Composite sliding mode position tracking control method for single-axis feed system

Abstract

The application discloses a composite sliding mode position tracking control method of a single-axis feeding system, and relates to the technical field of servo feeding control of numerical control equipment; the method comprises the following steps: establishing a double-freedom equivalent dynamic model of a ball screw single-axis feeding system and converting the model into a state space expression; acquiring the displacement of a motor side and a workbench side and a reference displacement, calculating the position error and the error derivative of the two sides, and constructing a composite error chain sliding mode variable; calculating a nominal equivalent control item according to the dynamic relationship, and calculating an approaching control item by using a non-homogeneous super-spiral algorithm containing a non-homogeneous power term, a linear term and an integral state; synthesizing the two control items into an equivalent axial control input, converting the input into a servo motor torque instruction and outputting the instruction to a servo driver; the application can take into account the position tracking precision, the anti-interference performance and the control input smoothness of the single-axis feeding system, and realizes high-precision and low-jitter position tracking control.

Description

Technical Field

[0001] This application relates to the field of servo feed control technology for CNC equipment, and more specifically, to a composite sliding mode position tracking control method for a single-axis feed system. Background Technology

[0002] Single-axis feed systems are crucial components of CNC machine tools, precision machining equipment, automated testing equipment, and linear motion actuators. They typically consist of a servo motor, coupling, ball screw pair, worktable, linear guide, and support components. The servo motor's output torque is transmitted to the ball screw via transmission components, and the ball screw pair converts the rotary motion into the linear motion of the worktable, thus achieving target position trajectory tracking. With increasing demands for high-precision machining, rapid positioning, and complex trajectory motion, single-axis feed systems not only need to have small steady-state position errors but also need to maintain fast dynamic response and relatively stable control input under the influence of reference trajectory changes, load fluctuations, and external disturbances.

[0003] Actual single-axis feed systems are not ideal rigid body structures. Components such as servo motors, couplings, ball screws, screw nuts, and worktables possess mass, moment of inertia, stiffness, and damping characteristics. Elastic deformation, vibration, and asynchronous motion on both sides may also exist in the transmission chain. Especially during high-speed start-stop, trajectory reversal, and changes in external load, dynamic differences can easily arise between the motion states on the motor side and the motion states on the worktable side. To improve the position tracking performance of the feed system, existing technologies typically employ methods such as proportional-integral-derivative control, feedforward control, sliding mode control, adaptive control, or disturbance compensation control, generating servo drive commands based on position or velocity errors.

[0004] In existing control methods, some schemes simplify the ball screw feed system into a single-mass rigid body model when establishing the controlled object model, or only use the position and velocity of the worktable side as the main control state. While this approach can reduce the complexity of the model and controller, it does not adequately consider the flexible connection, state coupling, and error propagation process between the motor side and the worktable side. When the system has significant transmission elasticity, damping variations, or dynamic asynchrony between the two sides, relying solely on the error of the worktable side is insufficient to fully reflect the actual motion state of the transmission chain, which may affect the accuracy of position tracking control and dynamic response performance.

[0005] Traditional sliding mode control is characterized by its simple structure and strong robustness, but it typically includes discontinuous switching control terms. Influenced by factors such as the controller sampling period, actuator bandwidth, model error, and state measurement noise, the switching control terms tend to change frequently near the sliding surface, causing significant high-frequency fluctuations in the control input and potentially leading to actuator chattering. While some higher-order sliding mode or superspiral control methods can mitigate these problems to some extent, under conditions of rapid changes in the reference trajectory, trajectory inflection points, or load disturbances, it remains difficult to simultaneously achieve optimal position tracking accuracy, dynamic response speed, and control input smoothness.

[0006] Furthermore, some disturbance compensation methods require real-time measurement of external loads or the addition of disturbance observers to estimate unknown disturbances. External disturbances are typically time-varying and uncertain, making accurate measurement difficult. The introduction of disturbance observers also increases the complexity of controller structure and parameter tuning and may be affected by model parameter deviations and measurement noise. Therefore, how to fully utilize the motion information from both sides of the drive train of a single-axis feed system without over-relying on external disturbance measurement or estimation, while improving position tracking accuracy and disturbance rejection performance and reducing high-frequency fluctuations in control input, remains a technical problem that needs to be solved in the position tracking control of existing single-axis feed systems.

[0007] In view of this, this application proposes a composite sliding mode position tracking control method for a single-axis feed system to solve the above problems. Summary of the Invention

[0008] To address the problems of insufficient dynamic state representation and difficulty in balancing position tracking accuracy and control input smoothness in existing single-axis feed systems under conditions of flexible transmission, external disturbances, and trajectory changes, this application proposes a composite sliding mode position tracking control method for single-axis feed systems. This method establishes an equivalent dynamic model that characterizes the coupling relationship between the motor side and the worktable side, comprehensively utilizes the position errors and their changing states on both sides to construct composite error chain sliding mode variables, and combines nominal equivalent control and non-homogeneous super-spiral approach control to generate servo motor torque commands. This improves position tracking accuracy and disturbance rejection performance, and reduces high-frequency fluctuations in control input.

[0009] This application proposes a composite sliding mode position tracking control method for a single-axis feed system, including:

[0010] A rotational-linear equivalent two-degree-of-freedom dynamic model of a single-axis ball screw feed system is established. Based on the ball screw transmission relationship, the rotational-linear equivalent two-degree-of-freedom dynamic model is converted into a linear-linear equivalent dynamic model, and the linear-linear equivalent dynamic model is converted into a state-space expression.

[0011] Based on the state-space expression, the equivalent linear displacement on the motor side, the linear displacement on the worktable side, and their corresponding reference displacements are obtained. The equivalent position error on the motor side, the position error on the worktable side, the derivative of the equivalent position error on the motor side, and the derivative of the position error on the worktable side are calculated to form a composite error state set. Based on the composite error state set, a composite error chain sliding mode variable is constructed.

[0012] The dynamic relationship of sliding mode variables in the composite error chain is determined based on the state-space expression. During the calculation process, the nominal equivalent control term is not used as the calculation input by the lumped external disturbance measurement value. For bounded lumped external disturbances, the integral state is updated and the non-homogeneous super-spiral approaching control term is calculated based on the non-homogeneous power term, linear term and integral state.

[0013] The nominal equivalent control term is added to the non-homogeneous super-spiral approach control term to obtain the equivalent axial control input; the equivalent axial control input is converted into a servo motor torque command according to the ball screw transmission relationship and output to the servo driver.

[0014] Furthermore, the methods for establishing the rotation-linear equivalent two-degree-of-freedom dynamic model include:

[0015] Using the lumped parameter method, the servo motor, coupling, and ball screw are equivalent to rotary motion components, and the screw nut and worktable are equivalent to linear motion components. The equivalent axial stiffness and equivalent axial damping coefficient are used to characterize the flexible connection relationship between the rotary motion components and the linear motion components.

[0016] Using the angular displacement of the rotating component and the linear displacement of the linear component as generalized coordinates, the system's kinetic energy, potential energy, energy dissipation function, and generalized external force vector are determined respectively. Substituting these into the Lagrange equation, a rotational-linear equivalent two-degree-of-freedom dynamic model is obtained, which includes the equivalent moment of inertia of the rotating component, the equivalent mass of the linear component, the equivalent viscous damping coefficient of the rotating component, the equivalent viscous damping coefficient of the linear component, the equivalent axial stiffness, the equivalent axial damping coefficient, the servo motor input torque, and the lumped external disturbances acting on the worktable side.

[0017] Furthermore, the methods for obtaining the state-space expression include:

[0018] The displacement conversion coefficient from rotary motion to linear motion is determined based on the ball screw lead. The product of the angular displacement of the rotary motion component and the displacement conversion coefficient is determined as the equivalent linear displacement on the motor side, and the linear displacement of the linear motion component is determined as the linear displacement on the worktable side.

[0019] Based on the displacement conversion coefficient, the servo motor input torque is converted into an equivalent axial control input, the equivalent moment of inertia of the rotating motion component is converted into an equivalent mass on the motor side, and the equivalent viscous damping coefficient of the rotating motion component is converted into an equivalent axial viscous damping coefficient on the motor side, thus obtaining a linear-linear equivalent dynamic model.

