Novel robot end force control actuator design method

Abstract

The invention discloses a novel robot tail end force control actuator design method, belongs to the technical field of robot machining chatter suppression, is suitable for high-end fields such as thin-wall part machining, and comprises the steps that a novel robot tail end force control actuator is designed, and a traditional motor driving method is abandoned for a joint of the novel robot tail end force control actuator; the elasticity of the connecting rod generated by the linear driving spring is adopted for simpler control, and the hydraulic shock absorber is additionally arranged at the position of the driving spring to prolong the service life of the driver. Aiming at the chatter suppression problem, on the basis of designing a novel robot end effector, the robot end effector is integrated into a robot thin-wall grinding machining system for chatter suppression, the dynamic characteristics of a robot grinding thin-wall workpiece are analyzed, and the function relation between grinding force and vibration response is established. According to the robot end force control actuator, the designed novel robot end force control actuator is integrated into a robot grinding machining system for flutter suppression, the problems that a traditional motor is high in driving cost, difficult to integrate, heavy, high in maintenance requirement and the like are solved, and effective flutter control can be conducted.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot machining chatter suppression, and in particular relates to a novel design method of a robot end force control actuator, and integrating the actuator into a robot grinding machining system for chatter suppression. Technical Background

[0002] Industrial robots have been widely used in various applications in many industries. One of the main obstacles limiting the application of industrial robots in high-precision machining is the lack of robot stiffness. The low stiffness characteristic may cause chatter in the robot during grinding, thereby reducing the accuracy of the machining dimensions. Therefore, a new force-controlled end effector is proposed to be integrated into the robot grinding work unit, which can effectively control the workpiece vibration. Summary of the invention

[0003] The purpose of the present invention is to propose a new design method for a robot end effector, which has three advantages: (1) The joint abandons the traditional motor drive method and adopts the elasticity of the connecting rod generated by the linear drive spring for simpler control; (2) The connecting rod parameters are optimized to keep the spring stiffness and the rigid connecting rod stiffness level, thereby balancing the insufficient stiffness of the PAM and the poor anti-interference ability; (3) The force-controlled end effector can not only output a constant normal grinding force, but also reduce the impact excitation and change the dynamic characteristics of the system due to the connection of the subsystem. (4) The installation of a hydraulic shock absorber at the drive spring can increase the life of the drive.

[0004] The technical solution of the present invention is

[0005] A novel robot end effector design method includes the following steps:

[0006] Step 1: Design a new type of force-controlled end effector, a parallel-structured force-controlled joint, which consists of three identical four-bar mechanisms. The output end of the joint is pushed upward by a passive linear spring, and the output end of the joint is pulled downward by a pneumatic artificial muscle PAM connected in parallel.

[0007] Step 2: Optimize the connecting rod parameters so that the passive linear spring is arranged in parallel with the active part of the pneumatic artificial muscle PAM, that is, their force-displacement curves remain parallel to each other. When the end effector is running, the input air pressure of the pneumatic artificial muscle PAM is adjusted to control the output force, and the displacement sensor records the displacement of the grinding tool relative to the workpiece in real time. The total output force generated by the end effector is equal to the difference between its two pneumatic artificial muscles PAM and the connecting rod.

[0008] Step 3: Attach the base of the force-controlled end effector to the end of the robot, and connect its output end to the grinding spindle. When this robot system is applied to the grinding of thin-walled workpieces, the characteristics of the force-controlled grinding process can be described as follows: (1) The workpiece and the grinding wheel maintain stable surface contact, assuming that the end effector subsystem is connected to the workpiece main system to form a new vibration system; (2) The mass of the thin-walled workpiece is comparable to that of the grinding wheel, and its dynamic characteristics change with its continuous and stable contact with the grinding wheel.

[0009] Step 4: Establish the function of contact force and vibration response to further study the correlation between contact force and vibration response.

[0010] The output force of the proposed force-controlled joint when executing step 1 is described by the following equation:

[0011] F J (p) = F s (x)-F p (x,p) (1)

[0012] Among them, F J (p) is the output force of the robot end effector, F S (x) is the resultant force of the linkage, F p (x, p) is the tension of the pneumatic artificial muscle PAM, x is the displacement of the joint, and p is the input pressure of the pneumatic artificial muscle PAM.

