Robot dynamics modeling method based on spinor theory and natural orthogonal complement

Abstract

The invention discloses a robot dynamic modeling method based on a spinor theory and natural orthogonal complements.The robot dynamic modeling method relates to the technical field of parallel robot dynamic modeling and comprises the steps that the mapping relation from an active joint speed vector to a rod piece speed spinor is determined, the constraint force spinor in an equation set is eliminated, and a dynamic model with the minimum freedom degree is established; and displaying that the model is applied to motor output torque prediction under a specific track, and verifying the effectiveness of the model. According to the method, the conciseness and the calculation efficiency of the model are remarkably improved, the problems encountered when a traditional method processes complex motion constraints are solved, the influence of non-acting constraint force in the calculation process can be effectively reduced, the complexity of the model can be greatly reduced, the real-time control performance is remarkably enhanced, and in addition, the real-time control performance is improved. The dynamic model established based on the method has higher universality and systematicness, the design process of the control system is greatly simplified, the error rate in the design process is reduced, and the development efficiency is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of parallel robot dynamics modeling, and particularly to a robot dynamics modeling method based on screw theory and natural orthogonal complement. Background Art

[0002] The robot dynamics model describes the relationship between joint motion, position, velocity, and acceleration and the output torque of the actuator. The establishment of the parallel robot dynamics model is not only crucial for system simulation but also very important for establishing an accurate and efficient model-based control system. The establishment of the multi-rigid-body system dynamics model is generally based on Newton-Euler equations, Euler-Lagrange equations, virtual work principle, Hamilton principle, or Kane equations.

[0003] The Newton-Euler method is the most classic and continuously active dynamics modeling method. Force analysis is performed on each moving link to establish the Newton-Euler equations to describe the relationship between the velocity screw and the force screw of the link. Then, the non-work-producing constraint force screws between adjacent links are eliminated through the natural orthogonal complement, thereby deriving the minimum-degree-of-freedom dynamic equation of the system. The natural orthogonal complement is defined as the velocity screw generation matrix, that is, the transformation matrix that maps from the active joint rate vector to the link velocity screw. This velocity screw generation matrix is naturally orthogonal to the robot motion constraint matrix, so it can be an effective tool for eliminating the non-work-producing constraint force helices between adjacent links. The concept of natural orthogonal complement was first proposed by Angeles and Lee, and its effectiveness has been proven in the dynamics modeling of serial robots. However, for parallel robots, the complex constraints caused by the closed-loop structure pose challenges in the derivation of the dynamics model using the natural orthogonal complement.

[0004] Currently, in the parallel robot dynamics modeling of the prior art, problems such as high model complexity, low computational efficiency, and difficulty in accurately describing complex motion constraints are usually faced. Especially for parallel robots with a closed-loop structure, their complex kinematic and dynamic relationships make it difficult for traditional modeling methods to efficiently and accurately establish the dynamics model. At the same time, when dealing with non-work-producing constraint forces, a large amount of computational resources are often required to eliminate these constraint forces, which affects the real-time control performance and simulation efficiency. In addition, for parallel robots with a specific configuration, there is a lack of a systematic method to simplify the dynamics equations, making the control system design process cumbersome and error-prone. These defects limit the performance of parallel robots in high-speed and high-precision application scenarios. Summary of the Invention

[0005] In view of the problems existing in the existing robot dynamics modeling methods based on screw theory and natural orthogonal complement, the present invention is proposed.

[0006] Therefore, in view of the defects of high complexity and low computational efficiency in the dynamic modeling of parallel robots in the prior art, the present invention adopts a dynamic modeling technique based on screw theory and natural orthogonal complement method, which effectively simplifies the model and improves the computational efficiency.

[0007] To solve the above technical problems, the present invention provides the following technical solutions:

[0008] In a first aspect, an embodiment of the present invention provides a robot dynamic modeling method based on screw theory and natural orthogonal complement, which includes:

[0009] Define the structure of a 3-PRR planar parallel robot, define the displacement screw and velocity screw of the moving platform, and determine the mapping relationship from the active joint rate vector to the bar velocity screw by analyzing the kinematics and geometry of the robot.

[0010] According to the inertial parameters and force conditions of each rigid moving bar, list the Newton-Euler dynamic equations of a single bar, combine the dynamic equations of all bars to complete the dynamic equation set of the entire robot, and use the velocity screw generation matrix as a tool to eliminate the constraint force screw in the equation set and establish a dynamic model with the minimum degree of freedom.

[0011] Verify the established dynamic model, compare it with the results of ADAMS simulation software, and demonstrate the prediction of the motor output torque under a specific trajectory by applying this model to confirm the effectiveness of the model.

