Five-axis machine tool dynamics modeling method and system based on per-unit group coding

Abstract

The invention discloses a five-axis machine tool dynamics modeling method and system based on per-unit group coding. The method comprises the following steps: S1, carrying out series five-axis machine tool modular kinematics derivation based on a local exponential product; s2, linear modeling of modular kinematics about kinematics parameters is completed; s3, acquiring linear representation of an inertia item, a centrifugal force, a Coriolis force item and a gravity item of the kinetic model about a multivariate polynomial; s4, describing the kinetic model of linear representation as a bilinear representation form about inertial parameters and multivariate polynomials; s5, eliminating redundant multivariate polynomials by utilizing matrix line correlation, and eliminating redundant parameters by utilizing matrix decomposition to obtain a minimum inertial parameter set; and S6, carrying out reverse coding on the multivariate polynomial coding numerical matrix and the parameter matrix after redundancy elimination to form a symbolic expression in a Homer form, and realizing efficient calculation of the model. The algorithm development and algorithm complexity of the five-axis machine tool are reduced, and the method is suitable for rapid deployment of a general five-axis machine tool algorithm.

Description

Technical Field

[0001] The invention belongs to the technical field of robot manufacturing, and in particular relates to a five-axis machine tool dynamics modeling method and system based on monoid coding. Background Art

[0002] The dynamic model is an important foundation for the mechatronic design and optimization, trajectory planning and motion control of multi-axis machine tools, and directly affects important performance indicators such as dynamic response capability and positioning accuracy. In order to cover complex blank processing and centralized process processing, the five-axis machine tool has a high degree of freedom configuration and a design with component offsets. However, the increase in the number of components and the complex design bring about problems such as low model derivation efficiency and high model calculation complexity, which greatly increases the difficulty of dynamic response analysis of machine tool components. Without changing the structure, the model can be simplified to improve the efficiency of model derivation and calculation. At present, there are several problems with the method of model simplification:

[0003] (1) The dynamic modeling of some multi-axis machine tools adopts a data-driven model learning method (such as patents CN111639422B, CN111639422A, and CN103425811A). This method first establishes a back-propagation deep neural network model, establishes a data set for the neural network model by collecting machine tool command trajectories and driving force information, and uses the deviation between the neural network prediction value and the actual driving force of the machine tool to update the network hidden layer connection weights. In addition, the trial and error method is required to adjust the update step size, the number of network layers, and the number of hidden layer neuron nodes. The process is time-consuming, resulting in low efficiency, and the generalization ability of modeling accuracy is difficult to guarantee.

[0004] (2) The second method is to use a reduced-order dynamic model (such as patents CN104950804A and CN106407528A). This method ignores the nonlinear dynamics of the whole machine dynamics, establishes a transfer function model corresponding to the simplified linear system, and reduces the order of the transfer function model while allowing the system to have an allowable output error. This method ignores the multi-axis coupling dynamic characteristics of the machine tool, resulting in a significant decrease in accuracy, making it difficult to use for high-speed and high-precision motion control.

[0005] (3) For multi-axis systems in series, most existing methods are to reduce redundant parameters (such as patents CN110539302A and CN110532581A). This method uses modeling methods such as Euler-Lagrangian dynamics to derive the whole-body dynamics of the multi-axis system, and reduces unnecessary parameter calculations by solving the number of identifiable parameters in the dynamics. However, this method is difficult to eliminate redundant nonlinear terms and dynamic parameters in the model at the same time, resulting in low computational efficiency. Summary of the invention

[0006] In order to solve the problems of low derivation efficiency and calculation efficiency of existing multi-axis machine tool dynamics modeling methods, the present invention utilizes monoid coding and redundant multivariate polynomials and dynamics parameter double feature elimination method to provide a universal and efficient five-axis machine tool dynamics efficient modeling and calculation method based on monoid coding for multi-axis machine tool systems.