[0020] The equivalent linear displacement on the motor side, the linear displacement on the worktable side, and their corresponding equivalent linear velocities are used to form the system state vector, and the state space expression is obtained based on the linear-linear equivalent dynamic model.

[0021] Furthermore, the methods for forming the composite error state set include:

[0022] The equivalent linear velocity on the motor side and the linear velocity on the worktable side are obtained based on the state-space expression. The time derivatives of the reference equivalent linear displacement on the motor side and the reference linear displacement on the worktable side are calculated to obtain the reference equivalent linear velocity on the motor side and the reference linear velocity on the worktable side.

[0023] The difference between the reference equivalent linear displacement on the motor side and the equivalent linear displacement on the motor side is defined as the equivalent position error on the motor side, and the difference between the reference linear displacement on the worktable side and the linear displacement on the worktable side is defined as the position error on the worktable side.

[0024] The difference between the reference equivalent linear velocity on the motor side and the equivalent linear velocity on the motor side is determined as the derivative of the equivalent position error on the motor side, and the difference between the reference linear velocity on the worktable side and the linear velocity on the worktable side is determined as the derivative of the position error on the worktable side.

[0025] The equivalent position error on the motor side, the position error on the worktable side, the derivative of the equivalent position error on the motor side, and the derivative of the position error on the worktable side are combined to form a composite error state set.

[0026] Furthermore, based on the set of composite error states, a sliding mode variable for the composite error chain is constructed, including:

[0027] Set the sliding surface gain corresponding to the table-side position error, the motor-side equivalent position error, the table-side position error derivative, and the motor-side equivalent position error derivative, respectively;

[0028] Each error state in the composite error state set is combined with its corresponding sliding surface gain, and the combination results are combined to obtain the sliding mode variables of the composite error chain.

[0029] Furthermore, the nominal equivalent control terms that do not use lumped external disturbance measurements as calculation inputs during the calculation process include:

[0030] The dynamic relationship of the sliding mode variables in the composite error chain is expressed as the sum of the nominal dynamic term and the equivalent disturbance corresponding to the lumped external disturbance, minus the product of the equivalent axial control input and the input gain. The nominal dynamic term is calculated based on the motor-side reference equivalent linear acceleration, the table-side reference linear acceleration, the composite error state set, the table-side dynamic state term, the motor-side dynamic state term, and the sliding surface gain. The input gain is determined based on the sliding surface gain corresponding to the derivative of the motor-side equivalent position error and the motor-side equivalent mass. Under nominal dynamic conditions, the equivalent disturbance is set to zero, and the first derivative of the composite error chain sliding mode variables is set to zero. The ratio of the nominal dynamic term to the input gain is then determined as the nominal equivalent control term.

[0031] Furthermore, the integral state is updated and the non-homogeneous superspinning approach control term is calculated, including:

[0032] The non-homogeneous superspiral approach control gain, integral state update gain, non-homogeneous power term adjustment parameter and linear term adjustment parameter are all positive, and a non-homogeneous power exponent greater than 0 and less than 1 is set.

[0033] The sign function term of the composite error chain sliding mode variable is adjusted according to the non-homogeneous power term adjustment parameter, and the composite error chain sliding mode variable is linearly adjusted according to the linear term adjustment parameter. The two adjustment results are combined and the integral state with the initial value of zero is updated through the integral state update gain.

[0034] A nonhomogeneous power term is constructed based on the absolute value, sign function, and nonhomogeneous power exponent of the sliding mode variable in the composite error chain. This nonhomogeneous power term is then adjusted using the adjustment parameter. The adjusted nonhomogeneous power term is combined with the linear adjustment result and processed through the nonhomogeneous superspiral approach control gain. Finally, it is added to the updated integral state and multiplied by the reciprocal of the input gain to obtain the nonhomogeneous superspiral approach control term.

[0035] Furthermore, based on the ball screw transmission relationship, the equivalent axial control input is converted into a servo motor torque command, including:

[0036] The ratio of the ball screw lead to 2π is determined as the displacement conversion coefficient. The equivalent axial control input is multiplied by the displacement conversion coefficient to obtain the servo motor torque command.

[0037] Furthermore, the sliding mode variables of the composite error chain satisfy:

[0038] ;

[0039] in, For the sliding mode variables of the composite error chain, This is the gain for the position error on the worktable side. This is the equivalent position error gain on the motor side. The derivative gain of the position error on the worktable side. The derivative gain of the equivalent position error on the motor side. , , and All values ​​are positive; For the positional error of the worktable side, This refers to the equivalent position error on the motor side. The derivative of the position error on the worktable side. This is the derivative of the equivalent position error on the motor side.

[0040] Furthermore, the sliding mode variable of the composite error chain is denoted as... Record the integration state as The integral state satisfies:

[0041] ;

[0042] ;

[0043] The nonhomogeneous superspiral approach control term satisfies:

[0044] ;

[0045] in, This is the non-homogeneous superspiral approach control term. For the equivalent mass on the motor side, This represents the sliding surface gain corresponding to the derivative of the equivalent position error on the motor side. For non-homogeneous superspiral approach control gain, Update the gain for the integral state. For non-homogeneous power terms, adjust the parameter. For adjusting parameters of the linear term, It is a non-homogeneous exponent that is greater than 0 and less than 1.

[0046] The composite sliding mode position tracking control method for a single-axis feed system proposed in this application has the following technical effects and advantages:

[0047] This application establishes a rotational-linear equivalent two-degree-of-freedom dynamic model for a single-axis ball screw feed system, and transforms it into a linear-linear equivalent dynamic model based on the ball screw transmission relationship. This allows the motor side and the worktable side to be described using a unified linear motion form. This enables the simultaneous characterization of the influence of motor-side inertia, worktable-side mass, transmission chain flexibility, viscous damping, and external loads on the system's motion state, improving the matching degree between the dynamic model and the actual single-axis feed system transmission process, and providing a more accurate model basis for position tracking control.

[0048] This application constructs a composite error state set by combining the equivalent position error on the motor side, the position error on the worktable side, and their corresponding position error derivatives, and then constructs sliding mode variables for the composite error chain using the corresponding sliding mode surface gain. Compared to constructing sliding mode variables solely based on the position error on the worktable side, this approach simultaneously utilizes motion information from both the motor and worktable sides to reflect the tracking deviations and error trends on both sides of the transmission chain. This enhances the characterization ability of state coupling and dynamic asynchrony phenomena caused by flexible transmission, and is beneficial for improving position tracking accuracy.

[0049] This application calculates the nominal equivalent control term based on the dynamic relationship of sliding mode variables in a composite error chain. This nominal equivalent control term does not directly include the lumped external disturbance term, but is determined based on the system state, reference motion state, sliding surface gain, and nominal dynamic model parameters. Therefore, it eliminates the need to use real-time measurements of the lumped external disturbance as controller input, and also eliminates the need for an additional disturbance observer, simplifying the controller structure and reducing dependence on the accuracy of external disturbance measurement and estimation.

[0050] For bounded lumped external disturbances, this application employs a nonhomogeneous superspiral algorithm, comprising nonhomogeneous power terms, linear terms, and integral states, to calculate the approaching control term. The nonhomogeneous power terms provide nonlinear regulation based on the amplitude changes of the sliding mode variables in the composite error chain; the linear terms enhance the regulation capability when the sliding mode variables deviate from the sliding surface; and the integral states continuously accumulate relevant information about the sliding mode variables. Through the synergistic effect of these three components, the system's disturbance rejection capability can be improved while reducing high-frequency control fluctuations caused by switching control in traditional sliding mode control, thus improving the continuity and smoothness of the control input.

[0051] This application also performs proportional matching on the non-homogeneous super-spiral approaching control term based on the input gain in the dynamic relationship of sliding mode variables in the composite error chain, enabling the nominal equivalent control term and the non-homogeneous super-spiral approaching control term to be combined in a unified equivalent axial control input form. Subsequently, based on the ball screw transmission relationship, the equivalent axial control input is converted into a servo motor torque command, forming a complete closed-loop control process from reference displacement, state feedback, error calculation, control term generation to torque command output, thereby improving the adaptability between the control law and the actual actuator.

