Cross coupling multi-axis synchronous control method based on EtherCAT

Abstract

The invention belongs to the field of EtherCAT bus control, and discloses a cross coupling multi-axis synchronous control method based on EtherCAT, which directly reduces the contour error through a cross coupling method, and also can calculate the deviation between the expected feeding speed and the actual speed of each axis so as to improve the accuracy of the cross coupling multi-axis synchronous control. And finally, calculating the total speed deviation of each shaft as an input signal to be input into each slave station so as to improve the synchronization performance of each shaft of the system.

Description

Technical Field

[0001] The invention belongs to the field of EtherCAT bus control, and in particular relates to a cross-coupling multi-axis synchronous control method based on EtherCAT. Background Art

[0002] In modern manufacturing, multi-axis control systems are increasingly used, especially in the field of high-precision machining. These systems usually consist of multiple axes, each of which is responsible for performing a specific motion task. In multi-axis systems, accurate synchronization control is crucial to ensure the coordination and accuracy of motion between the axes. Traditional synchronization control methods are usually based on pulse or other serial communication protocols, but these methods have some limitations, such as communication delays and low synchronization accuracy. In order to overcome these limitations, the industry has gradually turned to Ethernet-based control technology, among which the EtherCAT protocol is a highly regarded option.

[0003] In the industrial field, EtherCAT (Ethernet for Control Automation Technology) is a high-performance real-time Ethernet communication protocol with extremely low communication latency and high-speed transmission capabilities. It adopts an innovative master-slave structure to achieve real-time control by adding additional control information to the data packet. This structure enables EtherCAT to achieve highly flexible network topology and fast data transmission, which is very suitable for multi-axis control systems. However, in practical applications, multi-axis systems usually require cross-coupled multi-axis synchronous control, that is, the movement between multiple axes needs to be accurately synchronized in time and space. However, the current traditional cross-coupled multi-axis synchronous control methods often cannot meet this demand, because they often cannot effectively reduce the contour error of the system operation and the synchronization performance is not high enough, so that the target interpolation position of each axis in the next cycle cannot be accurately calculated. Summary of the invention

[0004] The purpose of the present invention is to address the deficiencies of the prior art and propose a cross-coupling multi-axis synchronous control method based on EtherCAT, which improves the synchronization accuracy of multi-axis collaborative operation by reducing the contour error of the multi-axis system.

[0005] To achieve the above object, the technical solution adopted by the present invention is:

[0006] A cross-coupling multi-axis synchronous control method based on EtherCAT is used to control the coordinated operation of a multi-axis system based on the EtherCAT protocol. The EtherCAT protocol involves a master station and multiple slave stations. The master station is a main controller, and the slave stations are multiple servo controllers. The cross-coupling multi-axis synchronous control method based on EtherCAT is implemented in the master station, and includes the following steps:

[0007] According to the contour line segment trajectory executed by each axis, the feed speed is calculated and interpolation processing is performed, and the contour line segment trajectory is discretized into multiple small straight line segments, each of which contains the feed speed and the target position;

[0008] Take out each small straight line segment in turn as the current interpolation segment;

[0009] Periodically send synchronization trigger signals to all slave stations, and receive the actual position and actual speed of each axis fed back by the slave stations after receiving the synchronization trigger signals;

[0010] The distance from the actual position to the target position of the interpolation line segment is calculated as the contour error. The corresponding contour errors of each slave station are coupled to the corresponding output through the cross-coupling controller, and the contour error compensation value of each axis is obtained according to the output in combination with the contour cross-coupling gain.

[0011] The speed deviation is calculated by subtracting the actual speed of each axis feedback from the feed speed of the interpolation line segment. The calculated speed deviation corresponding to each slave station is passed through the cross-coupling controller to obtain the cross-coupling speed deviation. Combined with the weighted proportional factor, the cross-coupling weighted speed deviation is obtained according to the cross-coupling speed deviation. The total speed deviation of each axis is obtained based on the speed deviation and the cross-coupling weighted speed deviation.

[0012] The contour error compensation value of each axis is compensated to the corresponding slave station, and the total speed deviation of each axis is input as an input signal to the corresponding slave station for motion control of the next synchronization cycle of the slave station.

