A rigid-flexible coupling modeling and analysis method for vacuum manipulators

Abstract

This invention belongs to the field of dynamic modeling of vacuum manipulators, specifically a rigid-flexible coupling modeling and analysis method for vacuum manipulators. The method includes: using a rigid body model to describe the arm and a string model from the continuum domain to describe the steel strip. The rigid body is a finite-degree-of-freedom system, while the continuum is an infinite-degree-of-freedom system. A global space discretization method is used to discretize the infinite-degree-of-freedom system into a finite-degree-of-freedom system. Then, the two finite-degree-of-freedom systems are systematically integrated to obtain the rigid-flexible coupling dynamic equations of the vacuum manipulator. This invention establishes a system dynamic model based on the energy method, with a clear approach, and provides a good solution for dynamic modeling of rigid-flexible coupling systems. It solves the problem that vacuum manipulators lack end-point real-time sensing and cannot perceive the system state in real time to provide state feedback for control.

Description

Technical Field

[0001] This invention relates to the field of vacuum manipulator dynamics modeling technology, and more specifically, to a rigid-flexible coupling modeling and analysis method for vacuum manipulators. Background Technology

[0002] In the integrated circuit chip manufacturing process, robotic arms are responsible for wafer transfer and are crucial components of key process equipment such as photolithography, etching, metrology, and inspection. Atmospheric robotic arms offer ample space and strong adhesion, requiring less stringent trajectory tracking accuracy. Furthermore, the addition of sensors at the transfer endpoint facilitates accurate repeatability, thus reducing the need for high-precision dynamic models; only the rigidity of the components needs to be considered for control accuracy. However, as integrated circuit chip manufacturing processes advance towards the nanometer scale, many processes must be performed in a vacuum environment, necessitating the use of vacuum robotic arms. The confined space and lack of active gripping in a vacuum environment demand extremely high tracking accuracy from the robotic arms, and the inability to install endpoint sensors means that repeatability accuracy relies heavily on accurate dynamic models. Therefore, the dynamic models of vacuum robotic arms must consider the flexible modal characteristics of key components. Summary of the Invention

[0003] In view of this, the purpose of this invention is to provide a rigid-flexible coupling dynamic modeling and analysis method that considers the flexibility of key components of a vacuum manipulator. This invention establishes a rigid-flexible coupling dynamic model that considers both the rigidity of the arm and the flexibility of the steel strip by taking into account the flexible characteristics of the steel strip. The model is simple in form and the modeling steps are clear, which can provide a basis for the establishment of an accurate dynamic model of a vacuum manipulator and subsequent control design.

[0004] The technical solution of the present invention is described in detail below.

[0005] This invention provides a rigid-flexible coupling modeling and analysis method for the flexibility of a vacuum manipulator. It performs dynamic modeling and analysis from the perspective of the energy method, using a rigid body model to describe the arm and a string model from the continuum domain to describe the steel strip. The rigid body is a finite-degree-of-freedom system, and the continuum is an infinite-degree-of-freedom system. A global space discretization method is used to discretize the infinite-degree-of-freedom system into a finite-degree-of-freedom system, and then the two finite-degree-of-freedom systems are systematically integrated to obtain the rigid-flexible coupling dynamic equations of the vacuum manipulator. The method includes the following steps:

[0006] 1) Establish a dynamic model of the rigid arm, treating the upper arm, lower arm, and end effector as rigid body linkage models, considering the inertial product effect of rigid body rotation, considering the connecting joints between the linkages, treating the joint modules as rigid body models with the center of mass distributed at the connection, calculating the potential energy and kinetic energy of each rigid body, and obtaining the dynamic equation of the rigid arm using the second type of Lagrange method.

