A robotic end-effector force control actuator design method

Abstract

The application discloses a robot end force control actuator design method, belongs to the technical field of robot machining chatter suppression, and is suitable for high-end fields such as thin-walled part machining and the like. The robot end force control actuator comprises a joint, and the joint discards the traditional motor driving method, adopts the elasticity of a connecting rod generated by a linear driving spring to perform simpler control, and adds a hydraulic shock absorber at the driving spring to increase the service life of the driver. In view of the chatter suppression problem, the robot end actuator is integrated into a robot thin-walled grinding machining system to perform chatter suppression on the basis of the design of the robot end actuator, the dynamic characteristics of robot grinding thin-walled workpieces are analyzed, and a functional relationship between the grinding force and the vibration response is established. The robot end force control actuator designed in the application is integrated into the robot grinding machining system to perform chatter suppression, so that the problems of high cost, difficulty in integration, weight, high maintenance requirement and the like of the traditional motor driving method can be overcome, and effective chatter control can be performed.

Description

Technical Field

[0001] This invention belongs to the field of chatter suppression technology in robot machining, and specifically relates to a design method for a robot end effector force control actuator, which is then integrated into a robot grinding system for chatter suppression. Technical Background

[0002] Industrial robots are widely used in various applications across many industries. One of the main obstacles limiting their application in high-precision machining is insufficient robot stiffness. Low stiffness can cause chattering during grinding, thus reducing the accuracy of dimensional accuracy. Therefore, a force-controlled end effector integrated into the robot's grinding cell is proposed to effectively control workpiece vibration. Summary of the Invention

[0003] The purpose of this invention is to propose a design method for a robot end effector, which has three advantages: (1) the joint abandons the traditional motor drive method and uses the elasticity of the linkage generated by the linear drive spring for simpler control; (2) the linkage parameters are optimized to keep the spring stiffness and rigid linkage stiffness at the same level, thereby balancing the insufficient stiffness and poor anti-interference ability of PAM; (3) the force-controlled end effector can not only output a constant normal grinding force, but also reduces the impact excitation and changes the dynamic characteristics of the system due to the connection of the subsystems; (4) the addition of a hydraulic damper at the drive spring can increase the life of the actuator.

[0004] The technical solution of the present invention is as follows:

[0005] A method for designing a robot end effector includes the following steps:

[0006] Step 1: Design a force-controlled end effector, a force-controlled joint with a parallel structure, which consists of three identical four-bar linkages. Driven by a passive linear spring, the output end of the joint is pushed upward, and the output end of the joint is pulled downward by a pneumatic artificial muscle (PAM) connected in parallel.

[0007] Step 2: Optimize the linkage parameters so that the passive linear spring is arranged parallel to the active component of the pneumatic artificial muscle (PAM), i.e., 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 regulated 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 in force between its two pneumatic artificial muscles (PAMs) and the linkage.

[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, a new vibration system is formed; (2) The mass of the thin-walled workpiece and the grinding wheel are comparable, 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, and further study the correlation between contact force and vibration response.

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

[0011]

[0012] in (p) is the output force of the robot's end effector. It is the resultant force of the linkage mechanism. is the tension of the pneumatic artificial muscle (PAM), x is the joint displacement, and p is the input pressure of the pneumatic artificial muscle (PAM).

[0013] To simplify control operations, the linkage design is optimized to ensure that the suggested best design... (p) remains constant with x, that is, at a certain value of p, The curve should be as close as possible to Parallel. When p is given. The curve has been determined through physical experiments and published by the PAM manufacturer. The research objective is to determine... and The optimal approximation within the expected tolerance requires the approximation theory of functions in the optimization design of connecting rods.

[0014] In step 2, the orthogonal polynomial approximation method is used to optimize the mechanism parameters. Expanded into the following orthogonal polynomial:

[0015]

[0016] It is the vertical thrust applied to the linkage mechanism, while h and These refer to the current displacement and zero displacement of the joint, respectively; therefore, x = h - .

[0017] According to the law of sinusoids, the deformation of a spring can be described as follows:

[0018]

[0019] Where l0 is the initial length of the spring, and k is the stiffness coefficient.

[0020] According to Hooke's Law, the elasticity F of a spring... k It can be represented as follows:

[0021]

[0022] During robotic machining, deformation of the end effector can affect machining stability. Furthermore, the rebound force of the spring is weakened by the hydraulic damper, reducing the damping force F provided by the hydraulic damper. d It can be represented as:

[0023]

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

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

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

[0027]

[0028] F L It is the force acting on the connecting rod, d L It is F L If the lever arm is d, then d L It can be represented as:

[0029]

[0030] According to the lever principle:

[0031]

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

[0033]

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

[0035]

[0036] The current displacement is given by the following formula:

[0037]

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

[0039]

[0040] Combining (9) to (11) yields F. s The expression:

[0041]

[0042] The coefficients of the orthogonal polynomials can be obtained using the best square approximation method, while... The coefficients of the relevant orthogonal polynomials depend on the coefficients of the link.

