Fuzzy uncertainty analysis method for dynamics optimization of parallel mechanism containing gap space

Abstract

The invention provides a fuzzy uncertainty analysis method for dynamics optimization of a parallel mechanism containing a gap space, and belongs to the technical field of industrial robot modeling and optimization, and the method comprises the following steps: S1, building a spherical pair model containing a dry friction gap; and S2, according to constraint conditions of the spatial parallel mechanism containing the spherical pair dry friction gap, obtaining a dynamic model of the three-degree-of-freedom spatial parallel mechanism containing the spherical pair dry friction gap. S3, performing optimization according to the dynamic model of the spatial parallel mechanism; and S4, agenting the optimized data into the dynamic model of the mechanism, selecting fuzzy parameters, establishing a dynamic model of the spatial parallel mechanism containing the fuzzy parameters, and carrying out solution analysis. According to the method, the adverse effect of the dry friction gap on the dynamic characteristics of the mechanism can be improved, and the method is provided for predicting the influence of the fuzzy parameters of the mechanism on the dynamic characteristics of the mechanism.

Description

Technical Field

[0001] The 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 space parallel mechanism containing a gap. Background Art

[0002] The parallel mechanism can be used as the main mechanism of parallel machine tools and parallel robots due to its advantages of high rigidity, high precision and good stability. In parallel machine tools, due to factors such as part errors caused by insufficient equipment processing accuracy and assembly errors that meet the normal operation of parts, there are gaps in the kinematic pairs of the main mechanism of parallel machine tools. The existence of dry friction gaps leads to frequent contact and collision between the elements of the kinematic pairs, destroying the surface texture structure of the kinematic pair elements, greatly affecting the connecting rod size error and the positioning accuracy of the moving platform, thereby causing changes in the positioning accuracy of the tool tip of the parallel machine tool and affecting the quality of the workpiece.

[0003] As the parallel machine tool works for a long time, the collision between the moving elements in the main mechanism will intensify, which may cause material loss and damage. Parameters such as the gap value may change. The change of parameters makes the dynamic response of the main mechanism uncertain, which will directly affect the positioning accuracy of the parallel machine tool and the quality of the processed 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 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 solve 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, which 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, and provide 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 space parallel mechanism containing a gap, 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 dry friction gap, and obtain the dynamic model of the three-degree-of-freedom spatial parallel machine tool with spherical auxiliary dry friction gap as follows:

[0009]

[0010] In the formula, 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 with 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: It 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, determining the optimization variables and the constraints of the optimization variables, obtaining the optimal optimization variables, and optimizing the dynamic model of the spatial parallel mechanism, specifically including the following sub-steps:

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

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

[0014] In the formula, w A represents the displacement response of the moving platform under ideal clearance-free conditions, 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 of 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] In the formula, 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, selecting fuzzy parameters, establishing a dynamic model of the spatial parallel mechanism containing fuzzy parameters and solving it, specifically including 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] In the formula, b is the upper boundary corresponding to the membership degree 0, d is the lower boundary corresponding to the membership degree 0, and c is the midpoint and the value corresponding to the membership degree 1;

[0029] S42. The dynamic model of the spatial parallel mechanism with dry friction spherical surface clearance containing fuzzy parameters is constructed as follows:

[0030]

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

[0032] S43, using the Kriging proxy model method to solve the kinetic 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 follows a normal distribution; n is the number of different gap values;

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

[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. The modulus of the gap eccentricity vector is constructed 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 as follows:

[0043] δ=e ij -c

[0044] In the formula, e ij is the modulus of the clearance eccentricity vector, c is the clearance value of the spherical pair;

[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] In the formula, F N is the normal contact force, F T is the tangential contact force, t t is the tangent 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 secondary dry friction gap is:

[0058]

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

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

[0061]

[0062] In the formula, 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 with Lagrange multipliers containing spherical pair clearance of the spatial parallel machine tool is established as follows:

[0064]

[0065] In the formula, 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: It 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 clearance.

