A stiffness optimization method for series elastic actuator in compliant grinding and polishing of complex curved parts

Abstract

The present invention discloses a method for optimizing the stiffness of a series elastic actuator for compliant grinding and polishing of parts with complex curved surfaces. The method aims to prevent interference between a linear bearing of the series elastic actuator and a baffle, selects the lowest spring stiffness that meets the normal working conditions of the series elastic actuator as the first minimum value of the spring stiffness; selects the lowest spring stiffness that meets the high-force bandwidth requirement as the second minimum value of the spring stiffness according to the large-force bandwidth required for grinding and polishing; selects the lowest spring stiffness that meets the minimum open-loop bandwidth requirement as the third minimum value of the spring stiffness according to the requirement for the open-loop bandwidth of a series elastic actuator control system in an actual grinding and polishing process; and uses the minimum value as the optimal spring stiffness of the series elastic actuator; by comprehensively selecting the optimal spring stiffness, the force control compliance of the series elastic actuator can be improved while meeting the actual grinding and polishing conditions, thereby improving the surface quality of the parts with complex curved surfaces after grinding and polishing.

Description

Technical Field

[0001] The invention belongs to the field of material grinding and polishing, and more specifically, relates to a method for optimizing the stiffness of a series elastic actuator for compliant grinding and polishing of complex curved surface parts. Background Art

[0002] Complex curved surface parts are widely used as core components of energy conversion devices such as aircraft engines and steam turbines. These parts have complex shapes, small machining allowances, and high requirements for contour accuracy and surface quality. In order to improve surface quality and accuracy, the grinding and polishing mechanism needs to have a certain degree of flexibility for flexible polishing. By adding a spring between the motor and the load in the grinding and polishing mechanism to form a series elastic force-controlled actuator with good flexibility, applying it to the grinding and polishing of complex curved surface parts can reduce overshoot, reduce steady-state force error, and improve the surface quality of parts after grinding and polishing.

[0003] Generally speaking, the better the flexibility of the grinding and polishing mechanism, the smaller the force control overshoot, the smaller the steady-state force error, and the better the surface quality after grinding and polishing; at the same time, the better the flexibility of the grinding and polishing mechanism, the smaller its impedance and the stronger its impact resistance, and it is not easy to cause damage to the grinding and polishing mechanism and the workpiece.

[0004] Therefore, the selection of spring stiffness in the series elastic force control actuator is particularly important. If the spring stiffness is too large, it will lead to poor compliance, easy overshoot, large steady-state force error, and poor force control effect; if the spring stiffness is too small, it will affect the bandwidth of the control system and the force control effect. Therefore, it is necessary to optimize the spring stiffness to ensure the surface quality of the complex surface after grinding and polishing. Summary of the invention

[0005] In response to the above defects or improvement needs of the prior art, the present invention provides a stiffness optimization method for a series elastic actuator for flexible grinding and polishing of complex curved surface parts, which can improve the flexibility of the grinding and polishing mechanism while meeting the grinding and polishing force control bandwidth, thereby improving the surface quality of complex curved surface parts after grinding and polishing.

[0006] To achieve the above object, according to a first aspect of the present invention, a method for optimizing the stiffness of a series elastic actuator for compliant grinding and polishing of complex curved surface parts is provided, comprising:

[0007] S1, establishing a transfer function of the series elastic actuator according to the control mode of the servo motor, and establishing an open-loop transfer function of the series elastic actuator control system based on the transfer function;

[0008] S2, calculating a first minimum value of the spring stiffness with the goal of preventing the linear bearing of the series elastic actuator from interfering with the baffle;

[0009] S3, determining a second minimum value of the spring stiffness according to the grinding head stiffness, the minimum value of the large force bandwidth required for grinding and polishing, and the relationship between the large force bandwidth and the spring stiffness;

[0010] S4, setting the amplitude of the open-loop transfer function to 1, and determining the third minimum value of the spring stiffness according to the preset minimum value of the open-loop bandwidth and the grinding head stiffness;

[0011] S5: Taking the minimum value among the first minimum value, the second minimum value and the third minimum value as the optimal spring stiffness of the series elastic actuator.

[0012] Preferably, when the control mode of the servo motor is a speed control mode or a position control mode, the high force bandwidth is related to the spring stiffness k s The relationship between them is:

[0013]

[0014] Among them, k e is the grinding head stiffness, k is the transmission ratio between the motor and the load, w0 is the high force bandwidth, F sat is the maximum force that the motor can output, V sat is the maximum speed that the motor can output;

[0015] When the control mode of the servo motor is the torque control mode, the large force bandwidth is related to the spring stiffness k s The relationship between them is:

[0016]

[0017] Where m is the load mass.

[0018] Preferably, the open-loop transfer function of the series elastic actuator control system is:

[0019] G v (s) = G(s)G d (s);

[0020] Where G(s) is the transfer function of the series elastic actuator. When the control mode of the servo motor is position control mode, When the servo motor is in speed control mode, When the servo motor is in torque control mode, Among them, k e is the grinding head stiffness, k s is the spring stiffness, k is the transmission ratio between the motor and the load, m is the load mass, and b is the load friction damping; G d (s) is the transfer function between the motor input and output.

