Dynamics sliding mode control method for multi-link mechanism with lubrication clearance and rigid-flexible effect

Abstract

The application discloses a kind of dynamics sliding mode control methods of multi-link mechanism containing lubrication gap and rigid-flexible effect, it is related to industrial robot modeling and control technical field, comprising: S1, the model of rotating pair containing lubrication gap, contact force model, rigid-flexible coupling model;S2, according to the constraint condition of rotating pair plane multi-link mechanism, obtain dynamics model;S3, according to dynamics model, establish error compensation control model;S4, according to error compensation control model, build simulation model, compare and analyze the parameters of displacement before and after control, and optimize error compensation control model established in step S3.The application can correct the motion trajectory of actuator in real time by establishing error compensation control model according to the dynamics model of plane multi-link mechanism, compensate the trajectory deviation of multi-link mechanism containing lubrication gap pair, and can improve the adverse effects of lubrication gap on the dynamics characteristics of rigid-flexible coupling mechanism.

Description

Technical Field

[0001] This invention relates to the field of industrial robot modeling and control technology, specifically to a sliding mode control method for the dynamics of multi-link mechanisms containing lubrication clearance and rigidity-flexibility effects. Background Technology

[0002] Multi-link mechanisms, as core main transmission components of critical mechanical equipment such as punch presses and presses, directly determine the forming quality of stamped parts and the operational stability of the equipment through the displacement accuracy of their end effectors. This is especially true in high-end manufacturing fields such as precision stamping and automotive body panel processing, where even micron-level trajectory deviations can lead to product scrap or mold damage. However, in actual operating conditions, the presence of clearances in the moving parts of the mechanism becomes a key bottleneck restricting accuracy. Since the fit clearance between bearings and journals is unavoidable, continuous collisions and friction occur at these clearances under cyclic loads. This nonlinear impact not only generates high-frequency vibration noise but also leads to microscopic wear on the contact surfaces, creating a vicious cycle of "clearance widening—collision intensification." Although grease lubrication or solid lubricants can temporarily alleviate friction and wear, under high-speed, heavy-load conditions, the lubricating oil film is easily damaged, and the lubrication effect decays over time, failing to fundamentally eliminate dynamic response errors. During high-speed operation, the elastic deformation of flexible links further affects the system's motion accuracy and stability.

[0003] Currently, the main methods for suppressing the adverse effects of clearance in moving parts include three aspects: first, optimizing the design level by reducing fit tolerances and adopting self-compensating clearance structures to reduce initial clearance; second, physical lubrication level by selecting high-temperature and high-pressure grease or oil-air lubrication systems to extend the effective lubrication cycle; and third, control strategy level by correcting trajectory deviations through feedforward compensation or PID feedback control.

[0004] However, the above methods are mostly passive suppression or local optimization: the optimization design is limited by the processing cost and assembly process, physical lubrication cannot adapt to extreme working conditions, and traditional control strategies are difficult to deal with the dynamic coupling of gap nonlinearity and component elastic deformation in real time. Summary of the Invention

[0005] To address the shortcomings of the prior art, the present invention aims to provide a sliding mode control method for the dynamics of multi-link mechanisms with lubrication gaps and rigid-flexible effects. By establishing an error compensation control model, the motion trajectory of the actuator can be corrected in real time, thereby improving the adverse effects of lubrication gaps on the dynamic characteristics of rigid-flexible coupling mechanisms and solving the problem of limited optimization design of multi-link mechanisms with lubrication gaps and rigid-flexible effects in the prior art.

[0006] Specifically, the present invention provides a dynamic sliding mode control method for a multi-link mechanism containing lubrication clearance and rigidity-flexibility effect, comprising the following steps:

[0007] S1. Establish the rotating pair model, contact force model, and rigid-flexible coupling model with lubrication gap;

[0008] S2. Based on the constraint conditions of the planar multi-link mechanism with a revolute joint containing a lubrication clearance, the dynamic model of the two-degree-of-freedom planar multi-link mechanism with a revolute joint containing a lubrication clearance is obtained.

[0009] S3. Establish an error compensation control model based on the dynamic model of the planar multi-link mechanism in step S2; including the following sub-steps:

[0010] S31. The displacement error of the slider of the gap mechanism is used as the main input signal of the sliding mode controller:

[0011] e(t) = x(t) - x e (t)

[0012] In the formula, e(t) represents the displacement error; x(t) represents the slider displacement of the system with gaps; x e (t) represents the slider displacement without gap;

[0013] S32. Sliding surface design of the sliding mode controller, the sliding surface function s(t) is as follows:

[0014]

[0015] In the formula, c is a positive real number; This is the speed error amount;

[0016] S33, the design control law is...

[0017]

[0018] In the formula, k and b are the coefficients of the system model; the equivalent control part of the control law is... The switching control part is ks(t); sign(s(t)) is a sign function that provides a high-frequency switching mechanism to ensure that the system state reaches the sliding surface within a finite time.

[0019] S34. Let a Lyapunov function be...

[0020]

[0021] In the formula, V(S)≥0, and V(S)=0 if and only if s(t)=0, which satisfies positive definiteness;

[0022] S35. Using the total cumulative error of the slider displacement as the objective function, the particle swarm optimization algorithm is used to search for the optimal solution of the three control parameters that satisfy the minimum value of the objective function within the search interval. The expression of the objective function is as follows:

[0023]

[0024] S4. Based on the error compensation control model established in step S3, build a simulation model of the sliding mode control system, compare and analyze the parameters of the displacement before and after control, and optimize the error compensation control model established in step S3.

[0025] Further: Step S1 includes the following sub-steps:

[0026] S11. Construct a model of a revolute pair with clearance;

[0027] S12. Construct a collision depth model at the contact point;

[0028] S13. The relative motion state between the bearing and the shaft in a rotating pair is reflected by the change in the collision depth, specifically as follows:

[0029]

[0030] S14. Establish the collision force model under dry friction at the clearance of the rotating pair and the oil film bearing capacity model under lubrication.

