A robot full-flexible rigid-flexible coupling parameterized modeling method and system

Abstract

The application provides a robot full-flexible rigid-flexible coupling parameterized modeling method and system, and belongs to the technical field of robot modeling; the method first establishes a full-flexible kinematic model containing base, joint and flexible deformation of a connecting rod based on MDH flexible expansion parameters, in combination with a floating coordinate method, a finite element method and a Rayleigh beam model; secondly, velocity and acceleration of key nodes and any point of the flexible connecting rod are recursively calculated; thirdly, local dynamic equations of each lumped mass component and distributed mass component are constructed based on a virtual power principle, and are converted to a generalized coordinate space through a generalized Jacobian matrix; and finally, a unified robot rigid-flexible coupling dynamic equation is assembled. The application realizes high efficiency and automation of the modeling process, improves the accuracy of the model in describing dynamic characteristics of the robot, and provides an accurate basic model for high-precision dynamic analysis and control of the robot.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of robot modeling, and particularly relates to a robot full-flexibility rigid-flexibility coupling parameterized modeling method and system. BACKGROUND

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

[0003] High-precision control and dynamic performance of a robot largely depend on the accuracy of its kinematic and dynamic models. In high-speed, heavy-load or high-precision operation scenarios, the robot links, joints and even the base, which are regarded as rigid bodies, will produce elastic deformations that cannot be ignored. Such deformations will cause positioning errors of the end effector and significantly change the dynamic characteristics of the system. If the traditional rigid body model is continued to be used for system analysis and control algorithm design, the operation precision and stability of the robot will be seriously affected. Therefore, establishing a high-precision rigid-flexibility coupling dynamic model and applying it to control has become one of the key technologies to improve the performance of high-end robots.

[0004] However, the existing robot flexibility modeling methods still have many limitations, mainly in the following aspects:

[0005] First, the model construction process is complex, and the automation and practicality are low. Most of the existing methods are based on energy-based modeling methods such as Lagrange method, and the derivation process involves complex partial differential operations. The calculation amount increases sharply with the increase of the degrees of freedom of the robot, resulting in low efficiency of model derivation. At the same time, the existing methods lack a parameterization mechanism to specify specific components as rigid, and usually require flexible modeling of all links or joints, further deepening the complexity and computational burden of the model. In addition, the modeling parameters used by many methods are incompatible with the MDH parameters commonly used in the industry, and often require the modal shape of the robot link to be obtained in advance through experiments or simulations, which brings a lot of additional pre-work. The above problems together result in a complex and tedious flexible modeling process, low automation, and make the rigid-flexibility coupling modeling practice currently mostly limited to simple planar two-link robot models in academic research, which seriously limits its engineering application in complex industrial robots.

[0006] Second, the flexibility factor is not fully considered, and the model accuracy is limited. Most existing flexible modeling methods can be classified as "joint flexibility modeling" or "link flexibility modeling", that is, only single joint flexibility or link flexibility is considered, and other structures are idealized as absolute rigid. However, in real applications, the base, joints and links of the robot may have different degrees of elastic deformation, and since the robot is a serial structure, the deformation of these components will be accumulated through the kinematic chain and will simultaneously affect the final pose response of the end effector. This one-sided consideration of the source of flexibility leads to the fact that the established model cannot fully reflect the actual deformation of the robot system, and introduces inevitable systematic modeling errors.

[0007] Third, the coupling effect is oversimplified, and the dynamic description is distorted. A few studies attempt to model the flexibility of both links and joints, but the coupling effect between the system's rigid large motion and the flexible elastic deformation is often not fully reflected. At the kinematic level, the flexibility deformation of the link is generally not correctly introduced into the position recursion formula, or is simply simplified as an equivalent additional angle at the joint, which makes the mathematical representation of the end effector position inaccurate. At the dynamic level, the influence of joint and link deformation on the system inertia matrix, Coriolis force / momentum matrix and other key dynamic parameters is often ignored or linearized, and the additional angular velocity caused by deformation is often incorrectly discarded, which brings problems such as the lack of "rotational stiffening" effect and instability of model numerical simulation. In addition, for almost all flexible modeling methods based on the assumed modal method, the influence of robot body attitude changes and end load changes on link vibration modes is not considered, and the modeling process is always based on a fixed set of link vibration mode functions that do not change, which further reduces the dynamic accuracy of the model under different configurations and loads. SUMMARY

[0008] To overcome the shortcomings of the prior art described above, the present application provides a robot full-flexibility rigid-flexible coupling parameterized modeling method and system, which aims to solve the problems of existing robot flexible modeling, such as complex model construction process, low automation degree, incompatibility with industry standard parameters, incomplete consideration of flexibility factors, and oversimplified deformation coupling effect, and to realize efficient, accurate and comprehensive description of the dynamic characteristics of the robot rigid-flexible coupling modeling.

