CN116619351A - Design method of interval observer of mechanical arm system based on event trigger mechanism - Google Patents

Abstract

The invention relates to a design method of an interval observer of a manipulator system based on an event trigger mechanism. According to the position and angular velocity at the joints of the manipulator, establish the mathematical model of the first manipulator system, obtain the second manipulator system mathematic model through state selection, set the event trigger condition according to the second manipulator system mathematic model, according to The event triggering condition is based on the positive system theory, combined with the mathematical model of the manipulator system, to construct an interval observer of the manipulator system under the event trigger mechanism. The present invention models a class of manipulator systems containing disturbances and nonlinearities, can overcome the disadvantage that the model does not fit the reality, provides event trigger conditions according to the model, reduces the use of network resources, and designs interval observation based on event trigger conditions The controller ensures the final bounded convergence of the error system, and at the same time ensures that the final state estimation interval is valid.

Description

Design Method of Interval Observer for Manipulator System Based on Event-triggered Mechanism

technical field

The present invention relates to the technical field of interval observers, in particular to a design method for an interval observer of a manipulator system based on an event trigger mechanism.

Background technique

The robotic arm is an automatic operating device that can imitate certain movement functions of the human hand and arm to grab, carry objects or operate tools according to a fixed program. It can replace human heavy labor to realize the mechanization and automation of production, and can operate in harmful environments to protect personal safety, so it is widely used in machinery manufacturing, metallurgy, electronics, light industry and atomic energy and other departments. The research on the manipulator began in the middle of the 20th century. With the development of computer and automation technology, especially since the first digital electronic computer came out in 1946, the calculation has made amazing progress, and it has developed in the direction of high speed, large capacity and low price. . At the same time, the urgent need for mass production has promoted the progress of automation technology and laid the foundation for the development of robotic arms. Due to its wide application and importance in many fields, the research related to manipulators has always attracted the attention of professional technicians in the field of control.

In the actual operation of the manipulator system, the state of the system may not be directly available due to external interference such as noise. At this time, it is necessary to design an observer for the system. Most of the manipulator system will be affected by external disturbances, and the traditional Luenberger observer cannot accurately observe the system state under external disturbances. The extended observer constructs the error system and substitutes it into the Lyapunov function to derive sufficient conditions to solve the Luenberger Problems with the observer. However, the extended observer cannot realize the state tracking of the original system, and there are too many parameters, and the integration is difficult, so it cannot theoretically guarantee its convergence, nor can it guarantee that the final state estimation interval is valid. Traditional observers still have problems such as interval observers are not stable enough.

Contents of the invention

Therefore, the technical problem to be solved by the present invention is to overcome the interference and non-linear factors that are not considered in the mechanical arm system in the prior art.

In order to solve the above-mentioned technical problems, the present invention provides a method for designing an interval observer of a manipulator system based on an event-triggered mechanism, including:

S1: According to the position and angular velocity of the joints of the manipulator, establish the mathematical model of the first manipulator system, and obtain the mathematical model of the second manipulator system through state selection;

S2: Set an event trigger condition according to the mathematical model of the second manipulator system;

S3: According to the event triggering condition, based on the positive system theory, combined with the mathematical model of the manipulator system, construct an interval observer of the manipulator system under the event trigger mechanism.

In one embodiment of the present invention, the method for establishing the mathematical model of the first mechanical arm system in S1 is:

The mathematical model of the first mechanical arm system is:

where q and are the position and angular velocity at the joints of the manipulator, M(q) is the inertia matrix, /> is the centrifugal force matrix, G(q) is the gravity matrix, u is the control input, and τ _d is the system disturbance.

