Robot full-flexible rigid-flexible coupling parametric modeling method and system

Abstract

The invention provides a robot full-flexible rigid-flexible coupling parametric modeling method and system, and belongs to the technical field of robot modeling. According to the method, firstly, based on MDH flexible expansion parameters, a floating coordinate method, a finite element method and a Rayleigh beam model are combined, and a full-flexible kinematics model containing flexible deformation of a base, joints and connecting rods is established; secondly, the speed and acceleration of key nodes of the model and any point of the flexible connecting rod are calculated in a recursive mode; constructing a local kinetic equation of each concentrated mass component and each distributed mass component based on a virtual power principle, and converting the local kinetic equations into a generalized coordinate space through a generalized Jacobian matrix; and finally, assembling to form a unified robot rigid-flexible coupling kinetic equation. According to the method, high efficiency and automation of the modeling process are realized, the accuracy of the model for describing the dynamic characteristics of the robot is improved, and an accurate basic model is provided for high-precision dynamic analysis and control of the robot.

Description

Technical Field

[0001] This invention belongs to the field of robot modeling technology, and in particular relates to a parametric modeling method and system for fully flexible rigid-flexible coupling of robots. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] The high-precision control and dynamic performance of robots largely depend on the accuracy of their kinematic and dynamic models. In high-speed, heavy-load, or high-precision operation scenarios, the robot's links, joints, and even base, which are treated as rigid bodies, undergo non-negligible elastic deformation. This deformation causes positioning errors in the end effector and significantly alters the system's dynamic characteristics. If traditional rigid-body models are continued for system analysis and control algorithm design, the robot's operational accuracy and stability will be severely affected. Therefore, establishing a high-precision rigid-flexible coupling dynamic model and applying it to control has become one of the key technologies for improving the performance of high-end robots.

[0004] However, existing methods for modeling robot flexibility still have many limitations, mainly in the following aspects: First, the model construction process is complex, with low automation and practicality. Most existing methods are based on energy-based modeling methods such as the Lagrange method, whose derivation involves complex partial differential calculations. The computational load increases dramatically with the robot's degrees of freedom, leading to inefficient model derivation. Simultaneously, existing methods lack a parameterization mechanism to specify rigid components, typically requiring flexible modeling of all links or joints, further increasing model complexity and computational burden. Furthermore, the modeling parameters used by many methods are incompatible with the MDH parameters commonly used in industry, and often require prior experimental or simulation acquisition of the robot's link mode shapes, resulting in significant additional manual work. These issues collectively lead to the current complex and cumbersome flexible modeling process with low automation, limiting the practice of rigid-flexible coupling modeling to simple planar two-link robot models in academic research, severely restricting its engineering application in complex industrial robots.

[0005] Second, the flexibility factor is not fully considered, resulting in limited model accuracy. Most existing flexibility modeling methods can be categorized as "joint flexibility modeling" or "link flexibility modeling," meaning they only consider joint or link flexibility and ideally assume other structures are absolutely rigid. However, in real-world applications, the robot's base, joints, and links may all exhibit varying degrees of elastic deformation. Furthermore, because robots are serial structures, the deformation of these components accumulates through the kinematic chain and simultaneously affects the final pose response of the end effector. This one-sided consideration of the sources of flexibility leads to models that cannot fully reflect the actual deformation of the robot system, introducing unavoidable systematic modeling errors.

[0006] Third, the coupling effect is oversimplified, leading to distorted dynamic descriptions. A few studies attempt to model the flexibility of both links and joints simultaneously, but the coupling effect between the system's rigid large motion and flexible elastic deformation is often not fully reflected. At the kinematic level, the flexible deformation of links is generally not correctly incorporated into the position recursion formula, or is simply simplified to an equivalent additional rotation angle at the joint, resulting in inaccurate mathematical representations of the end effector position. At the dynamic level, the influence of joint and link deformation on key dynamic parameters such as the system's inertia matrix and Coriolis / centrifugal force matrix is ​​often ignored or linearized, and the additional angular velocity caused by deformation is often incorrectly discarded, leading to problems such as the lack of "rotational stiffening" effect and instability in numerical simulations. Furthermore, for almost all flexible modeling methods based on the assumed modal method, the influence of robot body posture changes and end-effector load changes on link vibration modes is not considered. The modeling process is always based on a fixed and invariant set of link mode shape functions, which further reduces the dynamic accuracy of the model under different configurations and loads. Summary of the Invention

[0007] To overcome the shortcomings of the existing technologies, this invention provides a parametric modeling method and system for fully flexible rigid-flexible coupling of robots. It aims to solve the problems existing in the flexible modeling of robots, such as cumbersome model construction process, low degree of automation, incompatibility with industrial standard parameters, incomplete consideration of flexible factors, and oversimplification of deformation coupling effects. The invention achieves efficient, accurate, and comprehensive rigid-flexible coupling modeling that describes the dynamic characteristics of robots.

