Single-connecting-rod mechanical arm system modeling and double-asynchronous event trigger control design method

Abstract

The invention discloses a single-connecting-rod mechanical arm system modeling and double-asynchronous event trigger control design method. According to the method, random fluctuation of effective load mass and rotational inertia of a single-connecting-rod mechanical arm system, system interference and bandwidth limitation and time delay phenomena in a signal transmission process are considered, a single-connecting-rod mechanical arm model is established by utilizing a Semi-Markov theory, and the double-asynchronous event trigger control problem of the single-connecting-rod mechanical arm system is solved; the method comprises the following specific steps that firstly, random changes of effective load mass and rotational inertia are considered, and a model reflecting state changes of a single-connecting-rod mechanical arm system is established based on the Semi-Markov theory; then, designing an event generator and a controller which are asynchronous with the system modal information; furthermore, a Lyapunov function depending on the system mode and the residence time in the mode is constructed, stability analysis is conducted on the closed-loop single-connecting-rod mechanical arm system by means of a probability density function, and finally the obtained system random stability condition and the controller existence condition are expressed through a set of linear matrix inequality LMI.

Description

Technical Field

[0001] The present invention relates to a single-link robotic arm system technology, and in particular to a single-link robotic arm system modeling and dual asynchronous event trigger control design method, and more specifically to a single-link robotic arm system modeling and dual asynchronous event trigger control design method based on Semi-Markov theory. Background Art

[0002] The manipulator is a component of the entire robot system. It is a mechanical arm that operates under computer control. It is widely used in many industries such as transportation, aerospace, and industrial manufacturing. The manipulator can perform tasks that are highly repetitive or dangerous, which can reduce labor costs, improve production efficiency, and ensure the safety of the production process. However, the changes in the payload mass and moment of inertia in the single-link manipulator system are often accompanied by randomness and multimodality, making system modeling very difficult and the state difficult to control. At present, most of the stability analysis studies on single-link manipulator systems are based on Markov theory. The main tool for random stability analysis of single-link manipulator systems is Lyapunov stability theory. By constructing Lyapunov functions, the stability conditions and controller existence conditions of single-link manipulators based on linear matrix inequalities (LMI) are derived. However, in Markov theory, the dwell time of the mode has a memoryless exponential distribution, and the transfer rate is independent of time. This leads to a certain degree of conservatism in the system stability conditions and controller design schemes based on Markov theory. At the same time, the various components of the robotic arm or different robotic arms usually need to realize data transmission and remote control through the network, and are accompanied by external disturbances, time delays, data packet loss and other phenomena. This puts forward higher requirements on the mastery of modal information between the system, event generator and controller, the utilization of limited network resources and the realization of system stable control. Therefore, it is urgent to propose a more accurate modeling method and a less conservative stability criterion to ensure the effective control of the operating state of the single-link robotic arm system. Summary of the invention

[0003] The purpose of the present invention is to provide a single-link robotic arm system modeling and dual asynchronous event trigger control design method, which can achieve accurate and rapid judgment of whether the balance is leveled, narrow the error range, reduce human errors, and solve the problems existing in the above-mentioned prior art.

[0004] The present invention discloses a single-link robotic arm system modeling and dual asynchronous event-triggered control design method based on Semi-Markov theory, comprising the following steps:

[0005] (1) Considering the random changes of payload mass and moment of inertia during the modeling process of the single-link robotic arm system, a single-link robotic arm system model is established;

[0006]

[0007] In the formula, is the system state variable; Output for system control; It is the control input signal of the system; is the disturbance of the system; η(t) is the Semi-Markov jump process that describes the random changes of the payload mass and moment of inertia in the system; let η(t) = m, m = 1, 2, A m , B m , C m and D m is a system parameter matrix with appropriate dimensions.