[0009] To achieve the above object, one or more embodiments of the present application provide the following technical solutions:

[0010] The first aspect of the present application provides a robot full-flexibility rigid-flexible coupling parameterized modeling method;

[0011] A robot full-flexibility rigid-flexible coupling parameterized modeling method comprises:

[0012] Based on the MDH flexible extended parameters, combined with the floating coordinate method, the finite element method and the Rayleigh beam model, a kinematics model is established which contains the flexible deformation of the robot base, joints and links at the same time;

[0013] Based on the established kinematics model, the symbolic velocity and acceleration of each key node are recursively calculated, and the velocity and acceleration of any point on the flexible link with distributed mass are calculated through the velocity mapping operator;

[0014] Based on the virtual power principle, the dynamic equilibrium equations of each concentrated mass component and distributed mass component in the kinematics model in the Cartesian space are established respectively, and the dynamic equilibrium equations are converted to the generalized coordinate space by using the generalized Jacobian matrix;

[0015] In the generalized coordinate space, the local dynamic equations of all components are assembled to form a unified rigid-flexible coupled dynamic equation of the robot system.

[0016] As a further technical solution, based on the MDH flexible extended parameters, combined with the floating coordinate method, the finite element method and the Rayleigh beam model, a kinematics model is established which contains the flexible deformation of the robot base, joints and links at the same time, including:

[0017] Based on the MDH parameters, flexible binary switch parameters and joint flexibility type parameters are introduced for flexible modeling extension of rigid kinematics, and MDH flexible extended parameters are obtained;

[0018] 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 using the linear approximation of the deformation Lie algebra, and the homogeneous transformation matrix of the base, joint and link considering the flexible deformation is obtained;

[0019] Based on the homogeneous transformation matrix with small deformation, the flexible modeling of each component of the robot is carried out, the base and the joint are modeled as linear elastic models with concentrated mass for describing their small deformation characteristics; the robot link is modeled as a Rayleigh beam model with distributed mass and considering the moment of inertia, and the finite element method is used for discretization processing of the Rayleigh beam model, the link is divided into several beam elements, and the overall link deformation is represented by the element node deformation;

[0020] Based on the above defined MDH flexible extended parameters, homogeneous transformation matrix and component flexible model, the recursive relationship from the base coordinate system to the end effector coordinate system is established; according to the series order of the robot joints and links, the coordinate system transformation relationship containing all flexible deformation items of the base, joints and links is recursively calculated, and finally the programmed position recursive expression is obtained, and the full flexible kinematics model of the robot which contains the flexible deformation of the base, joints and links at the same time is obtained.

[0021] As a further technical solution, the deformation increment matrix is derived from a spatial deformation vector, the spatial deformation vector including a torsion vector and a linear deformation vector; a homogeneous transformation matrix of each component considering flexible deformation is a linear superposition of a rigid transformation homogeneous matrix and a deformation increment matrix, and a second-order and above small deformation term is ignored in position recursion.

[0022] As a further technical solution, based on the established kinematic model, the symbolic velocity and acceleration of each key node are recursively propagated, and the velocity and acceleration of any point on the flexible link with distributed mass are calculated through a velocity mapping operator, including:

[0023] The origin of each coordinate system in the kinematic model is set as a key node, and the recursive sequence is determined according to the coordinate system sequence from the robot base to the end effector;

[0024] Taking the base coordinate system fixed to the earth as the recursive starting point, for any two adjacent coordinate systems, the absolute angular velocity and linear velocity recursive relationship between the origins of the two coordinate systems is established, and the coordinate transformation is carried out in combination with the rotation matrix and translation vector between the two coordinate systems; according to the recursive sequence, the absolute angular velocity and linear velocity of all key nodes in the base coordinate system are calculated in turn;

