Ball screw drive error compensation method based on composite integral control

Abstract

A kind of ball screw drive error compensation method based on composite integral control, due to the influence of manufacturing and assembly error and other factors, ball screw system has certain error and deviation, in order to reduce the influence on the precision and performance of machine tool system, a kind of error compensation method of composite integral control is designed, considering the influence of friction, stiffness, lead error and elastic deformation, first, the corresponding model is established to estimate the corresponding factors;Second, by designing integral sliding surface and error observer to reduce the error in ball screw composite control system and improve the anti-interference ability of system, ensure the stability of closed-loop system, can effectively reduce the error of ball screw drive, has robustness to the change of friction, cutting force and other interference.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of mechanical engineering, and specifically relates to a ball screw transmission error compensation method based on composite integral control. BACKGROUND

[0002] In the field of mechanical engineering, ball screw transmission systems are widely used in various precision positioning and motion control systems. Traditional ball screw transmission system control methods mainly include proportional-integral-derivative (PID) control, empirical mode decomposition (EMD) method, etc. However, these traditional methods have certain limitations and cannot fully consider the nonlinear factors, dynamic response characteristics, and dynamic changes of transmission errors in the transmission system, resulting in a decline in the control performance of the transmission system and the inability to meet high-demand application scenarios, especially high-speed, high-precision, and high-reliability industrial automation systems.

[0003] In the traditional PID control method, only the current error and the integral and differential terms of the error are considered, while the nonlinear factors and the dynamic changes of the transmission error are ignored. This leads to the performance of the PID control method being limited when facing nonlinear friction, stiffness, lead error, and elastic deformation in the ball screw transmission system, and the transmission error cannot be effectively suppressed. On the other hand, the transmission system control based on the empirical mode decomposition (EMD) method also has certain limitations. The EMD method decomposes the signal into multiple intrinsic mode functions (IMF) to extract different frequency components in the signal. Then each IMF is analyzed and processed. However, in actual applications, the EMD method has problems such as mode aliasing and noise reduction difficulty, and its performance is unstable for non-stationary and nonlinear systems.

[0004] The prior art is as follows:

[0005] Comparison with the technology of patent CN105305913A "a new anti-interference following controller for ball screw feeding system"

[0006] The self-disturbance technology and proportional-integral control in patent CN105305913A are used to achieve accurate control of the motor position and load position. The integral and sliding mode control used in the present application achieves accurate control of the motor position and load position. In the self-disturbance technology and proportional-integral control, a large anti-interference gain needs to be introduced to achieve anti-interference and robust performance, which will slow down the response speed of the system. In contrast, the integral and sliding mode control used in the present application can achieve faster system response through appropriate sliding mode surface design and the introduction of integral terms.

[0007] The patent CN105305913A uses an extended state observer to evaluate the total disturbance. Compared with the extended state observer, the total disturbance observer used in the present application is simpler in form. The performance and robustness of the extended state observer are highly dependent on the selection and adjustment of parameters. Improper parameter selection can lead to performance degradation or stability problems of the observer. The parameters of the observer used in the present application can be adjusted by experience or some standard methods. The adjustment of these parameters is relatively simple, which reduces the difficulty of system debugging and parameter adjustment. It is easier to achieve robustness design for different disturbances and system changes, making the integral and sliding mode control more flexible in dealing with system uncertainties and disturbances.

[0008] Comparison with the technology of patent CN103837339A "A dynamic servo force driving system";

[0009] The patent CN103837339A uses linear sliding mode surface sliding mode control to achieve dynamic high-precision loading, while the present application uses integral sliding mode surface sliding mode control to achieve dynamic high-precision loading. The compensation ability of the linear sliding mode surface is limited, and it may not be able to provide sufficient compensation force in the case of large changes in system parameters, especially in applications requiring high-precision control. The maximum error in the patent is 15%, while the maximum error of the integral sliding mode control used in the present application is 0.012%. The integral sliding mode surface can accumulate errors and exert stronger compensation force on the sliding mode surface. This helps to reduce system errors and improve control precision and stability.