In one embodiment of the present invention, the method for obtaining the mathematical model of the second manipulator system in S1 is:

selection status Find the derivative of x ₁ , x ₂ :

Among them, d=M(x ₁ ) ^-1 τ _d , H(x ₁ )=M(x ₁ ) ^-1 , f(x ₁ ,x ₂ )=-M(x ₁ ) ^-1 (C(x ₁ ,x ₂ )x ₂ +G(x ₁ );

when

C＝[I _n×n 0 _n×n ]∈R ^n×2n

Get the mathematical model expression of the second manipulator system:

y(t)=Cx(t)

Among them, x(t)=[x ₁ (t),x ₂ (t)] ^T , y(t) is the output, u(t) is the control input, F(x(t)) is the nonlinear item, the matrix A, B, C, D are constant matrices, and d is an interference term.

In one embodiment of the present invention, according to the mathematical model of the second manipulator system in S2, the method for setting event triggering conditions is as follows:

The event trigger conditions are:

in, η is the event trigger time, /> is the sampling time triggered by the event, y(t) is the actual output value, and e _y (t) is the error value between the sampling time triggered by the event and the actual output value.

In one embodiment of the present invention, the method for constructing the interval observer of the manipulator system under the event-triggered mechanism in S3 is:

Determine whether the state matrix of the error system is a Metzler matrix;

If it is a Metzler matrix, build an interval observer and error system under the event trigger mechanism. If it is not a Metzler matrix, by introducing a linear constant coordinate transformation, the state matrix of the error system becomes a Metzler matrix, and a new vector is defined. The second manipulator system The mathematical model becomes the mathematical model of the third manipulator system, and the interval observer and error system under the event trigger mechanism are constructed according to the mathematical model of the third manipulator system.

In one embodiment of the present invention, when the state matrix of the error system is the Metzler matrix, the method for constructing the interval observer and the error system is:

Construct an upper bound observer and a lower bound observer. The upper bound observer always observes a value greater than the actual state, and the lower bound observer always observes a value smaller than the actual state. The initial condition satisfies have Build the observer as follows:

Among them, x ⁺ is the upper bound function of a vector function, is the lower bound observation function of a vector function, x ^- is the lower bound function of a vector function, /> is the derivative function of the lower bound observation function of a vector function, /> is the upper bound observation function of a vector function, and is /> the derivative of the upper-bounded observation function of a vector function,

Construct the interval observer of the manipulator system under the event trigger mechanism according to the event trigger conditions:

Build the error system:

in,

A-LC is the state matrix of the error system.

In one embodiment of the present invention, when the state matrix of the error system is not the Metzler matrix, the method for constructing the interval observer and the error system under the event trigger mechanism is:

When the state matrix of the error system is not a Metzler matrix, a linear constant coordinate transformation w(t)=Mx(t) is introduced so that the state matrix of the error system is a Metzler matrix, thus:

when satisfied definition /> The mathematical model of the second robotic arm system becomes the mathematical model of the third robotic arm system:

Construct an interval observer according to the mathematical model of the third manipulator system:

Build the error system:

in, is the state matrix of the error system.

In one embodiment of the present invention, before constructing the interval observer of the manipulator system under the event-triggered mechanism, it includes:

Consider a system:

where R is a known constant matrix, and the nonlinear function Then when R is a Metzler matrix, the system is a positive system.

Correspondingly, an embodiment of the present invention also provides an interval observer, including:

memory for storing computer programs;

The processor is configured to implement the steps of the above-mentioned method for designing an interval observer of a manipulator system based on an event-triggered mechanism when executing the computer program.

Correspondingly, the present invention also provides a computer-readable non-volatile storage medium, including:

The computer-readable instruction, when the computer reads the computer instruction, causes the computer to execute the above-mentioned method for designing an interval observer of a manipulator system based on an event-triggered mechanism.