[0007] In order to achieve the aforementioned object of the invention, the present invention adopts the following scheme:

[0008] One aspect of the present invention provides a five-axis machine tool dynamics modeling method based on monoid coding, comprising the following steps:

[0009] S1. Define each screw axis of the tandem five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the motion homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the tandem five-axis machine tool based on the local exponential product;

[0010] S2. Using the Kronecker product operator, extract multivariate polynomials from the homogeneous transformation matrix, the product of two homogeneous transformation matrices, and the single homogeneous transformation matrix about the motion displacement differential, and complete the linearization modeling of modular kinematics about kinematic parameters; numerically encode the multivariate polynomials in the monoid space to obtain the multivariate polynomial numerical matrix;

[0011] S3: According to the multivariate polynomial numerical matrix, matrix trace operator and column vector quantization operator, obtain the linear representation of the inertia term, centrifugal force and Coriolis force term, and gravity term of the dynamic model with respect to the multivariate polynomial, so as to realize the dynamic model representation as the linear representation of the multivariate polynomial vector and the parameter matrix;

[0012] S4: The linearly represented dynamic model is described as a bilinear representation of inertial parameters and multivariate polynomials;

[0013] S5: Eliminate redundant multivariate polynomials using matrix row correlation, and eliminate redundant parameters using matrix decomposition to obtain the minimum inertia parameter set;

[0014] S6: Reversely encode the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy elimination into a symbolic form in Horner form to achieve efficient calculation of the model.

[0015] Another aspect of the present invention provides a five-axis machine tool dynamics modeling system based on monoid coding, comprising:

[0016] The modular kinematic linearization modeling module is used to define each screw axis of the tandem five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the motion homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the tandem five-axis machine tool based on the local exponential product;

[0017] The multivariate polynomial encoding module is used to extract multivariate polynomials from a homogeneous transformation matrix, a product of two homogeneous transformation matrices, and a single homogeneous transformation matrix about the motion displacement differential using the Kronecker product operator, so as to complete the linearization modeling of modular kinematics about kinematic parameters; the multivariate polynomials are numerically encoded in the monoid space to obtain the multivariate polynomial numerical matrix;

[0018] A linear representation module, for obtaining the linear representation of the inertia term, centrifugal force and Coriolis force term, and gravity term of the dynamic model with respect to the multivariate polynomial according to the multivariate polynomial numerical matrix, the matrix trace operator, and the column vector quantization operator, so as to realize the linear representation of the dynamic model as the multivariate polynomial vector and the parameter matrix;

[0019] A bilinear representation module, used to describe a linearly represented dynamic model as a bilinear representation of inertial parameters and multivariate polynomials;

[0020] A minimum inertia parameter set generation module is used to eliminate redundant multivariate polynomials by using matrix row correlation, and to eliminate redundant parameters by using matrix decomposition to obtain a minimum inertia parameter set;

[0021] The model calculation module is used to reversely encode the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy elimination into a symbolic form in Horner form to achieve efficient calculation of the model.

[0022] Compared with the prior art, the present invention has at least the following advantages: the present invention uses monoid coding to derive models in a lower memory consumption than pure symbolic methods, including in terms of model derivation and model calculation, significantly improving the efficiency of algorithm development and execution. Secondly, the method is suitable for efficient dynamic model derivation and calculation of general tandem five-axis machine tools, reduces the algorithm development and algorithm complexity of five-axis machine tools, and is suitable for rapid deployment of general five-axis machine tool algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0024] Figure 1 It is a schematic diagram of a framework of a five-axis machine tool dynamic modeling method and system based on monoid coding provided by a typical implementation case of the present invention;

[0025] Figure 2It is a modular kinematic linearization modeling block diagram provided by a typical implementation case of the present invention;

[0026] Figure 3 It is a block diagram of a multivariate polynomial encoding based on a monoid provided by a typical implementation case of the present invention;

[0027] Figure 4 It is a kinetic modeling block diagram based on bilinear representation provided by a typical implementation case of the present invention;

[0028] Figure 5 It is a block diagram of a method for eliminating redundant multivariate polynomials and parameters of a dynamic model provided by a typical implementation case of the present invention. DETAILED DESCRIPTION

[0029] In order to make the purpose, technical scheme and advantages of the present invention clearer, the specific embodiments of the present invention are described in detail below in conjunction with the accompanying drawings. Examples of these preferred embodiments are illustrated in the accompanying drawings. The embodiments of the present invention shown in the accompanying drawings and described according to the accompanying drawings are merely exemplary, and the present invention is not limited to these embodiments.