[0010] The patent CN103837339A and the method used in the present application both use sliding mode control with disturbance observer, but the disturbance observer used in the present application is simpler in form, can detect and compensate any external disturbance, and has only one adjustment parameter. The disturbance observer used in the patent CN103837339A is relatively complex, and has three parameters to adjust, which increases the difficulty of system design and debugging, and increases the complexity of implementing the control method. Complex observer structure and large number of parameters can lead to high computational load, which is particularly limited in real-time control systems.

[0011] Comparison with the technology of patent CN107505914A "A high-precision motion control method considering high-order dynamic characteristics of ball screw feeding system";

[0012] The patent CN107505914A adopts a high-order modal filter controller and a zero-phase difference tracking control algorithm to realize the tracking accuracy and motion stability of the feeding system, and the application adopts integral and sliding mode control to realize the tracking accuracy and motion stability of the feeding system. The method in the patent involves the design and parameter optimization of the high-order modal filter controller, and the design and embedding of the zero-phase difference tracking controller, which is relatively complex and requires more design and debugging time. The method also involves frequency domain analysis, modal superposition calculation and parameter optimization, which requires real-time performance and computing power of the control system. The integral and sliding mode control algorithm adopted by the application can also achieve high-precision control, and is relatively simple and relatively low in computing load, and is more suitable for resource-limited real-time control environment.

[0013] The patent CN107505914A needs to perform modal analysis on the ball screw feeding system and determine the modal parameters. This requires a good understanding of the dynamic characteristics of the system and accurate parameter estimation. In contrast, the integral and sliding mode control method of the application does not require detailed analysis of the dynamic characteristics of the system, but uses error observers and integral controllers to achieve error compensation, which is more flexible and easier to implement.

[0014] The application comprehensively considers the nonlinear factors, dynamic response characteristics and dynamic changes of the transmission error in the ball screw transmission system. By establishing the corresponding model to estimate various factors, the performance of the transmission system can be more comprehensively analyzed and controlled. The design of the integral sliding mode surface and the error observer can effectively reduce the ball screw transmission error and improve the anti-interference ability of the system. The integral sliding mode surface realizes more accurate error compensation by integrating the error, thereby reducing the influence of the error. The error observer can estimate various disturbances and changes in the system in real time, and maintain the stability of the system by compensating for the estimated error. This method has strong robustness and good adaptability to changes in friction, cutting force and other disturbances. Traditional methods often cannot effectively handle the influence of these disturbances on system performance, while the compound integral control method can track and compensate for these disturbances in real time through the design of the integral sliding mode surface and the error observer, improving the anti-interference ability of the system. SUMMARY

[0015] To reduce the tracking error of the ball screw transmission, the application proposes a ball screw transmission error compensation method based on compound integral control, which considers the influence of factors such as friction, lead error, elastic deformation and external disturbances in modeling, reduces the tracking error and improves the anti-interference ability of the system, and has robustness to changes in friction, cutting force and other disturbances.

[0016] To achieve the above-mentioned purposes, the technical solution adopted by the application is:

[0017] A ball screw drive error compensation method based on composite integral control, comprising the following steps:

[0018] Step 1, the nonlinear friction of the ball screw drive and the axial and torsional vibration are modeled;

[0019] Step 2, the pitch error is modeled, and the modeling includes a repeated sinusoidal mode and a position-dependent linear offset;

[0020] Step 3, a simplified model of the ball screw drive is established, and the elastic deformation is modeled according to the established simplified dynamic model:

[0021] Step 4, the relationship between the workbench position and the control signal is determined;

[0022] Step 5, define the integral sliding surface;

[0023] Step 6, assuming that the inertia and viscous damping of the ball screw drive change slowly and are not significant, the change of external disturbance is considered to be constant and bounded;