[0007] 2) Establish a dynamic model of flexible steel strip, considering the flexibility of the steel strip and the geometric and force constraints it receives. Treat the steel strip as a string model in the continuum domain, establish a system of partial differential equations with infinite degrees of freedom using the generalized Hamiltonian principle, discretize it using the Galerkin global space discretization method, and obtain a system of ordinary differential equations with finite degrees of freedom by satisfying the boundary conditions.

[0008] 3) Establish a rigid-flexible coupling dynamic model for the vacuum manipulator, and obtain the rigid-flexible coupling dynamic equation set by orderly integrating the equations by increasing the degrees of freedom;

[0009] Furthermore, step 1) above specifically refers to:

[0010] 11) Establish coordinate systems for each axis: World coordinate system O0: Origin is the vertex of the front end of the upper arm, Z0 is perpendicular to the plane of the upper arm and upward, X0 is horizontal to the right, and Y0 is established according to the right-hand screw rule; 1-axis coordinate system O1: Z1 coincides with Z0, X1 is along the direction of the upper arm, and Y1 is established according to the right-hand screw rule; 2-axis coordinate system O2: Origin is the vertex of the front end of the forearm, Z2 is perpendicular to the plane of the forearm and upward, X2 is along the direction of the forearm, and Y2 is established according to the right-hand screw rule; 3-axis coordinate system O3: Origin is the vertex of the front end of the end effector, Z3 is perpendicular to the plane of the end effector and upward, X3 is along the direction of the end effector, and Y3 is established according to the right-hand screw rule; 4-axis coordinate system: Origin is the vertex of the rear end of the end effector, Z4 is perpendicular to the plane of the end effector and upward, X4 is along the direction of the end effector, and Y4 is established according to the right-hand screw rule;

[0011] 12) Based on the coordinate system established in step 11), the MDH parameter table is established according to the MDH (Modified Denavit-Hartenberg) parameter definition rules. The homogeneous transformation matrix between each coordinate system is determined by the MDH parameters, and then the homogeneous transformation matrix of each coordinate system relative to the world coordinate system is established.

[0012] 13) The steps for calculating potential energy and kinetic energy are as follows:

[0013] Let the homogeneous coordinates of the centroid of rod i in coordinate system i be represented by the transformation matrix of coordinate system i relative to the world coordinate system. Then, we obtain the homogeneous coordinates of the centroid of rod i relative to the world coordinate system. Calculate the potential energy of rod i m i Let g be the mass of the i-th rod, g be the gravitational constant, and the potential energy of the entire rigid arm be obtained by summing them up. n is the number of rods.

[0014] Let the homogeneous coordinates of any one-dimensional particle in rod i be in coordinate system i. Then, take the transformation matrix of coordinate system i relative to the world coordinate system to obtain the homogeneous coordinates of rod i relative to the world coordinate system. calculate The velocity of the element particle is obtained by the total differential. Calculate the kinetic energy of an elementary particle The kinetic energy T of rod i is obtained by integration. i =∫ 杆i dT i The kinetic energy of the entire rigid arm is obtained by summing them up. n is the number of rods.

[0015] 14) Subtract the potential energy T and kinetic energy V obtained in step 13) to obtain the Lagrange quantity L. Substitute L into the second kind of Lagrange equation for calculation. The second kind of Lagrange equation is as follows:

[0016]

[0017] Where q is the generalized coordinate and Q is the generalized force. The symbol for total differential is . For variational notation;

[0018] The dynamic equations of the rigid arm can be obtained by solving the second kind of Lagrange equations.

[0019]

[0020] Where M1 is the mass matrix, C1 is the damping matrix, G1 is the stiffness matrix, and τ1 is the torque matrix;

[0021] Furthermore, step 2) above specifically refers to:

[0022] 21) Establish the steel strip coordinate system and derive the position transformation matrix of the steel strip coordinate system relative to the world coordinate system. The steel strip is attached to the connecting rod to which it is located, and the coordinates of the connecting rod are the coordinate system of the steel strip.