[0013] Several optional methods are also provided below, but they are not intended to be additional limitations on the above-mentioned overall solution, but are merely further supplements or preferences. Under the premise that there are no technical or logical contradictions, each optional method can be combined with the above-mentioned overall solution separately, and multiple optional methods can also be combined.

[0014] Preferably, the calculation process of the contour error is as follows:

[0015] ε=E y Cy-E x Cx

[0016] E x =x r -x o

[0017] E y =y r -y o

[0018] Where ε is the contour error, E x is the tracking error of the x-axis, Ey is the tracking error of the y-axis, x r is the target position on the x-axis, x o is the actual position of the x-axis, y r is the target position on the y-axis, y o is the actual position of the y-axis, Cx is the profile cross-coupling gain coefficient of the x-axis, and Cx=sinθ, Cy is the profile cross-coupling gain coefficient of the y-axis, and Cy=cosθ, θ is the angle between the tangent of the interpolation segment and the x-axis.

[0019] Preferably, the step of coupling the calculated contour errors corresponding to the slave stations to obtain corresponding output quantities through a cross-coupling controller comprises:

[0020] ε′=K p ε+K i ∫εdt

[0021] In the formula, ε′ is the output, ε is the contour error, K p is the proportionality coefficient, K i Integration coefficient.

[0022] Preferably, the contour error compensation value of each axis is obtained according to the output in combination with the contour cross-coupling gain, including:

[0023] U x =x r +ε′Cx

[0024] U y =y r +ε′Cy

[0025] Where U x is the contour error compensation value of the x-axis, U y is the contour error compensation value of the y-axis, ε′ is the output, and x r is the target position on the x-axis, y r is the target position of the y-axis, Cx is the profile cross-coupling gain coefficient of the x-axis, and Cx=sinθ, Cy is the profile cross-coupling gain coefficient of the y-axis, and Cy=cosθ, θ is the angle between the tangent of the interpolation segment and the x-axis.

[0026] As a preference, the speed deviation is calculated by subtracting the actual speed of each axis feedback from the feed speed of the interpolation line segment, including:

[0027] E1(k)=E1(k-1)+ω1(k)-P1(k)

[0028] E2(k)=E2(k-1)+ω2(k)-P2(k)

[0029] Wherein, W1(k) is the speed deviation of axis 1 at the kth sampling period, E1(k-1) is the speed deviation of axis 1 at the k-1 moment, ω1(k) is the feed speed of axis 1 at the kth moment, P1(k) is the actual speed of axis 1 at the kth sampling period, E2(k) is the speed deviation of axis 2 at the kth sampling period, E2(k-1) is the speed deviation of axis 2 at the k-1 moment, ω2(k) is the feed speed of axis 2 at the kth moment, and P2(k) is the actual speed of axis 2 at the kth sampling period.

[0030] Preferably, the step of obtaining a cross-coupling speed deviation by passing the calculated speed deviation corresponding to each slave station through a cross-coupling controller comprises:

[0031] E(k)=E(k-1)+k1P1(k)-k2P2(k)

[0032]

[0033] k1+k2=1

[0034] Wherein, E(k) is the cross-coupling speed deviation at time k, E(k-1) is the cross-coupling speed deviation at time k-1, k1 is the speed cross-coupling gain coefficient of axis 1, k2 is the speed cross-coupling gain coefficient of axis 2, ω1(k) is the feed speed of axis 1 at the kth sampling period, P1(k) is the actual speed of axis 1 at the kth sampling period, ω2(k) is the feed speed of axis 2 at time k, and P2(k) is the actual speed of axis 2 at the kth sampling period.

[0035] Preferably, the combining weighted proportional factor obtains a cross-coupling weighted speed deviation according to the cross-coupling speed deviation, and obtains a total speed deviation of each axis based on the speed deviation and the cross-coupling weighted speed deviation, including:

[0036] C1(k)=E1(k)-αE(k)

[0037] C2(k)=E2(k)-αE(k)

[0038] Where C1(k) is the total speed deviation of axis 1 at time k, C2(k) is the total speed deviation of axis 2 at time k, E1(k) is the speed deviation of axis 1 at the kth sampling period, E2(k) is the speed deviation of axis 2 at the kth sampling period, E(k) is the cross-coupling speed deviation at time k, α is the weighted proportional factor, and αE(k) is the cross-coupling weighted speed deviation.