[0021] Preferably, in step S2, according to the maximum grinding and polishing contact force F0 required for grinding and polishing and the formula Determine the spring stiffness k s where G is the load gravity and H1 is the maximum distance the bearing can move at the initial moment.

[0022] Preferably, the grinding head stiffness ranges from 3 to 5 N / mm.

[0023] According to a second aspect of the present invention, there is provided a stiffness optimization system for a series elastic actuator for compliant grinding and polishing of a complex curved surface part, comprising: a computer readable storage medium and a processor;

[0024] The computer-readable storage medium is used to store executable instructions;

[0025] The processor is used to read the executable instructions stored in the computer-readable storage medium and execute the optimization method as described in the first aspect.

[0026] According to a third aspect of the present invention, a series elastic actuator for compliant grinding and polishing of parts with complex curved surfaces is provided, wherein the spring stiffness of the series elastic actuator is optimized using the optimization method described in the first aspect.

[0027] Preferably, the series elastic actuator adopts a belt grinding head.

[0028] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects compared with the prior art:

[0029] The method provided by the present invention aims at preventing the linear bearing of the series elastic actuator from interfering with the baffle, selects the minimum spring stiffness that meets the normal working conditions of the series elastic actuator as the first minimum value of the spring stiffness; selects the minimum spring stiffness that meets the large force bandwidth requirement as the second minimum value of the spring stiffness according to the large force bandwidth required for grinding and polishing; selects the minimum spring stiffness that meets the minimum open-loop bandwidth requirement as the third minimum value of the spring stiffness according to the requirement for the open-loop bandwidth of the series elastic actuator control system in the actual grinding and polishing process; finally, the minimum value among the first minimum value, the second minimum value and the third minimum value is used as the optimal spring stiffness of the series elastic actuator; by comprehensively selecting the optimal spring stiffness, the force control compliance of the series elastic actuator can be improved while meeting the actual grinding and polishing conditions, thereby improving the surface quality of complex curved surface parts after grinding and polishing.

[0030] Furthermore, the method provided by the present invention can further improve the force control compliance by selecting a grinding head with low stiffness (3-5N / mm). BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1One of the flow diagrams of the stiffness optimization method of a series elastic actuator for compliant grinding and polishing of complex curved surface parts provided by an embodiment of the present invention;

[0032] Figure 2 A second flow chart of a method for optimizing stiffness of a series elastic actuator for compliant grinding and polishing of complex curved surface parts provided by an embodiment of the present invention;

[0033] Figure 3 It is a schematic diagram of the structure of the series elastic actuator;

[0034] Figure 4 Schematic diagram of the contact model of the series elastic actuator;

[0035] Figure 5 A schematic diagram of a system identification model provided by an embodiment of the present invention;

[0036] Figure 6 A pseudo-random binary sequence signal diagram for system identification provided by an embodiment of the present invention;

[0037] Figure 7 A comparison diagram of the identification output and the model output during system identification provided by an embodiment of the present invention;

[0038] Figure 8 Open-loop Nyquist plots of different spring stiffnesses provided by embodiments of the present invention;

[0039] Fig. 9 Bode plots of different spring stiffnesses provided for embodiments of the present invention;

[0040] Fig.10 An open-loop Nyquist diagram of different grinding head stiffnesses provided by an embodiment of the present invention;

[0041] Fig.11 Bode diagrams of different grinding head stiffnesses provided by embodiments of the present invention;

[0042] Fig.12 A comparison diagram of force control of different spring stiffnesses provided in an embodiment of the present invention;

[0043] Fig.13 A comparison diagram of different grinding head stiffness force control provided by an embodiment of the present invention;

[0044] Fig.14 (a), (b), and (c) are schematic diagrams of abrasive belt grinding heads, fiber grinding heads, and grinding wheel grinding heads, respectively;

[0045] Fig.15 A schematic diagram of selecting spring stiffness according to a model provided in an embodiment of the present invention;

[0046] Fig.16 The saturation curve of the servo motor provided by the embodiment of the present invention.

[0047] Throughout the drawings, the same reference numerals are used to denote the same elements or structures, wherein:

[0048] 1- flange, 2- servo platform, 3- spring, 4- linear bearing, 5- baffle, 6- force sensor, 7- grinding head. DETAILED DESCRIPTION

[0049] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0050] When grinding and polishing parts with complex curved surfaces, the high contour accuracy requirement requires the grinding and polishing mechanism to have good flexibility. The series elastic force control actuator has good flexibility. Applying it to the grinding and polishing of parts with complex curved surfaces can reduce overshoot, reduce steady-state force error, and improve the surface quality of parts after grinding and polishing. However, the selection of spring stiffness in the series elastic force control actuator is particularly important. If the spring stiffness is too large, it will lead to poor flexibility and poor force control effect; if the spring stiffness is too small, it will affect the bandwidth of the control system and the force control effect will also be affected. Therefore, it is necessary to optimize the spring stiffness to ensure the surface quality of complex surfaces after grinding and polishing.