[0012] As a preferred solution of the robot dynamic modeling method based on screw theory and natural orthogonal complement of the present invention, wherein: the definition of the 3- P The structure of the RR planar parallel robot includes using the symmetric structure of the robot to define the equilateral triangles of the moving platform and the base, and connecting the series 3-PRR planar structure through three identical linkages. This robot can output planar motion, that is, the calculation formula is:

[0013]

[0014] Among them, p represents the position vector of the robot, θ represents the rotation angle of the revolute pair, c represents the position coordinate of the center of the moving platform on the motion plane, R 3 represents three-dimensional real space, x c represents the coordinate of the center of the moving platform on the X-axis direction of the plane, y c represents the coordinate of the center of the moving platform on the Y-axis direction of the plane.

[0015] As a preferred solution of the robot dynamics modeling method based on screw theory and natural orthogonal complement according to the present invention, wherein: the defined displacement screw and velocity screw of the moving platform include that the velocity screw of the moving platform is the derivative of the displacement screw with respect to time, and the specific calculation formula is:

[0016]

[0017] where, ω represents the angular velocity of the moving platform rotating around the Z axis, represents the velocity vector of the center of the moving platform on the motion plane, or ω represents the angular velocity of the moving platform around the Z axis, represents the time derivative of the displacement screw, t M represents the velocity screw of the moving platform;

[0018] The analysis of the joint rates of the velocity screw includes that the robot consists of three identical leg chains, and each leg chain is a serial 3- P RR kinematic chain. For the jth leg chain, the velocity of the moving platform is determined by the joint rates of the leg chain, and the velocity screw t M of the moving platform is obtained by linear transformation from the joint rate vector of the leg chain, and the specific calculation formula is:

[0019]

[0020] where, t M represents the velocity screw of the moving platform, J j represents the Jacobian matrix of the jth leg chain, represents the rate vector of the jth active joint;

[0021] By eliminating the passive joint rates, the direct relationship between the active joint rate vector and the velocity screw t M of the moving platform is obtained, and the specific calculation formula is:

[0022]

[0023] where, D and H respectively represent the forward and inverse Jacobian matrices of the robot, represents the active joint rate vector.

[0024] As a preferred solution of the robot dynamics modeling method based on screw theory and natural orthogonal complement according to the present invention, wherein: the listing of the equations according to its inertial parameters and force conditions includes the dynamic equations of a single rigid rod based on the Newton-Euler method, considering the inertia tensor, angular velocity, center-of-mass velocity of each rod, and the forces and torques applied to it;

[0025] In the planar case, the inertia tensor of each moving rigid rod is represented by a 3×3 matrix, and the specific formula is as follows:

[0026]

[0027] where I i represents the moment of inertia of the i-th moving rod rotating about an axis passing through its center of mass and parallel to the Z-axis, m i represents the mass of the i-th moving rod, 0 represents the two-dimensional zero vector, 1 represents the 2×2 identity matrix, M i represents the inertia matrix of the i-th rod, and 0 T represents the transpose of the zero vector.

[0028] As a preferred solution of the robot dynamics modeling method based on screw theory and natural orthogonal complement according to the present invention, wherein: the combination of the dynamic equations of all rods includes a dynamic equation set describing the entire robot, and the specific steps are as follows:

[0029] Combining the Newton-Euler equations of each rigid moving rod, the dynamics of the robot is shown by a 21-dimensional uncoupled equation set:

[0030]

[0031] wherein, represents the 21-dimensional uncoupled equation set, w A represents the active force screw, and w C represents the constraint force screw;

[0032] The elimination of the constraint force screw in the equation set includes eliminating the constraint force screw w C of the robot, and at the same time reducing the 21-dimensional uncoupled equation set to three dimensions.

[0033] As a preferred solution of the robot dynamics modeling method based on screw theory and natural orthogonal complement according to the present invention, wherein: the verification of the established dynamics model includes a 3- P RR parallel robot prototype with desktop size, the moving platform includes a lower-layer moving platform, an upper-layer moving platform and a connecting platform between the two, and the three parts are rigidly connected and belong to a rigid body. The side length of the triangle of the moving platform is represented by a, the side length of the triangle of the base is represented by b, and the rod length of the connecting rod is represented by l. The lengths are respectively:

[0034] a = 110mm, b = 400mm, l = 159.15mm

[0035] wherein, a represents the side length of the triangle of the moving platform, b represents the side length of the triangle of the base, and l represents the rod length of the connecting rod.