[0013] In order to simplify the control work, the optimal design of the connecting rod is to ensure that the best design is proposed to make F J (p) remains unchanged with x, that is, at a certain p value, F S The curve should be as close to F as possible. p When p is given, F p The curves have been determined by physical tests and published by PAM manufacturers. The research goal is to determine the F S (x) and F p (x) The best approximation within the desired tolerance requires the approximation theory of functions in the optimal design of connecting rods.

[0014] When executing step 2, the orthogonal polynomial approximation method is used to optimize the mechanism parameters. p (x) is expanded into the following orthogonal polynomial:

[0015] F p (x) = 0.003422x 4 -0.2002x 3 +4.33x 2 -52.31x+392.6 (2)

[0016] F Sis the vertical thrust added to the linkage, and h and h 0 They are the current displacement and zero displacement of the joint, so x = hh 0 .

[0017] According to the sine law, the deformation of the spring can be expressed as follows:

[0018]

[0019] Among them l 0 is the initial length of the spring and k is the stiffness coefficient.

[0020] According to Hooke's law, the elasticity F of the spring k It can be expressed as follows:

[0021] F k =kΔl (4)

[0022] During robot processing, the deformation of the end effector will also affect the stability of processing. In addition, the rebound force of the spring behind the hydraulic shock absorber is weakened, and the damping force F provided by the hydraulic shock absorber d It can be expressed as:

[0023] F d =cv (5)

[0024] Where c is the damping coefficient of the hydraulic shock absorber and v is the velocity of the piston.

[0025] The total damping force is F 总 =F k +F d

[0026] Then the restoring force d k The arm can be expressed as:

[0027] d k =bsinθ (6)

[0028] F L is the force acting on the connecting rod, d L Yes F L The force arm is d L It can be expressed as:

[0029]

[0030] According to the leverage principle:

[0031] (kΔl+cv)d k =F L d L (8)

[0032] F s It can be expressed as follows:

[0033] F s =F L cosδ (9)

[0034] Combined (3)~(8)F s It can be expressed as follows:

[0035]

[0036] The current displacement is given by:

[0037]

[0038] therefore, Figure 2 The geometric relationship shown can be expressed as follows:

[0039]

[0040] Combining (9) to (11) we can get F s The expression is:

[0041]

[0042] F S The coefficients of the orthogonal polynomial of (x) can be obtained by the best square approximation method, and S (x) The coefficients of the associated orthogonal polynomials depend on the coefficients of the connecting rods.

[0043] In order to make F in formula (2) p The coefficients of the polynomial of (x) are the same as F in formula (12) S The coefficients of the polynomials are equal, resulting in the following nonlinear equations:

[0044]

[0045] Given l 0 , k, d, c, v, The value of θ, θ∈(-60°,0°), the optimal connecting rod length can be obtained.

[0046] When executing step 3, the grinding system is simplified as follows: (1) While maintaining continuous and stable contact between the tool and the workpiece, the end effector subsystem is regarded as attached to the workpiece main system for grinding, and the vibration responses of the two subsystems in the normal direction of the grinding surface are consistent. (2) The dynamic model of the robot grinding system is simplified to a single degree of freedom (SDF) system with the above-specified vibration response direction difference. (3) The robot and fixture of the grinding system are both understood to be able to provide ideal stiffness support.

[0047] On a simplified basis when executing step 4, the vibration reduction mechanism based on force control can be explained as follows: When force control is turned on by the end effector, the end effector and the workpiece are integrated into a new vibration system, which changes the dynamic parameters of the grinding system, resulting in an increase in equivalent mass and equivalent damping. In addition, the vibration excitation of force control becomes a cyclic load with constant frequency and amplitude.

[0048] After continuous fully elastic collisions between the workpiece (mass block M) and the output end of the end effector (mass block m), the dynamic model of forceless control can be equivalent to a single degree of freedom system under continuous pulse excitation.

