CN116619351B - Design method of interval observer of mechanical arm system based on event trigger mechanism - Google Patents

Abstract

The present 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 of the manipulator joint, a mathematical model of the first manipulator system is established, and through state selection, a mathematical model of the second manipulator system is obtained. According to the mathematical model of the second manipulator system, an event trigger condition is set, and according to the event trigger condition, based on positive system theory and in combination with the mathematical model of the manipulator system, an interval observer of the manipulator system under the event trigger mechanism is constructed. The present invention models a type of manipulator system containing disturbances and nonlinearity, which can overcome the disadvantage that the model is not practical, gives event trigger conditions according to the model, reduces the use of network resources, and designs an interval observer based on the event trigger conditions, which ensures that the error system eventually converges in a bounded manner, 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 triggering mechanism

Technical Field

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

Background Art

A robotic arm is an automatic operating device that can imitate certain movements and functions of human hands and arms, and is used to grasp, carry objects or operate tools according to a fixed program. It can replace heavy labor to achieve mechanization and automation of production, and can operate in harmful environments to protect personal safety. Therefore, it is widely used in machinery manufacturing, metallurgy, electronics, light industry and atomic energy. The research on robotic arms began in the mid-20th century. With the development of computers and automation technology, especially since the advent of the first digital electronic computer in 1946, computing has made amazing progress and 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, research related to robotic arms has always attracted the attention of professional and technical personnel in the field of control.

In the actual operation of the robotic arm system, the state of the system may become inaccessible due to external interference such as noise. At this time, it is necessary to design an observer for the system. Most robotic arm systems are affected by external interference, and the traditional Luenberger observer cannot accurately observe the system state under external interference. The extended observer constructs an error system and substitutes it into the Lyapunov function to derive sufficient conditions to solve the problems of the Luenberger observer. However, the extended observer cannot track the state of the original system, and there are too many parameters, which makes integration difficult. It is impossible to guarantee its convergence in theory, and it is impossible to guarantee that the final state estimation interval is valid. Traditional observers also have problems such as interval observers are not stable enough.

Summary of the invention

Therefore, the technical problem to be solved by the present invention is to overcome the problem that the robot arm system in the prior art does not take interference and nonlinear factors into consideration.

In order to solve the above technical problems, the present invention provides a design method of an interval observer of a manipulator system based on an event trigger mechanism, comprising:

S1: According to the position and angular velocity of the joints of the robot arm, a mathematical model of the first robot arm system is established, and by state selection, a mathematical model of the second robot arm system is obtained;

S2: setting event triggering conditions according to the mathematical model of the second robotic arm system;

S3: According to the event triggering condition, based on the positive system theory and combined with the mathematical model of the robotic arm system, an interval observer of the robotic arm system under the event triggering mechanism is constructed.

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

The mathematical model of the first robotic arm system is:

where q and are the position and angular velocity of the robot joint, M(q) is the inertia matrix, is the centrifugal force matrix, G(q) is the gravity matrix, u is the control input, and _τd is the system disturbance.

In one embodiment of the present invention, the method for obtaining the mathematical model of the second robotic arm 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

The mathematical model expression of the second robotic arm system is obtained:

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 term, matrices A, B, C, D are constant matrices, and d is the interference term.

In one embodiment of the present invention, the method for setting the event triggering condition according to the second robotic arm system mathematical model in S2 is:

The event triggering conditions are:

in, η is the event triggering time, is the sampling time of event triggering, y(t) is the actual output value, and e _y (t) is the error between the sampling time of event triggering and the actual output value.

In one embodiment of the present invention, the method of constructing an interval observer of the robotic arm system under the event triggering mechanism in S3 is:

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

If it is a Metzler matrix, construct the 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 is transformed into a Metzler matrix, and a new vector is defined. The mathematical model of the second robotic arm system is transformed into the mathematical model of the third robotic arm system. According to the mathematical model of the third robotic arm system, the interval observer and error system under the event trigger mechanism are constructed.

