A time-delay sliding mode based three-axis contour error control method

Abstract

The present application relates to a kind of time delay-sliding mode-based three-axis contour error control method, belong to electromechanical control technical field.First, a kind of time delay-sliding mode controller under the consideration of model error and friction is designed to the movement of each motion axis is controlled, output actual position;Then consider the model deviation under the action of Kalman state prediction, the actual position of next time is accurately predicted;Finally, consider the search step empirical formula of tracking error and curvature, the optimal search range under each time is solved in real time, the contour error under each time is quickly estimated by a kind of efficient contour error estimation method, and the instruction trajectory is calculated and modified contour error.The present application simultaneously considers the three-axis contour error control under the model deviation and friction effect, so that the present method has more accurate contour error estimation and better control effect.

Description

Technical Field

[0001] This invention belongs to the field of electromechanical control technology, and relates to a three-axis contour error control method, particularly a high-efficiency contour error estimation and contour error control method based on time delay-sliding mode. Background Technology

[0002] Reference 1, “SH Kim, BK Min, Real-time tool path modification for machinetool contour error reduction, The International Journal of Advanced Manufacturing Technology, 2022, 120: 6969–6981,” discloses a contour error pre-compensation method using Kalman filtering. This method uses a Kalman filter to predict the system state and then reduces the contour error by modifying the trajectory. However, this method does not consider the influence of model error when predicting the state, which can lead to significant prediction bias when using the Kalman filter for state prediction.

[0003] Reference 2, “PR Ouyang, J. Acob, V. Pano, PD with sliding mode control for trajectory tracking of robotic system, Robotics and Computer-Integrated Manufacturing, 2014, 30: 190-200,” discloses an improved proportional-derivative (PD) based sliding mode control method. This method modifies traditional first-order sliding mode control and combines it with feedforward control to optimize the first-order sliding mode control. However, this method does not consider the effects of model bias and friction, resulting in limited practical control effectiveness.

[0004] The typical characteristic of the above literature is that neither contour error prediction nor contour error compensation takes into account the influence of model bias, resulting in limited contour error compensation effect. Summary of the Invention

[0005] To overcome the problems existing in current three-axis contour error control, this invention provides a three-axis contour error control method based on time delay-sliding mode.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0007] A three-axis contour error control method based on time delay-sliding mode, characterized in that it includes:

[0008] S1: Design a time-delay sliding mode controller that takes into account model error and friction to control the motion of each motion axis and output the actual position;

[0009] S2: Design a Kalman state predictor that takes into account the effects of model bias to predict the actual position at the next moment;

[0010] S3: Design an iterative search method that considers the search step size of tracking error and curvature to solve the optimal search range at each moment in real time;

[0011] S4: Use the contour error estimation method to estimate the contour error at each moment and modify the command trajectory.

[0012] A further technical solution of the present invention: The time-delay sliding mode controller considering model error and friction as described in S1 is specifically as follows:

[0013]

[0014] in, This is an equivalent control law. For delay estimator, To switch control laws.

[0015] A further technical solution of the present invention: The iterative search method for the search step size considering tracking error and curvature described in S3 is as follows:

[0016] Based on experience, the determination of the search range is related to the velocity of the reference point and the tracking error of the actual point. The following empirical formula for determining the search range is designed:

[0017]

[0018] in, This is an empirical coefficient. Let $\mathbf{k}$ be the tracking error at the $k$-th sampling time. Let Ts be the command speed at the k-th sampling time, and Ts be the sampling time. This represents the calculated search interval.

[0019] A further technical solution of the present invention: The specific details of solving the optimal search range at each time step in S3 are as follows:

[0020] Calculate the first k Reference point at each sampling time and The actual points obtained from the prediction The distances between them are respectively and Compare the sizes of the two, assuming The new search range is [ k , k +m(k)], otherwise the search interval becomes [ k -m( k ), k ].