[0031] Furthermore, step S1 also includes the following sub-steps:

[0032] S15. The elastic force of the flexible member in the rigid-flexible coupling mechanism is obtained as follows:

[0033] F f =F t +F l ;

[0034] The expression for the bending elastic force of a flexible beam element is as follows:

[0035]

[0036] In the formula, S″ is the second derivative of the shape function S with respect to ξ, E is the Young's modulus of the beam element, and I... f The moment of inertia of the cross section;

[0037] The expression for the axial elastic force of a flexible beam element is as follows:

[0038]

[0039] In the formula, A is the cross-sectional area of ​​the beam element, where ε f Let S be the strain tensor of the beam element, expressed as:

[0040]

[0041] In the formula, S f =S′ T S′, where S′ is the first derivative of the shape function S with respect to ξ.

[0042] Further: Step S14 includes the following sub-steps:

[0043] S141. Construct the normal collision force model and the tangential collision force model under dry friction conditions;

[0044] S142. Construct an oil film load-bearing capacity model under lubrication conditions:

[0045] radial velocity At that time, the normal oil film bearing capacity F LN Tangential oil film bearing capacity F LT They are respectively

[0046]

[0047] In the formula, k0 is the parameter in the boundary conditions of the Reynolds equation; L is the bearing length; μ is the dynamic viscosity of the lubricant; ω is the angular velocity of the shaft relative to the bearing;

[0048] radial velocity At that time, the normal and tangential oil film bearing capacities are respectively

[0049]

[0050] Furthermore: the dynamic model in step S2 is

[0051]

[0052] In the formula, M fr Φ is the mass matrix of the rigid-flexible coupled system; q The Jacobian matrix of the constraint equations. λ is the generalized acceleration vector; λ is the Lagrange multiplier; Q fr For a rigid-flexible coupled system, F represents the generalized external force. f For the elastic force of flexible rods; The time derivative of the displacement constraint equation is... α and β are the correction parameters for Baumgarte's default stability algorithm.

[0053] Further: Step S2 includes the following sub-steps:

[0054] S21. For a rigid-flexible coupling mechanism considering the lubrication gaps at rotating joints A and B, the constraint equations of the system are obtained.

[0055] S22. Based on the Lagrange multiplier method, the dynamic equations of rigid-flexible coupling are obtained as follows:

[0056]

[0057] M fr =diag(m i,m i J i );

[0058] In the formula, M fr Q is the mass matrix of the rigid-flexible coupled system; fr λ represents the generalized external force of the rigid-flexible coupling system; mi and Ji represent the mass and moment of inertia of component i, respectively; λ is the Lagrange multiplier; g is the generalized force of the system, including the inertial force of the system, the interaction force between kinematic elements generated by the gap kinematic pair, and the average mixed transition force composed of the oil film bearing capacity.

[0059] S23. Derive the dynamic model of the rigid-flexible coupling mechanism with lubrication gap.

[0060] Furthermore: the constraint equations of the system in step S21 are:

[0061]

[0062] In the formula, Φ(q,t) is the displacement constraint equation;

[0063] Then, by taking the second derivative of the displacement constraint equation with respect to time, the acceleration constraint equation is obtained as follows:

[0064]

[0065] In the formula, Φ qt Φ is the partial derivative of the Jacobian matrix with respect to time. tt This represents the second-order partial derivative of the displacement constraint equation with respect to time. γ is the generalized acceleration vector; γ is the right-hand side of the acceleration constraint equation.

[0066] Further: Step S33 includes the following sub-steps:

[0067] S331. Design Approach Law: The form of the exponential approach law is as follows:

[0068]

[0069] Further: Step S35 includes the following sub-steps:

[0070] S351. Initialize the particle swarm algorithm by generating initial particle swarm sample points based on optimization variables and objective function;

[0071] S352. Solve the dynamic model to calculate the eccentricity vector, normal velocity, tangential velocity, and contact point position;

[0072] S353, obtain the displacement, velocity, acceleration response data of the slider and the fitness function;

[0073] S354. Update particle velocity and position, and search for the globally optimal particle;

[0074] S355. Iterate over the particles until they meet the objective function to obtain the optimal optimization variables.

[0075] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0076] (1) This invention provides a sliding mode control method for the dynamics of a multi-link mechanism with lubrication gap and rigid-flexible effect. This method establishes an error compensation control model based on the dynamic model of the planar multi-link mechanism, which can correct the motion trajectory of the actuator in real time, compensate for the trajectory deviation of the multi-link mechanism with lubrication gap, improve the adverse effect of lubrication gap on the dynamic characteristics of rigid-flexible coupling mechanism, correct the effect of the motion trajectory of the actuator on the dynamic characteristics of the mechanism in real time, and eliminate the hidden dangers of collision and friction caused by the gap of the mechanism's motion pair.

[0077] (2) This invention optimizes the key parameters in the sliding mode controller by using the particle swarm optimization algorithm, finds the optimal parameters under the objective function, improves the accuracy of the dynamic control model of the rigid-flexible coupling multi-link mechanism with lubrication gap rotating pairs, optimizes the dynamic performance of the dynamic control system, makes the balance approach stability quickly, ensures the stable operation of the multi-link mechanism, and achieves precise control of the motion trajectory of the rigid-flexible coupling multi-link mechanism with lubrication gap rotating pairs.

[0078] (3) The present invention builds a sliding mode control system simulation model by establishing an error compensation control model, designs and builds a seven-bar linkage test platform considering the clearance of the rotating joint. The test device and its scheme are strictly constructed based on the contact collision theory and mechanism dynamics principle to ensure that the parameters are highly consistent with the previously established theoretical model. The accuracy of the sliding mode control system simulation model is verified by experiments, providing a theoretical basis for the dynamic sliding mode control method. Attached Figure Description

[0079] Figure 1 This is a schematic diagram of the overall process of the present invention;

[0080] Figure 2 The rotating pair clearance model provided by this invention;

[0081] Figure 3 A diagram illustrating the modeling method for a rigid beam mechanism provided by this invention;

[0082] Figure 4 A diagram illustrating the modeling method for the flexible beam model of the mechanism provided by this invention;

[0083] Figure 5 A schematic diagram of a seven-bar rigid-flexible coupling mechanism with lubrication gap provided by the present invention;

[0084] Figure 6 A planar schematic diagram of the clearance shaft at points A and B of the rotating joint used in the test platform provided by the present invention;

[0085] Figure 7 A plan view of the clearance shaft of the clearanceless rotating pair used in the test platform provided by the present invention;

[0086] Figure 8 A comparison diagram of experimental and theoretical results of the seven-bar linkage with lubrication clearance provided by the present invention;

[0087] Figure 9 Simulink diagram of the rigid-flexible coupling dynamic sliding mode control model of the mechanism with lubrication gap provided by the present invention;

[0088] Figure 10 Comparison diagram of slider displacement of rigid-flexible coupling mechanism with lubrication gap provided by the present invention;

[0089] Figure 11 Comparison diagram of slider displacement error of rigid-flexible coupling mechanism with lubrication gap provided by the present invention;

[0090] Figure 12 This invention provides information on the Simulink simulation model solver.