[0008] To achieve the above objectives, one or more embodiments of the present invention provide the following technical solutions: The first aspect of this invention provides a parametric modeling method for fully flexible rigid-flexible coupling of robots; A parametric modeling method for fully flexible rigid-flexible coupling of robots includes: Based on the MDH flexible extension parameters, and combined with the floating coordinate method, finite element method and Rayleigh beam model, a kinematic model that simultaneously includes the flexible deformation of the robot base, joints and links is established. Based on the established kinematic model, symbolic velocities and accelerations of each key node are recursively calculated, and the velocity and acceleration of any point on a flexible link with distributed mass are calculated using a velocity mapping operator. Based on the principle of virtual power, the dynamic equilibrium equations of each concentrated mass component and distributed mass component in the kinematic model are established in Cartesian space, and the dynamic equilibrium equations are transformed to generalized coordinate space using the generalized Jacobian matrix. In the generalized coordinate space, the local dynamic equations of all components are assembled to form a unified rigid-flexible coupling dynamic equation for the robot system.

[0009] As a further technical solution, based on the MDH flexible extension parameters, and combining the floating coordinate method, finite element method, and Rayleigh beam model, a kinematic model is established that simultaneously includes the flexible deformation of the robot base, joints, and links, including: Based on the MDH parameters, flexible binary switch parameters and joint flexibility type parameters are introduced to extend the flexible modeling of rigid kinematics, resulting in the MDH flexible extended parameters. The floating coordinate method is used to describe the flexible deformation of each component, and the spatial position of the deformed component is regarded as the superposition of rigid transformation and flexible deformation. The deformation increment matrix of each component is constructed by the first-order linear approximation of the Lie algebra representation of the deformation, and then the homogeneous transformation matrix of the base, joint and link considering the flexible deformation is obtained. Flexible modeling of robot components is performed based on homogeneous transformation matrix with small deformation. The base and joints are modeled as linear elastic models with concentrated mass to describe their small deformation characteristics. The robot links are modeled as Rayleigh beam models with distributed mass and considering rotational inertia. The Rayleigh beam model is discretized using the finite element method, and the links are divided into several beam elements. The deformation of the entire link is characterized by the deformation of the element nodes. Based on the MDH flexible extension parameters, homogeneous transformation matrix, and component flexibility model defined above, a recursive relationship is established from the base coordinate system to the end effector coordinate system. According to the serial sequence of robot joints and links, the coordinate system transformation relationship including all flexible deformation terms of the base, joints, and links is recursively calculated, and finally a stylized position recursive expression is formed, resulting in a fully flexible kinematic model of the robot that simultaneously covers the flexible deformation of the base, joints, and links.

[0010] As a further technical solution, the deformation increment matrix is ​​derived from the spatial deformation vector, which includes a torsional vector and a linear deformation vector; the homogeneous transformation matrix of each component considering flexible deformation is a linear superposition of the rigid transformation homogeneous matrix and the deformation increment matrix, and second-order and higher small deformation terms are ignored in the position recursion.

[0011] As a further technical solution, based on the established kinematic model, symbolic velocities and accelerations are recursively calculated for each key node, and the velocity and acceleration of any point on the flexible link with distributed mass are calculated using a velocity mapping operator, including: The origin of each coordinate system in the kinematic model is set as the key node, and the recursive sequence is determined according to the coordinate system order from the robot base to the end effector. Starting from the base coordinate system fixed to the ground, for any two adjacent coordinate systems, establish the recursive relationship between their origins for absolute angular velocity and linear velocity, and perform coordinate transformation by combining the rotation matrix and translation vector between the two coordinate systems; calculate in sequence according to the recursive sequence to obtain the absolute angular velocity and linear velocity of all key nodes in the base coordinate system. The system's generalized coordinates are a set of rigid joint angular vectors and flexible deformation vectors. The absolute angular velocity and linear velocity of each key node are expressed as linear functions of the generalized velocity using the generalized Jacobian matrix. The acceleration of the key node is obtained by differentiating the linearized expression of the key node velocity with respect to time. Based on the recursive results of velocity and acceleration of key nodes, a calculation relationship is defined for a specified point in the axis direction of the beam element coordinate system. For a specified point on any beam element in a flexible link, the velocity and acceleration of that point in the link coordinate system are derived using the calculation relationship and the velocity and acceleration of the relevant key nodes.

[0012] As a further technical solution, the velocity mapping operator consists of an antisymmetric matrix of the position of a specified point in the link coordinate system, an antisymmetric matrix of the link angular velocity, a homogeneous transformation matrix from the beam element coordinate system to the link coordinate system, a Rayleigh beam shape function, and a deformation selection matrix of the beam element; the velocity and acceleration of any point on the flexible link are obtained by multiplying the velocity mapping operator with the velocities of relevant key nodes.

[0013] As a further technical solution, based on the principle of virtual power, the dynamic equilibrium equations of each concentrated mass component and distributed mass component in the kinematic model are established in Cartesian space, and the dynamic equilibrium equations are transformed to a generalized coordinate space using the generalized Jacobian matrix, including: The base and joints in the kinematic model are set as concentrated mass components, and the flexible links in the kinematic model are set as distributed mass components, and discretized into several beam elements; Construct the mass matrix and stiffness matrix of the lumped mass component in Cartesian space, and the mass matrix per unit length of the beam element, respectively. Based on the principle of virtual power, the dynamic equilibrium equations in Cartesian space and the dynamic equilibrium equations of a single beam element in Cartesian space are established respectively. The Cartesian dynamic equilibrium equations of the lumped mass component are transformed to the generalized coordinate space using the generalized Jacobian matrix, yielding the mass matrix, Coriolis force matrix, gravity matrix, stiffness matrix, active force matrix, and external force matrix in the generalized coordinate space. Based on the deformation selection matrix of the beam element, the Cartesian dynamic equilibrium equations of a single beam element are transformed to the generalized coordinate space using the generalized Jacobian matrix. The transformation results of all beam elements are then summarized to form the local dynamic equations of the distributed mass component in the generalized coordinate space.