[0043] In order to make the equation (2) The coefficients of the polynomial and F in equation (12) S With the coefficients of the polynomials being equal, we obtain the following system of nonlinear equations:

[0044]

[0045] Given l0, k, d, c, v, The value, (-60°, 30°) This allows us to obtain the optimal link length.

[0046] When performing 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 being attached to the main workpiece system for grinding, and the vibration response of the two subsystems in the direction normal to the grinding surface is consistent; (2) The dynamic model of the robot grinding system is simplified as a single-degree-of-freedom system with the vibration response direction difference specified above; (3) The robot and the fixture of the grinding system are both understood to be able to provide ideal stiffness support.

[0047] Based on the equivalent single-free vibration system, the force-controlled vibration reduction mechanism can be explained as follows: when force control is activated through 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, thereby suppressing the vibration response in the normal direction; in addition, the vibration excitation of force control becomes a cyclic load with constant frequency and amplitude.

[0048] After a series of perfectly elastic collisions between the workpiece (mass M) and the output of the end effector (mass m), the uncontrollable dynamic model can be equivalent to a single-degree-of-freedom system under continuous pulse excitation. At the instant of collision, mass m...

[0049] A mass block m impacts a mass block M with velocity v1, and then the mass blocks m and m bounce off each other with velocities V2 and v2, respectively. Based on the "subsystem connection" assumption, after a fully plastic collision between the mass blocks M and m, the force-controlled dynamic system is modeled as a single-degree-of-freedom system with equivalent mass, damping, and stiffness.

[0050] In an uncontrollable system, if the velocity of the mass block M before the collision is set to 0, and assuming no energy loss during the collision, then v2 and V2 can be expressed as:

[0051]

[0052] Based on the definition of impulse, the excitation of a powerless control system can be expressed as:

[0053]

[0054] The differential equation for a system without force control can then be written as:

[0055]

[0056] Where x is the displacement response, c w It is the damping value, k w It is the workpiece stiffness. (t) represents the Diclave function. Applying the Laplace transform to the equation, the system displacement response at the moment of collision in equation (17) is:

[0057]

[0058] in, =c w / 2m n It is the damping ratio. It is the natural frequency. It is a damped natural frequency.

[0059] According to the superposition principle of linear systems, the contact vibration displacement response of a powerless control system can be described as follows:

[0060]

[0061] In a system with force control, the dynamic model of robotic grinding, where the force is controlled at the moment of collision, can be expressed as:

[0062]

[0063] Where c t k represents the damping of the end effector. t This indicates 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 equation (20) is:

[0065]

[0066] Where v1 is the collision velocity, the displacement response of the system at the instant of collision is:

[0067]

[0068] in, It's a new damping ratio. It is the new natural frequency. Under stable contact conditions, the excitation period of the system is T, and the amplitude is a cyclic load of the normal grinding force F.

[0069] After the collision, the excitation remains constant 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 the 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 and the equivalent damping ratio End effector m and end effector stiffness k t All of these will affect the vibration response of the parts. Since these parameters are coupled with each other, the vibration response of the workpiece cannot be controlled by formula (23).

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

[0073]

[0074] Where a is the contact radius, J1 is the Bessel function of the first kind, M0 is the ordinary point admittance, and B is the bending wavenumber of the thin-walled section. a Let M0 and B be the Helmholtz numbers of the Bessel function variables, which depend only on the material properties.

[0075]

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

[0077] In formula (24), the contact radius a has a non-linear relationship with the grinding force, which can be calculated according to Hertz contact theory, i.e.:

[0078]

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

[0080]

[0081] Therefore, the optimal grinding force set is composed of the local minimum points of surface mobility – the grinding force distribution. When calculating the vibration response, the real parts of the surface admittance corresponding to each point in the set are used as weighting coefficients for the vibration response. The vibration displacement response can be rewritten as:

[0082]

[0083] This model establishes a grinding force-vibration response function, which can effectively control chatter.