[0049] At speed v 1 Impact mass block M, then mass block m and mass block M move at speed V 2 and v 2 Based on the “subsystem connection” assumption, the dynamic system with force control is modeled as a single degree of freedom with equivalent mass, damping and stiffness after a fully plastic collision between mass block M and mass block m.

[0050] In a system without force control, the velocity of the mass block M before the collision is set to 0. Assuming that there is no energy loss in the collision, then v 2 and V 2 It can be expressed as:

[0051]

[0052] Based on the definition of impulse, the excitation of the force-free control system can be expressed as:

[0053]

[0054] Then the differential equation of the system without force control can be written as:

[0055]

[0056] Where x is the displacement response, c w is the damping value, k w is the workpiece stiffness, δ(t) represents the Dirker function, and the Laplace transform is applied to equation (17), where the system displacement response at the moment of collision is:

[0057]

[0058] Where ξ=c w / 2mω n is the damping ratio, ωn=(k w / m)1 / 2 is the natural frequency, ω d =ω n (1-ξ2 ) 1 / 2 is the damped natural frequency.

[0059] According to the superposition principle of linear systems, the post-contact vibration displacement response of the force-free control system can be described as:

[0060]

[0061] Then, in a force-controlled system, the dynamic model of robot grinding that controls the force at the moment of collision can be expressed as:

[0062]

[0063] where c t represents the damping of the end effector, k t represents the stiffness of the end effector.

[0064] When the system velocity changes but there is no system displacement at the moment of collision, the initial condition of formula (20) is:

[0065]

[0066] where v 1 is the collision velocity, then the displacement response of the system at the moment of collision is:

[0067]

[0068] Where ξ' is the new damping ratio, ω' n is the new natural frequency. In the case of stable contact, the system excitation period is T, and the amplitude is the cyclic load of the normal grinding force F.

[0069] After the collision, the excitation remains unchanged from 0 to T. Therefore, the excitation function can be equivalent to the sum of a step function with an excitation amplitude of +F at t = 0 and a step function with an excitation amplitude of -F at t = T. Based on Duhamel integral, the vibration displacement response of the system under the above excitation can be expressed as:

[0070]

[0071] As shown in formula (23), the normal grinding force F, the equivalent damping ratio ξ', the end effector m and the end effector stiffness k t All of these will have an impact on the vibration response of the part. Since these parameters are coupled to each other, formula (23) cannot be used to control the vibration response of the workpiece.

[0072] The parameter of the ratio of vibration response speed to vibration excitation is introduced to describe the vibration energy transfer characteristics, which quantitatively represents the energy transferred from the vibration source to the system through a certain contact area. For a circular contact area, the real part of the surface mobility can be expressed as:

[0073]

[0074] Where a is the contact radius, J 1 is the Bessel function of the first kind, M 0 is the ordinary point admittance, B is the bending wave number of the thin-walled part, and B a is the Helmholtz number of the Bessel function variable, M 0 and B are only related to the properties of the material, where:

[0075]

[0076] Where ω = 2π / T is the excitation circumference, B p is the bending stiffness of the workpiece, M d is the mass of the workpiece at the contact surface, E is the elastic modulus, h is the workpiece thickness, and v is the Poisson's ratio of the workpiece.

[0077] In formula (24), the contact radius a has a nonlinear relationship with the grinding force and can be calculated according to the Hertz contact theory, that is:

[0078] F=4 / 3·E * ·R -3 / 4 ·a 3 (26)

[0079] Where E * =E / (1-v 2 ) is Young's modulus, R is the equivalent radius of curvature, and by combining formulas (24) and (26), the function between the actual surface mobility and the normal grinding force can be obtained as follows:

[0080]

[0081] It can be concluded that the optimal grinding force set consists of the local minimum point of surface mobility-grinding force distribution. When calculating the vibration response, the real part of the surface admittance corresponding to each point in the set is used as the weighting coefficient of the vibration response, and the vibration displacement response can be rewritten as:

[0082] X'(t)=Re(M s )·x' 2 (t) (28)

[0083] The model establishes the grinding force-vibration response function, which can be used for effective chatter control.