[0014] As a further technical solution, in the generalized coordinate space, the local dynamic equations of all components are assembled to form a unified rigid-flexible coupling dynamic equation for the robot system. This includes: summarizing the mass, Coriolis force, gravity, stiffness and generalized force matrix of each component in the generalized coordinate space to construct an overall dynamic matrix, taking the joint driving force as the main driving force, eliminating the mutual canceling support reaction forces between components, and assembling to form an overall rigid-flexible coupling dynamic equation.

[0015] The second aspect of this invention provides a parametric modeling system for fully flexible rigid-flexible coupling of robots.

[0016] A parametric modeling system for fully flexible rigid-flexible coupling of robots includes: The kinematic model construction module is configured to: establish a kinematic model that simultaneously includes the flexible deformation of the robot base, joints and links, based on the MDH flexible extension parameters and combined with the floating coordinate method, finite element method and Rayleigh beam model. The velocity and acceleration recursion module is configured to: based on the established kinematic model, perform symbolic velocity and acceleration recursion on each key node, and calculate the velocity and acceleration of any point on the flexible link with distributed mass through the velocity mapping operator; The module for establishing dynamic equilibrium equations is configured to: establish dynamic equilibrium equations for each concentrated mass component and distributed mass component in the kinematic model in Cartesian space based on the principle of virtual power, and transform the dynamic equilibrium equations to generalized coordinate space using the generalized Jacobian matrix; The overall dynamic equation assembly module is configured to assemble the local dynamic equations of all components in the generalized coordinate space, forming a unified rigid-flexible coupling dynamic equation for the robot system.

[0017] A third aspect of the present invention provides a computer-readable storage medium having a program stored thereon, which, when executed by a processor, implements the steps of a parametric modeling method for fully flexible rigid-flexible coupling of a robot as described in the first aspect of the present invention.

[0018] A fourth aspect of the present invention provides an electronic device, including a memory, a processor, and a program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps in the parametric modeling method for fully flexible rigid-flexible coupling of a robot as described in the first aspect of the present invention.

[0019] The above one or more technical solutions have the following beneficial effects: (1) This invention defines flexible extension parameters for MDH, making the rigid-flexible coupling modeling parameters fully compatible with the MDH parameters for rigid robot modeling. Flexible modeling can be performed by supplementing necessary material property parameters, avoiding cumbersome parameter conversion. Users only need to supplement a small number of parameters such as flexible switches, joint types, and material properties on the basis of traditional rigid MDH parameters to perform flexible modeling, avoiding cumbersome parameter system conversion and repeated definition, greatly reducing the threshold for use. By introducing flexible switch parameters and joint flexibility type parameters, users can selectively specify the base, joint, or link as rigid or flexible according to actual accuracy requirements and computing resources, realizing "flexible on demand" parametric modeling. This effectively avoids unnecessary computational overhead, fundamentally controls model complexity, and solves the problems of model degree of freedom expansion and large computational load in existing methods. The finite element method is used to discretize the flexible link, eliminating the need for complex experiments or simulations to obtain the link's mode shape as in the hypothetical modal method, saving a lot of preliminary manpower work and simplifying the modeling preparation work. By directly constructing the dynamic equations based on the principle of virtual power, the cumbersome partial differential and derivative calculations in energy methods such as the Lagrange method are avoided, simplifying the derivation process and making it easier to automate the model export through symbolic computation software.

[0020] (2) This invention overcomes the limitations of existing methods that only focus on a single flexibility source by simultaneously considering the flexibility of the base, joints, and links within a unified framework. Through kinematic recursion, it accurately reflects the cumulative effect of the deformation of all components on the end-effector pose, thereby more completely reflecting the true deformation state of the robot system and eliminating the systematic errors introduced by the incomplete consideration of flexibility sources.

[0021] In the modeling process, this invention fully reflects the coupling effect between rigid motion and elastic deformation. At the kinematic level, the deformation of the link is introduced into the position recursion; at the dynamic level, the influence of deformation on key dynamic parameters such as the inertia matrix and Coriolis force matrix is ​​retained, and the angular velocity caused by torsion is considered, which correctly reflects nonlinear effects such as "rotational stiffening" and improves the realism and numerical stability of the dynamic simulation.

[0022] Because the finite element method is used instead of the fixed modal method, the mode shape function of the link is no longer fixed, but dynamically changes with the robot's pose and end effector load. This allows the dynamic model established by this method to adapt to different working states of the robot, significantly improving the dynamic accuracy of the model in the entire workspace, and is especially suitable for simulation and control under large-range motion and high dynamic conditions.

