A method for modeling dynamics of a flexible variable topology mechanism based on constraint analysis

Abstract

The application discloses a flexible variable topology mechanism dynamics modeling method based on constraint analysis, and mainly comprises the following steps: firstly, numbering the components of the flexible variable topology mechanism, establishing a topology graph by analyzing the topology sequence of the flexible variable topology mechanism, refining the adjacency matrix under different topology structures and performing logical operation, and extracting the basic constraint and conditional constraint in the flexible variable topology mechanism; secondly, removing the constraint of all components in the flexible variable topology mechanism, establishing the unconstrained dynamics equation of the flexible variable topology mechanism, introducing the Jacobian matrix of the basic constraint equation and the conditional constraint equation by using the Lagrange multiplier method on the basis, and establishing the dynamics equation of the flexible variable topology mechanism which can describe different topology structures; finally, analyzing the change of the constraint condition of the flexible component caused by the change of the topology structure, constructing a compatibility equation, modifying the dynamics equation of the flexible variable topology mechanism, and making the floating coordinate system of the flexible component adapt to the state before and after the change of the topology structure, so that the dynamics modeling method has good adaptability.

Description

Technical Field

[0001] This invention relates to a dynamic modeling method for flexible variable topology mechanisms based on constraint analysis. This method, grounded in topology and multibody dynamics theory, proposes a unified dynamic modeling framework for flexible variable topology mechanisms. It utilizes the Lagrange multiplier method to introduce refined basic and conditional constraints into the unconstrained dynamic equations of the flexible variable topology mechanism, thereby establishing a dynamic model that can describe different topologies. This modeling method analyzes constraints during topology changes, making the modeling process clearer and exhibiting good adaptability to modeling any type of flexible variable topology mechanism. Background Technology

[0002] Variable topology mechanisms are a new type of mechanism that can actively change its topology according to environmental characteristics and functional requirements, with the topology changing at least once during motion. With the development of aerospace technology, various aerospace missions are becoming increasingly complex. Due to their reconfigurable characteristics, variable topology mechanisms exhibit good adaptability in more complex and variable working environments, and are therefore widely used in important structures such as large satellite antennas, truss deployment mechanisms, and space robotic arms. However, as aerospace mechanisms continue to trend towards lightweight and large-scale designs, variable topology mechanisms applied in the aerospace field inevitably possess large-scale and high-flexibility structural characteristics. This means that large-scale flexible components in aerospace mechanisms, in addition to experiencing rigid movement and rotation, also undergo non-negligible flexible deformation. Rigid motion and deformation motion interact, generating strong coupling, leading to the mechanism becoming a time-varying, highly nonlinear complex system. Furthermore, when the topology of the mechanism changes, the boundary conditions and constraint relationships in the flexible variable topology mechanism also change. These problems make the dynamic modeling of flexible variable topology mechanisms extremely challenging. Therefore, how to analyze the topology and establish a dynamic model for flexible variable topology mechanisms has become a key issue of concern in the field of mechanism science.

[0003] Currently, significant progress has been made in modeling rigid variable topology mechanisms, particularly in the areas of structural composition principles, topological representation, and motion analysis, yielding fruitful results. However, modeling flexible variable topology mechanisms presents two major challenges: topological changes and flexible deformation. This makes the dynamic modeling process more complex and cumbersome. Furthermore, flexible variable topology mechanisms involve multiple topological structures during operation, making the construction of dynamic models capable of describing various topological structures a crucial issue that urgently needs to be addressed. Therefore, this invention proposes a constraint analysis-based dynamic modeling method for flexible variable topology mechanisms. First, through topological sequence analysis, the basic constraints and conditional constraints of the flexible variable topology mechanism are extracted. Second, all constraints in the flexible variable topology mechanism are removed, establishing an unconstrained dynamic model. Based on this, the Lagrange multiplier method is used to introduce the basic constraints into the unconstrained dynamic model, establishing a basic constraint dynamic model that remains unchanged throughout the entire motion of the flexible variable topology mechanism. Further, for different topological structures, different conditional constraints are added to the basic constraint dynamic model, establishing a dynamic model capable of describing any different topological structures. Finally, the changes in the conditional constraints on the flexible components are analyzed when the topology of the flexible variable topology mechanism changes. A motion compatibility equation is established, and the dynamic equation of the flexible variable topology mechanism is modified so that the floating coordinate system of the flexible components can adapt to the state before and after the topology change. Summary of the Invention

[0004] The technical problem this invention addresses is the shortcomings of existing technologies. To streamline the modeling process of flexible variable topology mechanisms, it proposes a constraint analysis-based dynamic modeling method for such mechanisms. This method, based on topology and multibody system dynamics theory, proposes a unified framework for dynamic modeling of flexible variable topology mechanisms, enabling the establishment of dynamic models that can describe different topologies. First, the invention analyzes the topological sequence of the flexible variable topology mechanism, extracting its fundamental and conditional constraints. Second, all constraints on the components are removed, establishing an unconstrained dynamic model of the flexible variable topology mechanism. Based on this, the Lagrange multiplier method is used to introduce the fundamental and conditional constraints into the dynamic equations, establishing a dynamic model that can describe all topologies. Finally, by analyzing the compatibility conditions of the flexible components, compatibility equations for the flexible components are established when the topology changes, thus refining the dynamic model of the flexible variable topology mechanism. This invention can be used for dynamic modeling of any flexible variable topology mechanism, with a streamlined modeling process and good adaptability.