[0084] The present invention designs a new type of robot end force control actuator, whose joints abandon the traditional motor drive method and adopt the elasticity of the connecting rod generated by the linear drive spring for simpler control, and add a hydraulic shock absorber at the drive spring to increase the service life of the driver. In view of the problem of chatter suppression, a new type of robot end effector is designed and integrated into the robot thin-wall grinding system for chatter suppression. The dynamic characteristics of the robot grinding thin-walled workpieces are analyzed, and the functional relationship between the grinding force and the vibration response is established. The patent of the present invention integrates the designed new type of robot end force control actuator into the robot grinding system for chatter suppression, which not only overcomes the problems of high cost, difficulty in integration, heavy weight, high maintenance requirements and other problems of traditional motor drive, but also can perform effective chatter control. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] Figure 1 This is a diagram of the force-controlled grinding system of the present invention.

[0086] Figure 2 This is a control schematic diagram of the robot end effector of the present invention.

[0087] Figure 3 This is the thin-walled parts robot grinding analysis model based on force control of the present invention.

[0088] Figure 4 It is a connecting rod mechanism diagram of the present invention. DETAILED DESCRIPTION

[0089] In view of this, the present invention provides a novel robot end effector design method. This method designs a novel force-controlled end effector to replace the traditional motor drive design, integrates it into the robot thin-wall grinding system, establishes a thin-wall grinding dynamics grinding profile based on force control, and establishes a grinding force-vibration response function, which can perform effective chatter control.

[0090] like Figure 1-Figure 4 The novel robot end effector design method comprises the following steps:

[0091] Step 1: Design a new robot end effector such as Figure 4 As shown, a parallel structure force control joint is composed of three identical six-bar mechanisms, which are driven by a passive linear spring to push the output end of the joint upward and pull the output end of the joint downward through a pneumatic artificial muscle (PAM) connected in parallel.

[0092] The output force of a force-controlled joint is described by the following equation:

[0093] F J (p) = F s (x)-F p(x,p) (1)

[0094] where F J (p) is the output force of the robot end effector, F S (x) is the resultant force of the linkage, F p (x, p) is the tension of the pneumatic artificial muscle PAM, x is the displacement of the joint, and p is the input pressure of the pneumatic artificial muscle PAM.

[0095] Step 2: Optimize the connecting rod parameters so that the passive linear spring is arranged in parallel with the active components of the pneumatic artificial muscle PAM. Figure 2 As shown, their force-displacement curves remain parallel to each other. When the end effector is running, the input air pressure of the pneumatic artificial muscle PAM is adjusted to control the output force, and the displacement sensor records the displacement of the grinding tool relative to the workpiece in real time. The total output force generated by the end effector is equal to the difference between the forces of its two pneumatic artificial muscles PAM and the connecting rod.

[0096] The orthogonal polynomial approximation method is used to optimize the mechanism parameters. p (x) is expanded into the following orthogonal polynomial:

[0097] F p (x) = 0.003422x 4 -0.2002x 3 +4.33x 2 -52.31x+392.6 (2)

[0098] F S is the vertical thrust added to the linkage, and h and h 0 They are the current displacement and zero displacement of the joint, so x = hh 0 .

[0099] According to the sine law, the deformation of the spring can be expressed as follows:

[0100]

[0101] Among them l 0 is the initial length of the spring and k is the stiffness coefficient.

[0102] According to Hooke's law, the elasticity F of the spring k It can be expressed as follows:

[0103] F k =kΔl (4)

[0104] During robot processing, the deformation of the end effector will also affect the stability of processing. In addition, the rebound force of the spring behind the hydraulic shock absorber is weakened, and the damping force F provided by the hydraulic shock absorber d It can be expressed as:

[0105] F d =cv (5)

[0106] Where c is the damping coefficient of the hydraulic shock absorber and v is the velocity of the piston.