[0030] The present invention proposes an efficient modeling and calculation method for the dynamics of a five-axis machine tool based on monoid coding, which includes a kinematic modeling module based on local exponential product, a monoid coding module, a dynamic linearization modeling module, a dynamic bilinearization modeling module, a redundant parameter and multivariate polynomial elimination module, and a dynamic decoding module. Figure 1 shown.

[0031] One aspect of the present invention provides a five-axis machine tool dynamics modeling method based on monoid coding, comprising the following steps:

[0032] S1. Define each screw axis of the tandem five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the motion homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the tandem five-axis machine tool based on the local exponential product;

[0033] S2. Using the Kronecker product operator, extract multivariate polynomials from the homogeneous transformation matrix, the product of two homogeneous transformation matrices, and the single homogeneous transformation matrix about the motion displacement differential, and complete the linearization modeling of modular kinematics about kinematic parameters; numerically encode the multivariate polynomials in the monoid space to obtain the multivariate polynomial numerical matrix;

[0034] S3: According to the multivariate polynomial numerical matrix, matrix trace operator and column vector quantization operator, obtain the linear representation of the inertia term, centrifugal force and Coriolis force term, and gravity term of the dynamic model with respect to the multivariate polynomial, so as to realize the dynamic model representation as the linear representation of the multivariate polynomial vector and the parameter matrix;

[0035] S4: The linearly represented dynamic model is described as a bilinear representation of inertial parameters and multivariate polynomials;

[0036] S5: Eliminate redundant multivariate polynomials using matrix row correlation, and eliminate redundant parameters using matrix decomposition to obtain the minimum inertia parameter set;

[0037] S6: Reversely encode the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy elimination into a symbolic form in Horner form to achieve efficient calculation of the model.

[0038] In one embodiment, the kinematic modeling based on the local exponential product in step S1 includes the initial pose modeling and relative motion pose modeling of adjacent components, and the mathematical formulas are respectively expressed as follows:

[0039]

[0040]

[0041] in, i-1 T i (0) represents the homogeneous transformation matrix of the initial pose, represents the spinor coordinate of the i-th component relative to the i-1-th component, q i is the rotation displacement of the ith component, E i represents the rotation of the i-th joint The homogeneous transformation matrix of the instantaneous motion of the spinor; SE(3) represents the three-dimensional special Euclidean group, and its physical meaning is to describe the movement and rotation characteristics of the rod; Based on the above mathematical derivation, the kinematic homogeneous transformation matrix of the entire machine tool is obtained by multiplying the initial posture and relative motion posture of multiple adjacent components, and the expression is as follows:

[0042]

[0043] In one embodiment, the linear modeling of modular kinematics with respect to kinematic parameters in step S2 is specifically: the motion homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters, and the mathematical formula is as follows:

[0044]

[0045] Among them, I 4 Represents the 4×4 identity matrix; represents the Kronecker product; w i Indicates from The extracted multivariate polynomial vector, V i Indicates from The extracted parameter matrix,the two constitute the linearized model of modular kinematics about kinematic parameters;

[0046] Kinematic homogeneous transformation matrix product form It is expressed as a multivariate polynomial and a linear expression of kinematic parameters. The mathematical formula is as follows:

[0047]

[0048] in, and Represents the linearized expression of a single motion homogeneous transformation matrix; V β Indicates from The extracted parameter matrix, V α Indicates from The extracted parameter matrix, w β Indicates from The extracted multivariate polynomial vector, w α Indicates from The extracted multivariate polynomial vector;

[0049] The partial differential form of the kinematic homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters:

[0050]

[0051] Among them, V i represents the kinematic parameter matrix, w′ i is a multivariate polynomial vector; vec means stacking the column vectors of the matrix vertically. represents the partial differential operation, and represents the continuous multiplication of Kronecker products.