The above technical solution of the present invention has the following advantages compared with the prior art:

The interval observer of a manipulator system based on an event-triggered mechanism according to the present invention establishes the mathematical model of the first manipulator system according to the position and angular velocity of the joints of the manipulator, and obtains the mathematical model of the second manipulator system through state selection. Model, according to the mathematical model of the second manipulator system, set event trigger conditions, according to the event trigger conditions, based on the positive system theory, combined with the mathematical model of the manipulator system, construct the manipulator system under the event trigger mechanism interval observer. The present invention models a class of manipulator systems containing disturbances and nonlinearities, can overcome the disadvantage that the model does not fit the reality, provides event trigger conditions according to the model, reduces the use of network resources, and designs interval observation based on event trigger conditions The device ensures the final bounded convergence of the error system, and at the same time ensures that the final state estimation interval is valid.

Description of drawings

In order to make the content of the present invention more easily understood, the present invention will be described in further detail below according to specific embodiments of the present invention in conjunction with the accompanying drawings, wherein

Fig. 1 is a flow chart of the present invention;

Fig. 2 is a schematic diagram of a two-degree-of-freedom mechanical arm model of the present invention;

Fig. 3 is a structural block diagram of the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and implement it, but the examples given are not intended to limit the present invention.

Embodiment one

As shown in FIG. 1, the design method of an interval observer of a manipulator system based on an event-triggered mechanism described in this embodiment specifically includes:

S1: Establish the mathematical model of the first manipulator system according to the position and angular velocity of the joints of the manipulator, and obtain the cooperative mathematical model of the second manipulator through state selection; S2: Set the event according to the mathematical model of the second manipulator system Trigger condition; S3: According to the event trigger condition, based on positive system theory and combined with the mathematical model of the manipulator system, construct an interval observer of the manipulator system under the event trigger mechanism.

The design method of the interval observer of the manipulator system based on the event-triggered mechanism described in this embodiment is based on the modeling method for the manipulator system with disturbance and nonlinearity, which is practical and reduces the network resources. Using the interval observer based on the event-triggered mechanism solves the problem that the extended observer cannot guarantee the final convergence, and at the same time ensures that the final state estimation interval is valid.

The method for establishing the mathematical model of the first manipulator system in S1 is: the mathematical model of the first manipulator system is:

where q and are the position and angular velocity at the joints of the manipulator, M(q) is the inertia matrix, /> is the centrifugal force matrix, G(q) is the gravity matrix, u is the control input, and τ _d is the system disturbance.

The method for obtaining the mathematical model of the second manipulator system in S1 is: select the state Find the derivative of x ₁ , x ₂ :

Among them, d=M(x ₁ ) ^-1 τ _d , H(x ₁ )=M(x ₁ ) ^-1 , f(x ₁ ,x ₂ )=-M(x ₁ ) ^-1 (C(x ₁ ,x ₂ )x ₂ +G(x ₁ );

when

C＝[I _n×n 0 _n×n ]∈R ^n×2n

Get the mathematical model expression of the second manipulator system:

y(t)=Cx(t)

Among them, x(t)=[x ₁ (t),x ₂ (t)] ^T , y(t) is the output, u(t) is the control input, F(x(t)) is the nonlinear item, the matrix A, B, C, D are constant matrices, and d is an interference term.

By establishing a mathematical model of the manipulator system with disturbance and nonlinearity, the problem that the existing extended observer does not contain system disturbance and nonlinear terms is not suitable for reality is solved.

In S2, according to the mathematical model of the second manipulator system, the method for setting the event triggering condition is: the event triggering condition is:

in, η is the event trigger time, /> is the sampling time triggered by the event, y(t) is the actual output value, and e _y (t) is the error value between the sampling time triggered by the event and the actual output value.

By setting event triggering conditions, the execution time and quantity of event triggering tasks can be effectively reduced, saving network bandwidth and resources, thus significantly saving communication network resources on the basis of fully ensuring the control performance of the closed-loop system.

The method of constructing the interval observer of the manipulator system under the event trigger mechanism in the described S3 is: judge whether the state matrix of the error system is a Metzler matrix; if it is a Metzler matrix, construct the interval observer and the error system under the event trigger mechanism, if It is not the Metzler matrix. By introducing linear constant coordinate transformation, the state matrix of the error system becomes a Metzler matrix, and a new vector is defined. The mathematical model of the second manipulator system becomes the third manipulator system mathematical model. According to the third manipulator system The mathematical model constructs the interval observer and error system under the event-triggered mechanism.