[0036] As a preferred solution of the robot dynamics modeling method based on screw theory and natural orthogonal complement according to the present invention, wherein: the comparison with the results of the ADAMS simulation software includes calculating the kinematics and dynamics of the robot in Matlab, based on the CAD model of the robot, simulating the kinematics and dynamics of the robot in ADAMS. The motion test trajectory of the robot is described by the displacement and rotation angle of the moving platform relative to the initial reference configuration in the O-XYZ coordinate system. By comparison, the calculation results based on the mathematical model and the simulation results based on ADAMS are obtained, and the control of robot dynamics modeling is completed.

[0037] In a second aspect, an embodiment of the present invention provides a robot dynamics modeling system based on screw theory and natural orthogonal complement, which includes: a definition module that defines the structure of a 3- P RR planar parallel robot, defines the displacement screw and velocity screw of the moving platform, and determines the mapping relationship from the active joint rate vector to the link velocity screw by analyzing the joint rate of the velocity screw; a construction module that, for a rigid motion link, lists a system of equations according to its inertial parameters and force conditions, combines the dynamic equations of all links to complete the dynamic equations of the entire robot, and uses the velocity screw generation matrix as a tool to eliminate the constraint force screw in the equations and establish a dynamic model with the minimum degree of freedom; a verification module that verifies the established dynamic model, compares it with the results of the ADAMS simulation software, and demonstrates the prediction of the motor output torque under a specific trajectory by applying the model to confirm the effectiveness of the model.

[0038] In a third aspect, an embodiment of the present invention provides a computer device, including a memory and a processor, where the memory stores a computer program, and wherein: when the processor executes the computer program, any step of the above-mentioned robot dynamics modeling method based on screw theory and natural orthogonal complement is implemented.

[0039] In a fourth aspect, an embodiment of the present invention provides a computer-readable storage medium, on which a computer program is stored, and wherein: when the computer program is executed by a processor, any step of the above-mentioned robot dynamics modeling method based on screw theory and natural orthogonal complement is implemented.

[0040] The beneficial effects of the present invention are: by adopting the screw theory and natural orthogonal complement method for 3 PThe dynamic modeling of the RR planar parallel robot significantly improves the simplicity and computational efficiency of the model, solves the problems encountered by traditional methods in dealing with complex motion constraints. The present invention can not only effectively reduce the influence of non-work done constraint forces in the calculation process, but also greatly reduce the model complexity, resulting in a significant enhancement of the real-time control performance. In addition, the dynamic model established based on this method has higher generality and systematicness, greatly simplifies the design process of the control system, reduces the error rate in the design process, and improves the development efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts. Among them:

[0042] Figure 1 It is a flowchart of the dynamic model of the minimum degree of freedom of a robot based on screw theory and natural orthogonal complement.

[0043] Figure 2 It is a 3 P Structural diagram of any configuration of the RR planar parallel robot.

[0044] Figure 3 It is a 3 P Structural diagram of the initial reference configuration of the RR planar parallel robot.

[0045] Figure 4 It is a schematic diagram of the motor displacement output force under the No. 1 test trajectory of a robot dynamic modeling method based on screw theory and natural orthogonal complement.

[0046] Figure 5 It is a schematic diagram of the motor displacement output force under the No. 2 test trajectory of a robot dynamic modeling method based on screw theory and natural orthogonal complement. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0047] In order to make the above objects, features, and advantages of the present invention more obvious and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings of the specification. Obviously, the described embodiments are part of the embodiments of the present invention, not all of them. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0048] In the following description, numerous specific details are set forth to provide a thorough understanding of the present invention. However, the present invention may be practiced in other ways than those specifically described herein. Those skilled in the art can make similar generalizations without departing from the spirit of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed below.

[0049] Secondly, the so-called "one embodiment" or "embodiment" herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation manner of the present invention. The appearances of "in one embodiment" in different places in this specification do not all refer to the same embodiment, nor are they separate or alternative embodiments that are mutually exclusive of other embodiments.

[0050] The present invention is described in detail in conjunction with schematic diagrams. When detailing the embodiments of the present invention, for ease of explanation, the cross-sectional views showing the device structure will be enlarged locally in a non-general proportion, and the schematic diagrams are only examples and should not limit the scope of protection of the present invention herein. In addition, in actual production, three-dimensional spatial dimensions including length, width, and depth should be included.

[0051] At the same time, in the description of the present invention, it should be noted that the orientation or positional relationship indicated by terms such as "upper, lower, inner, and outer" is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as a limitation to the present invention. In addition, the terms "first, second, or third" are only used for descriptive purposes and cannot be construed as indicating or implying relative importance.