[0091] Figure 13 The flowchart of the particle swarm optimization algorithm provided by this invention;

[0092] Figure 14 The flowchart shows the solution logic for the rigid-flexible coupling dynamic model of the seven-bar linkage with lubrication gap provided by this invention.

[0093] In the diagram, 1 is the bearing inner ring; 2 is the shaft; 3 is the lubricant; 4 is the contact position; 5 is the oil retainer ring mounting part; and 6 is the bearing mounting part. Detailed Implementation

[0094] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings.

[0095] This invention provides a dynamic sliding mode control method for a rigid-flexible coupled multi-link mechanism considering lubrication clearance rotating pairs, comprising the following steps:

[0096] S1. Establish the rotating pair model, contact force model, and rigid-flexible coupling model with lubrication gap;

[0097] Preferably, step S1 specifically includes the following sub-steps:

[0098] S11. Construct a model of a revolute joint with clearance. The eccentricity vector of the shaft's center of mass relative to the bearing's center of mass is:

[0099]

[0100] In the formula The centroids M of the bearing and shaft are respectively b M a The position vector in the global coordinate system.

[0101] S12. Construct a collision depth model for the contact point. When the bearing and shaft collide, the collision depth is...

[0102] δ ab =|e|-r;

[0103] In the formula, r represents the clearance between the bearing and the shaft, r = R b -R a R b and R a These are the radii of the bearing and the shaft, respectively.

[0104] S13. The relative motion state between the bearing and the shaft in a rotating pair can be reflected by the change in the collision depth. Specifically:

[0105]

[0106] S14. Modeling of collision force at the clearance of rotating pairs, including the collision force model under dry friction and the oil film bearing capacity model under lubrication.

[0107] Preferably, step S14 specifically includes the following sub-steps:

[0108] S141. The normal collision force model under dry friction conditions, its expression is:

[0109]

[0110] In the formula, F N is the normal impact force; K is the contact stiffness coefficient; n is the force exponent, the value of which depends on the material properties and is usually taken as 1.5; It is the relative penetration speed; c e The coefficient of recovery; The initial collision velocity.

[0111] The tangential collision force model under dry friction is expressed as follows:

[0112]

[0113] In the formula, c f c is the coefficient of friction; d For dynamic correction coefficients; v t This represents the relative sliding speed.

[0114] Based on the analysis of the normal collision force and tangential friction force above, the resultant contact force F at the clearance of the rotating joint can be obtained. D It can be represented as follows

[0115]

[0116] In the formula, and These are the normal unit vector and the tangential unit vector, respectively.

[0117] S142. Oil film load-bearing capacity model under lubrication conditions:

[0118] radial velocity At that time, the normal oil film bearing capacity F LN Tangential oil film bearing capacity F LT They are respectively

[0119]

[0120] In the formula, k0 is a parameter in the boundary conditions of the Reynolds equation; L is the bearing length; μ is the dynamic viscosity of the lubricant; and ω is the angular velocity of the shaft relative to the bearing.

[0121] radial velocity At that time, the normal and tangential oil film bearing capacities are respectively

[0122]

[0123] S15, the elastic force of the flexible member in the rigid-flexible coupling mechanism is

[0124] F f =F t +F l ;

[0125] The expression for the bending elastic force of a flexible beam element is as follows:

[0126]

[0127] In the formula, S″ is the second derivative of the shape function S with respect to ξ, and E is the Young's modulus of the beam element. f Let be the moment of inertia of the cross section.

[0128] The expression for the axial elastic force of a flexible beam element is as follows:

[0129]

[0130] In the formula, A is the cross-sectional area of ​​the beam element. Wherein, ε f Let S be the strain tensor of the beam element, expressed as:

[0131]

[0132] In the formula, S f =S′ T S′, where S′ is the first derivative of the shape function S with respect to ξ.

[0133] S2. Based on the constraint conditions of the planar multi-link mechanism with a revolute joint containing a lubrication clearance, the dynamic model of the two-degree-of-freedom planar multi-link mechanism with a revolute joint containing a lubrication clearance is obtained:

[0134]

[0135] In the formula, M fr The mass matrix of the rigid-flexible coupled system; Φ q The Jacobian matrix of the constraint equations. λ is the generalized acceleration vector; λ is the Lagrange multiplier; Q fr For a rigid-flexible coupled system, F represents the generalized external force. f For the elastic force of flexible rods; This is the time derivative of the displacement constraint equation. α and β are correction parameters for the Baumgarte default stability algorithm. When dealing with the constraint equations, having the same α and β means that the penalties for position and velocity violations are consistent, which can improve the stability of the numerical solution and allow the equations to converge quickly. α and β are usually taken as positive real numbers in the range [0, 50], and in this invention, they are all taken as 50.

[0136] Preferably, step S2 specifically includes the following sub-steps:

[0137] S21. For a rigid-flexible coupling mechanism considering the lubrication clearances at rotating joints A and B, the system's constraint equations are:

[0138]

[0139] In the formula, Φ(q,t) is the displacement constraint equation.

[0140] Then, by taking the second derivative of the displacement constraint equation with respect to time, the acceleration constraint equation is obtained as follows:

[0141]

[0142] In the formula, Φ qt Φ is the partial derivative of the Jacobian matrix with respect to time. tt This represents the second-order partial derivative of the displacement constraint equation with respect to time. γ is the generalized acceleration vector; γ is the right-hand side of the acceleration constraint equation.

[0143] S22. Based on the Lagrange multiplier method, the equations for rigid-flexible coupling dynamics can be obtained as follows:

[0144]

[0145] M fr =diag(m i ,m i J i );

[0146] In the formula, M fr Q is the mass matrix of the rigid-flexible coupled system; fr For a rigid-flexible coupled system, the generalized external force is m. i J i Let be the mass and moment of inertia of component i, respectively; λ be the Lagrange multiplier; g be the generalized force of the system, including the inertial force of the system, the interaction force between kinematic elements generated by the gap kinematic pair, and the average mixed transition force composed of the oil film bearing capacity.