[0025] The system generalized coordinates are the collection of rigid joint angle vectors and flexible deformation vectors, and the absolute angular velocity and linear velocity of each key node are expressed as a linear function of generalized velocity through a generalized Jacobian matrix; the key node acceleration is obtained by taking the derivative of the linearized expression of the key node velocity with respect to time;

[0026] Based on the recursive results of the velocity and acceleration of the key nodes, a calculation relationship of the specified point in the beam element coordinate system axis direction is defined; for a specified point on any beam element in the flexible link, the velocity and acceleration of the point in the link coordinate system are derived using the calculation relationship and the velocity and acceleration of the related key nodes.

[0027] As a further technical solution, the velocity mapping operator is composed of the skew-symmetric matrix of the position of the specified point in the link coordinate system, the skew-symmetric matrix of the angular velocity of the link, the homogeneous transformation matrix from the beam element coordinate system to the link coordinate system, the Rayleigh beam type function, and the 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 and the related key node velocity.

[0028] As a further technical solution, based on the virtual power principle, the dynamic equilibrium equations of each lumped mass component and distributed mass component in the kinematic model in the Cartesian space are established respectively, and the dynamic equilibrium equations are converted to the generalized coordinate space using the generalized Jacobian matrix, including:

[0029] setting the base and joints in the kinematics model as lumped mass components, setting the flexible link in the kinematics model as a distributed mass component, and discretizing into a plurality of beam units;

[0030] respectively constructing a mass matrix and a stiffness matrix of the lumped mass component in the Cartesian space and a unit length mass matrix of the beam unit;

[0031] based on the virtual power principle, respectively establishing the dynamic equilibrium equation in the Cartesian space and the dynamic equilibrium equation of a single beam unit in the Cartesian space;

[0032] using the generalized Jacobian matrix to convert the dynamic equilibrium equation of the lumped mass component in the Cartesian space to the generalized coordinate space, obtaining the mass matrix, the Coriolis force matrix, the gravity matrix, the stiffness matrix, the active force matrix and the external force matrix in the generalized coordinate space; based on the deformation selection matrix of the beam unit, converting the dynamic equilibrium equation of a single beam unit in the Cartesian space to the generalized coordinate space through the generalized Jacobian matrix, and collecting the conversion results of all beam units to form the local dynamic equation of the distributed mass component in the generalized coordinate space.

[0033] As a further technical solution, in the generalized coordinate space, the local dynamic equations of all components are assembled to form a unified robot system rigid-flexible coupling dynamic equation, including: collecting the mass, Coriolis force, gravity, stiffness and generalized force matrix of each component in the generalized coordinate space to construct the overall dynamic matrix, taking the joint driving force as the active force, eliminating the mutual counteracting support force between components, and assembling to form the overall rigid-flexible coupling dynamic equation.

[0034] The second aspect of the application provides a robot full-flexible rigid-flexible coupling parameterized modeling system.

[0035] A robot full-flexible rigid-flexible coupling parameterized modeling system comprises:

[0036] The kinematics model construction module is configured to: based on the MDH flexible expansion parameter, combining the floating coordinate method, the finite element method and the Rayleigh beam model, establish a kinematics model containing the robot base, joint and flexible deformation of the link at the same time;

[0037] The velocity and acceleration recursion module is configured to: based on the established kinematics model, recursively symbolize the velocity and acceleration of each key node, and calculate the velocity and acceleration of any point on the flexible link with distributed mass through the velocity mapping operator;

[0038] The dynamic equilibrium equation establishment module is configured to: based on the virtual power principle, respectively establish the dynamic equilibrium equation of each lumped mass component and the distributed mass component in the kinematics model in the Cartesian space, and convert the dynamic equilibrium equation to the generalized coordinate space by using the generalized Jacobian matrix.

[0039] The whole dynamics equation assembling module is configured to assemble the local dynamics equations of all components in the generalized coordinate space to form the unified robot system rigid-flexible coupling dynamics equation.

[0040] The third aspect of the present application provides a computer readable storage medium, which stores a program, and the program is executed by a processor to implement the steps of the robot full-flexible rigid-flexible coupling parameterized modeling method according to the first aspect of the present application.

