Heavy-load forming robot dynamic modeling method considering spherical pair clearance and connecting rod deformation

Abstract

The invention relates to a heavy-load forming robot dynamic modeling method considering spherical pair gaps and connecting rod deformation. The heavy-load forming robot dynamic modeling method comprises the following steps that S1, the basic structure of a heavy-load forming robot is determined; s2, the movement position relation of the heavy-load forming robot component is determined; s3, constructing a ball pair clearance contact model of the heavy-load forming robot; s4, a kinetic equation of the heavy-load forming robot sliding block is determined; s5, a kinetic equation of a heavy-load forming robot connecting rod is established; s6, a kinetic equation of the heavy-load forming robot moving platform is determined; and S7, determining a total kinetic equation of the heavy-load forming robot. According to the heavy-load forming robot dynamic modeling method considering the spherical pair gap and the connecting rod deformation, more accurate prediction of the real-time pose of the robot can be achieved. According to the method, the dynamic model considering the spherical pair gap and the connecting rod deformation is established, the dynamic model is verified through a loading experiment, and the accuracy of the error of the moving platform can reach 70%.

Description

Technical Field

[0001] This invention relates to the field of dynamic modeling of heavy-duty forming robots, and more specifically, to a dynamic modeling method for heavy-duty forming robots that considers ball joint clearance and link deformation. Background Technology

[0002] Heavy-duty forming robots, as a branch of robotics, are primarily used in the production and manufacturing of high-performance parts. They consist of multiple branches connecting a moving platform and a stationary platform. The movement of the moving platform is achieved through the synergistic action of these branches, exhibiting complex dynamic characteristics, especially under heavy-duty conditions. Molds are mounted on the moving platform and the base, respectively. By adjusting the movement of the slider, the motion of the moving platform is altered, causing the workpiece to undergo plastic deformation through localized contact with the mold, until the final net shape is achieved. With the rapid development of aerospace, medical devices, and automotive manufacturing, higher demands are placed on the accuracy of robot modeling. Due to the large forming loads during operation, the collisions at the ball joint gaps are intense, and the connecting rods deform significantly, severely affecting the robot's working accuracy. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a dynamic modeling method for heavy-duty forming robots that considers ball joint clearance and link deformation, which can realize accurate prediction of the robot's real-time pose, so as to facilitate subsequent precise control.

[0004] The technical solution adopted by this invention to solve its technical problem is: to construct a dynamic modeling method for heavy-duty forming robots that considers ball joint clearance and link deformation, including the following steps:

[0005] S1. Determine the basic structure of the heavy-duty forming robot;

[0006] S2. Determine the motion position relationship of the heavy-duty forming robot components;

[0007] S3. Construct a gap contact model at the ball joint of the heavy-duty forming robot;

[0008] S4. Determine the dynamic equations of the slider of the heavy-duty forming robot;

[0009] S5. Dynamic equations of the linkage in a heavy-duty forming robot;

[0010] S6. Determine the dynamic equations of the moving platform of the heavy-duty forming robot;

[0011] S7. Determine the overall dynamic equations of the heavy-duty forming robot.

[0012] According to the above scheme, the heavy-duty robot includes six sets of branches. Each branch includes a slider, an upper ball joint, a connecting rod, and a lower ball joint connected in sequence. The six sets of branches are connected to the moving platform. The slider serves as the input end of the robot. Through the coordinated movement of the sliders of the six sets of branches, the corresponding moving platform achieves the desired movement. An upper mold and a lower mold are respectively installed on the moving platform and the base. When the moving platform performs complex spatial movements, the blank in the mold will gradually take shape under the combined action of the upper and lower molds.