[0107] The total damping force is F 总 =F k +F d

[0108] Then the restoring force d k The arm can be expressed as:

[0109] d k =bsinθ (6)

[0110] F L is the force acting on the connecting rod, d L Yes F L The force arm is d L It can be expressed as:

[0111]

[0112] According to the leverage principle:

[0113] (kΔl+cv)d k =F L d L (8)

[0114] F s It can be expressed as follows:

[0115] F s =F L cosδ (9)

[0116] Combined (3)~(8)F s It can be expressed as follows:

[0117]

[0118] The current displacement is given by:

[0119]

[0120] therefore, Figure 2 The geometric relationship shown can be expressed as follows:

[0121]

[0122] Combining (9) to (11) we can get F s The expression is:

[0123]

[0124] F S The coefficients of the orthogonal polynomial of (x) can be obtained by the best square approximation method, and S (x) The coefficients of the associated orthogonal polynomials depend on the coefficients of the connecting rods.

[0125] In order to make F in formula (2) p The coefficients of the polynomial of (x) are the same as F in formula (12) S The coefficients of the polynomials are equal, resulting in the following nonlinear equations:

[0126]

[0127] Given l 0 , k, d, c, v, The value of θ, θ∈(-60°,0°), the optimal connecting rod length can be obtained.

[0128] Step 3: Attach the base of the force-controlled end effector to the end of the robot and connect its output end to the grinding spindle as shown in the figure. Figure 1 When this robot system is applied to thin-walled workpiece grinding, the characteristics of the force-controlled grinding process can be described as follows: (1) The workpiece and the grinding wheel maintain a stable surface contact, assuming that the end effector subsystem is connected to the workpiece main system as shown in Figure 3 As shown, a new vibration system is formed; (2) The mass of the thin-walled workpiece is comparable to that of the grinding wheel, and its dynamic characteristics change with its continuous and stable contact with the grinding wheel.

[0129] Step 4: Establish the function of contact force and vibration response to further study the correlation between contact force and vibration response.

[0130] In a system without force control, the velocity of the mass block M before the collision is set to 0. Assuming that there is no energy loss in the collision, then v 2 and V 2 It can be expressed as:

[0131]

[0132] Based on the definition of impulse, the excitation of the force-free control system can be expressed as:

[0133]

[0134] Then the differential equation of the system without force control can be written as:

[0135]

[0136] Where x is the displacement response, c wis the damping value, k w is the workpiece stiffness, δ(t) represents the Dirker function, and the Laplace transform is applied to equation (17), where the system displacement response at the moment of collision is:

[0137]

[0138] Where ξ=c w / 2mω n is the damping ratio, ωn=(k w / m)1 / 2 is the natural frequency, ω d =ω n (1-ξ 2 ) 1 / 2 is the damped natural frequency.

[0139] According to the superposition principle of linear systems, the post-contact vibration displacement response of the force-free control system can be described as:

[0140]

[0141] Then, in a force-controlled system, the dynamic model of robot grinding that controls the force at the moment of collision can be expressed as:

[0142]

[0143] where c t represents the damping of the end effector, k t represents the stiffness of the end effector.

[0144] When the system velocity changes but there is no system displacement at the moment of collision, the initial condition of formula (20) is:

[0145]

[0146] where v 1 is the collision velocity, then the displacement response of the system at the moment of collision is:

[0147]

[0148] Where ξ' is the new damping ratio, ω' n is the new natural frequency. In the case of stable contact, the system excitation period is T, and the amplitude is the cyclic load of the normal grinding force F.

[0149] After the collision, the excitation remains unchanged from 0 to T. Therefore, the excitation function can be equivalent to the sum of a step function with an excitation amplitude of +F at t = 0 and a step function with an excitation amplitude of -F at t = T. Based on Duhamel integral, the vibration displacement response of the system under the above excitation can be expressed as:

[0150]

[0151] As shown in formula (23), the normal grinding force F, the equivalent damping ratio ξ', the end effector m, and the end effector stiffness k t will all affect the vibration response of the part. Since these parameters are coupled with each other, formula (23) cannot be used to control the vibration response of the workpiece.