[0023] 22) The steps for calculating the kinetic energy, potential energy, and virtual work done by non-conservative forces on the steel strip are as follows:

[0024] The formula for calculating the kinetic energy T of a steel particle, arbitrarily selected within a chord of length L, is as follows:

[0025]

[0026] Where ρ(x) is the mass distribution function of the string along its length, y(x,t) is the transverse displacement function of the string at time t, and dx is the length of the elementary particle.

[0027] The formula for calculating the kinetic energy V of a steel particle, arbitrarily selected within a chord of length L, is as follows:

[0028]

[0029] Where T(x) is the distribution function of the tension within the string along its length;

[0030] Calculate the virtual work W done by the nonconservative forces in any single-element particle within a chord of length L. nc The formula is as follows:

[0031]

[0032] Where f(x,t) is the tension force on the string at time t;

[0033] 23) The potential energy V, kinetic energy T, and virtual work W done by the nonconservative forces obtained in step 22) nc Substituting these values ​​into the generalized Hamiltonian equation for calculation, the generalized Hamiltonian equation is as follows:

[0034]

[0035] Where t1 and t2 are time intervals, and dt is the integral over time. The dynamic equation y(x,t) of the flexible steel strip is obtained by solving the generalized Hamiltonian equation.

[0036] Taking one example, the following system of partial differential equations for y(x,t) is obtained:

[0037]

[0038] Where T is the tension on the string, f is the external force, and ρ is the mass per unit length of the string.

[0039] 24) The Galerkin global space discretization method is used to discretize y(x,t). The Galerkin discretization formula is as follows:

[0040]

[0041] Where N is the order of truncation, ψ i (x) is a basis function, β i (t) is the displacement function based on the basis functions. The infinite-degree-of-freedom system is discretized into a finite-degree-of-freedom system through discretization. The dynamic equations of the finite-degree-of-freedom system are as follows:

[0042]

[0043] Where β is the generalized coordinate, M2 is the equivalent mass matrix, C2 is the equivalent damping matrix, G2 is the equivalent stiffness matrix, and τ2 is the equivalent torque matrix.

[0044] 25) Transform the boundary conditions into basis functions ψ i The initial conditions of (x) are transformed as follows:

[0045] y(0,t)→ψ i (0)

[0046] y(L,t)→ψ i (L)

[0047] Furthermore, step 3) above specifically refers to:

[0048] 31) The dynamic equations of the rigid arm and the discretized flexible steel strip are systematically integrated into the dynamic equations of the rigid-flexible coupled system by increasing the degrees of freedom. The integration method is shown in the following formula:

[0049]

[0050] Where M is the equivalent mass matrix, C is the equivalent damping matrix, G is the equivalent stiffness matrix, and τ is the equivalent torque matrix;

[0051] 32) Order The formula in step 31) is transformed into the following formula:

[0052]

[0053] This formula is the dynamic equation for the rigid-flexible coupling of a vacuum manipulator.

[0054] Compared with the prior art, the advantages and positive effects of this invention are as follows:

[0055] This invention addresses the flexible characteristics of steel strips. Based on the distributed parameter method and the principles of elasticity, it equates the steel strip to a string model within the continuum domain. Starting with refined modeling, it fully considers the flexible characteristics and rigid-flexible coupling properties of the steel strip to establish a rigid-flexible coupling dynamic model for a vacuum manipulator. This improves the accuracy of the vacuum manipulator's dynamic modeling and predicts the system state based on the refined model. It solves the problem of vacuum manipulators lacking end-point real-time sensing, thus failing to provide real-time state feedback for control. This invention provides theoretical guidance for engineering applications of vacuum manipulators, employing dynamic modeling and analysis from the perspective of the energy method. The approach is clear, easy to implement, and possesses versatility and universality. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the structure and transmission of a vacuum manipulator.

[0057] Figure 2 Flowchart for modeling rigid-flexible coupling of a vacuum manipulator.

