A Five-Axis Machine Tool Dynamics Modeling Method and System Based on Monoid Coding

Abstract

This invention discloses a method and system for modeling the dynamics of a five-axis machine tool based on monoid coding. The method includes: S1, deriving the modular kinematics of a cascaded five-axis machine tool based on local exponential products; S2, completing the linearization modeling of the modular kinematics with respect to the kinematic parameters; S3, obtaining the linear representations of the inertia, centrifugal force, Coriolis force, and gravity terms of the dynamic model with respect to a multivariate polynomial; S4, describing the linearly represented dynamic model as a bilinear representation with respect to the inertia parameters and the multivariate polynomial; S5, eliminating redundant multivariate polynomials using matrix row correlation and eliminating redundant parameters using matrix decomposition to obtain the minimum set of inertia parameters; S6, inversely encoding the multivariate polynomial encoding numerical matrix and the parameter matrix after redundancy elimination into a symbolic representation with Homer form to achieve efficient model computation. This invention reduces the algorithm development and complexity of five-axis machine tools and is suitable for the rapid deployment of general-purpose five-axis machine tool algorithms.

Description

Technical Field

[0001] This invention belongs to the field of robot manufacturing technology, specifically relating to a five-axis machine tool dynamics modeling method and system based on monoid coding. Background Technology

[0002] Dynamic models are a crucial foundation for the electromechanical integration design and optimization, trajectory planning, and motion control of multi-axis machine tools, directly impacting important performance indicators such as dynamic response and positioning accuracy. To cover complex blank machining and concentrated process machining, five-axis machine tools feature high-degree-of-freedom configurations and designs with component offsets. However, the increased number of components and complex designs lead to problems such as low model derivation efficiency and high computational complexity, significantly increasing the difficulty of analyzing the dynamic response of machine tool components. Without altering the structure, model simplification can improve model derivation and computational efficiency. Current methods for model simplification suffer from the following problems:

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

[0004] (2) Secondly, a reduced-order dynamic model is used (e.g., patents CN104950804A and CN106407528A). This method establishes a simplified linear system transfer function model by ignoring the nonlinear dynamics of the whole machine. Under the condition that the system allows for permissible output errors, the order of this transfer function model is reduced. This method ignores the multi-axis coupling dynamic characteristics of the machine tool, resulting in a significant decrease in accuracy and making it difficult to use for high-speed, high-precision motion control.

[0005] (3) For multi-axis systems in series, most existing methods reduce redundant parameters (such as patents CN110539302A and CN110532581A). This method uses modeling methods such as Euler-Lagrange dynamics to derive the whole-body dynamics of the multi-axis system and reduces unnecessary parameter calculations by solving for 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] This invention addresses the problems of low derivation and computational efficiency in existing multi-axis machine tool dynamics modeling methods. It utilizes monoid coding and a method for eliminating redundant multivariate polynomials and dual-feature dynamic parameters to provide a general and efficient five-axis machine tool dynamics modeling and computation method based on monoid coding for multi-axis machine tool systems.

[0007] To achieve the aforementioned objectives, the present invention employs the following solution:

[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 serial five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the serial 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 of motion, the product of two homogeneous transformation matrices of motion, and a single homogeneous transformation matrix of motion with respect to the differential of motion displacement, and complete the linearization modeling of modular kinematics with respect to kinematic parameters; encode the multivariate polynomials numerically in the monoid space to obtain the multivariate polynomial encoding numerical matrix.

[0011] S3: Based on the multivariate polynomial encoding numerical matrix, matrix trace operator, and column vectorization operator, obtain the linear representation of the inertial term, centrifugal force, Coriolis force, and gravity term of the dynamic model with respect to the multivariate polynomial, and realize the dynamic model as a linear representation of multivariate polynomial vectors and parameter matrices;

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

[0013] S5: Redundant multivariate polynomials are eliminated by using matrix row correlation, and redundant parameters are eliminated by matrix decomposition to obtain the minimum set of inertial parameters.