[0023] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0024] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0025] Figure 1 This is a flowchart of the method in the first embodiment.

[0026] Figure 2 This is a system structure diagram of the second embodiment. Detailed Implementation

[0027] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0028] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations of the present invention.

[0029] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0030] Example 1 This embodiment discloses a parametric modeling method for fully flexible rigid-flexible coupling of robots; like Figure 1 As shown, a parametric modeling method for fully flexible rigid-flexible coupling of robots includes: Step S1: Based on the MDH flexible extension parameters, and combining the floating coordinate method, finite element method and Rayleigh beam model, establish a kinematic model that simultaneously includes the flexible deformation of the robot base, joints and links. Step S2: Based on the established kinematic model, symbolic velocity and acceleration are recursively calculated for each key node, and the velocity and acceleration of any point on the flexible link with distributed mass are calculated using the velocity mapping operator. Step S3: Based on the principle of virtual power, establish the dynamic equilibrium equations of each concentrated mass component and distributed mass component in the kinematic model in Cartesian space, and use the generalized Jacobian matrix to transform the dynamic equilibrium equations to the generalized coordinate space. Step S4: In the generalized coordinate space, assemble the local dynamic equations of all components to form a unified rigid-flexible coupling dynamic equation for the robot system.

[0031] Specifically, it also includes the following: In step S1, a flexible modeling extension of rigid kinematics is performed based on widely used MDH parameters. In MDH modeling, the base is numbered 0, starting from the base... The MDH nodes are numbered from 1 to 1. Use number and This represents any two adjacent MDH nodes, connected by a first link. ,joint Second Link Sequential connection. Any MDH node. A coordinate system is fixed at each location. Its coordinate axes Along the joint The direction of the axis, coordinate axis At the same time perpendicular to The coordinate system is fixed with the previous MDH node. coordinate axes .

[0032] In the case of full rigidity, coordinate system arrive Transformation matrix for: (1) in, , , , All are MDH parameters. for along Direction to distance, for along Direction to distance, for Around arrive The corner for Around arrive The corner. and These represent a rigid rotation or translation of a given magnitude along a specified coordinate axis, respectively. , , , These are all homogeneous transformation matrices corresponding to a single rigid rotation or translation. A coordinate system is defined for ease of subsequent explanation. , and They are respectively fixed to the first link. Second link and joints starting point.

[0033] Based on the MDH parameters, a flexible binary switch parameter and a joint flexibility type parameter are introduced to extend the flexible modeling of rigid kinematics, selectively performing flexible modeling on the base, joints, and links. A binary switch parameter is introduced into the homogeneous transformation matrix of each component. Implementation, in which Indicates rigidity. This represents flexibility. The homogeneous transformation matrix and spatial deformation vector of each component can be uniformly represented as: (2) In the formula To consider the flexible homogeneous transformation matrix, For rigid parts, For spatially deformable vectors, operators This represents the vertical splicing of various elements. and These are the torsion vector and the deformation vector, respectively. This is the flexible deformation increment matrix, and its specific definition and calculation method are given below. For the base and the first link... Second link and joints The corresponding switching parameters are as follows: , , and .

[0034] It also introduces a joint flexibility type parameter. To define different joint flexibility modeling methods, among which Indicates joint It is a rigid joint. This indicates that only torsion along the joint axis is considered. This represents modeling the six-dimensional deformation of the joint, which is done by modifying the joint. Spatial deformation vector The deformation parameters included in the implementation are as follows: (3) Furthermore, this method ignores second-order and higher small deformations in the position recursion to further simplify the model.

[0035] In line with the aforementioned MDH flexible scaling modeling, when Flexible homogeneous transformation matrix of each component The definition is derived as follows: Deformation is represented using the floating coordinate method based on a rigid reference frame defined by MDH parameters, treating the deformed spatial position as a superposition of rigid transformation and deformation. The coordinate system for small flexible deformations can be obtained through a first-order linear approximation of the deformation's Lie algebraic representation. arrive Homogeneous transformation matrix The general form is: (4) In the formula It is the identity matrix. and It can represent any two coordinate systems. and Let be the homogeneous transformation matrix for rigid transformation and deformation between the two coordinate systems. For the deformation increment matrix, For the transformed Lie algebra representation: (5) in For spatial deformation vectors, and These are the twist vector and the deformed vector, respectively, and the operators are... Represents the vertical concatenation of elements, operator This represents an antisymmetric matrix. Since the modeling deformations are all small deformations, the product of elements of second-order and higher small deformations can be considered to be 0.

[0036] Flexible modeling of each robot component is performed based on the general form of the homogeneous transformation matrix with small deformation to obtain the deformation increment matrix. This method models the base and joints as linear elastic models with lumped mass, and their rigid position transformations are respectively determined by the identity matrix. and homogeneous transformation matrix Definition, deformation increment matrix , They are defined as follows: (6) The base and joints are obtained when flexible deformation is considered. Homogeneous transformation matrix of spatial transformation: (7) Next, flexible modeling of the link is performed. Robot links have distributed mass, and simple lumped parameter modeling may result in significant deviations in deformation calculations. In this embodiment, a Rayleigh beam model with distributed mass and considering rotational inertia is used to model the flexible link, and the finite element method is employed for discretization of the link to achieve efficient and accurate calculations. Because the first link in the MDH model... Second Link Located at the joints Before and after rotation, the first link exhibits different dynamic characteristics; therefore, this method will... Second Link The model is created as two separate flexible beams to achieve differentiation. This addresses the problem that existing robot flexibility modeling methods do not consider structural flexibility parallel to the joint axis, thus failing to meet the flexibility modeling requirements of complex serial spatial structures in robots.