[0039] Compared with the prior art, the cross-coupling multi-axis synchronous control method based on EtherCAT provided by the present invention has the following significant advantages: the method can not only directly reduce the contour error by the cross-coupling method, but also can calculate the deviation between the expected feed speed and the actual speed of each axis, then enter the cross-coupling controller to obtain the cross-coupling weighted speed deviation, and finally calculate the total speed deviation of each axis as an input signal to each slave station to improve the synchronization performance of each axis of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 A flow chart of a cross-coupling multi-axis synchronous control method based on EtherCAT of the present invention;

[0041] Figure 2 A straight line profile error model diagram of a cross-coupling multi-axis synchronous control method based on EtherCAT of the present invention;

[0042] Figure 3 This is a schematic diagram of the overall structure based on EtherCAT and a cross coupler of the present invention. DETAILED DESCRIPTION

[0043] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0044] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art of the present invention. The terms used in the specification of the present invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention.

[0045] The present invention provides a cross-coupling multi-axis synchronous control method based on EtherCAT, which can be applied to synchronously control the x-axis and y-axis motion trajectories of multiple slave servers to reduce the contour error of the motion trajectory. For ease of understanding, this application takes the linear synchronous interpolation of the xy axis as an example for explanation.

[0046] The communication principle of the EtherCAT protocol in this embodiment is to adopt a master-slave structure, in which the EtherCAT Master acts as a host to control the communication process, usually referred to as a master station, and the EtherCAT Slave acts as a slave to process task instructions and data issued by the host, usually referred to as a slave station. The master station here is generally a main control board, and the slave stations are usually multiple servo controllers for driving AC servo motors.

[0047] The AC servo motor is set to a synchronous position mode, in which the position is adjusted according to the pulse signal sent by the servo controller, and the photoelectric encoder value, angular velocity value and torque value are output. The controller system includes: a SOEM EtherCAT master station control board, which includes a display module, an Ethernet control module, an SD card module and a U disk module. Each module is an existing module and will not be described in detail in this embodiment.

[0048] In actual applications, users can operate CNC machine tools through the display module to achieve human-computer interaction. The SD card module and the USB flash drive module are used to store the graphics and position information of the required cutting materials. The main control board is connected to the servo controller through the Ethernet control module and the network cable. It is responsible for initializing the EtherCAT network, realizing the clock synchronization between the master control board and the servo controller, sending position information to each servo controller, and receiving the reference clock, angular velocity value, position information and other data fed back from the servo controller.

[0049] like Figure 1 As shown, a cross-coupling multi-axis synchronous control method based on EtherCAT provided in this embodiment is implemented in a master station, comprising the following steps:

[0050] Step 1: According to the contour line segment trajectory executed by each axis, the feed speed is calculated and preliminary interpolation processing is performed, and the contour line segment trajectory is discretely divided into multiple small straight line segments, each of which contains the feed speed and the target position. The interpolation processing used in this embodiment is a time-division interpolation algorithm, that is, the entire processing process is divided into contour steps that need to be executed within each interpolation cycle, and the entire contour processing process is divided into interpolation cycles as unit time.

[0051] Step 2: Take out each small straight line segment in turn as the current interpolation segment.

[0052] Step 3: Periodically send a synchronization trigger signal to all slave stations to trigger the synchronization operation of the slave stations. After receiving the synchronization trigger signal, the slave station feeds back the actual position and actual speed of each axis to the master station, and the master station receives the actual position and actual speed fed back by the slave station.

[0053] When the controller system starts, the controller of the EtherCAT master station first initializes the Ethernet control module. This includes setting SOME and binding the socket to the specified network card name. Subsequently, the controller sends a broadcast packet to identify the number of slave servo controller devices in the network, and configures the slave station to put it in the pre-operation state. In the pre-operation state, PDO data is bound to each servo controller and SDO parameters are set and read. Next, the master controller completes clock synchronization and communication synchronization cycle settings with the first servo controller, and switches the servo controller to safe operation mode. Then, RPDO and TPDO data are mapped to global variables for data setting and reading and writing. Subsequently, the slave station switches to operation mode, starts the reading and writing of PDO data, and performs master-slave clock synchronization.