[0041] The fourth aspect of the present application provides an electronic device, which comprises a memory, a processor and a program stored in the memory and executable on the processor, and the processor implements the steps of the robot full-flexible rigid-flexible coupling parameterized modeling method according to the first aspect of the present application when executing the program.

[0042] The above one or more technical solutions have the following beneficial effects:

[0043] (1) The present application defines the MDH flexible expansion parameter, so that the rigid-flexible coupling modeling parameter is fully compatible with the MDH parameter of the rigid robot modeling, and the flexible modeling can be performed by supplementing the necessary material property parameters, avoiding the cumbersome parameter conversion. Users only need to supplement a small number of parameters such as flexible switch, joint type and material property on the basis of traditional rigid MDH parameters, so as to perform flexible modeling, avoid cumbersome parameter system conversion and repeated definition, greatly reduce the use threshold. By introducing the flexible switch parameter and the joint flexibility type parameter, the user can selectively specify the base, joint or connecting rod as rigid or flexible according to the actual precision requirement and calculation resource, realizing the "on-demand flexibility" parameterized modeling. This effectively avoids unnecessary calculation overhead, fundamentally controls the model complexity, and solves the problems of model degree of freedom expansion and large calculation amount in the prior art. The finite element method is used for discretization of the flexible connecting rod, without the need for complex experiments or simulations in advance to obtain the modal shape of the connecting rod as in the assumed modal method, a large amount of preliminary manpower is saved, and the modeling preparation work is simplified. The dynamics equation is constructed directly based on the virtual power principle, avoiding the cumbersome partial differential and derivation operation in the energy method such as Lagrange method, simplifying the derivation process, and being more easy to realize automatic model export through symbolic calculation software.

[0044] (2) The present application simultaneously considers the base flexibility, joint flexibility and connecting rod flexibility in the unified framework, overcoming the limitation of the prior art which only focuses on a single flexible source. Through kinematic recursion, the cumulative effect of deformation of all components on the end pose is accurately reflected, so that the real deformation state of the robot system can be more completely reflected, and the systematic error introduced due to incomplete consideration of the flexible source is eliminated.

[0045] In the modeling process, the coupling effect between rigid motion and elastic deformation is fully embodied. In the kinematics level, the link deformation is introduced into the position recursion; in the dynamics level, the influence of deformation on the inertia matrix, Coriolis force matrix and other key dynamic parameters is reserved, and the angular velocity caused by torsion is considered, the nonlinear effects such as 'rotational rigidification' are correctly reflected, and the reality and numerical stability of the dynamics simulation are improved.

[0046] Due to the adoption of the finite element method instead of the fixed modal method, the vibration mode function of the connecting rod is no longer fixed, but dynamically changes with the robot pose and end load. This makes the dynamic model established by the method adaptive to different working states of the robot, significantly improving the dynamic accuracy of the model in the whole working space, and being especially suitable for simulation and control in large-range motion and high-dynamic working conditions.

[0047] Advantages of additional aspects of the application will be partially given in the following description, partially will become obvious from the following description, or will be understood by the practice of the application. BRIEF DESCRIPTION OF DRAWINGS

[0048] The accompanying drawings, which form a part of this specification, are included to provide a further understanding of the application, and are incorporated by reference herein. The embodiments depicted herein are provided by way of example only, and are not intended to limit the present application unless otherwise specified.

[0049] Figure 1 A method flowchart of the first embodiment.

[0050] Figure 2 A system structure diagram of the second embodiment. DETAILED DESCRIPTION

[0051] It should be noted that the following detailed description is merely exemplary in nature and is intended to provide further description of the application. Unless otherwise defined, 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 application belongs.

[0052] It should be noted that the terms used herein are merely intended to describe specific embodiments and are not intended to limit the exemplary embodiments according to the present application.

[0053] In the case of no conflict, the embodiments in the present application and the features in the embodiments can be combined with each other.