[0051] Based on this, an embodiment of the present invention provides a method for optimizing the stiffness of a series elastic actuator for smooth grinding and polishing of complex curved surface parts. Figure 1 As shown, including:

[0052] S1, establishing a transfer function of the series elastic actuator according to the control mode of the servo motor; and establishing an open-loop transfer function and a closed-loop transfer function of the series elastic actuator control system according to the transfer function.

[0053] Specifically, S11, for a specific grinding and polishing scenario, a series elastic actuator (ie, a series elastic force-controlled actuator) contact model is established.

[0054] In order to further improve the compliance of force control, the grinding head stiffness is taken into account in the contact model of the series elastic actuator in the embodiment of the present invention. Generally speaking, there are three control modes for servo motors, namely position mode, speed mode and torque mode. A suitable servo motor control mode needs to be selected according to the actual needs of grinding and polishing. Then, the series elastic actuator contact model is established according to the selected servo motor control mode. If the servo motor selects the position control mode, the series elastic actuator contact model can be established in the form of a relationship between the motor end displacement and the grinding and polishing contact force, and its transfer function model is as follows:

[0055]

[0056] Where X m is the motor output displacement, F0 is the grinding contact force, and G1(s) contains the spring stiffness k. s and contact stiffness k e .

[0057] If the servo motor selects the speed control mode, the contact model of the series elastic actuator can be established in the form of the relationship between the motor end output speed and the grinding and polishing contact force. The transfer function model is as follows:

[0058]

[0059] Where V m is the motor output speed, F0 is the grinding contact force, k is the transmission ratio between the motor and the load, and the spring stiffness k is included in the G2(s) relationship. s and contact stiffness k e .

[0060] If the servo motor selects the torque control mode, the contact model of the series elastic actuator can be established in the form of the relationship between the output force at the motor end and the grinding and polishing contact force. The transfer function model is as follows:

[0061]

[0062] Among them, F m is the motor output force, F0 is the grinding contact force, m is the load mass, b is the load friction damping, and G3(s) contains the spring stiffness k. s and contact stiffness k e .

[0063] S12, for specific grinding and polishing scenarios, establish open-loop transfer function and closed-loop transfer function of the series elastic actuator control system.

[0064] The open-loop transfer function and the closed-loop transfer function are established in the following way:

[0065] Since the transfer function of the series elastic actuator contact model has been established above, it is also necessary to establish the transfer function between the motor input and the motor output when the motor is controlled. Therefore, the open-loop transfer function of the series elastic actuator can be obtained by the series elastic actuator contact model and the transfer function between the motor input and the motor output. After adding the controller and closed-loop feedback, the closed-loop transfer function of the series elastic actuator can be obtained.

[0066] Usually, the transfer function between the motor input and the motor output can be obtained by an offline system identification experiment. Assume that the transfer function between the motor input and the motor output obtained by identification is:

[0067]

[0068] Where m and n are the orders of the numerator and denominator in the transfer function, a n and b m are coefficients of different orders in the transfer function.

[0069] If the servo motor selects the position control mode, the open-loop transfer function and closed-loop transfer function of the series elastic actuator control system are

[0070]

[0071]

[0072] If the servo motor selects the speed control mode, the open-loop transfer function and closed-loop transfer function of the series elastic actuator control system are

[0073] G ol2 (s) = G2(s)G d (s)

[0074]

[0075] If the servo motor selects the torque control mode, the open-loop transfer function and closed-loop transfer function of the series elastic actuator control system are respectively

[0076]

[0077]

[0078] Furthermore, for specific grinding and polishing scenarios, the stability analysis of the series elastic actuator is carried out, and the influence of spring stiffness and grinding head stiffness on the series elastic actuator system is analyzed to provide a basis for the selection of grinding head stiffness and spring stiffness.

[0079] The stability analysis is carried out in the following way:

[0080] According to the open-loop and closed-loop transfer function models of the series elastic actuator control system established in S12, the influence of spring stiffness and grinding head stiffness on the stability of the series elastic actuator control system (including open-loop system and closed-loop system) is analyzed by using Nyquist diagram, Bode diagram and frequency domain characteristic indicators, providing a basis for the selection of spring stiffness and grinding head stiffness.

[0081] Then, for the specific grinding and polishing scenario, first select the grinding head stiffness, and then select the optimal spring stiffness.

[0082] The specific selection is as follows:

[0083] According to the results of the above stability analysis, in order to maximize the compliance of the series elastic actuator, first select the grinding head with the lowest stiffness among the most commonly used grinding head types. Then, according to the stiffness of the selected grinding head, select the lowest spring stiffness that meets the requirements of the grinding and polishing force control bandwidth according to S2-S4.

[0084] S2, calculating a first minimum value of the spring stiffness with the goal of preventing the linear bearing of the series elastic actuator from interfering with the baffle.