In one embodiment of the present invention, when the state matrix of the error system is a 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 values greater than the actual state, and the lower bound observer always observes values less than the actual state. The initial conditions satisfy have Construct 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 of the lower bound observation function of a vector function, is the upper bound observation function of a vector function, The upper bound of a vector function is the derivative of the observation function,

Construct an interval observer for the robotic arm system under the event trigger mechanism according to the event trigger conditions:

Construct 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 a Metzler matrix, the method for constructing the interval observer and the error system under the event triggering 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 to make the state matrix of the error system 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 based on the mathematical model of the third robotic arm system:

Construct the error system:

in, is the state matrix of the error system.

In one embodiment of the present invention, before constructing an interval observer of a robotic arm system under an event trigger mechanism, the following steps are included:

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.

Accordingly, an embodiment of the present invention further provides an interval observer, including:

Memory for storing computer programs;

A processor is used to implement the steps of the above-mentioned method for designing an interval observer for a robotic arm system based on an event trigger mechanism when executing the computer program.

Accordingly, the present invention further provides a computer-readable non-volatile storage medium, comprising:

Computer-readable instructions, when a computer reads the computer instructions, enable the computer to execute the above-mentioned design method of an interval observer for a robotic arm system based on an event trigger 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 trigger mechanism described in the present invention establishes a mathematical model of a first manipulator system according to the position and angular velocity of the manipulator joint, obtains a mathematical model of a second manipulator system through state selection, sets an event trigger condition according to the mathematical model of the second manipulator system, and constructs an interval observer of the manipulator system under the event trigger mechanism based on the positive system theory and the mathematical model of the manipulator system according to the event trigger condition. The present invention models a type of manipulator system containing disturbances and nonlinearity, which can overcome the disadvantage that the model is not practical, gives an event trigger condition according to the model, reduces the use of network resources, designs an interval observer based on the event trigger condition, ensures the final bounded convergence of the error system, and ensures that the final state estimation interval is valid.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to make the content of the present invention more clearly understood, the present invention is further described in 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;

FIG2 is a schematic diagram of a two-degree-of-freedom mechanical arm model of the present invention;

FIG3 is a structural block diagram of the present invention.

DETAILED DESCRIPTION

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 embodiments are not intended to limit the present invention.

Embodiment 1

As shown in FIG1 , the design method of an interval observer for a robotic arm system based on an event trigger mechanism described in this embodiment specifically includes:

S1: According to the position and angular velocity of the robot arm joints, a mathematical model of the first robot arm system is established, and the collaborative mathematical model of the second robot arm is obtained through state selection; S2: According to the mathematical model of the second robot arm system, event trigger conditions are set; S3: According to the event trigger conditions, based on the positive system theory and combined with the mathematical model of the robot arm system, an interval observer of the robot arm system under the event trigger mechanism is constructed.

The design method of the interval observer of the robotic arm system based on the event trigger mechanism described in this embodiment, in terms of modeling method, models the robotic arm system containing disturbances and nonlinearities, which is in line with reality and reduces the use of network resources. The interval observer based on the event trigger 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 robotic arm system in S1 is as follows: The mathematical model of the first robotic arm system is:

where q and are the position and angular velocity of the robot joint, M(q) is the inertia matrix, is the centrifugal force matrix, G(q) is the gravity matrix, u is the control input, and _τd is the system disturbance.

The method for obtaining the mathematical model of the second robotic arm 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

The mathematical model expression of the second robotic arm system is obtained:

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 term, matrices A, B, C, D are constant matrices, and d is the interference term.

By establishing a mathematical model of the robotic arm system containing disturbances and nonlinearities, the problem that the existing extended observer does not contain system disturbances and nonlinear terms and is not practical is solved.

The method of setting the event triggering condition according to the second robotic arm system mathematical model in S2 is: the event triggering condition is:

in, η is the event triggering time, is the sampling time of event triggering, y(t) is the actual output value, and e _y (t) is the error between the sampling time of event triggering and the actual output value.