[0014] S6: The multivariate polynomial encoded numerical matrix and the parameter matrix after redundancy removal are reverse encoded into a symbolic form with Horner form to achieve efficient model computation.

[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 a serial five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the serial five-axis machine tool based on the local exponential product.

[0017] The multivariate polynomial encoding module is used to extract multivariate polynomials from the homogeneous transformation matrix of motion, the product of two homogeneous transformation matrices of motion, and a single homogeneous transformation matrix of motion with respect to the differential of motion displacement using the Kronecker product operator, thereby completing the linearization modeling of modular kinematics with respect to kinematic parameters; and to numerically encode the multivariate polynomials in the monoid space to obtain the multivariate polynomial numerical matrix.

[0018] The linear representation module is used to obtain the linear representation of the inertial, centrifugal, Coriolis, and gravity terms of the dynamic model with respect to the multivariate polynomial based on the multivariate polynomial encoding numerical matrix, matrix trace operator, and column vectorization operator, thereby realizing the dynamic model as a linear representation of multivariate polynomial vectors and parameter matrices.

[0019] The bilinear representation module is used to describe a linearly represented dynamic model as a bilinear representation of inertial parameters and a multivariate polynomial.

[0020] The minimum inertial parameter set generation module is used to eliminate redundant multivariate polynomials by utilizing matrix row correlation and to eliminate redundant parameters by using matrix decomposition to obtain the minimum inertial parameter set.

[0021] The model computation module is used to reverse encode the multivariate polynomial encoded numerical matrix and the parameter matrix after redundancy removal into a symbolic representation with Horner form, thereby enabling efficient model computation.

[0022] Compared with existing technologies, this invention has at least the following advantages: Firstly, the monoid coding method used in this invention, compared to the purely symbolic method, results in lower memory consumption, significantly improving the efficiency of algorithm development and execution, including both model derivation and computation. Secondly, this method is suitable for the efficient derivation and computation of dynamic models for general-purpose serial five-axis machine tools, reducing the algorithm development and complexity for five-axis machine tools, and facilitating the rapid deployment of general-purpose five-axis machine tool algorithms. Attached Figure Description

[0023] To more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0024] Figure 1 This is a schematic diagram of the framework of a five-axis machine tool dynamic modeling method and system based on monoid coding, provided in a typical embodiment of the present invention.

[0025] Figure 2This is a modular kinematic linearization modeling block diagram provided in a typical embodiment of the present invention;

[0026] Figure 3 This is a block diagram of multivariate polynomial encoding based on monoids provided in a typical embodiment of the present invention;

[0027] Figure 4 This is a block diagram of dynamic modeling based on bilinear representation provided in a typical embodiment of the present invention;

[0028] Figure 5 This is a block diagram of a dynamic model redundancy multivariate polynomial and parameter elimination method provided in a typical embodiment of the present invention. Detailed Implementation

[0029] To make the objectives, technical solutions, and advantages of the present invention clearer, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Examples of these preferred embodiments are illustrated in the drawings. The embodiments of the present invention shown in and described with reference to the drawings are merely exemplary, and the present invention is not limited to these embodiments.

[0030] This invention proposes an efficient modeling and calculation method for five-axis machine tool dynamics based on monoid coding, comprising 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, such as... Figure 1 As 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 serial five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the serial 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 of motion, the product of two homogeneous transformation matrices of motion, and a single homogeneous transformation matrix of motion with respect to the differential of motion displacement, and complete the linearization modeling of modular kinematics with respect to kinematic parameters; encode the multivariate polynomials numerically in the monoid space to obtain the multivariate polynomial encoding numerical matrix.

[0034] S3: Based on the multivariate polynomial encoding numerical matrix, matrix trace operator, and column vectorization operator, obtain the linear representation of the inertial term, centrifugal force, Coriolis force, and gravity term of the dynamic model with respect to the multivariate polynomial, and realize the dynamic model as a linear representation of multivariate polynomial vectors and parameter matrices;

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

[0036] S5: Redundant multivariate polynomials are eliminated by using matrix row correlation, and redundant parameters are eliminated by matrix decomposition to obtain the minimum set of inertial parameters.