[0013] According to the above scheme, the displacement calculation method for each slider in step S1 includes the following steps:

[0014] A1, A2, A3, A4, A5, and A6 are the centers of the slider and the upper ball joint. The distance of the slider movement is given by S; B1, B2, B3, B4, B5, and B6 are the centers of the lower ball joint; coordinate system S O Fixed on the bed surface; coordinate system S O In the diagram, the position vector r of points A1, A2, A3, A4, A5, and A6 is... a1 ,r a2 ,r a3 ,r a4 ,r a5 ,r a6 The corresponding settings are as follows:

[0015]

[0016] In the formula, It is the distribution angle corresponding to each point A1, A2…A6; r A It is the radius corresponding to all points A;

[0017] Coordinate system S P On the lower plane of the moving platform, the coordinate system value of the origin P is s. p (α,β,γ,x,y,z); the position vectors B1, B2, B3, B4, B5, B6 of each point are respectively set as The calculation formula is as follows:

[0018]

[0019] in, It is the distribution angle corresponding to each point B1, B2...B6, r B d is the radius corresponding to all points B. h It is the plane of the lower spherical joint to S P Distance to the (xoy) plane;

[0020] According to the coordinate transformation matrix and r bi The relationship between them is as follows:

[0021]

[0022] in, and T P These are the moving platform and moving coordinate system S. P To a fixed coordinate system S O The rotation matrix and translation matrix, and r bi These are the position vectors of points B1, B2, B3, B4, B5, and B6 in the moving coordinate system and the fixed coordinate system, respectively.

[0023] Since the length of the connecting rod remains constant, the displacement of each slider... Solve using the following formula:

[0024]

[0025] Among them, l g It is the length of all the links;

[0026] Point r ai and r bi Substituting into the constraint equations, we get:

[0027]

[0028] Where c is the abbreviation for cos and s is the abbreviation for sin; the equation can be solved as follows:

[0029]

[0030] In the formula,

[0031] According to the above scheme, in step S2, the coordinate system of the slider is S Si Since the slider can only move along the guide rail, its generalized coordinates can be expressed using the Newton-Euler method as q. Si =[x si ,y si ,0,0,0,0,] T ,in These represent the positions of the slider's center of mass in the x and y directions, respectively;

[0032] The connecting rod is considered as two mass blocks, and the coordinate system of the connecting rod is divided into an upper coordinate system. and lower coordinate system The generalized coordinates of the link are expressed using the Newton-Euler method as follows: in This indicates the change in position of the upper mass block. This indicates the change in angle of the upper mass block; This indicates the positional change of the lower mass block. This indicates the change in angle of the lower mass block;

[0033] The coordinate system of the moving platform is S P Its generalized coordinates are represented by the Newton-Euler method as q P =[x p ,y p ,z p ,α p ,β p ,γ p ] T , where x p ,y p ,z p α represents the change in position of the moving platform. p ,β p ,γ p This indicates the change in the angle of the moving platform;

[0034] In step S2, the six branches of the heavy-duty robot are represented by generalized coordinates as follows:

[0035]

[0036] According to the above scheme, in step S3, the LN contact model is used to establish the gap contact model at the ball joint.

[0037] According to the above scheme, in step S3, the gap vector e is introduced. i Let e ​​represent the position of the center of the sphere relative to the center of the sphere base, and let e be the position vector. Ai and e Bi This represents the clearance vector at the top and bottom of each ball joint; its magnitude is expressed as... The normal vector is represented as The deformation δ that occurs when the ball head collides with the ball seat i With position vector e i Related; can be represented as

[0038] δ i =|e i |-c (8)

[0039] Where c = R o -R i R represents the gap between the ball head and the ball seat. o and R i These are the radii of the ball head and the ball seat, respectively;

[0040] There is a relative offset between the center of the ball head and the center of the ball seat; relative position offset e Ai ,e Bi Represented as:

[0041]

[0042] In the formula, e Ai It is the position vector of the center of the upper ball head relative to the center of the ball seat;

[0043]

[0044] In the formula e Bi It is the position vector of the lower ball head center relative to the ball seat center;