[0037] Specifically, with the first link The following example illustrates the linkage flexibility modeling process of this method. The linkage length is... Divided into Each beam element has There are finite element nodes, and the length of each beam element is... , This refers to the element numbering of the beam element in the linkage. In the Rayleigh beam model, all points on the same beam section deform identically, and the spatial position of each point on the beam is a single-valued function of the coordinates along the axial direction. In the undeformed state, an element coordinate system is defined at the starting point of each beam element. Define its Axis pointing link Along the axial direction, the coordinate systems of each beam element are parallel to each other, and a link coordinate system is established at the starting point of the flexible link. Its origin and The points coincide, and their directions are determined based on actual needs. Assume any point in this link... Located in the Within a beam element, the point lies in the element coordinate system. The position in the middle It can be represented as: (8) in Let the coordinates be the axis directions of the beam element. For this point in the element coordinate system Rigid position in For this point in the element coordinate system The deformation amounts in the figure are expressed as follows: (9) In the formula To describe the shape function of beam element deformation, Normalized coordinates within the cell. Beam element The deformation vector, and They are located in the beam element Deformation of the first and last finite element nodes. Under the assumption of small deformation, there is an approximate relationship between deformation and bending within the beam element. , Based on this relationship, a type function is defined. for: (10) In addition to location, a complete description of the state of a point on a beam element also requires the angles of its bending and torsion: (11) in The shape functions for bending and torsion of beam elements are defined as follows: (12) Second link The calculation method for any point in the middle is exactly the same; you only need to calculate the first link. The relevant parameters are changed to the second link. The corresponding value. Based on the aforementioned definition, the first link Second Link Any point on the beam element coordinate system The position in the middle and reversal for: (13) Subscript represent or .definition Let be the homogeneous transformation matrix from the link coordinate system to the beam element coordinate system, and then we can obtain the coordinates of the point in the link coordinate system. The position in the middle and reversal for: (14) In the formula Determined based on the actual coordinate system. arrive The homogeneous transformation matrix, The coordinate systems of the beam elements are parallel to each other. arrive The homogeneous transformation matrix is ​​given. The above equation describes the deformation at any point inside the link. To complete the kinematic recursion, the homogeneous transformation matrices at both ends of the link are also needed. The deformation at the end of the link is defined as follows: ,in , Link coordinate system To the end unit coordinate system The rotation matrix, The symbol for a diagonal matrix is... Let be the deformation of the last finite element node of the link. Based on the general form of equation (2), and considering that there is no attitude change in the rigid transformation of the link, the homogeneous transformation matrix of the two links considering flexibility is simplified as follows: (15) in and Link and The deformation increment matrix. Combining the above flexible modeling of the base, joints, and links, the time coordinate system considering flexibility is obtained. arrive homogeneous transformation matrix for: (16) In the formula MDH parameters Determined coordinate system arrive The homogeneous transformation matrix is ​​assumed to remain unchanged in this method. Combined with the recursive formulas for each coordinate system, the flexible time-base coordinate system can be recursively transformed. To coordinate system homogeneous transformation matrix Represented as: (17) The above recursive formula yields a stylized position recursive expression containing all deformation terms, realizing kinematic modeling of a fully flexible robot based on MDH flexible extended parameters. The model contains all possible deformation terms and allows users to selectively perform flexible modeling of various robot components according to actual needs. Furthermore, the flexible kinematic model of the robot can be directly generated through the MDH parameter table, solving the pain points of complex model construction process, incomplete consideration of flexible factors, and oversimplification of coupling effects in current robot flexible modeling.

[0038] In step S2, velocity and acceleration are recursively calculated based on the fully flexible kinematic model established in step 1. First, a unified notation convention is adopted for the velocity and angular velocity vectors. The superscript of the vector represents the reference coordinate system of the velocity, and the subscript defines the measurement object and reference object of the velocity. When representing the absolute velocity of the measured object relative to the stationary reference frame, the subscript... Abbreviated as First, for each key node (i.e., the origin of each coordinate system defined in step S1), along... The velocity is recursively calculated in the order of the preceding and following coordinate systems. and Its velocity recursion relationship is as follows: (18) In the formula and , and Coordinate systems and The absolute angular velocity and linear velocity of the origin relative to the stationary coordinate system. coordinate system arrive The rotation matrix, coordinate system arrive The translation vector, , The relative angular velocity and linear velocity generated by rigid motion between the two coordinate systems. , These are the relative angular velocity and linear velocity generated by deformation. The velocity recursion originates from a base coordinate system fixed to the earth. Initially, its initial velocity was , By using the velocity recursion relationship between adjacent coordinate systems and coordinate transformation, the velocities of all key nodes can be obtained sequentially, and the angular velocities of each key node in the base coordinate system can be obtained. and speed .