[0037] S6: The multivariate polynomial encoded numerical matrix and the parameter matrix after redundancy removal are reverse encoded into a symbolic form with Horner form to achieve efficient model computation.

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

[0039]

[0040] in, i-1 T i (0) represents the homogeneous transformation matrix of the initial pose. q represents the screw coordinate of the i-th component relative to the (i-1)-th component. i Let E be the rotational displacement of the i-th component. i Indicates that the i-th component wraps around The homogeneous transformation matrix of the spinor for instantaneous motion; SE(3) represents the three-dimensional special Euclidean group, which physically describes the movement and rotation characteristics of the rod; Based on the above mathematical derivation, the homogeneous transformation matrix of the entire machine tool is obtained by multiplying the initial poses and relative motion poses of multiple adjacent components, and the expression is as follows:

[0041]

[0042] In one embodiment, the linearization modeling of modular kinematics with respect to kinematic parameters in step S2 specifically involves: representing the homogeneous kinematic transformation matrix as a linear expression of a multivariate polynomial and kinematic parameters, as shown in the following mathematical formula:

[0043]

[0044] Where I4 represents a 4×4 identity matrix; Indicates the Kronecker product; w i Indicates from The extracted multivariate polynomial vector, V i Indicates from The extracted parameter matrices together form a linearized model of modular kinematics with respect to the kinematic parameters.

[0045] The product form of two homogeneous transformation matrices of motion It can be expressed as a linear representation of a multivariate polynomial and kinematic parameters, and its mathematical expression is as follows:

[0046]

[0047] in, and V represents the linearized expression for the homogeneous transformation matrix of a single motion; 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;

[0048] The partial differential form of the homogeneous kinematic transformation matrix is ​​expressed as a linear expression of a multivariate polynomial and kinematic parameters:

[0049]

[0050] Among them, V i Let w' represent the kinematic parameter matrix. i is a multivariate polynomial vector; vec indicates stacking the column vectors of the matrix vertically. This represents partial differential operations. and This represents the product of the Kronecker products.

[0051] In one embodiment, step S2 involves numerically encoding the multivariate polynomial in a monoid space, including multivariate polynomial encoding, the definition of elements and binary operations in the monoid, and the definition of the Kronecker product. The notation of the multivariate polynomial is expressed as follows:

[0052]

[0053] in, Let c represent the i-th multivariate polynomial in the set of variables v. i Let e ​​represent a real number. i,t Denotes the order of a multivariate polynomial. This indicates a cumulative multiplication operation. Represents the set of real numbers. Let v represent the set of natural numbers, and let v represent the set containing k variables;

[0054] Define a mapping The coefficients and degrees of a single multivariate polynomial are extracted and encoded into the following vector:

[0055]

[0056] in, The expression represents the transpose of a real vector, and T represents the transpose operation of a vector.

[0057] For the set Φ0 consisting of all individual multivariate polynomial encoded vectors, a binary operation satisfying closure, associativity, and identity is defined as follows:

[0058]

[0059] Furthermore, This forms a monoid; to implement the Kronecker product operation between a vector containing multiple multivariate polynomials and another vector containing multiple multivariate polynomials in the encoding space, the corresponding Kronecker product operation needs to be defined in the monoid, as follows:

[0060]

[0061] Where Φ0 represents the set of encoded vectors. This represents the multiplication operation between any two encoded vectors that belong to the encoded vector set. The Kronecker product operation is represented between any two encoded vectors belonging to the set of encoded vectors.

[0062] In one embodiment, the dynamic model in step S3 is represented as a linear representation of a multivariate polynomial vector and parameter matrix, including inertial terms, Coriolis force and centrifugal force terms, and gravity terms, which are transformed into a linear representation of the multivariate polynomial vector and inertial parameter matrix. Through the Kronecker product operation defined in the monoid, the kinematic homogeneous transformation matrix of the entire machine tool in linear representation, along with the general Euler-Lagrange dynamic equations, is used to obtain the following dynamics in linear representation of the multivariate polynomial using the cyclic property of the matrix trace:

[0063]

[0064] in, and These are from the inertia term Coriolis force and centrifugal force terms And the gravity term G(q), and the extracted multivariate polynomial vector; at the same time V d V c and V g It includes kinematic parameters and inertial parameters.