[0045] During robot operation, due to the gap in the ball joint, the ball seat and ball head will inevitably collide, thus generating a contact force; the contact force F C Decompose it into force components along the normal and tangential directions; considering the hysteresis damping, the LN nonlinear contact force model can be used to decompose A i B i The contact force at the ball joint is expressed as follows:

[0046]

[0047] In the formula, K C The generalized stiffness coefficient (N / m) R i and R o σ represents the contact radius of the ball head and the ball seat, respectively; i and σ o These are the material coefficients for the ball head and the ball seat, respectively. u k and E k These are Poisson's ratio and elastic modulus of the material, respectively; D Ai D Bi For generalized damping coefficient, Where c e The coefficient of restitution is the material recovery factor. The initial collision velocity; δ Ai ,δ Bi The relative normal collision depth is n; n is the force exponent. Relative collision velocity;

[0048] Using step function To represent contact force In the ball sub-A i and B i The contact force at the point is expressed as:

[0049]

[0050] Action on ball pair A i and B iThe contact force on the surface is expressed as:

[0051]

[0052] According to the above scheme, in step S4, the equilibrium equation of the slider is:

[0053]

[0054] In the formula, F Ai The driving force of the slider, m S This indicates the mass of the slider.

[0055] According to the above scheme, in step S5, the dynamic equation of the connecting rod is expressed as:

[0056] Mass block on connecting rod:

[0057]

[0058] lower mass block of connecting rod:

[0059]

[0060] In the formula, This represents the constraint force exerted by the slider on the connecting rod. This represents the constraint forces exerted by the moving platform on the connecting rods. This indicates the weight of the connecting rod; and These represent the generalized coordinates of the upper and lower mass blocks of the connecting rod, respectively. and These are the stiffness matrix and damping matrix of the connecting rod, respectively, and the calculation formula is as follows: in χ and ε are linear damping coefficients. and This represents the angular velocity of the upper and lower mass blocks of the connecting rod. and It is the inertia matrix of the upper and lower mass blocks of the connecting rod in a fixed coordinate system. R ω It is the mapping matrix from Euler angular velocity to angular velocity in a fixed coordinate system.

[0061] According to the above scheme, in step S6, the dynamic equation of the moving platform can be expressed as:

[0062]

[0063] In the formula, F represents the constraint internal force of the link on the moving platform. P Indicates external overload on the dynamic platform, m Pg represents the gravity of the moving platform. I represents the angular velocity of the moving platform. P It is the inertia matrix of the moving platform in the fixed coordinate system. The moving platform has a moving coordinate system S. P To a fixed coordinate system S O The rotation matrix.

[0064] According to the above scheme, in step S7, the overall dynamic equation is expressed as:

[0065]

[0066] The dynamic modeling method for heavy-duty forming robots that considers ball joint clearance and link deformation according to the present invention has the following advantages:

[0067] (1) This invention proposes a dynamic modeling method for heavy-duty forming robots that considers ball joint clearance and link deformation, which can achieve more accurate prediction of robot real-time pose.

[0068] (2) The present invention establishes a dynamic model that considers the ball joint clearance and connecting rod deformation, and verifies the dynamic model through loading experiments. The accuracy of the dynamic platform error can reach 70%. Attached Figure Description

[0069] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0070] Figure 1 This is a schematic diagram of the basic structure of a heavy-duty forming robot;

[0071] Figure 2 This is a schematic diagram showing the positional relationships of components in a heavy-duty forming robot.

[0072] Figure 3 This is a schematic diagram of the ball joint gap contact model of a heavy-duty forming robot;

[0073] Figure 4 This is a force diagram of the heavy-duty forming robot as a whole;

[0074] Figure 5 This is the overall dynamic equation for the heavy-duty forming robot;

[0075] Figure 6 This is a schematic diagram of the input displacement of a heavy-duty forming robot;

[0076] Figure 7 This is a schematic diagram of the error of the moving platform of the heavy-duty forming robot;

[0077] Figure 8 This is a schematic diagram illustrating the error accuracy of the moving platform of a heavy-duty forming robot under loading experiments. Detailed Implementation

[0078] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0079] The dynamic modeling method for heavy-load forming robots that considers ball joint clearance and link deformation in this invention includes the following steps:

[0080] S1. Determine the basic structure of the heavy-duty forming robot.

