Fuzzy uncertainty analysis method for dynamic optimization of spatial parallel mechanism with gap

Abstract

The present invention provides a fuzzy uncertainty analysis method for the dynamic optimization of a spatial parallel mechanism containing a gap, belonging to the technical field of industrial robot modeling and optimization, and comprising the following steps: S1, establishing a spherical pair model containing a dry friction gap; S2, obtaining a dynamic model of a three-degree-of-freedom spatial parallel mechanism containing a spherical pair dry friction gap based on the constraint conditions of the spatial parallel mechanism containing a spherical pair dry friction gap. S3, performing optimization based on the dynamic model of the spatial parallel mechanism; S4, using the optimized data as the dynamic model of the mechanism and selecting fuzzy parameters, establishing a dynamic model of the spatial parallel mechanism containing fuzzy parameters and performing solution analysis. The present invention can improve the adverse effects of dry friction gaps on the dynamic characteristics of a mechanism, and provides a method for predicting the effects of fuzzy parameters of a mechanism on its dynamic characteristics.

Description

Technical Field

[0001] The present invention relates to the technical field of industrial robot modeling and optimization, and in particular to a fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism containing a gap space. Background Art

[0002] Parallel mechanisms, with their advantages of high rigidity, precision, and stability, serve as the core mechanisms of parallel machine tools and parallel robots. However, in parallel machine tools, factors such as component errors caused by insufficient machining precision and assembly errors required to ensure proper functioning of the parts can lead to gaps in the kinematic pairs of the parallel machine's main mechanisms. The presence of dry friction gaps leads to frequent contact and collisions between kinematic pair elements, disrupting their surface textures and significantly affecting connecting rod dimensional errors and the positioning accuracy of the moving platform. Consequently, this can alter the positioning accuracy of the parallel machine tool's tool tip, impacting workpiece quality.

[0003] Over extended operation, the kinematic elements of a parallel machine tool (PMT) can collide with each other, potentially causing material loss and damage. Parameters such as clearances can change, creating uncertainty in the dynamic response of the PMT, directly impacting the PMT's positioning accuracy and the quality of the workpiece.

[0004] Therefore, how to improve the adverse effects of dry friction clearance on the dynamic characteristics of the mechanism through optimization and predict the influence of the mechanism fuzzy parameters on the dynamic characteristics of the mechanism, that is, the machining accuracy of the parallel machine tool, is an urgent problem to be solved. Summary of the Invention

[0005] In order to address the deficiencies of the above-mentioned prior art, the purpose of the present invention is to provide a fuzzy uncertainty analysis method for the dynamic optimization of a parallel mechanism containing a gap space. This method can improve the adverse effects of dry friction gap on the dynamic characteristics of the parallel mechanism of the robot body and improve the working accuracy of the parallel mechanism. It also provides a new analysis method for predicting the influence of fuzzy parameters on the dynamic characteristics of the mechanism, thereby improving the accuracy of dynamic analysis prediction and better predicting the life of the parallel mechanism.

[0006] Specifically, the present invention provides a fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism containing a gap space, which comprises the following steps:

[0007] S1. Establish a spherical pair model with dry friction gap;

[0008] S2. Establish the constraint conditions of the spatial parallel machine tool with spherical auxiliary friction gap, and obtain the dynamic model of the three-degree-of-freedom spatial parallel machine tool with spherical auxiliary friction gap as follows:

[0009]

[0010] Where, α b and β b Indicates a correction factor greater than 0, Q c represents the system generalized force vector of the parallel mechanism, Φ c Represents the constraint equation of the mechanism containing spherical pair clearance, represents the time derivative of the constraint equation of the mechanism containing the spherical pair clearance; Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, M is the mass matrix, λ is the Lagrange multiplier, and γ is the right side of the acceleration constraint equation;

[0011] S3. Determine the optimization variables and the constraints of the optimization variables, obtain the optimal optimization variables, and optimize the dynamic model of the spatial parallel mechanism. Specifically, the steps include the following:

[0012] S31. Take the displacement error of the moving platform as the optimization objective function:

[0013] f(x)=||w A -w B ||

[0014] Where w A represents the displacement response of the dynamic platform under the ideal state without gap, w B =q a It represents the displacement response of the moving platform with clearance in the x direction, which is obtained from the dynamic model of the mechanism:

[0015]

[0016] S32. Determine the optimization variable x, select the mass and moment of inertia of the end effector of the mechanism as the optimization variables, and the array of the optimization variable x is expressed as:

[0017] x=[m d ,J xx x,J yy y,J zz z] T

[0018] In the formula, x is the optimization variable, m d is the mass of the end effector, J xx x,J yy y,J zz z is the moment of inertia of the end effector;

[0019] S33. Construct the constraints for the optimization variable x:

[0020] x i =[(md ) min ,(J xx x) min ,(J yy y) min ,(J zz z) min ] T

[0021] x s =[(m d ) max ,(J xx x) max ,(J yy y) max ,(J zz z) max ] T

[0022] x i ≤x≤x s

[0023] Where x i and x s are the upper and lower bounds of the optimization variable x respectively;

[0024] S34, using a particle swarm optimization algorithm to solve the established dynamic model of the space parallel mechanism with gaps to obtain the optimal optimization variable x;

[0025] S4. Select fuzzy parameters, establish a dynamic model of the spatial parallel mechanism containing fuzzy parameters, and solve it, which specifically includes the following sub-steps:

[0026] S41, the spherical surface clearance value c is set as a fuzzy parameter and obeys the Gaussian fuzzy number, which is specifically expressed as follows:

[0027]

[0028] Where b is the upper boundary corresponding to membership 0, d is the lower boundary corresponding to membership 0, and c is the midpoint and the value corresponding to membership 1;

[0029] S42. Construct the dynamic model of the spatial parallel mechanism with fuzzy parameters and dry friction spherical pair clearance as follows:

[0030]

[0031] In the formula, the superscript ~ indicates that the mechanism parameters become fuzzy parameters;

[0032] S43, using the Kriging surrogate model method to solve the dynamic model obtained in step S42 with the optimization variables, and determining the solution formula as follows:

[0033] H(n)=f(n)β a +z(n)

[0034] Where H(n) is the predicted value, f(n) is the basis function of variable n, and β a is the regression coefficient, z(n) is a random process that obeys the normal distribution; n is the number of different gap values;

[0035] S44. Determine the dynamic response value of the known sample point based on the dynamic response value of the known sample point using the solution formula determined in step S43:

[0036]

[0037] Where r is the correlation vector between the sample point and the point to be measured; F is the basis function f(n), Y is the dynamic response value of the known sample point, and R is the correlation matrix between the dynamic response values of the known sample point and the unknown sample point; is the predicted kinetic response value interval.

[0038] Preferably, step S1 specifically includes the following steps:

[0039] S11. Construct the modulus of the gap eccentricity vector as follows:

[0040] e ij =P Zj -P Zi

[0041] Among them, P Zi is the position vector of the center point of the spherical shell in the local coordinate system, P Zj is the position vector of the center point of the sphere in the global coordinate system;

[0042] S12. Construct a collision depth model at the contact point:

[0043] δ=e ij -c

[0044] Where, e ij is the modulus of the gap eccentricity vector, and c is the spherical pair clearance value;

[0045] S13. By judging the collision depth δ, the contact collision state between the sphere and the spherical shell is determined. The judgment formula is as follows:

[0046]

[0047] 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.

[0048] Preferably, in step S13, the relative normal velocity V at the potential contact point is n and the relative tangential velocity V t It is expressed as follows:

[0049]

[0050] Where V Dj , V Cj are the positions of the contact points on the sphere and spherical shell in the global coordinate system, n n is the normal unit vector at the gap contact point;

[0051] The contact force of the sphere on the shell is expressed as:

[0052] F i =F N ·n n +F T ·t t

[0053] Where, F N is the normal contact force, F T is the tangential contact force, t t is the tangential unit vector of the contact point;

[0054] The reaction force of the spherical shell on the sphere is expressed as:

[0055] F j =-F i .

[0056] Preferably, step S2 specifically includes the following sub-steps:

[0057] S21. The constraint equation for constructing a spatial parallel machine tool with a spherical auxiliary friction gap is:

[0058]

[0059] Where, Φ 1...n is the motion constraint;

[0060] After that, we can get the acceleration constraint equation by taking the derivative twice with respect to time:

[0061]

[0062] Where, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, and γ represents the right side of the constraint equation;

[0063] S22. The dynamic equation of the spatial parallel machine tool with spherical pair clearance and Lagrange multiplier is established as follows:

[0064]

[0065] Where Q c Represents the system generalized force vector of the parallel machine tool, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, M is the mass matrix, and λ is the Lagrange multiplier;

[0066] M=diag[mmm J xx J yy J zz ]

[0067] Where m is the mass of the rod and J is the moment of inertia;

[0068] S23. Establish a dynamic model of the spatial parallel machine tool with a spherical pair containing clearances.

[0069] Preferably, in step S44, the DACE toolkit and the OD45 solver in MATLAB are used for solving the problem.

[0070] Preferably, the dynamic response values of the known sample points in step S44 are dynamic response values of the mechanism under multiple sets of fuzzy parameters.

[0071] Preferably, step S34 specifically includes the following sub-steps:

[0072] S341, initializing the particle swarm algorithm, generating initial particle swarm sample points based on the optimization variables and the objective function;

[0073] S342, solving the dynamic model to calculate the eccentricity vector, normal velocity, tangential velocity, and contact point position;

[0074] S343, obtaining the displacement, velocity, acceleration response data and fitness function of the moving platform;

[0075] S344, updating particle speed and position, searching for the global optimal particle;

[0076] S345. Iterate the particles until the objective function is met and the optimal optimization variables are obtained.

[0077] Compared with the prior art, the present invention has the following beneficial effects:

[0078] (1) The present invention provides a fuzzy uncertainty analysis method for the dynamic optimization of a parallel mechanism containing a gap space. The method can improve the adverse effects of dry friction gap on the dynamic characteristics of the mechanism, and provide a new analysis method for predicting the influence of the fuzzy parameters of the mechanism on the dynamic characteristics of the mechanism, thereby improving the accuracy of dynamic analysis prediction and better predicting the life of the parallel mechanism.

[0079] (2) The present invention optimizes the parameters of the kinematic model through the particle swarm algorithm, greatly improving the accuracy of the dynamic optimization model of the spatial parallel mechanism with gaps, improving the accuracy of the parallel mechanism and reducing the collision force generated by the operation of the mechanism, and making the prediction data interval smaller and more accurate.