[0065] Another aspect of the present invention provides a five-axis machine tool dynamics modeling system based on monoid coding, comprising:

[0066] The modular kinematic linearization modeling module is used to define each screw axis of a serial five-axis machine tool, list the homogeneous transformation matrix of the initial coordinate system and the homogeneous transformation matrix of the current screw axis relative to the initial coordinate system, and derive the modular kinematics of the serial five-axis machine tool based on the local exponential product.

[0067] The multivariate polynomial encoding module is used to extract multivariate polynomials from the homogeneous transformation matrix of motion, the product of two homogeneous transformation matrices of motion, and a single homogeneous transformation matrix of motion with respect to the differential of motion displacement using the Kronecker product operator, thereby completing the linearization modeling of modular kinematics with respect to kinematic parameters; and to numerically encode the multivariate polynomials in the monoid space to obtain the multivariate polynomial numerical matrix.

[0068] The linear representation module is used to obtain the linear representation of the inertial, centrifugal, Coriolis, and gravity terms of the dynamic model with respect to the multivariate polynomial based on the multivariate polynomial encoding numerical matrix, matrix trace operator, and column vectorization operator, thereby realizing the dynamic model as a linear representation of multivariate polynomial vectors and parameter matrices.

[0069] The bilinear representation module is used to describe a linearly represented dynamic model as a bilinear representation of inertial parameters and a multivariate polynomial.

[0070] The minimum inertial parameter set generation module is used to eliminate redundant multivariate polynomials by utilizing matrix row correlation and to eliminate redundant parameters by using matrix decomposition to obtain the minimum inertial parameter set.

[0071] The model computation module is used to reverse encode the multivariate polynomial encoded numerical matrix and the parameter matrix after redundancy removal into a symbolic representation with Horner form, thereby enabling efficient model computation.

[0072] In one embodiment, the kinematic modeling based on local exponential products in the modular kinematic linearization modeling module includes initial pose modeling of adjacent components and relative motion pose, which are mathematically represented as follows:

[0073]

[0074] in, i-1 T i (0) represents the homogeneous transformation matrix of the initial pose. q represents the screw coordinate of the i-th component relative to the (i-1)-th component. i E represents the rotational displacement of the i-th component; i Indicates that the i-th component wraps around The screw is used as the instantaneous homogeneous transformation matrix; SE(3) represents the three-dimensional special orthogonal group, which physically describes 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 derived. It is obtained by multiplying the initial poses and relative motion poses of multiple adjacent components, and the expression is as follows:

[0075]

[0076] In one embodiment, the linearization modeling of modular kinematics with respect to kinematic parameters in the multivariate polynomial encoding module specifically involves: the homogeneous kinematic transformation matrix of the entire machine tool is represented as a linear expression of a multivariate polynomial and kinematic parameters, as shown in the following mathematical formula:

[0077]

[0078] Where I4 represents a 4×4 identity matrix; Indicates the Kronecker product; w i Indicates from The extracted multivariate polynomial vector, V i Indicates from The extracted parameter matrices together form a linearized model of modular kinematics with respect to the kinematic parameters.

[0079] The product form of two homogeneous transformation matrices of motion It can be expressed as a linear representation of a multivariate polynomial and kinematic parameters, and its mathematical expression is as follows:

[0080]

[0081] in, and V represents the linearized expression for the homogeneous transformation matrix of a single motion; 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;

[0082] The partial differential form of the homogeneous kinematic transformation matrix is ​​expressed as a linear expression of a multivariate polynomial and kinematic parameters:

[0083]

[0084] Among them, V i Let w' represent the kinematic parameter matrix. i is a multivariate polynomial vector; vec indicates stacking the column vectors of the matrix vertically. This represents partial differential operations. and This represents the product of the Kronecker products.