[0039] Redefine the length as System generalized coordinates Rigid joint angle vector and flexible deformation vector The set of: (19) In the formula This is the set of rigid rotation angles of all joints of the robot, and the flexible deformation vector. This includes the deformation vectors of all flexible components: (20) in For the base deformation vector, Let be the deformation vector for all joints. and The first link Second Link The link deformation vector is composed of the deformations of all finite element nodes. The velocities of each key node can be expressed as generalized velocities using the generalized Jacobian matrix. Linear functions: (twenty one) in and These are the angular velocity and linear velocity of the key nodes, respectively. and These are the angular velocity Jacobi and the velocity Jacobi, respectively. Their subscripts and superscripts have the same meaning as the velocity subscript. The symbols used in the subscripts are... , and Represents any defined coordinate system. The acceleration of each key node, obtained by differentiating the velocity recursion over time, is: (twenty two) In the formula and The derivative of Jacobi with respect to time is obtained using the chain rule: (twenty three) The velocity of any point in the flexible link can be obtained based on the velocity recursion results of each key node. Taking the first link as an example... Beam elements in Taking as an example, the velocity at any point on it and angular velocity It can be represented as: (twenty four) In the formula This specifies the rigid position of a point along the axis of the beam element coordinate system. and The deformation velocity and torsional angular velocity of a specified point in the link coordinate system. and This is a mapping operator from the velocity at a critical node to the velocity and angular velocity at any point on the link. This is the set of velocities for relevant key nodes. (Definition) To select the matrix, the variables are represented as follows: (25) In the formula Beam element The time derivative of the deformation of the finite element nodes at both ends. for Jacobian matrix, Choose the deformation matrix for the beam element. The acceleration at that point can then be calculated as follows: (26) The formula contains: (27) The connecting rod can be obtained in a similar way. The velocity and acceleration at any point in the system. The velocity and acceleration of all key nodes and any point on the links in the rigid-flexible coupling model can be obtained recursively.

[0040] In step S3, local dynamics modeling can be performed on each lumped mass node and the link with distributed mass using the kinematic recursion results. For components such as joints and bases that are considered to have lumped mass, their Cartesian mass matrix is... and stiffness matrix It can be written directly as: (28) Symbols in the formula Can be used as a base and joints , For the quality of the components, Let be the inertial tensor of the component about its center of mass. coordinate system The vector to the centroid of the component. and These represent the torsional and translational stiffness of the component, respectively. Based on the principle of virtual power, the local dynamic equation of the component is written as: (29) In the formula For the spatial velocity of the concentrated mass component, Virtual velocity, For spatial acceleration, The virtual velocity generated by deformation, For spatial support reaction force and active force, For external forces, It is the acceleration due to gravity. The local dynamic equations of lumped masses are transformed into a generalized coordinate space using the Jacobian matrix: (30) In the formula , , , , The mass, Coriolis force, gravity, stiffness, active force, and external force matrices of a lumped mass component in generalized coordinate space are defined as follows: (31) in Let Jacobian matrix be the spatial velocity. Let be the derivative of the spatial velocity Jacobian with respect to time. Choose a matrix for the transformation.

[0041] Now consider a flexible link with distributed mass, where the symbol... Take it as a connecting rod or For the link numbered... For any beam element, its mass matrix per unit length for: (32) In the formula The density of the beam element. Let be the cross-sectional area of ​​the beam element. Let the length be the beam element length. Let be the second moment of the cross section of the beam element.

[0042] The stiffness matrix of the beam element is derived from the stress-strain relationship, defining the strain-displacement matrix. for: (33) The material stiffness matrix of the beam element is redefined as follows: ,in and Given Young's modulus and shear modulus, respectively, the strain at any point in the beam element can be obtained. and stress They are respectively: (34) The dynamic equations of the beam element are written using the principle of virtual power: (35) In the formula Let the virtual velocity of the beam element be... For the spatial acceleration of the beam element, For the spatial support reactions and active forces of the beam element, For external force, Here is the unit mass matrix of the beam element based on the link coordinate system. The integral expression for the beam element is extracted as follows: (36) In the formula , The mass, Coriolis force, gravity, stiffness, and generalized force matrices of the beam element in the generalized coordinate space are as follows: (37) in The deformation selection matrix for the beam element is given. The local dynamic equations of the beam element in the generalized coordinate space are then obtained as follows: (38) In step S4, based on the aforementioned local dynamic equations of the lumped mass and distributed mass components, the overall dynamic matrices of the rigid-flexible coupled robot system can be obtained as follows: (39) The main driving force in the system is the joint driving force, and the reaction forces of each component cancel each other out, resulting in a generalized force for the entire system. and external forces They are respectively: (40) In the formula Let be the driving force of each joint. Substituting into the local dynamics equations and eliminating the imaginary velocity vectors, we finally obtain the overall dynamics equations of the rigid-flexible coupled robot: (41) This completes the modeling of the rigid-flexible coupled robot using this method. The robot's dynamics can be simulated and controlled using this model.

