is obtained：
Formula (3) is substituted into formula (Kc), merges similar terms and obtains：
Vb(t) first derivative is asked to the time, obtains：
Formula (7)~(9) are substituted into formula (Ke), are obtained：
Vc(t) first derivative is asked to the time, obtains：
Formula (Kc)~(Kg) is substituted into V (t)=Va(t)+Vb(t)+Vc(t), it obtains：
I.e.Negative definiteness be verified.
It can be obtained by formula (Kh)：
It can be obtained by formula (10)：
So as to have：
Convolution (4) and (5) can obtain φ ' (L, t)=0, φ (L, t)=w ' (L, t), this shows
Convolution (6) and (9) and above analysis, may finally obtain e (t)=0, illustrate the flexibility
Timoshenko beam mechanical arm systems have preferable angleonly tracking performance.
S105, according to the simulation result, verify that apply control to the flexibility Timoshenko beams mechanical arm system dynamic
Whether the control effect after work meets preset requirement；If the control effect meets the preset requirement, the boundary is preserved
The gain parameter of controller terminates the operation；If the control effect does not meet the preset requirement, the boundary control is corrected
The gain parameter of device processed, re-starts Digital Simulation.
It should be noted that please referring to Fig. 2 and Fig. 3, Fig. 2 is that another flow based on first embodiment of the invention is shown
Figure.Fig. 3 is the structure diagram of the flexible Timoshenko beams mechanical arm operation in first embodiment of the invention.As shown in Fig. 2,
If the control effect does not meet the preset requirement, the gain parameter of the boarder controller is corrected, is re-started
Digital Simulation, specially：
Correct the gain parameter of the boarder controller, according to the gain parameter verify the Lyapunov functions and
The orthotropicity and negative definiteness of Lyapunov function first derivatives, and using MATLAB simulation softwares to the flexibility Timoshenko
Beam mechanical arm system carries out Digital Simulation.It is understood that flexibility Timoshenko beams machinery is judged according to simulation result
Whether vibration, shearing deformation amount and the angle of arm meet the requirements, if cannot meet the requirements, readjust the increasing of boarder controller
Beneficial parameter alpha1,α2,α3,α4,k1,k2,k3,k4.If met the requirements, terminate.
In the present embodiment, the simulation result includes the vibration amplitude of flexibility Timoshenko beam mechanical arms, shearing shape
Variable and angle value.The boarder controller includes anti-saturation controller and angle controller.The boarder controller includes moving
Dynamic sensor, disturbance observer, central controller and driving device.
In the present embodiment, Fig. 4 and Fig. 5 are please referred to, wherein, Fig. 4 is not apply control in first embodiment of the invention
Flexible Timoshenko beams mechanical arm elastic deformation w (x, t) simulation result schematic diagram.Fig. 5 is first embodiment of the invention
In the flexible Timoshenko beams mechanical arm for not applying control shearing deformation amount φ (x, t) simulation result schematic diagram.Such as Fig. 4
Shown in Fig. 5, when not adding control, there is vibration (lateral displacement) and shearing deformation amount in mechanical arm everywhere.
In the present embodiment, Fig. 7 and Fig. 8 are please referred to, wherein, Fig. 7 is after the application in first embodiment of the invention controls
Flexible Timoshenko beams mechanical arm amount of elastic deformation w (x, t) simulation result schematic diagram.Fig. 8 is that the present invention first is implemented
Shearing deformation amount φ (x, t) simulation result schematic diagram of the flexible Timoshenko beams mechanical arm after application control in example.Such as
Shown in Fig. 7 and Fig. 8, vibration suppression, after t=2s, the flexibility are carried out using flexible Timoshenko beams mechanical arm
The amplitude of Timoshenko beam mechanical arms tends to be relatively steady, and amplitude is near equilbrium position.
