Gap parallel mechanism rigid-flexible coupling dynamics control method considering uncertain parameters
Patent Information
- Application Number
- CN202611204723.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-08-10
- Publication Date
- 2026-10-16
AI Technical Summary
关键参数的不确定性会使机构的动力学也具有不确定性，变得难以预测，这会直接影响机构运动的精度和准确性
本发明通过建立含间隙的刚柔耦合空间并联机构模型，得到并联机构刚柔耦合动力学模型，并在此基础上建立考虑模糊参数的空间并联机构刚柔耦合动力学模型，可以反馈间隙对并联机构动力学特性产生的不良影响。
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Figure CN122926795A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of parallel mechanism modeling and optimization technology, specifically to a rigid-flexible coupling dynamic control method for gap parallel mechanisms that considers uncertain parameters. Background Technology
[0002] Parallel mechanisms are closed-loop mechanical structures that connect a movable "moving platform" and a fixed "stationary platform" in parallel through multiple independent motion chains. They have advantages such as high rigidity, large load-bearing capacity, high precision, no cumulative error, and fast dynamic response. They can be applied in fields such as motion simulators, industrial manufacturing, precision operation and medical care, and cutting-edge exploration.
[0003] Key parameters such as clearance and friction coefficient at the kinematic pairs of parallel mechanisms may change under the influence of various factors, including machining and assembly deviations, friction and wear, material properties, and operating conditions. The uncertainty of these key parameters also introduces uncertainty into the mechanism's dynamics, making them difficult to predict, which directly affects the precision and accuracy of the mechanism's motion.
[0004] Therefore, in order to ensure the precision and accuracy of parallel mechanisms, improve their dynamic performance, and make them better and more perfectly approximate the theoretical motion, it is urgent to establish a dynamic control method for gap parallel mechanisms that considers uncertain parameters, and to study how to reduce the influence of uncertain parameters on the dynamic response of the mechanism through feedback control. Summary of the Invention
[0005] To address the shortcomings of the prior art, the present invention aims to provide a rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters. This method establishes a rigid-flexible coupling dynamic model and a control model for the spatial parallel mechanism considering fuzzy parameters, and feeds back the adverse effects of the gap on the dynamic characteristics of the parallel mechanism. This provides a new control method for the influence of fuzzy parameters on the dynamic characteristics of the mechanism, which can improve the accuracy of dynamic analysis and prediction, and better predict the motion of the parallel mechanism.
[0006] Specifically, on the one hand, the present invention provides a rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters, comprising the following steps: S1. Establish and combine a spherical pair model with dry friction gap and a flexible beam element model to form a rigid-flexible coupled spatial parallel mechanism model with gap; S2. Based on the constraints of the parallel mechanism model established in S1, and using the Lagrange multiplier method, the rigid-flexible coupling dynamic model of the parallel mechanism is obtained. S3. Based on the dynamic model in S2, establish a rigid-flexible coupling dynamic model of the spatial parallel mechanism with fuzzy parameters: S31. Select fuzzy parameters, and set the spherical pair gap value c as the fuzzy parameter, which follows the Gaussian fuzzy number: ; In the formula, b is the upper boundary corresponding to membership degree 0, d is the lower boundary corresponding to membership degree 0, and a is the value corresponding to membership degree 1. S32. Construct a rigid-flexible coupled dynamic model of a spatial parallel mechanism with fuzzy parameters: ; In the formula, The mass matrix of the mechanism, For constraint equations on generalized coordinates Jacobian matrix, For the generalized acceleration matrix, This is the matrix formed by the generalized force vectors acting on each component in a rigid-flexible coupling dynamic model containing dry friction gaps. These are the constraint equations for the mechanism; For Lagrange multiplier terms, superscript Represents fuzzy and uncertain variables; S4. Establish a rigid-flexible coupling dynamic control model for a spatial parallel mechanism using sliding mode control: Construct the active arm driving function: ; In the formula, To adjust the angle of the active arm, The feedback control law of the system in the X, Y, and Z directions; the system driving function. Represented as: ; In the formula, For the driving function of the spatial parallel mechanism; By introducing the control post-drive function into the dynamic model in S3, a rigid-flexible coupled dynamic control model of the parallel mechanism with gaps is established: ; Wherein, the superscript r represents the function with flexible parameters, and the superscript k represents the driving function after the introduction of control; S5. The dynamic control model obtained in S4 is solved using the Kriging surrogate model method. Based on the gap parameter and the dynamic response value of the known sample points, the mapping relationship between the gap uncertainty parameter and the dynamic response is established.
[0007] Furthermore: S1 establishes a spherical pair model containing dry friction gaps by constructing a model of the gap eccentricity vector and a collision depth model at the contact point. By judging the collision depth, the contact collision state between the sphere and the shell is identified. When a collision occurs and deformation occurs, the relative normal velocity and relative tangential velocity at the potential contact point, the contact force of the sphere on the shell, and the reaction force of the shell on the sphere are calculated.