[0005] This invention relates to a dynamic modeling method for flexible variable topology mechanisms based on constraint analysis. The basic principles and specific steps of this method are as follows:

[0006] This invention analyzes the topological sequence of flexible variable topology mechanisms to obtain adjacency matrices under different topological structures, and then extracts the basic constraints and conditional constraints in the flexible variable topology mechanisms. Based on the Lagrange multiplier method, the basic constraints and conditional constraints are introduced into the dynamic equations of unconstrained flexible variable topology mechanisms, thereby establishing a dynamic model of flexible variable topology mechanisms that can describe arbitrary topological structures.

[0007] Step 1: Make reasonable assumptions about the model. Since the mass of the joint is much smaller than the mass of the component, the mass of the joint is ignored, and it is assumed that the topological change process of the flexible variable topology mechanism is smooth.

[0008] Step 2: Analyze the different topologies of the flexible variable topology mechanism and number all components that appear during the entire working process.

[0009] Step 3: Establish a topology graph, analyze the topology sequence, and construct the adjacency matrix of different topologies of the flexible variable topology mechanism.

[0010] Step 4: Perform matrix operations on the adjacency matrices under different topologies to extract the basic constraints and conditional constraints of the flexible variable topology mechanism.

[0011] Step 5: Analyze the deformation in the sensitive direction of the flexible component and discretize the flexible component.

[0012] Step 6: Remove the constraints on all components in the flexible variable topology mechanism and establish the dynamic equations of the flexible variable topology mechanism in the unconstrained state.

[0013] Step 7: Using the Lagrange multiplier method, establish the dynamic equations of the flexible variable topology mechanism with basic constraints.

[0014] Step 8: Based on the dynamic equations of the flexible variable topology mechanism with basic constraints, add conditional constraints under different topologies to establish dynamic equations for different topologies.

[0015] Step 9: Analyze the compatibility conditions of the flexible components before and after the constraint changes, establish motion compatibility equations, and describe the relationship between the elements before and after the change in the system topology.

[0016] Preferably, for large-scale and highly flexible variable topology mechanisms, the size and mass of the components are much larger than the size and mass of the joints. Therefore, the mass of the joints is ignored in the modeling process of flexible variable topology mechanisms. In addition, in order to avoid collisions and impacts that could damage large-scale components, most flexible variable topology mechanisms operate smoothly during topological changes. Therefore, it is assumed that the topological change process of flexible variable topology mechanisms proceeds smoothly.

[0017] Preferably, since the flexible variable topology mechanism undergoes multiple topological changes during operation, the number and connection methods of components may change under different topological structures. Therefore, it is necessary to analyze the different topological structures of the flexible variable topology mechanism, extract all components that appear in the entire operation process, and number each component accordingly in order to analyze the topological sequence of the flexible variable topology mechanism and lay the foundation for subsequent dynamic modeling.

[0018] Preferably, the elements in the adjacency matrix can indicate the adjacency relationships between components in the flexible variable topology mechanism, that is, whether there are constraints between components. Therefore, by analyzing the topology sequence and establishing the adjacency matrix under different topologies of the flexible variable topology mechanism, the existence of constraints between components under different topologies can be obtained. The elements of the adjacency matrix of the flexible variable topology mechanism can be represented as follows:

[0019]

[0020] In the formula, D ij This represents the element in the i-th row and j-th column of the adjacency matrix of the flexible variable topology mechanism.

[0021] Preferably, logical operations are performed on the elements in the adjacency matrix under different topologies to extract the basic constraint adjacency matrix and condition constraint adjacency matrix of the flexible variable topology mechanism. Further, the basic constraints that remain unchanged during the overall motion of the flexible variable topology mechanism and the condition constraints that change with the topology can be obtained.

[0022] Preferably, since flexible components have an infinite number of degrees of freedom, it is difficult to obtain an exact solution to their dynamics problem. Therefore, it is necessary to analyze the different characteristics of the flexible component, simplify it appropriately to obtain the deformation in the sensitive direction of the component, and then discretize it. Let the deformation vector in the sensitive direction of the flexible component be:

[0023] u f = Φq f (2)

[0024] In the formula, u f This represents the deformation vector in the sensitive direction of the component. Φ =[Φ1 … Φ N ] represents the modal vector matrix. q f =[q f1 … q fN ] T denoted as modal coordinate array, where N indicates that the vibration of the flexible component is retained up to the Nth mode.

[0025] Preferably, by performing kinematic analysis on any flexible component, the dynamic equations for the free motion of a single flexible component are established. Based on this, the unconstrained dynamic equations for a flexible variable topology mechanism containing all components are established:

[0026]

[0027] In the formula, M For the generalized mass matrix, For generalized coordinate arrays, K Generalized stiffness matrix, Q F Q is the generalized active force vector. v It is the generalized force vector of the second-order velocity term.