[0054] Embodiment one

[0055] The embodiment discloses a robot full-flexible rigid-flexible coupling parameterized modeling method;

[0056] As shown in Figure 1 A robot full-flexible rigid-flexible coupling parameterized modeling method comprises the following steps:

[0057] Step S1, based on the MDH flexible expansion parameters, combining the floating coordinate method, finite element method and Rayleigh beam model, a kinematics model is established which contains the flexible deformation of robot base, joints and connecting rods at the same time;

[0058] Step S2, based on the established kinematics model, the symbolic velocity and acceleration of each key node are recursively calculated, and the velocity and acceleration of any point on the flexible connecting rod with distributed mass are calculated through the velocity mapping operator;

[0059] Step S3, based on the virtual power principle, the dynamic equilibrium equations of each concentrated mass component and distributed mass component in the kinematics model in the Cartesian space are established respectively, and the dynamic equilibrium equations are converted to the generalized coordinate space by using the generalized Jacobian matrix;

[0060] Step S4, in the generalized coordinate space, the local dynamic equations of all components are assembled to form a unified robot system rigid-flexible coupled dynamic equation.

[0061] Specifically, the following contents are included:

[0062] In step S1, the flexible modeling extension of rigid kinematics is based on the widely used MDH parameters. In MDH modeling, the base is numbered 0, and the MDH nodes starting from the base are numbered 1 to . The numbers and are used to represent any two adjacent MDH nodes, and the two nodes are connected by the first connecting rod , the joint and the second connecting rod in sequence. Any MDH node is fixedly connected with a coordinate system , the coordinate axis of which is along the axis direction of the joint , and the coordinate axis is perpendicular to the coordinate axis of the MDH node fixedly connected with the coordinate system . In the case of full rigidity, the transformation matrix of the coordinate system

[0063] to is:

[0064] (1)

[0065] wherein, , , , ​​are MDH parameters, is is is is is is is is is is is is is is is is and represent rigid rotation or translation along the specified coordinate axis by the given magnitude, respectively, , , , are homogeneous transformation matrices corresponding to a single rigid rotation or translation. For the convenience of subsequent description, coordinate systems , and are defined, which are fixed to the first link , the second link and the joint origin, respectively.

[0066] On the basis of MDH parameters, flexible binary switch parameters and joint flexible type parameters are introduced for flexible modeling extension of rigid kinematics, and the base, joint and link are selectively flexibly modeled. By introducing a binary switch parameter is realized, where represents rigidity, represents flexibility. The homogeneous transformation matrix and the spatial deformation vector of each component can be uniformly represented as:

[0067] (2)

[0068] In the formula, is the homogeneous transformation matrix considering flexibility, is the rigid part, is the spatial deformation vector, and the operator represents longitudinal splicing of each element, and are the twist vector and the deformation vector, respectively, is the flexible deformation increment matrix, the specific definition and calculation method of which are given below. For the base, the first link , the second link and the joint , the corresponding switch parameters are 、 、 and .

[0069] A joint flexibility type parameter is also introduced to define different joint flexibility modeling approaches, where denotes a joint is a rigid joint, denotes torsion considering only the joint axis direction, denotes modeling of six-dimensional deformation of a joint, which is achieved by modifying the included deformation parameters in the spatial deformation vector of the joint :

[0070] (3)

[0071] In addition, the present method neglects second-order and higher small deformations in position propagation to further simplify the model.

[0072] To match the aforementioned MDH flexibility extension modeling, when the definition of the homogeneous transformation matrix of each component flexibility is derived by the following method: using the floating coordinate method to represent the deformation on the basis of the rigid reference frame defined by the MDH parameters, and regarding the spatial position after deformation as the superposition of rigid transformation and deformation. By the Lie algebra representation of deformation, a first-order linear approximation can be obtained to get the general form of the homogeneous transformation matrix of coordinate system to with flexibility small deformation:

[0073] (4)

[0074] where is the unit matrix, and may represent any two coordinate systems, and are the homogeneous transformation matrices of rigid transformation and deformation between the two coordinate systems, is the deformation increment matrix, is the Lie algebra representation of deformation:

[0075] (5)

[0076] where is the spatial deformation vector, and are the torsion vector and the deformation vector, respectively, and the operator represents the longitudinal splicing of each element, and the operator ​represents the inverse symmetric matrix. Since the deformation is small, the product of the second and higher order small deformation elements is considered to be 0.