[0008] (2) Design asynchronous event trigger conditions and controllers;

[0009] (2.1) Based on the Semi-Markov random process, a single-link robotic arm system is constructed, and the asynchronous event triggering conditions are designed as follows:

[0010] e T (t)Ω α(t) e(t)≤δ α(t) x T (t k d)Ω α(t) x(t k d)

[0011] In the formula, e(t) = x(t k d+cd)-x(t k d), x(t k d+cd) represents the current sampling data, x(t k d) indicates the latest trigger data, δ α(t) ∈[0,1) is a parameter related to the system mode, Ω α(t) >0 indicates the event trigger weight matrix that needs to be determined, and α(t) indicates the modal information actually received by the event trigger condition. Let α(t) = p, p = 1, 2, η(t) is asynchronous with α(t) and satisfies the conditional probability P r {α(t)=p|η(t)=m}=ψ mp ,0≤ψ mp ≤1,

[0012] The holding interval time of the zero-order holder is where t k+1 d=t kd+cd, Consider data delay in network transmission Where 0≤τ(t)≤τ M , the event triggering condition is transformed into the following form:

[0013] e T (t)Ω p e(t)≤δ p [x(t-τ(t))-e(t)] T Ω p [x(t-τ(t))-e(t)]

[0014] (2.2) Further, the asynchronous event trigger controller is designed as follows:

[0015] u(t)=K β(t) x(t k d)

[0016] In the formula, K β(t) is the controller gain to be determined, and β(t) represents the modal information actually received by the controller. Let β(t) = q, q = 1, 2, η(t) and β(t) be asynchronous and satisfy the conditional probability P r {β(t)=q|η(t)=m}=μ mq , 0≤μ mq ≤1,

[0017] (2.3) Combining the single-link robotic arm system model with the asynchronous event triggering conditions and controller, a closed-loop single-link robotic arm system model is obtained:

[0018]

[0019] (3) Construct a Lyapunov function that depends on the system modal changes and the dwell time within the mode; give the stability criterion of the single-link robotic arm system, and then realize the stability analysis of the single-link robotic arm system;

[0020] Consider a single-link robot system model. All conversion rates depend on the dwell time h. When the system mode changes, h is updated to 0. For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m , Q, R, Ω p , the positive constant δ p , τ M ,γ,matrix K q This makes the following matrix inequality hold:

[0021]

[0022] in,

[0023]

[0024] I represents the identity matrix, then the single-link robotic arm system is said to be stochastically stable and satisfies H ∞ Performance.

[0025] (4) Based on the derived stability criterion of the single-link robot system and related lemmas, the existence condition of the asynchronous controller is derived, and the controller gain that ensures the stability of the single-link robot system is obtained, and then the asynchronous controller is designed;

[0026] For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a set of positive definite matrices P with appropriate dimensions m , Z, Q, R, Ω p , matrix Y q , and the positive constant δ p , τ M , σ, ε1, ε2, so that the following inequality holds:

[0027]

[0028] in,

[0029]

[0030] The single-link robotic arm system is said to be stochastically stable and satisfies H ∞ performance, and determine the controller gain as: K q =OS -1 M -1 SO T Y q .

[0031] (5) Controller performance test;

[0032] The linear matrix inequality toolbox in Matlab is used to determine whether the system parameter matrix under all given modes satisfies the controller design conditions given in step (4). If so, it is determined that the single-link robot arm system is stochastically stable based on the designed asynchronous controller.

[0033] Beneficial effects: Compared with the prior art, the present invention has the following advantages: 1. By introducing the Semi-Markov process, the random changes of the payload mass and moment of inertia of the single-link robotic arm system can be described more accurately and reasonably; 2. By performing stability analysis and controller design on the established single-link robotic arm system model based on the Semi-Markov theory, the dual asynchronous event triggering control problem of the single-link robotic arm system can be solved. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0035] Figure 1 is a flow chart of the steps of the present invention;

[0036] Figure 2 It is a schematic diagram of a single-link robotic arm system;

[0037] Figure 3 It is a single-link robot system based on Semi-Markov theory, event triggering conditions, and mode change diagram of the controller;