[0043] Example 2 This embodiment discloses a parametric modeling system for fully flexible rigid-flexible coupling of robots; like Figure 2 As shown, a parametric modeling system for fully flexible rigid-flexible coupling of robots includes: The kinematic model construction module is configured to: establish a kinematic model that simultaneously includes the flexible deformation of the robot base, joints and links, based on the MDH flexible extension parameters and combined with the floating coordinate method, finite element method and Rayleigh beam model. The velocity and acceleration recursion module is configured to: based on the established kinematic model, perform symbolic velocity and acceleration recursion on each key node, and calculate the velocity and acceleration of any point on the flexible link with distributed mass through the velocity mapping operator; The module for establishing dynamic equilibrium equations is configured to: establish dynamic equilibrium equations for each concentrated mass component and distributed mass component in the kinematic model in Cartesian space based on the principle of virtual power, and transform the dynamic equilibrium equations to generalized coordinate space using the generalized Jacobian matrix; The overall dynamic equation assembly module is configured to assemble the local dynamic equations of all components in the generalized coordinate space, forming a unified rigid-flexible coupling dynamic equation for the robot system.

[0044] Example 3 The purpose of this embodiment is to provide a computer-readable storage medium.

[0045] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in a parametric modeling method for fully flexible rigid-flexible coupling of a robot as described in Example 1.

[0046] Example 4 The purpose of this embodiment is to provide an electronic device.

[0047] An electronic device includes a memory, a processor, and a program stored in the memory and executable on the processor. When the processor executes the program, it implements the steps in the parametric modeling method for fully flexible rigid-flexible coupling of a robot as described in Embodiment 1.

[0048] The steps and methods involved in the apparatuses of Embodiments 2, 3, and 4 above correspond to those in Embodiment 1. For specific implementation details, please refer to the relevant description section of Embodiment 1. The term "computer-readable storage medium" should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying an instruction set for execution by a processor and enabling the processor to perform any of the methods in this invention.

[0049] Those skilled in the art will understand that the modules or steps of the present invention described above can be implemented using general-purpose computer devices. Optionally, they can be implemented using computer-executable program code, thereby allowing them to be stored in a storage device for execution by a computer device, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. The present invention is not limited to any particular combination of hardware and software.

[0050] While the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A parametric modeling method for fully flexible rigid-flexible coupling of robots, characterized in that, include: Based on the MDH flexible extension parameters, and combined with the floating coordinate method, finite element method and Rayleigh beam model, a kinematic model that simultaneously includes the flexible deformation of the robot base, joints and links is established. Based on the established kinematic model, symbolic velocities and accelerations of each key node are recursively calculated, and the velocity and acceleration of any point on a flexible link with distributed mass are calculated using a velocity mapping operator. Based on the principle of virtual power, the dynamic equilibrium equations of each concentrated mass component and distributed mass component in the kinematic model are established in Cartesian space, and the dynamic equilibrium equations are transformed to generalized coordinate space using the generalized Jacobian matrix. In the generalized coordinate space, the local dynamic equations of all components are assembled to form a unified rigid-flexible coupling dynamic equation for the robot system.

2. The parametric modeling method for fully flexible rigid-flexible coupling of robots as described in claim 1, characterized in that, Based on the MDH flexible extension parameters, and combining the floating coordinate method, finite element method, and Rayleigh beam model, a kinematic model is established that simultaneously includes the flexible deformation of the robot base, joints, and links, including: Based on the MDH parameters, flexible binary switch parameters and joint flexibility type parameters are introduced to extend the flexible modeling of rigid kinematics, resulting in the MDH flexible extended parameters. The floating coordinate method is used to describe the flexible deformation of each component, and the spatial position of the deformed component is regarded as the superposition of rigid transformation and flexible deformation. The deformation increment matrix of each component is constructed by the first-order linear approximation of the Lie algebra representation of the deformation, and then the homogeneous transformation matrix of the base, joint and link considering the flexible deformation is obtained. Flexible modeling of robot components is performed based on homogeneous transformation matrix with small deformation. The base and joints are modeled as linear elastic models with concentrated mass to describe their small deformation characteristics. The robot links are modeled as Rayleigh beam models with distributed mass and considering rotational inertia. The Rayleigh beam model is discretized using the finite element method, and the links are divided into several beam elements. The deformation of the entire link is characterized by the deformation of the element nodes. Based on the MDH flexible extension parameters, homogeneous transformation matrix, and component flexibility model defined above, a recursive relationship is established from the base coordinate system to the end effector coordinate system. According to the serial sequence of robot joints and links, the coordinate system transformation relationship including all flexible deformation terms of the base, joints, and links is recursively calculated, and finally a stylized position recursive expression is formed, resulting in a fully flexible kinematic model of the robot that simultaneously covers the flexible deformation of the base, joints, and links.

3. The parametric modeling method for fully flexible rigid-flexible coupling of robots as described in claim 2, characterized in that, The deformation increment matrix is ​​derived from the spatial deformation vector, which includes the torsional vector and the linear deformation vector. The homogeneous transformation matrix of each component considering flexible deformation is a linear superposition of the rigid transformation homogeneous matrix and the deformation increment matrix, and second-order and higher small deformation terms are ignored in the position recursion.