By setting event trigger conditions, the execution time and number of event-triggered tasks can be effectively reduced, saving network bandwidth and resources, thereby significantly saving communication network resources while fully ensuring the control performance of the closed-loop system.

The method for constructing the interval observer of the robotic arm system under the event trigger mechanism in S3 is: determine 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 a Metzler matrix, introduce a linear constant coordinate transformation to transform the state matrix of the error system into a Metzler matrix, define a new vector, transform the mathematical model of the second robotic arm system into the mathematical model of the third robotic arm system, and construct the interval observer and the error system under the event trigger mechanism according to the mathematical model of the third robotic arm system.

When the state matrix of the error system is a Metzler matrix, the method of constructing the interval observer and the error system under the event trigger mechanism is: construct an upper bound observer and a lower bound observer, the upper bound observer always observes values greater than the actual state, the lower bound observer always observes values less than the actual state, and the initial conditions satisfy have Construct 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 of the lower bound observation function of a vector function, is the upper bound observation function of a vector function, The upper bound of a vector function is the derivative of the observation function,

Construct an interval observer for the robotic arm system under the event trigger mechanism according to the event trigger conditions:

Construct 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 the error system under the event trigger 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 to make the state matrix of the error system 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 based on the mathematical model of the third robotic arm system:

Construct the error system:

in, is the state matrix of the error system.

Through event triggering conditions and combined with the mathematical model of the robotic arm system, an interval observer is constructed. The coordinate transformation method is added to the basic event-triggered observer design method to ensure the final bounded convergence of the error system and also ensure that the final state estimation interval is valid.

The stability of the interval observer of the manipulator system based on the event trigger mechanism described in this embodiment is proved by lemmas and definitions.

Lemma 1: If the nonlinear function 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), that is,

F(x)＝ _g1 (x) _-g2 (x)

Based on this there is a new function So that

a.

b.

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

Lemma 2: Based on Lemma 1, let the Jacobian matrix of F(x) be bounded, so that we can further obtain

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

Lemma 3: For two given vectors W and Y of suitable dimensions, the following inequality holds

Where P＞0.

Lemma 4: If the matrix H ₂ is a real matrix, Then the following equations (i), (ii) and (iii) are equivalent:

Definition 1: Under zero initial condition δ(0) = 0, the K-type 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-type function.

Definition 2: For the error system, if the initial value is known and there exists a K-type function δ and a KL-type function ε, then under the event triggering condition (3), the following inequality holds:

||e(t)||<ε(||e(0)||,t)+δ(||d[0,t]||),

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

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

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 to a bounded state.

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 a Metzler matrix:

Next, we select the Lyapunov function V(t) = ξ ^T (t) Pξ(t) and take the time derivative of V(t) to obtain

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

According to the event triggering conditions, we can get:

Right now

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

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

in

So we can get:

From Definition 1, we can get the sufficient condition of the theorem, and the proof is complete.

Embodiment 2

This embodiment provides an interval observer, including:

Memory for storing computer programs;

The processor is used to implement the steps of the method for designing an interval observer of a robotic arm system based on an event trigger mechanism as described in the first embodiment when executing the computer program.

Embodiment 3

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

Computer-readable instructions, when a computer reads the computer instructions, enable the computer to execute the method for designing an interval observer of a robotic arm system based on an event trigger mechanism as described in Example 1.

Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code.

The present application is described with reference to the flowchart and/or block diagram of the method, device (system) and computer program product according to the embodiment of the present application. It should be understood that each process and/or box in the flowchart and/or block diagram, and the combination of the process and/or box in the flowchart and/or block diagram can be realized by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device produce a device for realizing the function specified in one process or multiple processes in the flowchart and/or one box or multiple boxes in the block diagram.