[0085] In one embodiment, the multivariate polynomial encoding module numerically encodes the multivariate polynomial in the monoid space, including multivariate polynomial encoding, the definition of elements and binary operations in the monoid, and the definition of the Kronecker product. The multivariate polynomial notation is expressed as follows:

[0086]

[0087] in, Let c represent the i-th multivariate polynomial in the set of variables v. i Let e ​​represent a real number. i,t Denotes the order of a multivariate polynomial. This indicates a cumulative multiplication operation. Represents the set of real numbers. Let v represent the set of natural numbers, and let v represent the set containing k variables;

[0088] Define a mapping The coefficients and degrees of a single multivariate polynomial are extracted and encoded into the following vector:

[0089]

[0090] in, The expression represents the transpose of a real vector, and T represents the transpose operation of a vector.

[0091] For the set Φ0 consisting of all individual multivariate polynomial encoded vectors, a binary operation satisfying closure, associativity, and identity is defined as follows:

[0092]

[0093] Furthermore, This forms a monoid; to implement the Kronecker product operation between a vector containing multiple multivariate polynomials and another vector containing multiple multivariate polynomials in the encoding space, the corresponding Kronecker product operation needs to be defined in the monoid, as follows:

[0094]

[0095] Where Φ0 represents the set of encoded vectors. This represents the multiplication operation between any two encoded vectors that belong to the encoded vector set. The Kronecker product operation is represented between any two encoded vectors belonging to the set of encoded vectors.

[0096] 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 inertial terms, Coriolis force and centrifugal force terms, and gravity terms, which are transformed into a linear representation of a multivariate polynomial vector and an inertial parameter matrix. Through the Kronecker product operation defined in the monoid, the kinematic homogeneous transformation matrix of the entire machine tool in linear representation, along with the general Euler-Lagrange dynamic equations, is used to obtain the following dynamics in the multivariate polynomial linear representation using the matrix trace cyclic property:

[0097]

[0098] in, and These are from the inertia term Coriolis force and centrifugal force terms And the gravity term G(q), and the extracted multivariate polynomial vector; at the same time V d V c and V g It includes kinematic parameters and inertial parameters.

[0099] In one embodiment, the multivariate polynomial vectors and parameter matrices in the linear expressions of the inertial, Coriolis, centrifugal, and gravity terms are extracted using the following methods:

[0100] (a) Inertia term:

[0101]

[0102] Among them, e ij and f ij Represented as:

[0103]

[0104] (b) Coriolis force and centrifugal force terms:

[0105]

[0106] Among them, e i,jk and f i,jk Represented as:

[0107]

[0108] (c) Gravity term:

[0109]

[0110] Here This indicates that multiple matrices or vector blocks are stacked on the diagonal of a larger matrix to form a block diagonal matrix.

[0111] In one embodiment, the removal of redundant multivariate polynomials and inertial parameters from the dynamic model includes the bilinear representation of the dynamic model, and the removal of redundant multivariate polynomials and inertial parameters, such as... Figure 5 As shown. The bilinear dynamic model is as follows:

[0112]

[0113] in It is an index matrix containing kinematic parameters. This is the standard inertia parameter matrix. Secondly, matrix row correlation analysis is used to eliminate [the problematic elements]. The redundant multivariate polynomials in the equation are obtained.

[0114]

[0115] in Finally, the full-rank decomposition method was used to... By decomposing the parameters and removing redundant inertial parameters, a simplified parameter vector θ is obtained. b This leads to the simplified parametric linearized dynamic model:

[0116]

[0117] It is worth noting that the redundancy removal process here is implemented in the coding space.

[0118] In one embodiment, the step of reverse encoding the multivariate polynomial encoded numerical matrix and the parameter matrix after redundancy removal into a parametric linearized dynamic model symbol with Horner form specifically involves encoding the multivariate polynomial after redundancy removal. Sort the encoding matrices in descending order based on their row numbers and size. Finally, combine the constant matrices L and P. i and the minimum inertial parameter vector θ b By using the definition of multivariate polynomials and Horner notation, we obtain the following simplified parametric linearized dynamic model:

[0119] τ T =Yθ b

[0120] Where Y represents the regression matrix.