[0052] In one embodiment, in step S2, the multivariate polynomial is numerically encoded in the monoid space, including the multivariate polynomial encoding, the definition of elements and binary operations in the monoid, and the definition of the Kronecker product, and the multivariate polynomial is symbolically expressed as follows:

[0053]

[0054] in, represents a multivariate polynomial with respect to a set of variables v, c i represents a real number, e i，t represents the order of the multivariate polynomial, represents the cumulative multiplication operation, represents the set of real numbers, represents a set of natural numbers, and v represents a set of variables;

[0055] Defining a mapping The coefficients and degree of a single multivariate polynomial are extracted and encoded into a vector as follows:

[0056]

[0057] in, represents the transpose of a real vector, and T represents the transpose operation of a vector;

[0058] For the set of all single multivariate polynomial encoding vectors Φ 0 , define a binary operation that satisfies closure, associativity, and identity, as follows:

[0059]

[0060] Further, Form a monoid; in order to realize the Kronecker product operation of a vector containing multiple multivariate polynomials and another vector containing multiple multivariate polynomials in the coding space, it is necessary to define the corresponding Kronecker product operation in the monoid, as follows:

[0061]

[0062] Among them, Φ 0 represents the set of encoding vectors, represents the multiplication operation between any two encoding vectors that belong to the set of encoding vectors, represents the Kronecker product operation between any two encoding vectors that belong to the set of encoding vectors.

[0063] In one embodiment, the dynamic model in step S3 is represented as a linear representation of a multivariate polynomial vector and a parameter matrix, including inertia terms, Coriolis force and centrifugal force terms, and gravity terms converted into linear representations of multivariate polynomial vectors and inertia parameter matrices; the following dynamics about the multivariate polynomial linear representation is obtained by using the Kronecker product operation defined in the monoid, the linearly represented motion homogeneous transformation matrix, and the general Euler-Lagrange dynamics equation using the matrix trace cyclic property:

[0064]

[0065] in, and From the inertia term Coriolis and centrifugal force terms And the gravity term G(q), extract the multivariate polynomial vector; at the same time V d , V c and V g Contains kinematic parameters and inertial parameters.

[0066] Another aspect of the present invention provides a five-axis machine tool dynamics modeling system based on monoid coding, comprising:

[0067] The modular kinematic linearization modeling module is used to define each screw axis of the tandem five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the motion homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the tandem five-axis machine tool based on the local exponential product;

[0068] The multivariate polynomial encoding module is used to extract multivariate polynomials from a homogeneous transformation matrix, a product of two homogeneous transformation matrices, and a single homogeneous transformation matrix about the motion displacement differential using the Kronecker product operator, so as to complete the linearization modeling of modular kinematics about kinematic parameters; the multivariate polynomials are numerically encoded in the monoid space to obtain the multivariate polynomial numerical matrix;

[0069] A linear representation module, for obtaining the linear representation of the inertia term, centrifugal force and Coriolis force term, and gravity term of the dynamic model with respect to the multivariate polynomial according to the multivariate polynomial numerical matrix, the matrix trace operator, and the column vector quantization operator, so as to realize the linear representation of the dynamic model as the multivariate polynomial vector and the parameter matrix;

[0070] A bilinear representation module, used to describe a linearly represented dynamic model as a bilinear representation of inertial parameters and multivariate polynomials;

[0071] A minimum inertia parameter set generation module is used to eliminate redundant multivariate polynomials by using matrix row correlation, and to eliminate redundant parameters by using matrix decomposition to obtain a minimum inertia parameter set;

[0072] The model calculation module is used to reversely encode the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy elimination into a symbolic form in Horner form to achieve efficient calculation of the model.

[0073] In one embodiment, the kinematic modeling based on the local exponential product in the modular kinematic linearization modeling module includes the initial pose modeling and the relative motion pose modeling of adjacent components, and the mathematical formulas are respectively expressed as follows:

[0074]

[0075]

[0076] in, i-1 T i (0) represents the homogeneous transformation matrix of the initial pose, represents the spinor coordinate of the i-th component relative to the i-1-th component, q i is the rotation of the i-th component located at; E irepresents the rotation of the i-th joint The homogeneous transformation matrix of the instantaneous motion of the spinor; SE(3) represents the three-dimensional special orthogonal group, and its physical meaning is to describe the movement and rotation characteristics of the rod; Based on the above mathematical derivation, the kinematic homogeneous transformation matrix of the entire machine tool is obtained by multiplying the initial posture and relative motion posture of multiple adjacent components, and the expression is as follows:

[0077]

[0078] In one embodiment, the linear modeling of modular kinematics about kinematic parameters in the multivariate polynomial coding module is specifically: the motion homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters, and the mathematical formula is as follows:

[0079]

[0080] Among them, I 4 Represents the 4×4 identity matrix; represents the Kronecker product; w i Indicates from The extracted multivariate polynomial vector, V i Indicates from The extracted parameter matrix,the two constitute the linearized model of modular kinematics about kinematic parameters;

[0081] Kinematic homogeneous transformation matrix product form It is expressed as a multivariate polynomial and a linear expression of kinematic parameters. The mathematical formula is as follows:

[0082]

[0083] in, and Represents the linearized expression of a single motion homogeneous transformation matrix; V β Indicates from The extracted parameter matrix, V α Indicates from The extracted parameter matrix, w β Indicates from The extracted multivariate polynomial vector, w α Indicates from The extracted multivariate polynomial vector;

[0084] The partial differential form of the kinematic homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters:

[0085]

[0086] Among them, V irepresents the kinematic parameter matrix, w′ i is a multivariate polynomial vector; vec means stacking the column vectors of the matrix vertically. represents the partial differential operation, and represents the continuous multiplication of Kronecker products.

[0087] In one embodiment, the multivariate polynomial encoding module numerically encodes the multivariate polynomial in the monoid space, including the multivariate polynomial encoding, the elements and binary operations in the monoid, and the definition of the Kronecker product, and the symbolic expression of the multivariate polynomial is as follows:

[0088]

[0089] in, represents a multivariate polynomial with respect to a set of variables v, c i represents a real number, e i，t represents the order of the multivariate polynomial, represents the cumulative multiplication operation, represents the set of real numbers, represents a set of natural numbers, and v represents a set of variables;

[0090] Defining a mapping The coefficients and degree of a single multivariate polynomial are extracted and encoded into a vector as follows:

[0091]

[0092] in, represents the transpose of a real vector, and T represents the transpose operation of a vector;

[0093] For the set of all single multivariate polynomial encoding vectors Φ 0 , define a binary operation that satisfies closure, associativity, and identity, as follows:

[0094]

[0095] Further, Form a monoid; in order to realize the Kronecker product operation of a vector containing multiple multivariate polynomials and another vector containing multiple multivariate polynomials in the coding space, it is necessary to define the corresponding Kronecker product operation in the monoid, as follows:

[0096]

[0097] Among them, Φ 0 represents the set of encoding vectors, represents the multiplication operation between any two encoding vectors that belong to the set of encoding vectors, represents the Kronecker product operation between any two encoding vectors that belong to the set of encoding vectors.

[0098] In one embodiment, the dynamic model in the linear representation module is represented as a linear representation of a multivariate polynomial vector and a parameter matrix, including inertia terms, Coriolis force and centrifugal force terms, and gravity terms converted into linear representations of multivariate polynomial vectors and inertia parameter matrices; the following dynamics about the multivariate polynomial linear representation is obtained by using the Kronecker product operation defined in the monoid, the linearly represented motion homogeneous transformation matrix, and the general Euler-Lagrange dynamics equations using the matrix trace cyclic property:

[0099]

[0100] in, and From the inertia term Coriolis and centrifugal force terms And the gravity term G(q), extract the multivariate polynomial vector; at the same time V d , V c and V g Contains kinematic parameters and inertial parameters.

[0101] In one embodiment, the multivariate polynomial vectors and parameter matrices in the linear expressions of the inertia term, the Coriolis force and the centrifugal force term, and the gravity term are extracted by the following methods:

[0102] (a) Inertia term:

[0103]

[0104]

[0105] Among them, e ij and f ij It is expressed as:

[0106]

[0107]

[0108] (b) Coriolis force and centrifugal force terms:

[0109]

[0110]

[0111] Among them, e i,jk and fi,jk It is expressed as:

[0112]

[0113]

[0114] (c) Gravity term:

[0115]

[0116]

[0117] Here It means stacking multiple matrices or vector blocks onto the diagonal of a large matrix to form a block diagonal matrix.