[0084] This invention designs a robot end effector with force control. Its joints abandon traditional motor-driven methods, instead employing the elasticity of a linkage generated by a linear drive spring for simpler control. Furthermore, a hydraulic damper is added to the drive spring to increase the actuator's lifespan. Addressing chatter suppression, based on the designed robot end effector, it is integrated into a robot thin-walled grinding system for chatter suppression. The dynamic characteristics of robot grinding of thin-walled workpieces are analyzed, and a functional relationship between grinding force and vibration response is established. This invention integrates the designed robot end effector with a robot grinding system for chatter suppression, overcoming the problems of high cost, difficulty in integration, heavy weight, and high maintenance requirements of traditional motor-driven systems, while also achieving effective chatter control. Attached Figure Description

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

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

[0087] Figure 3 This invention provides a force-controlled robotic grinding analysis model for thin-walled parts.

[0088] Figure 4 This is a diagram of the linkage mechanism of the present invention. Detailed Implementation

[0089] In view of this, the present invention provides a design method for a robot end effector. This method designs a force-controlled end effector to replace the traditional motor-driven design and integrates it into a robot thin-wall grinding system. It establishes a force-controlled thin-wall grinding dynamics model and a grinding force-vibration response function, enabling effective chatter control.

[0090] like Figures 1-4 The robot end effector design method includes the following steps:

[0091] Step 1: Design a robot end effector such as Figure 4 As shown, the force-controlled joint has a parallel structure, which consists of three identical six-bar linkages. Driven by a passive linear spring, the joint's output end is pushed upward, and the output end is pulled downward by a pneumatic artificial muscle (PAM) connected in parallel.

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

[0093]

[0094] in (p) is the output force of the robot's end effector. It is the resultant force of the linkage mechanism. is the tension of the pneumatic artificial muscle (PAM), x is the joint displacement, and p is the input pressure of the pneumatic artificial muscle (PAM).

[0095] Step 2: Optimize the linkage parameters so that the passive linear spring is arranged parallel to the active component 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 regulated 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 in force between its two pneumatic artificial muscles (PAMs) and the connecting rod.

[0096] The mechanism parameters are optimized using an orthogonal polynomial approximation method. Expanded into the following orthogonal polynomial:

[0097]

[0098] It is the vertical thrust applied to the linkage mechanism, while h and These refer to the current displacement and zero displacement of the joint, respectively; therefore, x = h - .

[0099] According to the law of sinusoids, the deformation of a spring can be described as follows:

[0100]

[0101] Where l0 is the initial length of the spring, and k is the stiffness coefficient.

[0102] According to Hooke's Law, the elasticity F of a spring... k It can be represented as follows:

[0103]

[0104] During robotic machining, deformation of the end effector can affect machining stability. Furthermore, the rebound force of the spring is weakened by the hydraulic damper, reducing the damping force F provided by the hydraulic damper. d It can be represented as:

[0105]

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

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

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

[0109]

[0110] F L It is the force acting on the connecting rod, d L It is F L If the lever arm is d, then d L It can be represented as:

[0111]

[0112] According to the lever principle:

[0113]

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

[0115]

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

[0117]

[0118] The current displacement is given by the following formula:

[0119]

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

[0121]

[0122] Combining (9) to (11) yields F. s The expression:

[0123]

[0124] The coefficients of the orthogonal polynomials can be obtained using the best square approximation method, while... The coefficients of the relevant orthogonal polynomials depend on the coefficients of the link.

[0125] In order to make the equation (2) The coefficients of the polynomial and F in equation (12) S With the coefficients of the polynomials being equal, we obtain the following system of nonlinear equations:

[0126]

[0127] Given l0, k, d, c, v, The value, (-60°, 30°) This allows us to obtain the optimal link length.

[0128] Step 3: Attach the base of the force-controlled end effector to the robot end, and connect its output end to the grinding spindle as follows: Figure 1 As shown. 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 maintains stable surface contact with the grinding wheel, assuming that the end effector subsystem is connected to the workpiece main system as follows. 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, and further study the correlation between contact force and vibration response.

[0130] In an uncontrollable system, if the velocity of the mass block M before the collision is set to 0, and assuming no energy loss during the collision, then v2 and V2 can be expressed as:

[0131]

[0132] Based on the definition of impulse, the excitation of a powerless control system can be expressed as:

[0133]

[0134] The differential equation for a system without force control can then be written as:

[0135]

[0136] Where x is the displacement response, c w It is the damping value, k w It is the workpiece stiffness. (t) represents the Diclave function. Applying the Laplace transform to the equation, the system displacement response at the moment of collision in equation (17) is:

[0137]

[0138] in, =c w / 2m n It is the damping ratio. It is the natural frequency. It is a damped natural frequency.