[0147] S23. Derive the dynamic model of the rigid-flexible coupling mechanism with lubrication gap.

[0148] S3. Establish an error compensation control model based on the dynamic model of the planar multi-link mechanism. This includes the following sub-steps:

[0149] S31. The displacement error of the slider of the gap mechanism is used as the main input signal of the sliding mode controller:

[0150] e(t) = x(t) - x e (t);

[0151] In the formula, e(t) represents the displacement error; x(t) represents the slider displacement of the system with gaps; x e (t) represents the slider displacement of the ideal system without gaps.

[0152] S32. Sliding surface design of the sliding mode controller: The sliding surface is typically designed as a linear combination of system state variables (error quantities), and the dynamic characteristics of the system need to be considered. To ensure that the system on the sliding surface can converge, the designed sliding surface function s(t) is as follows:

[0153]

[0154] In the formula, c is a positive real number; The derivative of e(t) with respect to time t represents the velocity error. This sliding surface design is based on the system's first derivative and can effectively reflect the system's dynamic characteristics.

[0155] S33. To ensure the system state is driven to the sliding surface s(t) = 0, and slides on the sliding surface until it converges to the equilibrium point and stabilizes the system, the control law is designed as follows:

[0156]

[0157] In the formula, k and b are the coefficients of the system model; the equivalent control part of this control law is: The switching control part is ks(t); sign(s(t)) is a sign function that provides a high-frequency switching mechanism to ensure that the system state reaches the sliding surface within a finite time.

[0158] Preferably, step S33 specifically includes the following sub-steps:

[0159] S331. Design of the reaching law: The reaching law enables the system state to rapidly approach the sliding surface at an exponential rate, while avoiding chattering problems in traditional sliding mode control. The control law used in this invention implicitly incorporates an exponential reaching law in the form of...

[0160]

[0161] S34. To determine the stability of the control system, let a Lyapunov function be...

[0162]

[0163] In the formula, V(S)≥0, and V(S)=0 if and only if s(t)=0, which obviously satisfies positive definiteness. The system is asymptotically stable on the sliding surface s(t=0).

[0164] S35. In a sliding mode control system, parameters c, b, and k are key control parameters, and their values ​​significantly affect the control effect. Using the total cumulative error of the slider displacement as the objective function, a particle swarm optimization algorithm is employed to search for the optimal solution of the three control parameters that satisfies the minimum value of the objective function within the specified interval. The expression for the objective function is as follows:

[0165]

[0166] Preferably, step S35 specifically includes the following sub-steps:

[0167] S351. Initialize the particle swarm algorithm by generating initial particle swarm sample points based on optimization variables and objective function;

[0168] S352. Solve the dynamic model to calculate the eccentricity vector, normal velocity, tangential velocity, and contact point position;

[0169] S353, obtain the displacement, velocity, acceleration response data of the slider and the fitness function;

[0170] S354. Update particle velocity and position, and search for the globally optimal particle;

[0171] S355. Iterate over the particles until they meet the objective function to obtain the optimal optimization variables.

[0172] S4. Based on the established mathematical model, build a Simulink simulation model of the sliding mode control system, and compare and analyze the parameters and control effects of the displacement before and after control. The sliding mode control system of the seven-bar rigid-flexible coupling mechanism with lubrication clearance mainly consists of 6 parts, including a dynamic subsystem without control, an error comparison subsystem 1, an error comparison subsystem 2, a control subsystem, a subsystem with control, and a controller parameter optimization subsystem.

[0173] Preferably, the dynamic modeling in step S4 is solved using the ode15s solver in MATLAB.

[0174] This embodiment uses a common planar two-degree-of-freedom multi-link mechanism, commonly used in key equipment such as punch presses and presses, as the research object for analysis. During operation, as the running time increases, the collisions and vibrations between the kinematic pairs intensify, leading to wear and damage of the materials at the contact points between bearings and shafts, thus reducing the motion accuracy of the mechanism. At high speeds, the elastic deformation of the flexible links further affects the system's motion accuracy and stability. To comprehensively address this issue, this example proposes a motion accuracy compensation modeling method based on active control. By establishing a sliding mode control model of a rigid-flexible coupling mechanism with clearance, the motion trajectory deviation of the slider is corrected in real time to improve the accuracy of the actuator's motion trajectory.

[0175] like Figure 5 The diagram shows a seven-bar rigid-flexible coupling mechanism with lubrication gap. This mechanism employs a double-crank drive configuration, consisting of a frame, two drive cranks, three sets of intermediate flexible connecting rods, and an actuator slider. The mechanism innovatively features a dual-drive system: the main drive motor at the first crank serves as the primary power source, while the auxiliary regulating motor at the second crank enables precise control of the main motion. The two motors, through the coordinated transmission of the connecting rod system, ultimately drive the slider to complete the working stroke. Compared to traditional single-degree-of-freedom mechanisms, this mechanism can achieve independent or coordinated motion of two degrees of freedom in a plane, exhibiting smoother and more reliable dynamic characteristics, and can meet diverse processing needs under complex working conditions.

[0176] In this system, O-XY is the global coordinate system, with the X-axis pointing vertically downwards and the Y-axis pointing horizontally to the right. The position coordinates of component i (i = 1, 2, 3, 4, 6, 7) in the global coordinate system are determined by (x... i ,y i The orientation is indicated by the angle θ between the x-axis and the positive x-axis. i Confirmed. S0 represents the distance between the origin of the coordinate system and the lower limit of the actuator slider.

[0177] Since the clearance at the cranks has a significant impact on the mechanism, the clearances A and B located on the two cranks are analyzed. Clearance A is located at the connection between the first driving crank and the second connecting rod, while clearance B exists between the second driving crank and the third connecting rod. Both clearance locations are situated at critical nodes in the power transmission path and have a significant impact on the dynamic performance of the mechanism.

[0178] like Figure 2 The diagram shows the clearance model of the rotating joint at points A and B of the mechanism. Lubricant 3 is added to the clearance of the rotating joint. Due to the clearance between the shaft 2 and the inner ring 1 of the bearing, the shaft 2 exists at contact position 4 where it collides with the inner ring of the bearing when it rotates. O—XY is the global coordinate system of the system, b represents the bearing, and a represents the shaft. The centroids M of the bearing and shaft are respectively b M a The position vector in the global coordinate system.