[0038] Figure 4 It is the release time and release interval diagram of the event triggering condition;

[0039] Figure 5 It is the system state response diagram of the single-link manipulator closed-loop system based on Semi-Markov theory;

[0040] Figure 6 This is the system state response diagram of the single-link robotic arm open-loop system based on Semi-Markov theory. DETAILED DESCRIPTION

[0041] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0042] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] like Figure 1 As shown, a single-link robotic arm system modeling and dual asynchronous event-triggered control design method based on Semi-Markov theory includes the following steps:

[0044] (1) Modeling of single-link robotic arm system based on Semi-Markov theory

[0045] Combination Figure 2The single-link robotic arm system shown in the figure, the dynamic equation of the single-link robotic arm system can be described in the following form:

[0046]

[0047] Where θ(t) is the angular position of the manipulator, L is the length of the manipulator, M is the mass of the payload, g is the acceleration due to gravity, J is the moment of inertia, ν(t) is the viscous friction coefficient, and u(t) is the control input.

[0048] Considering the random variation of payload mass and moment of inertia, the single-link manipulator model is established through a Semi-Markov process with two modes. Based on the Semi-Markov theory, considering the existence of external disturbance ω(t), the state space model of the single-link manipulator system is established as follows:

[0049]

[0050] Where m (η(t) = m) represents the Semi-Markov process and takes values ​​of 1 and 2, L takes a value of 0.5 m, M1 takes a value of 0.12 kg, M2 takes a value of 0.39 kg, and g takes a value of 9.81 m / s 2 , ν(t)=ν(0) takes the value of 2kg·m 2 / s, J1 is 6.67kg·m 2 , J2 is 10kg·m 2 The variation of this Semi-Markov process is based on the following transition probabilities:

[0051]

[0052] In the formula, Furthermore, the system parameters can be expressed as:

[0053]

[0054] D1=D2=[0 1]

[0055] (2) Asynchronous event triggering conditions and controller design

[0056] For the single-link robotic arm system, the asynchronous event triggering conditions are designed as follows:

[0057] e T (t)Ω α(t) e(t)≤δ α(t) x T (t k d)Ω α(t) x(t kd) (3)

[0058] In the formula, e(t) = x(t k d+cd)-x(t k d), x(t k d+cd) represents the current sampling data, x(t k d) indicates the latest trigger data, δ α(t) ∈[0,1) is a parameter related to the system mode, Ω α(t) >0 indicates the event trigger weight matrix that needs to be determined, and α(t) indicates the modal information actually received by the event trigger condition. Let α(t) = p, p = 1, 2, η(t) is asynchronous with α(t) and satisfies the conditional probability P r {α(t)=p|η(t)=m}=ψ mp ,0≤ψ mp ≤1,

[0059] The holding interval time of the zero-order holder is where t k+1 d=t k d+cd, Consider data delay in network transmission Where 0≤τ(t)≤τ M , the event triggering condition is transformed into the following form:

[0060] e T (t)Ω p e(t)≤δ p [x(t-τ(t))-e(t)] T Ω p [x(t-τ(t))-e(t)] (4)

[0061] Furthermore, the asynchronous event triggering controller is designed as follows:

[0062] u(t)=K β(t) x(t k d) (5)

[0063] In the formula, K β(t) is the controller gain to be determined, and β(t) represents the modal information actually received by the controller. Let β(t) = q, q = 1, 2, η(t) and β(t) be asynchronous and satisfy the conditional probability P r {β(t)=q|η(t)=m}=μ mq , 0≤μ mq ≤1,

[0064] Combining the single-link robotic arm system model with the asynchronous event triggering conditions and controller, a closed-loop single-link robotic arm system model is obtained:

[0065]

[0066] (3) Stability criterion method for single-link robotic arm system

[0067] Criteria: Consider the single-link robot system model. All conversion rates depend on the dwell time h. When the system mode changes, h is updated to 0. For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m , Q, R, Ω p , the positive constant δ p , τ M ,γ,matrix K q This makes the following matrix inequality hold:

[0068]

[0069] in,

[0070]

[0071] I represents the identity matrix, * represents the symmetric terms in the matrix, then the single-link robotic arm system is said to be stochastically stable and satisfies H ∞ Performance.