4. The parametric modeling method for fully flexible rigid-flexible coupling of robots as described in claim 1, characterized in that, Based on the established kinematic model, symbolic velocities and accelerations are recursively calculated for each key node. The velocity and acceleration of any point on a flexible link with distributed mass are then calculated using a velocity mapping operator, including: The origin of each coordinate system in the kinematic model is set as the key node, and the recursive sequence is determined according to the coordinate system order from the robot base to the end effector. Starting from the base coordinate system fixed to the ground, for any two adjacent coordinate systems, establish the recursive relationship between their origins for absolute angular velocity and linear velocity, and perform coordinate transformation by combining the rotation matrix and translation vector between the two coordinate systems; calculate in sequence according to the recursive sequence to obtain the absolute angular velocity and linear velocity of all key nodes in the base coordinate system. The system's generalized coordinates are a set of rigid joint angular vectors and flexible deformation vectors. The absolute angular velocity and linear velocity of each key node are expressed as linear functions of the generalized velocity using the generalized Jacobian matrix. The acceleration of the key node is obtained by differentiating the linearized expression of the key node velocity with respect to time. Based on the recursive results of velocity and acceleration of key nodes, a calculation relationship is defined for a specified point in the axis direction of the beam element coordinate system. For a specified point on any beam element in a flexible link, the velocity and acceleration of that point in the link coordinate system are derived using the calculation relationship and the velocity and acceleration of the relevant key nodes.

5. The parametric modeling method for fully flexible rigid-flexible coupling of robots as described in claim 1, characterized in that, The velocity mapping operator consists of an antisymmetric matrix of the position of a specified point in the link coordinate system, an antisymmetric matrix of the link angular velocity, a homogeneous transformation matrix from the beam element coordinate system to the link coordinate system, a Rayleigh beam shape function, and a deformation selection matrix of the beam element. The velocity and acceleration of any point on the flexible link are obtained by multiplying the velocity mapping operator with the velocities of relevant key nodes.

6. The parametric modeling method for fully flexible rigid-flexible coupling of robots as described in claim 1, characterized in that, Based on the principle of virtual power, the dynamic equilibrium equations for each concentrated mass component and distributed mass component in the kinematic model are established in Cartesian space, and the dynamic equilibrium equations are transformed to a generalized coordinate space using the generalized Jacobian matrix, including: The base and joints in the kinematic model are set as concentrated mass components, and the flexible links in the kinematic model are set as distributed mass components, and discretized into several beam elements; Construct the mass matrix and stiffness matrix of the lumped mass component in Cartesian space, and the mass matrix per unit length of the beam element, respectively. Based on the principle of virtual power, the dynamic equilibrium equations in Cartesian space and the dynamic equilibrium equations of a single beam element in Cartesian space are established respectively. The Cartesian dynamic equilibrium equations of the lumped mass component are transformed to the generalized coordinate space using the generalized Jacobian matrix, yielding the mass matrix, Coriolis force matrix, gravity matrix, stiffness matrix, active force matrix, and external force matrix in the generalized coordinate space. Based on the deformation selection matrix of the beam element, the Cartesian dynamic equilibrium equations of a single beam element are transformed to the generalized coordinate space using the generalized Jacobian matrix. The transformation results of all beam elements are then summarized to form the local dynamic equations of the distributed mass component in the generalized coordinate space.

7. The parametric modeling method for fully flexible rigid-flexible coupling of robots as described in claim 1, characterized in that, In the generalized coordinate space, the local dynamic equations of all components are assembled to form a unified rigid-flexible coupling dynamic equation for the robot system. This includes: summarizing the mass, Coriolis force, gravity, stiffness and generalized force matrix of each component in the generalized coordinate space to construct an overall dynamic matrix, taking the joint driving force as the main driving force, eliminating the mutual canceling support reaction forces between components, and assembling to form an overall rigid-flexible coupling dynamic equation.

8. A parametric modeling system for fully flexible rigid-flexible coupling of robots, characterized in that, include: The kinematic model construction module is configured to: establish a kinematic model that simultaneously includes the flexible deformation of the robot base, joints and links, based on the MDH flexible extension parameters and combined with the floating coordinate method, finite element method and Rayleigh beam model. The velocity and acceleration recursion module is configured to: based on the established kinematic model, perform symbolic velocity and acceleration recursion on each key node, and calculate the velocity and acceleration of any point on the flexible link with distributed mass through the velocity mapping operator; The module for establishing dynamic equilibrium equations is configured to: establish dynamic equilibrium equations for each concentrated mass component and distributed mass component in the kinematic model in Cartesian space based on the principle of virtual power, and transform the dynamic equilibrium equations to generalized coordinate space using the generalized Jacobian matrix; The overall dynamic equation assembly module is configured to assemble the local dynamic equations of all components in the generalized coordinate space, forming a unified rigid-flexible coupling dynamic equation for the robot system.

9. A computer-readable storage medium having a program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps in the parametric modeling method for fully flexible rigid-flexible coupling of a robot as described in any one of claims 1-7.

10. An electronic device comprising a memory, a processor, and a program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps in the parametric modeling method for fully flexible rigid-flexible coupling of a robot as described in any one of claims 1-7.

Images