[0179] like Figure 1 As shown, the sliding mode control method for the dynamics of a multi-link mechanism with lubrication clearance and rigidity-flexibility effect includes the following steps:

[0180] S1. Establish the rotating pair model, contact force model, and rigid-flexible coupling model with lubrication gap;

[0181] S2. Based on the constraint conditions of the planar multi-link mechanism with a rotating pair containing lubrication clearance, the dynamic model of the two-degree-of-freedom planar multi-link mechanism with a rotating pair containing lubrication clearance is obtained.

[0182] S3. Establish an error compensation control model based on the dynamic model of the planar multi-link mechanism.

[0183] S4. Based on the established mathematical model, build a simulation model of the sliding mode control system, compare and analyze the parameters of the displacement before and after control, and optimize the error compensation control model in step S3.

[0184] S1. Establish the rotating pair model, contact force model, and rigid-flexible coupling model with lubrication gap.

[0185] Model of a rotating pair including lubrication clearance:

[0186] The eccentricity vector of the shaft's center of mass relative to the bearing's center of mass is:

[0187]

[0188] The corresponding unit vector is

[0189] n = e / |e| (2);

[0190] Eccentricity is the offset of the bearing center relative to the shaft center.

[0191]

[0192] In the formula, r is the clearance value, which is the difference between the radii of the bearing and the shaft. b R a These are the radii of the bearing and the shaft, respectively.

[0193] The first derivative of equation (3) with respect to time is:

[0194]

[0195] The offset angle, which is the angle between the direction of the eccentric vector and the positive X-axis, is

[0196]

[0197] In the formula, e x e y These are the components of the eccentric vector e in the X and Y directions, respectively.

[0198] The first derivative of equation (5) with respect to time is:

[0199]

[0200] In the formula, e x e y The first derivative with respect to time.

[0201] When the bearing and shaft collide, the collision depth is...

[0202] δ ab =|e|-r (7);

[0203] In the formula, r represents the clearance between the bearing and the shaft, r = R b -R a R b and R a These are the radii of the bearing and the shaft, respectively.

[0204] Therefore, the relative motion state between the bearing and the shaft in a rotating pair can be reflected by the change in the collision depth. Specifically:

[0205]

[0206] Contact force model with dry friction gap:

[0207] The normal collision force is modeled using the LN model, and its expression is:

[0208]

[0209] In the formula, F Nis the normal impact force; K is the contact stiffness coefficient; n is the force exponent, the value of which depends on the material properties and is usually taken as 1.5; It is the relative penetration speed; c e The coefficient of recovery; The initial collision velocity.

[0210] In equation (9), the solution for the stiffness coefficient K is:

[0211]

[0212] In the formula, K b and K a These are the equivalent stiffness coefficients of the bearing and the shaft at the contact point, respectively.

[0213] E i (i = b, a) is the elastic modulus; u i It is Poisson's ratio.

[0214] A modified Coulomb friction model is used to model the tangential collision force to improve computational stability; its expression is as follows:

[0215]

[0216] In the formula, c f c is the coefficient of friction; d For dynamic correction coefficients; v t For relative sliding velocity, the expression is as follows:

[0217]

[0218] In the formula, v s and v d These are the preset static friction critical speed threshold and dynamic friction critical speed threshold, respectively.

[0219] Based on the analysis of the normal collision force and tangential friction force above, the resultant contact force F at the clearance of the rotating joint can be obtained. D It can be represented as follows

[0220]

[0221] In the formula, and These are the normal unit vector and the tangential unit vector, respectively.

[0222] The expression for the oil film bearing capacity model including lubrication gaps is:

[0223] radial velocity At that time, the normal oil film bearing capacity F LN Tangential oil film bearing capacity F LT They are respectively

[0224]

[0225] In the formula, k0 is a parameter in the boundary conditions of the Reynolds equation; L is the bearing length; μ is the dynamic viscosity of the lubricant; and ω is the angular velocity of the shaft relative to the bearing.

[0226] radial velocity At that time, the normal and tangential oil film bearing capacities are respectively

[0227]

[0228] In practical applications, due to the high degree of overlap of the centroids of the shaft-bearing system (ε→0), directly solving the oil film bearing capacity model can easily lead to abrupt changes in the solution results. Therefore, an improved algorithm needs to be introduced. The improved normal and tangential oil film bearing capacity model is as follows:

[0229]

[0230] In the formula, ε0 is a constant greater than 0, used to correct the interval; m is the correction coefficient, which is a positive real number and ranges from 0 to 5.

[0231] Oil film bearing capacity F at the gap L The components in the X and Y directions are

[0232]

[0233] To accurately characterize the dynamic behavior of the kinematic pairs, a transition force model is established, and a hybrid model capable of accurately describing the lubrication-contact state transition is constructed, expressed as follows:

[0234]

[0235] In the formula, F C F L These are the interstitial force and the oil film bearing capacity, respectively; F D This refers to the contact force under dry friction conditions.

[0236] like Figure 4 As shown, this is a modeling method for accurately describing the large deformation of a flexible beam using the absolute nodal coordinate method. First, the flexible beam element is divided into one-dimensional two-node components. In the global coordinate system, the position vector r of any point on the flexible beam element can be expressed as...

[0237]

[0238] In the formula, r x ,r yLet be the components of the position vector of any point on the flexible beam element in the X and Y directions, respectively; x is the coordinate of any point on the central axis of the beam element in the beam element coordinate system; S is the shape function of the large deformation flexible beam element; q f For the generalized coordinates of the flexible beam element.

[0239] The generalized coordinates of a flexible beam element can be expressed as:

[0240]

[0241] In the formula, (r ix ,r iy ) and (r jx ,r jy ) are the position vectors of nodes i and j, respectively. and It is the slope vector.

[0242] Let the natural coordinates of the beam element be ξ = x / l, where l is the length of the flexible beam element. Then the shape function S of the large deformation flexible beam element can be expressed as follows:

[0243]

[0244] In the formula, s1, s2, s3, and s4 are polynomial functions, and s1 = 1 - 3ξ 2 +2ξ 3 s2=ξ-2ξ 2 +ξ 3 s3=3ξ 2 -2ξ 3 s4=-ξ 2 +ξ 3 .