[0152] Introduce a parameter that is the ratio of the vibration response velocity to the vibration excitation to describe the vibration energy transfer characteristics, and quantitatively represent the energy transferred from the vibration source to the system through a certain contact area. For a circular contact area, the real part of the surface mobility can be expressed as:

[0153]

[0154] where a is the contact radius, J 1 is the Bessel function of the first kind, M 0 is the ordinary point admittance, B is the flexural wave number of the thin-walled part, B a is the Helmholtz number of the Bessel function variable, M 0 and B are only related to the properties of the material, where:

[0155]

[0156] where ω = 2π / T is the excitation circular frequency, B p is the bending stiffness of the workpiece, M d is the mass of the workpiece at the contact surface, E is the elastic modulus, h is the thickness of the workpiece, and v is the Poisson's ratio of the workpiece.

[0157] In formula (24), the contact radius a has a non-linear relationship with the grinding force and can be calculated according to the Hertz contact theory, that is:

[0158] F = 4 / 3·E * ·R -3 / 4 ·a 3 (26)

[0159] where E * = E / (1 - v 2 ) is the Young's modulus, R is the equivalent curvature radius. Combining formulas (24) and (26), the function between the actual surface mobility and the normal grinding force can be obtained as follows:

[0160]

[0161] It can be concluded that the optimal grinding force set consists of the local minimum points of the surface mobility - the grinding force distribution. When calculating the vibration response, the real part of the surface admittance corresponding to each point in the set is used as the weighting coefficient of the vibration response, and the vibration displacement response can be rewritten as:

[0162] X'(t)=Re(M s )·x' 2 (t) (28)

[0163] The model establishes the grinding force-vibration response function, which can be used for effective chatter control.

[0164] The present invention integrates the designed new robot end force control actuator into the robot grinding system: its joint abandons the traditional motor drive method, and adopts the elasticity of the connecting rod generated by the linear drive spring for simpler control; the drive spring is equipped with a hydraulic shock absorber, which can not only enhance the stability of the robot grinding process, but also increase the life of the driver; the connecting rod parameters are optimized to keep the spring stiffness and the rigid connecting rod stiffness at the same level, so as to balance the insufficient PAM stiffness and poor anti-interference ability; the force-controlled end actuator can not only output a constant normal grinding force, but also reduce the impact excitation and change the dynamic characteristics of the system due to the connection of the subsystem; the grinding force-vibration response function is established, and effective vibration control can be performed. The patent of the present invention integrates the designed new robot end force control actuator into the robot grinding system for vibration suppression, which not only overcomes the problems of high cost, difficulty in integration, weight, high maintenance requirements, etc. of traditional motor drive, but also can perform effective vibration control.

Claims

1. A novel design method for a robot end force control actuator, characterized in that: The steps include: Step 1: Design a new type of robot end force control actuator, a parallel structure force control joint, which consists of three identical four-bar mechanisms. The output end of the joint is pushed upward by a passive linear spring drive, and the output end of the joint is pulled downward by a pneumatic artificial muscle PAM connected in parallel; Step 2: Optimize the connecting rod parameters so that the passive linear spring is arranged in parallel with the active component of the pneumatic artificial muscle PAM, that is, their force-displacement curves remain parallel to each other. When the end effector is running, the input air pressure of the pneumatic artificial muscle PAM is adjusted to control the output force, and the displacement sensor records the displacement of the grinding tool relative to the workpiece in real time. The total output force generated by the end effector is equal to the difference between the forces of its two pneumatic artificial muscles PAM and the connecting rod; Step 3: Attach the base of the force-controlled end effector to the end of the robot, and connect its output end to the grinding spindle. When this robot system is applied to the grinding of thin-walled workpieces, the characteristics of the force-controlled grinding process can be described as follows: (1) The workpiece and the grinding wheel maintain stable surface contact, assuming that the end effector subsystem is connected to the workpiece main system to form a new vibration system; (2) The mass of the thin-walled workpiece is comparable to that of the grinding wheel, and its dynamic characteristics change with its continuous and stable contact with the grinding wheel; Step 4: Establish the function of contact force and vibration response to further study the correlation between contact force and vibration response.

