Truss structure distributed vibration control method based on topology communication network

Abstract

The invention discloses a truss structure distributed vibration control method based on a topological communication network, and the method comprises the following steps: 1, building a finite element model of a truss structure, carrying out the modal analysis of the truss structure, and obtaining the inherent attribute information of the truss structure; 2, performing dynamic analysis on the inherent attribute information of the truss structure by using a modal superposition method; 3, dividing subsystems for the truss structure, and defining a topology communication network among the subsystems; 4, establishing a distributed vibration control system by using a state space method according to the kinetic analysis result; and 5, designing a control law of the distributed vibration control system by adopting an LQR control algorithm. According to the invention, the stability of the control system can be greatly improved, and good working capability can be maintained during normal operation and fault.

Description

Technical Field

[0001] The present invention belongs to the technical field of intersection of aerospace and architecture, and in particular relates to a distributed vibration control method for a truss structure based on a topological communication network. Background Art

[0002] Space truss structures, with their light weight, high load-bearing efficiency, and strong scalability, are an effective solution for the large-scale, large-scale deployment of future spacecraft. With the rapid development of aerospace technology, the diversity and size of spacecraft structures are increasing, and the demand for large truss structures is expected to continue to grow. The vibration of spacecraft structures has always been a critical issue in the aerospace field. Due to the large size, low stiffness, low damping, and dense modal distribution of large truss structures, structural vibrations are difficult to rapidly attenuate. In practical engineering, structural vibration and structure-borne noise caused by truss structures are becoming increasingly common. Failure to effectively and promptly control the vibration of the structure will affect the pointing accuracy of the satellite payload and may even cause the payload to fail.

[0003] In the field of structural vibration control, structural vibration control methods include passive vibration control methods, active vibration control methods, and integrated active and passive vibration control methods. Passive vibration control methods do not require an additional energy source and have the advantages of simplicity, reliability, and low cost. However, due to material properties, passive vibration control methods can generally only suppress medium- and high-frequency vibrations of the structure. Active vibration control methods offer significant control effects and high control flexibility. Depending on the control system structure, active vibration control methods can be further divided into centralized, decentralized, and distributed vibration control methods. Currently, for active vibration control of structures, most researchers use traditional centralized vibration control methods. These methods require higher computational performance from individual control systems. In large spacecraft, if the central controller fails, the vibration control system of the large spacecraft structure will completely fail, directly affecting the performance of the spacecraft payload.

[0004] To address these issues, researchers have proposed decentralized vibration control methods. These methods divide engineering structures with large degrees of freedom into control regions based on their structural characteristics, creating numerous smaller subsystems. Each subsystem has its own independent and complete control loop, enabling it to relatively independently complete its control tasks. However, decentralized vibration control methods fail to account for information changes between subsystems. If the controllers of one or more subsystems fail, the control performance of the decentralized control system may be unsatisfactory, and may even affect the stability of the entire system. Distributed vibration control methods incorporate topological communication modules between subsystems. These subsystems communicate with each other through a given topological communication network, allowing all subsystems to work together to achieve the control objectives of the global system. This control method not only effectively reduces the computational cost of individual controllers but also offers excellent global stability. If the control of one or more subsystems in a distributed vibration control system fails, the overall structure can still be controlled through the control of other subsystems and topological communication between subsystems. Currently, research on distributed vibration control is largely in its infancy, with no systematic control methodology established. Most research focuses on simple structural models, and research and application of control to large truss structures is limited. Summary of the Invention

[0005] To address the above technical issues, this paper provides a distributed vibration control method for truss structures based on a topological communication network. Taking a truss structure as the research object, after completing the structural dynamics modeling, a topological communication network is designed based on the information interaction between subsystems. The distributed vibration control system is modeled using the state-space method, and a control law for distributed vibration control is developed. This method can be applied to vibration control simulations and experiments for large-scale engineering truss structures, effectively suppressing truss vibrations.