[0080] (3) The present invention takes into account the uncertain parameter gap value when outputting the prediction data, and obtains the prediction value interval by simulating the actual changes in uncertainty, predicting the mechanism motion error, and preventing the mechanism accuracy from decreasing beyond the working range of the parallel mechanism. By predicting the life of the kinematic pair, the kinematic pair can be replaced in time to avoid affecting the working accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 It is a schematic diagram of the overall process of the present invention;

[0082] Figure 2 A schematic diagram of a spherical pair collision with a gap provided by an embodiment of the present invention;

[0083] Figure 3 Schematic diagram of a 3-PSR spatial parallel mechanism with spherical pair clearance provided by an embodiment of the present invention;

[0084] Figure 4 A schematic diagram of the structure of a 3-PSR mechanism including a spherical pair clearance provided in an embodiment of the present invention;

[0085] Figure 5 Schematic diagram of the X-direction displacement optimization simulation of the parallel mechanism provided by an embodiment of the present invention;

[0086] Figure 6 A schematic diagram of a simulation of X-direction speed optimization for a parallel mechanism provided by an embodiment of the present invention;

[0087] Figure 7 A schematic diagram of the X-direction acceleration optimization simulation of the parallel mechanism provided by an embodiment of the present invention;

[0088] Figure 8 A schematic diagram of the X-direction displacement simulation of a parallel mechanism with fuzzy parameters provided by an embodiment of the present invention;

[0089] Figure 9 A schematic diagram of the X-direction velocity simulation of a parallel mechanism with fuzzy parameters provided by an embodiment of the present invention;

[0090] Figure 10 A schematic diagram of the X-direction acceleration simulation of a parallel mechanism with fuzzy parameters provided by an embodiment of the present invention;

[0091] Figure 11 A schematic diagram of the X-direction displacement optimization simulation of a parallel mechanism with fuzzy parameters provided by an embodiment of the present invention;

[0092] Figure 12 A schematic diagram of a simulation of X-direction speed optimization of a parallel mechanism with fuzzy parameters provided by an embodiment of the present invention;

[0093] Figure 13 A schematic diagram of the X-direction acceleration optimization simulation of a parallel mechanism with fuzzy parameters provided by an embodiment of the present invention;

[0094] Figure 14 Schematic diagram of optimization of a 3-DOF parallel robot provided by an embodiment of the present invention;

[0095] Figure 15 Flowchart of the 3-PSR parallel robot provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0096] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings.

[0097] The present invention provides a fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism with a gap space, such as Figure 1 As shown, it includes the following steps:

[0098] S1. Establishing a spherical pair model with dry friction gap; specifically comprising the following steps:

[0099] S11. Construct the modulus of the gap eccentricity vector as follows:

[0100] e ij =P Zj -P Zi

[0101] Among them, P Zi is the position vector of the center point of the spherical shell in the local coordinate system, P Zj is the position vector of the sphere center in the global coordinate system.

[0102] S12. Construct a collision depth model at the contact point:

[0103] δ=e ij -c

[0104] Where, e ij is the modulus of the gap eccentricity vector, and c is the spherical pair clearance value.

[0105] S13. By judging the collision depth δ, the contact collision state between the sphere and the spherical shell is determined. The judgment formula is as follows:

[0106]

[0107] 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.

[0108] In step S13, the relative normal velocity V at the potential contact point n and the relative tangential velocity V t It is expressed as follows:

[0109]

[0110] Where V Dj , V Cj are the positions of the contact points on the sphere and spherical shell in the global coordinate system, n n is the normal unit vector at the gap contact point.

[0111] The contact force of the sphere on the shell is expressed as:

[0112] F i =F N ·n n +F T ·t t

[0113] Where, F N is the normal contact force, F T is the tangential contact force, t t is the tangential unit vector of the contact point.

[0114] The reaction force of the spherical shell on the sphere is expressed as:

[0115] F j =-F i .

[0116] S2. Establishing the constraint conditions of the spatial parallel mechanism with a spherical auxiliary dry friction gap and obtaining the dynamic model of the three-degree-of-freedom spatial parallel mechanism with a spherical auxiliary dry friction gap, specifically including the following sub-steps:

[0117] S21. The constraint equation for constructing a spatial parallel mechanism with a spherical auxiliary dry friction gap is:

[0118] Φ c (q)=[Φ1Φ2.......Φ n ] T =0

[0119] Where, Φ1...n is the motion constraint;

[0120] Taking the time derivative twice yields the acceleration constraint equation:

[0121]

[0122] Where, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, and γ represents the right side of the constraint equation.

[0123] S22. The dynamic equation of the spatial parallel mechanism with spherical clearance and Lagrange multipliers is established as follows:

[0124]

[0125] Where Q c represents the system generalized force vector of the parallel mechanism, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, M is the mass matrix, and λ is the Lagrange multiplier.

[0126] M=diag[mmm J xx J yy J zz ]

[0127] Where m is the mass of the rod and J is the moment of inertia.

[0128] S23. The dynamic model of the spatial parallel mechanism with a spherical pair containing clearance is established as follows:

[0129]

[0130] Where, α b and β b Indicates a correction factor greater than 0, Q c represents the system generalized force vector of the parallel mechanism, Φ c Represents the constraint equation of the mechanism containing spherical pair clearance, represents the time derivative of the constraint equation of the mechanism containing the spherical pair clearance; Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, M is the mass matrix, λ is the Lagrange multiplier, and γ is the right side of the acceleration constraint equation.