[0058] Figure 3 This is a schematic diagram of the dynamic model of a rigid arm.

[0059] Figure 4 This is a diagram of the dynamic model of the flexible steel strip.

[0060] Figure 5 This is a schematic diagram of the coordinate system for each axis.

[0061] Figure 6 This is a schematic diagram of a string model.

[0062] Markings in the diagram: 1—Z-axis motor, 2—θ-axis motor, 3—R-axis motor, 4—upper arm housing, 5—first pulley inside the upper arm, 6—second pulley inside the upper arm, 7—lower arm housing, 8—first pulley inside the lower arm, 9—second pulley inside the lower arm, 10—end effect actuator housing, 11—first steel belt, 12—second steel belt. Detailed Implementation

[0063] The present invention will now be described with reference to specific embodiments.

[0064] This invention provides a method for establishing a rigid-flexible coupled dynamic model of a robotic arm with a flexible steel strip. The method includes: describing the arm using a rigid body model and the steel strip using a string model from the continuum domain. The rigid body is a finite-degree-of-freedom system, and the continuum is an infinite-degree-of-freedom system. A global space discretization method is used to discretize the infinite-degree-of-freedom system into a finite-degree-of-freedom system. Then, the two finite-degree-of-freedom systems are systematically integrated to obtain the rigid-flexible coupled dynamic equations of the vacuum manipulator. This invention establishes the system dynamic model based on the energy method, with a clear approach, and provides a good solution for dynamic modeling of rigid-flexible coupled systems.

[0065] Figure 1 This is a schematic diagram of the structure and transmission of a vacuum manipulator provided in one embodiment, such as... Figure 1 As shown, in this embodiment, the vacuum manipulator consists of a Z-axis motor 1, a θ-axis motor 2, an R-axis motor 3, a main arm housing 4, a first inner pulley 5 inside the main arm, a second inner pulley 6 inside the main arm, a forearm housing 7, a first inner pulley 8 inside the forearm, a second inner pulley 9 inside the forearm, an end effector, a first steel belt 11, and a second steel belt 12, wherein the first steel belt 11 and the second steel belt 12 are as follows: Figure 4 As shown. Z-axis motor 1 is responsible for Z-axis translation, and the Z-axis motion is not coupled with the planar motion of the arm. The output shaft of θ-axis motor 2 is fixed to the upper arm housing 4. When θ-axis motor 2 rotates, it drives the entire arm to rotate. The output shaft of R-axis motor 3 is fixed to the first pulley 5 inside the upper arm. One end of the first steel belt 11 is fixed to the first pulley 5 inside the upper arm, and the other end is fixed to the second pulley 6 inside the upper arm. At the same time, the second pulley 6 inside the upper arm is fixed to the first pulley 8 inside the forearm. One end of the second steel belt 12 is fixed to the first pulley 8 inside the forearm, and the other end is fixed to the second pulley 9 inside the forearm. At the same time, the second pulley 9 inside the forearm is fixed to the end effector housing 10. Thus, when R-axis motor 3 rotates, it drives the forearm and end effector 10 to rotate through the first steel belt 11 and the second steel belt 12.

[0066] Figure 2 Here is a flowchart of the rigid-flexible coupling modeling process for a vacuum manipulator in one embodiment, such as... Figure 2As shown in the figure, this embodiment proposes a rigid-flexible coupling modeling and analysis method for vacuum manipulators, which may specifically include the following steps:

[0067] S1) Establish a dynamic model of the rigid arm, such as Figure 3 As shown, the upper arm housing 4, lower arm housing 7, and end effector housing 10 are considered as rigid body linkage models. Considering the product of inertia effect of rigid body rotation and the connecting joints between the linkages, the first pulley 5 inside the upper arm is considered as the shoulder joint, the second pulley 6 inside the upper arm and the first pulley 8 inside the lower arm are considered as elbow joints, and the second pulley 9 inside the lower arm is considered as the wrist joint. The shoulder joint, elbow joint, and wrist joint are considered as rigid body models with their centers of mass distributed at the connection points. The potential energy and kinetic energy of each rigid body are calculated, and the dynamic equations of the rigid arm are obtained using the second type of Lagrange method; θ1 is the upper arm housing 4, lower arm housing 7, and end effector housing 10. The angle between the arm housing 4 and the x-axis of the world coordinate system, θ2 is the angle between the upper arm housing 4 and the lower arm housing 7, θ3 is the angle between the lower arm housing 7 and the end effector housing 10, L1 is the length of the upper arm housing 4, L2 is the length of the lower arm housing 7, L3 is the length of the end effector housing 10, O1 is the front vertex of the upper arm housing 4, O2 is the front vertex of the lower arm housing 7, O3 is the front vertex of the end effector 10, and A, B, C, D, E, F, G, H, A2, and A3 are the endpoints of the line segments used in calculating potential energy and kinetic energy.

[0068] S2) Establish a dynamic model for the flexible steel strip, such as Figure 4 As shown, one end of the first steel belt 11 is fixed to the first pulley 5 inside the boom, and the other end is fixed to the second pulley 6 inside the boom. One end of the second steel belt 12 is fixed to the first pulley 8 inside the forearm, and the other end is fixed to the second pulley 9 inside the forearm. Combined with... Figure 1 The fixed connections of the various components described herein are as follows: when the R-axis motor 3 rotates, it sequentially drives the first pulley 5, the first steel belt 11, the second pulley 6, the first pulley 8, the second steel belt 12, the second pulley 9, and the end effector housing 10 inside the upper arm to rotate. Considering the flexibility of the steel belt and the geometric and force constraints it receives, the steel belt is regarded as a string model in the continuum domain. An infinite degree of freedom partial differential equation system is established using the generalized Hamiltonian principle. The system is then discretized using the Galerkin global space discretization method, and a finite degree of freedom ordinary differential equation system is obtained by satisfying the boundary conditions.

[0069] S3) Establish a rigid-flexible coupling dynamic model of the vacuum manipulator, and orderly integrate the rigid-flexible coupling dynamic equations by increasing the degrees of freedom;

[0070] Furthermore, step S1) above specifically includes:

[0071] S1001) Establish coordinate systems for each axis, such as Figure 5As shown, first, establish a world coordinate system O0: the origin is the vertex of the front end of the upper arm, Z0 is perpendicular to the plane of the upper arm and upward, X0 is horizontal to the right, and Y0 is established according to the right-hand screw rule; 1-axis coordinate system O1: Z1 coincides with Z0, X1 is along the direction of the upper arm, and Y1 is established according to the right-hand screw rule; 2-axis coordinate system O2: the origin is the vertex of the front end of the forearm, Z2 is perpendicular to the plane of the forearm and upward, X2 is along the direction of the forearm, and Y2 is established according to the right-hand screw rule; 3-axis coordinate system O3: the origin is the vertex of the front end of the end effector, Z3 is perpendicular to the plane of the end effector and upward, X3 is along the direction of the end effector, and Y3 is established according to the right-hand screw rule; 4-axis coordinate system: the origin is the vertex of the rear end of the end effector housing, Z4 is perpendicular to the plane of the end effector and upward, X4 is along the direction of the end effector, and Y4 is established according to the right-hand screw rule.

[0072] S1002) Based on the coordinate system established in step S1001), an MDH parameter table is established according to the MDH parameter definition rules. The homogeneous transformation matrix between each coordinate system is determined by the MDH parameters, and then the homogeneous transformation matrix of each coordinate system relative to the world coordinate system is established.