In the present embodiment, Fig. 6 and Fig. 9 are please referred to, wherein, Fig. 6 is not apply control in first embodiment of the invention
Flexible Timoshenko beams mechanical arm wheel hub angular position (t) simulation result schematic diagram.Fig. 9 is that the present invention first is implemented
Angular position (t) simulation result schematic diagram of the flexible Timoshenko beams mechanical arm wheel hub after application control in example.It please join
Figure 10 is read, Figure 10 is the angle of the flexible Timoshenko beams mechanical arm wheel hub after the application control in first embodiment of the invention
Site error e (t) simulation result schematic diagrams.As shown in Figure 10, angle is carried out using to the flexibility Timoshenko beams mechanical arm
Degree tracking inhibits, and the angle of flexible T beams mechanical arm has preferable tracking performance.
A kind of boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations is present embodiments provided, is wrapped
It includes：The dynamic characteristic of flexibility Timoshenko beam mechanical arm systems is obtained, and according to the dynamic characteristic, structure flexibility
Timoshenko beam mechanical arm system models；According to the flexibility Timoshenko beam mechanical arm system models, the control of structure boundary
Device processed；The stability of the flexibility Timoshenko beam mechanical arm systems is verified under anti-saturation control action；Utilize MATLAB
Simulation software carries out Digital Simulation to the flexibility Timoshenko beams mechanical arm system, obtains simulation result；According to described imitative
It is true as a result, verification whether the control effect after flexibility Timoshenko beams mechanical arm system application control action is met it is pre-
If it is required that；If the control effect does not meet the preset requirement, the boarder controller is adjusted according to the simulation result
Gain parameter, with preferable anti-saturation control and tracking performance.The present invention can realize more stable, smart to mechanical arm
True tracking and control.
The above is the preferred embodiment of the present invention, it is noted that for those skilled in the art
For, without departing from the principle of the present invention, several improvement and deformation can also be made, these are improved and deformation is also considered as
Protection scope of the present invention.
One of ordinary skill in the art will appreciate that realizing all or part of flow in above-described embodiment method, being can be with
Relevant hardware is instructed to complete by computer program, the program can be stored in a computer read/write memory medium
In, the program is when being executed, it may include such as the flow of the embodiment of above-mentioned each method.Wherein, the storage medium can be magnetic
Dish, CD, read-only memory (Read-Only Memory, ROM) or random access memory (Random Access
Memory, RAM) etc..
Claims (9)
1. a kind of boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations, which is characterized in that including as follows
Step：
The dynamic characteristic of flexibility Timoshenko beam mechanical arm systems is obtained, and according to the dynamic characteristic, structure flexibility
Timoshenko beam mechanical arm system models；
According to the flexibility Timoshenko beam mechanical arm system models, boarder controller is built；
Based on the flexibility Timoshenko beam mechanical arm system models, the flexibility Timoshenko beam mechanical arm systems are built
Lyapunov functions；
According to the Lyapunov functions, the stability of the flexibility Timoshenko beam mechanical arm systems is verified；
When judging that the flexibility Timoshenko beam mechanical arm systems meet preset stability requirement, emulated using MATLAB
Software carries out Digital Simulation to the flexibility Timoshenko beams mechanical arm system, obtains simulation result；
According to the simulation result, the control after applying control action to the flexibility Timoshenko beams mechanical arm system is verified
Whether effect meets preset requirement；
If the control effect meets the preset requirement, the gain parameter of the boarder controller is preserved, terminates the operation；
If the control effect does not meet the preset requirement, the gain parameter of the boarder controller is corrected, is re-started
Digital Simulation.