[0077] The flexible modeling of each part of the robot is based on the general form of the small deformation homogeneous transformation matrix to obtain the deformation increment matrix The base and joint are modeled as linear elastic models with concentrated mass in the method, and their rigid position transformations are respectively defined by unit matrix and homogeneous transformation matrix The deformation increment matrix , is respectively defined as:

[0078] (6)

[0079] The homogeneous transformation matrix of the base and joint space transformation considering flexible deformation is obtained:

[0080] (7)

[0081] Flexible modeling of the connecting rod is then performed. The connecting rod of the robot has distributed mass, and simple concentrated parameter modeling method may have large deviation in deformation calculation. In the embodiment, the Rayleigh beam model with distributed mass and considering rotational inertia is used to model the flexible connecting rod, and the finite element method is used for discrete processing of the connecting rod to realize efficient and accurate calculation. Since the first connecting rod and the second connecting rod are respectively located before and after the joint rotation, they have different dynamic characteristics, so the first connecting rod and the second connecting rod are modeled as two separate flexible beams to realize the distinction. This will solve the problem that the existing robot flexible modeling method does not consider the structural flexibility parallel to the joint axis direction and does not meet the flexible modeling requirements of the complex serial space structure of the robot.

[0082] Specifically, the flexible modeling process of the connecting rod of the method is described taking the first connecting rod as an example. The connecting rod has a length of and is divided into beam units, with finite element nodes, and the length of each beam unit is , is the number of beam units in the connecting rod. In the Rayleigh beam model, the deformations of the points in the same beam cross section are the same, and the spatial positions of the points on the beam are single-valued functions of the axis direction coordinates. In the undeformed state, the element coordinate system is defined at the starting point of each beam unit, and the axis thereof points to the connecting rod 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:

[0083] (8)

[0084] in Let be the coordinates of the beam element's axis. 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:

[0085] (9)

[0086] 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:

[0087] (10)

[0088] 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:

[0089] (11)

[0090] in The shape functions for bending and torsion of beam elements are defined as follows:

[0091] (12)

[0092] 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. corresponding value of the point. Based on the foregoing definitions, the position and twist of any point in the beam element coordinate system are:

[0093] (13)

[0094] where the subscript represents or . Defining as the homogeneous transformation matrix of the link coordinate system to the beam element coordinate system, the position and twist of the point in the link coordinate system are:

[0095] (14)

[0096] where is the homogeneous transformation matrix from to determined according to the actual coordinate system setting, is the homogeneous transformation matrix from the parallel beam element coordinate system to . The above formula describes the deformation of any point inside the link, and to complete the kinematic recursion, the homogeneous transformation matrices of the two ends of the link are also needed. Defining the deformation of the end of the link as where , is the rotation matrix of the link coordinate system to the end element coordinate system , is the diagonal matrix symbol, is the deformation of the last finite element node of the link. Based on the general form of formula (2) and considering that there is no pose transformation in the rigid transformation of the link, the homogeneous transformation matrix of the two links when considering flexibility is simplified as:

[0097] (15)

[0098] where and are the deformation increment matrices of the links and . Integrating the above flexible modeling of the base, joint and link, the homogeneous transformation matrix of the coordinate system to when considering flexibility is:

[0099] ​​​(16)

[0100] where is the MDH parameter The determined coordinate system to The homogeneous transformation matrix, the method assumes that it does not deform. Combined with the recursive formula of each coordinate system, the flexible time base coordinate system to the coordinate system The homogeneous transformation matrix is expressed as:

[0101] (17)

[0102] Through the above recursive formula, the programmed position recursive expression containing all deformation terms is obtained, the kinematics modeling of the full flexible robot based on the MDH flexible expansion parameter is realized, all possible deformation terms are contained in the model, and the user can selectively model the flexibility of each component of the robot according to the actual demand, and the flexible kinematics model of the robot can also be generated directly through the MDH parameter table, solving the pain point problems of complex model construction process, incomplete consideration of flexible factors and excessive simplification of coupling effect in current robot flexible modeling.

[0103] In step S2, the velocity and acceleration recursion is carried out based on the full flexible kinematics model established in step 1. First, a unified symbol convention is adopted for the velocity and angular velocity vectors, the upper subscript of the vector is the reference coordinate system of the velocity, and the lower subscript defines the measured object and the reference object of the velocity. When the measured object is relative to the static reference system, the subscript is abbreviated as . First, the velocity recursion is carried out on the key nodes (i.e. the coordinate system origins defined in step S1) along the order of For any two adjacent coordinate systems and , the velocity recursive relationship is:

[0104] (18)

[0105] where and , and are the absolute angular velocity and linear velocity of the coordinate system and origin relative to the static coordinate system, is the rotation matrix of the coordinate system to , is the translation vector of the coordinate system to , , 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 .