[0024] Step 7, define the Lyapunov function;

[0025] Step 8, define the first derivative of the Lyapunov function:

[0026] Step 9, get the form of the composite integral control law;

[0027] Step 10, in order to ensure that the secondary feedback loop established for the elastic deformation estimation does not endanger the stability of the entire system, a low-pass filter needs to be introduced to reduce the elastic deformation estimation speed,

[0028] Step 11, in order to prevent the closed-loop system from entering the limit cycle when the zero-speed position command is applied, a dead zone filter is also needed;

[0029] Step 12, in order to ensure that the closed-loop control does not endanger the safe operation of the ball screw drive, the loop transfer function needs to be analyzed in the frequency domain for stability, and the Nyquist stability criterion is used.

[0030] As a further improvement of the application, the step 1 is specifically as follows:

[0031] 1.1 The model of nonlinear friction is as follows:

[0032]

[0033] Where ω is the angular speed of the motor, Ω is the speed constant, K a is the amplifier gain, K t is the motor torque constant, T s is the static friction torque, T dis the total dynamic friction torque related to the motor speed, and T s +T d is the total Coulomb friction torque.

[0034] 1.2 Assuming the first axial mode of the drive is f a , and the first three torsional modes are f t1 , f t2 and f t3 , a notch filter is designed to attenuate the first and second torsional modes, and the form of the notch filter is as follows:

[0035]

[0036] where ω1 = 2πf t1 represents the natural frequency of the first torsional mode, ω2 = 2πf t2 represents the natural frequency of the second torsional mode, and ζ is the damping ratio of the notch filter, which is determined according to the selection of the notch width.

[0037] The step 2 establishes a model as follows:

[0038]

[0039] where x is the nominal axis position, A is the pitch error size, h p is the ball screw pitch, Ф is the adjustment position phase, and c0 and c1 are linear interpolation parameters for adjusting the position-related offset.

[0040] As a further improvement of the application, the step 3 is specifically as follows:

[0041] 3.1 The simplified model of the ball screw drive is represented by the following formula:

[0042]

[0043] where m1 and m2 are the equivalent inertias of the motor and the workbench respectively, b1 represents the viscous damping at the motor and the bearing, b2 represents the viscous damping at the guide rail, u T is the transmission torque from the motor to the workbench, u is the control signal, r c is the friction force distribution factor, and 0 ≤ r c ≤ 1, d is the external disturbance, x1, are the displacement, velocity and acceleration of the motor respectively, and x2, are the displacement, velocity and acceleration of the workbench side respectively.

[0044] 3.2 The equivalent axial compliance of the drive, i.e. the inverse of the equivalent axial stiffness, is expressed as follows:

[0045]

[0046] where P1, P2 are determined by static loading test, P1 represents the constant term in the equivalent axial compliance function. It represents the reference compliance value when the table position changes. P2 represents the coefficient in the equivalent axial compliance function which is linearly related to the table position. It shows the degree of contribution of the table position change to the equivalent axial compliance.

[0047] 3.3 Elastic deformation is obtained by dividing the transmitted torque by the equivalent axial stiffness, which is represented by the following formula:

[0048]

[0049] u T From equation (4), we have:

[0050]

[0051] Substituting equation (5) and equation (7) into equation (6), the elastic deformation model is obtained as follows:

[0052]

[0053] As a further improvement of the present application, the step 7 Lyapunov function is defined as follows:

[0054]

[0055] The first derivative of the step 8 Lyapunov function is as follows:

[0056]

[0057] Substituting equation (11) and equation (12) into equation (15) gives:

[0058]

[0059] where And Since the limit of equation (13) is always less than or equal to 0, the following expression will guarantee asymptotic stability, i.e.

[0060]

[0061] where, K s is the feedback gain.