[0052] Unless otherwise clearly defined and limited in the present invention, the terms "mounted, connected, and coupled" should be understood in a broad sense. For example, it may be a fixed connection, a detachable connection, or an integral connection; it may also be a mechanical connection, an electrical connection, or a direct connection, and may also be indirectly connected through an intermediate medium, or may be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0053] Embodiment 1

[0054] Referring to Figures 1 to 5 , which is the first embodiment of the present invention. This embodiment provides a robot dynamics modeling method based on screw theory and natural orthogonal complement, including:

[0055] S1: Define the structure of a 3- P RR planar parallel robot, define the displacement screw and velocity screw of the moving platform, and determine the mapping relationship from the active joint rate vector to the link velocity screw by analyzing the kinematics and geometry of the robot.

[0056] Among them, define 3- P The structure of the RR planar parallel robot includes using the symmetric structure of the robot to define an equilateral triangle of the moving platform and the base, and connecting a series of 3-PRR planar structures through three identical linkages. This robot can output planar motion, that is, the calculation formula is:

[0057]

[0058] Among them, p represents the position vector of the robot, θ represents the rotation angle of the revolute pair, c represents the position coordinate of the center of the moving platform, and R 3 represents three-dimensional real space, and x c represents the coordinate of the center of the moving platform in the X-axis direction on the plane, and y c represents the coordinate of the center of the moving platform in the Y-axis direction on the plane.

[0059] Driven by a linear motor fixed on the base, the design purpose of this robot is to generate high-frequency and small-amplitude vibrations on the plane for rigid body inertia parameter identification and earthquake simulation, etc. The axis of the revolute pair is perpendicular to the motion plane, and the axis of the prismatic pair is coplanar with the motion plane. The linear motor drives the slider to move along the three sides of the equilateral triangle with B i as the vertex, and the revolute pair moves passively accordingly. The center of the revolute pair closer to the driving motor is A i , and the center of the revolute pair farther from the driving motor is M i , and the moving platform is an equilateral triangle with M i as the vertex. The center C of the moving platform is selected as the operating point. Driven by 3 linear motors, for the convenience of analysis, the global reference coordinate system O-XYZ is fixed on the base, the origin O is located at the center of the base triangle, the X-axis is along the direction of B3B1, the Y-axis is along the direction of OB2, and the Z-axis is perpendicular to the base plane and passes through point O.

[0060] Define the displacement screw and velocity screw of the moving platform. The velocity screw of the moving platform is the derivative of the displacement screw with respect to time, and the specific calculation formula is:

[0061]

[0062] Among them, ω represents the angular velocity of the moving platform rotating around the Z-axis, represents the linear velocity of the center of the moving platform, or ω represents the angular velocity of the moving platform around the Z-axis, represents the time derivative of the displacement screw, and t M represents the velocity screw of the moving platform.

[0063] By analyzing the joint rates of the velocity screw, including for the robot with three identical linkages, each linkage is a series of 3- PRR kinematic chain. For the j-th limb, the velocity of the moving platform is determined by the limb joint rates. The velocity screw t of the moving platform is obtained through a linear transformation from the joint rate vector of the limb, and the specific calculation formula is: Specifically, the calculation formula is:

[0064]

[0065] where t M represents the velocity screw of the moving platform, and J j represents the Jacobian matrix of the j-th limb, represents the rate vector of the j-th active joint;

[0066] The velocity screw t of the moving platform is calculated through the joint rate vectors of each limb , where J M is the Jacobian matrix that converts the joint rate to the velocity screw, and the specific calculation formula is: i Specifically, the calculation formula is:

[0067]

[0068] where J i represents the Jacobian matrix of the j-th limb, represents the rate vector of the j-th active joint, e j represents the unit vector, E represents the basic screw matrix, p Aj represents the translation vector of the moving platform relative to the base, p Mj represents the translation vector of the moving platform relative to the base, R 3×3 represents a 3×3 real matrix, represents the translational velocity of the j-th limb, and represent the rotational velocity of the j-th limb, and T represents the transpose of the vector.

[0069] By eliminating the passive joint rates, the direct relationship between the active joint rate vector and the velocity screw t of the moving platform M is obtained, and the specific calculation formula is:

[0070]

[0071] where D and H respectively represent the forward and inverse Jacobian matrices of the robot, represents the active joint rate vector.

[0072] S2: According to the inertial parameters and force conditions of each rigid moving rod, list the Newton-Euler dynamics equations for a single rod, combine the dynamics equations of all rods to complete the dynamics equations of the entire robot, and use the velocity screw generation matrix as a tool to eliminate the constraint force screw in the equations and establish a dynamics model with the minimum degrees of freedom.