[0028] Preferably, based on the unconstrained dynamic equations of the flexible variable topology mechanism, and according to the Lagrange multiplier method, the dynamic equations of the flexible variable topology mechanism with basic constraints are established by introducing the basic constraint equations and their Jacobian matrices extracted in step four:

[0029]

[0030] In the formula, C b (q,t) represents the basic constraint equations of the flexible variable topology mechanism. C bq Let λ represent the Jacobian matrix of the fundamental constraint equations. b This represents the Lagrange multiplier vector of the fundamental constraint equations.

[0031] Preferably, based on the dynamic equations of the flexible variable topology mechanism with basic constraints, similar to step eight, the Lagrange multiplier method is used to introduce the conditional constraint equations and their Jacobian matrices under different topologies, thus establishing dynamic equations for the flexible variable topology mechanism that can describe different topologies:

[0032]

[0033] In the formula, The conditional constraint equations represent the flexible variable topology mechanism. Represents the Jacobian matrix of the conditional constraint equations. Let ξ be the Lagrange multiplier vector representing the conditional constraint equations, and let ξ represent the flexible variable topology mechanism under the topology ξ, with time t∈(t ξ-1 ,t ξ ] represents the time interval under the topology ξ of the flexible variable topology mechanism.

[0034] Preferably, because the topology changes during the movement of the flexible variable topology mechanism, the constraint relationships of the flexible components change, thus altering the boundary conditions of individual flexible components. Therefore, the floating coordinate system of the flexible component before the topology change is no longer suitable for the changed state. To make the floating coordinate system of the flexible component adapt to the changed topology, a motion compatibility equation is established to describe the relationships between elements of the system's spatial components when the constraint relationships change. It is assumed that the operating time domain of the flexible variable topology mechanism is T = [t0, t...]. e This includes n topologies, requiring n-1 topological changes. Therefore, the time domain T contains t1, ..., t... ξ ,…,t n-1 One topological time; take any topological time t. ξ Perform compatibility analysis, at t ξ Within its smallest neighborhood, that is:

[0035]

[0036] In the formula,

[0037]

[0038] At the topological change time t ξ Within the neighborhood of , the position compatibility equation and velocity compatibility equation of the flexible variable topology mechanism are respectively expressed as:

[0039]

[0040]

[0041] In the formula, This represents the position vector of the floating coordinate system in the inertial coordinate system after the topological change. This represents the angle vector of the floating coordinate system relative to the inertial coordinate system after the topological changes. This represents the velocity vector of the floating coordinate system in the inertial coordinate system after the topological change. This represents the angular velocity vector of the floating coordinate system relative to the inertial coordinate system after the topology changes;

[0042] Solutions to the compatibility equations of equations (59) and (60) and It can uniquely determine the position, direction and velocity vector of the floating coordinate system in the inertial frame after the topology changes, thereby ensuring the continuity and stability of the system topology changes; combined with equations (47), (59) and (60), they together constitute the dynamic equation of the flexible variable topology mechanism.

[0043] The advantages of this invention compared to the prior art are:

[0044] Dynamic modeling of flexible variable topology mechanisms integrates two major challenges: topological changes and flexible deformation, making the modeling process more cumbersome and complex. Therefore, there is an urgent need to construct a dynamic modeling framework that can describe the different topological structures of flexible variable topology mechanisms. This invention addresses the current insufficient research on modeling methods for flexible variable topology mechanisms by proposing a constraint analysis-based dynamic modeling method. This method, based on the topological graph method in topology, analyzes the topological sequence of the flexible variable topology mechanism under different topological structures, obtaining an adjacency matrix containing all components. Through logical operation analysis of the adjacency matrix, the basic constraints and conditional constraints of the flexible variable topology mechanism are extracted, transforming the description of topological changes into a description of constraint changes, making the modeling process clearer. By discretizing the flexible components in the flexible variable topology mechanism, the flexible deformation motion is introduced into the dynamic equations, and a dynamic model of the flexible variable topology mechanism under constrained conditions is established. Based on the Lagrange multiplier method, the basic constraints and conditional constraints are introduced into the unconstrained dynamic equations, constructing a dynamic modeling framework for flexible variable topology mechanisms that can describe different topological structures. To ensure that the floating coordinate system of the flexible component can adapt to the state before and after the topology change, the dynamic model is modified using the compatibility equation, which further guarantees the accuracy and completeness of the modeling method. Attached Figure Description

[0045] Figure 1 This is a flowchart of the overall modeling framework in this invention;

[0046] Figure 2 This is a schematic diagram of the numbering rules for the tree-shaped flexible variable topology mechanism in this invention;

[0047] Figure 3 This is a schematic diagram of the numbering rules for non-tree-shaped flexible variable topology mechanisms in this invention;

[0048] Figure 4 This is a simplified diagram of an example of a three-topology flexible variable topology mechanism in this invention;

[0049] Figure 5 This is a monochrome topology diagram illustrating the three-topology flexible variable topology mechanism of this invention.