[0062] As a further improvement of the present application, the step 9 compound integral control law form is obtained from equation (17), equation (11) and equation (12) as follows:

[0063]

[0064] The control law is written in the form of feedforward and feedback, the feedforward is The feedback is in the form of:

[0065]

[0066] As a further improvement of the application, the step 11 dead zone filter is in the form of:

[0067]

[0068] Where, f c is the cut-off frequency, obtained by stability analysis and tracking test.

[0069] As a further improvement of the application, the step 11 dead zone filter is in the form of:

[0070]

[0071] Where, v min is the threshold of speed, when the absolute value of speed command is less than v min , the output of dead zone filter is set to zero, so as to prevent the system from entering limit cycle during very small speed command.

[0072] As a further improvement of the application, the step 12 loop transfer function affecting the stability of closed-loop control system is in the form of:

[0073] L(s)=GG N G F -FG f (G F +J e s 2 +B e s) (22)

[0074] Where, G is the transfer function of actuator, F is the axial flexibility of elastic deformation compensation.

[0075] Compared with the prior art, the application has the following technical effects:

[0076] The application provides a ball screw transmission error compensation method based on composite integral control, various factors are estimated through a friction force u fric , a lead error LE and an elastic deformation ED, and an integral sliding surface S and an error observer are designed to obtain a composite integral control law u C , so as to reduce errors and improve the anti-interference ability of the system, and ensure the stability of the closed-loop system. The method can effectively reduce the ball screw transmission error, has robustness to changes of interference such as friction force and cutting force, and improves the precision and performance of the transmission system. BRIEF DESCRIPTION OF DRAWINGS

[0077] Figure 1 The overall control scheme block diagram of a composite integral control ball screw drive error compensation method of the present application;

[0078] Figure 2 For a given sinusoidal reference trajectory with an amplitude of 0.35 and a frequency of 80Hz;

[0079] Figure 3 For a given sinusoidal reference trajectory with an amplitude of 0.35 and a frequency of 80Hz, and without added disturbance, the tracking curve of the driver output;

[0080] Figure 4 For a given sinusoidal reference trajectory with an amplitude of 0.35 and a frequency of 80Hz, and without added disturbance, the tracking error curve of the driver; DETAILED DESCRIPTION

[0081] The present application will be further described in conjunction with the accompanying drawings and specific embodiments:

[0082] Reference Figures 1-4 A composite integral control ball screw drive error compensation method, comprising the following steps:

[0083] Step 1, the nonlinear friction of the ball screw driver and the axial and torsional vibration are modeled, the process is as follows:

[0084] 1.1 The model of nonlinear friction is as follows:

[0085]

[0086] Where ω is the motor angular velocity, Ω is the speed constant, K a is the amplifier gain, K t is the motor torque constant, T s is the static friction torque, T d is the motor speed related dynamic friction torque, so that T s +T d is the total coulomb friction torque;

[0087] 1.2 Assuming that the first axial mode of the driver is f a , the first three torsional modes are f t1 , f t2 and f t3 , a notch filter is designed to attenuate the first and second torsional modes, and the form of the notch filter is as follows:

[0088]

[0089] Where ω1=2πf t1ω1= 2πf1 t2 ω2= 2πf2, ζ is the damping ratio of the notch filter, which is determined by the selection of the notch width.

[0090] Step 2, modeling the lead error, which includes a repeated sinusoidal pattern and a position-dependent linear offset, the model is established as follows:

[0091]

[0092] where x is the nominal axis position, A is the lead error size, h p is the ball screw pitch, Ф is the adjustment position phase, c0 and c1 are linear interpolation parameters, used to adjust the position-dependent offset.