[0121] It should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This way of describing the specification is only for clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions of each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for five-axis machine tool dynamics modeling based on monoid coding, characterized by, Comprise: S1, each rotational axis of the serial five-axis machine tool is defined, the initial coordinate system homogeneous transformation matrix and the current rotational axis relative to the initial coordinate system homogeneous transformation matrix are listed, and the local exponential product is used for modular kinematics derivation of the serial five-axis machine tool; S2, using the kronecker product operator, extracting multivariate polynomials from motion homogeneous transformation matrix, product of two motion homogeneous transformation matrices and single motion homogeneous transformation matrix about motion displacement differential, completing linear modeling of modular kinematics about kinematic parameters; numerical coding of multivariate polynomials in monoid space is carried out to obtain multivariate polynomial coding numerical matrix; S3: according to the multivariate polynomial coding numerical matrix, matrix trace operator and column vectorization operator, the inertia term, centrifugal force and coriolis force term and gravity term of the dynamic model are obtained Linear representation about multivariate polynomial, realize the dynamic model to be represented as linear representation of multivariate polynomial vector and parameter matrix; S4: the linear representation of the dynamic model is described as a bilinear representation about the inertia parameter and the multivariate polynomial; S5: using matrix row correlation to eliminate redundant multivariate polynomials, using matrix decomposition to eliminate redundant parameters, and obtaining the minimum inertia parameter set; S6: the multivariate polynomial coding numerical matrix and the parameter matrix after eliminating redundancy are decoded into symbolic expressions with Horner form, and the efficient calculation of the model is realized.

2. The monoid-based encoding method for five-axis machine tool dynamics modeling according to claim 1, characterized in that, The kinematics modeling based on local exponential product in step S1 includes initial pose modeling of adjacent components and relative motion pose, and the mathematical expressions are as follows: wherein, i-1 T i (0) denotes a homogeneous transformation matrix of an initial pose, denotes the screw coordinates of the i-th component relative to the i-1-th component, q i is the rotational displacement of the i-th component; E i denotes the motion homogeneous transformation matrix of the i-th component around screw doing instantaneous motion; SE(3) denotes a three-dimensional special orthogonal group, which has a physical meaning of describing the moving and rotating characteristics of a rod; based on the above mathematical formula derivation, the kinematics homogeneous transformation matrix of the entire machine tool is obtained by multiplying the initial pose and the relative motion pose of multiple adjacent components, and the expression is as follows:

3. The monoid-based encoding method for five-axis machine tool dynamics modeling according to claim 2, characterized in that, The linear modeling of modular kinematics about kinematic parameters in step S2 is as follows: the kinematics homogeneous transformation matrix of the whole machine tool is represented as linear expression of multivariate polynomial and kinematic parameters, and the mathematical expression is as follows: where I4 denotes a 4x4 identity matrix; denotes the Kronecker product; w i denotes the vector of extracted polynomial vectors, V i denotes the vector of extracted parameters, both constituting a linearized model of the modular kinematics with respect to the kinematic parameters; Product of two motion homogeneous transformation matrices is expressed as a multivariate polynomial and a linear representation of kinematic parameters, which is mathematically expressed as follows: wherein, and denotes a linearized representation of a single motion homogeneous transformation matrix; V β denotes a parameter matrix extracted from V α denotes a parameter matrix extracted from w β denotes a vector of multivariate polynomials extracted from w α denotes a vector of multivariate polynomials extracted from ​ The partial differential form of the kinematics homogeneous transformation matrix is represented as linear expression of multivariate polynomial and kinematic parameters: where V i denotes the kinematic parameter matrix, w i is a multivariate polynomial vector; vec denotes the vertical stacking of the column vectors of a matrix, denotes the partial differential operation, and denotes the Kronecker product.