[0006] To achieve the above object, the present invention adopts the following technical solutions:

[0007] In a first aspect, the present invention provides a distributed vibration control method for a truss structure based on a topological communication network, comprising the following steps:

[0008] Step 1: Establish a finite element model of the truss structure, perform modal analysis on the truss structure, and obtain the inherent property information of the truss structure;

[0009] Step 2: Use the modal superposition method to perform dynamic analysis on the inherent property information of the truss structure;

[0010] Step 3: Divide the truss structure into subsystems and define the topological communication network between subsystems;

[0011] Step 4: Based on the dynamic analysis results, a distributed vibration control system is established using a state space method;

[0012] Step 5: Use the LQR control algorithm to design the control law of the distributed vibration control system.

[0013] In a second aspect, the present invention provides an electronic device comprising: one or more processors; a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned distributed vibration control method of a truss structure based on a topological communication network.

[0014] In a third aspect, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enables the processor to implement the aforementioned distributed vibration control method for a truss structure based on a topological communication network.

[0015] The beneficial effects of the present invention are:

[0016] On the basis of multiple subsystems, the present invention adds a topological communication network capable of information exchange, so that the subsystems can work together, and a good control effect can still be maintained when a controller fails, and the system stability is strong.

[0017] The present invention can be applied to control experiments of actual structures, and for relatively complex large-scale truss structures, the present invention has a good control effect and strong system stability when encountering faults.

[0018] This invention is applicable to a variety of truss-based structures and has broad applications in aerospace, construction, and other fields, demonstrating its significant practical value. This invention is particularly effective for structures such as space telescopes and truss-based satellites, which are difficult to maintain in real time. It significantly increases the stability of their control systems, maintaining good performance both during normal operation and during fault conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 It is a finite element model of a triangular prism truss structure;

[0020] Figure 2 This is a physical picture of the triangular prism truss built in the laboratory;

[0021] Figure 3 It is a schematic diagram of distributed vibration control of triangular prism truss;

[0022] Figure 4 This is a schematic diagram of a distributed control system built using Simulink;

[0023] Figure 5 This is a comparison chart between distributed control experiments and simulations;

[0024] Figure 6 It is the displacement curve diagram of the normal operation of the control system;

[0025] Figure 7 It is the displacement curve of partial controller failure. DETAILED DESCRIPTION

[0026] The present invention will be further described below with reference to the accompanying drawings and examples.

[0027] This paper proposes a distributed vibration control method for truss structures based on a topological communication network. By establishing communication relationships between subsystems, this method addresses the issues of poor system stability and the inability to effectively suppress truss vibrations in the event of a fault in traditional centralized / distributed control methods. Specifically, the method includes the following steps:

[0028] Step 1: Establish a finite element model of the truss structure, perform modal analysis on the truss structure, and obtain the inherent property information of the truss structure;

[0029] Step 2: Use the modal superposition method to perform dynamic analysis on the inherent property information of the truss structure;

[0030] Step 3: Divide the truss structure into subsystems and define the topological communication network between subsystems;

[0031] Step 4: Based on the dynamic analysis results, a distributed vibration control system is established using the state space method;

[0032] Step 5: Use the LQR control algorithm to design the control law of the distributed vibration control system .

[0033] Furthermore, based on the analysis results of step one and the design methods of steps two to five, a distributed control system is built in Simulink to perform control simulation or experiments on the target truss.

[0034] Wherein, in step 1, the inherent attribute information includes the mass matrix, stiffness matrix, damping matrix, eigenvector matrix and each order modal frequency of the truss structure.

[0035] Among them, step 2 includes: assuming the number of degrees of freedom of the truss structure model is , the number of actuator rods is , the dynamic equation of the truss structure considering active control force is:

[0036] ,

[0037] in, is the mass matrix; is the stiffness matrix; is the damping matrix; is the displacement of the truss structure node; is the position matrix of the actuator; is the control force of the actuator, the superscript · represents the first-order time derivative, and the superscript ·· represents the second-order time derivative;

[0038] Performing modal analysis on the truss structure can obtain the structure's The non-rigid mode. At this time, the displacement of the truss structure node and modal displacement The relationship is:

[0039] ,

[0040] in, Before the truss structure model The eigenvector matrix composed of the eigenvectors of the order mode; is the modal displacement in the modal coordinate system.