[0093] Step 3, a simplified model of the ball screw drive is established, and the elastic deformation is modeled according to the established simplified dynamic model, the process is as follows:

[0094] 3.1 The simplified model of the ball screw drive is represented by the following formula:

[0095]

[0096] where m1, m2 are the equivalent inertia of the motor and the workbench respectively, b1 represents the viscous damping at the motor and bearing, b2 represents the viscous damping at the guide rail, u T is the motor-to-workbench transmission torque, u is the control signal, r c is the friction force distribution factor, and 0 ≤ r c ≤ 1, d is the external disturbance. x1, are the displacement, velocity, and acceleration of the motor, respectively, x2, are the displacement, velocity, and acceleration of the workbench side, respectively.

[0097] 3.2 The equivalent axial compliance of the drive, which is the inverse of the equivalent axial stiffness, is expressed as follows:

[0098]

[0099] where P1, P2 are determined by static loading tests, P1 represents the constant term in the equivalent axial compliance function. It represents the baseline compliance value when the workbench position changes. P2 represents the coefficient in the equivalent axial compliance function that is linearly related to the workbench position. It shows the degree of contribution of the workbench position change to the equivalent axial compliance.

[0100] 3.3 The elastic deformation is obtained by dividing the transmission torque by the equivalent axial stiffness, which is represented by the following formula:

[0101]

[0102] u T From equation (4), we have:

[0103]

[0104] Substituting equation (5) and equation (7) into equation (6), the elastic deformation model is given as follows:

[0105]

[0106] Step 4, the relationship between the worktable position and the control signal is as follows:

[0107]

[0108] where, J e = m1+ m2 is the equivalent inertia of the actuator, B e = b1+ b2 is the viscous damping of the actuator.

[0109] The velocity of the actuator can be expressed by equation (9) as follows:

[0110]

[0111] Step 5, the integral sliding surface is defined as follows:

[0112]

[0113] where, γ is the proportional gain parameter of the sliding surface. It controls the slope of the sliding surface, i.e., the sensitivity of the sliding surface. λ is the integral gain parameter of the sliding surface. It is used to adjust the integral term of the position error to eliminate the steady-state error of the system. cr = x r + ED-LE is the corrected position command, x is the actual position of the worktable.

[0114] Step 6, it is assumed that the inertia and viscous damping of the ball screw transmission change slowly and are not significant. The change of external disturbance is considered to be constant and bounded. The external disturbance can be tracked by the following observer:

[0115]

[0116] where, T is the control period, k is the control interval counter, ρ is the parameter adaptive gain, is the observed first-order derivative of the disturbance, κ is used to limit the integral control action of the disturbance:

[0117]

[0118] Step 7, the Lyapunov function is defined as follows:

[0119]

[0120] Step 8, the first derivative of the defined Lyapunov function is as follows:

[0121]

[0122] Substitute equation (11) and equation (12) into equation (15) to get:

[0123]

[0124] Where And Since the restriction of equation (13) is always less than or equal to 0, the following expression will guarantee asymptotic stability, i.e.

[0125]

[0126] Where, K s is the feedback gain.

[0127] Step 9, the compound integral control law form can be obtained from equation (17), equation (11) and equation (12) as follows:

[0128]

[0129] Write the control law in the form of feedforward and feedback, the feedforward is The feedback form is as follows:

[0130]

[0131] Step 10, in order to ensure that the secondary positive feedback loop established for the elastic deformation estimation will not endanger the stability of the whole system, it is necessary to introduce a low-pass filter to reduce the speed of elastic deformation estimation, the form of the low-pass filter is as follows:

[0132]

[0133] Where, f c is the cut-off frequency, which is obtained by stability analysis and tracking test.

[0134] Step 11, in order to prevent the closed-loop system from entering the limit cycle when the zero speed position command is applied, a dead zone filter is also needed, which is in the form as follows:

[0135]

[0136] Where, v min is the threshold value of speed, when the absolute value of speed command is less than v minAt this time, the output of the dead-band filter is set to zero, thus preventing the system from entering the limit cycle during very small speed commands.

[0137] Step 12, in order to ensure that the closed-loop control does not endanger the safe operation of the ball screw transmission, the loop transfer function needs to be analyzed in the frequency domain stability, and the Nyquist stability criterion is adopted.