[0085] Specifically, based on the physical model of the series elastic actuator (i.e., the contact model) and the contact force of the polishing plan, the minimum spring stiffness that the series elastic actuator needs to meet is calculated so that the series elastic actuator can work normally and ensure that the linear bearing in the series elastic actuator does not interfere with the baffle.

[0086] Assume that the load mass is M, H1 is the maximum distance the bearing can move at the initial moment, and H2 is the distance the load moves under the action of the load gravity G and the planned maximum grinding and polishing contact force F0. In order to ensure that the load does not interfere, the following relationship must be satisfied:

[0087] H1≥H2

[0088]

[0089] So the spring stiffness should satisfy

[0090]

[0091] That is, according to the maximum grinding contact force F0 required for grinding and polishing and the formula Determine the spring stiffness k s where G is the load gravity and H1 is the maximum distance the bearing can move at the initial moment.

[0092] S3, determine the spring stiffness k according to the minimum value of the grinding head stiffness, the large force bandwidth required for grinding and polishing, and the relationship between them. s The second minimum value of .

[0093] Specifically, since the servo motor has a saturation characteristic, that is, the servo motor has an instantaneous limit on the force and speed output capacity, as long as the motor moves within these limits, the overall linear bandwidth of the series elastic actuator will not be affected by the spring stiffness, because when the spring stiffness decreases, the same control bandwidth can be guaranteed by increasing the control gain. However, the large force bandwidth will be affected. The "large force bandwidth" refers to the maximum frequency at which the series elastic actuator can vibrate at the force amplitude of the motor's maximum output force. In some grinding and polishing requirements, the material removal of the workpiece is required to be relatively large, so the series elastic actuator needs to have the ability to output a large force (for example, more than 50N), so the series elastic actuator must meet certain large force bandwidth conditions to meet the grinding and polishing requirements. Because when the motor outputs the maximum force, if the series elastic actuator wants to output a large force, the spring displacement must be large, but the motor is limited by force and speed saturation. When the motor reaches the corresponding saturation speed, the speed cannot be increased. Therefore, the spring stiffness has a limit on the large force bandwidth of the series elastic actuator. The larger the spring stiffness, the larger the corresponding large force bandwidth; the smaller the spring stiffness, the smaller the corresponding large force bandwidth. Therefore, according to the large force bandwidth required for grinding and polishing, the lowest spring stiffness that meets the large force bandwidth requirement is selected to improve the force control compliance.

[0094] It should be pointed out that if the servo motor selects position or speed control mode, since the motor position is the integral of the motor speed, there is no essential difference between the motor position and the motor speed. Fig.16 As shown,

[0095]

[0096] Among them, F m is the motor output force, v m is the motor output speed, F sat is the maximum force that the motor can output, V sat The maximum speed that the motor can output.

[0097] Therefore, the transfer function between the grinding and polishing contact force and the motor output force is:

[0098]

[0099] Therefore, the relationship between the maximum grinding and polishing contact force and the maximum output force of the motor can also be written as

[0100]

[0101] Therefore, its bandwidth expression is:

[0102]

[0103] Therefore, by choosing the spring stiffness k s The corresponding large force bandwidth can be calculated, and the minimum spring stiffness that meets the large force bandwidth requirement can be selected to improve force control compliance.

[0104] If the servo motor is in torque control mode, the relationship between the maximum grinding and polishing contact force and the maximum output force of the motor can be directly established based on the transfer function established in step 1.

[0105]

[0106] Therefore, its bandwidth expression is:

[0107]

[0108] By choosing the spring stiffness k in the transfer function s The high force bandwidth can be calculated, and the minimum spring stiffness that meets the high force bandwidth requirement can be selected to improve force control compliance.

[0109] S4, let the open-loop transfer function G of the series elastic actuator be v The amplitude margin of (s) is equal to 1. The spring stiffness k is determined according to the grinding head stiffness, the minimum value of the preset open-loop bandwidth and the grinding head stiffness. s The third smallest value of .

[0110] The amplitude margin of the open-loop frequency characteristic A(ω)=|G v (jω)|, phase margin Where j is a complex number and w is the open-loop bandwidth.

[0111] It is understandable that due to G v The amplitude expression of (s) includes the grinding head stiffness, spring stiffness and open loop bandwidth. Therefore, in G v When the amplitude of (s) is equal to 1, the third minimum value of the spring stiffness can be calculated based on the known grinding head stiffness and the minimum value of the preset open-loop bandwidth.

[0112] Specifically, the minimum spring stiffness that meets the minimum open-loop bandwidth requirement is selected based on the requirements for the open-loop bandwidth of the series elastic actuator control system during the actual grinding and polishing process. This is because the spring stiffness has a great influence on the open-loop bandwidth of the series elastic actuator. The larger the spring stiffness, the larger the open-loop bandwidth of the series elastic actuator control system; the smaller the spring stiffness, the smaller the open-loop bandwidth of the series elastic actuator control system.