[0041] Therefore, the dynamic equation of the truss structure with active control force is converted to:

[0042] ,

[0043] Multiply both sides by the left ,have to:

[0044] ,

[0045] in:

[0046] ,

[0047] The superscript T represents transposition. is the modal mass matrix of the structure; is the modal stiffness matrix of the structure; is the modal damping matrix of the structure; I represents the unit matrix; is the i-th order modal stiffness of the structure; , is the frequency of the i-th mode shape.

[0048] At this point, the dynamic model of the truss structure in the modal coordinate system was obtained through the modal superposition method.

[0049] Wherein, step three includes:

[0050] Divide the entire truss structure into multiple subsystems based on the layout of the actuator rods. In this step, generally adhere to the following principles: each subsystem contains at least one actuator rod and one sensor; all subsystems contain the same number of actuator rods and sensors; and the rods within each subsystem are interconnected, with no crossover between subsystems.

[0051] After the subsystem division is completed, the topological communication network between subsystems is defined according to the information interaction requirements between subsystems. The topological communication network consists of two parts: is the topological communication relationship matrix of the distributed vibration control system, indicating whether each subsystem communicates with each other. Its expression is:

[0052] ,

[0053] Its row blocks is the topological communication relationship matrix of each subsystem, Representation and Subsystem The number of subsystems that have communication relationships. When receiving information from a subsystem, The column matrix where the subsystem is located is , otherwise The number of topological communication relationships depends on the subsystem. The subsystem will have possible topological communication relationships.

[0054] is the topological communication weight coefficient matrix of the distributed vibration control system, which represents the weight of the communication signal between each subsystem. Its diagonal blocks are the topological communication weight coefficient matrices of each subsystem. , whose expression is:

[0055] ,

[0056] Set up separately and After that, the product That is, it is an assembled topological communication network that can simultaneously characterize the presence and strength of communication between subsystems.

[0057] Wherein, step four includes:

[0058] Assume that the number of degrees of freedom of the truss structure model is , the number of actuator rods is . It is divided into subsystems, and the number of actuators in each subsystem is , the structural dynamics equation considering active control force is:

[0059] ,

[0060] in, is the position matrix of the i-th actuator located in the i-th structure; is the control force of the actuator of part i.

[0061] The structural dynamics equation of the truss structure considering the active control force is written in the form of a state space equation, and the state equation of the distributed vibration control system is obtained:

[0062] ,

[0063] in:

[0064] ,

[0065] in, is the state matrix of the control system; is the control matrix of the control system; is the state quantity of the control system; is the control quantity of the control system.

[0066] Assume the number of sensors in the control system is , the sensor position matrix is , the number of sensors in each subsystem is , the position matrix of each subsystem sensor is The observation equation of the distributed vibration control system can be obtained:

[0067] ,

[0068] in,

[0069] ,

[0070] in, is the observation matrix of the control system; is the measurement quantity of the control system.

[0071] According to the idea of distributed vibration control and the definition of observation equation, the control quantity of distributed vibration control system can be obtained The detailed expression is as follows:

[0072] ,

[0073] Control volume By independent control and additional control It consists of two parts: the independent control quantity is the control feedback of each subsystem itself, and the additional control quantity is the control feedback obtained by each subsystem after topological communication with other subsystems through the topological communication network.

[0074] ,

[0075] in, It is the measurement information feedback matrix of the distributed vibration control system; is the measurement information feedback matrix of the i-th subsystem of the distributed vibration control system, and Already defined in step 3.

[0076] The control quantity of the control system The expression is brought into the state equation of the control system, and the dynamic response of the truss structure under distributed vibration control is obtained by solving the state equation of the control system.

[0077] Among them, step five includes:

[0078] Assume that the number of degrees of freedom of the truss structure model is , the modal analysis order is , the number of actuator rods is , the number of subsystems is The structural dynamic equation of the truss structure in the modal coordinate system is:

[0079] ,

[0080] in, is the mass matrix; is the stiffness matrix; is the damping matrix; Before the structure The eigenvector matrix composed of the eigenvectors of the order mode; is the modal displacement in the modal coordinate system; is the position matrix of the actuator; It is the control force of the actuator.