These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory produce a manufactured product including an instruction device that implements the functions specified in one or more processes in the flowchart and/or one or more boxes in the block diagram.

These computer program instructions may also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, whereby the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one or more processes in the flowchart and/or one or more boxes in the block diagram.

Obviously, the above embodiments are merely examples for clear explanation and are not intended to limit the implementation methods. For those skilled in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the implementation methods here. The obvious changes or modifications derived from these are still within the protection scope of the invention.

Claims (8)

1. A design method for an interval observer of a robotic arm system based on an event trigger mechanism, characterized in that: S1: According to the position and angular velocity of the joints of the robot arm, a mathematical model of the first robot arm system is established, and by state selection, a mathematical model of the second robot arm system is obtained; S2: setting event triggering conditions according to the mathematical model of the second robotic arm system; S3: According to the event triggering condition, based on the positive system theory and combined with the mathematical model of the second manipulator system, an interval observer of the manipulator system under the event triggering mechanism is constructed; The method of setting the event triggering condition according to the mathematical model of the second robotic arm system in S2 is: The event triggering conditions are: in, η is the event triggering time, is the sampling time of event triggering, y(t) is the actual output value, e _y (t) is the error value between the sampling time of event triggering and the actual output value; The method for constructing the interval observer of the manipulator system under the event triggering mechanism in S3 is: Determine whether the state matrix of the error system is a Metzler matrix; If it is a Metzler matrix, construct the 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 is transformed into a Metzler matrix, and a new vector is defined. The mathematical model of the second robotic arm system is transformed into the mathematical model of the third robotic arm system. According to the mathematical model of the third robotic arm system, the interval observer and error system under the event trigger mechanism are constructed. 2. The method for designing an interval observer of a manipulator system based on an event trigger mechanism according to claim 1, characterized in that the method for establishing the mathematical model of the first manipulator system in S1 is: The mathematical model of the first robotic arm system is: where q and are the position and angular velocity of the robot joint, M(q) is the inertia matrix, is the centrifugal force matrix, G(q) is the gravity matrix, u is the control input, and _τd is the system disturbance. 3. The method for designing an interval observer of a manipulator system based on an event trigger mechanism according to claim 1, characterized in that 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 The mathematical model expression of the second robotic arm system is obtained: 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 term, matrices A, B, C, D are constant matrices, and d is the interference term. 4. The design method of the interval observer of the manipulator system based on the event trigger mechanism according to claim 1 is characterized in that when the state matrix of the error system is a Metzler matrix, the method of constructing the interval observer and the error system under the event trigger mechanism is: Construct an upper bound observer and a lower bound observer. The upper bound observer always observes values greater than the actual state, and the lower bound observer always observes values less than the actual state. The initial conditions satisfy have Construct 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 of the lower bound observation function of a vector function, is the upper bound observation function of a vector function, The upper bound of a vector function is the derivative of the observation function, Construct an interval observer for the robotic arm system under the event trigger mechanism according to the event trigger conditions: Construct the error system: in, A-LC is the state matrix of the error system. 5. The method for designing an interval observer for a manipulator system based on an event trigger mechanism according to claim 1, characterized in that when the state matrix of the error system is not a 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 to make the state matrix of the error system 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 based on the mathematical model of the third robotic arm system: Construct the error system: in, is the state matrix of the error system. 6. The method for designing an interval observer for a manipulator system based on an event trigger mechanism according to claim 1, characterized in that before constructing the interval observer for the manipulator system under the event trigger mechanism, the method comprises: 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. 7. An interval observer, comprising: Memory for storing computer programs; A processor is used to implement the steps of a method for designing an interval observer for a robotic arm system based on an event trigger mechanism as described in any one of claims 1 to 6 when executing the computer program. 8. A computer-readable non-volatile storage medium, comprising: Computer-readable instructions, when a computer reads the computer-readable instructions, enable the computer to execute the design method of an interval observer for a robotic arm system based on an event-triggered mechanism as described in any one of claims 1-6.