[0050] Figure 6 This is a schematic diagram of the kinematic analysis of a single flexible component in this invention;

[0051] Figure 7 A diagram illustrating the connection relationship between any two flexible components in this invention;

[0052] Figure 2 and Figure 3 In the diagram, B0 represents the root component, and B1, ..., B nThis represents components 1 to n in a flexible variable topology mechanism; Figure 4 and Figure 5 In the diagram, J1,…,J4 represent joints 1 to 4; Figure 6 In the middle, o and o i Represent the inertial coordinate system o-xyz and component B respectively. i floating coordinate system o i -x i y i z i The origin, r i Indicates component B i any point P i Relative to the position vector of the inertial coordinate system o-xyz, R i Represents the floating coordinate system o of the flexible component i -x i y i z i The origin o i The position vector P relative to the origin o of the inertial coordinate system o-xyz is... ri Point P represents the point when the flexible component is undeformed. i In the floating coordinate system o i -x i y i z i The position of u ri Point P represents the point when the flexible component is undeformed. ri In the floating coordinate system o i -x i y i z i The position vector in, u fi Point P represents the deformation caused by the flexible component. ri Point P i The deformation vector; Figure 7 China j Represents a floating coordinate system o j -x j y j z j The origin, R j Represents the floating coordinate system o of the flexible component j -x j y j z j The origin o j The position vector P relative to the origin o of the inertial coordinate system o-xyz is... rj Point P represents the point when the flexible component is undeformed. j In the floating coordinate system o j -x j y j z j The position of urj Point P represents the point when the flexible component is undeformed. rj In the floating coordinate system o i -x i y i z i The position vector in, u fj Point P represents the deformation caused by the flexible component. rj Point P j The deformation vector, h ij Indicates component B i Point P i To component B j Point P j The position vector. Detailed implementation method:

[0053] The present invention will be further described in detail below with reference to the accompanying drawings and specific examples, but the implementation of the present invention is not limited thereto.

[0054] This invention proposes a dynamic modeling method for flexible variable topology mechanisms based on constraint analysis. Combining topology and multi-body dynamics theory, it extracts the basic constraints and conditional constraints in the flexible variable topology mechanism through adjacency matrix. After discretizing the flexible components, an unconstrained dynamic model of the flexible variable topology mechanism is established. On this basis, the basic constraints and conditional constraints are introduced into the dynamic model to establish a dynamic model of a flexible variable topology mechanism that can describe any topology.

[0055] The specific steps of this method are as follows:

[0056] Step 1: Make reasonable assumptions about the model. Since the mass of the joint is much smaller than the mass of the component, the mass of the joint is ignored. This invention mainly targets large-scale and highly flexible variable topology mechanisms in the aerospace field. In most cases, the size and mass of the component are much larger than the size and mass of the joint. Therefore, in the dynamic modeling of the flexible variable topology mechanism proposed in this invention, the mass of the joint is ignored. In addition, to avoid collisions and impacts that could damage large-scale components, most flexible variable topology mechanisms operate smoothly during topological changes. Therefore, it is assumed that the topological change process of the flexible variable topology mechanism proceeds smoothly.

[0057] Step 2: Analyze the different topologies of the flexible variable topology mechanism and number all components that appear during the entire working process.

[0058] Flexible variable topology mechanisms can be classified into tree-like systems and non-tree-like systems. Due to the different topologies, the numbering rules also differ. A tree-like system is a system without any closed loops and with only one root component. For example... Figure 2The tree-like system shown, due to its simple structure, can be numbered using a regular numbering method. The regular numbering method is a tree-like growth-style numbering method. First, the root component B0 is determined. Then, starting from the root component, the components are numbered layer by layer in a left-to-right, bottom-to-top order, ensuring that the numbers of external components are greater than the numbers of internal components. This completes the numbering of the entire mechanism. Figure 2 Taking the tree-shaped flexible variable topology mechanism shown as an example, components B0 to B6 were numbered according to the method described above. If the system contains closed loops or multiple root components, the system is called a non-tree-shaped system. This method requires numbering all components appearing during the operation of the flexible variable topology mechanism, which often results in some components existing alone in the topology diagram; that is, a single component becomes a root component itself. Therefore, the vast majority of flexible variable topology mechanisms are non-tree-shaped systems. For example... Figure 3 The diagram shows a non-tree system containing a closed loop and two root components. When numbering this system, it's necessary to select any two components in the closed loop and cut their hinges, temporarily releasing the constraints between them, thus transforming the closed-loop system into a derived tree system. Based on this, each subsystem in the non-tree system is numbered according to the numbering rules for tree systems. After numbering, the two components whose constraints were previously released are merged, resulting in the final numbering of the non-tree system. Figure 3 Taking the non-tree-shaped flexible variable topology mechanism shown as an example, when numbering the components, any two components are selected to release the constraints. Figure 3 It transforms into a flexible variable topology mechanism composed of two tree systems, and the components are numbered according to the tree system method (B0~B1). 10 After numbering, restore the released constraints on components B4 and B7 to complete the process. Figure 3 The numbering of the non-tree-shaped system flexible variable topology mechanism is shown.

[0059] Step 3: Establish a topology graph, analyze the topology sequence, and construct the adjacency matrix for different topologies of the flexible variable topology mechanism. To analyze the constraint changes during the operation of the flexible variable topology mechanism, the established topology graph needs to include all components under different topologies. In current topology establishment methods, if a component is separated from the overall mechanism, it will not be included in the topology graph. However, this invention, in order to obtain a topology graph containing all components, requires the established topology graph to include these components separated from the overall mechanism. Figure 4 For example, consider a flexible variable topology mechanism with a three-topology structure. Its topology diagram is shown in Figure 5. In this mechanism, component B3 is separated from the overall mechanism under a certain topology structure, but is still included in the topology diagram.