[0081] Will 、 and Divide into blocks in the form of a matrix piece,

[0082] The structural dynamic equation of the block matrix of the truss structure in the modal coordinate system is:

[0083] ,

[0084] in, is the position matrix of the i-th actuator located in the i-th structure; is the control force of the actuator rod of part i; is the eigenvector matrix of the i-th part.

[0085] set up , ,get:

[0086] ,

[0087] Separate each line of the formula to get Independent dynamic equations:

[0088] ,

[0089] set up , At this time, The ith independent dynamic equation among the ith independent dynamic equations is written in the form of state space, and the state equation of the ith subsystem of the distributed vibration control system is obtained:

[0090] ,

[0091] in:

[0092] ,

[0093] ,

[0094] is the state matrix of the i-th subsystem; is the control matrix of the i-th subsystem; is the state quantity of the i-th subsystem; is the control variable of the ith subsystem.

[0095] Assume that the measurement information of all subsystems of the distributed vibration control system is node displacement and velocity, and the number of sensors in the i-th subsystem is , the sensor position matrix is The observation equation of the i-th subsystem of the distributed vibration control system can be obtained:

[0096] ,

[0097] in:

[0098] ,

[0099] in, is the observation matrix of the i-th subsystem; is the measurement quantity of the ith subsystem.

[0100] Assume that the quadratic performance index of the i-th subsystem of the distributed vibration control system is:

[0101] ,

[0102] in, and is the positive definite weight coefficient matrix.

[0103] make:

[0104] , ,

[0105] in, is the LQR control weight coefficient.

[0106] but:

[0107] ,

[0108] The LQR control algorithm can be used to calculate the control gain matrix of the i-th subsystem of the distributed vibration control system: .

[0109] The relationship between the independent control quantity and measurement quantity of the i-th subsystem of the distributed vibration control system is: After considering the topological communication network, the i-th subsystem of the distributed vibration control system will add additional control quantity. The additional control quantity of the i-th subsystem is defined as .in, is the topological communication weight coefficient of the ith subsystem. At this time, the independent control quantity of the ith subsystem is added to the additional control quantity, and the relationship between the total control quantity and the measurement quantity of the ith subsystem can be obtained. .

[0110] After that, write code in MATLAB, extract structural property information, input control parameters, and then build the control loop in Simulink.

[0111] Example

[0112] The following is a scheme for performing distributed vibration control on a triangular prism truss structure using the present invention.

[0113] In the finite element software, Figure 1 The finite element model of a triangular prism truss structure shown in Figure 1 consists of 43 nodes and 84 rods, three of which are actuators. The three bottom nodes are fixed to the coordinate system, and a counterweight is located at the top intermediate node 43. Modal analysis of the structure was performed, and structural property information was read using a custom MATLAB program to complete the dynamic modeling of the triangular prism truss.

[0114] In the laboratory, we set up Figure 2The triangular prism truss shown in the figure is used for simulation and experimental comparison to verify the correctness of the proposed method. The truss structure is constructed using 45-gauge steel and has a total height of 2.08 meters. The mass of each node is 0.03407 kg, and the top counterweight weighs 1.314 kg. Its first-order mode is bending, with a fundamental frequency of 1.4317 Hz.

[0115] The triangular prism truss is divided into three subsystems along its height. Each subsystem is equipped with an actuator and a sensor. The actuator replaces the original rod after installation, and the sensor is installed at the node to measure the node's x-direction displacement. In this scheme, the topological communication relationship between subsystems adopts the same method as adjacent subsystems communicating with each other. That is, subsystem 1 only receives information from subsystem 2, subsystem 2 receives information from subsystems 1 and 3, and subsystem 3 only receives information from subsystem 2.

[0116] According to the topological communication relationship of the three subsystems, the topological communication relationship matrix of the control system is:

[0117] ,

[0118] The topological communication weight coefficient matrix of the three subsystems is set as follows:

[0119] ,

[0120] Schematic diagram of distributed vibration control of triangular prism truss Figure 3 shown.