[0118] In one embodiment, the elimination of redundant multivariate polynomials and inertia parameters of the dynamic model includes bilinear representation of the dynamic model, elimination of redundant multivariate polynomials and inertia parameters, such as Figure 5 As shown. The bilinear dynamic model is as follows:

[0119]

[0120] in is the indicator matrix containing the kinematic parameters, is the standard inertia parameter matrix. Secondly, the matrix row correlation analysis is used to eliminate The redundant multivariate polynomial in

[0121] in Finally, the full rank decomposition method is used to Decompose and remove redundant inertia parameters to obtain a simplified parameter vector θ b , and then the simplified parameter linearized dynamic model is obtained:

[0122]

[0123] It is worth noting that the redundancy elimination process here is implemented in the encoding space.

[0124] In one embodiment, the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy removal are reversely encoded into a parameter linearized dynamic model symbolic form in Homer form, specifically, the encoded multivariate polynomial after redundancy removal is Arrange the encoding matrix in descending order according to its row and size. Finally, combine the constant matrix L and P i , and the minimum inertia parameter vector θ b, through the multivariate polynomial definition and Homer symbolic expression, the following simplest parameter linearized dynamic model is obtained as follows:

[0125] τ T =Yθ b

[0126] Where Y represents the regression matrix.

[0127] It should be understood that although this specification is described according to implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions of each embodiment can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A five-axis machine tool dynamics modeling method based on monoid coding, characterized in that: include: S1. Define each screw axis of the tandem five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the motion homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the tandem five-axis machine tool based on the local exponential product; S2. Using the Kronecker product operator, extract multivariate polynomials from the homogeneous transformation matrix, the product of two homogeneous transformation matrices, and the single homogeneous transformation matrix about the motion displacement differential, and complete the linearization modeling of modular kinematics about kinematic parameters; numerically encode the multivariate polynomials in the monoid space to obtain the multivariate polynomial numerical matrix; S3: According to the multivariate polynomial numerical matrix, matrix trace operator and column vector quantization operator, obtain the linear representation of the inertia term, centrifugal force and Coriolis force term, and gravity term of the dynamic model with respect to the multivariate polynomial, so as to realize the dynamic model representation as the linear representation of the multivariate polynomial vector and the parameter matrix; S4: The linearly represented dynamic model is described as a bilinear representation of inertial parameters and multivariate polynomials; S5: Eliminate redundant multivariate polynomials using matrix row correlation, and eliminate redundant parameters using matrix decomposition to obtain the minimum inertia parameter set; S6: Reversely encode the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy elimination into a symbolic form in Horner form to achieve efficient calculation of the model.

2. The five-axis machine tool dynamics modeling method based on monoid coding according to claim 1 is characterized in that: The kinematic modeling based on the local exponential product in step S1 includes the initial pose modeling and relative motion pose modeling of adjacent components, and the mathematical formulas are respectively expressed as follows: in, i-1 T i (0) represents the homogeneous transformation matrix of the initial pose, represents the spinor coordinate of the i-th component relative to the i-1-th component, q i is the rotation of the i-th component located at; E i represents the i-th joint around The homogeneous transformation matrix of the instantaneous motion of the spinor; SE(3) represents the three-dimensional special orthogonal group, and its physical meaning is to describe the movement and rotation characteristics of the rod; Based on the above mathematical derivation, the kinematic homogeneous transformation matrix of the entire machine tool is obtained by multiplying the initial posture and relative motion posture of multiple adjacent components, and the expression is as follows:

3. The five-axis machine tool dynamics modeling method based on monoid coding according to claim 1 is characterized in that: The linear modeling of modular kinematics with respect to kinematic parameters in step S2 is specifically: the motion homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters, and the mathematical formula is as follows: Where I4 represents the 4×4 identity matrix; represents the Kronecker product; w i Indicates from The extracted multivariate polynomial vector, V i Indicates from The extracted parameter matrix,the two constitute the linearized model of modular kinematics about kinematic parameters; Kinematic homogeneous transformation matrix product form It is expressed as a multivariate polynomial and a linear expression of kinematic parameters. The mathematical formula is as follows: in, and Represents the linearized expression of a single motion homogeneous transformation matrix; V β Indicates from The extracted parameter matrix, V α Indicates from The extracted parameter matrix, w β Indicates from The multivariate polynomial vector extracted, wα represents the vector from The extracted multivariate polynomial vector; The partial differential form of the kinematic homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters: Among them, V i represents the kinematic parameter matrix, w′ i is a multivariate polynomial vector; vec means stacking the column vectors of the matrix vertically. represents the partial differential operation, and represents the continuous multiplication of Kronecker products.

4. The five-axis machine tool dynamics modeling method based on monoid coding according to claim 1 is characterized in that: In step S2, the multivariate polynomial is numerically encoded in the monoid space, including the multivariate polynomial encoding, the elements and binary operations in the monoid, and the definition of the Kronecker product, and the multivariate polynomial is symbolically expressed as follows: in, represents a multivariate polynomial with respect to a set of variables v, c i represents a real number, e i，t represents the order of the multivariate polynomial, represents the cumulative multiplication operation, represents the set of real numbers, represents a set of natural numbers, and v represents a set of variables; Defining a mapping The coefficients and degree of a single multivariate polynomial are extracted and encoded into a vector as follows: in, represents the transpose of a real vector, and T represents the transpose operation of a vector; For the set Φ0 of all single multivariate polynomial encoding vectors, define a binary operation that satisfies closure, associativity, and identity as follows: Further, Form a monoid; in order to realize the Kronecker product operation of a vector containing multiple multivariate polynomials and another vector containing multiple multivariate polynomials in the coding space, it is necessary to define the corresponding Kronecker product operation in the monoid, as follows: Among them, Φ0 represents the set of encoding vectors, represents the multiplication operation between any two encoding vectors that belong to the set of encoding vectors, represents the Kronecker product operation between any two encoding vectors that belong to the set of encoding vectors.

5. The five-axis machine tool dynamics modeling method based on monoid coding according to claim 1 is characterized in that: In step S3, the dynamic model is represented as a linear representation of a multivariate polynomial vector and a parameter matrix, including inertia terms, Coriolis force and centrifugal force terms, and gravity terms converted into linear representations of multivariate polynomial vectors and inertia parameter matrices; the following dynamics of the multivariate polynomial linear representation are obtained by using the Kronecker product operation defined in the monoid, the linearly represented homogeneous transformation matrix of motion, and the general Euler-Lagrange dynamics equations using the matrix trace cyclic properties: in, and From the inertia term Coriolis and centrifugal force terms And the gravity term G(q), extract the multivariate polynomial vector; at the same time V d , V c and V g Contains kinematic parameters and inertial parameters.

6. A five-axis machine tool dynamics modeling system based on monoid coding, characterized in that: include: The modular kinematic linearization modeling module is used to define each screw axis of the tandem five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the motion homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the tandem five-axis machine tool based on the local exponential product; The multivariate polynomial encoding module is used to extract multivariate polynomials from a homogeneous transformation matrix, a product of two homogeneous transformation matrices, and a single homogeneous transformation matrix about the motion displacement differential using the Kronecker product operator, so as to complete the linearization modeling of modular kinematics about kinematic parameters; the multivariate polynomials are numerically encoded in the monoid space to obtain the multivariate polynomial numerical matrix; A linear representation module, for obtaining the linear representation of the inertia term, centrifugal force and Coriolis force term, and gravity term of the dynamic model with respect to the multivariate polynomial according to the multivariate polynomial numerical matrix, the matrix trace operator, and the column vector quantization operator, so as to realize the linear representation of the dynamic model as the multivariate polynomial vector and the parameter matrix; A bilinear representation module, used to describe a linearly represented dynamic model as a bilinear representation of inertial parameters and multivariate polynomials; A minimum inertia parameter set generation module is used to eliminate redundant multivariate polynomials by using matrix row correlation, and to eliminate redundant parameters by using matrix decomposition to obtain a minimum inertia parameter set; The model calculation module is used to reversely encode the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy elimination into a symbolic form in Homer form to achieve efficient calculation of the model.