[0081] This heavy-duty robot comprises six sets of branches, each consisting of a slider, an upper ball joint, a connecting rod, and a lower ball joint. These six sets of branches are connected to the moving platform in a specific arrangement. The slider serves as the robot's input; through the coordinated movement of the six sets of branch sliders, the corresponding moving platform achieves the desired motion. An upper mold and a lower mold are mounted on the moving platform and the base, respectively. When the moving platform performs complex spatial movements, the blank in the mold gradually takes shape under the combined action of the upper and lower molds. Figure 1 The diagram shown is its kinematic schematic. A1, A2, A3, A4, A5, and A6 are the centers of the slider and the upper ball joint. Let S be the distance the slider moves, and B1, B2, B3, B4, B5, and B6 be the centers of the lower ball joint. Coordinate system S. O Fixed to the bed surface. Coordinate system S O In the diagram, the position vector r of points A1, A2, A3, A4, A5, and A6 is... a1 ,r a2 ,r a3 ,r a4 ,r a5 ,r a6 The corresponding settings are as follows:

[0082]

[0083] In the formula, It is the distribution angle corresponding to each point A1, A2…A6; r A It is the radius corresponding to all points A.

[0084] Coordinate system S P On the lower plane of the moving platform, the coordinate system value of the origin P is s. p (α, β, γ, x, y, z). The position vectors B1, B2, B3, B4, B5, and B6 of each point are respectively set as... The calculation formula is as follows:

[0085]

[0086] in, It is the distribution angle corresponding to each point B1, B2...B6, r Bd is the radius corresponding to all points B. h It is the plane of the lower spherical joint to S P The distance to the (xoy) plane.

[0087] According to the coordinate transformation matrix and r bi The relationship between them is as follows:

[0088]

[0089] in, and T P These are the moving platform and moving coordinate system S. P To a fixed coordinate system S O The rotation matrix and translation matrix, and r bi These are the position vectors of points B1, B2, B3, B4, B5, and B6 in the moving coordinate system and the fixed coordinate system, respectively.

[0090] Since the length of the connecting rod remains constant, the displacement of each slider... The solution can be obtained using the following formula:

[0091]

[0092] Among them, l g It is the length of all the links.

[0093] Point r ai and r bi Substituting into the constraint equations, we get:

[0094]

[0095] Where c is the abbreviation for cosine and s is the abbreviation for sinine. The equation can be solved as follows:

[0096]

[0097] S2. Determine the motion position relationship of the heavy-duty forming robot components.

[0098] The heavy-duty forming robot consists of 6 sliders, 6 links, and a moving platform, totaling 13 components. Since the links undergo significant deformation under heavy loads, while other components deform less, it is assumed that each link comprises two mass blocks, and the deformation of other components is equivalent to that of one mass block. Disregarding deformation, there are a total of 19 mass blocks. The sliders' coordinate system is S... Si Since the slider can only move along the guide rail, its generalized coordinates can be expressed using the Newton-Euler method as q. Si =[x si ,y si ,0,0,0,0,]T ,in These represent the positions of the slider's center of mass in the x and y directions, respectively.

[0099] Since the connecting rod is considered as two mass blocks, the coordinate system of the connecting rod is divided into an upper coordinate system. and lower coordinate system The generalized coordinates of the link can be expressed using the Newton-Euler method as follows: in This indicates the change in position of the upper mass block. This indicates the change in angle of the upper mass block; This indicates the positional change of the lower mass block. This indicates the change in the angle of the lower mass block.