[0131] S3. Determine the optimization variables and the constraints of the optimization variables, obtain the optimal optimization variables, and optimize the dynamic model of the spatial parallel mechanism. Specifically, the steps include the following:

[0132] S31. Take the displacement error of the moving platform as the optimization objective function:

[0133] f(x)=||w A -w B ||

[0134] Where w A represents the displacement response of the dynamic platform under the ideal state without gap, w B =q a It represents the displacement response of the moving platform with clearance in the x direction, which is obtained from the dynamic model of the mechanism:

[0135]

[0136] S32. Determine the optimization variable x, select the mass and moment of inertia of the end effector of the mechanism as the optimization variables, and the array of the optimization variable x is expressed as:

[0137] x=[m d ,J xx x,J yy y,J zz z] T

[0138] In the formula, x is the optimization variable, m d is the mass of the end effector, J xx x,J yy y,J zz z is the moment of inertia of the end effector.

[0139] S33. Construct the constraints for the optimization variable x:

[0140] x i =[(m d ) min ,(J xx x) min ,(J yy y) min ,(J zz z) min ] T

[0141] x s =[(m d ) max ,(J xx x) max ,(J yy y) max ,(J zz z)max ] T

[0142] x i ≤x≤x s

[0143] Where x i and x s are the upper and lower bounds of the optimization variable x, respectively.

[0144] S34. Use the particle swarm optimization algorithm to solve the established dynamic model of the parallel mechanism with gap space to obtain the optimal optimization variable x.

[0145] This step specifically includes the following sub-steps: S341, initializing the particle swarm algorithm, generating initial particle swarm sample points based on the optimization variables and the objective function.

[0146] S342. Solve the dynamic model and calculate the eccentricity vector, normal velocity, tangential velocity and contact point position.

[0147] S343. Obtain the displacement, velocity, and acceleration response data of the dynamic platform to calculate the constraint reaction force and fitness function at the corresponding gap node.

[0148] S344. Update particle velocity and position, and search for the global optimal particle.

[0149] S345. Iterate the particles until the objective function is met and the optimal optimization variables are obtained.

[0150] S4. Select fuzzy parameters, establish a dynamic model of the spatial parallel mechanism containing fuzzy parameters, and solve it, which specifically includes the following sub-steps:

[0151] S41, the spherical surface clearance value c is set as a fuzzy parameter and obeys the Gaussian fuzzy number, which is specifically expressed as follows:

[0152]

[0153] Where b is the upper boundary corresponding to membership 0, d is the lower boundary corresponding to membership 0, and c is the midpoint and the value corresponding to membership 1.

[0154] S42. Construct the dynamic model of the spatial parallel mechanism with fuzzy parameters and dry friction spherical pair clearance as follows:

[0155]

[0156] In the formula, the superscript ~ indicates that the mechanism parameters become fuzzy parameters.

[0157] S43, using the Kriging surrogate model method to solve the dynamic model obtained in step S42 with the optimization variables, and determining the solution formula as follows:

[0158] H(n)=f(n)β a +z(n)

[0159] Where H(n) is the predicted value, f(n) is the basis function of variable n, and β a is the regression coefficient, z(n) is a random process that obeys the normal distribution; n is the number of different gap values.

[0160] S44. Determine the dynamic response value of the known sample point based on the dynamic response value of the known sample point using the solution formula determined in step S43.

[0161]

[0162] Where r is the correlation vector between the sample point and the point to be measured; F is the basis function f(n), Y is the dynamic response value of the known sample point, and R is the correlation matrix between the dynamic response values of the known sample point and the unknown sample point; is the predicted dynamic response value range. In step S44, the DACE toolkit and ode45 solver in MATLAB are used to solve the problem. The dynamic response value of the known sample point is the dynamic response value of the mechanism under multiple sets of fuzzy parameters. Specific embodiments

[0164] This embodiment uses a very common three-degree-of-freedom spatial parallel mechanism that can be used as the main mechanism of a parallel drilling robot as the research object for analysis. During the use of the drilling robot, as the drilling robot works for a long time, the mutual collision between the kinematic sub-elements in the main parallel mechanism intensifies, which may cause the loss and damage of materials, thereby causing parameters such as the gap value to change. The change in parameters makes the dynamic response of the drilling robot's main mechanism uncertain, which will directly affect the accuracy of the drilling robot and make it impossible to guarantee the accuracy and quality of the hole. This embodiment can predict and analyze the life of the drilling robot by optimizing the dynamics of the parallel drilling robot and analyzing it based on fuzzy uncertainty, thereby ensuring its working accuracy.

[0165] like Figure 2 and Figure 3As shown in the figure, the drilling robot's 3-PSR parallel mechanism consists of a frame, a moving platform, and three identical PSR branches, where P represents a moving pair, S represents a spherical pair, and R represents a revolute pair. The three moving pairs are defined as P1, P2, and P3, the three spherical pairs are defined as S1, S2, and S3, and the three revolute pairs are defined as R1, R2, and R3. P1P2 is perpendicular to the plane where the spherical pair and the moving pair are always perpendicular to the plane where P1P2P3 lies. The rotation axes of the revolute pairs R1 and R2 are parallel to each other and perpendicular to the rotation axis of R3. The plane formed by the three revolute pairs is parallel to the plane formed by the three moving pairs. The rotation axes of the revolute pairs R1 and R2 are parallel to OP3, and the rotation axis of R3 is parallel to P1P2.