[0008] Furthermore: S1 obtains the elastic force expression for the flexible beam element model as follows: ; In the formula, , , , G and G are Lamé constants.
[0009] Furthermore, S2 includes the following sub-steps: S21. The coordinates of the component using the centroid coordinate method are described using the centroid coordinate method for the active arm, the lower short rod of the parallelogram branch, and the moving platform. ; In the formula, For position coordinates, The pose of the centroid coordinate system of the moving component in the fixed coordinate system; The coordinates of the long branch of the parallelogram are described using the natural point coordinate method: ; In the formula, and These represent the position vectors of the two endpoints of the component in a fixed coordinate system; S22. Based on the Lagrange multiplier method, the dynamic model of the rigid-flexible coupled spatial parallel mechanism with dry friction gap is obtained: ; In the formula, The mass matrix of the mechanism, For constraint equations on generalized coordinates Jacobian matrix, For the generalized acceleration matrix, For Lagrange multipliers, This represents the generalized force vector experienced by each component in a rigid-flexible coupling dynamic model with gaps.
[0010] Furthermore, S21 also includes the following steps: After describing a single coordinate, define the generalized coordinates of the spatial parallel mechanism rigid-flexible coupling dynamic model system, establish the constraint equations between the drive arm and the frame, introduce the flexible rod and clearance, and obtain the constraint equations of the rigid-flexible coupling parallel mechanism with clearance: .
[0011] Furthermore: The method for constructing the active arm driving function in S4 is as follows: Designed sliding surface function: ; Design control law: ; Based on the positional relationship between the active arm drive function and the end effector, the feedback control law of the dynamic response of the end effector is mapped onto the active arm drive function.
[0012] Furthermore, S4 also includes determining the optimal controller parameters for the control model. The method involves manually narrowing the parameters down to a range as the search variable constraint range, using the total cumulative error of the end effector as the objective function, and employing a particle swarm optimization algorithm to search for the optimal solution of control parameters within the range that satisfies the minimum value of the objective function. The expression for the search objective function is: .
[0013] Furthermore: the solution formula for the dynamic control model in S5 is: ; In the formula, It is a predicted value. It is a variable basis functions, It is the regression coefficient. It is a random process that follows a normal distribution. This represents the number of different gap values.
[0014] Furthermore, the method in S5 for establishing the mapping relationship between the uncertain clearance parameters and the dynamic response based on the clearance parameters and the known dynamic response values of sample points includes: setting parameters for the mechanism, including mechanism parameters and fuzzy parameters; calculating the contact force, using the ode45 solver to solve for the displacement and velocity, and obtaining the displacement and velocity response functions of the moving platform; feeding the obtained response functions back to the mechanism, updating the relevant variables of the mechanism through the control model until they meet the objective function, reaching the required number of iterations, and outputting the response.
[0015] Furthermore: the method for calculating the relative normal velocity and relative tangential velocity at the potential contact point, the contact force of the sphere on the shell, and the reaction force of the shell on the sphere is as follows: A normal collision force model is established based on the LN model: ; In the formula, The initial collision velocity, For collision speed, n is the restoration coefficient, and n is the correction coefficient; ; In the formula, For elastic modulus, Poisson's ratio; A tangential friction model is established using the modified Coulomb friction model, which is expressed as follows: ; In the formula, and These are the friction coefficient and the dynamic correction coefficient, respectively. and These are the speed limits for dynamic and static friction.
[0016] The contact force generated by the clearance on the bushing is: ; In the formula, Let be the unit direction vector along the normal direction of the contact surface. It is the unit direction vector along the tangential direction of the contact surface.
[0017] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention establishes a rigid-flexible coupled spatial parallel mechanism model with gaps, obtains a rigid-flexible coupled dynamic model of the parallel mechanism, and on this basis, establishes a rigid-flexible coupled dynamic model of the spatial parallel mechanism considering fuzzy parameters, which can reflect the adverse effects of gaps on the dynamic characteristics of the parallel mechanism.