[0072] Proof: For the single-link robot system model, choose a Lyapunov function that depends on the system mode change and the dwell time within the mode:

[0073]

[0074] Where P m is a positive definite matrix to be determined.

[0075] Then, we find the infinitesimal operator of equation (8) and use the cumulative distribution function and probability density function to get:

[0076]

[0077] By using Jensen's inequality, we can get:

[0078]

[0079] Considering the event triggering condition (4), we can get:

[0080]

[0081] In the formula, Based on formula (7), the system satisfies H ∞ Performance indicators.

[0082] When ω(t) = 0, based on formula (7), we can get:

[0083]

[0084] In the formula, matrix It can be deduced from formula (7). Then, it can be deduced:

[0085]

[0086] According to the Dynkin formula, we can get:

[0087]

[0088] Finally, when T approaches infinity, we can get:

[0089]

[0090] Get the certificate.

[0091] (4) Controller gain solution

[0092] First, the lemma used to solve the event-triggering related parameters and controller gains is given.

[0093] Lemma 1: Given real matrices A, B, C, X, W1, W2, W3 of appropriate dimensions, there exists a matrix P = P T >0 makes:

[0094]

[0095] The matrix P = P if and only if there exists a scalar σ > 0. T >0, the matrix Z is such that:

[0096]

[0097] Lemma 2: For a full-rank matrix B, rank(B) = n u , B∈ nx×nu , the singular value decomposition of B can be described as Among them, O·O T =I,V·V T =I. Matrix Z T >0,M∈ nx×nx , N∈ nu×nx , there exists a matrix Make If and only if the following conditions hold:

[0098]

[0099] Criteria: For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m , Z, Q, R, Ω p , matrix Y q , and the positive constant δ p , τ M , σ, ε1, ε2, so that the following inequality holds:

[0100]

[0101] in,

[0102]

[0103] The single-link robotic arm system is said to be stochastically stable and satisfies H ∞ performance, and the controller gain can be determined as: K q =OS -1 M -1 SO T Y q .

[0104] Proof: By Lemma 1, inequality (7) is equivalent to:

[0105]

[0106] in,

[0107]

[0108] Multiply both sides of inequality (13) by the matrix diag{I,I,I,I,I,I,Z,I}, considering the inequality Go to Processing Item-ZR -1 Z, definition According to Lemma 2, for exist Make Z T =OS -1 MSO T .make We can get: Then we can get (11) and (12). Proof.

[0109] Embodiments of the present invention are described below:

[0110] Single-link robotic arm system Figure 2 As shown, the modeled related system matrix is ​​given as follows:

[0111]

[0112] D1=D2=[0 1],τ M =0.01, δ1=0.1,δ2=0.2

[0113] ε1=ε2=0.5, φ 12 = φ 21 =0.5, σ=0.01

[0114] Set the perturbation to:

[0115]

[0116] Given the conditional probability matrix Γ = [ψ mp ] and Υ=[μ mq ]as follows:

[0117]

[0118] Based on the above parameters, the method of the present invention is used to perform simulation tests on the single-link robotic arm system. Figure 3 Describes the single-link robotic arm system based on Semi-Markov theory, event triggering conditions, and modal changes of the controller. Figure 4 Describes the release time and release interval of the event trigger scheme, Figure 5 The system state response of a single-link robotic arm closed-loop system based on Semi-Markov theory is depicted. Figure 6 The system state response of a single-link robotic arm open-loop system based on Semi-Markov theory is depicted.