[0052] In summary, this application can balance the position tracking accuracy, disturbance rejection performance, and control input smoothness of a single-axis feed system, achieving high-precision, low-jitter position tracking control. Attached Figure Description

[0053] Figure 1 This is a flowchart of a composite sliding mode position tracking control method for a single-axis feed system according to this application;

[0054] Figure 2This is a schematic diagram of the structure of the single-axis feed system of this application;

[0055] Figure 3 This is a block diagram of the composite sliding mode position tracking control of the single-axis feed system of this application;

[0056] Figure 4 This is the position tracking curve under the step reference displacement of this application;

[0057] Figure 5 This is the tracking error curve under the step reference displacement of this application;

[0058] Figure 6 This is the position tracking curve under the triangular wave reference displacement of this application;

[0059] Figure 7 This is the tracking error curve under the triangular wave reference displacement of this application. Detailed Implementation

[0060] The technical solutions in the embodiments of this application will be described in detail, clearly and completely below with reference to the accompanying drawings.

[0061] Example 1:

[0062] like Figure 1 As shown, this embodiment discloses a composite sliding mode position tracking control method for a single-axis feed system, including:

[0063] A rotary-linear equivalent two-degree-of-freedom dynamic model of a single-axis ball screw feed system is established. Based on the ball screw transmission relationship, the rotary-linear equivalent two-degree-of-freedom dynamic model is converted into a linear-linear equivalent dynamic model, and then the linear-linear equivalent dynamic model is converted into a state-space expression. The specific implementation method is as follows:

[0064] This embodiment uses a single-axis ball screw feed system as the controlled object. The single-axis feed system includes a servo motor, coupling, bearings, ball screw, screw nut, worktable, linear guide, and base. The output torque of the servo motor is transmitted to the ball screw via the coupling. The ball screw and screw nut cooperate to convert the rotary motion into the linear motion of the worktable.

[0065] It should be noted that, Figure 2 This is a schematic diagram of the structure of a single-axis feed system.

[0066] The single-axis feed system is simplified into a rotational-linear equivalent two-degree-of-freedom dynamic model using the lumped parameter method. The servo motor, coupling, and ball screw are equated as rotational motion components, while the screw nut and worktable are equated as linear motion components. Equivalent axial stiffness and equivalent axial damping coefficient are used to characterize the flexible connection between the rotational and linear motion components.

[0067] It should be noted that the lumped parameter method referred to in this application refers to the method of equivalently concentrating the physical parameters such as mass, moment of inertia, stiffness and damping that are continuously distributed in the single-axis feed system of the ball screw into a finite number of motion units according to the motion relationship of each component. Each motion unit and its connection relationship are characterized by equivalent mass, equivalent moment of inertia, equivalent axial stiffness and equivalent axial damping coefficient, thereby simplifying the actual feed system into a dynamic model with finite degrees of freedom.

[0068] The generalized coordinate vector of the rotation-linear equivalent two-degree-of-freedom dynamic model is defined as follows:

[0069] ;

[0070] in, For generalized coordinate vectors, This refers to the angular displacement of the rotating moving part. For linear displacement of linearly moving parts, superscript This represents the transpose of a vector.

[0071] The system kinetic energy of the rotational-linear equivalent two-degree-of-freedom dynamic model is:

[0072] ;

[0073] in, As the system's kinetic energy, The equivalent moment of inertia of the rotating component. The equivalent mass of the linearly moving component. The angular velocity of the rotating moving part. Let be the linear velocity of the linearly moving component, and the point above represents the first derivative of the corresponding variable with respect to time.

[0074] The system potential energy of the rotational-linear equivalent two-degree-of-freedom dynamic model is:

[0075] ;

[0076] in, For the system potential energy, This is the equivalent axial stiffness of a single-axis feed system. This is the displacement conversion coefficient from rotational motion to linear motion. It is the relative displacement between the equivalent linear displacement of the rotating component and the linear displacement of the linear component.

[0077] The energy dissipation function of the rotation-linear equivalent two-degree-of-freedom dynamic model is:

[0078] ;

[0079] in, Let be the energy dissipation function. Let be the equivalent viscous damping coefficient of the rotating component. is the equivalent viscous damping coefficient of the linearly moving component. This is the equivalent axial damping coefficient for the flexible connection between the rotary and linear motion components. The relative velocity between two equivalent moving parts.

[0080] The generalized external force vector of the rotational-linear equivalent two-degree-of-freedom dynamic model is:

[0081] ;

[0082] in, For generalized external force vectors, This refers to the input torque of the servo motor. The negative sign indicates that the direction of the lumped external disturbance acting on the worktable side is opposite to the positive direction of the linear displacement on the worktable side.

[0083] According to the Lagrange equation:

[0084] ;

[0085] in, For Lagrange functions, Let be the dissipation function. For the first A generalized coordinate system For the first The first derivative of a generalized coordinate with respect to time In order to be with the first Generalized external forces corresponding to each generalized coordinate For generalized coordinate sequence number.

[0086] Substituting the system kinetic energy, system potential energy, energy dissipation function, and generalized external force vector into the Lagrange equations, we obtain a rotational-linear equivalent two-degree-of-freedom dynamic model:

[0087] ;

[0088] in, For the angular acceleration of the rotating moving part, The linear acceleration of the linearly moving component is represented by the two points above, which indicate the second derivative of the corresponding variable with respect to time. Including the rotational inertia of servo motors, couplings, and ball screws. Including the mass of the lead screw nut and the worktable; displacement conversion factor satisfy:

[0089] ;

[0090] in, This refers to the lead of the ball screw.

[0091] To ensure that two equivalent moving parts use a unified linear displacement representation, the following definition is provided:

[0092] ;

[0093] ;

[0094] ;

[0095] ;

[0096] in, This is the equivalent linear displacement on the motor side. This represents the linear displacement of the worktable side. The equivalent axial control input is obtained by converting the input torque of the servo motor. For the equivalent mass on the motor side, For the equivalent mass of the workbench side, This is the equivalent axial viscous damping coefficient on the motor side. It is the equivalent axial viscous damping coefficient on the worktable side.

[0097] This leads to the linear-linear equivalent dynamic model:

[0098] ;

[0099] in, For the lumped external disturbance acting on the workbench side, and These are the equivalent linear velocity on the motor side and the linear velocity on the worktable side, respectively. and These are the equivalent linear acceleration on the motor side and the linear acceleration on the worktable side, respectively. The equivalent linear velocity on the motor side and the linear velocity on the worktable side are calculated based on the changes in the sampled values ​​of the equivalent linear displacement on the motor side and the linear displacement on the worktable side within adjacent control cycles, and the calculation results are obtained after filtering.

[0100] Define the state vector and output vector as follows:

[0101] ;

[0102] ;

[0103] The state-space expression of the linear-linear equivalent dynamic model is:

[0104] ;

[0105] ;

[0106] in:

[0107] ;

[0108] ;

[0109] ;

[0110] in, Let be the system state vector. The first derivative of the system state vector with respect to time. For the system output vector, The state matrix, To control the input matrix, The perturbation input matrix, This is the output matrix. The state variables in the state-space expression include the equivalent linear displacement on the motor side, the linear displacement on the worktable side, and their corresponding equivalent linear velocities. The state-space expression is used to subsequently calculate the position and velocity errors on the motor side and the worktable side, and to construct the sliding mode variables of the composite error chain.

[0111] Based on the state-space expression, the equivalent linear displacement on the motor side, the linear displacement on the worktable side, and their corresponding reference displacements are obtained. The equivalent position error on the motor side, the position error on the worktable side, the derivative of the equivalent position error on the motor side, and the derivative of the position error on the worktable side are calculated to form a composite error state set. Based on this composite error state set, a composite error chain sliding mode variable is constructed. The specific implementation method is as follows:

[0112] Based on the state-space expression of the aforementioned single-axis feed system, the equivalent linear displacement on the motor side is obtained. Linear displacement of the worktable side Equivalent linear velocity on the motor side and linear velocity on the worktable side Simultaneously, the equivalent linear displacement of the motor side is obtained. and the reference linear displacement of the worktable side .

[0113] Among them, the equivalent linear displacement on the motor side and the linear displacement on the worktable side are used as the position feedback input controller of the single-axis feed system; the equivalent linear velocity on the motor side and the linear velocity on the worktable side are obtained according to the aforementioned sampling value difference and filtering method, and are used to calculate the derivative of the equivalent position error on the motor side and the derivative of the position error on the worktable side.