[0139] According to the superposition principle of linear systems, the contact vibration displacement response of a powerless control system can be described as follows:

[0140]

[0141] In a system with force control, the dynamic model of robotic grinding, where the force is controlled at the moment of collision, can be expressed as:

[0142]

[0143] Where c t k represents the damping of the end effector. t This indicates 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 equation (20) is:

[0145]

[0146] Where v1 is the collision velocity, the displacement response of the system at the instant of collision is:

[0147]

[0148] in, It's a new damping ratio. It is the new natural frequency. Under stable contact conditions, the excitation period of the system is T, and the amplitude is a cyclic load of the normal grinding force F.

[0149] After the collision, the excitation remains constant 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 the 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 and the equivalent damping ratio End effector m and end effector stiffness k t All of these will affect the vibration response of the parts. Since these parameters are coupled with each other, the vibration response of the workpiece cannot be controlled by formula (23).

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

[0153]

[0154] Where a is the contact radius, J1 is the Bessel function of the first kind, M0 is the ordinary point admittance, and B is the bending wavenumber of the thin-walled section. a Let M0 and B be the Helmholtz numbers of the Bessel function variables, which depend only on the material properties.

[0155]

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

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

[0158]

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

[0160]

[0161] Therefore, the optimal grinding force set is composed of the local minimum points of surface mobility – the grinding force distribution. When calculating the vibration response, the real parts of the surface admittance corresponding to each point in the set are used as weighting coefficients for the vibration response. The vibration displacement response can be rewritten as:

[0162]

[0163] This model establishes a grinding force-vibration response function, which can effectively control chatter.

[0164] This invention integrates a designed robotic end effector with force control into a robotic grinding system. Its joints abandon traditional motor-driven methods, instead employing the elasticity of links generated by linear drive springs for simpler control. Adding hydraulic dampers to the drive springs not only enhances the stability of the robotic grinding process but also increases the lifespan of the actuator. Optimized link parameters maintain the spring stiffness and rigid link stiffness at a horizontal level, thus balancing the insufficient stiffness and poor anti-interference capability of the PAM (Power Actuation Mechanism). The force-controlled end effector not only outputs a constant normal grinding force but also reduces impact excitation and alters the system's dynamic characteristics due to the subsystem's connection. A grinding force-vibration response function is established, enabling effective chatter control. This invention integrates the designed robotic end effector with force control into a robotic grinding system for chatter suppression, overcoming the problems of high cost, difficulty in integration, heavy weight, and high maintenance requirements of traditional motor-driven systems, while also providing effective chatter control.

Claims

1. A design method for a robot end effector with force control, characterized in that, Includes the following steps: Step 1: Design a robot end effector with a parallel force control joint. The joint consists of three identical four-bar linkages. The output end of the joint is pushed upward by a passive linear spring and pulled downward by a pneumatic artificial muscle (PAM) connected in parallel. Step 2: Optimize the linkage parameters so that the passive linear spring is arranged parallel to 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 in force between its two pneumatic artificial muscle (PAM) components and the linkage. 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, a new vibration system is formed; (2) The mass of the thin-walled workpiece and the grinding wheel are comparable, 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, and further study the correlation between contact force and vibration response.

2. The design method for a robot end effector according to claim 1, characterized in that: The output force of the force-controlled joint proposed in step 1 is described by the following equation: ； in (p) is the output force of the robot's end effector. It is the resultant force of the linkage mechanism. is the tension of the pneumatic artificial muscle (PAM), x is the joint displacement, and p is the input pressure of the pneumatic artificial muscle (PAM). To simplify control operations, the linkage design is optimized to ensure that the suggested best design... (p) remains constant with x, that is, at a certain value of p, The curve should be as close as possible to Parallel; when p is given, The curve has been determined through physical experiments and published by the manufacturer of pneumatic artificial muscles (PAM); the research objective is to determine... and The optimal approximation within the expected tolerance requires the approximation theory of functions in the optimization design of connecting rods.