[0021] A further technical solution of the present invention: In S4, the contour error is estimated at each moment using a contour error estimation method, specifically as follows:

[0022] Based on the determined final interval, at the actual point With reference outline and The lines connecting them and the actual points and and The line connecting the two points is approximated by two perpendicular line segments, represented by the first line and the second line respectively; the actual point The distances to the first and second lines are respectively expressed as: and ;calculate and The minimum value between them is used as the predicted actual point. Contour error at the location .

[0023] A further technical solution of the present invention: The calculation of contour error and modification of command trajectory in S4 specifically refers to: the calculated contour error , for the k Modify the command position at +1 sampling time: The modified instruction trajectory is then used as the instruction input.

[0024] A computer system is characterized by comprising: one or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to implement the method described above.

[0025] A computer-readable storage medium is characterized by storing computer-executable instructions, which, when executed, are used to implement the above-described method.

[0026] The beneficial effects of this invention are as follows:

[0027] This invention first designs a time-delay sliding mode controller, considering model error and friction, to control the motion of each axis and output the actual position. Then, it proposes a Kalman state prediction method considering model deviation to accurately predict the actual position at the next moment. Finally, it uses an empirical formula for the search step size based on tracking error and curvature to solve for the optimal search range at each moment in real time. An efficient contour error estimation method is then used to quickly estimate the contour error at each moment, calculate the contour error, and modify the command trajectory. This invention simultaneously considers three-axis contour error control under model deviation and friction, resulting in more accurate contour error estimation and better control performance. Attached Figure Description

[0028] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts.

[0029] Figure 1 This is a diagram of the tool tip trajectory of the three-axis machining path in an embodiment of the method of the present invention.

[0030] Figure 2 This is a comparison chart of the estimated and actual contour error values ​​in the embodiments of the present invention.

[0031] Figure 3 This is a search range diagram in an embodiment of the method of the present invention.

[0032] Figure 4 This is an embodiment of the method of the present invention. X , Y , Z Axis tracking error result diagram.

[0033] Figure 5 This is a Kalman state prediction diagram in an embodiment of the method of the present invention.

[0034] Figure 6 This is a flowchart of the present invention. Detailed Implementation

[0035] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0036] The method proposed in this invention was tested on an open-type five-axis CNC machining platform. The sampling time interval of the control board... The time is 0.001s. The experimental trajectory is a three-axis rose trajectory, where the trajectory of the knife tip is as follows: Figure 1As shown. The actual position of each axis is obtained from feedback by an incremental encoder. The friction compensation control algorithm in this invention is implemented in Matlab / Simulink 2015b.

[0037] This invention provides an efficient contour error estimation and control method based on time delay-sliding mode, the specific steps of which are as follows:

[0038] Step 1: Design an equivalent control law to perform motion control on the nominal model of the machine tool servo system:

[0039]

[0040] This is the nominal value of the equivalent inertia. This is the nominal value of the viscous friction coefficient. For a continuous model used to approximate the Stribeck friction model, Let be the trend function of friction changes. The friction amplitude, , , Each is a real point Position, velocity, acceleration To control the input voltage.

[0041] Step 2: Design the following first-order proportional-differential sliding mode controller to control the nominal model:

[0042]

[0043] A positive value designed by humans. , These are the derivatives of the tracking error and the tracking error, respectively.

[0044] Differentiate the sliding surface:

[0045]

[0046] The feedforward compensation term is designed as follows:

[0047]

[0048] Based on the average value theory, the following feedback control law is designed:

[0049]

[0050] in, , It is a set positive value.

[0051] Based on the above derivation, a two-degree-of-freedom control structure with feedforward and proportional-derivative feedback is finally established for the nominal system model. The final equivalent control law can be expressed in the following form:

[0052]

[0053] Step 3: Design a time delay estimator to estimate the model deviation caused by friction, changes in cutting force, and disturbances during machine tool operation. The system's model deviation can be expressed as:

[0054]

[0055] in These are the equivalent inertia deviation, viscosity coefficient deviation, and friction amplitude coefficient deviation caused by interference during machine tool operation, respectively.