When the state matrix of the error system is the Metzler matrix, the method of constructing the interval observer and the error system under the event trigger mechanism is as follows: construct the upper bound observer and the lower bound observer, the observation value of the upper bound observer is greater than the actual state at all times, and the lower bound observer The observed value of the device is smaller than the actual state at all times, and the initial condition satisfies have Build the observer as follows:

Among them, x ⁺ is the upper bound function of a vector function, is the lower bound observation function of a vector function, x ^- is the lower bound function of a vector function, /> is the derivative function of the lower bound observation function of a vector function, /> is the upper bound observation function of a vector function, and is /> the derivative of the upper-bounded observation function of a vector function,

Construct the interval observer of the manipulator system under the event trigger mechanism according to the event trigger conditions:

Build the error system:

in,

A-LC is the state matrix of the error system.

When the state matrix of the error system is not a Metzler matrix, the method of constructing the interval observer and error system under the event trigger mechanism is: when the state matrix of the error system is not a Metzler matrix, introduce a linear constant coordinate transformation w(t)=Mx( t), so that the state matrix of the error system is the Metzler matrix, thus:

when satisfied definition /> The mathematical model of the second robotic arm system becomes the mathematical model of the third robotic arm system:

Construct an interval observer according to the mathematical model of the third manipulator system:

Build the error system:

in, is the state matrix of the error system.

Through the event trigger conditions, combined with the mathematical model of the manipulator system, an interval observer is constructed, and the method of coordinate transformation is added to the design method of the basic event trigger observer to ensure the final bounded convergence of the error system and the final state. Estimate intervals are valid.

The stability of the interval observer of the event-triggered mechanism-based manipulator system described in this embodiment is proved below through lemmas and definitions.

Lemma 1: The nonlinear F(x(t)) satisfies the Lipschitz condition and is globally differentiable, then it can be represented by two increasing Lipschitz functions g ₁ (x) and g ₂ (x), namely

F(x)＝g ₁ (x)-g ₂ (x)

Based on this there is a new function thus making

a.

b.

From this we can get the bound of F(x):

Lemma 2: On the basis of Lemma 1, let the Jacobian matrix of F(x) be bounded, so that we can further get

Among them, F ₁ , F ₂ , F ₃ , and F ₄ are all constant matrices.

Lemma 3: For given two vectors W and Y with appropriate dimensions, the following inequalities hold

where P>0.

Lemma 4: If the matrix _H2 is a real matrix, Then the following equations (i), (ii) and (iii) are valence:

Definition 1: Under the zero initial condition δ(0)=0, the class K function δ is continuous and monotonically increasing. If the function ε(·, t) is a K-type function, and ε(s, ) is monotonically decreasing, then when t tends to zero, ε(s, t) also tends to zero, and ε is A KL-like function.

Definition 2: For the error system, if the initial value is known, and there is a K-type function δ and a KL-type function ε, then under the action of the event trigger condition (3), the following inequalities hold:

||e(t)||<ε(||e(0)||,t)+δ(||d[0,t]||),

At this time, the error system is input to the state stable under the influence of the external disturbance d(t).

Theorem: If there are coefficients γ _i (i=1,2,...,6), λ>0, η>0, matrix T>0, symmetric matrix P>0, L makes the following conditions true

Where ∑ ₁ = γ ₁ + γ ₂ + γ ₃ + γ ₄ + γ ₅ + γ ₆ , ∑ ₂ = G ^T P + PG, Then the upper and lower bound errors will eventually converge boundedly.