[0114] The motor-side reference equivalent linear displacement and the table-side reference linear displacement are pre-set reference displacement trajectories. The reference displacement trajectory used for control calculation is at least second-order differentiable. Time derivatives are obtained for each trajectory to yield the motor-side reference equivalent linear velocity, table-side reference linear velocity, motor-side reference equivalent linear acceleration, and table-side reference linear acceleration. When the original reference displacement command contains step changes, triangular wave inflection points, or other non-differentiable positions, the corresponding positions are first smoothed using a trajectory smoothing process to generate continuous reference displacement, reference velocity, and reference acceleration, which are then used for calculating the composite error state and nominal equivalent control terms.

[0115] Calculate the position error of the worktable side based on the reference linear displacement and the linear displacement of the worktable side:

[0116] ;

[0117] Calculate the equivalent position error on the motor side based on the reference equivalent linear displacement and the equivalent linear displacement on the motor side:

[0118] ;

[0119] Taking the time derivatives of the table-side position error and the motor-side equivalent position error respectively, we get:

[0120] ;

[0121] ;

[0122] in, For the positional error of the worktable side, This refers to the equivalent position error on the motor side. This is the derivative of the position error on the worktable side, which is also the speed error on the worktable side. This is the derivative of the equivalent position error on the motor side, which is also the equivalent speed error on the motor side; superscript The first dot represents the reference value of the corresponding variable, the second dot above it represents the first derivative of the corresponding variable with respect to time, and the two dots above it represent the second derivative of the corresponding variable with respect to time.

[0123] The composite error state set is composed of the worktable-side position error, the motor-side equivalent position error, the worktable-side position error derivative, and the motor-side equivalent position error derivative, and is represented as a composite error state vector:

[0124] ;

[0125] in, For the composite error state vector, the superscript... This represents the transpose of a vector.

[0126] Based on the state-space expression of the single-axis feed system, the dynamic relationship of the composite error state vector is as follows:

[0127] ;

[0128] in:

[0129] ;

[0130] ;

[0131] in, The first derivative of the composite error state vector with respect to time; The table-side dynamic state term consists of state-related terms in the table-side dynamic equations, excluding the lumped external disturbance term. The motor-side dynamic state term consists of state-related terms in the motor-side dynamic equations, excluding the equivalent axial control input term. For the equivalent mass on the motor side, For the equivalent mass of the workbench side, For equivalent axial stiffness, This is the equivalent axial damping coefficient for the flexible connection between the rotary and linear motion components. This represents the equivalent axial viscous damping coefficient on the motor side. This is the equivalent axial viscous damping coefficient on the worktable side. This is the equivalent axial control input. This refers to the lumped external disturbance acting on the workbench side.

[0132] Construct a composite error chain sliding mode variable based on the aforementioned composite error state set:

[0133] ;

[0134] in, For the sliding mode variables of the composite error chain, satisfying The set of states constitutes the sliding surface of the composite error chain; This is the gain for the position error on the worktable side. This is the equivalent position error gain on the motor side. The derivative gain of the position error on the worktable side. The derivative gain of the equivalent position error on the motor side, and , , and All values ​​are positive.

[0135] It should be noted that the sliding surface gain , , and All values ​​are positive. The setting methods include: under the same single-axis feed system dynamic model and other controller parameters, inputting sinusoidal reference displacement, smoothed step reference displacement, and triangular wave reference displacement respectively to obtain the table-side position error. Equivalent position error on the motor side Derivative of the position error on the workbench side Derivative of equivalent position error on the motor side The range of variation; set the initial value of the corresponding sliding surface gain according to the numerical range of each error state, so that each error state participates in the calculation of the sliding variable of the composite error chain after being processed by the corresponding sliding surface gain.

[0136] To ensure that the positional error on the worktable side, the equivalent positional error on the motor side, and their corresponding positional error derivatives can be combined within the same composite error chain sliding mode variable, the sliding surface gain is set according to the units used for each error state. In this embodiment, the positional error is expressed in millimeters, and the positional error derivative is expressed in millimeters per second. Therefore, the positional error gain on the worktable side... Equivalent position error gain on the motor side The units are all per millimeter, and the derivative gain of the worktable side position error is... Equivalent position error derivative gain on the motor side The units are all seconds per millimeter. Therefore, the product of each error state and the corresponding sliding surface gain is dimensionless, and the constructed composite error chain sliding mode variables... It is a dimensionless variable.

[0137] Based on the initial values, keeping the non-homogeneous superspin control parameters and the controlled object parameters unchanged, for , , and Adjustments were made, and changes in position tracking error and control input were recorded. Position tracking performance was evaluated using steady-state root mean square error, steady-state average absolute error, steady-state maximum absolute error, overshoot, rise time, and settling time. Control input smoothness was evaluated using the root mean square value, peak-to-peak value, and high-frequency residual of the control input. When increasing the corresponding sliding surface gain reduced the position tracking error, and the peak-to-peak value and high-frequency residual of the control input were not higher than the corresponding evaluation index values ​​before adjustment, the adjusted parameter values ​​were retained. When the position tracking error evaluation index did not decrease, or the peak-to-peak value or high-frequency residual of the control input was higher than the corresponding evaluation index values ​​before adjustment, the parameter values ​​before adjustment were used. The sliding surface gain was determined by comprehensively comparing the evaluation results under three reference displacement input conditions.

[0138] In the data example of this embodiment, the simulation time is set to 10 seconds, and a fixed step size is used. The fourth-order Runge-Kutta algorithm was used to solve the problem. At t=3s, a step-type lumped external disturbance with an amplitude of 100 N was applied to the worktable side. Tests were conducted using a sinusoidal reference displacement, a step reference displacement varying from 0 mm to 500 mm and smoothed by a speed-limiting circuit, and a triangular wave reference displacement with a frequency of 0.5 Hz. Based on the above settings, the following parameters were determined: For 150, For 50, 0.4 The value is 4.

[0139] The sliding mode variable of the composite error chain As the calculation input for the nominal equivalent control term and the non-homogeneous superspiral approaching control term.

[0140] It should be noted that, Figure 3 This is a block diagram of the composite sliding mode position tracking control of the single-axis feed system of this application.

[0141] The dynamic relationship of sliding mode variables in the composite error chain is determined based on the state-space expression. During the calculation, the nominal equivalent control term is not used as the input of the lumped external disturbance measurement. For bounded lumped external disturbances, the integral state is updated and the non-homogeneous superspiral approaching control term is calculated based on a non-homogeneous power term, a linear term, and an integral state. The specific implementation is as follows:

[0142] Based on the state-space expression of the aforementioned single-axis feed system, the set of composite error states, and the dynamic relationship of the sliding mode variables of the composite error chain, the nominal equivalent control term and the non-homogeneous superspiral approaching control term are calculated.

[0143] To characterize the relationship between nominal dynamics, lumped external disturbances, and equivalent axial control inputs in the dynamic relationship of the aforementioned composite error chain sliding mode variables, the dynamic relationship of the aforementioned composite error chain sliding mode variables is summarized as follows:

[0144] ;

[0145] in:

[0146] ;

[0147] ;

[0148] in, For nominal dynamic items; This is the equivalent disturbance of the lumped external disturbance in the dynamic relationship of sliding mode variables in the composite error chain; The input gain is the equivalent axial control input in the dynamic relationship.

[0149] It should be noted that the nominal dynamic item Equivalent disturbance and input gain None of these are control parameters that require separate tuning. Among them, the nominal dynamic term... The equivalent disturbance is calculated based on the composite error state, reference equivalent linear acceleration, reference linear acceleration, table-side dynamics state term, motor-side dynamics state term, and the aforementioned sliding surface gain obtained within the current control cycle. This is used to characterize the effect of lumped external disturbances on the dynamic relationship of sliding mode variables in a composite error chain. It is only used for system modeling and control law derivation and does not participate in the online calculation of the actual controller; input gain Sliding surface gain corresponding to the derivative of the equivalent position error on the motor side Equivalent mass of the motor side They are jointly determined, and are subsequently determined after the sliding surface gain and dynamic model parameters are determined.

[0150] In this embodiment, the lumped external disturbance satisfies a bounded condition within the predetermined operating range of the single-axis feed system, that is, there exists a positive real number. , so that:

[0151] ;

[0152] in, This is the upper bound of the amplitude of the lumped external disturbance. The lumped external disturbance is an external input to the controlled object acting on the workbench side, and is not used as a direct calculation input for the nominal equivalent control term, the non-homogeneous superspiral integral state, or the non-homogeneous superspiral approaching control term.