[0056] Design the following time delay estimator to estimate the model bias at the current time step:

[0057] Where L is the time extension, .

[0058] Define the time delay estimation bias as: .

[0059] Step 4: Design the following switching control law to compensate for the time delay estimation error in Step 3:

[0060]

[0061] in It is a set positive value.

[0062] The symbolic function is represented as:

[0063]

[0064] Sign function This can cause chattering on the sliding surface, which is detrimental to the normal operation of the controller. Therefore, the following saturation function is designed. This makes the changes in the control law smoother.

[0065] ,in

[0066] in The boundary layer thickness is a set value.

[0067] Combining steps 1-4, the time-delay sliding mode control law designed for the controlled machine tool servo system can be obtained:

[0068]

[0069] Step 5: Design a Kalman state predictor that considers model bias for the first... k The actual location at +1 sampling time The state-space equations of the system can be written in the following form:

[0070]

[0071] in The result of the time delay estimation in step 3. i express x, y, z Three axes of motion. The above equation can be further simplified to the following form:

[0072]

[0073] in, For system status,

[0074]

[0075] By discretizing the state equations of a continuous system using the zero-order hold method, the following results can be obtained:

[0076]

[0077] in , These are the discrete quantities. , These are discrete error and measurement error, respectively. , For the first k The sampling time and the first sampling time k +1 state variables at sampling time, For the first k The calculation results at each sampling time point.

[0078] Step 6: The specific implementation process of two-step prediction using Kalman filtering:

[0079] Using inaccurate calculation results and inaccurate measurement results Perform data fusion to achieve state variables Posterior estimation:

[0080]

[0081] in For the Kalman gain that needs to be designed, The prior estimates of state variables under conditions free from measurement noise are defined as follows:

[0082] Define prior estimation bias: ;

[0083] Define the posterior estimation error: ;

[0084] The definitions of the covariance of the prior estimation error and the posterior estimation error are also given:

[0085]

[0086] in , Let be the covariance of the prior estimation error and the covariance of the posterior estimation error at the (k+1)th sampling time, respectively. Discrete error The covariance matrix.

[0087] In order to make the posterior estimated state Closer to the actual state quantity , variance It needs to be further reduced, that is: ,in for The traces.

[0088] Therefore, a more suitable Kalman gain needs to be selected: ;

[0089] Where R is the measurement error The covariance.

[0090] No. k The predicted value of the actual position at +1 sampling time is:

[0091] ,in , , The state prediction results are for the three motion axes x, y, and z, respectively.

[0092] Step 7: Based on experience, it was found that the determination of the search range is related to the speed of the reference point and the tracking error of the actual point. The following empirical formula for determining the search range is designed:

[0093]

[0094] This is an empirical coefficient. For the first k The tracking error at each sampling time, For the first k The command speed at each sampling moment, where Ts is the sampling time. This represents the calculated search interval.

[0095] Step 8: Calculate the first... k Reference point at each sampling time and The actual points obtained from the prediction The distances between them are respectively and Compare the sizes of the two, assuming The new search range is [ k , k +m(k)], otherwise the search interval becomes [ k -m( k ), k ].

[0096] Step 9: Then, divide the search interval obtained in Step 8 into segments with a length of n=8, and calculate the distance from the actual point. The location of the nearest split point, assuming that the point is located at... a ( k The reference point corresponding to this point is... Repeat step 8, at this time m( k )=8, further narrowing the search range.

[0097] Step 10: Then, divide the search interval obtained in Step 9, setting the length of the division to n=4, and calculate the distance from the actual point. The location of the nearest split point, assuming that the point is located at... b ( k The reference point corresponding to this point is... Repeat step 8, at this time m( k )=4, further narrowing the search range.