[0018] This invention is based on a rigid-flexible coupling dynamic model of a spatial parallel mechanism considering fuzzy parameters. By establishing a rigid-flexible coupling dynamic control model for the spatial parallel mechanism through sliding mode control, it can control adverse effects, reduce motion deviations caused by uncertain parameters, and ensure and improve the motion accuracy of the parallel mechanism. Simultaneously, it provides a new method for predicting the influence of fuzzy parameters, clearances, and component elasticity on the dynamic characteristics of the mechanism, thereby improving the accuracy of dynamic analysis and prediction. Attached Figure Description
[0019] Figure 1 A flowchart illustrating the rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters, as described in this invention. Figure 2 A structural diagram of the 3-RRPaR spatial parallel mechanism provided in an embodiment of the present invention; Figure 3 The rotating joint clearance model provided in the embodiments of the present invention; Figure 4 This is a three-dimensional two-node beam element model provided in an embodiment of the present invention; Figure 5 This is a schematic diagram of the rigid-flexible coupling structure with gap provided in an embodiment of the present invention; Figure 6 The solution process for the rigid-flexible coupling dynamic control model of a parallel mechanism with gaps provided in the embodiments of the present invention; Figure 7 This is a simulation diagram of the X-direction displacement of the parallel mechanism provided in an embodiment of the present invention; Figure 8 This is a simulation diagram of the displacement difference in the X direction of the parallel mechanism provided in an embodiment of the present invention; Figure 9 This is a simulation diagram of the Y-direction displacement of the parallel mechanism provided in an embodiment of the present invention; Figure 10 This is a simulation diagram of the displacement difference in the Y direction of the parallel mechanism provided in an embodiment of the present invention; Figure 11 This is a simulation diagram of the X-direction velocity of the parallel mechanism provided in an embodiment of the present invention; Figure 12 This is a simulation diagram of the velocity difference in the X direction of the parallel mechanism provided in an embodiment of the present invention; Figure 13 This is a simulation diagram of the Y-direction velocity of the parallel mechanism provided in an embodiment of the present invention; Figure 14 This is a simulation diagram of the velocity difference in the Y direction of the parallel mechanism provided in an embodiment of the present invention. Detailed Implementation
[0020] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings.
[0021] Figure 1 The present invention provides a detailed flowchart of the rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters, comprising the following steps: S1. Establish a model of a rigid-flexible coupled spatial parallel mechanism with gaps; S11. Establish a spherical pair model containing dry friction gap; S12. Establish a flexible beam element model, whose elastic force expression is: ; By combining a flexible beam with a spherical pair, a model of a rigid-flexible coupled spatial parallel mechanism with gaps is established. S2. Based on the constraint conditions of the rigid-flexible coupled spatial parallel mechanism with dry friction gap, the rigid-flexible coupled dynamic model of the parallel mechanism is obtained. S21. The coordinates of the active arm, the lower short rod of the parallelogram branch, and the moving platform of the component are described using the centroid coordinate method, and are expressed as follows: ; In the formula, For position coordinates, The pose of the centroid coordinate system of the moving component in the fixed coordinate system; The coordinates of the remaining components are described using the natural point coordinate method, expressed as follows: ; In the formula, and These represent the position vectors of the two endpoints of the component in a fixed coordinate system; S22. Based on the Lagrange multiplier method, the dynamic equations of a rigid-flexible coupled spatial parallel mechanism with dry friction gap are as follows: ; In the formula, The mass matrix of the mechanism, For constraint equations on generalized coordinates Jacobian matrix, For the generalized acceleration matrix, For Lagrange multipliers, This represents the generalized force vector acting on each component in a rigid-flexible coupling dynamic model with gaps. These are the constraint equations for the mechanism; S3. Establish a rigid-flexible coupling dynamic model of a spatial parallel mechanism considering fuzzy parameters: S31. Select fuzzy parameters, set the spherical pair gap value c as the fuzzy parameter and make it conform to the Gaussian fuzzy number, expressed as: ; In the formula, b is the upper boundary corresponding to membership degree 0, d is the lower boundary corresponding to membership degree 0, and a is the midpoint, which is also the value corresponding to membership degree 1. S32. Construct the rigid-flexible coupling dynamic model of the spatial parallel mechanism with fuzzy parameters as follows: ; Since some parameters at the gap become fuzzy and uncertain parameters, the quantities related to them in the dynamic equation all become fuzzy and uncertain variables. Adding them... To represent; S4. Establish a rigid-flexible coupling dynamic control model for a spatial parallel mechanism using sliding mode control: The active arm drive function is expressed as: ; In the formula, To adjust the angle of the active arm, This represents the feedback control law of the system in the X, Y, and Z directions. In the control model, the system driving function is expressed as: ; By incorporating the controlled driving function into the dynamic model considering the gap, a rigid-flexible coupled dynamic control model for the parallel mechanism with gap is established as follows: .
[0022] S5. Dynamic control analysis of rigid-flexible coupling spatial parallel mechanisms considering fuzzy parameters: S51. The Kriging surrogate model method is used to solve the dynamic model obtained in S4. The solution formula is as follows: ; In the formula, It is a predicted value. It is a variable basis functions, It is the regression coefficient. It is a random process that follows a normal distribution. This represents the number of different gap values.