[0166] A global coordinate system OA-XYZ is established on the frame, denoted as {A}, with the origin OA located directly below point P, the X-axis pointing vertically downward, the Y-axis pointing to the spherical pair S3, and the Z-axis satisfying the right-hand rule. A local coordinate system o'-xyz is established on the moving platform, denoted as {B}, with the origin o'

[0167] Located at the midpoint of the line connecting the revolute pairs R1 and R2, the direction of the x-axis is vertically downward, the direction of the y-axis points to the revolute pair R3, and the direction of the z-axis satisfies the right-hand rule.

[0168] In the mechanism, the initial length between the movable pair Pi and point P is hi, the fixed length between point P and the coordinate origin OA is a1, the length between the movable pair Pi and the spherical pair Si is l1, the length between the spherical pair Si and the revolute pair Ri is l2, and the length between the revolute pair Ri and the coordinate origin o' is b1. L11 represents the first link of the first branch, and L12 represents the second link of the first branch.

[0169] The dynamic analysis and optimization methods of the three-DOF spatial parallel mechanism with spherical dry friction clearance include:

[0170] S1. Establish a spherical pair model with dry friction gap.

[0171] S2. According to the constraint conditions of the spatial parallel mechanism with spherical auxiliary dry friction gap, the dynamic model of the three-degree-of-freedom spatial parallel mechanism with spherical auxiliary dry friction gap is obtained.

[0172] S3. Optimize based on the dynamic model of the spatial parallel mechanism.

[0173] S4. The optimized data is used as the agent mechanism dynamic model and fuzzy parameters are selected to establish the dynamic model of the spatial parallel mechanism containing fuzzy parameters and perform solution analysis.

[0174] Figure 1 The figure shows the collision diagram of a spherical pair with clearance. i and Zj are the center points of the spherical shell and the sphere, R and r are the radii of the spherical shell and the sphere, respectively. i and D j are the potential contact collision points on the spherical shell and the sphere, F N and F T are the normal contact force and tangential contact force at the collision point, A n n and A t t are the normal unit vector and tangential unit vector at the collision point, respectively.

[0175] Step 1: Establish a spherical pair model with dry friction gap.

[0176] The position vector of the center point of the spherical shell in the global coordinate system is:

[0177] P Zi =r ki +R ki ·p Zi (k=1,2,3) (1)

[0178] Where r ki is the position vector of the center of mass of the connecting rod connected to the spherical shell in the global coordinate system, R ki is the change matrix of the local coordinate system fixed to the connecting rod relative to the global coordinate system, p Zi is the position vector of the center point of the spherical shell in the local coordinate system.

[0179]

[0180] Where c represents cosine and s represents sine.

[0181] The position vector of the center point of the sphere in the global coordinate system is:

[0182] P Zj =r kj +R kj ·p Zj (k=4,5,6) (3).

[0183] The eccentricity vector between the sphere and the center point of the spherical shell is:

[0184] e ij =P Zj -P Zi =r kj +R kj ·p Zj -r ki +R ki ·p Zi (4).

[0185] The modulus of the eccentric vector at the gap is:

[0186]

[0187] Where, e ijx Indicates e ij The component in the x direction, e ij (1,1) represents e ij The first number in .

[0188] The derivative of the eccentricity vector is:

[0189]

[0190] The normal unit vector at the gap contact point is:

[0191]

[0192] The clearance value in a spherical pair can be expressed by the radius of the shell and the sphere:

[0193] c=Rr (8).

[0194] When the sphere and the shell collide during motion, the collision depth at the contact point is:

[0195] δ=e ij -c (9).

[0196] By judging the collision depth δ, the contact collision state between the sphere and the spherical shell can be determined:

[0197]

[0198] Potential contact point C i and D j The position vector in the global coordinate system is:

[0199]

[0200] Potential contact point C i and D j The relative contact velocity can be derived from the above formula:

[0201]

[0202] in

[0203]

[0204] Where, ω i is the velocity transformation matrix relative to the global coordinate system, is the velocity vector of the connecting rod center of mass in the global coordinate system.

[0205] The relative contact velocity at the potential contact point can be decomposed into the relative normal velocity and the relative tangential velocity:

[0206]

[0207] The improved Flores normal contact force model is expressed as follows:

[0208]

[0209] Where K is the stiffness coefficient, and its value is related to the contact surface shape and material properties; n is the power exponent, which is usually 1.5; δ is the collision depth, is the collision velocity, which is the derivative of δ, is the initial collision velocity; c e is the coefficient of restitution, which is related to the material properties.

[0210] The improved Coulomb tangential friction model can be expressed as:

[0211]

[0212] Where c f is the coefficient of sliding friction; c g is the dynamic correction coefficient, and its specific expression is:

[0213] The contact force of the sphere on the shell can be expressed as:

[0214] F i =F N ·n n +F T ·t t (17).

[0215] The reaction force of the spherical shell on the sphere can be expressed as:

[0216] F j =-F i (18).

[0217] The moment generated by the spherical shell and the sphere at the center of mass of the connecting rod is:

[0218]

[0219] Step 2: According to the constraint conditions of the spatial parallel mechanism containing a spherical auxiliary dry friction gap, a dynamic model of the three-degree-of-freedom spatial parallel mechanism containing a spherical auxiliary dry friction gap is obtained.