[0245] Absolute velocity vector of flexible beam element It can be obtained by differentiating equation (24) with respect to time, that is Thus, the kinetic energy T of the flexible beam element is obtained. f The expression is as follows

[0246]

[0247] From equation (25), the mass matrix M of the flexible beam element can be obtained. f The expression is

[0248]

[0249] In the formula, ρ, A, and V represent the material density, cross-sectional area, and volume of the flexible beam element, respectively.

[0250] The elastic force of the flexible rod is:

[0251] F f =Ft +F l (27);

[0252] The expression for the bending elastic force of a flexible beam element is as follows:

[0253]

[0254] In the formula, S″ is the second derivative of the shape function S with respect to ξ, and E is the Young's modulus of the beam element. f Let be the moment of inertia of the cross section.

[0255] The expression for the axial elastic force of a flexible beam element is as follows:

[0256]

[0257] In the formula, A is the cross-sectional area of ​​the beam element. Wherein, ε f Let S be the strain tensor of the beam element, expressed as:

[0258]

[0259] In the formula, S f =S′ T S′, where S′ is the first derivative of the shape function S with respect to ξ.

[0260] The expression for the generalized gravity distributed load of a flexible beam element is:

[0261]

[0262] In the formula, m f and g are the constant mass matrix and gravitational acceleration, respectively.

[0263] S2. Based on the constraint conditions of the planar multi-link mechanism with a rotating pair containing lubrication clearance, the dynamic model of the two-degree-of-freedom planar multi-link mechanism with a rotating pair containing lubrication clearance is obtained.

[0264] The generalized coordinates of a rigid member can be expressed as:

[0265]

[0266] The generalized coordinates of a flexible component can be represented as:

[0267]

[0268] Therefore, the generalized coordinates of the system are represented as

[0269] q = (q1 q2 q3 q4 q6 q7) T (34);

[0270] For a rigid-flexible coupling mechanism considering the lubrication clearances at rotating joints A and B, the system constraint equations are as follows:

[0271]

[0272] Taking the first derivative of equation (35) with respect to time, we obtain the velocity constraint equation as follows:

[0273]

[0274] In the formula, Φ t Φ is the partial derivative of the displacement constraint equation with respect to time; q The Jacobian matrix of the constraint equations.

[0275] υ is the generalized velocity vector; υ is the right-hand side of the velocity constraint equation.

[0276] Taking the second derivative of equation (35) with respect to time, we obtain the acceleration constraint equation as follows:

[0277]

[0278] In the formula, Φ qt Φ is the partial derivative of the Jacobian matrix with respect to time. tt This represents the second-order partial derivative of the displacement constraint equation with respect to time. γ is the generalized acceleration vector; γ is the right-hand side of the acceleration constraint equation.

[0279] Based on the Lagrange multiplier method, the dynamic equations for rigid-flexible coupling can be obtained as follows:

[0280]

[0281] M fr =diag(m i ,m i J i (41);

[0282] In the formula, M fr Q is the mass matrix of the rigid-flexible coupled system; fr For a rigid-flexible coupled system, the generalized external force is m. i J i Let be the mass and moment of inertia of component i, respectively; λ be the Lagrange multiplier.

[0283] Combining equations (38) and (40), we obtain the dynamic equation of the rigid-flexible coupling mechanism with lubrication gap as follows:

[0284]

[0285] From equation (42), we can obtain And λ. However, this equation does not include displacement and velocity constraints, which may lead to constraint violation during the numerical solution process. The Baumgarte violation stabilization algorithm can be used to correct this, applying these two constraints to the acceleration constraint equation, resulting in the equation:

[0286]

[0287] In the formula, This is the time derivative of the displacement constraint equation. α and β are correction parameters for the Baumgarte default stability algorithm. When dealing with the constraint equations, having the same α and β means that the penalties for position and velocity violations are consistent, which can improve the stability of the numerical solution and allow the equations to converge quickly. α and β are usually taken as positive real numbers in the range [0, 50], and in this invention, they are all taken as 50.

[0288] S3. Establish an error compensation control model based on the dynamic model of the planar multi-link mechanism.

[0289] Sliding surfaces are typically designed as linear combinations of system state variables (errors), and the dynamic characteristics of the system must be considered. To ensure the convergence of the system on the sliding surface, the designed sliding surface function s(t) is as follows:

[0290]

[0291] In the formula, c is a positive real number, and e(t) is the displacement error. The derivative of e(t) with respect to time t represents the velocity error. This sliding surface design is based on the system's first derivative and can effectively reflect the system's dynamic characteristics.

[0292] When the system is on the sliding surface s(t) = 0, we have

[0293]

[0294] Therefore, differentiating with respect to the sliding surface s(t) yields

[0295]

[0296] In the formula, for The derivative with respect to time t is the acceleration error.

[0297] To ensure the system state is driven to the sliding surface s(t) = 0, and slides on the sliding surface until it converges to the equilibrium point and stabilizes, the control law is designed as follows:

[0298]

[0299] In the formula, k and b are the coefficients of the system model; the equivalent control part of this control law is: The switching control part is ks(t); sign(s(t)) is a sign function that provides a high-frequency switching mechanism to ensure that the system state reaches the sliding surface within a finite time.

[0300] Substituting equation (47) into equation (46), since it is on the sliding surface s(t) = 0, we have

[0301]

[0302] This indicates that on the sliding surface, the error e(t) converges exponentially to zero. To determine the stability of the system, let a Lyapunov function be...

[0303]

[0304] In the formula, V(S)≥0, and V(S)=0 if and only if s(t)=0, which obviously satisfies positive definiteness.

[0305] Differentiating the Lyapunov function V(S) yields

[0306]

[0307] Substituting equation (4.30) into equation (4.32) yields

[0308]

[0309] Since k > 0 and b > 0, therefore It is negative definite. Therefore, according to Lyapunov's stability theorem, since V(S) is positive definite and It is negative definite, and the system is asymptotically stable on the sliding surface s(t)=0.

[0310] The control law used in this section implicitly incorporates the form of an exponential reaching law.

[0311]

[0312] This approach law enables the system state to rapidly approach the sliding surface at an exponential rate, while avoiding chattering problems in traditional sliding mode control.