[0023] S52. Using the solution formula determined in S51, establish the mapping relationship between the gap uncertainty parameter and the dynamic response based on the gap parameter and the known dynamic response value of the sample point. Specific implementation examples: This implementation example uses a three-degree-of-freedom spatial parallel mechanism of the main body of a parallel machining platform as the research object for analysis. During the movement of the machining platform, with the long-term operation of the robotic arm, the collision phenomenon between the kinematic pairs in the main parallel mechanism intensifies, which may cause material loss and damage. This leads to changes in parameters such as gap value and friction force. The changes in parameters make the dynamic response of the main body of the machining platform uncertain, which directly affects the accuracy of the parallel mechanism. This implementation example provides a rigid-flexible coupling dynamic control method for spatial parallel mechanisms that consider uncertain parameters and gaps. It can correct the accuracy of the parallel mechanism's motion and ensure its working precision.
[0025] S1: Establish a spherical pair model with gaps.
[0026] S11: Description of the 3-RRPaR spatial parallel mechanism structure.
[0027] 3-RRPaR spatial parallel mechanism structure as follows Figure 2 As shown, it consists of an end effector and three identical kinematic branches, denoted by I, II, and III, and has three degrees of freedom.
[0028] S12: Modeling the clearance of the rotating joint.
[0029] A simplified diagram of the radial clearance of the rotating joint is shown below. Figure 3As shown. Due to the clearance of the rotating joint, the collision and friction between the shaft and the bushing will generate corresponding contact forces. , These are the center coordinates of the bearing and the shaft, respectively. , Let be the radii of the bearing and the shaft, respectively, and E be the eccentric vector. The expression is: (1) In the formula, Let be the lower position vector of the bushing center in the global coordinate system. It is the position vector of the axis center in the global coordinate system, and the chosen coordinate system is the global coordinate system.
[0030] The magnitude of the eccentricity vector is: (2) The radial embedding depth is expressed as: (3) In the formula, This is the clearance value at the rotating joint.
[0031] A normal collision force model is established based on the LN model. The LN model considers energy dissipation and local deformation during the collision friction process and is widely used in the dynamics study of mechanisms with gaps. It can be expressed as: (4) In the formula, The initial collision velocity, For collision speed, is the restoration coefficient, and n is the correction coefficient.
[0032] (5) In the formula, For elastic modulus, It is Poisson's ratio.
[0033] A tangential friction model is established using the modified Coulomb friction model, expressed as follows: (6) In the formula, and These are the friction coefficient and the dynamic correction coefficient, respectively. and These are the speed limits for dynamic and static friction; the contact force generated by the clearance on the bushing is: (7) In the formula, Let be the unit direction vector along the normal direction of the contact surface. It is the unit direction vector along the tangential direction of the contact surface.
[0034] S2: Establish a flexible beam element model.
[0035] The flexible component is discretized using a three-dimensional two-node beam element based on the ANCF method. Nodes Q and H are located at the centers of the two end faces of the beam element, respectively. Direction vectors are established at the two nodes, ensuring that they are perpendicular to each other. and Let H and Q represent the position vectors of nodes in a fixed coordinate system, respectively. The three-dimensional two-node beam element model is as follows: Figure 4 As shown.
[0036] Each node is represented using 12 generalized coordinates, therefore a beam element has a total of 24 generalized coordinates. The coordinates of the element nodes are represented as follows: (8) In the formula, express The partial derivative vector in the x-direction is defined the same for all other vectors. For a three-dimensional two-node beam element model, solving for the shape function is essential; its expression is: (9) In the formula, Indicates the Kronecker product. This represents a third-order identity matrix, with the following internal parameter values: (10) In the formula, , , , This represents the length of the beam element before deformation.
[0037] In summary, the displacement field of the three-dimensional two-node beam element is: (11) Before solving for the mass matrix of the beam element, the first derivative of the displacement field matrix of the three-dimensional two-node beam element needs to be obtained. Its velocity expression is as follows: (12) In the formula, Find the first derivative of the generalized coordinates of the beam element with respect to time.
[0038] The kinetic energy expression for the beam element model is: (13) In the formula, For the density of the beam, Let V be the volume of the beam, and let M be the mass matrix of the beam element. Represented as: (14) The generalized external force expression can be obtained from the principle of virtual work: (15) In the formula, Let w be the force exerted at any point w on the beam element.
[0039] The gravitational force acting on the beam element is expressed as: (16) In the formula, g is the acceleration due to gravity.
[0040] Based on continuum mechanics, the Lagrange strain tensor is as follows: (17) In the formula, The value can be: (18) In the formula, , , , For shape function matrix From the a-th row, the expression for the elastic force of the beam element is further obtained as follows: (19) In the formula, , , , G and G are Lamé constants.
[0041] S3: Dynamic modeling of rigid-flexible coupling with clearance of rotating joint.
[0042] S31: Select fuzzy parameters, set the spherical pair gap value c as the fuzzy parameter and make it conform to the Gaussian fuzzy number, expressed as: ; In the formula, b is the upper boundary corresponding to membership degree 0, d is the lower boundary corresponding to membership degree 0, and a is the midpoint, which is also the value corresponding to membership degree 1.
[0043] S32: Construct a rigid-flexible coupling dynamic model of a spatial parallel mechanism with fuzzy parameters.