[0100] The coordinate system of the moving platform is S P Its generalized coordinates can be represented by the Newton-Euler method as q P =[x p ,y p ,z p ,α p ,β p ,γ p ] T , where x p ,y p ,z p α represents the change in position of the moving platform. p ,β p ,γ p This indicates the change in the angle of the moving platform.

[0101] The 6-PSS parallel mechanism equipment can be represented by generalized coordinates as follows:

[0102]

[0103] Based on the above description, the relationships between the components can be established as follows: Figure 2 As shown. r Si , and The upper and lower mass blocks of the slider and connecting rod are in a fixed coordinate system S. O The position vector in r. P The moving platform is in the fixed coordinate system S O The position vector in the middle. The lower ball subcenter is in the moving coordinate system S of the moving platform. P The position vector in the middle.

[0104] S3. Construct a gap contact model at the ball joint of the heavy-duty forming robot.

[0105] Due to the significant changes in clearance under heavy loads, the LN contact model is used to establish the clearance contact model at the ball joint. Based on this method, a clearance vector e is introduced. i To indicate the position of the ball's center relative to the center of the spherical base, such as Figure 3 As shown. Position vector e Ai and e Bi This represents the clearance vector at the top and bottom of each ball joint link. Its magnitude is expressed as... The normal vector is represented as The deformation δ that occurs when the ball head collides with the ball seat i With position vector e i Related. Can be represented as

[0106] δ i =|e i |-c (8)

[0107] Where c = R o -R i R represents the gap between the ball head and the ball seat. o and R i These are the radii of the ball head and the ball seat, respectively.

[0108] The robot has 12 ball joints. Due to the gaps in the joints, there is a relative offset between the center of the ball head and the center of the ball seat. This relative positional offset... , It can be represented as: (9) In the formula, It is the position vector of the center of the upper ball head relative to the center of the ball seat.

[0109] (10) In the formula , It is the position vector of the lower ball head center relative to the ball seat center.

[0112] During robot operation, due to the gap in the ball joint, the ball seat and ball head will inevitably collide, generating a contact force F. C This can usually be decomposed into force components along the normal and tangential directions. Here, only the influence of perpendicular contact force is considered. The LN nonlinear contact force model, taking into account hysteresis damping, can be used to decompose A... i B i The contact force at the ball joint is expressed as follows:

[0113]

[0114] In the formula, K CThe generalized stiffness coefficient (N / m) R i and R o σ represents the contact radius of the ball head and the ball seat, respectively; i and σ o These are the material coefficients for the ball head and the ball seat, respectively. u k and E k These are Poisson's ratio and elastic modulus of the material, respectively. D Ai D Bi The generalized damping coefficient (N·s / m) Where c e The coefficient of restitution for materials is taken as 0.9; The initial collision velocity (m / s); δ Ai ,δ Bi The relative normal collision depth is (m); n is the force exponent, which is 1.5, depending on the material of the contact surface. The relative collision velocity is (m / s).

[0115] Since there are multiple contact states between the ball and the ball seat when considering the ball joint gap, a step function is used. To represent contact force In the ball sub-A i and B i The contact force at the point can be expressed as:

[0116]

[0117] Action on ball pair A i and B i The contact force on the surface can be expressed as

[0118]

[0119] S4. Determine the dynamic equations of the slider of the heavy-duty forming robot.

[0120] In this robot, all six sliders move along guide rails, and the constraints in other directions are provided by the guide rails (fixed platform). Therefore, when performing force analysis on the sliders, only the forces along the guide rails need to be considered. The force situation of the sliders is as follows: Figure 4 As shown, F Ai The driving force of the slider, m S This represents the mass of the slider. The equilibrium equation for the slider can be determined as:

[0121]

[0122] S5. Construct the dynamic equations of the linkage of the heavy-duty forming robot.