7. The five-axis machine tool dynamics modeling system based on monoid coding according to claim 6 is characterized in that: The kinematic modeling based on the local exponential product in the modular kinematic linearization modeling module includes the initial pose modeling and relative motion pose modeling of adjacent components, and the mathematical formulas are respectively expressed as follows: in, i-1 T i (0) represents the homogeneous transformation matrix of the initial pose, represents the spinor coordinate of the i-th component relative to the i-1-th component, q i is the rotation of the i-th component located at; E i represents the i-th joint around The homogeneous transformation matrix of the instantaneous motion of the spinor; SE(3) represents the three-dimensional special orthogonal group, and its physical meaning is to describe the movement and rotation characteristics of the rod; Based on the above mathematical derivation, the kinematic homogeneous transformation matrix of the entire machine tool is obtained by multiplying the initial posture and relative motion posture of multiple adjacent components, and the expression is as follows:

8. The five-axis machine tool dynamics modeling system based on monoid coding according to claim 6 is characterized in that: The linear modeling of modular kinematics about kinematic parameters in the multivariate polynomial coding module is specifically: the motion homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters, and the mathematical formula is as follows: Kinematic homogeneous transformation matrix product form It is expressed as a multivariate polynomial and a linear expression of kinematic parameters. The mathematical formula is as follows: in, and Represents the linearized expression of a single motion homogeneous transformation matrix; V β Indicates from The extracted parameter matrix, V α Indicates from The extracted parameter matrix, w β Indicates from The extracted multivariate polynomial vector, w a Indicates from The extracted multivariate polynomial vector; The partial differential form of the kinematic homogeneous transformation matrix is ​​expressed as a multivariate polynomial and a linear expression of the kinematic parameters: Among them, V i represents the kinematic parameter matrix, w′ i is a multivariate polynomial vector; vec means stacking the column vectors of the matrix vertically. represents the partial differential operation, and represents the continuous multiplication of Kronecker products.

9. The five-axis machine tool dynamics modeling system based on monoid coding according to claim 6 is characterized in that: The multivariate polynomial encoding module numerically encodes the multivariate polynomial in the monoid space, including the multivariate polynomial encoding, the elements and binary operations in the monoid, and the definition of the Kronecker product, and the symbolic expression of the multivariate polynomial is as follows: in, represents a multivariate polynomial with respect to a set of variables v, c i represents a real number, e i，t represents the order of the multivariate polynomial, represents the cumulative multiplication operation, represents the set of real numbers, represents a set of natural numbers, and v represents a set of variables; Defining a mapping The coefficients and degree of a single multivariate polynomial are extracted and encoded into a vector as follows: in, represents the transpose of a real vector, and T represents the transpose operation of a vector; For the set Φ0 of all single multivariate polynomial encoding vectors, define a binary operation that satisfies closure, associativity, and identity as follows: Further, Form a monoid; in order to realize the Kronecker product operation of a vector containing multiple multivariate polynomials and another vector containing multiple multivariate polynomials in the coding space, it is necessary to define the corresponding Kronecker product operation in the monoid, as follows: Among them, Φ0 represents the set of encoding vectors, represents the multiplication operation between any two encoding vectors that belong to the set of encoding vectors, represents the Kronecker product operation between any two encoding vectors that belong to the set of encoding vectors.

10. The five-axis machine tool dynamics modeling system based on monoid coding according to claim 6, characterized in that: The dynamic model in the linear representation module is represented as a linear representation of a multivariate polynomial vector and a parameter matrix, including inertia terms, Coriolis force and centrifugal force terms, and gravity terms converted into linear representations of multivariate polynomial vectors and inertia parameter matrices; through the Kronecker product operation defined in the monoid, the linearly represented motion homogeneous transformation matrix and the general Euler-Lagrange dynamics equation, the following dynamics about the multivariate polynomial linear representation is obtained by using the matrix trace cyclic property: in, and From the inertia term Coriolis and centrifugal force terms And the gravity term G(q), extract the multivariate polynomial vector; at the same time V d , V c and V g Contains kinematic parameters and inertial parameters.

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