[0106] Redefine the length as System generalized coordinates Rigid joint angle vector and flexible deformation vector The set of:

[0107] (19)

[0108] 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:

[0109] (20)

[0110] 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:

[0111] (twenty one)

[0112] 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:

[0113] (twenty two)

[0114] In the formula and The derivative of Jacobi with respect to time is obtained using the chain rule:

[0115] (twenty three)

[0116] 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:

[0117] (twenty four)

[0118] 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:

[0119] (25)

[0120] 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:

[0121] (26)

[0122] The formula contains:

[0123] (27)

[0124] 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.

[0125] In step S3, the local dynamics of each lumped mass node and link with distributed mass can be modeled by kinematic recursion results. For components with lumped mass such as joints and base, their mass matrix and stiffness matrix in Cartesian space can be directly written as:

[0126] (28)

[0127] where the symbols can be taken as the base and joint , the mass of the component, the inertia tensor of the component with respect to the center of mass, the vector from the origin of the coordinate system to the center of mass of the component, and the torsional and translational stiffness of the component, respectively. The local dynamics equation of the component is written based on the principle of virtual power as:

[0128] (29)

[0129] where is the spatial velocity of the lumped mass component, is the virtual velocity, is the spatial acceleration, is the virtual velocity due to deformation, is the spatial reaction force and active force, is the external force, is the gravitational acceleration, . The local dynamics equation of the lumped mass is converted to the generalized coordinate space through the Jacobian matrix:

[0130] (30)

[0131] where , , , , are the mass, Coriolis force, gravity, stiffness, active force, and external force matrices of the lumped mass component in the generalized coordinate space, which are defined as:

[0132] (31)

[0133] where is the Jacobian matrix of the spatial velocity, is the derivative of the spatial velocity Jacobian with respect to time, is the deformation selection matrix.

[0134] Consider a flexible link with distributed mass, the symbol is taken as the link or . For any beam element numbered in the link, its unit length mass matrix is:

[0135] (32)

[0136] where is the density of the beam element, is the cross-sectional area of the beam element, is the length of the beam element, is the second moment of the cross-section of the beam element.

[0137] The stiffness matrix of the beam element is derived from the stress-strain relationship, defining the strain-displacement matrix as:

[0138] (33)

[0139] The material stiffness matrix of the beam element is defined as where and are the Young's modulus and the shear modulus, respectively. The strain and the stress at any point in the beam element are:

[0140] (34)

[0141] The dynamic equation of the beam element is written from the virtual work principle:

[0142] (35)

[0143] where is the virtual velocity of the beam element, is the spatial acceleration of the beam element, is the spatial support reaction and active force of the beam element, is the external force, is the unit mass matrix of the beam element based on the link coordinate system. The integral form of the beam element is:

[0144] (36)

[0145] where , . The mass, Coriolis force, gravity, stiffness, and generalized force matrices of the beam element in the generalized coordinate space are:

[0146] (37)

[0147] wherein is the deformation selection matrix of the beam element. The local dynamic equation of the beam element in the generalized coordinate space is obtained as:

[0148] (38)

[0149] In step S4, based on the local dynamic equations of the aforementioned lumped mass and distributed mass components, the dynamic matrix of the overall rigid-flexible coupled robot system is obtained as:

[0150] (39)

[0151] The active force in the system is only the joint driving force, and the component counter forces cancel each other out. The generalized force of the overall system and the external force are respectively:

[0152] (40)

[0153] wherein is the driving force of each joint. The local dynamic equation is substituted and the virtual velocity vector is eliminated. The overall dynamic equation of the rigid-flexible coupled robot is finally obtained as:

[0154] (41)

[0155] Thus, the rigid-flexible coupled robot modeling of the method is completed. The dynamic simulation and control of the rigid-flexible coupled robot can be performed through the model.

[0156] Embodiment Two

[0157] The embodiment discloses a robot full-flexible rigid-flexible coupled parameterized modeling system.