[0054] After the master-slave clock synchronization is completed, the master station sends the position parameters of the desired trajectory and the reference feed speed to the servo controller in the periodic synchronous position mode every 1ms. After receiving it, the servo controller will feedback the actual position and actual speed to the master station in real time, so that the master station can calculate the contour error and speed deviation.

[0055] Step 4: Calculate the distance from the actual position to the target position of the interpolation line segment as the contour error, couple the calculated contour errors corresponding to each slave station through the cross-coupling controller to obtain the corresponding output, and combine the contour cross-coupling gain to obtain the contour error compensation value of each axis according to the output.

[0056] like Figure 2 As shown in the figure, the calculation process of contour error is as follows:

[0057] ε=E y Cy-E x Cx

[0058] E x =x r -x o

[0059] E y =y r -y o

[0060] Where ε is the contour error, E x is the tracking error of the x-axis, E y is the tracking error of the y-axis, x r is the target position on the x-axis, x o is the actual position of the x-axis, y r is the target position on the y-axis, y ois the actual position of the y-axis, Cx is the profile cross-coupling gain coefficient of the x-axis, and Cx=sinθ, Cy is the profile cross-coupling gain coefficient of the y-axis, and Cy=cosθ, θ is the angle between the tangent of the interpolation segment and the x-axis.

[0061] By adding cross-coupling control and using the cross-coupling controller, the corresponding contour errors of each slave station are coupled to the corresponding output through the cross-coupling controller. The cross-coupling controller adopts PI control, and its control law is:

[0062] ε′=K p ε+K i ∫εdt

[0063] In the formula, ε′ is the output, K p is the proportionality coefficient, K i Integration coefficient.

[0064] The essence of cross-coupling control is to use a cross-coupling controller to change the original open-loop contour control of the system into a closed-loop control, so that each axis has a coupling relationship. The application of this embodiment is mainly to compare the position feedback signal with the set target trajectory position to calculate the contour error and the contour cross-coupling gain. The present invention adopts an adaptive adjustment strategy to adjust the cross-coupling gain parameters according to real-time feedback to optimize the multi-axis synchronous control effect.

[0065] The ε′ obtained after the cross-coupling controller is multiplied by the contour cross-coupling gain, and superimposed with the target trajectory of each axis to obtain the contour error compensation value. The calculation formula is as follows:

[0066] U x =x r +ε′Cx

[0067] U y =y r +ε′Cy

[0068] Where U x is the contour error compensation value of the x-axis, U y is the contour error compensation value of the y-axis.

[0069] Step 5. Calculate the speed deviation by subtracting the actual speed fed back by each axis from the feed speed of the interpolation segment. Pass the calculated speed deviation corresponding to each slave station through a cross-coupling controller to obtain a cross-coupling speed deviation. Combined with the weighted proportional factor, a cross-coupling weighted speed deviation is obtained according to the cross-coupling speed deviation. The total speed deviation of each axis is obtained based on the speed deviation and the cross-coupling weighted speed deviation.

[0070] After the master station obtains the actual speed P1(k) and P2(k) of each axis fed back by the slave station, the speed deviations of each axis when it is independently controlled can be obtained in the kth sampling period:

[0071] E1(k)=E1(k-1)+ω1(k)-P1(k)

[0072] E2(k)=E2(k-1)+ω2(k)-P2(k)

[0073] In the formula, E1(k) is the speed deviation of axis 1 at the kth sampling period, E1(k-1) is the speed deviation of axis 1 at the k-1 moment, ω1(k) is the feed speed of axis 1 at the kth moment, P1(k) is the actual speed of axis 1 at the kth sampling period, E2(k) is the speed deviation of axis 2 at the kth sampling period, E2(k-1) is the speed deviation of axis 2 at the k-1 moment, ω2(k) is the feed speed of axis 2 at the kth moment, and P2(k) is the actual speed of axis 2 at the kth sampling period. Axis 1 is the x-axis / y-axis, and axis 2 is the y-axis / x-axis.