[0044] In parallel mechanisms, the deformation of the branches is directly transmitted to the moving platform, affecting the position and orientation of the end effector. Parallelogram-shaped branches, due to their small slenderness, are more prone to elastic deformation. Therefore, considering the elasticity of the six long links of three parallelogram branches, with the remaining components being rigid, and considering the clearance between the radial revolute joints at points R1 and R3 on the moving platform (with clearance values C1 and C2 respectively), and the remaining revolute joints being ideal, a rigid-flexible coupling dynamic model of a 3-RRPaR spatial parallel mechanism with clearances is established.
[0045] To distinguish it from rigid body dynamics modeling with gaps, a superscript 'r' is added after the relevant variables in the rigid-flexible coupling dynamics model with gaps. This represents the position vector of each component in the global coordinate system. Based on this, the generalized coordinates of the rigid-flexible coupling dynamic model system of the 3-RRPaR spatial parallel mechanism with gaps can be obtained as follows: (20) use The mass matrix of each component is represented as follows: ;(twenty one) The process of establishing the constraint equations is illustrated using branch 1 and the constraints at the moving platform as examples. The simplified structural diagram of branch 1 and the establishment of its local coordinate system are shown below. Figure 5 As shown.
[0046] The constraint equations between the drive arm and the frame are: ;(twenty two) In the formula, This represents the position vector of the centroid of component 1 in a fixed coordinate system. This represents the transformation matrix of drive arm 1. , and These represent the rotation angles of drive arm 1 relative to the X-axis, Y-axis, and Z-axis of the fixed coordinate system, respectively. Let be the rotation angle of the active arm on branch 1 on the Y-axis, which is a function of time t.
[0047] After introducing the flexible rod, the constraint equations for the parallelogram branch and each component on the moving platform are as follows: ;(twenty three) In the formula, This represents the position vector of the centroid of component i in a fixed coordinate system. The transformation matrix represents component i. This indicates the positions of the two endpoints of component i in the global coordinate system. This represents the transformation matrix of the joint coordinate system at the nodes on the flexible rod of branch 1. The length of the lever arm. Let the length of the short rod of the parallelogram-shaped chain be denoted as . This indicates the length of the long branch of the parallelogram.
[0048] For the platform R1 and R3, which are connected to hinges 1 and 3, the radial revolute joint clearance is considered. Therefore, the eccentric vector generated by the collision at the revolute joint clearance is introduced into the constraint equation. To solve the positional relationship between the moving platform and connected components with the revolute joint clearance, the constraint equations of the parallelogram branch and each component on the moving platform after introducing the clearance are as follows: ;(twenty four) In the formula, This represents the position vector of the center of mass of the moving platform in global coordinates when considering the clearance of the rotating joint. This represents the vector of the center position of the clearance axis at point R1 of the rotating joint when considering the clearance of the rotating joint.
[0049] The constraint equations for the rigid-flexible coupled parallel mechanism with gaps are obtained as follows: (25) use Let represent the generalized forces acting on each component in a rigid-flexible coupled dynamic model with gaps. Then, the generalized force vector of the system can be expressed as: (26) The rigid-flexible coupling dynamic model of the parallel mechanism with gaps, based on the Lagrange multiplier method, is as follows: (27) In the formula, The mass matrix of the mechanism, For constraint equations on generalized coordinates Jacobian matrix, For the generalized acceleration matrix, For Lagrange multipliers.
[0050] S4: Establish a rigid-flexible coupling dynamic model of the parallel mechanism considering fuzzy parameters.
[0051] S41: Fuzzy parameter.
[0052] To provide a mathematically formalized expression for fuzzy concepts, fuzzy set theory introduces membership functions as its fundamental descriptive method, with the domain of discourse... Fuzzy sets on yes A mapping to 1: (28) In the formula, x is One of the elements, when , For x pairs Membership degree. Membership function. The x-value pairs are represented by the values in the [0,1] interval. The degree of subordination. The closer it is to 1, the higher the degree of membership. The closer a value is to 0, the lower the degree of membership. When the membership function is 0 or 1, it is a classic set, meaning it either does not belong to or is completely part of the set. Therefore, the selection of the membership function type and interval not only involves objectivity but also reflects subjectivity.
[0053] S42: Kriging model.
[0054] fuzzy parameters Substituting the values, the Kriging equation can be expressed as: (29) In the formula, As basis functions, It is the regression coefficient. It is a random process that follows a normal distribution.
[0055] The covariance is: (30) In the formula, and For two sample points, It is the correlation function of sample points, which is related to the distance between sample points. This represents the process variance.
[0056] Suppose there are i sample points in total, and their corresponding response values are... Based on the sample points and response values, the following can be inferred: and Right now: (31) (32) In the formula, This is the correlation matrix between sample points and response values.