[0121] Using the control system design method of this method, a control simulation model is built in Simulink. The schematic diagram after the construction is completed is as follows: Figure 4 As shown in Figure 1. The state-space system defines the initial state of the structure and the transfer relationship between the control variable and the sensor observation. The observation is output by the state-space system. It is discretized and converted into a state variable, which is multiplied by the calculated feedback gain to obtain an independent control variable. Furthermore, an additional control variable is calculated through the topological communication network. The sum of these two components constitutes the total control variable for the system, which is then input into the state space. After the simulation is complete, the displacement curve of the target node can be observed on an oscilloscope connected to the state-space output to determine the effectiveness of the control.

[0122] An initial external force of about -3N is applied to the 43rd node of the truss. After the triangular prism truss reaches a stable state, the initial external force is unloaded and the control system starts working. At this time, the x-direction displacement of the node 43 in the simulation and experiment is as follows: Figure 5 The displacement of node 43 converges quickly, and the simulation and experimental results are basically consistent, which verifies the effectiveness of this method.

[0123] Under the condition that the control system is working normally, a pulse excitation signal with a peak value of 1N and a duration of 0.2s is applied to the x direction of node 43. The centralized, decentralized and distributed vibration control systems of the present invention are used for simulation respectively. The displacement response of the node is as follows: Figure 6 As shown in Figure 3, all three vibration control methods can effectively suppress the vibration of the triangular prism truss structure. The distributed vibration control method can achieve a similar control effect as the centralized vibration control method.

[0124] When one controller in the control system fails and cannot provide gain, a pulse excitation signal with a peak value of 1N and a duration of 0.2s is applied to the x direction of node 43. The simulation is performed using centralized, decentralized and distributed vibration control systems of the present invention respectively. The displacement response of the node is as follows: Figure 7 In this case, the centralized control method cannot control the structure, and the decentralized control has poor control effect. However, the distributed control of the present invention can effectively suppress the vibration of the truss, and the control effect is significantly better than other methods.

[0125] In a second aspect, the present invention provides an electronic device comprising: one or more processors; a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned distributed vibration control method of a truss structure based on a topological communication network.

[0126] In a third aspect, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enables the processor to implement the aforementioned distributed vibration control method for a truss structure based on a topological communication network.

[0127] The specific embodiments described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above are only specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A distributed vibration control method for a truss structure based on a topological communication network, characterized by: The steps include: Step 1: Establish a finite element model of the truss structure, perform modal analysis on the truss structure, and obtain the inherent property information of the truss structure; Step 2: Use the modal superposition method to perform dynamic analysis on the inherent property information of the truss structure; Step 3: Divide the truss structure into subsystems and define the topological communication network between subsystems; Step 4: Based on the dynamic analysis results, a distributed vibration control system is established using a state space method; Step 5: Use the LQR control algorithm to design the control law of the distributed vibration control system.

2. The distributed vibration control method for a truss structure based on a topological communication network according to claim 1, characterized in that: In the step 1, the inherent attribute information includes the mass matrix, stiffness matrix, damping matrix, eigenvector matrix and modal frequencies of each order of the truss structure.

3. The distributed vibration control method for a truss structure based on a topological communication network according to claim 1, characterized in that: The second step includes: Assume that the number of degrees of freedom of the truss structure model is , the number of actuator rods is , the dynamic equation of the truss structure considering active control force is: ， in, is the mass matrix; is the stiffness matrix; is the damping matrix; is the displacement of the truss structure node; is the position matrix of the actuator; is the control force of the actuator, the superscript · represents the first-order time derivative, and the superscript ·· represents the second-order time derivative; Perform modal analysis on the truss structure and obtain the front The non-rigid mode, at this time, the displacement of the truss structure node and modal displacement The relationship is: ， in, For the truss structure model The eigenvector matrix composed of the eigenvectors of the order mode; is the modal displacement in the modal coordinate system; Therefore, the dynamic equation of the truss structure with active control force is converted to: ， Multiply both sides by the left ,have to: ， in: ， The superscript T represents transposition. is the modal mass matrix of the structure; is the modal stiffness matrix of the structure; is the modal damping matrix of the structure, I represents the unit matrix; is the i-th order modal stiffness of the structure; , is the frequency of the i-th mode shape.