[0069] Preferably, in step S44, the solution is obtained by using the dace toolkit and ode45 solver in matlab.

[0070] Preferably, the dynamic response values ​​of the known sample points in step S44 are dynamic response values ​​of the mechanism under multiple groups 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 variable is 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 dynamic optimization of a space parallel mechanism containing a gap. 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 effects of 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, preventing the mechanism accuracy from decreasing beyond the working range of the parallel mechanism, and by predicting the life of the kinematic pair, timely replacing the kinematic pair 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 A schematic diagram of a 3-PSR spatial parallel mechanism with spherical pair clearance provided in 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 A schematic diagram of a simulation of X-direction displacement optimization of a parallel mechanism provided by an embodiment of the present invention;

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

[0087] Figure 7 A schematic diagram of the acceleration optimization simulation in the X direction 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 in an embodiment of the present invention;

[0089] Fig. 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] Fig.10 A schematic diagram of acceleration simulation in the X direction of a parallel mechanism with fuzzy parameters provided in an embodiment of the present invention;

[0091] Fig.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] Fig.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] Fig.13 A schematic diagram of the acceleration optimization simulation in the X direction of a parallel mechanism with fuzzy parameters provided in an embodiment of the present invention;

[0094] Fig.14 Optimization schematic diagram of a 3-DOF parallel robot provided in an embodiment of the present invention;

[0095] Fig.15 A flow chart of a 3-PSR parallel robot provided in an embodiment of the present invention. DETAILED DESCRIPTION

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

[0097] The present invention provides a fuzzy uncertainty analysis method for dynamic optimization of a space parallel mechanism with a gap, 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. The modulus of the gap eccentricity vector is constructed 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 center point of the sphere in the global coordinate system.

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

[0103] δ=e ij -c

[0104] In the formula, e ij is the modulus of the clearance 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] In the formula, F N is the normal contact force, F T is the tangential contact force, t t is the tangent 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 spherical auxiliary dry friction gap, and obtaining the dynamic model of the three-degree-of-freedom spatial parallel mechanism with 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 secondary dry friction gap is:

[0118] Φ c (q)=[Φ1Φ2.......Φ n ] T =0

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

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

[0121]

[0122] In the formula, 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 with Lagrange multipliers of the spatial parallel mechanism containing spherical pair clearance is established as follows:

[0124]

[0125] In the formula, 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: It 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 spherical pair with clearance of the spatial parallel mechanism is established as follows:

[0129]

[0130] In the formula, 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 with 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: It 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, determining the optimization variables and the constraints of the optimization variables, obtaining the optimal optimization variables, and optimizing the dynamic model of the spatial parallel mechanism, specifically including the following sub-steps:

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

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

[0134] In the formula, w A represents the displacement response of the moving platform under ideal clearance-free conditions, 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 of 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] In the formula, 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 space parallel mechanism with gaps 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, solving the dynamic model, calculating the eccentricity vector, normal velocity, tangential velocity and contact point position.

[0147] 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 nodes.

[0148] S344, update particle speed and position, and search for the global optimal particle.

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

[0150] S4, selecting fuzzy parameters, establishing a dynamic model of the spatial parallel mechanism containing fuzzy parameters and solving it, specifically including 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] In the formula, b is the upper boundary corresponding to the membership degree 0, d is the lower boundary corresponding to the membership degree 0, and c is the midpoint and the value corresponding to the membership degree 1.