[0073] S1003) Let the homogeneous coordinates of the centroid of rod i in coordinate system i be denoted by the transformation matrix of coordinate system i relative to the world coordinate system. Then, we obtain the homogeneous coordinates of the centroid of rod i relative to the world coordinate system. Calculate the potential energy of rod i m i Let g be the mass of the i-th rod, g be the gravitational constant, and the potential energy of the entire rigid arm be obtained by summing them up. n is the number of rods.

[0074] Let the homogeneous coordinates of any one-dimensional particle in rod i be in coordinate system i. Then, take the transformation matrix of coordinate system i relative to the world coordinate system to obtain the homogeneous coordinates of rod i relative to the world coordinate system. calculate The velocity of the element particle is obtained by the total differential. Calculate the kinetic energy of an elementary particle The kinetic energy T of rod i is obtained by integration. i =∫ 杆i dT i The kinetic energy of the entire rigid arm is obtained by summing them up.

[0075] S1004) Subtract the potential energy T and kinetic energy V obtained in step S1003) to obtain the Lagrange quantity L. Substitute L into the second kind of Lagrange equation for calculation. The second kind of Lagrange equation is as follows:

[0076]

[0077] Where q is the generalized coordinate and Q is the generalized force.

[0078] The dynamic equations of the rigid arm can be obtained by solving the second kind of Lagrange equations.

[0079]

[0080] Where M1 is the mass matrix, C1 is the damping matrix, G1 is the stiffness matrix, and τ1 is the torque matrix; the specific expansions of M1, C1, G1, and τ1 are as follows:

[0081]

[0082] Furthermore, step S2) above specifically includes:

[0083] S2001) Establish the steel strip coordinate system and derive the position transformation matrix of the steel strip coordinate system relative to the world coordinate system. The steel strip is attached to the connecting rod to which it is located, and the coordinates of the connecting rod are the coordinate system of the steel strip.

[0084] S2002) String model schematic diagram as follows Figure 6 As shown, the steps for calculating the kinetic energy, potential energy, and virtual work done by non-conservative forces on the steel strip are as follows:

[0085] The formula for calculating the kinetic energy T of a steel particle, arbitrarily selected within a chord of length L, is as follows:

[0086]

[0087] Where ρ(x) is the mass distribution function of the string along its length, and y(x,t) is the transverse displacement function of the string at time t;

[0088] The formula for calculating the kinetic energy V of a steel particle, arbitrarily selected within a chord of length L, is as follows:

[0089]

[0090] Where T(x) is the distribution function of the tension within the string along its length;

[0091] Calculate the virtual work W done by the nonconservative forces in any single-element particle within a chord of length L. nc The formula is as follows:

[0092]

[0093] Where f(x,t) is the tension force on the string at time t;

[0094] S2003) The potential energy V, kinetic energy T, and virtual work W done by the nonconservative force obtained in step S2002) nc Substituting these values ​​into the generalized Hamiltonian equation for calculation, the generalized Hamiltonian equation is as follows:

[0095]

[0096] Solving the generalized Hamiltonian equation yields the dynamic equation y(x,t) for the flexible steel strip;

[0097] S2004) The Galerkin global space discretization method is used to discretize y(x,t). The Galerkin discretization formula is as follows:

[0098]

[0099] Where N is the order of truncation, ψ i (x) is a basis function, β i (t) is the displacement function based on the basis functions. The infinite-degree-of-freedom system is discretized into a finite-degree-of-freedom system through discretization. The dynamic equations of the finite-degree-of-freedom system are as follows:

[0100]

[0101] Where β is the generalized coordinate, M2 is the equivalent mass matrix, C2 is the equivalent damping matrix, G2 is the equivalent stiffness matrix, and τ2 is the equivalent torque matrix; M2, C2, G2, and τ2 are expanded as follows:

[0102]

[0103] Taking the truncation order N=2 as an example, we can obtain

[0104]

[0105] S2005) transforms the boundary conditions into basis functions ψ i The initial conditions of (x) are transformed as follows:

[0106] y(0,t)→ψ i (0)