2. A novel design method for a robot end force control actuator according to claim 1, characterized in that: The output force of the proposed force-controlled joint when executing step 1 is described by the following equation: F J (p)=F s (x)-F p (x,p) (1) where F J (p) is the output force of the robot end effector, F S (x) is the resultant force of the linkage, F p (x, p) is the tension of the pneumatic artificial muscle PAM, x is the displacement of the joint, and p is the input pressure of the pneumatic artificial muscle PAM. In order to simplify the control work, the optimal design of the connecting rod is to ensure that the best design is proposed to make F J (p) remains unchanged with x, that is, at a certain p value, F S The curve should be as close to F as possible. p When p is given, F p The curve of F has been determined through physical experiments and published by the manufacturer of pneumatic artificial muscles (PAM); the research goal is to determine S (x) and F p (x) The best approximation within the desired tolerance requires the approximation theory of functions in the optimal design of connecting rods.

3. A novel robot end force control actuator method according to claim 2, characterized in that: When executing step 2, the orthogonal polynomial approximation method is used to optimize the mechanism parameters. p (x) is expanded into the following orthogonal polynomial: F p (x)=0.003422x 4 -0.2002x 3 +4.33x 2 -52.31x+392.6 (2) F S is the vertical thrust added to the linkage, while h and h0 refer to the current displacement and zero displacement of the joint, respectively, so x = h-h0; According to the sine law, the deformation of the spring can be expressed as follows: Where l0 is the initial length of the spring and k is the stiffness coefficient; According to Hooke's law, the elasticity F of the spring k It can be expressed as follows: F k =kΔl (4) During robot processing, the deformation of the end effector will also affect the stability of processing. In addition, the rebound force of the spring behind the hydraulic shock absorber is weakened, and the damping force F provided by the hydraulic shock absorber d It can be expressed as: F d =cv (5) Where c is the damping coefficient of the hydraulic shock absorber, and v is the velocity of the piston; The total damping force is F 总 =F k +F d Then the restoring force d k The arm can be expressed as: d k =bsinθ (6) F L is the force acting on the connecting rod, d L Yes F L The force arm is d L It can be expressed as: According to the leverage principle: <h2 style=";text-align:left;direction:ltr">(kΔl+cv)d<h2 style=";text-align:left;direction:ltr"> k <h2 style=";text-align:left;direction:ltr"> =F<h2 style=";text-align:left;direction:ltr"> L <h2 style=";text-align:left;direction:ltr"> d<h2 style=";text-align:left;direction:ltr"> L <h2 style=";text-align:left;direction:ltr"> (8) F s It can be expressed as follows: F s =F L cosδ (9) Combined (3)~(8)F s It can be expressed as follows: The current displacement is given by: Therefore, the geometric relationship can be expressed as follows: Combining (9) to (11) we can get F s The expression is: F S The coefficients of the orthogonal polynomial of (x) can be obtained by the best square approximation method, and S (x) The coefficients of the associated orthogonal polynomials depend on the coefficients of the connecting rods; In order to make F in formula (2) p The coefficients of the polynomial of (x) are the same as F in formula (12) S The coefficients of the polynomials are equal, resulting in the following nonlinear equations: Given l0, k, d, c, v, The value of θ, θ∈(-60°,0°), the optimal connecting rod length can be obtained.

4. A novel robot end force control actuator method according to claim 2, characterized in that: When executing step 3, the grinding system is simplified as follows: (1) While maintaining continuous and stable contact between the tool and the workpiece, the end effector subsystem is regarded as attached to the workpiece main system for grinding, and the vibration responses of the two subsystems in the normal direction of the grinding surface are consistent; (2) The dynamic model of the robot grinding system is simplified to a single degree of freedom (SDF) system with the above-specified vibration response direction difference; (3) The robot and fixture of the grinding system are understood to be able to provide ideal stiffness support.