Proof: Definition It can be obtained from the mathematical model of the third manipulator system and the error system when the state matrix of the error system is not the Metzler matrix:

Next, select the Lyapunov function V(t)= ^ξT (t)Pξ(t), and perform time derivative on V(t) to get

According to Lemma 3, the above formula can be transformed into

Triggered by the event, you can get:

Right now

in Then combined with Lemma 2, we can also get:

Put the two inequalities obtained above into (1), and let Then it can be proved that:

in

So you can get:

From definition 1, the sufficient conditions of the theorem can be obtained, and the proof is completed.

Embodiment two

This embodiment provides an interval observer, including:

memory for storing computer programs;

The processor is configured to implement the steps of the method for designing an interval observer of a manipulator system based on an event-triggered mechanism described in Embodiment 1 when executing the computer program.

Embodiment three

This embodiment also provides a computer-readable non-volatile storage medium, including:

The computer-readable instructions, when the computer reads the computer instructions, make the computer execute the method for designing an interval observer of a manipulator system based on an event-triggered mechanism described in Embodiment 1.

Those skilled in the art should understand that the embodiments of the present application may be provided as methods, systems, or computer program products. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) having computer-usable program code embodied therein.

The present application is described with reference to flowcharts and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present application. It should be understood that each procedure and/or block in the flowchart and/or block diagram, and a combination of procedures and/or blocks in the flowchart and/or block diagram can be realized by computer program instructions. These computer program instructions may be provided to a general purpose computer, special purpose computer, embedded processor, or processor of other programmable data processing equipment to produce a machine such that the instructions executed by the processor of the computer or other programmable data processing equipment produce a An apparatus for realizing the functions specified in one or more procedures of the flowchart and/or one or more blocks of the block diagram.

These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing apparatus to operate in a specific manner, such that the instructions stored in the computer-readable memory produce an article of manufacture comprising instruction means, the instructions The device realizes the function specified in one or more procedures of the flowchart and/or one or more blocks of the block diagram.

These computer program instructions can also be loaded onto a computer or other programmable data processing device, causing a series of operational steps to be performed on the computer or other programmable device to produce a computer-implemented process, thereby The instructions provide steps for implementing the functions specified in the flow chart or blocks of the flowchart and/or the block or blocks of the block diagrams.

Apparently, the above-mentioned embodiments are only examples for clear description, and are not intended to limit the implementation. For those of ordinary skill in the art, on the basis of the above description, other changes or changes in various forms can also be made. It is not necessary and impossible to exhaustively list all the implementation manners here. However, the obvious changes or changes derived therefrom are still within the scope of protection of the present invention.

Claims (10)

1. A design method of an interval observer of a mechanical arm system based on an event trigger mechanism is characterized by comprising the following steps of:

s1: establishing a first mechanical arm system mathematical model according to the position and the angular velocity of the mechanical arm joint, and obtaining a second mechanical arm system mathematical model through state selection;

s2: setting event triggering conditions according to the mathematical model of the second mechanical arm system;

s3: and according to the event triggering condition, based on a positive system theory, combining the mathematical model of the mechanical arm system to construct an interval observer of the mechanical arm system under an event triggering mechanism.

2. The method for designing an interval observer of a mechanical arm system based on an event triggering mechanism according to claim 1, wherein the method for establishing the mathematical model of the first mechanical arm system in S1 is as follows:

the first mechanical arm system mathematical model is as follows:

wherein q andthe position and the angular velocity of the mechanical arm joint are respectively represented by M (q) which is an inertial matrix,>is a centrifugal force matrix, G (q) is a gravity matrix, u is a control input, τ _d Is a system disturbance.

3. The method for designing an interval observer of a mechanical arm system based on an event triggering mechanism according to claim 1, wherein the method for obtaining the mathematical model of the second mechanical arm system in S1 is as follows:

selecting state x ₁ ＝q,Obtaining x ₁ ,x ₂ Is guided by:

wherein d=m (x ₁ ) ^-1 τ _d ，H(x ₁ )＝M(x ₁ ) ^-1 ，f(x ₁ ,x ₂ )＝-M(x ₁ ) ^-1 (C(x ₁ ,x ₂ )x ₂ +G(x ₁ )；

When (when)

C＝[I _n×n 0 _n×n ]∈R ^n×2n

Obtaining a mathematical model expression of a second mechanical arm system:

y(t)＝Cx(t)

wherein x (t) = [ x ] ₁ (t),x ₂ (t)] ^T Y (t) is the output, u (t) is the control input, F (x (t)) is the nonlinear term, matrices A, B, C, D are constant matrices, and D is the disturbance term.