[0060] Based on the topology diagram, the adjacency matrix D under various topological structures can be established during the operation of the flexible variable topology mechanism. ξ Adjacency matrix D ξ The elements in can be represented as:

[0061]

[0062] by Figure 5 For example, based on the topology graph, the adjacency matrices D1, D2, and D3 of the three topological structures can be obtained, as follows:

[0063]

[0064]

[0065]

[0066] Step 4: Perform matrix operations on the adjacency matrices under different topologies to extract the basic constraints and conditional constraints of the flexible variable topology mechanism.

[0067] Constraints that remain unchanged throughout the motion of a flexible variable topology mechanism are called fundamental constraints, while constraints that change when the topology changes are called conditional constraints. To make the modeling process of flexible variable topology mechanisms clearer, it is necessary to extract the fundamental and conditional constraints. The elements in the adjacency matrix reflect whether a constraint relationship exists between two components; therefore, the adjacency matrix D under different topologies of the flexible variable topology mechanism is... ξ By performing logical operations, the basic constraint adjacency matrix and conditional constraint adjacency matrix of the flexible variable topology mechanism can be obtained. Based on these matrices, the basic and conditional constraints of the flexible variable topology mechanism can be extracted, and thus the basic and conditional constraint equations can be established. (Adjacency matrix) D ξ It is a symmetric matrix, and when the topology does not contain self-loops, D ξ All elements on the main diagonal are zero. Therefore, after the adjacency matrix completes the logical operation, the basic constraints and conditional constraints of the flexible variable topology mechanism can be obtained only through the upper triangular region of the main diagonal.

[0068] By performing a logical AND operation on all adjacency matrices under different topologies, the basic constrained adjacency matrix can be obtained:

[0069] D b = D 1& D 2&… D ξ &… D n (14)

[0070] In the formula, D brepresents the basic constraint adjacency matrix, and n represents the number of topologies that the flexible variable topology mechanism has during operation.

[0071] Based on the fundamental constraints, the adjacency matrix of each topology mechanism of the flexible variable topology mechanism is used. D ξ Reduce the basic constraint matrix D b This allows us to obtain the conditional constraint adjacency matrix for each topology of the flexible variable topology mechanism:

[0072] D cξ = D ξ - D b (ξ=1,2,…,n) (15)

[0073] In the formula, D cξ Let represent the conditional adjacency matrix under the topological structure ξ.

[0074] by Figure 3 Taking the flexible variable topology mechanism shown as an example, the basic constraint adjacency matrix and the condition constraint adjacency matrix are as follows:

[0075]

[0076]

[0077]

[0078]

[0079] According to the basic constraint adjacency matrix shown in equation (16), the constraints between frame B0 and component B1, and between component B1 and component B2, are the basic constraints during the operation of the flexible variable topology mechanism. According to the conditional basic constraint adjacency matrices under different topologies shown in equations (17)-(19), in topology 1, the complete constraint relationship of topology 1 can be obtained by adding the constraints between frame B0 and component B3 and between component B2 and component B3 on the basis of the basic constraints; in topology 2, the complete constraint relationship of topology 2 can be obtained by adding the constraints between component B2 and component B3 on the basis of the basic constraints; in topology 2, no constraints need to be added, and the basic constraints are the complete constraint relationship.

[0080] Step 5: Analyze the deformation in the sensitive direction of the flexible component and discretize the flexible component.

[0081] Since flexible components possess an infinite number of degrees of freedom, obtaining an exact solution to their dynamics problem is difficult. Therefore, it is necessary to analyze the different characteristics of the flexible component, obtain the deformation in the sensitive direction, and then perform discretization after appropriate simplification. Let the deformation vector in the sensitive direction of the flexible component be:

[0082] u f = Φq f (20)

[0083] In the formula, u f This represents the deformation vector in the sensitive direction of the component. Φ =[Φ1 … Φ N ] represents the modal vector matrix. q f =[q f1 … q fN ] T denoted as modal coordinate array, where N indicates that the vibration of the flexible component is retained up to the Nth mode.

[0084] Step 6: Remove the constraints on all components in the flexible variable topology mechanism and establish the dynamic equations of the flexible variable topology mechanism in the unconstrained state.

[0085] After determining the contact constraint relationships of all components in the flexible variable topology mechanism, the free motion dynamic equations of individual components are established. Then, the free motion dynamic equations of all components are synthesized into the unconstrained dynamic equations of the flexible variable topology mechanism. For example... Figure 5 As shown, this is an arbitrary flexible component B that moves freely. i Before establishing the dynamic equations, a kinematic analysis of the flexible component is first performed. In the figure, the deformation of the flexible component can be decomposed into rigid motion and flexible deformation. To facilitate the motion description, an inertial coordinate system o-xyz is established, and a floating coordinate system o is established when the flexible component is not deformed. i -x i y i z i Floating coordinate system o i -x i y i z i It moves only with the rigid motion of the flexible component and is unaffected by the deformation motion within the flexible component. Therefore, flexible component B... i The position vector r relative to the inertial coordinate system o-xyz i It can be written as:

[0086] r i =R i + A i (u ri +u fi ) (twenty one)

[0087] In the formula, R i Represents the floating coordinate system o of the flexible component i -x i y i z i Relative to the position vector o-xyz in the inertial coordinate system, u ri P represents the state when the flexible component is undeformed. ri Position vector, u fi Point P of the flexible component after deformation ri Offset to point P i position vector, A i Floating coordinate system o i -x i y i z i The rotation transformation matrix to the inertial coordinate system o-xyz.