5. A novel robot end force control actuator method according to claim 2, characterized in that: On a simplified basis when executing step 4, the vibration reduction mechanism based on force control can be explained as follows: when force control is turned on by the end effector, the end effector and the workpiece are integrated into a new vibration system, which changes the dynamic parameters of the grinding system, resulting in an increase in equivalent mass and equivalent damping; in addition, the vibration excitation of force control becomes a cyclic load with constant frequency and amplitude; After continuous fully elastic collisions between the workpiece mass block M and the mass block m at the output end of the end effector, the dynamic model with force control can be equivalent to a single degree of freedom system under continuous pulse excitation. At the moment of collision, the mass block m hits the mass block M at a speed of v1, and then the mass blocks m and the mass blocks M rebound against each other at speeds of V2 and v2 respectively; based on the "subsystem connection" assumption, after a fully plastic collision between the mass blocks M and the mass blocks m, the dynamic system with force control is modeled as a single degree of freedom with equivalent mass, damping and stiffness; In a system without force control, the velocity of the mass block M before the collision is set to 0. Assuming that there is no energy loss in the collision, v2 and V2 can be expressed as: Based on the definition of impulse, the excitation of the force-free control system can be expressed as: Then the differential equation of the system without force control can be written as: Where x is the displacement response, c w is the damping value, k w is the workpiece stiffness, δ(t) represents the Dirker function, and the Laplace transform is applied to equation (17), where the system displacement response at the moment of collision is: Where ξ=c w / 2mω n is the damping ratio, ωn=(k w / m)1 / 2 is the natural frequency, ω d =ω n (1-ξ 2 ) 1 / 2 is the damped natural frequency; According to the superposition principle of linear systems, the post-contact vibration displacement response of the force-free control system can be described as: Then, in a force-controlled system, the dynamic model of robot grinding that controls the force at the moment of collision can be expressed as: where c t represents the damping of the end effector, k t represents the stiffness of the end effector; When the system velocity changes but there is no system displacement at the moment of collision, the initial condition of formula (20) is: Where v1 is the collision velocity, the displacement response of the system at the moment of collision is: Where ξ' is the new damping ratio, ω' n is the new natural frequency; in the case of stable contact, the system excitation period is T, and the amplitude is the cyclic load of the normal grinding force F; After the collision, the excitation remains unchanged from 0 to T. Therefore, the excitation function can be equivalent to the sum of a step function with an excitation amplitude of +F at t = 0 and a step function with an excitation amplitude of -F at t = T. Based on Duhamel integral, the vibration displacement response of the system under the above excitation can be expressed as: As shown in formula (23), the normal grinding force F, the equivalent damping ratio ξ', the end effector m and the end effector stiffness k t All of them will affect the vibration response of the parts. Since these parameters are coupled with each other, formula (23) cannot be used to control the vibration response of the workpiece. The parameter of the ratio of vibration response speed to vibration excitation is introduced to describe the vibration energy transfer characteristics, which quantitatively represents the energy transferred from the vibration source to the system through a certain contact area. For a circular contact area, the real part of the surface mobility can be expressed as: Where a is the contact radius, J1 is the first kind of Bessel function, M0 is the ordinary point admittance, B is the bending wave number of the thin-walled part, and B a is the Helmholtz number of the Bessel function variable, M0 and B are only related to the properties of the material, where: Where ω = 2π / T is the excitation circumference, B p is the bending stiffness of the workpiece, M d is the mass of the workpiece at the contact surface, E is the elastic modulus, h is the workpiece thickness, and v is the Poisson's ratio of the workpiece; In formula (24), the contact radius a has a nonlinear relationship with the grinding force and can be calculated according to the Hertz contact theory, that is: F=4 / 3·E * ·R -3 / 4 ·a 3 (26) Where E * =E / (1-v 2 ) is Young's modulus, R is the equivalent radius of curvature, and by combining formulas (24) and (26), the function between the actual surface mobility and the normal grinding force can be obtained as follows: It can be concluded that the optimal grinding force set consists of the local minimum point of surface mobility-grinding force distribution. When calculating the vibration response, the real part of the surface admittance corresponding to each point in the set is used as the weighting coefficient of the vibration response, and the vibration displacement response can be rewritten as: X'(t)=Re(M s )·x'2(t)。 (28)。

Images