[0113] Through the frequency characteristic indicators in the results of the above stability analysis, we can know the open-loop bandwidth of the series elastic actuator control system corresponding to a given spring stiffness value (i.e., open-loop cut-off bandwidth), and then we can choose the minimum spring stiffness that meets the minimum open-loop bandwidth requirement, which can ensure that while meeting the bandwidth requirements, the greatest possible force control compliance is obtained.

[0114] S5: taking the minimum value among the first minimum value, the second minimum value and the third minimum value as the optimal spring stiffness.

[0115] Preferably, when the control mode of the servo motor is a speed control mode or a position control mode, the high force bandwidth is related to the spring stiffness k s The relationship between them is:

[0116]

[0117] Among them, k e is the grinding head stiffness, w0 is the high force bandwidth, F sat is the maximum force that the motor can output, V sat is the maximum speed that the motor can output;

[0118] When the control mode of the servo motor is the torque control mode, the large force bandwidth is related to the spring stiffness k s The relationship between them is:

[0119]

[0120] Preferably, the grinding head stiffness ranges from 3 to 5 N / mm.

[0121] like Figure 2 As shown, the method provided by the present invention is further illustrated by taking a specific example as follows.

[0122] Step 1: Establish the contact model of series elastic actuator.

[0123] The structural model of the series elastic actuator used in the present invention is as follows: Figure 3 As shown, it includes a flange 1, a servo platform 2, a spring 3, a linear bearing 4, a baffle 5, a force sensor 6 and a grinding head 7.

[0124] As a technical solution of the present invention, the servo motor adopts the speed control mode, then the contact model of the series elastic actuator is as follows Figure 4 The contact model of the series elastic actuator is established as follows:

[0125] K s (X m -X0)=F0

[0126] F0=K e ·X0

[0127] Where X m is the motor output displacement, X0 is the load displacement, K s is the spring stiffness coefficient, K e is the contact stiffness coefficient, F0 is the contact force, and the spring compression is the difference between the displacement at the motor end and the displacement at the load end.

[0128] It can be obtained that the transfer function from the motor end displacement to the contact force is:

[0129]

[0130] Therefore, the transfer function from motor end velocity to contact force is:

[0131]

[0132] Step 2: Establish the open-loop transfer function and closed-loop transfer function of the series elastic actuator control system.

[0133] according to Figure 5 The system identification model in the servo motor is used to identify the servo motor speed loop transfer function. As a technical solution of the present invention, the speed loop transfer function model is a second-order model. Since the actual servo system is a fixed discrete cycle control, the present invention identifies the discrete model of the speed loop transfer function from the input / output measurement. The discrete model of the system can be used to express the second-order difference equation

[0134] y(k)+a1y(k-1)+a2y(k-2)=b1u(k-1)+b2u(k-2)

[0135] Among them, a1, a2, b1 and b2 are the model parameters to be identified, u(k) is the input vector corresponding to the input speed command, and y(k) is the output vector corresponding to the speed output command. In order to ensure the reliability of identification, the whole process is carried out under load. In addition, the pseudo-random binary sequence signal is used as the speed input command (such as Figure 6 As shown), the speed output command is used as the output of system identification.

[0136] According to the input speed command V c And output speed data V m , the speed loop transfer function identified by the least squares method is

[0137]

[0138] In order to verify the accuracy of the speed loop transfer function identification, the present invention uses the identified transfer function with the same pseudo-random binary sequence signal as input, and the simulated speed output command can match the actual speed output well (such as Figure 7Therefore, the accuracy of the speed loop transfer function identification is verified.

[0139] Therefore, the open-loop transfer function of the series elastic actuator control system is

[0140]

[0141] Wherein, k is the lead of the ball screw, and its value is 5 mm.

[0142] The PID controller is used for closed-loop control of the series elastic actuator, so the closed-loop transfer function is

[0143]

[0144] Step three, perform stability analysis on the series elastic actuator.

[0145] The stiffness of the flexible element is a key parameter in the series elastic actuator system. Reasonable selection of the stiffness coefficient of the flexible element has an important influence on the stability of the entire system. The Nyquist diagram, Bode diagram and frequency domain characteristic index are used to analyze the influence of the spring stiffness coefficient and the grinding head stiffness coefficient on the system stability, providing a theoretical basis for the selection of the stiffness coefficients of the spring and the grinding head.

[0146] (1) Effect of spring stiffness on the stability of the open-loop system of a series elastic actuator

[0147] Take the grinding head stiffness coefficient k e =3N / mm. When the spring stiffness coefficient k s When 0.5N / mm, 1N / mm, 1.5N / mm, 3N / mm, 4N / mm and 6N / mm are selected respectively, the Nyquist diagram is drawn according to the transfer function (such as Figure 8 The changing pattern of system stability is analyzed by using the Bode diagram (as shown in 9) and the Bode diagram (as shown in 9).

[0148] According to the Nyquist stability criterion, it can be seen that the system is stable. As the spring stiffness coefficient increases, the frequency characteristic curve is closer to (-1, j0), and the system stability becomes worse.