[0088] Discretized u fi By introducing equation (21), we can obtain:

[0089] r i =R i + A i (u ri + Φ i q fi ) (twenty two)

[0090] To establish a dynamic model using the Lagrange method, it is necessary to take the first derivative of equation (22) with respect to time t to obtain the velocity at any point in the flexible component, as follows:

[0091]

[0092] In the formula, A i Represents a floating coordinate system o i -x i y i z i The rotation transformation matrix to the inertial coordinate system o-xyz can be represented by Euler angles, Euler parameters, and Rodriguez parameters. To simplify the dynamic model, avoid singularity problems, and optimize the modeling process, Euler parameters are selected as the generalized coordinates to describe the floating coordinate system in this invention.

[0093] Equation (23) can be written as:

[0094]

[0095] In the formula, Indicates component Bi The derivative of the four-dimensional Euler parameter vector with respect to time, E i =[e i0 e i1 e i2 e i3 ] T e i0 e i1 e i2 and e i3 Represents Euler parameters,

[0096]

[0097] Based on the above kinematic analysis, an unconstrained dynamic model of a flexible variable topology mechanism is established using the Lagrange method. This requires determining the generalized mass matrix, generalized stiffness matrix, and generalized force vector of any flexible component.

[0098] The generalized mass matrix of any flexible component can be obtained using kinetic energy. The kinetic energy of any flexible component is:

[0099]

[0100] In the formula, T i Represents any flexible component B i kinetic energy, ρ i V represents the density of the flexible component. i Represents the volume of a flexible component. q For generalized coordinate arrays, M For the generalized mass matrix, the generalized coordinates q and generalized mass array M They are as follows:

[0101]

[0102]

[0103] By calculating the flexible member B i The virtual work of the elastic force can be used to obtain the generalized stiffness matrix of the flexible component. The virtual work of the elastic force is:

[0104]

[0105] In the formula, δW ei Represents any flexible component B i The virtual work of elastic force, σ i Represents the stress vector, ε i Represents the strain vector. K i Let σ represent the generalized stiffness matrix. i ε i as well asK i as follows:

[0106] ε i = DΦ i q fi (30)

[0107] σ i =Eε i (31)

[0108]

[0109] In the formula, D Let denote the differential operator matrix, and E denote the elastic modulus matrix.

[0110] Arbitrary flexible component B i The generalized force vector can be obtained by calculating virtual work, such as... Figure 6 As shown, if any point P on the flexible member i Concentrated force F acting on top i Then the work done in vain can be expressed as:

[0111]

[0112] In the formula, Q Fi F represents the concentrated force i The corresponding generalized force,

[0113]

[0114] According to the Lagrange method, the unconstrained dynamic equation of a flexible component is:

[0115]

[0116] In the formula,

[0117]

[0118] By merging the matrices corresponding to the dynamic equations of all individual components in the flexible variable topology mechanism, we can obtain the unconstrained dynamic equations of the flexible variable topology mechanism, as follows:

[0119]

[0120] In the formula,

[0121]

[0122]

[0123]

[0124]

[0125]

[0126] Step 7: Using the Lagrange multiplier method, establish the dynamic equations of the flexible variable topology mechanism with basic constraints.

[0127] like Figure 7 As shown, this represents any two flexible components B in a flexible variable topology mechanism. i and B j The connection diagram shows any two points P between the two components. i and P j Through vector h ij Therefore, the constraint equations between the two components can be expressed as:

[0128] C(q,t)=R i + A i (u ri + Φ i q fi )+h ij -R j - A j (u rj + Φ j q fj )=0 (43)

[0129] Based on the basic constraints extracted in step three, the corresponding basic constraint equations are established, and the Jacobian matrix of each basic constraint equation is extracted, as follows:

[0130]

[0131] In the formula, κ represents the number of fundamental constraint equations.

[0132] Using the Lagrange multiplier method, based on equation (35), the basic constraint equations and their Jacobian matrices are added to establish the dynamic equations of the flexible variable topology mechanism containing basic constraints, as follows:

[0133]

[0134] In the formula, This represents the generalized constraint force acting on the flexible member. Let C be a Lagrange multiplier vector. b (q,t) represents the basic constraint equation.

[0135] Step 8: Based on the dynamic equations of the flexible variable topology mechanism with basic constraints, add conditional constraints under different topologies to establish dynamic equations for different topologies.

[0136] Based on the constraints extracted in step three, constraint equations are established for multiple topologies. During the flexible variable topology operation, the set of constraint equations for different topologies is as follows:

[0137]

[0138] In the formula, This represents the set of all conditional constraint equations that have occurred during the operation of the flexible variable topology. Let ξ represent the constraint equations under the topological structure ξ.