[0153] It should be noted that the upper bound of the amplitude of the lumped external disturbance is... The amplitude upper bound is determined based on the total external disturbances that may act on the worktable side under the predetermined operating conditions of the single-axis feed system. Specifically, when simulating using a preset external disturbance signal, the absolute value of the preset external disturbance signal is obtained over the entire simulation time interval, and the maximum value of its absolute value is determined as the upper bound of the amplitude. When using an actual single-axis feed system for control, the maximum value of the absolute value of the lumped external disturbance is determined based on the allowable external load range of the worktable or pre-acquired external load test data, and this maximum value is defined as the upper limit of the amplitude. The upper limit of the amplitude. Bounded conditions used to limit lumped external disturbances are not used as input parameters for the actual controller's online calculations.

[0154] In this embodiment, the applied step-type lumped external disturbance is 0N before the simulation time reaches 3s and 100N after the simulation time reaches 3s. Therefore, the maximum absolute value of the lumped external disturbance over the entire simulation time interval is 100N. Accordingly, the upper bound of the amplitude of the lumped external disturbance is set. Set to 100 N.

[0155] The upper limit of amplitude With the equivalent disturbance They have different meanings, among which, Used to denote the upper bound of the absolute value of lumped external disturbances. It is the equivalent effect of the lumped external disturbance in the dynamic relationship of sliding mode variables in the composite error chain after being converted by the equivalent mass on the workbench side and the corresponding sliding surface gain.

[0156] Furthermore, the equivalent disturbance satisfies the conditions of bounded amplitude and bounded rate of change within the predetermined operating range of the single-axis feed system. Specifically, within each continuous time interval of the equivalent disturbance, there exists a preset positive real number... and , so that:

[0157] ;

[0158] in, This is the upper bound of the equivalent disturbance amplitude. The upper bound of the equivalent disturbance rate of change is defined as follows: When the lumped external disturbance is a step-type disturbance, the bounded condition of the rate of change applies to all continuous time intervals other than the step jump moment; the equivalent disturbance amplitude before and after the step jump still satisfies the bounded condition of the amplitude. The upper bound of the equivalent disturbance amplitude and the upper bound of the rate of change are used to determine the non-homogeneous superspinner control parameters offline and are not used as direct inputs for online calculation of the controller.

[0159] The nominal equivalent control term is calculated using the equivalent control method. Under nominal dynamic conditions, the equivalent disturbance corresponding to the lumped external disturbance is set to zero, and the first derivative of the sliding mode variable of the composite error chain with respect to time is set to zero, i.e.:

[0160] ;

[0161] ;

[0162] Based on the dynamic relationship of the sliding mode variables in the composite error chain, the nominal equivalent control term is obtained:

[0163] ;

[0164] After expanding the nominal dynamic term, the nominal equivalent control term is expressed as:

[0165] ;

[0166] in, This refers to the nominal equivalent control term. The nominal equivalent control term is calculated based on the reference equivalent linear acceleration, reference linear acceleration, composite error state, system state, sliding surface gain, and nominal dynamic model parameters, and does not directly include lumped external disturbance terms.

[0167] The nominal dynamic conditions are only used to determine the calculation form of the nominal equivalent control term and do not indicate that there are no lumped external disturbances in the actual controlled object. The actual lumped external disturbances still act as inputs to the controlled object on the workbench side and are reflected in the sliding mode variables of the composite error chain through changes in the linear displacement, linear velocity and corresponding error states of the workbench side.

[0168] For the bounded lumped external disturbance, a non-homogeneous superspiral algorithm, which includes non-homogeneous power terms, linear terms, and non-homogeneous superspiral integral states, is used to calculate the non-homogeneous superspiral approach control term.

[0169] To determine the proportional coefficient in the nonhomogeneous superhelical approach control term, assume that the equivalent axial control input consists of the nominal equivalent control term and the nonhomogeneous superhelical approach control term:

[0170] ;

[0171] in, It is the non-homogeneous superspiral approach control term.

[0172] Substituting the nominal equivalent control term and the non-homogeneous superspiral approaching control term into the dynamic relationship of the sliding mode variables in the composite error chain, we obtain:

[0173] ;

[0174] Therefore, it can be seen that the non-homogeneous superspiral approach control term acts on the dynamic relationship of the sliding mode variables of the composite error chain through the aforementioned input gain. To ensure that the non-homogeneous power term, the linear term, and the non-homogeneous superspiral integral state act on the dynamic relationship according to their respective gains, the non-homogeneous superspiral approach control term is set as follows:

[0175] ;

[0176] in, The proportional gain is the proportional coefficient of the nonhomogeneous superspinning approach control term, which is the input gain in the dynamic relationship of the sliding mode variables in the composite error chain. The reciprocal of this is used to offset the input gain. The scaling factor... It is not a separately set control parameter, but rather an equivalent mass on the motor side. and sliding surface gain The result was obtained after determination. For non-homogeneous superspiral approach control gain, For non-homogeneous power terms, adjust the parameter. For adjusting parameters of the linear term, It is a non-homogeneous power exponent. It is a non-homogeneous superspiral integral state.

[0177] The non-homogeneous power exponent satisfies:

[0178] ;

[0179] The , and All values ​​are positive. The non-homogeneous superspiral control parameters are determined before controller operation based on the bounded range of the equivalent disturbance and the position tracking test results under preset reference displacement input conditions. This ensures that the approaching effects generated by the non-homogeneous power term, linear term, and non-homogeneous superspiral integral state can cover the equivalent disturbance influence within the predetermined operating range. Once determined, the non-homogeneous superspiral control parameters remain unchanged during the same position tracking control process.

[0180] The nonhomogeneous superspinning control parameters are used to adjust the influence of different control components in the nonhomogeneous superspinning algorithm. Specifically, the nonhomogeneous superspinning approach control gain is used to adjust the influence of the combination of the nonhomogeneous power term and the linear term on the nonhomogeneous superspinning approach control term; the nonhomogeneous superspinning integral state update gain... The cumulative velocity used to adjust the non-homogeneous superspinning integral state; non-homogeneous power term adjustment parameter. Used to adjust the influence of the sign function term in non-homogeneous power terms and integral state update relations; linear term adjustment parameter. Used to adjust the linear control component corresponding to the sliding mode variable of the composite error chain; non-homogeneous power exponent Used to determine the degree of nonlinearity of nonhomogeneous power terms as the absolute value of the sliding mode variable in the composite error chain changes.

[0181] It should be noted that, due to the composite error chain sliding mode variable For dimensionless variables, , and All are dimensionless quantities. To ensure that all components in the nonhomogeneous superspiral approach control term have consistent units, the nonhomogeneous superspiral approach control gain... The unit is per second, that is Integral state update gain The unit is per square second, that is ; Adjustment parameter for non-homogeneous power terms Linear term adjustment parameters and non-homogeneous power exponents All parameters are dimensionless; non-homogeneous superscrew integral state The unit is per second, that is Therefore, the non-homogeneous power term and linear term after processing the non-homogeneous superspiral approach control gain have the same units as the integral state. After multiplying by the reciprocal of the input gain, a non-homogeneous superspiral approach control term in Newtons is obtained, which can be added to the nominal equivalent control term to form an equivalent axial control input.

[0182] The non-homogeneous power term in the non-homogeneous superspiral approach control term is expressed as:

[0183] ;

[0184] The linear term is represented as:

[0185] ;

[0186] in, Let be the absolute value of the sliding mode variable in the composite error chain. For a symbolic function, its value is determined as follows:

[0187] ;

[0188] The nonhomogeneous superspiral integral state is represented as follows:

[0189] ;

[0190] in, For integration time variable, The gain is updated for the non-homogeneous superspiral integral state, and The value is positive.

[0191] Rewriting the integral expression as a state update relation yields:

[0192] ;

[0193] The initial value of the non-homogeneous superspin integral state is set as follows:

[0194] ;

[0195] Given an initial value of zero, the above integral expression and the state update relation have the same computational meaning. Using the state update relation allows for the continuous storage of the cumulative value of the combination of symbolic function terms and linear terms during the control process, eliminating the need to recalculate the complete integral from the initial time to the current time in each control cycle.