4. The distributed vibration control method for a truss structure based on a topological communication network according to claim 1, characterized in that: The step three includes: Based on the layout of the actuator rods, the entire truss structure is divided into multiple subsystems to construct a distributed vibration control system. Each subsystem contains at least one actuator rod and one sensor, and each subsystem contains the same number of actuator rods and sensors. The actuator rods in each subsystem are interconnected, and there is no crossover between subsystems. After the subsystem division is completed, the topological communication network between subsystems is defined according to the information interaction requirements between subsystems. The topological communication network consists of two parts, among which: is the topological communication relationship matrix of the distributed vibration control system, indicating whether each subsystem communicates; is the topological communication weight coefficient matrix of the distributed vibration control system, which represents the weight of the communication signal between each subsystem; Set up separately and After that, the product That is the assembled topological communication network.

5. The distributed vibration control method for truss structures based on a topological communication network according to claim 1, characterized in that: The fourth step includes: Assume that the number of degrees of freedom of the truss structure model is , the number of actuator rods is , which is divided into subsystems, and the number of actuators in each subsystem is , the structural dynamics equation of the truss structure considering the active control force is written in the form of a state space equation, and the state equation of the distributed vibration control system is obtained: ， in, is the state matrix of the control system; is the control matrix of the control system; is the state quantity of the control system; is the control quantity of the control system; Assume the number of sensors in the control system is , the sensor position matrix is , divide the sensors into p groups, and the number of sensors in each group is , the position matrix of each group of sensors is , the observation equation of the distributed vibration control system is obtained: ， in, is the observation matrix of the control system; is the measurement quantity of the control system; According to the idea of distributed vibration control and the observation equation, the control quantity of the distributed vibration control system is obtained ; The control quantity of the control system The expression is brought into the state equation of the control system, and the dynamic response of the truss structure under distributed vibration control is obtained by solving the state equation of the control system.

6. The distributed vibration control method for truss structures based on a topological communication network according to claim 5, characterized in that: The control quantity of the distributed vibration control system for: ， Where, represents an independent control quantity, Indicates the additional control amount, It is the measurement information feedback matrix of the distributed vibration control system; ， in, is the measurement information feedback matrix of the i-th subsystem of the distributed vibration control system.

7. The distributed vibration control method for truss structures based on a topological communication network according to claim 6, characterized in that: The step five includes: Assume that the number of degrees of freedom of the truss structure model is , the modal analysis order is , the number of actuator rods is , the number of subsystems is , the structural dynamics equation of the truss structure in the modal coordinate system is: ， in, is the mass matrix; is the stiffness matrix; is the damping matrix; Before the structure The eigenvector matrix composed of the eigenvectors of the order mode; is the modal displacement in the modal coordinate system; is the position matrix of the actuator; is the control force of the actuator; Divide the matrix in the formula into blocks and split it by row to obtain the state equation and observation equation of the i-th subsystem of the distributed vibration control system: ， ， is the state matrix of the i-th subsystem; is the control matrix of the i-th subsystem; is the state quantity of the i-th subsystem; is the control quantity of the ith subsystem, is the observation matrix of the i-th subsystem; is the measurement quantity of the i-th subsystem; The quadratic performance index of the i-th subsystem is: ， in, and is the positive definite weight coefficient matrix; According to the calculation results of the quadratic performance index, the LQR control algorithm is used to calculate the control gain matrix of the i-th subsystem of the distributed vibration control system. , add the independent control quantity of the ith subsystem and the additional control quantity to obtain the relationship between the total control quantity and the measurement quantity of the ith subsystem, .

8. The distributed vibration control method for truss structures based on a topological communication network according to claim 1, characterized in that: The method further comprises extracting structural intrinsic attribute information by writing code in MATLAB, inputting the topological communication network and the positive definite weight coefficient matrix 、 , and then build a control loop in Simulink to realize system simulation.

9. An electronic device, characterized in that: include: one or more processors; a memory for storing one or more programs; When one or more programs are executed by the one or more processors, the one or more processors implement the distributed vibration control method for a truss structure based on a topological communication network as described in any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that Executable instructions are stored thereon, and when the instructions are executed by the processor, the processor can implement the distributed vibration control method of a truss structure based on a topological communication network as described in any one of claims 1 to 8.

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