[0123] Because the deformation of the connecting rod under heavy loads needs to be considered, the elastic and damping forces generated by the deformation must be taken into account. The force situation of the connecting rod is as follows: Figure 4 As shown, This represents the constraint force exerted by the slider on the connecting rod. This represents the constraint forces exerted by the moving platform on the connecting rods. This represents the weight of the connecting rod. The dynamic equations of the connecting rod can be expressed as follows:

[0124] Mass block on connecting rod:

[0125]

[0126] lower mass block of connecting rod:

[0127]

[0128] In the formula, and These represent the generalized coordinates of the upper and lower mass blocks of the connecting rod, respectively. and These are the stiffness matrix and damping matrix of the connecting rod, respectively, and the calculation formula is as follows: in χ and ε are linear damping coefficients. and This represents the angular velocity of the upper and lower mass blocks of the connecting rod. and It is the inertia matrix of the upper and lower mass blocks of the connecting rod in a fixed coordinate system. R ω It is the mapping matrix from Euler angular velocity to angular velocity in a fixed coordinate system.

[0129] S6. Construct the dynamic equations of the heavy-duty forming robot's moving platform.

[0130] The moving platform is subjected to its own weight, external heavy loads, and the constraint forces of six links. The force situation of the moving platform is as follows: Figure 4 As shown, F represents the constraint internal force of the link on the moving platform. P Indicates external overload on the dynamic platform, m P g represents the gravity of the moving platform. The dynamic equation of the moving platform can be expressed as:

[0131]

[0132] In the formula, I represents the angular velocity of the moving platform. P It is the inertia matrix of the moving platform in the fixed coordinate system. The moving platform has a moving coordinate system S. PTo a fixed coordinate system S O The rotation matrix.

[0133] S7. Determine the overall dynamic equations of the heavy-duty forming robot.

[0134] By performing force analysis on the robot, the dynamic equations of each component are obtained. Combining these equations yields the overall dynamic equation, which can be expressed as:

[0135]

[0136] Based on the above model, the variables that need to be determined include 12 slider driving forces, 78 mass block poses, and 144 constraint forces, totaling 234 variables. In addition, there are 36 contact force constraints, 36 elastic and damping force constraints, 72 force constraints, and 90 force and moment balance equations, totaling 234 equations. For example... Figure 5 As shown, the number of unknown variables in the dynamic model is the same as the number of equations. By solving the above set of equations, the dynamic state of the heavy-duty forming robot at any time can be obtained.

[0137] Based on the above-mentioned dynamic modeling method for heavy-duty forming robots that considers ball joint clearance and link deformation, a dynamic model of a 6-PSS heavy-duty forming robot was established. The design parameters are shown in Table 1, and the basic structure is as follows. Figure 1 As shown. Given the load F acting on the 6-PSS heavy-duty forming robot. P =[0,0,6]MN and the position of action r e =r e [sin(πt),cos(πt),0]. Each slider is pressed as follows: Figure 6 The given motion is input, i.e. Given that, based on the above dynamic modeling method, the moving platform error at different times within a single cycle of the robot can be calculated as follows: Figure 7 As shown. A loading experiment was conducted on a real machine using a disc-shaped blank with a radius of 100 mm and a height of 50 mm. The comparison between the experimental and theoretical attitude errors is as follows: Figure 8 As shown, the prediction accuracy of the moving platform in the x, y, z, α, β and γ directions can reach 68.82%, 69.57%, 68.69%, 69.79%, 69.57% and 69.42% respectively. This verifies the correctness of the above dynamic model that considers the ball joint clearance and connecting rod deformation, and is applicable to heavy-load conditions.

[0138] Table 1 Design parameters of 6-PSS parallel robot

[0139]

[0140] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A dynamic modeling method for heavy-duty forming robots considering ball joint clearance and link deformation, characterized in that, Includes the following steps: S1. Determine the basic structure of the heavy-duty forming robot; S2. Determine the motion position relationship of the heavy-duty forming robot components; S3. Construct a gap contact model at the ball joint of the heavy-duty forming robot; S4. Determine the dynamic equations of the slider of the heavy-duty forming robot; S5. Dynamic equations of the linkage in a heavy-duty forming robot; S6. Determine the dynamic equations of the moving platform of the heavy-duty forming robot; S7. Determine the overall dynamic equations of the heavy-duty forming robot.