[0074] The speed error signal is used as the input signal of the cross-coupling controller. Figure 3 In the cross-coupled controller shown, the cross-coupled speed error can be obtained as:

[0075] E(k)=E(k-1)+k1P1(k)-k2P2(k)

[0076]

[0077] k1+k2=1

[0078] Wherein, E(k) is the cross-coupling speed deviation at time k, E(k-1) is the cross-coupling speed deviation at time k-1, k1 is the speed cross-coupling gain coefficient of axis 1, and k2 is the speed cross-coupling gain coefficient of axis 2.

[0079] Then, the weighted proportional factor α is introduced to obtain the cross-coupling weighted velocity deviation αE(k), and the total velocity deviation of each axis is obtained through calculation:

[0080] C1(k)=E1(k)-αE(k)

[0081] C2(k)=E2(k)-αE(k)

[0082] Where C1(k) is the total speed deviation of axis 1 at time k, C2(k) is the total speed deviation of axis 2 at time k, and α is the weighted proportional factor.

[0083] Step 6: Compensate the contour error compensation value of each axis to the corresponding slave station, and input the total speed deviation of each axis as an input signal to the corresponding slave station for motion control of the next synchronization cycle of the slave station.

[0084] In this embodiment, the total speed deviation is input as an input signal to each slave station, and the obtained contour error compensation value is compensated to each slave station servo controller respectively, and finally the position of the target interpolation of the next cycle is jointly obtained. The process of the slave station obtaining the position of the target interpolation of the next cycle according to the input signal and the compensation value is a prior art, which will not be described in detail in this embodiment.

[0085] The method of the present invention can not only directly reduce the contour error through the cross-coupling method, but also calculate the deviation between the expected feed speed of each axis and the actual speed, then enter the cross-coupling controller to obtain the cross-coupling weighted speed deviation, and finally calculate the total speed deviation of each axis as an input signal to each slave station to improve the synchronization performance of each axis of the system.

[0086] The technical features of the above-described embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0087] The above-mentioned embodiments only express several implementation modes of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the invention. It should be pointed out that, for a person of ordinary skill in the art, several modifications and improvements can be made without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the attached claims.

Claims

1. A cross-coupling multi-axis synchronous control method based on EtherCAT, which is used to control the coordinated operation of a multi-axis system based on the EtherCAT protocol. The EtherCAT protocol involves a master station and multiple slave stations, wherein the master station is a main controller and the slave stations are multiple servo controllers. The method is characterized in that: The cross-coupling multi-axis synchronous control method based on EtherCAT is implemented in the master station, comprising the following steps: According to the contour line segment trajectory executed by each axis, the feed speed is calculated and interpolation processing is performed, and the contour line segment trajectory is discretized into multiple small straight line segments, each of which contains the feed speed and the target position; Take out each small straight line segment in turn as the current interpolation segment; Periodically send synchronization trigger signals to all slave stations, and receive the actual position and actual speed of each axis fed back by the slave stations after receiving the synchronization trigger signals; The distance from the actual position to the target position of the interpolation line segment is calculated as the contour error. The corresponding contour errors of each slave station are coupled to the corresponding output through the cross-coupling controller, and the contour error compensation value of each axis is obtained according to the output in combination with the contour cross-coupling gain. The speed deviation is calculated by subtracting the actual speed of each axis feedback from the feed speed of the interpolation line segment. The calculated speed deviation corresponding to each slave station is passed through the cross-coupling controller to obtain the cross-coupling speed deviation. Combined with the weighted proportional factor, the cross-coupling weighted speed deviation is obtained according to the cross-coupling speed deviation. The total speed deviation of each axis is obtained based on the speed deviation and the cross-coupling weighted speed deviation. The contour error compensation value of each axis is compensated to the corresponding slave station, and the total speed deviation of each axis is input as an input signal to the corresponding slave station for motion control of the next synchronization cycle of the slave station.