[0220] The constraint equation of the spatial parallel mechanism with spherical auxiliary dry friction gap is:

[0221] Φ c (q)=[Φ P1 Φ P2 Φ P3 Φ R1 Φ R2 Φ R3 Φ D ] T =0 33×1 (20)

[0222] Where, Φ P is the moving joint constraint, Φ R is the rotational joint constraint, Φ D is the driving constraint.

[0223] Taking the time derivative of equation (20) twice yields the acceleration constraint equation:

[0224]

[0225] Where, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, and γ represents the right side of the constraint equation.

[0226] The generalized coordinate matrix of the 3-PSR mechanism is:

[0227]

[0228] Where q k =[x k ,y k ,z k ,α k ,β k ,γ k ], x k ,y k ,z k is the position vector of the origin of the local coordinate system of component k (k=1, 2…7) in the global coordinate system, α k ,β k ,γ k The Euler angle vector of the local coordinate system origin in the global coordinate system.

[0229] The acceleration constraint equation is obtained by taking the time sphere derivative twice of equation (20).

[0230] Where, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, and γ represents the right side of the constraint equation.

[0231] The dynamic equation of the 3-PSR spatial parallel mechanism with spherical clearance and Lagrange multipliers is established as follows:

[0232]

[0233] In the process of solving the dynamics of the mechanism with clearance, the position constraint of the ideal spherical pair is replaced by the contact force constraint in the clearance. Therefore, the contact force and torque in the spherical pair with clearance need to be substituted into Q c middle.

[0234] According to formulas (35)((17) and (36)(18), the contact force in the spherical pair with clearance can be obtained as:

[0235]

[0236] Where n n S1，2，3 and t t S1，2，3 F represents the normal unit vector and tangent unit vector of the contact surface where the collision point is located in the spherical pair S1, S2, and S3; N S1，2，3 and F T S1，2，3 represents the normal contact force and tangential contact force in the spherical pair; F i S1，2，3 and They represent the forces exerted by the sphere on the shell and the forces exerted by the shell on the sphere, which are the action and reaction forces respectively.

[0237] According to formulas (35) and (19), the torque generated by the contact force at the collision point in the spherical pair with clearance in the 3-PSR spatial parallel mechanism at the center of mass of the connecting rod is:

[0238]

[0239] Where, and They represent the contact force F acting on the spherical shell in the spherical pair with clearance. i S1，2，3 , the contact force on the sphere The moment generated at the center of mass of the connecting rod connected to the spherical pair; and Represents the position vector of the contact collision point in the spherical pair S1 in the global coordinate system; r i S1，2，3 and They represent the position vector of the center of mass of the connecting rod connected to the spherical shell in the global coordinate system and the position vector of the center of mass of the connecting rod connected to the sphere in the global coordinate system respectively.

[0240] Therefore, the generalized force vector Q on the 3-PSR spatial parallel mechanism is c Expressed as:

[0241]

[0242] Where H k =[m k g 0 0 0 0 0] T (k=1,2,…,7), g is the acceleration due to gravity,

[0243] According to formulas (21), (23) and the Baumgarte default stability algorithm, the dynamic formula of the spherical pair with clearance of the 3-PSR spatial parallel mechanism can be established as follows:

[0244]

[0245] Step 3: Optimize the spatial parallel mechanism based on its dynamic model. This design approach integrates the spatial parallel mechanism's dynamic model with an optimization algorithm to optimize the parameters of key components within the mechanism. A multi-objective optimization strategy is employed to design the objective function, design variables, and constraints.

[0246] Since there is a gap in the joints of the main parallel mechanism, the working accuracy of the drilling robot is affected. Therefore, this paper takes the displacement error of the moving platform, that is, the drilling position error, as the objective function, which is expressed as follows:

[0247] f(x)=||w x -w y || (28)

[0248] Where w x represents the displacement response of the dynamic platform under the ideal state without gap, w B =q a It represents the displacement response of the moving platform with clearance in the x direction, which is obtained from the dynamic model of the mechanism:

[0249]

[0250] The end effector, as an actuator, is often designed according to specific working conditions. By adopting the mass distribution method, the topology, structure type, and material of the parallel mechanism are not changed. Therefore, the mass and moment of inertia of the end effector of the mechanism are selected as optimization variables. The design variable x can be expressed as:

[0251] m7 J xx x J yy y J zz z

[0252] x=[m7,Jxx x,J yy y,J zz z] T (30)

[0253] Where x is the optimized design variable array, m7 is the mass of the end effector, J xx x,J yy y,J zz z is the moment of inertia of the end effector.

[0254] The constraints mainly consider the range of variation of the optimization variables. For this study, it can be expressed as:

[0255]

[0256] Where x i and x s are the upper and lower bounds of the design variable x, which are 0.7 and 1.3 times the constant, respectively. The numerical ranges are shown in Table 3.

[0257] The solution of the dynamic optimization model of the spatial parallel mechanism with gaps requires the help of optimization algorithms. This patent uses the particle swarm algorithm with fast convergence speed, few parameters, simple and easy-to-implement algorithm to solve the problem. The main solution process is as follows: Figure 14 shown.

[0258] Step 4: Use the optimized data as the agent mechanism dynamics model and select fuzzy parameters to establish the dynamics model of the spatial parallel mechanism containing fuzzy parameters and perform solution analysis.

[0259] The spherical sub-gap value c is set as the fuzzy parameter and obeys the Gaussian fuzzy number.