4. The monoid-based encoding method for five-axis machine tool dynamics modeling according to claim 1, characterized in that, The numerical coding of multivariate polynomials in monoid space in step S2 includes multivariate polynomial coding, element definition in monoid and binary operation definition, and kronecker product definition, and the symbolic expression of multivariate polynomial is as follows: wherein denotes the i-th multivariate polynomial with respect to the set of variables v, c i denotes a real number, e i,t denotes the degree of the multivariate polynomial, denotes a multiplication operation, denotes a set of real numbers, denotes a set of natural numbers, v denotes a set containing k variables; Defining a mapping The coefficients and degrees of the individual multivariate polynomials are extracted and encoded into a vector as follows: wherein denotes the transpose of a real vector, T denotes the transpose operation of a vector; For the set Φ0 composed of all single multivariate polynomial coding vectors, a binary operation satisfying closure, associativity and identity is defined as follows: Further, The set of vectors forms a monoid; in order to implement the Kronecker product operation of a vector containing multivariate polynomials with another vector containing multivariate polynomials in the code space, the corresponding Kronecker product operation needs to be defined in the monoid as follows: where Φ0denotes the set of code vectors, denotes the multiplication between any two code vectors belonging to the set of code vectors, denotes the Kronecker product between any two code vectors belonging to the set of code vectors.

5. The monoid-based encoding method for five-axis machine tool dynamics modeling according to claim 1, characterized in that, In step S3, the dynamic model is represented as linear representation of multivariate polynomial vector and parameter matrix, including inertia term, coriolis force and centrifugal force term, and gravity term into linear representation about multivariate polynomial vector and inertia parameter matrix; through the kronecker product operation defined in monoid, the kinematics homogeneous transformation matrix of the whole machine tool and the general Euler-Lagrange dynamic equation are linearly represented, and the dynamics about multivariate polynomial linear representation is obtained by using the cyclic property of matrix trace as follows: where and are the multivariate polynomial vectors extracted from the inertial terms Coriolis and centrifugal force terms and the gravitational term G(q); while V d , V c and V g contain the kinematic and inertial parameters.

6. A five-axis machine tool dynamics modeling system based on monoid coding, characterized by, Comprise: The modular kinematics linear modeling module is used for defining each rotational axis of the serial five-axis machine tool, listing the initial coordinate system homogeneous transformation matrix and the current rotational axis relative to the initial coordinate system homogeneous transformation matrix, and deriving the modular kinematics of the serial five-axis machine tool based on local exponential product; A multi-variate polynomial encoding module is configured to extract multi-variate polynomials from a motion homogeneous transformation matrix, a product of two motion homogeneous transformation matrices, and a single motion homogeneous transformation matrix with respect to motion displacement differential, complete linear modeling of modular kinematics with respect to kinematic parameters by using a Kronecker product operator, and perform numerical encoding of the multi-variate polynomials in a monoid space to obtain a multi-variate polynomial encoding numerical matrix; A linear representation module is configured to obtain linear representations of inertia terms, centrifugal and Coriolis force terms, and gravity terms of a dynamic model with respect to multi-variate polynomials according to the multi-variate polynomial encoding numerical matrix, a matrix trace operator, and a column vectorization operator, and realize linear representation of the dynamic model as a multi-variate polynomial vector and a parameter matrix; A bilinear representation module is configured to describe the linear representation of the dynamic model as a bilinear representation form with respect to inertia parameters and multi-variate polynomials; A minimum inertia parameter set generation module is configured to remove redundant multi-variate polynomials by using matrix row correlation and remove redundant parameters by using matrix decomposition to obtain a minimum inertia parameter set; A model calculation module is configured to perform reverse encoding of the multi-variate polynomial encoding numerical matrix and the parameter matrix after removal of redundancy into a symbolic expression with a Horner form to realize efficient calculation of the model.