[0154] S42. The dynamic model of the spatial parallel mechanism with dry friction spherical surface clearance containing fuzzy parameters is constructed as follows:

[0155]

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

[0157] S43, using the Kriging proxy model method to solve the kinetic 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 a 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 interval. In step S44, the DACE toolkit and ode45 solver in MATLAB are used to solve. The dynamic response value of the known sample point is the dynamic response value of the mechanism under multiple groups 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 motion 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 main mechanism of the drilling robot uncertain, which will directly affect the accuracy of the drilling robot, making it impossible to guarantee the accuracy and quality of the hole. This embodiment can predict and analyze the life of the drilling robot by dynamically optimizing 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 3-PSR parallel mechanism of the punching robot 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 rotating pair. The three moving pairs are defined as P1, P2 and P3, the three spherical pairs are defined as S1, S2 and S3 respectively, and the three rotating pairs are defined as R1, R2 and R3 respectively. P1P2 is perpendicular to the plane where the line connecting the spherical pair and the moving pair is always perpendicular to the plane where P1P2P3 is located. The rotation axes of the rotating pairs R1 and R2 are parallel to each other and perpendicular to the rotation axis of R3 at the same time. The plane formed by the three rotating pairs is parallel to the plane formed by the three moving pairs. The rotation axes of the rotating 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 direction meeting 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 moving pair Pi and point P is hi, the fixed length between point P and the coordinate origin OA is a1, the length between the moving 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 chain, and L12 represents the second link of the first branch chain.

[0169] The dynamic analysis and optimization methods of the three-DOF spatial parallel mechanism with spherical auxiliary friction gap 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 according to the dynamic model of the spatial parallel mechanism.

[0173] S4. Use the optimized data as the agent mechanism dynamics model and select fuzzy parameters, establish the dynamics model of the spatial parallel mechanism with 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, respectively, 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 tangent 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] In the formula, 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 magnitude of the eccentricity vector at the gap is:

[0186]

[0187] In the formula, 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 spherical shell and the sphere:

[0193] c=Rr (8).

[0194] When the sphere and the spherical 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 identified:

[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] In the formula, 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:

[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 index, 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] In the formula, 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 spherical 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 secondary 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] In the formula, Φ P is the moving joint constraint, Φ R is the rotational joint constraint, Φ D To drive constraints.

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

[0224]

[0225] In the formula, 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] In the formula, 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] In the formula, 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 with Lagrange multipliers of the 3-PSR spatial parallel mechanism with spherical clearance is established as:

[0232]

[0233] In the process of solving the dynamics of the mechanism with clearance, the contact force constraint in the clearance replaces the position constraint of the ideal spherical pair. 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, 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 respectively represent the forces exerted by the sphere on the spherical shell and the forces exerted by the spherical shell on the sphere, which are the action force and reaction force.

[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] In the formula, and They represent the contact force F 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 respectively 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.

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

[0241]

[0242] In the formula, 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 according to the dynamic model of the spatial parallel mechanism. The design method is to integrate the dynamic model of the spatial parallel mechanism with the optimization algorithm and optimize the parameters of the key components in the mechanism. A multi-objective optimization strategy is used to design the optimization objective function, design variables and constraints respectively.

[0246] Since there is a gap at the joint 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] In the formula, w x represents the displacement response of the moving platform under ideal clearance-free conditions, 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] As an actuator, the end effector is often designed according to specific working conditions. The mass distribution method does not change the topological configuration, structural type and material of the parallel mechanism. 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 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 optimization 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] In the formula, 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 range is shown in Table 3.

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

[0258] Step 4, the optimized data is used as the dynamic model of the agent mechanism and fuzzy parameters are selected to establish the dynamic model of the spatial parallel mechanism containing fuzzy parameters and perform solution analysis.

[0259] The spherical pair clearance 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 tested.

[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) can obtain 20 sets of mechanism dynamic response data, which can be substituted into formula (37), that is, the dace toolkit completes the solution of the spatial parallel mechanism dynamic model with dry friction spherical pair clearance considering fuzzy parameters. The overall flow chart is as follows Fig.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 is conducive to the long-term stable operation of the punching robot. Figure 8, Fig. 9 , Fig.10 , Fig.11 , Fig.12 , Fig.13 The prediction interval of the dynamic response of the mechanism under fuzzy parameters is displayed. The results show that as the parameters become fuzzy parameters, the dynamic response of the mechanism will change within a certain range. The prediction interval of the dynamic response is obtained through the kriging proxy model. It plays an important role in better grasping the changes in the dynamic characteristics of the mechanism and improving the motion accuracy in actual work.