[0073] Among them, listing the equations according to its inertial parameters and force conditions includes the dynamics equations of a single rigid rod based on the Newton-Euler method, considering the inertia tensor, angular velocity, center-of-mass velocity of each rod, and the forces and torques applied to it;

[0074] In the planar case, the inertia tensor of each moving rigid rod is represented by a 3×3 matrix, and the specific formula is:

[0075]

[0076] where, I i represents the moment of inertia of the i-th moving rod rotating about the axis passing through its center of mass and parallel to the Z-axis, m i represents the mass of the i-th moving rod, 0 represents the two-dimensional zero vector, 1 represents the 2×2 identity matrix, M i represents the inertia matrix of the i-th rod, 0 T represents the transpose of the zero vector;

[0077] The velocity screw t i of the i-th moving rod is represented by a three-dimensional vector, and the specific calculation formula is:

[0078]

[0079] where, ω i represents the angular velocity of the i-th moving rod, with the counterclockwise direction being positive, c i represents the position vector of the center of mass C i of the rod relative to the origin O, represents the linear velocity at the center of mass of the i-th moving rod, t i represents the velocity screw;

[0080] The planar force screw w i acting on the i-th moving rod, the specific calculation formula is:

[0081]

[0082] where, w i represents the planar force screw, n i and f i respectively represent the resultant torque and resultant force acting on the center of mass of the i-th moving rod, Denote the wrench screw of the work - done force exerted on the rod by the environment and the motor, Denote the non - work - done constraint force screw exerted on the rod by the adjacent rod;

[0083] In the case of neglecting all dissipative forces, the Newton - Euler equation of the \(i\) - th moving rod based on screw theory is expressed as:

[0084]

[0085] where, Denote the force screw exerted by the motor, that is, the active part of the work - done force screw For the \(j\) - th limb, \(j = 1,2,3,\cdots,t\) j Represents the velocity screw of the \(j\) - th slider, j+3 Represents the velocity screw of the \(j\) - th connecting rod, \(t_7=t\) M Represents the velocity screw of the moving platform, Denote the constraint force screw of the \(i\) - th rod, Denote the Newton - Euler equation of the \(i\) - th moving rod based on screw theory.

[0086] S2.1: Combine the dynamic equations of all rods to form a dynamic equation set describing the whole robot. The specific steps are as follows:

[0087] Combine the Newton - Euler equations of each rigid moving rod. The dynamics of the robot can be shown by a 21 - dimensional uncoupled equation set:

[0088]

[0089] where, Denote the 21 - dimensional uncoupled equation set, A Denote the active force screw, C Denote the constraint force screw;

[0090] where,

[0091]

[0092]

[0093] The \(t\), A and C in

[0094] are respectively called the velocity screw, active force screw and constraint force screw of the planar parallel robot. C Eliminating the constraint force screw in the equation set includes eliminating the constraint force screw \(w\) of the robot,

[0095] The motion constraint equation of the robot is represented in the homogeneous linear form of the robot's velocity screw as follows:

[0096] Kt = 0,

[0097] where Kt represents the velocity screw of the constraint matrix, t represents the velocity screw, and K represents the constraint matrix, and R 21×21 represents a 21×21 matrix representation;

[0098] The motion states of the robot's links are determined by the motor speeds, i.e., the active joint speeds. Then, the velocity screw t of the robot is represented by the linear transformation of the active joint speed vector as follows:

[0099]

[0100] where t represents the velocity screw, T represents the velocity screw generation matrix of the robot, and R 21×3 represents a 21×3 matrix representation.

[0101] Substituting the motion constraint equation of the robot in the homogeneous linear form of the robot's velocity screw into the linear transformation of the robot's velocity screw t through the active joint speed vector can be represented as:

[0102]

[0103] where the product of matrix K and matrix T is a zero matrix. The specific formula is:

[0104] KT = O 21×3

[0105] where the velocity screw generation matrix T of the robot is called the natural orthogonal complement of the motion constraint matrix K. Establishing the dynamic model with the minimum degrees of freedom includes establishing the dynamic model with the minimum degrees of freedom. The specific steps are as follows:

[0106] The specific formula for the constraint force screw of the robot not doing work is:

[0107]

[0108] where the velocity screw generation matrix T is the annihilator of the constraint force helix w C t T w C represents the constraint force screw of the transpose of the velocity screw, represents the transpose of the active joint speed vector, T T represents the transpose of the velocity generation matrix, and 0 represents the zero vector or zero value.