[0107] y(L,t)→ψ i (L)

[0108] Furthermore, step S3 above specifically includes:

[0109] S3001) The dynamic equations of the rigid arm and the discretized flexible steel strip are systematically integrated into the dynamic equations of the rigid-flexible coupled system by increasing the degrees of freedom. The integration method is shown in the following formula:

[0110]

[0111] Where M is the equivalent mass matrix, C is the equivalent damping matrix, G is the equivalent stiffness matrix, and τ is the equivalent torque matrix;

[0112] S3002) Order The formula in step 31) is transformed into the following formula:

[0113]

[0114] This formula is the dynamic equation for the rigid-flexible coupling of a vacuum manipulator.

[0115] Taking the results in steps S1004) and S2004) as examples, we can obtain

[0116]

Claims

1. A method for rigid-flexible coupling modeling and analysis of a vacuum manipulator, characterized in that, Dynamic modeling and analysis are performed from the perspective of the energy method. A rigid body model is used to describe the arm, and a string model from the continuum domain is used to describe the steel strip. The rigid body is a finite-degree-of-freedom system, and the continuum is an infinite-degree-of-freedom system. A global space discretization method is used to discretize the infinite-degree-of-freedom system into a finite-degree-of-freedom system. Then, the two finite-degree-of-freedom systems are integrated in an orderly manner to obtain the rigid-flexible coupling dynamic equations of the vacuum manipulator. The steps include: 1) Establish a dynamic model of the rigid arm, treating the upper arm, lower arm, and end effector as rigid body linkage models, considering the inertial product effect of rigid body rotation, considering the connecting joints between the linkages, treating the joint modules as rigid body models with the center of mass distributed at the connection, calculating the potential energy and kinetic energy of each rigid body, and obtaining the dynamic equation of the rigid arm using the second type of Lagrange method. 2) Establish a dynamic model of flexible steel strip, considering the flexibility of the steel strip and the geometric and force constraints it receives. Treat the steel strip as a string model in the continuum domain, establish a system of partial differential equations with infinite degrees of freedom using the generalized Hamiltonian principle, discretize it using the Galerkin global space discretization method, and obtain a system of ordinary differential equations with finite degrees of freedom by satisfying the boundary conditions. 3) Establish a rigid-flexible coupling dynamic model for the vacuum manipulator, and obtain the rigid-flexible coupling dynamic equations by orderly integrating them through increasing the degrees of freedom; where: Step 2) specifically involves: 21) Establish the steel strip coordinate system and derive the position transformation matrix of the steel strip coordinate system relative to the world coordinate system. The steel strip is attached to the connecting rod to which it is located, and the coordinates of the connecting rod are the coordinate system of the steel strip. 22) Calculate the kinetic energy, potential energy, and virtual work done by nonconservative forces on the steel strip; In length of L Calculate the kinetic energy of the steel by arbitrarily selecting a single-element particle in the string. T The formula is as follows: ， in It is the mass distribution function of the chord along its length. yes t The transverse displacement function of the chord at time t, It is the length of the elementary particle; In length of L Calculate the potential energy of the steel strip by arbitrarily selecting a single-element particle in the string. V The formula is as follows: ， in It is the distribution function of the tension within the string along its length; In length of L Calculate the virtual work done by nonconservative forces by arbitrarily selecting a single-element particle in the string. The formula is as follows: ， in for t The tension in the string at that moment; 23) The potential energy obtained in step 22) V ,kinetic energy T The work done by non-conservative forces Substitute it into the generalized Hamiltonian equation for calculation; The generalized Hamiltonian equation is as follows: ， in It is time. The dynamic equation of the flexible steel strip is obtained by integrating over time and solving the generalized Hamiltonian equation. ; 24) Using the Galerkin global space discretization method to perform... Discretize the data; The Galerkin discrete formula is as follows: ， Where N is the order of truncation, These are basis functions. It is a displacement function based on basis functions; By discretizing the infinite-degree-of-freedom system into a finite-degree-of-freedom system, the dynamic equations of the finite-degree-of-freedom system are as follows: ， in For generalized coordinates, For the equivalent quality matrix, The equivalent damping matrix, The equivalent stiffness matrix, This is the equivalent torque matrix; 25) Transform boundary conditions into basis functions The initial conditions and transformation relationships are as follows: ， ； Step 3) specifically refers to: 31) The dynamic equations of the rigid arm and the discretized flexible steel strip are systematically integrated into the dynamic equations of the rigid-flexible coupled system by increasing freedom. The integration method is shown in the following formula: ， in For the equivalent quality matrix, q For generalized coordinates, The equivalent damping matrix, The equivalent stiffness matrix, This is the equivalent torque matrix; 32) Order The formula in step 31) is transformed into the following formula: ， This formula is the dynamic equation for the rigid-flexible coupling of a vacuum manipulator.