[0149] like Fig. 9 As shown in the figure, as the spring stiffness coefficient increases, the amplitude in the frequency characteristic will also increase, and the cutoff frequency of the system will shift to the right. Therefore, the open-loop bandwidth of the system will increase and the system response will become faster.

[0150] according to Fig. 9 The frequency characteristic indicators of the analysis are shown in Table 1.

[0151] Table 1 Frequency domain characteristics of open-loop system under different spring stiffness coefficients

[0152]

[0153] (2) Effect of grinding head stiffness on the stability of the open-loop system of the series elastic actuator

[0154] Take the spring stiffness coefficient k s =1.5N / mm. When the grinding head stiffness coefficient k e When 0.5N / mm, 1N / mm, 3N / mm, 5N / mm, 10N / mm and 20N / mm are selected respectively, the Nyquist diagram is drawn according to the transfer function (such as Fig.10 ) and Bode plots (as shown Fig.11 The changing law of system stability is analyzed by using the

[0155] According to the Nyquist stability criterion, the system is stable. As the grinding head stiffness coefficient increases, the frequency characteristic curve is closer to (-1, j0), and the stability of the SEA system tends to deteriorate.

[0156] As the stiffness coefficient of the grinding head continues to increase, the amplitude in the frequency characteristic also increases, and the cutoff frequency of the system shifts to the right. Therefore, the open-loop bandwidth of the system increases and the system response becomes faster.

[0157] according to Fig.11 The frequency characteristic indicators of the analysis are shown in Table 2.

[0158] Table 2 Frequency domain characteristics of open-loop system under different grinding head stiffness coefficients

[0159]

[0160]

[0161] (3) Effect of spring stiffness on the stability of the series elastic actuator closed-loop system

[0162] In the closed-loop control of the series elastic actuator, the system PID control parameters are: K p =1, K i =0, K d = 0. Take the grinding head stiffness coefficient K e =3N / mm. When the spring stiffness coefficient k s When 0.5N / mm, 1N / mm, 1.5N / mm, 3N / mm, 4N / mm and 6N / mm are selected respectively, the calculated frequency characteristic indicators are shown in Table 3.

[0163] Table 3 Frequency domain characteristics of closed-loop systems with different spring stiffness

[0164]

[0165] It can be seen from Table 3 that with the increase of spring stiffness, the resonance peak of the series elastic actuator system increases, the overshoot of the system increases, and the relative stability of the system decreases. When the spring stiffness increases, the bandwidth frequency increases, and the response capability of the SEA system becomes faster.

[0166] In order to verify the accuracy of the influence of spring stiffness on the series elastic actuator, the present invention conducted force-controlled grinding and polishing experiments with different spring stiffnesses. The experimental results are as follows: Fig.12 As shown. The experimental results show that when the spring stiffness is k1=1.5N / mm, k1=2.7N / mm, k1=3.9N / mm, respectively, the steady-state force error is ±0.2N, ±0.4N, ±0.6N, respectively, and the overshoot is about 0%, 6%, and 10%, respectively. Therefore, the lower the spring stiffness, the smaller the steady-state error, the lower the overshoot, and the better the compliance, which effectively proves the accuracy of the present invention.

[0167] (4) Effect of grinding head stiffness on the stability of the series elastic actuator closed-loop system

[0168] Similarly, in the closed-loop control of the series elastic actuator, the system PID control parameters are: K p =1, K i =0, K d = 0. Take the spring stiffness coefficient k s =1.5N / mm. When the grinding head stiffness coefficient k e When 0.5N / mm, 1N / mm, 3N / mm, 5N / mm, 10N / mm and 20N / mm are selected respectively, the calculated frequency characteristic indicators are shown in Table 4.

[0169] Table 4 Frequency domain characteristics of closed-loop systems with different contact stiffness

[0170]

[0171] It can be seen from Table 4 that with the increase of grinding head stiffness, the resonance peak of the series elastic actuator system increases, the overshoot of the system increases, and the relative stability of the system decreases. When the grinding head stiffness increases, the bandwidth frequency increases, and the response capability of the SEA system becomes faster.

[0172] In order to verify the accuracy of the influence of grinding head stiffness on the series elastic actuator, the present invention conducted force-controlled grinding and polishing experiments with different grinding head stiffness. The experimental results are as follows: Fig.13As shown. The experimental results show that when the grinding heads are belt grinding heads, fiber grinding heads and grinding wheel grinding heads, the steady-state force errors are ±0.3N, ±0.4N and ±0.6N respectively, and the overshoots are 0%, 2% and 5% respectively. Therefore, the lower the grinding head stiffness, the smaller the steady-state error, the lower the overshoot, and the better the compliance, which effectively proves the accuracy of the present invention. It can be understood that although the present embodiment is a conclusion drawn from the influence of spring stiffness and grinding head stiffness on the stability of the series elastic actuator control system when the servo motor adopts the speed control mode (the smaller the spring stiffness and grinding head stiffness, the better the stability and compliance of the system), it can be seen from the series elastic actuator transfer function (contact model) of the position control mode, speed control mode and torque control mode that the influence of spring stiffness and grinding head stiffness on the stability of the series elastic actuator control system is consistent. Therefore, when the servo motor adopts the position control mode or the torque control mode, this conclusion can also be used as the basis for selecting spring stiffness and grinding head stiffness.