[0057] When inputting unknown points The response value can then be deduced, that is: (33) In the formula, This is the correlation matrix between sample points and unknown points.
[0058] The number of sample points is selected based on the complexity of the dynamic equations and the dimensionality of the uncertain parameters. .
[0059] (34) In the formula, K represents the complexity factor of the computation equation. Indicates the dimension of the independent variable.
[0060] S43: A rigid-flexible coupled dynamic model considering fuzzy uncertain parameters.
[0061] In the rigid-flexible coupling dynamic modeling of spatial parallel mechanisms considering fuzzy parameters, the gap value, the cross-sectional parameters of the parallelogram branch rod, and the elastic modulus of the parallelogram branch rod are set as fuzzy parameters and obey Gaussian fuzzy numbers.
[0062] In the rigid-flexible coupling dynamic model of the gap, the membership function of the fuzzy gap value is expressed as: (35) In the formula, 0.19mm is the upper boundary corresponding to membership degree 0, 0.21mm is the lower boundary corresponding to membership degree 0, and 0.20 is the midpoint, which is also the value corresponding to membership degree 1, with a standard deviation of 0.02.
[0063] Furthermore, the membership function of the fuzzy cross-section parameter is expressed as: (36) Similarly, the membership function of the fuzzy elastic modulus can be expressed as: (37) The Kriging model is used to solve the rigid-flexible coupled dynamic model with gaps, considering fuzzy uncertain parameters. The variables involved in the dynamic equations include physical and geometric parameters, and each fuzzy uncertain parameter transforms independently. Therefore, the rigid-flexible coupled dynamic model with gaps, considering fuzzy uncertain parameters, can be expressed as: (38) Since some parameters at the gap become fuzzy and uncertain parameters, the quantities related to them in the dynamic equation all become fuzzy and uncertain variables. Adding " "to indicate".
[0064] S5: Rigid-flexible coupling dynamic control model of spatial parallel mechanism.
[0065] S51: Sliding Mode Control Sliding mode control based on linear sliding surfaces typically involves designing the sliding surface as a linear combination of system state variables, while also considering the system's dynamic characteristics. To ensure convergence of the system on the sliding surface, the designed sliding surface function... as follows: (39) In the formula, c is a positive real number. This refers to the displacement error. for The derivative with respect to time t yields the velocity error. The design of the sliding surface is based on the first derivative of the system, effectively reflecting the dynamic characteristics of the system.
[0066] When the system is on the sliding surface: (40) further, Differentiating with respect to time, we get: (41) In the formula, for The derivative with respect to time t is the acceleration error.
[0067] To ensure the system state is driven to the sliding surface and slides there, eventually converging to an equilibrium point and stabilizing the system, the control law is designed as follows: (42) In the formula, k and b are the coefficients of the system model; the equivalent control part of this control law is: The switching control section is ; The sign function provides a high-frequency switching mechanism to ensure that the system state reaches the sliding surface within a finite time.
[0068] Substituting equation (42) into equation (41), since it is on the slip surface, we have: (43) This indicates that on the sliding surface, the error e(t) converges exponentially to zero. To determine the stability of the system, let a Lyapunov function be defined as: (44) In the formula, If and only if hour, It is clear that the expression satisfies positive definiteness.
[0069] Differentiating the Lyapunov function V(S) yields: (45) Substituting equation (43) into equation (45), we get: (46) Since k > 0 and b > 0, therefore It is negative definite. Therefore, according to Lyapunov's stability theorem, since V(S) is positive definite, and For negative constant, the system is on the sliding surface It is gradually stable.
[0070] The control law used in this section implicitly contains the form of an exponential reaching law: (47) This approach law enables the system state to rapidly approach the sliding surface at an exponential rate, while avoiding chattering problems.
[0071] In a sliding mode control system, parameters c, b, and k are key control parameters. Parameter b is related to the system's physical characteristics or mathematical model and reflects the system's sensitivity to the control input u. Increasing b decreases the amplitude of u but may reduce the response speed; decreasing b increases the amplitude of u and speeds up the response speed but may cause instability. Parameter c adjusts the slope of the sliding surface, affecting the system's speed and stability in reaching the sliding surface. Increasing c speeds up the arrival speed but increases chattering, while decreasing c slows down the arrival speed and reduces chattering. Parameter k adjusts the approach speed. Increasing k speeds up the approach speed but increases chattering, while decreasing k slows down the approach speed and reduces chattering.
[0072] S52: Consider the uncertainty of the rigid-flexible coupling dynamic control model of the spatial parallel mechanism.