2. The cross-coupling multi-axis synchronous control method based on EtherCAT according to claim 1, characterized in that: The calculation process of the contour error is as follows: ε=E y Cy-E x Box E x =x r -x o AND y =and r -and o Where ε is the contour error, E x is the tracking error of the x-axis, E y is the tracking error of the y-axis, x r is the target position on the x-axis, x o is the actual position of the x-axis, y r is the target position on the y-axis, y o is the actual position of the y-axis, Cx is the profile cross-coupling gain coefficient of the x-axis, and Cx=sinθ, Cy is the profile cross-coupling gain coefficient of the y-axis, and Cy=cosθ, θ is the angle between the tangent of the interpolation segment and the x-axis.

3. The cross-coupling multi-axis synchronous control method based on EtherCAT according to claim 1, characterized in that: The step of coupling the calculated contour errors corresponding to the slave stations to obtain corresponding output quantities through a cross-coupling controller includes: ε′=K p e+K i ∫εdt In the formula, ε′ is the output, ε is the contour error, K p is the proportionality coefficient, K i Integration coefficient.

4. The cross-coupling multi-axis synchronous control method based on EtherCAT according to claim 1, characterized in that: The combined contour cross-coupling gain is used to obtain the contour error compensation value of each axis according to the output, including: U x =x r +ε′Cx U y =y r +ε′Cy Where U x is the contour error compensation value of the x-axis, U y is the contour error compensation value of the y-axis, ε′ is the output, and x r is the target position on the x-axis, y r is the target position of the y-axis, Cx is the profile cross-coupling gain coefficient of the x-axis, and Cx=sinθ, Cy is the profile cross-coupling gain coefficient of the y-axis, and Cy=cosθ, θ is the angle between the tangent of the interpolation segment and the x-axis.

5. The cross-coupling multi-axis synchronous control method based on EtherCAT according to claim 1, characterized in that: The speed deviation is calculated by subtracting the actual speed of each axis feedback from the feed speed of the interpolation segment, including: E1(k)=E1(k-1)+ω1(k)-P1(k) E2(k)=E2(k-1)+ω2(k)-P2(k) Wherein, E1(k) is the speed deviation of axis 1 at the kth sampling period, E1(k-1) is the speed deviation of axis 1 at the k-1 moment, ω1(k) is the feed speed of axis 1 at the kth moment, P1(k) is the actual speed of axis 1 at the kth sampling period, E2(k) is the speed deviation of axis 2 at the kth sampling period, E2(k-1) is the speed deviation of axis 2 at the k-1 moment, ω2(k) is the feed speed of axis 2 at the kth moment, and P2(k) is the actual speed of axis 2 at the kth sampling period.

6. The cross-coupling multi-axis synchronous control method based on EtherCAT according to claim 1, characterized in that: The step of obtaining a cross-coupling speed deviation by passing the calculated speed deviation corresponding to each slave station through a cross-coupling controller comprises: E(k)=E(k-1)+k1P1(k)-k2P2(k) k1+k2=1 Wherein, E(k) is the cross-coupling speed deviation at time k, E(k-1) is the cross-coupling speed deviation at time k-1, k1 is the speed cross-coupling gain coefficient of axis 1, k2 is the speed cross-coupling gain coefficient of axis 2, ω1(k) is the feed speed of axis 1 at time k, P1(k) is the actual speed of axis 1 at the kth sampling period, ω2(k) is the feed speed of axis 2 at time k, and P2(k) is the actual speed of axis 2 at the kth sampling period.

7. The cross-coupling multi-axis synchronous control method based on EtherCAT according to claim 1, characterized in that: The combined weighted proportional factor is used to obtain a cross-coupling weighted speed deviation according to the cross-coupling speed deviation, and the total speed deviation of each axis is obtained based on the speed deviation and the cross-coupling weighted speed deviation, including: C1(k)=E1(k)-αE(k) C2(k)=E2(k)-αE(k) Where C1(k) is the total speed deviation of axis 1 at time k, C2(k) is the total speed deviation of axis 2 at time k, E1(k) is the speed deviation of axis 1 at the kth sampling period, E2(k) is the speed deviation of axis 2 at the kth sampling period, E(k) is the cross-coupling speed deviation at time k, α is the weighted proportional factor, and αE(k) is the cross-coupling weighted speed deviation.

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