[0260]

[0261] In the formula, 0.27mm is the upper boundary corresponding to the membership degree 0, 0.33mm is the lower boundary corresponding to the membership degree 0, and 0.3mm is the midpoint and the value corresponding to the membership degree 1.

[0262] The dynamic model of the spatial parallel mechanism with dry friction spherical pair clearance considering fuzzy parameters is:

[0263]

[0264] In the formula, the superscript ~ indicates that the mechanism parameters become fuzzy parameters.

[0265] The Kriging surrogate model method is used to solve the above equation as follows:

[0266] H(n)=f(n)β a +z(n) (34)

[0267] Where H(n) is the predicted value, f(n) is the basis function of variable n, and β a is the regression coefficient, and z(n) is a random process that obeys the normal distribution.

[0268] cov[z(n i ),z(k j )]=σ a 2 R(n i ,k j )i,j=1,2....m (35)

[0269] Where n i ,k j is the sample point, m represents the number of selected points, σ a 2 is the process variance, R(n i ,k j ) is the correlation coefficient between any two sample points, and the correlation coefficient is selected as the Gaussian coefficient.

[0270] Derived from the sample points and function values b and σ 2 The values are:

[0271]

[0272] When an unknown point x is input, the function response value can be inferred:

[0273]

[0274] Where r is the correlation vector between the sample point and the point to be measured.

[0275] In the fuzzy parameters Within the range of values, 20 groups of data are randomly selected and recorded as n 1....20 Substituting into formula (32) we can get 20 sets of mechanism dynamic response data, which can be substituted into formula (37) to complete the solution of the dynamic model of the spatial parallel mechanism with dry friction spherical pair clearance considering fuzzy parameters using the dace toolkit. The overall flow chart is as follows: Figure 15 shown.

[0276] Figure 5 、 Figure 6 and Figure 7 , respectively, shows the comparison of x-direction displacement, velocity, and acceleration before and after optimization of the 3-PSR mechanism. The results show that the displacement deviation of the optimized mechanism is significantly smaller than that before optimization, and the acceleration fluctuation is also significantly slowed down. This indicates that the optimization reduces the displacement deviation of the mechanism, improves the drilling accuracy of the punching robot, and reduces the collision force, which helps the punching robot to work stably for a long time. Figure 8、 Figure 9 、 Figure 10 、 Figure 11 、 Figure 12 、 Figure 13 The prediction intervals of the mechanism's dynamic response under fuzzy parameters are shown. The results show that as the parameters become fuzzy, the mechanism's dynamic response changes within a certain range. The prediction intervals of the dynamic response are obtained using the kriging surrogate model. This plays an important role in better understanding the dynamic characteristics of the mechanism and improving motion accuracy in practical applications.

[0277] The above results are simulated, and the structural parameters and dynamic simulation parameters of the 3-PSR mechanism, the dynamic simulation parameters of the 3-PSR mechanism, and the value ranges of the constraint conditions are shown in Table 1, Table 2, and Table 3, respectively.

[0278] Table 1 3-PSR mechanism structural parameters and dynamic simulation parameters

[0279]

[0280] Table 2 Dynamic simulation parameters of 3-PSR mechanism

[0281]

[0282] Table 3 Constraint value range

[0283]

[0284] Specifically, the ode45 function is used in MATLAB to simulate the mechanism dynamics model (27), and the simulation parameters are as follows:

[0285]

[0286] The above simulations demonstrate that model optimization can better predict the dynamic response of parallel mechanisms, improving the accuracy of prediction results. This method can mitigate the adverse effects of dry friction clearance on the dynamic characteristics of the mechanism, significantly improving the operating accuracy of parallel mechanisms such as drilling robots. It also provides a new analytical method for predicting the impact of fuzzy parameters on the dynamic characteristics of the mechanism, improving the accuracy of dynamic analysis predictions and better predicting the lifespan of parallel mechanisms.

[0287] The embodiments described above are merely descriptions of preferred implementations of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should fall within the scope of protection determined by the claims of the present invention.