[0109] Multiply both the left and right sides of the 21-dimensional uncoupled equation system by T on the leftT , substituting the linear transformation into the 21-dimensional uncoupled equations, eliminating the constraint force screw, and obtaining the dynamic equation of the joint space robot with the minimum degrees of freedom as follows:

[0110]

[0111] where, τ represents the torque vector, represents the acceleration vector of the inertia matrix, represents the velocity vector of the Coriolis force and centrifugal force matrix.

[0112] where,

[0113] Ⅰ = T T MT, τ = T T w A

[0114] where, M = diag(M1, …, M7) ∈ R 21×21 , I ∈ R 3×3 represents the inertia matrix of the robot in the joint space, C ∈ R 3×3 represents the Coriolis force and centrifugal force matrix of the robot in the joint space, τ = [τ1, τ2, τ3] T represents the motor torque vector, τ j represents the torque exerted by the j-th motor on the j-th slider. The robot inertia matrix I is symmetric positive definite and related to the pose of the robot. In addition, all elements in the inertia matrix I have the same dimension, and the unit is kg.

[0115] S3: Verify the established dynamic model and compare it with the results of the ADAMS simulation software, and show the prediction of the motor output torque under a specific trajectory by applying this model to confirm the effectiveness of the model.

[0116] where, verifying the established dynamic model includes using a 3- P RR parallel robot prototype with the desktop size. The moving platform includes a lower moving platform, an upper moving platform, and a connecting platform between the two. The three parts are rigidly connected and belong to a rigid body. The side length of the triangle of the moving platform is represented by a, the side length of the triangle of the base is represented by b, and the rod length of the connecting rod is represented by l. The lengths are respectively:

[0117] a = 110mm, b = 400mm, l = 159.15mm

[0118] where, a represents the side length of the triangle of the moving platform, b represents the side length of the triangle of the base, and l represents the rod length of the connecting rod.

[0119] Furthermore, a comparison with the results of the ADAMS simulation software includes calculating the kinematics and dynamics of the robot in Matlab. Based on the CAD model of the robot, the kinematics and dynamics of the robot are simulated in ADAMS. The motion test trajectory of the robot is described by the displacement and rotation angle of the moving platform relative to the initial reference configuration in the O-XYZ coordinate system. By comparison, the calculation results based on the mathematical model and the simulation results based on ADAMS are obtained, and the control of the robot dynamics modeling is completed.

[0120] The formula it describes is as follows:

[0121]

[0122] Among them, x1(t) and x2(t) represent the X coordinates of the first and second points at time t, y1(t) and y2(t) represent the Y coordinates of the first and second points at time t, and θ1(t) and θ2(t) represent the angles of the first and second points at time t;

[0123] By comparison, the calculation results based on the mathematical model and the simulation results based on ADAMS are obtained, and the control of the robot dynamics modeling is completed.

[0124] A high degree of consistency is maintained. Therefore, it shows that the proposed method for parallel robot dynamics modeling based on screw theory and natural orthogonal complement has high accuracy and effectiveness, and can be further used for the dynamics simulation of parallel robots and model-based real-time control.

[0125] In a preferred embodiment, a robot dynamics modeling system based on screw theory and natural orthogonal complement, the system includes a definition module, which defines the structure of a 3- P RR planar parallel robot, defines the displacement screw and velocity screw of the moving platform, and determines the mapping relationship from the active joint rate vector to the bar velocity screw by analyzing the kinematics and geometry of the robot; a construction module, which lists the equations according to the inertial parameters and force conditions of the rigid motion bars, and lists the Newton-Euler dynamics equations of a single bar according to the inertial parameters and force conditions of each rigid motion bar, combines the dynamics equations of all bars to complete the dynamics equations of the entire robot, and uses the velocity screw generation matrix as a tool to eliminate the constraint force screw in the equations and establish a dynamics model with the minimum degrees of freedom; a verification module, which verifies the established dynamics model, compares it with the results of the ADAMS simulation software, and demonstrates the prediction of the motor output torque under a specific trajectory by applying the model to confirm the effectiveness of the model.

[0126] Each of the above - mentioned unit modules can be embedded in the processor of a computer device in hardware form or be independent of it, or be stored in the memory of the computer device in software form, so that the processor can call and execute the operations corresponding to each of the above modules.

[0127] The computer device can be a terminal. The computer device includes a processor, a memory, a communication interface, a display screen, and an input device connected through a system bus. Among them, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non - volatile storage medium and an internal memory. The non - volatile storage medium stores an operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs in the non - volatile storage medium. The communication interface of the computer device is used to communicate with an external terminal in a wired or wireless manner. The wireless manner can be achieved through WIFI, a carrier network, NFC (Near Field Communication), or other technologies. The display screen of the computer device can be a liquid crystal display screen or an electronic ink display screen. The input device of the computer device can be a touch layer covering the display screen, or a button, a trackball, or a touchpad set on the housing of the computer device, or an external keyboard, touchpad, or mouse, etc.