[0196] Substituting the nonhomogeneous superspiral approach control term into the dynamic relationship of the sliding mode variables in the composite error chain, we obtain:

[0197] ;

[0198] Thus, the nonhomogeneous power term, the linear term, and the nonhomogeneous superspiral integral state jointly influence the dynamic relationship of the sliding mode variables in the composite error chain. The nonhomogeneous superspiral integral state and the nonhomogeneous superspiral approach control term are both calculated based on the sliding mode variables of the composite error chain, without requiring measurement or estimation of lumped external disturbances; the nominal equivalent control term is calculated based on the system state, reference equivalent linear acceleration, reference linear acceleration, sliding surface gain, and nominal dynamic model parameters.

[0199] Within each control cycle, the equivalent linear displacement on the motor side, the linear displacement on the worktable side, and their corresponding equivalent linear velocities are first acquired, and the composite error state set and composite error chain sliding mode variables are calculated. Then, based on the system state, reference equivalent linear acceleration, reference linear acceleration, and nominal dynamic model parameters, the dynamic state terms on the worktable side and the dynamic state terms on the motor side are calculated, and the nominal equivalent control term is calculated. Subsequently, the non-homogeneous superspiral integral state is updated based on the current composite error chain sliding mode variables, and the non-homogeneous superspiral approach control term is calculated based on the composite error chain sliding mode variables and the updated non-homogeneous superspiral integral state.

[0200] In the simulation implementation of this embodiment, each fixed simulation step size is used as a control update cycle, that is, the control cycle is the same as the fixed simulation step size. Within each control update cycle, the single-axis feed system state, the composite error state set, the composite error chain sliding mode variable, and the non-homogeneous superspiral integral state are updated sequentially, and the corresponding nominal equivalent control term and non-homogeneous superspiral approaching control term are calculated.

[0201] In this embodiment, the non-homogeneous superspiral control parameters are set as follows: , , , , .

[0202] The nonhomogeneous superspiral control parameters are determined through position tracking tests under preset reference displacement input conditions before the controller is run. Specifically, keeping the dynamic model parameters of the single-axis feed system, sliding surface gain, reference displacement input, lumped external disturbance conditions, and simulation solution conditions unchanged, a nonhomogeneous power exponent is first selected within a range greater than 0 and less than 1. Then , , and Set them to positive real initial values ​​respectively.

[0203] During parameter adjustment, a single-parameter adjustment method is used, adjusting only one parameter at a time. , , , and One parameter is selected, while keeping the other parameters unchanged. After each adjustment, the position tracking test is re-executed to obtain the position tracking error on the worktable side and the equivalent axial control input. The position tracking accuracy is evaluated using at least one of the steady-state root mean square error, steady-state average absolute error, and steady-state maximum absolute error, and the smoothness of the control input is evaluated using at least one of the control input root mean square value, peak-to-peak value, and high-frequency residual. For step reference displacement input, overshoot, rise time, and settling time are also obtained.

[0204] If the adjusted parameters reduce at least one of the steady-state root mean square error or the steady-state maximum absolute error, and the remaining position tracking error evaluation indicators and the high-frequency residual of the control input are not higher than the corresponding evaluation indicators before adjustment, the adjusted parameter values ​​are retained; otherwise, the parameter values ​​before adjustment are restored. Each non-homogeneous superspiral control parameter is adjusted sequentially in the above manner until one round of parameter adjustment is completed and all parameters no longer change, thus obtaining the setting result of the non-homogeneous superspiral control parameters.

[0205] Specifically, increasing the nonhomogeneous superspiral approach control gain correspondingly increases the influence of the combination of the nonhomogeneous power term and the linear term on the nonhomogeneous superspiral approach control term, thus increasing... Correspondingly increase the update rate of the non-homogeneous superspin integral state, increase Correspondingly increase the influence of non-homogeneous power terms and sign function terms, increase The effect of the linear term is increased accordingly. Non-homogeneous power exponent. The same evaluation method is used to determine the range that is greater than 0 and less than 1.

[0206] The above parameters are set before the controller runs and remain unchanged during the same position tracking control operation. Among them, the non-homogeneous power exponent... It lies within the range of values ​​greater than 0 and less than 1.

[0207] In the simulation process of this embodiment, the simulation time is set to A fixed-step fourth-order Runge-Kutta algorithm is used to simultaneously solve the state of the single-axis feed system and the non-homogeneous hyperspin integral state. The fixed simulation step size is set to... .

[0208] Simulation time reached At that time, an amplitude of [value] is applied to the worktable side. The step-type lumped external disturbance is used only as an external input to the controlled object and does not participate in the direct calculation of the nominal equivalent control term, the non-homogeneous hyperspinning integral state, or the non-homogeneous hyperspinning approaching control term.

[0209] Following the above processing, nominal equivalent control terms and non-homogeneous superspiral approaching control terms are obtained respectively.

[0210] The nominal equivalent control term is added to the non-homogeneous super-helical approach control term to obtain the equivalent axial control input; the equivalent axial control input is converted into a servo motor torque command according to the ball screw transmission relationship and output to the servo driver. The specific implementation method is as follows:

[0211] Adding the nominal equivalent control term to the non-homogeneous superhelical approaching control term yields the equivalent axial control input:

[0212] ;

[0213] in, This is the equivalent axial control input. For nominal equivalent control items, It is the non-homogeneous superspiral approach control term.

[0214] Based on the aforementioned ball screw transmission relationship, the equivalent axial control input is converted into a servo motor torque command:

[0215] ;

[0216] Based on the aforementioned displacement conversion coefficient ,get:

[0217] ;

[0218] in, This is the torque command for the servo motor.

[0219] In each control cycle, the calculated nominal equivalent control term is added to the non-homogeneous super-spiral approaching control term to obtain the equivalent axial control input. The equivalent axial control input is converted into a servo motor torque command according to the displacement conversion coefficient, and the servo motor torque command is output to the servo driver.

[0220] It should be noted that, in order to verify the position tracking performance, transient response performance and equivalent axial control input smoothness of the above-mentioned composite sliding mode position tracking control method, the composite sliding mode position tracking control method proposed in this application is compared with the proportional-integral-derivative control method, the traditional binomial sliding mode control method, the composite sliding mode control method using composite error chain sliding mode variables and the classical super-spiral sliding mode control method under the same single-axis feed system dynamic model, simulation solution conditions and lumped external disturbance conditions.

[0221] For ease of description, the proportional-integral-derivative control method is denoted as PID, the traditional binomial sliding mode control method is denoted as SMC, the composite sliding mode control method using composite error chain sliding mode variables is denoted as CSMC, the composite sliding mode position tracking control method based on composite error chain sliding mode variables and non-homogeneous superhelical algorithm proposed in this application is denoted as NHSTA-SMC, and the classical superhelical sliding mode control method is denoted as STASMC.

[0222] This embodiment uses the aforementioned sliding surface gain and non-homogeneous superspiral control parameters for simulation, and no separate control parameter table is set. The definitions and values ​​of the table-side position error gain, motor-side equivalent position error gain, table-side position error derivative gain, and motor-side equivalent position error derivative gain are all the same as those previously defined.

[0223] The simulation time was set to 10 seconds, and the fourth-order Runge-Kutta algorithm with a fixed step size was used for the solution. The fixed simulation step size was set to... When the simulation time reaches 3 seconds, a step-type lumped external disturbance with an amplitude of 100N is applied to the workbench side. Tests are conducted using a sinusoidal reference displacement, a step reference displacement ranging from 0mm to 500mm and smoothed by a speed-limiting circuit, and a triangular wave reference displacement with a frequency of 0.5Hz.

[0224] Position tracking performance is evaluated using corresponding indicators from steady-state root mean square error (RMSE), steady-state mean absolute error (MAE), steady-state maximum absolute error (MAE), amplitude error, phase lag, overshoot, rise time, and settling time. RMSE is denoted as RMSE, and MAE as MAE. The smoothness of the equivalent axial control input is evaluated using steady-state root mean square value (RMS), steady-state peak-to-peak value (MPS), high-frequency residual root mean square value (RMS), and local high-frequency residuals. RMS is denoted as RMS, and the high-frequency residuals are the difference between the equivalent axial control input and its smoothing trend term, used to characterize the high-frequency fluctuation components in the equivalent axial control input.