[0139] After obtaining the Jacobian matrix of different constraints under various topologies according to equation (46), the Lagrange multiplier method is used to add different constraint equations to equation (45), thus establishing the dynamic equations that can describe all topologies, as follows:

[0140]

[0141] In the formula, The conditional constraint equations represent the flexible variable topology mechanism. Represents the Jacobian matrix of the conditional constraint equations. Let Lagrange multiplier vectors represent the conditional constraint equations, where time t∈(t ξ-1 ,t ξ ] represents the time interval under the topology ξ of the flexible variable topology mechanism.

[0142] Step 9: Analyze the compatibility conditions of the flexible components before and after the constraint changes, establish motion compatibility equations, and describe the relationship between the elements before and after the change in the system topology.

[0143] When the topology of a flexible variable topology mechanism changes instantaneously, a set of constraints may be added or removed on the boundaries of the flexible component. This necessitates a new floating coordinate system to describe the vibration state after the constraint change. To ensure the continuity of the flexible component's motion and enable the flexible variable topology mechanism to smoothly transition from one topology to another, an interface condition for this smooth transition must be defined. This condition, called the compatibility condition, is used to define the position and orientation of the new floating coordinate system for the flexible component.

[0144] Assume the operating time domain of the flexible variable topology mechanism is T = [t0, t... eThis includes n topologies, requiring n-1 topological changes. Therefore, the time domain T contains t1, ..., t... ξ ,…,t n-1 Each topologically variable time t. ξ Perform compatibility analysis, at t ξ The smallest neighborhood Right now:

[0145]

[0146] In the formula,

[0147]

[0148] In t ξ The smallest neighborhood Inside, any flexible component B before and after the topological change. i The deformation of the shape does not change instantaneously, therefore any point P on it will not change instantaneously. ri The position vectors in an inertial frame of reference have the following relationship:

[0149]

[0150] In the formula,

[0151]

[0152]

[0153] Since the state of the system is known before the structural topology changes, that is... Given that, according to equations (50)-(52), we can obtain:

[0154]

[0155] Similarly, suppose that the point passes through point P ri There exists a rigid line segment whose vector in the floating coordinate system is like This can be expressed in an inertial frame as follows:

[0156]

[0157] In the formula, Let I be the vector of a line segment in the inertial frame, and let I be the identity matrix. for antisymmetric array, θ P The three components are vectors. At point P ri The small rotation relative to the floating coordinate system. for The opposition is called the "battlefield".

[0158] Before the topological change occurs in the flexible variable topology mechanism, equation (54) can be written as follows:

[0159]

[0160] After the topological structure changes, equation (54) can be written as

[0161]

[0162] Similarly, in t ξ The smallest neighborhood Inside, passing through point P ri The rigid line segment is No change, therefore

[0163]

[0164] Then there is

[0165]

[0166] In equations (53) and (58), and Determined by the dynamic equations before the topology change. as well as All are known quantities. Let the angle of rotation of the floating coordinate system within the inertial frame be... The function is such that, therefore, by combining equations (53) and (58), the position of the new floating coordinate system can be obtained. and direction The compatibility equation for the flexible variable topology mechanism can be expressed in the following form:

[0167]

[0168] Equation (59) is a set of nonlinear algebraic equations, and its solution is... and The position and orientation of the floating coordinate system in the inertial frame after the topological change can be uniquely determined. Differentiating equation (59) with respect to time t yields the velocity compatibility equation, as follows:

[0169]

[0170] The combined equations (47), (59) and (60) together constitute the dynamic equation of the flexible variable topology mechanism, which has good adaptability for modeling any flexible variable topology mechanism.

Claims

1. A dynamic modeling method for flexible variable topology mechanisms based on constraint analysis, the specific steps of which are as follows: Step 1: Make reasonable assumptions about the model. Since the mass of the joint is much smaller than the mass of the component, the mass of the joint is ignored, and it is assumed that the topological change process of the flexible variable topology mechanism is smooth. Step 2: Analyze the different topologies of the flexible variable topology mechanism and number all components that appear during the entire working process; Step 3: Establish a topology graph, analyze the topology sequence, and construct the adjacency matrix for different topological structures of the flexible variable topology mechanism; Step 4: Perform matrix operations on the adjacency matrices under different topologies to extract the basic constraints and conditional constraints of the flexible variable topology mechanism; Step 5: Analyze the deformation in the sensitive direction of the flexible component and discretize the flexible component; Step 6: Remove the constraints on all components in the flexible variable topology mechanism and establish the dynamic equations of the flexible variable topology mechanism in the unconstrained state; Step 7: Using the Lagrange multiplier method, establish the dynamic equations of the flexible variable topology mechanism with basic constraints; Step 8: Based on the dynamic equations of the flexible variable topology mechanism with basic constraints, add conditional constraints under different topologies to establish dynamic equations for different topologies; Step 9: Analyze the compatibility conditions of the flexible components before and after the constraint changes, establish motion compatibility equations, and describe the relationship between the elements before and after the change in the system topology.

2. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step one, for large-scale and highly flexible variable topology mechanisms, the size and mass of the components are much larger than the size and mass of the joints. Therefore, the mass of the joints is ignored in the modeling process of flexible variable topology mechanisms. In addition, to avoid collisions and impacts that could damage large-scale components, most flexible variable topology mechanisms operate smoothly during topological changes. Therefore, it is assumed that the topological change process of flexible variable topology mechanisms proceeds smoothly.

3. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step two, since the flexible variable topology mechanism undergoes multiple topological changes during operation, the number and connection methods of components may change under different topological structures. Therefore, it is necessary to analyze the different topological structures of the flexible variable topology mechanism, extract all components that have appeared in the entire operation process, and number each component accordingly in order to analyze the topological sequence of the flexible variable topology mechanism and lay the foundation for subsequent dynamic modeling.

4. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step three, the elements of the adjacency matrix indicate the adjacency relationships between components in the flexible variable topology mechanism, thus indicating whether constraints exist between components. Therefore, by analyzing the topology sequence and establishing adjacency matrices for different topological structures of the flexible variable topology mechanism, the existence of constraints between components under different topological structures can be obtained. The elements of the adjacency matrix of the flexible variable topology mechanism can be represented as follows: In the formula, D ij This represents the element in the i-th row and j-th column of the adjacency matrix of the flexible variable topology mechanism.

5. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step four, logical operations are performed on the elements in the adjacency matrix under different topologies to extract the basic constraint adjacency matrix and condition constraint adjacency matrix of the flexible variable topology mechanism. Further, the basic constraints that remain unchanged during the overall motion of the flexible variable topology mechanism and the condition constraints that change with the topology can be obtained.

6. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step five, since the flexible component has an infinite number of degrees of freedom, the exact solution to its dynamics problem is difficult to obtain. Therefore, it is necessary to analyze the different characteristics of the flexible component, simplify it appropriately to obtain the deformation in the sensitive direction of the component, and then discretize it. Let the deformation vector in the sensitive direction of the flexible component be: In the formula, u f This represents the deformation vector in the sensitive direction of the component. Φ =[Φ1 … Φ N ] represents the modal vector matrix. q f =[q f1 … q fN ] T denoted as modal coordinate array, where N indicates that the vibration of the flexible component is retained up to the Nth mode.

7. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step six, kinematic analysis is performed on any flexible component to establish the dynamic equations for the free motion of a single flexible component. Based on this, the unconstrained dynamic equations for the flexible variable topology mechanism containing all components are established: In the formula, M For the generalized mass matrix, q For generalized coordinate arrays, K Generalized stiffness matrix, Q F Q is the generalized active force vector. v It is the generalized force vector of the second-order velocity term.

8. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step seven, based on the unconstrained dynamic equations of the flexible variable topology mechanism, and according to the Lagrange multiplier method, the dynamic equations of the flexible variable topology mechanism with basic constraints are established by introducing the basic constraint equations and their Jacobian matrices extracted in step four: In the formula, C b (q,t) represents the basic constraint equations of the flexible variable topology mechanism. C bq Let λ represent the Jacobian matrix of the fundamental constraint equations. b This represents the Lagrange multiplier vector of the fundamental constraint equations.

9. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step eight, based on the dynamic equations of the flexible variable topology mechanism with basic constraints, similar to step eight, the Lagrange multiplier method is used to introduce the conditional constraint equations and their Jacobian matrices under different topologies, thus establishing the dynamic equations of the flexible variable topology mechanism that can describe different topologies: In the formula, The conditional constraint equations represent the flexible variable topology mechanism. Represents the Jacobian matrix of the conditional constraint equations. Let ξ be the Lagrange multiplier vector representing the conditional constraint equations, and let ξ represent the flexible variable topology mechanism under the topology ξ, with time t∈(t ξ-1 ,t ξ ] represents the time interval under the topology ξ of the flexible variable topology mechanism.

10. The dynamic modeling method for flexible variable topology mechanisms based on constraint analysis as described in claim 1, characterized in that... In step nine, the topology changes during the movement of the flexible variable topology mechanism, which alters the constraint relationships of the flexible components and thus changes the boundary conditions of individual flexible components. Therefore, the floating coordinate system of the flexible component before the topology change is no longer suitable for the changed state. In order to adapt the floating coordinate system of the flexible component to the changed state, a motion compatibility equation is established to describe the relationship between the elements of the system space components when the constraint relationships change. Assume the operating time domain of the flexible variable topology mechanism is T = [t0, t...]. e This includes n topologies, requiring n-1 topological changes. Therefore, the time domain T contains t1, ..., t... ξ ,…,t n-1 Each topologically variable time; for any topologically variable time t ξ Perform compatibility analysis and take t ξ The smallest neighborhood of , i.e.: In the formula, At the topological change time t ξ Within the neighborhood of , the position compatibility equation and velocity compatibility equation of the flexible variable topology mechanism are respectively expressed as: In the formula, This represents the position vector of the floating coordinate system in the inertial coordinate system after the topological change. This represents the angle vector of the floating coordinate system relative to the inertial coordinate system after the topological changes. This represents the velocity vector of the floating coordinate system in the inertial coordinate system after the topological change. This represents the angular velocity vector of the floating coordinate system relative to the inertial coordinate system after the topology changes; Solutions to the compatibility equations of equations (8) and (9) and It can uniquely determine the position, direction and velocity vector of the floating coordinate system in the inertial frame after the topology changes, thus ensuring the continuity and stability of the system topology changes; combined with equations (5), (8) and (9), they together constitute the dynamic equation of the flexible variable topology mechanism.

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