[0073] In a rigid body state, the driving function of the 3-RRPaR spatial parallel mechanism Coordinates of the centroid of the end effector The positional relationship between them can be represented by matrix J: (48) Based on the positional relationship between the active arm drive function and the end effector, the feedback control law of the end effector's dynamic response is mapped onto the active arm drive function, expressed as: (49) In the formula, To adjust the angle of the active arm, This represents the feedback control law of the system in the X, Y, and Z directions. In the control model, the system driving function can be expressed as: (50) By incorporating the controlled driving function into the dynamic model considering the gap, a rigid-flexible coupled dynamic control model for the parallel mechanism with gap is established as follows: (51) To simplify the calculation process and reduce the number of iterations, the range of controller parameter variations should first be narrowed down to a specific interval through manual adjustments. Then, using the total cumulative error of the end effector as the objective function, the optimal solution for control parameters that satisfies the minimum value of the objective function within the specified interval is searched. The expression for the objective function is as follows: (52) S6: Solving the dynamic control of rigid-flexible coupling in a parallel mechanism with gaps.
[0074] S61: Solution flow for rigid-flexible coupling dynamic control of parallel mechanisms with gaps.
[0075] The solution process for the established rigid-flexible coupling dynamic control model of the spatial parallel mechanism considering uncertain parameters and gaps is as follows: Figure 6 As shown.
[0076] The specific steps for solving the equation are as follows: (1) Input the initial parameters of the 3-RRPaR spatial parallel mechanism.
[0077] (2) In the rigid-flexible coupling dynamic model of the gap, determine the relative positional relationship between the shaft and the bushing, and further calculate the contact force at the gap rotating pair through the gap force model.
[0078] (3) Substitute the relevant variables from step (2) into the rigid-flexible coupling dynamic model. Use the ode45 solver to calculate the rigid-flexible coupling dynamic model with gaps.
[0079] (4) Calculate the error of the end effector in the X, Y, and Z directions, and further calculate... ,according to Calculate the angle compensation amount of the drive arm .
[0080] (5) Determine if the time variable has reached the termination time. If it has not reached the termination time, introduce the solved relevant variables into the gap rigid body dynamics model and return to step (2). If the termination time has reached the termination time, then use the calculated variables. Substitute this into the rigid-flexible coupling dynamic control model.
[0081] (6) The result obtained in step (5) Substitute it into the Kriging model and output the response after control.
[0082] S62: Dynamic control analysis of rigid-flexible coupling of parallel mechanisms with gaps.
[0083] The above results were simulated, and the specific parameters of each component are shown in Table 1. Table 1: Structural Parameters of 3-RRPaR Spatial Parallel Mechanism
[0084] The displacement and velocity responses of the end effector are direct indicators of its accuracy. Therefore, this section analyzes the displacement and velocity responses of the end effector before and after control to verify the effectiveness of the control strategy. The displacement and velocity responses of the end effector and their comparisons are also discussed. Figures 7-14 As shown.
[0085] Before and after control, the absolute values of the maximum displacement error of the end effector in the X direction were 0.258 mm and 0.230 mm, respectively, with small vibration amplitude; the absolute values of the maximum displacement error in the Y direction were 0.303 mm and 0.268 mm, respectively, with small vibration amplitude; the absolute values of the maximum velocity error of the end effector in the X direction were 0.046 mm / s and 0.031 mm / s, respectively; the absolute values of the maximum velocity error in the Y direction were 0.482 mm / s and 0.412 mm / s, respectively; at the same time, the displacement and velocity curves in the X and Y directions after control showed a more stable trend.
[0086] The above analysis results show that after control, the displacement of the end effector is more accurate and the motion precision of the mechanism is improved; at the same time, the speed fluctuation is effectively suppressed, making its motion process more stable. The displacement and speed changes of the end effector after control verify the effectiveness of this control strategy.
[0087] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.
Claims
1. A rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters, characterized in that: It includes the following steps: S1. Establish and combine a spherical pair model with dry friction gap and a flexible beam element model to form a rigid-flexible coupled spatial parallel mechanism model with gap; S2. Based on the constraints of the parallel mechanism model established in S1, and using the Lagrange multiplier method, the rigid-flexible coupling dynamic model of the parallel mechanism is obtained. S3. Based on the dynamic model in S2, establish a rigid-flexible coupling dynamic model of the spatial parallel mechanism with fuzzy parameters: S31. Select fuzzy parameters, and set the spherical pair gap value c as the fuzzy parameter, which follows the Gaussian fuzzy number: ； In the formula, b is the upper boundary corresponding to membership degree 0, d is the lower boundary corresponding to membership degree 0, and a is the value corresponding to membership degree 1. S32. Construct a rigid-flexible coupled dynamic model of a spatial parallel mechanism with fuzzy parameters: ； In the formula, The mass matrix of the mechanism, For constraint equations on generalized coordinates Jacobian matrix, For the generalized acceleration matrix, This is the matrix formed by the generalized force vectors acting on each component in a rigid-flexible coupling dynamic model containing dry friction gaps. These are the constraint equations for the mechanism; For Lagrange multiplier terms, superscript Represents fuzzy and uncertain variables; S4. Establish a rigid-flexible coupling dynamic control model for a spatial parallel mechanism using sliding mode control: Construct the active arm driving function: ； In the formula, To adjust the angle of the active arm, The feedback control law of the system in the X, Y, and Z directions; the system driving function. Represented as: ； In the formula, For the driving function of the spatial parallel mechanism; By introducing the control post-drive function into the dynamic model in S3, a rigid-flexible coupled dynamic control model of the parallel mechanism with gaps is established: ； Wherein, the superscript r represents the function with flexible parameters, and the superscript k represents the driving function after the introduction of control; S5. The dynamic control model obtained in S4 is solved using the Kriging surrogate model method. Based on the gap parameter and the dynamic response value of the known sample points, the mapping relationship between the gap uncertainty parameter and the dynamic response is established.
2. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 1, characterized in that, The method for establishing a spherical pair model with dry friction gap in S1 is as follows: construct a model of the gap eccentricity vector and a collision depth model at the contact point. By judging the collision depth, the contact collision state between the sphere and the shell is identified. When the collision occurs and deformation occurs, the relative normal velocity and relative tangential velocity at the potential contact point, the contact force of the sphere on the shell, and the reaction force of the shell on the sphere are calculated.
3. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 1, characterized in that, S1 yields the elastic force expression for the flexible beam element model as follows: ； In the formula, , , , G and G are Lamé constants.
4. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 1, characterized in that, S2 includes the following sub-steps: S21. The coordinates of the component using the centroid coordinate method are described using the centroid coordinate method for the active arm, the lower short rod of the parallelogram branch, and the moving platform. ； In the formula, For position coordinates, The pose of the centroid coordinate system of the moving component in the fixed coordinate system; The coordinates of the long branch of the parallelogram are described using the natural point coordinate method: ； In the formula, and These represent the position vectors of the two endpoints of the component in a fixed coordinate system; S22. Based on the Lagrange multiplier method, the dynamic model of the rigid-flexible coupled spatial parallel mechanism with dry friction gap is obtained: ； In the formula, The mass matrix of the mechanism, For constraint equations on generalized coordinates Jacobian matrix, For the generalized acceleration matrix, For Lagrange multipliers, This represents the generalized force vector experienced by each component in a rigid-flexible coupling dynamic model with gaps.
5. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 4, characterized in that, S21 also includes the following steps: After describing a single coordinate, define the generalized coordinates of the rigid-flexible coupling dynamic model system of the spatial parallel mechanism, establish the constraint equations between the drive arm and the frame, introduce the flexible rod and clearance, and obtain the constraint equations of the rigid-flexible coupling parallel mechanism with clearance: 。 6. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 1, characterized in that, The method for constructing the active arm driving function in S4 is as follows: Designed sliding surface function: ； Design control law: ； Based on the positional relationship between the active arm drive function and the end effector, the feedback control law of the dynamic response of the end effector is mapped onto the active arm drive function.
7. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 1, characterized in that, S4 also includes determining the optimal controller parameters for the control model. The method involves manually narrowing the parameters down to a range as the search variable constraint range, using the total cumulative error of the end effector as the objective function, and employing a particle swarm optimization algorithm to search for the optimal solution of control parameters within the range that satisfies the minimum value of the objective function. The expression for the search objective function is: 。 8. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 1, characterized in that, The solution formula for the dynamic control model in S5 is: ； In the formula, It is a predicted value. It is a variable basis functions, It is the regression coefficient. It is a random process that follows a normal distribution. This represents the number of different gap values.
9. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 1, characterized in that, The method in S5 for establishing the mapping relationship between the uncertain parameters of the clearance and the dynamic response based on the clearance parameters and the dynamic response values of known sample points includes: setting parameters for the mechanism, including mechanism parameters and fuzzy parameters; calculating the contact force, solving for the displacement and velocity using the ode45 solver to obtain the displacement and velocity response functions of the moving platform; feeding the obtained response functions back to the mechanism, updating the relevant variables of the mechanism through the control model until they meet the objective function, reaching the required number of iterations, and outputting the response.
10. The rigid-flexible coupling dynamic control method for a gap parallel mechanism considering uncertain parameters according to claim 2, characterized in that, The method for calculating the relative normal velocity and relative tangential velocity at the potential contact point, the contact force between the sphere and the shell, and the reaction force between the shell and the sphere is as follows: A normal collision force model is established based on the LN model: ； In the formula, The initial collision velocity, For collision speed, n is the restoration coefficient, and n is the correction coefficient; ； In the formula, For elastic modulus, Poisson's ratio; A tangential friction model is established using the modified Coulomb friction model, which is expressed as follows: ； In the formula, and These are the friction coefficient and the dynamic correction coefficient, respectively. and These are the speed limits for dynamic and static friction; the contact force generated by the clearance on the bushing is: ； In the formula, Let be the unit direction vector along the normal direction of the contact surface. It is a unit direction vector along the tangential direction of the contact surface.