[0225] It should be noted that, Figure 4 This is the position tracking curve under a step reference displacement. Figure 5 This is the tracking error curve under a step reference displacement. Figure 6 This is the position tracking curve under the triangular wave reference displacement. Figure 7 This is the tracking error curve under the triangular wave reference displacement.

[0226] Since the original step reference displacement command contains abrupt changes, the initial stage after trajectory smoothing still has large transient changes. Therefore, under step reference displacement input conditions, steady-state error, overshoot, rise time, and settling time are mainly used to evaluate transient response performance; the smoothness of equivalent axial control input is mainly evaluated under sinusoidal reference displacement and triangular wave reference displacement input conditions.

[0227] The position tracking performance of each control method under sinusoidal reference displacement input conditions is shown in Table 1.

[0228] Table 1. Comparison of position tracking performance of various control methods under sinusoidal reference displacement input.

[0229]

[0230] Table 1 shows that both PID and SMC exhibit significant position tracking errors under sinusoidal reference displacement input conditions. Specifically, SMC has a steady-state RMSE of 31.2686 mm, a maximum steady-state absolute error of 56.0199 mm, and a phase lag of -9.8529°. CSMC's steady-state RMSE decreases to 5.9933 mm, indicating that incorporating the position errors and their derivatives from both the table and motor sides into the construction of the composite error chain sliding mode variables can improve the characterization of the coupled error state of the single-axis feed system.

[0231] The steady-state RMSE of NHSTA-SMC is 0.0104 mm, the steady-state MAE is 0.0093 mm, the steady-state maximum absolute error is 0.0161 mm, the amplitude error is 0.0031 mm, and the phase lag is 0.0033°. The position tracking errors of NHSTA-SMC and STASMC are at the same level, indicating that both can achieve high position tracking accuracy under sinusoidal reference displacement input conditions.

[0232] Under sinusoidal reference displacement input conditions, the equivalent axial control input after the system enters steady state is analyzed, and the results are shown in Table 2.

[0233] Table 2 Comparison of Equivalent Axial Control Input Smoothness of Various Control Methods under Sinusoidal Reference Displacement Input

[0234]

[0235] As shown in Table 2, the steady-state RMS of NHSTA-SMC is 867.7951 N, which is basically consistent with the 868.1160 N of STASMC, indicating that the overall amplitude of the equivalent axial control input of the two is close; the steady-state peak-to-peak value of NHSTA-SMC is 2472.0718 N, which is lower than the 2537.2805 N of STASMC.

[0236] The high-frequency residual RMS of NHSTA-SMC is 1.7139 N, lower than that of STASMC (22.7230 N). The steady-state local high-frequency residual RMS decreases from 21.9862 N for STASMC to 1.4921 N, and the peak-to-peak value of the steady-state local high-frequency residual decreases from 66.5378 N for STASMC to 12.6820 N. Therefore, with essentially the same position tracking accuracy, NHSTA-SMC can reduce high-frequency fluctuations in the equivalent axial control input.

[0237] It should be noted that although the high-frequency residual index of PID is relatively small, its position tracking accuracy is significantly lower than that of NHSTA-SMC, as shown in Table 1, when considered in conjunction with the position tracking error. Therefore, the overall performance of a control method cannot be evaluated solely based on the numerical value of the high-frequency residual; a comprehensive judgment must also be made by considering the position tracking error and the equivalent axial control input amplitude.

[0238] The step reference displacement changes from 0 mm to 500 mm and is smoothed by a speed limiting circuit. The position tracking performance of each control method is shown in Table 3.

[0239] Table 3 Comparison of position tracking performance of various control methods under step reference displacement input

[0240]

[0241] As shown in Table 3, under the step reference displacement input condition, the steady-state RMSE of the PID is 0.9468 mm, the settling time is 0.80821 s, and it produces an overshoot of 10.94%. Although the SMC has a shorter rise time and settling time, its overshoot reaches 71.00%, indicating that it is prone to overshooting when the reference displacement changes.

[0242] The overshoot of both CSMC and STASMC is close to zero. The overshoot of NHSTA-SMC is also close to zero, with a rise time of 0.03754s and a settling time of 0.07357s, which are lower than those of CSMC (0.04737s and 0.08392s) and STASMC (0.04736s and 0.08390s), respectively.

[0243] The frequency of the triangular wave reference displacement was set to 0.5 Hz. Since the triangular wave reference displacement exhibits a change in velocity direction at the inflection point, in addition to the steady-state error, the maximum inflection point error and the average maximum inflection point error were used to evaluate the position tracking performance of each control method near the trajectory inflection point. The results are shown in Table 4.

[0244] Table 4. Comparison of position tracking performance of various control methods under triangular wave reference displacement input.

[0245]

[0246] Table 4 shows that under the triangular wave reference displacement input condition, the steady-state RMSE of SMC is 27.1813 mm, the steady-state MAE is 24.3674 mm, and the maximum error at the inflection point is 37.8426 mm. The steady-state RMSE of CSMC decreases to 9.2288 mm, the steady-state MAE decreases to 6.5849 mm, and the maximum error at the inflection point decreases to 17.3370 mm. This indicates that using a composite error chain sliding mode variable can improve the position tracking performance of the triangular wave ramp section and near the inflection point.

[0247] The steady-state RMSE of NHSTA-SMC is 0.2678 mm, the steady-state MAE is 0.0669 mm, the maximum absolute error in steady state and the maximum error at the inflection point are both 2.1303 mm, and the average maximum error at the inflection point is 2.1275 mm. Compared with STASMC, the steady-state RMSE of NHSTA-SMC decreased from 0.3122 mm to 0.2678 mm, and the maximum error at the inflection point decreased from 2.5249 mm to 2.1303 mm.

[0248] Because the triangular wave reference displacement exhibits a change in velocity direction at the inflection point, the equivalent axial control input is prone to local high-frequency fluctuations in the neighborhood of the inflection point. Therefore, steady-state RMS, high-frequency residual RMS, and inflection point local high-frequency residual RMS are used to evaluate the smoothness of the equivalent axial control input of each control method. The results are shown in Table 5.

[0249] Table 5. Comparison of the smoothness of the equivalent axial control input for each control method under triangular wave reference displacement input.

[0250]

[0251] Table 5 shows that the SMC exhibits large steady-state RMS and high-frequency residual RMS, indicating significant equivalent axial control input fluctuations under triangular wave reference displacement input conditions. Compared to the SMC, the CSMC reduces the steady-state RMS, high-frequency residual RMS, and inflection point local high-frequency residual RMS, suggesting that the composite error chain sliding mode variable has a certain improving effect on control input smoothness.

[0252] The steady-state RMS of NHSTA-SMC is 2399.6160 N, lower than that of STASMC (4871.9910 N); the high-frequency residual RMS is 1871.3847 N, lower than that of STASMC (4627.6387 N); and the local high-frequency residual RMS at the inflection point is 6090.1982 N, lower than that of STASMC (14700.7714 N). Therefore, NHSTA-SMC can reduce the high-frequency fluctuations of the equivalent axial control input in the neighborhood of the triangular wave reference displacement inflection point.

[0253] The high-frequency residual index of PID is lower than that of each sliding mode control method. However, as shown in Table 4, the steady-state RMSE, steady-state MAE and inflection point error of PID are significantly greater than those of NHSTA-SMC. Its smaller high-frequency residual cannot alone indicate that its overall control performance is better than that of NHSTA-SMC.

[0254] In summary, under sinusoidal reference displacement input conditions, NHSTA-SMC can reduce high-frequency fluctuations in the equivalent axial control input while maintaining high position tracking accuracy; under step reference displacement input conditions, NHSTA-SMC can achieve shorter rise and settling times without significant overshoot; and under triangular wave reference displacement input conditions, NHSTA-SMC can reduce steady-state position tracking error, inflection point position tracking error, and high-frequency fluctuations in the equivalent axial control input in the inflection point neighborhood.