[0158] As Figure 2 shown, a robot full-flexible rigid-flexible coupled parameterized modeling system comprises:

[0159] A kinematic model construction module is configured to: based on MDH flexible expansion parameters, combine the floating coordinate method, the finite element method, and the Rayleigh beam model to establish a kinematic model that simultaneously contains the flexible deformation of a robot base, joints, and connecting rods;

[0160] A velocity and acceleration recursion module is configured to: based on the established kinematic model, recursively symbolize the velocity and acceleration of each key node, and calculate the velocity and acceleration of any point on the flexible connecting rod with distributed mass through a velocity mapping operator;

[0161] The dynamic equilibrium equation establishing module is configured to: based on the virtual power principle, respectively establish dynamic equilibrium equations of each lumped mass component and distributed mass component in the kinematics model in the Cartesian space, and convert the dynamic equilibrium equations to the generalized coordinate space by using a generalized Jacobian matrix;

[0162] The overall dynamic equation assembling module is configured to: assemble the local dynamic equations of all components in the generalized coordinate space to form a unified robot system rigid-flexible coupling dynamic equation.

[0163] Embodiment three

[0164] An object of the embodiment is to provide a computer-readable storage medium.

[0165] A computer-readable storage medium, having stored thereon a computer program, which, when executed by a processor, implements the steps of the robot full-flexible rigid-flexible coupling parameterized modeling method according to embodiment 1.

[0166] Embodiment four

[0167] An object of the embodiment is to provide an electronic device.

[0168] An electronic device, comprising a memory, a processor, and a program stored in the memory and executable on the processor, wherein the processor implements the steps of the robot full-flexible rigid-flexible coupling parameterized modeling method according to embodiment 1 when executing the program.

[0169] The steps and methods involved in the above embodiments two, three and four correspond to embodiment one, and the specific embodiments can be referred to the relevant description part of embodiment one. The term "computer-readable storage medium" should be understood as including a single medium or multiple media of one or more instruction sets; it should also be understood as including any medium capable of storing, encoding or carrying instruction sets for execution by a processor and causing the processor to perform any method in the present application.

[0170] Those skilled in the art should understand that each module or step of the present application described above can be realized by a general computer device, alternatively, they can be realized by program codes executable by a computing device, so that they can be stored in a storage device for execution by a computing device, or they can be respectively manufactured into individual integrated circuit modules, or a plurality of modules or steps among them can be manufactured into a single integrated circuit module. The present application is not limited to any specific combination of hardware and software.

[0171] The above describes the specific embodiments of the present application in combination with the drawings, but is not a limitation on the protection scope of the present application. Those skilled in the art should understand that various modifications or variations made by those skilled in the art on the basis of the technical solutions of the present application without creative labor are still within the protection scope of the present application.

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 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, as follows: 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 extension 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. By using the first-order linear approximation of the deformed Lie algebra, the deformation increment matrix of each component is constructed, and then the homogeneous transformation matrix of the base, joint, and link considering flexible deformation is obtained. Flexible modeling of robot components is performed based on a 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, dividing the links into several beam elements. The deformation of the entire link is characterized by the deformation of the element nodes. Based on the defined MDH flexible extension parameters, homogeneous transformation matrix, and component flexible models, a recursive relationship is established from the base coordinate system to the end effector coordinate system. According to the serial connection order 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. 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, 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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 combining the floating coordinate method, finite element method, and Rayleigh beam model. Specifically: 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 extension 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. By using the first-order linear approximation of the deformed Lie algebra, the deformation increment matrix of each component is constructed, and then the homogeneous transformation matrix of the base, joint, and link considering flexible deformation is obtained. Flexible modeling of robot components is performed based on a 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, dividing the links into several beam elements. The deformation of the entire link is characterized by the deformation of the element nodes. Based on the defined MDH flexible extension parameters, homogeneous transformation matrix, and component flexible models, a recursive relationship is established from the base coordinate system to the end effector coordinate system. According to the serial connection order 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. 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.

8. A computer-readable storage medium having a program stored thereon, characterized in that, When executed by the processor, the program implements the steps in the parametric modeling method for fully flexible rigid-flexible coupling of a robot as described in any one of claims 1-6.

9. 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 1-6.

Images