2. The method for dynamic modeling of heavy-load forming robots considering ball joint clearance and link deformation according to claim 1, characterized in that, The heavy-duty robot includes six sets of branches. Each branch includes a slider, an upper ball joint, a connecting rod, and a lower ball joint connected in sequence. The six sets of branches are connected to the moving platform. The slider serves as the input end of the robot. Through the coordinated movement of the sliders of the six sets of branches, the corresponding moving platform achieves the desired movement. An upper mold and a lower mold are respectively installed on the moving platform and the base. When the moving platform performs complex spatial movements, the blank in the mold will gradually take shape under the combined action of the upper and lower molds.

3. The method for dynamic modeling of heavy-load forming robots considering ball joint clearance and link deformation according to claim 2, characterized in that, In step S1, the displacement of each slider is calculated using the following steps: A1, A2, A3, A4, A5, and A6 are the centers of the slider and the upper ball joint. The distance of the slider movement is given by S; B1, B2, B3, B4, B5, and B6 are the centers of the lower ball joint; coordinate system S O Fixed on the bed surface; coordinate system S O In the diagram, the position vector r of points A1, A2, A3, A4, A5, and A6 is... a1 ,r a2 ,r a3 ,r a4 ,r a5 ,r a6 The corresponding settings are as follows: In the formula, It is the distribution angle corresponding to each point A1, A2…A6; r A It is the radius corresponding to all points A; Coordinate system S P On the lower plane of the moving platform, the coordinate system value of the origin P is s. p (α,β,γ,x,y,z); the position vectors B1, B2, B3, B4, B5, B6 of each point are respectively set as The calculation formula is as follows: in, It is the distribution angle corresponding to each point B1, B2...B6, r B d is the radius corresponding to all points B. h It is the plane of the lower spherical joint to S P Distance to the (xoy) plane; According to the coordinate transformation matrix and r bi The relationship between them is as follows: in, and T P These are the moving platform and moving coordinate system S. P To a fixed coordinate system S O The rotation matrix and translation matrix, and r bi These are the position vectors of points B1, B2, B3, B4, B5, and B6 in the moving coordinate system and the fixed coordinate system, respectively. Since the length of the connecting rod remains constant, the displacement of each slider... Solve using the following formula: Among them, l g It is the length of all the links; Point r ai and r bi Substituting into the constraint equations, we get: Where c is the abbreviation for cos and s is the abbreviation for sin; the equation can be solved as follows: In the formula, 4. The method for dynamic modeling of heavy-load forming robots considering ball joint clearance and link deformation according to claim 2, characterized in that, In step S2, the coordinate system of the slider is S. Si Since the slider can only move along the guide rail, its generalized coordinates can be expressed using the Newton-Euler method as q. Si =[x si ,y si ,0,0,0,0,] T ,in These represent the positions of the slider's center of mass in the x and y directions, respectively; The connecting rod is considered as two mass blocks, and the coordinate system of the connecting rod is divided into an upper coordinate system. and lower coordinate system The generalized coordinates of the link are expressed using the Newton-Euler method as follows: in This indicates the change in position of the upper mass block. This indicates the change in angle of the upper mass block; This indicates the positional change of the lower mass block. This indicates the change in angle of the lower mass block; The coordinate system of the moving platform is S P Its generalized coordinates are represented by the Newton-Euler method as q P =[x p ,y p ,z p ,α p ,β p ,γ p ] T , where x p ,y p ,z p α represents the change in position of the moving platform. p ,β p ,γ p This indicates the change in the angle of the moving platform; In step S2, the six branches of the heavy-duty robot are represented by generalized coordinates as follows:

5. The method for dynamic modeling of heavy-duty forming robots considering ball joint clearance and link deformation according to claim 2, characterized in that, In step S3, the LN contact model is used to establish the gap contact model at the ball joint.