[0138] L(s)=GG N G F -FG f (G F +J e s 2 +B e s) (22)

[0139] Wherein, G is the transmission transfer function, F is the elastic deformation compensation axial flexibility.

[0140] In order to verify the feasibility of the method, the simulation results of the control method on the MATLAB platform are given:

[0141] The parameters are given as follows: in formula (1), Omega=85 rad / s, K a =1.7193A / V, K t =0.57Nm / A, T s =0.294Nm, T d =0.392Nm; in formula (2), f t1 =445Hz, f t2 =1080Hz, zeta=0.05; in formula (3), A=1.987um, h p =20mm, Phi=2.079rad, c0=[0,0.075,0.125,0.230,0.270,0.340], c1=[0,0.075,0.125,0.230,0.270,0.340]; in formula (8), m1=1.85136*10 -3 V / (rad / s 2 ), b1=9.6*10 -4 V / (rad / s), r c =1, P1=5.432*10 -3 rad / V, P2=1.625*10 -5 1 / V; in formula (18), J e =2.20410 -3 V / (rad / s 2 ), B e =1.20*10 -3V / (rad / s), γ=45000, K s =5.56×10 -4 λ=45000, ρ=5.56×10 -4 In equation (20), f c =80Hz; in equation (21), v min =10mm / s.

[0142] from Figure 4 It can be seen that the maximum absolute value of the tracking error without interference is 131.499um and the root mean square error is 66.428um. After interference, the maximum absolute value of the tracking error is 132.20um and the root mean square error is 66.746um. It can be seen that the composite integral control can effectively reduce the ball screw transmission error and is robust to changes in external interference.

[0143] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any modifications or equivalent changes made based on the technical essence of the present invention shall still fall within the scope of protection claimed by the present invention.

Claims

1. A method for compensation of ball screw drive error based on compound integral control, comprising the following steps, characterized in that: Step 1, modeling the nonlinear friction and axial and torsional vibration of the ball screw drive; Step 2, modeling the lead error, which includes a repeating sinusoidal pattern and a position-dependent linear offset; Step 3, establishing a simplified model of the ball screw drive, and modeling the elastic deformation according to the established simplified dynamic model; The step 3 is specifically as follows: 3.

1. The simplified model of the ball screw drive is represented by the following formula: where m1, m2 are the equivalent inertia of the motor and the worktable, b1 represents the viscous damping at the motor and the bearing, b2 represents the viscous damping at the guide, u T is the transmission torque from the motor to the worktable, u is the control signal, r c is the friction distribution factor, and 0≤r c ≤1, u fric is the nonlinear friction, d is the external disturbance, x1, are the displacement, velocity, and acceleration of the motor, respectively, x2, are the displacement, velocity, and acceleration of the worktable side, respectively. 3.

2. The equivalent axial compliance of the drive, that is, the inverse expression of the equivalent axial stiffness, is as follows: Where P1, P2 are determined by static loading test, P1 represents the constant term in the equivalent axial compliance function, which represents the reference compliance value when the position of the workbench changes, and P2 represents the coefficient in the equivalent axial compliance function which is linearly related to the position of the workbench, which indicates the degree of contribution of the position change of the workbench to the equivalent axial compliance; 3.