Claims

1. A fuzzy uncertainty analysis method for dynamic optimization of a spatial parallel mechanism with gaps, characterized by: It includes the following steps: S1. Establish a spherical pair model with dry friction gap; S2. Establish the constraint conditions of the spatial parallel machine tool with spherical auxiliary friction gap, and obtain the dynamic model of the three-degree-of-freedom spatial parallel machine tool with spherical auxiliary friction gap as follows: Where, α b and β b Indicates a correction factor greater than 0, Q c represents the system generalized force vector of the parallel mechanism, Φ c Represents the constraint equation of the mechanism containing spherical pair clearance, represents the time derivative of the constraint equation of the mechanism containing the spherical pair clearance; Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, M is the mass matrix, λ is the Lagrange multiplier, and γ is the right side of the acceleration constraint equation; S3. Determine the optimization variables and the constraints of the optimization variables, obtain the optimal optimization variables, and optimize the dynamic model of the spatial parallel mechanism. Specifically, the steps include the following: S31. Take the displacement error of the moving platform as the optimization objective function: f(x)=||w A -w B || Where w A represents the displacement response of the dynamic platform under the ideal state without gap, w B =q a It represents the displacement response of the moving platform with clearance in the x direction, which is obtained from the dynamic model of the mechanism; S32. Determine the optimization variable x, select the mass and moment of inertia of the end effector of the mechanism as the optimization variables, and the array of the optimization variable x is expressed as: x=[m d ,J xx x,J yy y,J zz z] T In the formula, x is the optimization variable, m d is the mass of the end effector, J xx x,J yy y,J zz z is the moment of inertia of the end effector; S33. Construct the constraints for the optimization variable x: x i =[(m d ) min ,(J xx x) min ,(J yy y) min ,(J zz z) min ] T x s =[(m d ) max ,(J xx x) max ,(J yy y) max ,(J zz z) max ] T x i ≤x≤x s Where x i and x s are the upper and lower bounds of the optimization variable x respectively; S34, using a particle swarm optimization algorithm to solve the established dynamic model of the space parallel mechanism with gaps to obtain the optimal optimization variable x; S4. Select fuzzy parameters, establish a dynamic model of the spatial parallel mechanism containing fuzzy parameters, and solve it, which specifically includes the following sub-steps: S41, the spherical surface clearance value c is set as a fuzzy parameter and obeys the Gaussian fuzzy number, which is specifically expressed as follows: c∈(b,d) is in mm Where b is the upper boundary corresponding to membership 0, d is the lower boundary corresponding to membership 0, and c is the midpoint and the value corresponding to membership 1; S42. Construct the dynamic model of the spatial parallel mechanism with fuzzy parameters and dry friction spherical pair clearance as follows: In the formula, the superscript ~ indicates that the mechanism parameters become fuzzy parameters; S43, using the Kriging surrogate model method to solve the dynamic model obtained in step S42 with the optimization variables, and determining the solution formula as follows: H(n)=f(n)β a +z(n) Where H(n) is the predicted value, f(n) is the basis function of variable n, and β a is the regression coefficient, z(n) is a random process that obeys the normal distribution; n is the number of different gap values; S44. Determine the dynamic response value of the known sample point based on the dynamic response value of the known sample point using the solution formula determined in step S43: Where r is the correlation vector between the sample point and the point to be measured; F is the basis function f(n), Y is the dynamic response value of the known sample point, and R is the correlation matrix between the dynamic response values of the known sample point and the unknown sample point; is the predicted kinetic response value interval.

2. The fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism with a gap according to claim 1 is characterized by: Step S1 specifically includes the following steps: S11. Construct the modulus of the gap eccentricity vector as follows: e ij =P Zj -P Zi Among them, P Zi is the position vector of the center point of the spherical shell in the local coordinate system, P Zj is the position vector of the center point of the sphere in the global coordinate system; S12. Construct a collision depth model at the contact point: δ=e ij -c Where, e ij is the modulus of the gap eccentricity vector, and c is the spherical pair clearance value; S13. By judging the collision depth δ, the contact collision state between the sphere and the spherical shell is determined. The judgment formula is as follows: 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.

3. The fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism with a gap according to claim 1 is characterized by: In step S13, the relative normal velocity V at the potential contact point n and the relative tangential velocity V t It is expressed as follows: Where V Dj , V Cj are the positions of the contact points on the sphere and spherical shell in the global coordinate system, n n is the normal unit vector at the gap contact point; The contact force of the sphere on the shell is expressed as: F i =F N ·n n +F T ·t t Where, F N is the normal contact force, F T is the tangential contact force, t t is the tangential unit vector of the contact point; The reaction force of the spherical shell on the sphere is expressed as: F j =-F i 。 4. The fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism with a gap according to claim 1, characterized in that: Step S2 specifically includes the following sub-steps: S21. The constraint equation for constructing a spatial parallel mechanism with a spherical auxiliary dry friction gap is: F c (q)=[Φ1 Φ2 ....... Φ n ] T =0 Where, Φ 1...n is the motion constraint; After that, we can get the acceleration constraint equation by taking the derivative twice with respect to time: Where, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, and γ represents the right side of the constraint equation; S22. The dynamic equation of the spatial parallel mechanism with spherical clearance and Lagrange multipliers is established as follows: Where Q c represents the system generalized force vector of the parallel mechanism, Φ cq The Jacobian matrix representing the constraint equation of the mechanism with spherical pair clearance is: represents the second derivative of the generalized coordinate q of the mechanism with respect to time, M is the mass matrix, and λ is the Lagrange multiplier; M=diag[m m m J xx J yy J zz ] Where m is the mass of the rod and J is the moment of inertia; S23. Establish a dynamic model of the spatial parallel mechanism with a spherical pair containing clearances.

5. The fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism with a gap according to claim 1 is characterized by: In step S44, the DACE toolkit and the OD45 solver in MATLAB are used to solve the problem.

6. The fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism with a gap according to claim 5, characterized in that: The dynamic response values of the known sample points in step S44 are the dynamic response values of the mechanism under multiple sets of fuzzy parameters.

7. The fuzzy uncertainty analysis method for dynamic optimization of a parallel mechanism with a gap according to claim 1 is characterized by: Step S34 specifically includes the following sub-steps: S341, initializing the particle swarm algorithm, generating initial particle swarm sample points based on the optimization variables and the objective function; S342, solving the dynamic model to calculate the eccentricity vector, normal velocity, tangential velocity, and contact point position; S343, obtaining the displacement, velocity, and acceleration response data of the moving platform to calculate the constraint reaction force and fitness function at the corresponding gap node; S344, updating particle speed and position, searching for the global optimal particle; S345. Iterate the particles until the objective function is met and the optimal optimization variables are obtained.

Images