[0098] Step 11: Then, divide the search interval obtained in Step 10, setting the length of the division to n=2, and calculate the distance from the actual point. The location of the nearest split point, assuming that the point is located at... c ( k The reference point corresponding to this point is... Repeat step 8, at this time m( k =2, determine the nearest reference contour point The final interval, let it be [ c ( k ), c ( k )+2).

[0099] Step 12: Based on the final interval determined in the previous step, at the actual point... With reference outline and The lines connecting them and the actual points and and The line connecting the two points can be approximated by two perpendicular line segments, represented as line 1 and line 2 respectively. Actual point The distances to line 1 and line 2 are respectively expressed as... and Finally, calculate. and The minimum value between them is used as the predicted actual point. Contour error at the location .

[0100] Step 13: Use the contour error calculated in Step 12. , for the k Modify the command position at +1 sampling time: The modified instruction trajectory is then used as the instruction input.

[0101] As can be seen, this example, by taking into account the effects of model deviation and friction, is able to estimate and compensate for contour errors more accurately.

[0102] Figure 2 This is a comparison chart of the contour error estimated using the contour error estimation method of the present invention and the actual contour error. Figure 3 The results obtained by using the method of this invention for search range estimation show that the method of this invention can estimate the optimal search range at every time step, achieving a balance between computational resources and search efficiency. Figure 4 The tracking errors of each motion axis obtained by the time-delay sliding mode control and the proportional-derivative sliding mode control proposed in this invention can be seen to be smaller by using the method of this invention. Figure 5 The prediction error values ​​obtained by Kalman filtering considering model bias and those not considering model bias show that the method of this invention has a smaller prediction error because it takes model bias into account. The tracking error control results for each motion axis are shown in Table 1, and the controller performance results are shown in Table 2.

[0103] Table 1 Tracking Error Control Results

[0104]

[0105] *Note: : Represents the mean value of the contour error; : Indicates the maximum value of the contour error.

[0106] Table 2 Contour Error Control Results

[0107]

[0108] *Note: ITAEs: represents the integral of the product of time and the absolute value of error with respect to time; ISVs: represents the integral of the square of the control input with respect to time.

[0109] As can be seen, this example, by taking into account the effects of model deviation and friction, is able to estimate and compensate for contour errors more accurately.

[0110] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the scope of the technology disclosed in the present invention, and such modifications or substitutions should all be covered within the scope of protection of the present invention.