3. The elastic deformation is obtained by dividing the transmitted torque by the equivalent axial stiffness, which is represented by the following formula: u T From equation (4) we get: The elastic deformation model obtained by bringing formula (5) and formula (7) into formula (6) is as follows: Step 4, determining the relationship between the position of the workbench and the control signal; Specifically as follows: where J e = m1+ m2 is the equivalent inertia of the transmission, B e = b1+ b2 is the viscous damping of the transmission; The velocity of the drive is represented by formula (9) as follows: Step 5, defining the integral sliding surface, which is specifically defined as follows: where γ is the proportional gain parameter of the sliding surface, which controls the slope of the sliding surface, i.e. the sensitivity of the sliding surface, and λ is the integral gain parameter of the sliding surface, which is used to adjust the integral term of the position error to eliminate the steady-state error of the system, x cr = x r + ED-LE is the corrected position command, x is the actual position of the worktable, LE is the lead error model, and ED is the elastic deformation model; Step 6, assuming that the inertia and viscous damping of the ball screw drive change slowly and are not significant, and the change of external disturbance is considered constant and bounded; The external disturbance is tracked by the following observer: where T is the control period, k is the control interval counter, p is the parameter adaptive gain, is the observed first derivative of the disturbance, and K is used to limit the integral control action of the disturbance: Step 7, defining the Lyapunov function; Step 8, defining the first derivative of the Lyapunov function: Step 9, obtaining the form of the compound integral control law as follows: where K s is the feedback gain; The control law is written in the form of feedforward and feedback, the feedforward being The feedback is in the form of: Step 10, in order to ensure that the secondary feedback loop established for the estimation of elastic deformation does not endanger the stability of the entire system, a low-pass filter needs to be introduced to reduce the estimation speed of the elastic deformation, Step 11, in order to prevent the closed-loop system from entering the limit cycle when the zero-speed position command is applied, a dead zone filter is also needed; Step 12, in order to ensure the safe operation of the ball screw drive under closed-loop control, the frequency domain stability analysis of the loop transfer function is needed, and the Nyquist stability criterion is adopted.

2. The ball screw drive error compensation method based on composite integral control according to claim 1, characterized in that, The step 1 is specifically as follows: 1.

1. The model of nonlinear friction is as follows: where ω is the motor angular velocity, Ω is the speed constant, K a is the amplifier gain, K t is the motor torque constant, T s is the static friction torque, T d is the motor speed dependent dynamic friction torque, such that T s + T d is the total Coulomb friction torque; 1.

2. Assume that the first axial mode of the drive is at f a , and the first three torsional modes are at f t1 , f t2 , and f t3 . Design a notch filter to attenuate the first and second torsional modes, in the form: where ω1= 2πf t1 represents the natural frequency of the first torsional mode, ω2= 2πf t2 represents the natural frequency of the second torsional mode, and ζ is the damping ratio of the notch filter, which value is determined according to the choice of the notch width.

3. The ball screw drive error compensation method based on composite integral control according to claim 1, characterized in that, The step 2 establishes the model as follows: where x is the nominal axis position, A is the pitch error magnitude, h p is the ball screw pitch, φ is the adjusted position phase, and c0and c1are linear interpolation parameters used to adjust the position dependent offset.

4. The ball screw drive error compensation method based on compound integral control according to claim 3, characterized in that, The step 7 defines the Lyapunov function as follows: The first derivative of the Lyapunov function in step 8 is as follows: Bring formula (11) and formula (12) into formula (15) to get: wherein And Since the restriction of formula (13) is always less than or equal to 0, the following expression will guarantee asymptotic stability, i.e. where K s is the feedback gain.

5. The ball screw drive error compensation method based on compound integral control according to claim 4, characterized in that, The form of the low-pass filter in step 10 is as follows: where f c is the cutoff frequency, obtained from stability analysis and tracking tests.

6. The ball screw drive error compensation method based on compound integral control according to claim 5, characterized in that, The form of the dead zone filter in step 11 is as follows: where v min is a threshold value for the speed, and when the absolute value of the speed command is less than v min the output of the dead-band filter is set to zero, thus preventing the system from entering the limit cycle during very small speed commands. The form of the loop transfer function affecting the stability of the closed-loop control system in step 12 is as follows: L(s) = GG N G F -FG f (G F +J e s 2 +B e s) (22) where G is the transmission function of the transmission, G N is a notch filter, and F is the axial flexibility of the elastic deformation compensation.

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