[0255] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A composite sliding mode position tracking control method for a single-axis feed system, characterized in that, include: A rotational-linear equivalent two-degree-of-freedom dynamic model of a single-axis ball screw feed system is established. Based on the ball screw transmission relationship, the rotational-linear equivalent two-degree-of-freedom dynamic model is converted into a linear-linear equivalent dynamic model, and the linear-linear equivalent dynamic model is converted into a state-space expression. Based on the state-space expression, the equivalent linear displacement on the motor side, the linear displacement on the worktable side, and their corresponding reference displacements are obtained. The equivalent position error on the motor side, the position error on the worktable side, the derivative of the equivalent position error on the motor side, and the derivative of the position error on the worktable side are calculated to form a composite error state set. Based on the composite error state set, a composite error chain sliding mode variable is constructed. The dynamic relationship of sliding mode variables in the composite error chain is determined based on the state-space expression, and the nominal equivalent control term is not used as the calculation input by the lumped external disturbance measurement value during the calculation process; For bounded lumped external disturbances, based on the non-homogeneous superspiral algorithm which includes non-homogeneous power terms, linear terms and integral states, the integral state is updated and the non-homogeneous superspiral approach control term is calculated. The nominal equivalent control term is added to the non-homogeneous super-spiral approach control term to obtain the equivalent axial control input; the equivalent axial control input is converted into a servo motor torque command according to the ball screw transmission relationship and output to the servo driver.

2. The composite sliding mode position tracking control method for a single-axis feed system according to claim 1, characterized in that, The methods for establishing a rotational-linear equivalent two-degree-of-freedom dynamic model include: Using the lumped parameter method, the servo motor, coupling, and ball screw are equivalent to rotary motion components, and the screw nut and worktable are equivalent to linear motion components. The equivalent axial stiffness and equivalent axial damping coefficient are used to characterize the flexible connection relationship between the rotary motion components and the linear motion components. Using the angular displacement of the rotating component and the linear displacement of the linear component as generalized coordinates, the system's kinetic energy, potential energy, energy dissipation function, and generalized external force vector are determined respectively. Substituting these into the Lagrange equation, a rotational-linear equivalent two-degree-of-freedom dynamic model is obtained, which includes the equivalent moment of inertia of the rotating component, the equivalent mass of the linear component, the equivalent viscous damping coefficient of the rotating component, the equivalent viscous damping coefficient of the linear component, the equivalent axial stiffness, the equivalent axial damping coefficient, the servo motor input torque, and the lumped external disturbances acting on the worktable side.

3. The composite sliding mode position tracking control method for a single-axis feed system according to claim 2, characterized in that, Methods for obtaining state-space expressions include: The displacement conversion coefficient from rotary motion to linear motion is determined based on the ball screw lead. The product of the angular displacement of the rotary motion component and the displacement conversion coefficient is determined as the equivalent linear displacement on the motor side, and the linear displacement of the linear motion component is determined as the linear displacement on the worktable side. Based on the displacement conversion coefficient, the servo motor input torque is converted into an equivalent axial control input, the equivalent moment of inertia of the rotating motion component is converted into an equivalent mass on the motor side, and the equivalent viscous damping coefficient of the rotating motion component is converted into an equivalent axial viscous damping coefficient on the motor side, thus obtaining a linear-linear equivalent dynamic model. The equivalent linear displacement on the motor side, the linear displacement on the worktable side, and their corresponding equivalent linear velocities are used to form the system state vector, and the state space expression is obtained based on the linear-linear equivalent dynamic model.

4. The composite sliding mode position tracking control method for a single-axis feed system according to claim 1, characterized in that, Methods for forming a composite error state set include: The equivalent linear velocity on the motor side and the linear velocity on the worktable side are obtained based on the state-space expression. The time derivatives of the reference equivalent linear displacement on the motor side and the reference linear displacement on the worktable side are calculated to obtain the reference equivalent linear velocity on the motor side and the reference linear velocity on the worktable side. The difference between the reference equivalent linear displacement on the motor side and the equivalent linear displacement on the motor side is defined as the equivalent position error on the motor side, and the difference between the reference linear displacement on the worktable side and the linear displacement on the worktable side is defined as the position error on the worktable side. The difference between the reference equivalent linear velocity on the motor side and the equivalent linear velocity on the motor side is determined as the derivative of the equivalent position error on the motor side, and the difference between the reference linear velocity on the worktable side and the linear velocity on the worktable side is determined as the derivative of the position error on the worktable side. The equivalent position error on the motor side, the position error on the worktable side, the derivative of the equivalent position error on the motor side, and the derivative of the position error on the worktable side are combined to form a composite error state set.

5. The composite sliding mode position tracking control method for a single-axis feed system according to claim 4, characterized in that, The sliding mode variables of the composite error chain are constructed based on the composite error state set, including: Set the sliding surface gain corresponding to the table-side position error, the motor-side equivalent position error, the table-side position error derivative, and the motor-side equivalent position error derivative, respectively; Each error state in the composite error state set is combined with its corresponding sliding surface gain, and the combination results are combined to obtain the sliding mode variables of the composite error chain.

6. The composite sliding mode position tracking control method for a single-axis feed system according to claim 1, characterized in that, The nominal equivalent control terms that do not use lumped external disturbance measurements as input during the calculation process include: The dynamic relationship of the sliding mode variables in the composite error chain is expressed as the sum of the nominal dynamic term and the equivalent disturbance corresponding to the lumped external disturbance, minus the product of the equivalent axial control input and the input gain. The nominal dynamic term is calculated based on the motor-side reference equivalent linear acceleration, the table-side reference linear acceleration, the composite error state set, the table-side dynamic state term, the motor-side dynamic state term, and the sliding surface gain. The input gain is determined based on the sliding surface gain corresponding to the motor-side equivalent position error derivative and the motor-side equivalent mass. Under nominal dynamic conditions, the equivalent disturbance is set to zero, and the first derivative of the composite error chain sliding mode variables is set to zero. The ratio of the nominal dynamic term to the input gain is determined as the nominal equivalent control term.

7. The composite sliding mode position tracking control method for a single-axis feed system according to claim 6, characterized in that, Update the integral state and calculate the non-homogeneous superspin approach control term, including: The non-homogeneous superspiral approach control gain, integral state update gain, non-homogeneous power term adjustment parameter and linear term adjustment parameter are all positive, and a non-homogeneous power exponent greater than 0 and less than 1 is set. The sign function term of the composite error chain sliding mode variable is adjusted according to the non-homogeneous power term adjustment parameter, and the composite error chain sliding mode variable is linearly adjusted according to the linear term adjustment parameter. The two adjustment results are combined and the integral state with the initial value of zero is updated through the integral state update gain. A nonhomogeneous power term is constructed based on the absolute value, sign function, and nonhomogeneous power exponent of the sliding mode variable in the composite error chain. This nonhomogeneous power term is then adjusted using the adjustment parameter. The adjusted nonhomogeneous power term is combined with the linear adjustment result and processed through the nonhomogeneous superspiral approach control gain. Finally, it is added to the updated integral state and multiplied by the reciprocal of the input gain to obtain the nonhomogeneous superspiral approach control term.

8. The composite sliding mode position tracking control method for a single-axis feed system according to claim 1, characterized in that, Based on the ball screw transmission relationship, the equivalent axial control input is converted into servo motor torque commands, including: The ratio of the ball screw lead to 2π is determined as the displacement conversion coefficient. The equivalent axial control input is multiplied by the displacement conversion coefficient to obtain the servo motor torque command.

9. The composite sliding mode position tracking control method for a single-axis feed system according to claim 5, characterized in that, The sliding mode variables of the composite error chain satisfy: ； in, For the sliding mode variables of the composite error chain, This is the gain for the position error on the worktable side. This is the equivalent position error gain on the motor side. The derivative gain of the position error on the worktable side. The derivative gain of the equivalent position error on the motor side. , , and All values ​​are positive; For the positional error of the worktable side, This refers to the equivalent position error on the motor side. The derivative of the position error on the worktable side. This is the derivative of the equivalent position error on the motor side.

10. The composite sliding mode position tracking control method for a single-axis feed system according to claim 7, characterized in that, Let the sliding mode variable of the composite error chain be denoted as... Record the integration state as The integral state satisfies: ； ； The nonhomogeneous superspiral approach control term satisfies: ； in, This is the non-homogeneous superspiral approach control term. For the equivalent mass on the motor side, This represents the sliding surface gain corresponding to the derivative of the equivalent position error on the motor side. For non-homogeneous superspiral approach control gain, Update the gain for the integral state. For non-homogeneous power terms, adjust the parameter. For adjusting parameters of the linear term, It is a non-homogeneous exponent that is greater than 0 and less than 1.

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