Claims

1. A three-axis contour error control method based on time delay-sliding mode, characterized in that, include: Step 1: Design an equivalent control law to perform motion control on the nominal model of the machine tool servo system: This is the nominal value of the equivalent inertia. This is the nominal value of the viscous friction coefficient. For fitting the Stribeck friction model, a continuous model is used. Let be the trend function of friction changes. The friction amplitude, , , Each is a real point Position, velocity, acceleration To control the input voltage; Step 2: Design the following first-order proportional-differential sliding mode controller to control the nominal model: A positive value designed by humans. , These are the derivatives of the tracking error and the tracking error, respectively. Differentiate the sliding surface: The feedforward compensation term is designed as follows: Based on the average value theory, the following feedback control law is designed: in, , It is a set positive value; Based on the above derivation, a two-degree-of-freedom control structure with feedforward and proportional-derivative feedback is finally established for the nominal system model. The final equivalent control law can be expressed in the following form: Step 3: Design a time delay estimator to estimate the model deviation caused by friction, changes in cutting force, and disturbances during machine tool operation; the system model deviation is expressed as: in These are the equivalent inertia deviation, viscosity coefficient deviation, and friction amplitude coefficient deviation caused by interference during machine tool operation, respectively. Design the following time delay estimator to estimate the model bias at the current time step: Where L is the time extension, ; Define the time delay estimation bias as: ; Step 4: Design the following switching control law to compensate for the time delay estimation error in Step 3: in It is a set positive value; The symbolic function is represented as: Sign function This can cause chattering on the sliding surface, which is detrimental to the normal operation of the controller. Therefore, the following saturation function is designed. This makes the changes in the control law smoother. ,in in The boundary layer thickness is a predetermined value. Combining steps 1-4, the time-delay sliding mode control law designed for the controlled machine tool servo system can be obtained: Step 5: Design a Kalman state predictor that considers model bias for the first... k +1 sampling time actual position The state-space equations of the system can be written in the following form: in The result of the time delay estimation in step 3. i express x, y, z Three axes of motion; the above equation can be further simplified to the following form: in, For system status, By discretizing the state equations of a continuous system using the zero-order hold method, the following results can be obtained: in , These are the discrete quantities. , These are discrete error and measurement error, respectively. , For the first k The sampling time and the first sampling time k +1 state variables at sampling time, For the first k Calculation results at each sampling time point; Step 6: The specific implementation process of two-step prediction using Kalman filtering: Using inaccurate calculation results and inaccurate measurement results Perform data fusion to achieve state variables Posterior estimation: in For the Kalman gain that needs to be designed, The prior estimates of state variables under conditions free from measurement noise are defined as follows: Define prior estimation bias: ; Define the posterior estimation error: ; The definitions of the covariance of the prior estimation error and the posterior estimation error are also given: in , Let be the covariance of the prior estimation error and the covariance of the posterior estimation error at the (k+1)th sampling time, respectively. Discrete error The covariance matrix; In order to make the posterior estimated state Closer to the actual state quantity , variance It needs to be further reduced, that is: ,in for traces; Therefore, a more suitable Kalman gain needs to be selected: ; Where R is the measurement error covariance; No. k The predicted value of the actual position at +1 sampling time is: ,in , , The state prediction results are for the three motion axes x, y, and z, respectively. Step 7: Based on experience, it was found that the determination of the search range is related to the speed of the reference point and the tracking error of the actual point. The following empirical formula for determining the search range is designed: This is an empirical coefficient. For the first k The tracking error at each sampling time. For the first k The command speed at each sampling moment, where Ts is the sampling time. The calculated search interval; Step 8: Calculate the first... k Reference point at each sampling time and The actual points obtained from the prediction The distances between them are respectively and Compare the sizes of the two, assuming The new search range is [ k , k +m(k)], otherwise the search interval becomes [ k -m( k ), k ]; Step 9: Then, divide the search interval obtained in Step 8 into segments with a length of n=8, and calculate the distance from the actual point. The location of the nearest split point, assuming that the point is located at... a ( k The reference point corresponding to this point is... Repeat step 8, at this time m( k The search range is further narrowed down to 8. Step 10: Then, divide the search interval obtained in Step 9, setting the length of the division to n=4, and calculate the distance from the actual point. The location of the nearest split point, assuming that the point is located at... b ( k The reference point corresponding to this point is... Repeat step 8, at this time m( k )=4, further narrowing the search interval; Step 11: Then, divide the search interval obtained in Step 10, setting the length of the division to n=2, and calculate the distance from the actual point. The location of the nearest split point, assuming that the point is located at... c ( k The reference point corresponding to this point is... Repeat step 8, at this time m( k =2, determine the nearest reference contour point The final interval, let it be [ c ( k ), c ( k )+2]; Step 12: Based on the final interval determined in the previous step, at the actual point... With reference outline and The lines connecting them and the actual points and and The line connecting the two points is approximated by two perpendicular line segments, represented as line 1 and line 2 respectively; the actual point The distances to line 1 and line 2 are respectively expressed as... and Finally, calculate and The minimum value between them is used as the predicted actual point. Contour error at the location ; Step 13: Use the contour error calculated in Step 12. , for the k Modify the command position at +1 sampling time: The modified instruction trajectory is then used as the instruction input.

2. A computer system, characterized in that... include: One or more processors, a computer-readable storage medium for storing one or more programs, wherein, when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to implement the method of claim 1.

3. A computer-readable storage medium, characterized in that... The device stores computer-executable instructions, which, when executed, are used to implement the method of claim 1.

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