3. The method for a robot end effector force control according to claim 2, characterized in that: In step 2, the orthogonal polynomial approximation method is used to optimize the mechanism parameters. Expanded into the following orthogonal polynomial: ； It is the vertical thrust applied to the linkage mechanism, while h and These refer to the current displacement and zero displacement of the joint, respectively; therefore, x = h - ; According to the law of sinusoids, the deformation of a spring can be described 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 a spring... k It can be represented as follows: ； During robotic machining, deformation of the end effector can affect machining stability. Furthermore, the rebound force of the spring is weakened by the hydraulic damper, reducing the damping force F provided by the hydraulic damper. d It can be represented as: ； Where c is the damping coefficient of the hydraulic shock absorber, and v is the piston speed; The total damping force is F 总 =F k +F d Then the restoring force d k The arm can be represented as: ； F L It is the force acting on the connecting rod, d L It is F L If the lever arm is d, then d L It can be represented as: ； According to the lever principle: ； F s It can be represented as follows: ； Combined (3)~(8)F s It can be represented as follows: ； The current displacement is given by the following formula: ； Therefore, the geometric relationship can be expressed as follows: ； Combining (9) to (11) yields F. s The expression: ； The coefficients of the orthogonal polynomials can be obtained using the best square approximation method, while... The coefficients of the relevant orthogonal polynomials depend on the coefficients of the link; In order to make the equation (2) The coefficients of the polynomial and F in equation (12) S With the coefficients of the polynomials being equal, we obtain the following system of nonlinear equations: ； Given l0, k, d, c, v, The value, (-60°, 30°) This allows us to obtain the optimal link length.

4. The method for a robot end effector according to claim 2, characterized in that: When performing 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 being attached to the main workpiece system for grinding, and the vibration response of the two subsystems in the direction normal to the grinding surface is consistent; (2) The dynamic model of the robot grinding system is simplified as a single-degree-of-freedom system with the vibration response direction difference specified above; (3) The robot and the fixture of the grinding system are both understood to be able to provide ideal stiffness support.

5. A robot end effector force control method according to claim 4, characterized in that: Based on the equivalent single-free vibration system, the force-controlled vibration reduction mechanism can be explained as follows: when force control is activated through 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, thereby suppressing the vibration response in the normal direction; in addition, the vibration excitation of force control becomes a cyclic load with constant frequency and amplitude. After continuous perfectly elastic collisions between the workpiece mass block M and the end effector output mass block m, the uncontrolled dynamic model can be equivalent to a single-degree-of-freedom system under continuous pulse excitation. At the moment of collision, mass block m impacts mass block M with velocity v1, and then mass blocks m and M bounce off each other with velocities V2 and v2, respectively. Based on the "subsystem connection" assumption, after a fully plastic collision between mass blocks M and m, the force-controlled dynamic system is modeled as a single-degree-of-freedom system with equivalent mass, damping, and stiffness. In an uncontrollable system, if the velocity of the mass block M before the collision is set to 0, and assuming no energy loss during the collision, then v2 and V2 can be expressed as: ； Based on the definition of impulse, the excitation of a powerless control system can be expressed as: ； The differential equation for a system without force control can then be written as: ； Where x is the displacement response, c w It is the damping value, k w It is the workpiece stiffness. (t) represents the Diclave function. Applying the Laplace transform to the equation, the system displacement response at the moment of collision in equation (17) is: ； in, =c w / 2m n It is the damping ratio. It is the natural frequency. It is a damped natural frequency; According to the superposition principle of linear systems, the contact vibration displacement response of a powerless control system can be described as follows: ； In a system with force control, the dynamic model of robotic grinding, where the force is controlled at the moment of collision, can be expressed as: ； Where c t k represents the damping of the end effector. t Indicates 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 equation (20) is: ； Where v1 is the collision velocity, the displacement response of the system at the instant of collision is: ； in, It's a new damping ratio. It is a new natural frequency; under stable contact conditions, the excitation period of the system is T, and the amplitude is a cyclic load of the normal grinding force F. After the collision, the excitation remains constant 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 the 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 and the equivalent damping ratio End effector m and end effector stiffness k t All of these will affect the vibration response of the parts. Since these parameters are coupled with each other, the vibration response of the workpiece cannot be controlled by formula (23). The ratio of vibration response velocity to vibration excitation is introduced as a parameter to describe the vibration energy transfer characteristics, quantitatively representing the energy transferred from the vibration source to the system through a certain contact area; for a circular contact region, the real part of the surface mobility can be expressed as: ； Where a is the contact radius, J1 is the Bessel function of the first kind, M0 is the ordinary point admittance, and B is the bending wavenumber of the thin-walled section. a Let M0 and B be the Helmholtz numbers of the Bessel function variables, which depend only on the material properties. ； in =2 / T is the excitation pi, B p It is the bending stiffness of the workpiece, M d E is the mass of the workpiece at the contact surface, h is the elastic modulus, v is the workpiece thickness, and v is the workpiece Poisson's ratio. In formula (24), the contact radius a has a non-linear relationship with the grinding force, which can be calculated according to Hertz contact theory, i.e.: ； Where E * =E / (1-v 2 Let ) be Young's modulus and R be the equivalent radius of curvature. Combining formulas (24) and (26), the function between the actual surface mobility and the normal grinding force can be obtained as follows: ； Therefore, the optimal grinding force set is composed of the local minimum point of 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: 。

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