[0173] Step 4: Select the grinding head stiffness and then select the optimal spring stiffness.

[0174] From the analysis of the effect of grinding head stiffness on the stability of the series elastic actuator system in step three, it can be seen that the greater the grinding head stiffness, the worse the compliance of the series elastic actuator, the greater the overshoot, and the more unstable the system.

[0175] Generally speaking, due to the high requirements for contour accuracy of complex curved surfaces and the direct contact between the grinding head and the workpiece, low-rigidity grinding heads are preferred among commonly used grinding heads to increase the flexibility of the series elastic actuator to improve the surface quality of complex curved surfaces. Currently, the grinding heads widely used for complex surface grinding and polishing are: belt grinding heads, fiber grinding heads and grinding wheel grinding heads, as shown in Figure 2. Fig.14 As shown in (a), (b), and (c) in the figure. Since the belt grinding head has a smaller rigidity and is more flexible when in contact with the workpiece, the belt grinding head is selected to grind complex curved surfaces to improve the surface quality of the parts. After experimental calibration, the rigidity of the selected belt grinding head is 3N / mm.

[0176] After selecting the grinding head stiffness, you also need to select the spring stiffness, as follows:

[0177] (1) According to the physical model of the series elastic actuator and the contact force of the grinding and polishing plan, the minimum spring stiffness that the series elastic actuator needs to meet is calculated so that the series elastic actuator can work normally and ensure that the linear bearing 4 in the series elastic actuator does not interfere with the baffle 5.

[0178] according to Fig.15 As shown, during the movement of the series elastic actuator, in order to ensure that the load does not interfere, the following relationship must be satisfied:

[0179] H1≥H2

[0180]

[0181] Where H1 is the maximum distance the load can move at the initial moment, and H2 is the distance the load moves under the action of the load gravity G and the planned maximum grinding and polishing contact force F0. According to the designed three-dimensional model of the series elastic actuator, H1 = 30mm, the load gravity is G = 29N, as a technical solution of the present invention, assuming that the planned maximum grinding and polishing contact force is F0 = 6N, then it can be calculated that the spring stiffness must meet

[0182] K s ≥1.2N / mm

[0183] That is, the spring stiffness k s The first minimum value is 1.2 N / mm.

[0184] It should be noted that the minimum spring stiffness calculated above does not take into account the influence of load motion inertia and friction. Therefore, in actual practice, the actual selected minimum spring stiffness should be slightly larger than 1.2N / mm, for example, 1.3N / mm~1.5N / mm.

[0185] (2) According to the large force bandwidth required for grinding and polishing, select the minimum spring stiffness that meets the large force bandwidth requirements to improve the force control compliance.

[0186] Because the transfer function from motor end displacement to grinding and polishing contact force is:

[0187]

[0188] As a technical solution of the present invention, it is assumed that the saturation characteristic curve of the servo motor is as follows: Fig.15 As shown, the motor output speed and output force must satisfy the following relationship

[0189]

[0190] Therefore, the transfer function between the grinding and polishing contact force and the motor output force is:

[0191]

[0192] Therefore, the relationship between the maximum grinding and polishing contact force and the maximum output force of the motor can also be written as

[0193]

[0194] Therefore, the high-energy bandwidth can be calculated as

[0195]

[0196] Therefore, we can select the minimum spring stiffness that meets the large force bandwidth requirements according to the large force bandwidth required in actual grinding and polishing to improve the force control compliance.

[0197] As a technical solution of the present invention, it is assumed that the minimum bandwidth required for actual grinding and polishing is ω0 = 5HZ, and the motor saturation coefficient Since the grinding head stiffness is k e =3N / mm, so according to the formula Calculate the spring stiffness k s ≥1.5N / mm. That is, the spring stiffness k s The second minimum value is 1.5 N / mm.

[0198] (3) According to the requirements for the open-loop bandwidth of the series elastic actuator control system during the actual grinding and polishing process, the minimum spring stiffness that meets the minimum open-loop bandwidth requirement is selected.

[0199] In the series elastic actuator control system, there is a requirement for the open-loop bandwidth of the system. Therefore, we can select the lowest spring stiffness according to the open-loop bandwidth requirement. As a technical solution of the present invention, a suitable spring stiffness can be selected according to the actual grinding head stiffness during the grinding and polishing process to meet the open-loop bandwidth requirement of the series elastic actuator control system.