[0128] In summary, the present invention uses screw theory and natural orthogonal complement method to perform dynamic modeling on a 3 P RR planar parallel robot, significantly improving the simplicity and computational efficiency of the model, solving the problems encountered by traditional methods in dealing with complex motion constraints. The present invention can not only effectively reduce the influence of non - working constraint forces in the calculation process, but also greatly reduce the model complexity, significantly enhancing the real - time control performance. In addition, the dynamic model established based on this method has higher generality and systematicness, greatly simplifying the design process of the control system, reducing the error rate in the design process, and improving the development efficiency.

[0129] Embodiment 2

[0130] Referring to Figures 1 to 5 , this is the second embodiment of the present invention. This embodiment provides a robot dynamic modeling method based on screw theory and natural orthogonal complement. In order to verify the beneficial effects of the present invention, scientific demonstration is carried out through simulation experiments.

[0131] This planar parallel robot is driven by three identical voice - coil motors. The slider of the active moving pair includes the moving voice - coil of the motor and the platform fixed on the moving voice - coil. The moving platform includes three parts, namely the bottom - layer moving platform, the upper - layer moving platform, and the connecting platform between the two. The three parts are rigidly connected and belong to a rigid body. The side length of the triangle of the moving platform is represented by a, and the side length of the triangle of the base is represented by b. In addition, the inertia parameters of different rods are shown in Table 1:

[0132] Table 1 Inertia Parameter Table of Different Rods

[0133] Rod Mass (g) <![CDATA[Moment of inertia (g·mm 2 )]]> Motor moving voice coil 1020 - Motor moving platform 449.66 - Connecting rod 638.63 1578606.60 Lower moving platform 417.35 716283.92 Moving platform connecting table 287.65 132353.74 Upper moving platform 1047.57 5324476.16

[0134] The experimental simulation results show that, compared with the prior art, the present invention can significantly improve the response speed and running stability of the robot while ensuring high precision, providing strong technical support for the parallel robot in high-speed and high-precision application scenarios. This indicates that in the field of parallel robots, the present invention provides a more efficient and accurate dynamic modeling and control scheme.

[0135] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A robot dynamics modeling method based on screw theory and natural orthogonal complement, characterized in that: including Define the structure of the 3-PRR planar parallel robot, define the displacement screw and velocity screw of the moving platform, and determine the mapping relationship from the active joint rate vector to the bar velocity screw by analyzing the kinematics and geometry of the robot; According to the inertial parameters and force conditions of each rigid moving bar, list the Newton-Euler dynamics equations of a single bar, combine the dynamics equations of all bars to complete the dynamics equations of the entire robot, and use the velocity screw generation matrix as a tool to eliminate the constraint force screw in the equations and establish a dynamic model with the minimum degree of freedom; Verify the established dynamic model, compare it with the results of the ADAMS simulation software, demonstrate the prediction of the motor output torque under a specific trajectory by applying the model, and confirm the effectiveness of the model.

2. The robot dynamics modeling method based on screw theory and natural orthogonal complement according to claim 1, characterized in that: The defined 3- P The structure of the RR planar parallel robot includes using the symmetric structure of the robot to define an equilateral triangle for the moving platform and the base, and connecting three identical chain branches in series to form a 3- P RR planar structure. This robot can output planar motion, and its calculation formula is: Among them, p represents the position vector of the robot, θ represents the rotation angle of the revolute pair, c represents the position coordinates of the center of the moving platform on the motion plane, and R 3 represents the three-dimensional real number space, x c represents the coordinate of the center of the moving platform in the X-axis direction on the plane, and y c represents the coordinate of the center of the moving platform in the Y-axis direction on the plane.

3. The robot dynamics modeling method based on screw theory and natural orthogonal complement according to claim 2, characterized in that: The above-mentioned definition of the displacement screw and velocity screw of the moving platform includes that the velocity screw of the moving platform is the derivative of the displacement screw with respect to time, and the specific calculation formula is: where ω represents the angular velocity of the moving platform rotating about the Z-axis, represents the translational velocity of the center of the moving platform on the motion plane, or ω represents the angular velocity of the moving platform about the Z-axis, represents the derivative of the displacement screw with respect to time, t M represents the velocity screw of the moving platform; The joint rates of the velocity twist are analyzed, including that the robot consists of three identical leg chains, and each leg chain is a serial 3- P RR kinematic chain. For the j-th leg chain, the velocity of the moving platform is determined by the joint rates of the leg chain, and the velocity twist t of the moving platform is obtained through linear transformation from the joint rate vector of the leg chain. The specific calculation formula is as follows: where, t M represents the velocity screw of the moving platform, J j represents the Jacobian matrix of the j-th branch chain, represents the rate vector of the j-th active joint; By eliminating the passive joint rates, a direct relationship is obtained from the active joint rate vector to the twist t of the moving platform M The specific calculation formula is as follows: where D and H represent the forward and inverse Jacobian matrices of the robot, respectively, denote the vector of active joint rates.