[0313] In a sliding mode control system, parameters c, b, and k are key control parameters, and their values ​​significantly affect the control performance. Using the total cumulative error of the slider displacement as the objective function, a particle swarm optimization algorithm is employed to search for the optimal solution of the three control parameters that satisfies the minimum value of the objective function within a given interval. The expression for the objective function is as follows:

[0314]

[0315] The following is the process of verifying the established mathematical model and the accuracy of the theory through experiments in step S4:

[0316] By comparing the model's predictions with actual measurement data, the model's accuracy and reliability can be evaluated, providing a solid basis for subsequent engineering design and optimization. This invention designs and constructs a seven-bar linkage test platform considering the clearance of rotating joints. The test device and its design are strictly based on contact collision theory and mechanism dynamics principles, ensuring a high degree of consistency in parameters with the previously established theoretical model.

[0317] The test platform consists of three parts: a structural system, a speed control system, and a data acquisition system. The structural system is based on section 1.4. Figure 4 The design of the experimental platform structure is based on a schematic diagram of a two-degree-of-freedom seven-bar linkage. The structural system includes a crank, connecting rod, slider, guide rail, pin, support plate, and frame. The frame and links are all made of 6061 aluminum alloy with a density of 2.80 g / cm³ and an elastic modulus of 70 GPa.

[0318] The clearance of a rotating joint can be achieved by machining shafts with different diameters. For an ideal shaft with an outer diameter of 12mm, a clearance of 0.3mm can be achieved by machining shafts with an outer diameter of... This is achieved through the clearance shaft, such as... Figure 6 and Figure 7 As shown. All other members are connected using 12mm standard shafts, with no clearance between the shafts and bearings. Specifically, as... Figure 6 As shown, the clearance shaft is located at points A and B. The inner ring 1 of the bearing has a diameter of 12 mm. The bearing is sleeved and installed on the shaft mounting part 6 of the rotating shaft 2, which has an outer diameter of 11.4 mm. There is a 0.6 mm gap between the inner ring 1 of the bearing and the rotating shaft 2. Figure 7 As shown, the inner ring 1 of the remaining backlash-free shaft bearings has a diameter of 12mm. The bearing sleeve is installed on the shaft mounting part 6 of the 12mm diameter shaft 2, with no gap between the inner ring 1 and the shaft 2. Each shaft 2 is equipped with an oil retainer ring mounting part 5, which has a small outer diameter for installing the oil retainer ring to prevent oil leakage.

[0319] As the end effector of the mechanism, the slider's operation plays a crucial role in the overall performance of the mechanism. Therefore, studying and analyzing the slider's motion characteristics is highly representative and of great significance. To verify the accuracy of the theoretical model, the actual acceleration of the slider was measured in the experiment and compared with the slider acceleration predicted by the theoretical model.

[0320] Based on a seven-bar linkage test platform considering the clearance of rotating joints, lubricating oil was dripped into rotating joints A and B respectively. After the mechanism stabilized, measurements were taken over three complete motion cycles. Figure 8As shown, this is a comparison of experimental and theoretical results for the slider acceleration of a rigid-flexible coupling seven-bar linkage with lubrication gap.

[0321] Comparing the slider acceleration curves reveals: Figure 8 The flexible lubrication curve shows that the slider acceleration is lower than that of the rigid body, which is due to the consideration of the rod's flexibility. The experimental curve exhibits significant fluctuations, while the theoretical curve is relatively stable, but the experimental data and theoretical derivation results generally match in their trends. The phase shift and amplitude inconsistency at the peak vibration response between theoretical and experimental data are mainly attributed to the nonlinear friction effect generated by the contact between the slider and the guide rail, the mechanical vibration interference generated by the motor drive system, the accumulation of dimensional chain errors during manufacturing and assembly, and the measurement uncertainty of the acceleration sensing system. Therefore, the experimental results basically verify the accuracy of the theoretical model.

[0322] S4. Based on the established mathematical model, build a simulation model of the sliding mode control system, and compare and analyze the parameters and control effects of the displacement before and after control.

[0323] The sliding mode control system for a seven-bar rigid-flexible coupling mechanism with lubrication clearance mainly consists of six parts: a dynamics subsystem (without control), an error comparison subsystem 1, an error comparison subsystem 2, a control subsystem, a subsystem with control, and a controller parameter optimization subsystem. Details are as follows: Figure 10 and Figure 11 As shown.

[0324] The role of the non-control dynamics subsystem is to provide the dynamic response output of the seven-bar linkage before control. It mainly includes the Integartor integration module, a time signal, and a Matlab Function module for storing Matlab functions. First, the Matlab functions for the seven-bar linkage considering backlash and those without backlash are stored in the function1 and function2 modules respectively. The displacement, velocity, and acceleration of the output sliders are then set in function1 and function2 modules respectively. Next, the Integartor integration module is added, and the initial integration value, motor drive speeds w1 and w4, and the Clock time module are set in the model resource manager to provide the time signal t.

[0325] like Figure 10 and 11 As shown in the displacement comparison diagram and displacement error comparison diagram, before control, due to the influence of the flexibility of the rod, the slider displacement error is larger than that of the rigid body mechanism. The error gradually increases over time, reaching a maximum value of 19 × 10⁻⁶ near 0.2 s. -5The error rate was initially measured in m, then reached its lowest point around 0.85 seconds, with the overall slider displacement error fluctuating significantly. However, the controlled slider displacement error curve only reached 17 × 10⁻⁶ at 0.5 seconds. -4 m, while in other time periods, the error was significantly reduced compared to before control, reaching as low as 2×10. -5 m. This indicates that the control measures effectively reduced the slider displacement error under lubrication conditions. For example... Figure 13 The diagram shows a flowchart of the particle swarm optimization algorithm. For key parameters in sliding mode controllers, optimization algorithms are needed to find the optimal solution more accurately. Figure 14 The diagram shown is a flowchart of the overall operation logic of the model.

[0326] The simulation parameters for the above results are as follows: the geometric parameters of the planar seven-bar linkage are shown in Table 1, the structural parameters are shown in Table 2, the parameters of the flexible components are shown in Table 3, and the lubrication clearance parameters are shown in Table 4.

[0327] Table 1 Geometric Parameters

[0328]

[0329] Table 2 Structural Parameters

[0330]

[0331] Table 3 Parameters of Flexible Components

[0332]

[0333]

[0334] Table 4 Lubrication clearance parameters

[0335]

[0336] Specifically, in Simulink, solver information such as simulation time, solver type, and integration step size is set. Details are as follows: Figure 14 As shown.