2. the boundary control method according to claim 1 for flexibility Timoshenko beam mechanical arm anti-saturations, special
Sign is that the dynamic characteristic includes kinetic energy, the flexibility Timoshenko of flexibility Timoshenko beam mechanical arm systems
The virtual work that the potential energy and nonconservative force of beam mechanical arm system do the flexibility Timoshenko beam mechanical arm systems；Its
In,
The kinetic energy is：
Wherein, x ∈ [0, L] be each position of flexibility Timoshenko beam mechanical arms, t ∈ [0, ∞) be the time, IhFor turning for wheel hub
Dynamic inertia, L are the length of flexible machine Timoshenko beam mechanical arms, and ρ is the unit length of flexibility Timoshenko beam mechanical arms
Homogeneous quality, IρFor the unit turn inertia of flexible Timoshenko beams mechanical arm, m is the quality of end load, and J is born for end
The rotary inertia of load, φ (x, t) for flexible mechanical arm under xoy coordinate systems in the cross torsion deformation of position x moment t, machinery
Absolute displacement y (x, t) of the arm under xoy coordinates is defined as y (x, t)=w (x, t)+x θ (t), and wherein w (x, t) is in xoy coordinates
Under system in time t position x flexibility Timoshenko beam mechanical arm systems elastic deformation, θ (t) be mechanical arm angle of rotation
Degree；
The potential energy is：
Wherein, EI is the bending stiffness of flexibility Timoshenko beam mechanical arms, and K=kGA, k are for one by soft and fine Timoshenko
The constant that beam mechanical arm cross-sectional shape determines, A are the cross-sectional area of flexibility Timoshenko beam mechanical arms, and G is flexibility
The coefficient of rigidity of Timoshenko beam mechanical arms；
The virtual work is：
δ W=u (t) δ y (L, t)+τ1(t)δφ(L,t)+τ2(t)δθ(t)；
Wherein, δ be variation symbol, u (t), τ1(t) and τ2(t) device in order to control.
The structure flexibility Timoshenko beam mechanical arm system models, specifically, by the kinetic energy, the potential energy, the void
Work(substitutes into Hamiton's principle, and obtaining flexible Timoshenko beams mechanical arm system model is：
Wherein,
W (0, t)=φ (0, t)=0；
Wherein,
3. the boundary control method according to claim 1 for flexibility Timoshenko beam mechanical arm anti-saturations, special
Sign is that the boarder controller is u (t), τ1(t) and τ2(t)；Wherein,
Wherein, α1,a2,a3,a4,k1,k2,k3,k4Gain parameter for the boarder controller；α1,a2,a3,a4,k1,k2,k3,k4
Value be more than 0；E (t) is angular error, and e (t)=θ (t)-θd。
4. the boundary control method according to claim 1 for flexibility Timoshenko beam mechanical arm anti-saturations, special
Sign is, described based on the flexibility Timoshenko beam mechanical arm system models, builds the flexibility Timoshenko beam machines
The Lyapunov functions of tool arm system, specially：
Based on the flexibility Timoshenko beam mechanical arm system models, the flexibility Timoshenko beam mechanical arm systems are designed
Lyapunov functions,
V (t)=Va(t)+Vb(t)+Vc(t)；
Wherein,
Represent energy term；
Represent auxiliary item；
Vc (t)=α3ln(cosh(k3E (t))), represent addition Item.
5. the boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations according to claim 1 or 3,
It is characterized in that, it is described according to the Lyapunov functions, verify the stability of the flexibility Timoshenko beam mechanical arm systems,
Specially：
It verifies the orthotropicity of Lyapunov functions, show that the flexibility Timoshenko beam mechanical arm systems meet Lyapunov meanings
Stabilization under justice；
It verifies the negative definiteness of Lyapunov function first derivatives, show that the flexibility Timoshenko beam mechanical arm systems meet gradually
Into stabilization.
6. the boundary control method for flexibility Timoshenko beam mechanical arm anti-saturations according to claim 1 or 3,
It is characterized in that, the boarder controller includes anti-saturation controller and angle controller.
7. the boundary control method according to claim 1 for flexibility Timoshenko beam mechanical arm anti-saturations, special
Sign is that the boarder controller includes movable sensor, disturbance observer, central controller and driving device.
8. the boundary control method according to claim 1 for flexibility Timoshenko beam mechanical arm anti-saturations, special
Sign is, if the control effect does not meet the preset requirement, corrects the gain parameter of the boarder controller, weight
It is new to carry out Digital Simulation, specially：
Correct the gain parameter of the boarder controller, according to the gain parameter verify the Lyapunov functions and
The orthotropicity and negative definiteness of Lyapunov function first derivatives, and using MATLAB simulation softwares to the flexibility Timoshenko
Beam mechanical arm system carries out Digital Simulation.
9. the boundary control method according to claim 1 for flexibility Timoshenko beam mechanical arm anti-saturations, special
Sign is that the simulation result includes vibration amplitude, shearing deformation amount and the angle value of flexibility Timoshenko beam mechanical arms.
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