4. The method for robot dynamics modeling based on screw theory and natural orthogonal complement according to claim 3, characterized in that: The above-mentioned listing of the dynamics equations according to its inertial parameters and force conditions includes the dynamics equations of a single rigid bar based on the Newton-Euler method, considering the inertia tensor, angular velocity, centroid velocity of each bar, and the forces and torques applied to it; In the planar case, the inertia tensor of each moving rigid bar is represented by a 3×3 matrix, and the specific formula is: Among them, I i represents the moment of inertia of the i-th moving rod rotating about an axis passing through its centroid and parallel to the Z-axis, m i represents the mass of the i-th moving rod, 0 represents a two-dimensional zero vector, 1 represents a 2×2 identity matrix, M i represents the inertia matrix of the i-th rod, 0 T represents the transpose of the zero vector.

5. The robot dynamics modeling method based on screw theory and natural orthogonal complement according to claim 4, characterized in that: The above-mentioned combination of the dynamics equations of all bars includes describing the dynamics equations of the entire robot, and the specific steps are as follows: Combine the Newton-Euler equations of each rigid moving bar, and the dynamics of the robot is shown by a 21-dimensional uncoupled equations: Among them, represents a 21-dimensional uncoupled equation system, and w A represents the active force screw, and w C represents the constraint force screw; The elimination of the constraint screws in the equations includes eliminating the constraint screw w of the robot C while reducing the 21-dimensional uncoupled equations to three dimensions.

6. The robot dynamics modeling method based on screw theory and natural orthogonal complement according to claim 5, wherein: The established kinetic model for verification includes making a desktop-sized 3- P RR parallel robot prototype. The moving platform includes a lower moving platform, an upper moving platform, and a connecting platform between the two. The three parts are rigidly connected and belong to a rigid body. The side length of the triangle of the moving platform is represented by a, the side length of the triangle of the base is represented by b, and the rod length of the connecting rod is represented by l. The lengths are respectively: a = 110mm, b = 400mm, l = 159.15mm Among them, a represents the side length of the triangle of the moving platform, b represents the side length of the triangle of the base, and l represents the bar length of the connecting bar.

7. The robot dynamics modeling method based on screw theory and natural orthogonal complement according to claim 6, characterized in that: The above-mentioned comparison with the results of the ADAMS simulation software includes calculating the kinematics and dynamics of the robot in Matlab, simulating the kinematics and dynamics of the robot in ADAMS based on the CAD model of the robot, and the motion test trajectory of the robot is described by the displacement and rotation angle of the moving platform relative to the initial reference configuration in the O-XYZ coordinate system. By comparing the calculation results based on the mathematical model with the simulation results based on ADAMS, the control of the robot dynamics modeling is completed.

8. A robot dynamics modeling system based on screw theory and natural orthogonal complement, based on the robot dynamics modeling method based on screw theory and natural orthogonal complement according to any one of claims 1 to 7, characterized in that: including A definition module that defines the structure of the 3-PRR planar parallel robot, defines the displacement screw and velocity screw of the moving platform, and determines the mapping relationship from the active joint rate vector to the bar velocity screw by analyzing the kinematics and geometry of the robot; A establishment module that lists the Newton-Euler dynamics equations of a single bar according to the inertial parameters and force conditions of each rigid moving bar, combines the dynamics equations of all bars to complete the dynamics equations of the entire robot, uses the velocity screw generation matrix as a tool to eliminate the constraint force screw in the equations, and establishes a dynamic model with the minimum degree of freedom; A verification module, which verifies the established dynamic model, compares it with the results of the ADAMS simulation software, demonstrates the prediction of the motor output torque under a specific trajectory by applying the model, and verifies the effectiveness of the model.

9. A computer device, comprising a memory and a processor, the memory storing a computer program, characterized in that: When the processor executes the computer program, the steps of the robot dynamics modeling method based on screw theory and natural orthogonal complement according to any one of claims 1 to 7 are implemented.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by the processor, the steps of the robot dynamics modeling method based on screw theory and natural orthogonal complement according to any one of claims 1 to 7 are implemented.

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