2. The rigid-flexible coupling modeling and analysis method for vacuum manipulators according to claim 1, characterized in that, Step 1) specifically involves: 11) Establish coordinate systems for each axis: World coordinate system O0: Origin is the vertex of the front end of the upper arm, Z0 is perpendicular to the plane of the upper arm and upward, X0 is horizontal to the right, and Y0 is established according to the right-hand screw rule; 1-axis coordinate system O1: Z1 coincides with Z0, X1 is along the direction of the upper arm, and Y1 is established according to the right-hand screw rule; 2-axis coordinate system O2: Origin is the vertex of the front end of the forearm, Z2 is perpendicular to the plane of the forearm and upward, X2 is along the direction of the forearm, and Y2 is established according to the right-hand screw rule; 3-axis coordinate system O3: Origin is the vertex of the front end of the end effector, Z3 is perpendicular to the plane of the end effector and upward, X3 is along the direction of the end effector, and Y3 is established according to the right-hand screw rule; 4-axis coordinate system: Origin is the vertex of the rear end of the end effector, Z4 is perpendicular to the plane of the end effector and upward, X4 is along the direction of the end effector, and Y4 is established according to the right-hand screw rule; 12) Based on the coordinate system established in step 11), the MDH parameter table is established according to the MDH parameter definition rules, the homogeneous transformation matrix between each coordinate system is determined by the MDH parameters, and then the homogeneous transformation matrix of each coordinate system relative to the world coordinate system is established. 13) Calculate the potential energy and kinetic energy; Let the homogeneous coordinates of the centroid of rod i in coordinate system i be represented by the transformation matrix of coordinate system i relative to the world coordinate system. Then, we obtain the homogeneous coordinates of the centroid of rod i relative to the world coordinate system. Calculate the potential energy of rod i. , Let g be the mass of the i-th rod, g be the gravitational constant, and the potential energy of the entire rigid arm be obtained by summing them up. ，n The number of rods; Let the homogeneous coordinates of any one-dimensional particle in rod i be in coordinate system i. Then, take the transformation matrix of coordinate system i relative to the world coordinate system to obtain the homogeneous coordinates of rod i relative to the world coordinate system. ,calculate The velocity of the element particle is obtained by the total differential. Calculate the kinetic energy of the elementary particle. The kinetic energy of rod i is obtained by integration. The kinetic energy of the entire rigid arm is obtained by summing them up. ; 14) The kinetic energy obtained in step 13) T and potential energy V Subtraction yields the Lagrange quantity L ,Will L Substitute these equations into the second kind of Lagrange equation for calculation. The second kind of Lagrange equation is as follows: ， in q For generalized coordinates, Q For generalized force, The symbol for total differential is . For variational notation; The dynamic equations of the rigid arm are obtained by solving the second kind of Lagrange equations. ， in For the quality matrix, Here is the damping matrix. Here is the stiffness matrix. This is the torque matrix.

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