7. The monoid-based encoding system for five-axis machine tool dynamics modeling of claim 6, wherein, The local exponential product-based kinematic modeling in the modular kinematics linear modeling module includes initial pose modeling and relative motion pose of adjacent components, and mathematical expressions are respectively as follows: wherein, i-1 T i (0) denotes a homogeneous transformation matrix of an initial pose, denotes the screw coordinates of the ith component relative to the (i-1)th component, q i is the rotational displacement of the ith component; E i denotes the motion homogeneous transformation matrix of the ith component around screw doing instantaneous motion; SE(3) denotes a three-dimensional special orthogonal group, which has a physical meaning of describing the moving and rotating characteristics of a rod; based on the above mathematical formula derivation, the kinematics homogeneous transformation matrix of the entire machine tool is obtained by multiplying the initial pose and the relative motion pose of multiple adjacent components, and the expression is as follows:

8. The monoid-based encoding system for five-axis machine tool dynamics modeling of claim 7, wherein, The linear modeling of modular kinematics with respect to kinematic parameters in the multi-variate polynomial encoding module is specifically that the kinematic homogeneous transformation matrix of the entire machine tool is represented as a linear expression of multi-variate polynomials and kinematic parameters, and a mathematical expression is as follows: Product of two motion homogeneous transformation matrices is expressed as a multivariate polynomial and a linear representation of kinematic parameters, which is mathematically expressed as follows: wherein, and denotes a linearized expression of a single motion homogeneous transformation matrix; V β denotes a parameter matrix extracted from V α denotes a parameter matrix extracted from w β denotes a multivariate polynomial vector extracted from w α denotes a multivariate polynomial vector extracted from ​ The partial differential form of the kinematic homogeneous transformation matrix is represented as a linear expression of multi-variate polynomials and kinematic parameters: where V i represents a kinematic parameter matrix, w i is a multivariate polynomial vector; vec represents the vertical stacking of column vectors of a matrix, denotes a partial differential operation, and denotes the Kronecker product.

9. The monoid-based encoding system for five-axis machine tool dynamics modeling of claim 6, wherein, The numerical encoding of the multi-variate polynomials in the monoid space in the multi-variate polynomial encoding module includes multi-variate polynomial encoding, definition of elements and binary operations in the monoid, and definition of the Kronecker product, and the multi-variate polynomial symbolic expression is as follows: wherein denotes the i-th multivariate polynomial with respect to the set of variables v, c i denotes a real number, e i,t denotes the degree of the multivariate polynomial, denotes a multiplication operation, denotes a set of real numbers, denotes a set of natural numbers, v denotes a set containing k variables; Defining a mapping The coefficients and degrees of the individual multivariate polynomials are extracted and encoded into a vector as follows: wherein denotes the transpose of a real vector, T denotes the transpose operation of a vector; For a set Φ0 composed of all single multi-variate polynomial encoding vectors, a binary operation satisfying closure, associativity, and identity is defined as follows: Further, The set of vectors forms a monoid; in order to implement the Kronecker product operation of a vector containing multivariate polynomials with another vector containing multivariate polynomials in the code space, the corresponding Kronecker product operation needs to be defined in the monoid as follows: where Φ0denotes the set of code vectors, denotes the multiplication between any two code vectors belonging to the set of code vectors, denotes the Kronecker product between any two code vectors belonging to the set of code vectors.

10. The monoid-based encoding system for five-axis machine tool dynamics modeling of claim 6, wherein, The linear representation of the dynamic model as a linear representation of a multi-variate polynomial vector and a parameter matrix in the linear representation module includes conversion of inertia terms, Coriolis force and centrifugal force terms, and gravity terms into a linear representation with respect to a multi-variate polynomial vector and an inertia parameter matrix; through the Kronecker product operation defined in the monoid, the kinematic homogeneous transformation matrix of the entire machine tool and the general Euler-Lagrange dynamic equation are linearly represented, and the dynamic with respect to the multi-variate polynomial linear representation is obtained by using the cyclic property of the matrix trace as follows: where and are the multivariate polynomial vectors extracted from the inertial terms Coriolis and centrifugal force terms and the gravitational term G(q); while V d , V c and V g contain the kinematic and inertial parameters.

Images