[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 3-PSR mechanism dynamics simulation parameters

[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] It can be seen from the above simulation that the dynamic response of the parallel mechanism can be better predicted by optimizing the model, and the accuracy of the prediction results can be improved. It can be seen that this method can improve the adverse effects of dry friction gap on the dynamic characteristics of the mechanism, and greatly improve the working accuracy of parallel mechanisms such as punching robots. It also provides a new analysis method for predicting the influence of mechanism fuzzy parameters on the dynamic characteristics of the mechanism, improves the accuracy of dynamic analysis prediction, and better predicts the life of the parallel mechanism.

[0287] The embodiments described above are only descriptions of the preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design 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 all fall within the protection scope 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 dry friction gap, and obtain the dynamic model of the three-degree-of-freedom spatial parallel machine tool with spherical auxiliary dry friction gap as follows: In the formula, 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 with 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: It 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, determining the optimization variables and the constraints of the optimization variables, obtaining the optimal optimization variables, and optimizing the dynamic model of the spatial parallel mechanism, specifically including the following sub-steps: S31. Take the displacement error of the moving platform as the optimization objective function: f(x)=‖w A -w B ‖ In the formula, w A represents the displacement response of the moving platform under ideal clearance-free conditions, w B =q a It represents the displacement response of the moving platform with clearance in the x direction, which is obtained by 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 of 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 In the formula, 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, selecting fuzzy parameters, establishing a dynamic model of the spatial parallel mechanism containing fuzzy parameters and solving it, specifically including 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: In the formula, b is the upper boundary corresponding to the membership degree 0, d is the lower boundary corresponding to the membership degree 0, and c is the midpoint and the value corresponding to the membership degree 1; S42. The dynamic model of the spatial parallel mechanism with dry friction spherical surface clearance containing fuzzy parameters is constructed as follows: In the formula, the superscript ~ indicates that the mechanism parameters become fuzzy parameters; S43, using the Kriging proxy model method to solve the kinetic 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 follows a normal distribution; n is the number of different gap values; S44, using the solution formula determined in step S43, the dynamic response value of the known sample point is determined based on the dynamic response value of the known sample point: 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 spatial parallel mechanism with gaps according to claim 1 is characterized in that: Step S1 specifically includes the following steps: S11. The modulus of the gap eccentricity vector is constructed 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 as follows: δ=e ij -c In the formula, e ij is the modulus of the clearance eccentricity vector, c is the clearance value of the spherical pair; 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 spatial parallel mechanism with gaps 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 In the formula, F N is the normal contact force, F T is the tangential contact force, t t is the tangent 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 spatial parallel mechanism with gaps according to claim 1 is characterized in that: Step S2 specifically includes the following sub-steps: S21. The constraint equation for constructing a spatial parallel mechanism with a spherical secondary dry friction gap is: F c (q)=[Φ1Φ2.......Φ n ] T =0 In the formula, Φ 1...n is the motion constraint; After that, we take the derivative twice with respect to time to get the acceleration constraint equation: In the formula, 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 with Lagrange multipliers of the spatial parallel mechanism containing spherical pair clearance is established as follows: In the formula, 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: It 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 spatial parallel mechanism with gaps according to claim 1 is characterized in that: In step S44, the solution is obtained by using the dace toolkit and ode45 solver in matlab.

6. The fuzzy uncertainty analysis method for dynamic optimization of a spatial parallel mechanism with gaps 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 groups of fuzzy parameters.

7. The fuzzy uncertainty analysis method for dynamic optimization of a spatial parallel mechanism with gaps according to claim 1, characterized in that: 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 variable is obtained.

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