[0200] According to the open-loop transfer function G of the series elastic actuator control system established in step 2 v (s), and obtain the amplitude of its open-loop frequency characteristic (i.e., the open-loop transfer function G v The amplitude of (s) is:

[0201]

[0202] Order |G v (jω)|=1, since the selected grinding head stiffness is k e =3N / mm, if the minimum open-loop cutoff frequency (i.e., open-loop bandwidth) is required to be ω=4.9rad / s, the spring stiffness should be k s ≥1.5N / mm. That is, the spring stiffness k s The third minimum value is 1.5N / mm. Therefore, the spring stiffness can be selected. In this example, the optimal spring stiffness is 1.2N / mm.

[0203] The present invention can comprehensively select the optimal spring stiffness through the three spring stiffness selection methods proposed above, and can maximize the compliance of the force control of the series elastic actuator on the basis of meeting the actual grinding and polishing conditions.

[0204] Through the spring stiffness optimization method of the series elastic actuator proposed in the present invention, the flexibility of the force control of the series elastic actuator can be maximized while meeting the grinding and polishing requirements, and the surface quality of complex curved surface parts after grinding and polishing can be improved.

[0205] In summary, the present invention establishes an open-loop and closed-loop transfer function model of a series elastic actuator control system, and uses Nyquist diagrams, Bode diagrams, and frequency domain characteristic indicators to analyze the influence of spring stiffness and grinding head stiffness on the stability of the series elastic actuator control system, and concludes that the smaller the spring stiffness and grinding head stiffness, the higher the flexibility and stability of the system, providing a basis for the selection of spring stiffness and grinding head stiffness. On this basis, a stiffness optimization method for a series elastic actuator for flexible grinding and polishing of complex curved surface parts is provided, which can improve the flexibility of the grinding and polishing mechanism while meeting the requirements of the grinding and polishing force control bandwidth.

[0206] The embodiment of the present invention provides a stiffness optimization system for a series elastic actuator for compliant grinding and polishing of complex curved surface parts, comprising: a computer-readable storage medium and a processor;

[0207] The computer-readable storage medium is used to store executable instructions;

[0208] The processor is used to read the executable instructions stored in the computer-readable storage medium and execute the optimization method described in any of the above embodiments.

[0209] The present invention provides a series elastic actuator for smooth grinding and polishing of parts with complex curved surfaces. The spring stiffness of the series elastic actuator is optimized by using the optimization method described in any of the above embodiments.

[0210] Preferably, the series elastic actuator adopts a belt grinding head.

[0211] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for optimizing the stiffness of a series elastic actuator for compliant grinding and polishing of complex curved surface parts, characterized in that: include: S1, establishing a transfer function of the series elastic actuator according to the control mode of the servo motor, and establishing an open-loop transfer function of the series elastic actuator control system based on the transfer function; S2, calculating a first minimum value of the spring stiffness with the goal of preventing the linear bearing of the series elastic actuator from interfering with the baffle; S3, determining a second minimum value of the spring stiffness according to the grinding head stiffness, the minimum value of the large force bandwidth required for grinding and polishing, and the relationship between the large force bandwidth and the spring stiffness; S4, setting the amplitude of the open-loop transfer function to 1, and determining the third minimum value of the spring stiffness according to the preset minimum value of the open-loop bandwidth and the grinding head stiffness; S5, taking the minimum value among the first minimum value, the second minimum value and the third minimum value as the optimal spring stiffness of the series elastic actuator; When the control mode of the servo motor is speed control mode or position control mode, the large force bandwidth is related to the spring stiffness k s The relationship between them is: Among them, k e is the grinding head stiffness, k is the transmission ratio between the motor and the load, w0 is the high force bandwidth, F sat is the maximum force that the motor can output, V sat is the maximum speed that the motor can output; When the control mode of the servo motor is the torque control mode, the large force bandwidth is related to the spring stiffness k s The relationship between them is: Where, m is the load mass; The open-loop transfer function of the series elastic actuator control system is: G v (s)=G(s)G d (s); Where G(s) is the transfer function of the series elastic actuator. When the control mode of the servo motor is position control mode, When the servo motor is in speed control mode, When the servo motor is in torque control mode, Among them, k e is the grinding head stiffness, k s is the spring stiffness, m is the load mass, b is the load friction damping, and k is the transmission ratio between the motor and the load; G d (s) is the transfer function between the motor input and output in the corresponding control mode; In step S2, according to the maximum grinding contact force F0 required for grinding and polishing and the formula Determine the spring stiffness k s where G is the load gravity and H1 is the maximum distance the bearing can move at the initial moment.

2. The method according to claim 1, characterized in that The value range of the grinding head stiffness is 3-5N / mm.

3. A stiffness optimization system for series elastic actuators for smooth grinding and polishing of complex curved surface parts, characterized in that: include: A computer readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is used to read the executable instructions stored in the computer-readable storage medium and execute the optimization method according to any one of claims 1-2.

4. A series elastic actuator for smooth grinding and polishing of complex curved surface parts, characterized in that: The spring stiffness of the series elastic actuator is optimized by using the optimization method described in any one of claims 1 to 2.

5. The series elastic actuator according to claim 4, characterized in that: The series elastic actuator adopts a sanding belt grinding head.

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