6. The method for dynamic modeling of heavy-duty forming robots considering ball joint clearance and link deformation according to claim 5, characterized in that, In step S3, the gap vector e is introduced. i Let e ​​represent the position of the center of the sphere relative to the center of the sphere base, and let e be the position vector. Ai and e Bi This represents the clearance vector at the top and bottom of each ball joint; its magnitude is expressed as... The normal vector is represented as The deformation δ that occurs when the ball head collides with the ball seat i With position vector e i Related; can be represented as d i =|e i |-c (8) Where c = R o -R i R represents the gap between the ball head and the ball seat. o and R i These are the radii of the ball head and the ball seat, respectively; There is a relative offset between the center of the ball head and the center of the ball seat; relative position offset e Ai ,e Bi Represented as: In the formula, e Ai It is the position vector of the center of the upper ball head relative to the center of the ball seat; In the formula e Bi It is the position vector of the lower ball head center relative to the ball seat center; During robot operation, due to the gap in the ball joint, the ball seat and ball head will inevitably collide, thus generating a contact force; the contact force F C Decompose it into force components along the normal and tangential directions; considering the hysteresis damping, the LN nonlinear contact force model can be used to decompose A i B i The contact force at the ball joint is expressed as follows: In the formula, K C The generalized stiffness coefficient (N / m) R i and R o σ represents the contact radius of the ball head and the ball seat, respectively; i and σ o These are the material coefficients for the ball head and the ball seat, respectively. u k and E k These are Poisson's ratio and elastic modulus of the material, respectively; D Ai D Bi For generalized damping coefficient, Where c e The coefficient of restitution is the material recovery factor. The initial collision velocity; δ Ai ,δ Bi The relative normal collision depth is n; n is the force exponent. Relative collision velocity; Using step function To represent contact force In the ball sub-A i and B i The contact force at the point is expressed as: Action on ball pair A i and B i The contact force on the surface is expressed as:

7. The method for dynamic modeling of heavy-duty forming robots considering ball joint clearance and link deformation according to claim 2, characterized in that, In step S4, the equilibrium equation of the slider is: In the formula, F Ai The driving force of the slider, m S This indicates the mass of the slider.

8. The method for dynamic modeling of heavy-load forming robots considering ball joint clearance and link deformation according to claim 2, characterized in that, In step S5, the dynamic equation of the connecting rod is expressed as: Mass block on connecting rod: lower mass block of connecting rod: In the formula, This represents the constraint force exerted by the slider on the connecting rod. This represents the constraint forces exerted by the moving platform on the connecting rods. This indicates the weight of the connecting rod; and These represent the generalized coordinates of the upper and lower mass blocks of the connecting rod, respectively. and These are the stiffness matrix and damping matrix of the connecting rod, respectively, and the calculation formula is as follows: in χ and ε are linear damping coefficients. and This represents the angular velocity of the upper and lower mass blocks of the connecting rod. and It is the inertia matrix of the upper and lower mass blocks of the connecting rod in a fixed coordinate system. R ω It is the mapping matrix from Euler angular velocity to angular velocity in a fixed coordinate system.

9. The method for dynamic modeling of heavy-duty forming robots considering ball joint clearance and link deformation according to claim 2, characterized in that, In step S6, the dynamic equation of the moving platform can be expressed as: In the formula, F represents the constraint internal force of the link on the moving platform. P Indicates external overload on the dynamic platform, m P g represents the gravity of the moving platform. I represents the angular velocity of the moving platform. P It is the inertia matrix of the moving platform in the fixed coordinate system. The moving platform has a moving coordinate system S. P To a fixed coordinate system S O The rotation matrix.

10. The method for dynamic modeling of heavy-load forming robots considering ball joint clearance and link deformation according to claim 2, characterized in that, In step S7, the overall dynamic equation is expressed as:

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