4. The method for designing a section observer of a mechanical arm system based on an event triggering mechanism according to claim 1, wherein the method for setting an event triggering condition according to the mathematical model of the second mechanical arm system in S2 is as follows:

the event triggering conditions are as follows:

wherein ,eta is the moment of event triggering->For the event-triggered sampling instant, y (t) is the actual output value, e _y And (t) is the error value between the sampling time triggered by the event and the actual output value.

5. The method for designing a section observer of a mechanical arm system based on an event triggering mechanism according to claim 1, wherein the method for constructing the section observer of the mechanical arm system under the event triggering mechanism in S3 is as follows:

judging whether a state matrix of the error system is a Metzler matrix or not;

if the state matrix of the error system is changed into the Metzler matrix by introducing linear constant coordinate transformation, defining a new vector, changing the mathematical model of the second mechanical arm system into the mathematical model of the third mechanical arm system, and constructing the interval observer and the error system under the event trigger mechanism according to the mathematical model of the third mechanical arm system.

6. The method for designing a section observer of a mechanical arm system based on a time trigger mechanism according to claim 5, wherein when the state matrix of the error system is a Metzler matrix, the method for constructing the section observer and the error system under the event trigger mechanism is as follows:

an upper-bound observer and a lower-bound observer are constructed, the time observation value of the upper-bound observer is larger than the actual state, the time observation value of the lower-bound observer is smaller than the actual state, and the initial condition is satisfiedThere is->The observer was constructed as follows:

wherein ,x⁺ Is the upper bound function of a vector function,is a lower-bound observation function of a vector function, x ^- Is a lower bound function of a vector function, < +.>Is the derivative of the lower observation function of a vector function,/for the vector function>An upper-bound observation function which is a vector function, is +.>The derivative of the upper-bound observation function of a vector function,

constructing an interval observer of the mechanical arm system under the event triggering mechanism according to the event triggering condition:

and (3) constructing an error system:

wherein ,

the a-LC is the state matrix of the error system.

7. The method for designing a section observer of a mechanical arm system based on an event triggering mechanism according to claim 5, wherein when the state matrix of the error system is not a Metzler matrix, the method for constructing the section observer and the error system under the event triggering mechanism is as follows:

when the state matrix of the error system is not a Metzler matrix, a linear constant coordinate transformation w (t) =mx (t) is introduced such that the state matrix of the error system is a Metzler matrix, so that there is:

when meeting the requirementsDefinitions->The second mechanical arm system mathematical model is changed into a third mechanical arm system mathematical model:

constructing an interval observer according to the mathematical model of the third mechanical arm system:

and (3) constructing an error system:

wherein ,is a state matrix of the error system.

8. The method for designing a section observer of a mechanical arm system based on an event triggering mechanism according to claim 5, wherein before constructing the section observer of the mechanical arm system under the event triggering mechanism, the method comprises:

consider a system:

wherein R is a known constant matrix and is a nonlinear functionThen when R is the Metzler matrix, the system is a positive system.

9. An interval observer, comprising:

a memory for storing a computer program;

a processor for implementing the steps of a method for designing an interval observer of a robotic arm system based on an event triggering mechanism as claimed in any one of claims 1-8 when executing said computer program.

10. A computer-readable non-volatile storage medium, comprising:

computer readable instructions which, when read by a computer, cause the computer to perform a method for designing a section observer of a robotic arm system based on an event triggering mechanism as claimed in any one of claims 1-8.