[0337] The simulation results show that controlling the model can effectively influence the dynamics of the actuator and improve the accuracy of the displacement trajectory. Therefore, this method can mitigate the adverse effects of lubrication gaps on the dynamic characteristics of rigid-flexible coupling mechanisms, and can correct the impact of the actuator's motion trajectory on the mechanism's dynamic characteristics in real time. This effectively improves the working stability and service life of multi-link mechanisms with lubrication gaps and rigid-flexible effects in industrial production, making it suitable for widespread application.

[0338] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for dynamic sliding mode control of a multi-link mechanism with lubricated clearance and rigid-flex effects, characterized by: The method comprises the following steps: S1, establishing a rotating pair model with a lubrication gap, a contact force model and a rigid-flexible coupling model; S2, obtaining a dynamic model of a two-degree-of-freedom planar multi-link mechanism with a lubrication gap rotating pair according to the constraint conditions of the planar multi-link mechanism with the lubrication gap rotating pair; S3, establishing an error compensation control model according to the dynamic model of the planar multi-link mechanism in step S2, comprising the following sub-steps: S31, taking the displacement error amount of the slider of the gap mechanism as the main input signal of the sliding mode controller: e(t) = x(t) - x e (t); where e(t) represents the displacement error quantity; x(t) represents the slider displacement of the system with a gap; and x e (t) represents the slider displacement without a gap. S32, designing a sliding surface of the sliding mode controller, and the sliding surface function s(t) is as follows: where c is a positive real number; is the velocity error quantity; S33, designing a control law as where k, b are coefficients of the system model; the equivalent control part of the control law is The switching control part is ks(t); the sign(s(t)) is a sign function providing a high-frequency switching mechanism to ensure that the system state reaches the sliding surface in a finite time. S34, setting a Lyapunov function as In the formula, V(S)≥0, V(S)=0 when and only when s(t)=0, and V(S) satisfies the positive definite; S35, taking the total amount of the cumulative error of the slider displacement as the objective function, and searching for the optimal solution of the three control parameters in the search interval range that satisfies the minimum value of the objective function by using a particle swarm optimization algorithm, and the expression of the search objective function is as follows: S4, building a sliding mode control system simulation model according to the error compensation control model established in step S3, comparing and analyzing the displacement parameters before and after control, and optimizing the error compensation control model established in step S3.

2. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flexible effect according to claim 1, characterized in that: Step S1 comprises the following sub-steps: S11, constructing a gap rotating pair model; S12, constructing a contact point collision depth model; S13, the relative motion state of the bearing and the shaft in the rotating pair is reflected by the change of the collision depth, and specifically When S14, establishing a collision force model in a dry friction state of the rotating pair gap and an oil film bearing force model in a lubrication state.

3. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flexible effect according to claim 2, characterized in that: Step S1 further comprises the following sub-steps: S15, obtaining the elastic force of the flexible link of the rigid-flexible coupling mechanism as F f = F t + F l ; The bending elastic force expression of the flexible beam unit is where S" is the second derivative of the shape function S with respect to ξ, E is the Young's modulus of the beam element, I is the moment of inertia of the cross-section f of the cross-section; The axial elastic force expression of the flexible beam unit is where A is the cross-sectional area of the beam element, where ε f is the strain tensor of the beam element, expressed as where S f = S' T S', S' is the first derivative of the shape function S with respect to ξ.

4. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flex effect according to claim 2, characterized in that: Step S14 comprises the following sub-steps: S141, constructing a normal collision force model in a dry friction state and a tangential collision force model in a dry friction state; S142, constructing an oil film bearing force model in a lubrication state: radial velocity normal oil film load F LN tangential oil film load F LT are respectively In the formula, k0 is a parameter in the boundary condition of the Reynolds equation; L is the length of the bearing; μ is the dynamic viscosity of the lubricant; ω is the angular velocity of the shaft relative to the bearing; Radial velocity At this time, the normal and tangential oil film carrying capacities are 5. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flex effect according to claim 1, characterized in that: The dynamic model in step S2 is where M fr is the mass matrix of the rigid-flexible coupling system; Φ q is the Jacobian matrix of the constraint equation, is the generalized acceleration vector; λ is the Lagrange multiplier; Q fr is the generalized external force of the rigid-flexible coupling system; F f is the elastic force of the flexible bar; is the time derivative of the displacement constraint equation, α, β are the correction parameters of the Baumgarte violation stabilization algorithm.

6. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flex effect according to claim 5, characterized in that: Step S2 comprises the following sub-steps: S21, for the rigid-flexible coupling mechanism considering the lubrication gaps of rotating pairs A and B, the constraint equation of the system is obtained; S22, based on the Lagrange multiplier method, the rigid-flexible coupling dynamics equation is obtained as M fr = diag(m i ,m i ,J i ); where M fr is the mass matrix of the rigid-flexible coupling system; Q fr is the generalized external force of the rigid-flexible coupling system; m i , J i are the mass and the moment of inertia of the member i, respectively. λ is the Lagrange multiplier; g is the system generalized force, including the inertia force of the system and the action force between the elements of the gap motion pair, as well as the average mixed transition force composed of the oil film bearing force; S23, the dynamic model of the rigid-flexible coupling mechanism with the lubrication gap is derived.

7. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flex effect according to claim 6, characterized in that: The constraint equation of the system in step S21 is: In the formula, Φ(q,t) is the displacement constraint equation; Then, the second-order derivative of the displacement constraint equation with respect to time is obtained, and the acceleration constraint equation is where Φ qt is the Jacobian matrix of the velocity constraint equations; Φ tt is the second-order partial derivative of the displacement constraint equations with respect to time; and is the generalized acceleration vector; and γ is the right-hand side of the acceleration constraint equations.

8. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flex effect according to claim 1, characterized in that: Step S33 comprises the following sub-steps: S331, designing a reaching law: the form of the exponential reaching law is 9. The dynamic sliding mode control method of multi-bar linkage mechanism with lubricated clearance and rigid-flex effect according to claim 1, characterized in that: Step S35 comprises the following sub-steps: S351, initialize the particle swarm algorithm, generate initial particle swarm sample points based on optimization variables and objective functions; S352, solve the dynamic model to calculate the eccentric vector, normal velocity, tangential velocity and contact point position; S353, obtain the displacement, velocity, acceleration response data and fitness function of the slider; S354, update the particle velocity and position, search for the global optimal particle; S355, iterate the particles until the target function is met, and obtain the optimal optimization variable.

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