[0119] The embodiments described above are only descriptions of the preferred modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A single-link robotic arm system modeling and dual asynchronous event-triggered control design method, characterized in that: The following steps are involved: Step S1: Establish a single-link robotic arm system model; Step S2: designing asynchronous event triggering conditions and controllers, and combining the single-link robotic arm system model to obtain a closed-loop single-link robotic arm system model; Step S3: Given a random stability condition of a single-link robotic arm system model; Step S4: solving the asynchronous controller gain and designing the asynchronous controller; Step S5: controller performance test.

2. A single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 1, characterized in that: The single-link robotic arm system model is established in step S1 as follows: In the formula, is the system state variable; Output for system control; It is the control input signal of the system; is the disturbance of the system; η(t) is the Semi-Markov jump process that describes the random changes of the payload mass and moment of inertia in the system; let η(t) = m, m = 1, 2, A m , B m , C m and D m is a system parameter matrix with appropriate dimensions.

3. The single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 1 is characterized in that: In step S2, the asynchronous event triggering conditions are designed as follows: e T (t)Ω α(t) e(t)≤δ α(t) x T (t k d)Ω α(t) x(t k d) In the formula, e(t) = x(t k d+cd)-x(t k d), x(t k d+cd) represents the current sampling data, x(t k d) indicates the latest trigger data, δ α(t) ∈[0,1) is a parameter related to the system mode, Ω α(t) >0 indicates the event trigger weight matrix to be determined, α(t) indicates the modal information actually received by the event trigger condition; let α(t) = p, p = 1, 2, η(t) is asynchronous with α(t) and satisfies the conditional probability P r {α(t)=p|η(t)=m}=ψ mp ,0≤ψ mp ≤1, The holding interval time of the zero-order holder is where t k+1 d=t k d+cd, Consider data delay in network transmission Where 0≤τ(t)≤τ M , the event triggering condition is transformed into the following form: e T (t)Ω p e(t)≤δ p [x(t-τ(t))-e(t)] T Ω p [x(t-τ(t))-e(t)]。 4. A single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 3, characterized in that: In step S2, the asynchronous event triggering controller is designed as follows: u(t)=K β(t) x(t k d) In the formula, K β(t) is the controller gain to be determined, β(t) represents the modal information actually received by the controller; let β(t) = q, q = 1, 2, η(t) and β(t) be asynchronous and satisfy the conditional probability P r {β(t)=q|η(t)=m}=μ mq , 0≤μ mq ≤1, 5. A single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 4, characterized in that: In step S2, the single-link robotic arm system model is combined with the asynchronous event triggering condition and the controller to obtain a closed-loop single-link robotic arm system model:

6. A single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 5, characterized in that: The step S3 is specifically as follows: Consider a single-link robot system model. All conversion rates depend on the dwell time h. When the system mode changes, h is updated to 0. For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a series of positive definite matrices P with appropriate dimensions m , Q, R, Ω p , the positive constant δ p , τ M ,γ,matrix K q This makes the following matrix inequality hold: in, I represents the identity matrix, * represents the symmetric terms in the matrix, then the single-link robotic arm system is said to be stochastically stable and satisfies H ∞ Performance.

7. A single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 6, characterized in that: The step S4 is specifically as follows: For all modes m = 1, 2, p = 1, 2, q = 1, 2, if there exists a set of positive definite matrices P with appropriate dimensions m , Z, Q, R, Ω p , matrix Y q , and the positive constant δ p , τ M , σ, ε1, ε2, so that the following inequality holds: in, The single-link robotic arm system is said to be stochastically stable and satisfies H ∞ performance, and determine the controller gain as: K q =OS -1 M - 1 SO T Y q .

8. The single-link robotic arm system modeling and dual asynchronous event-triggered control design method according to claim 7 is characterized in that: The step S5 is specifically as follows: The linear matrix inequality toolbox in Matlab is used to determine whether the system parameter matrix under all given modes satisfies the controller design conditions given in step S4. If so, it can be determined that the single-link robotic arm system is stochastically stable based on the designed asynchronous event trigger conditions and controller.

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