WO2019047457A1 - 一种等效平面交叉耦合控制方法 - Google Patents

一种等效平面交叉耦合控制方法 Download PDF

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WO2019047457A1
WO2019047457A1 PCT/CN2018/071688 CN2018071688W WO2019047457A1 WO 2019047457 A1 WO2019047457 A1 WO 2019047457A1 CN 2018071688 W CN2018071688 W CN 2018071688W WO 2019047457 A1 WO2019047457 A1 WO 2019047457A1
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axis
contour
equivalent plane
point
error
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French (fr)
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马建伟
贾振元
宋得宁
王福吉
刘巍
张宁
陈思宇
贺广智
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大连理工大学
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Priority to US16/311,922 priority Critical patent/US10921772B2/en
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/19—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B11/00—Automatic controllers
    • G05B11/01—Automatic controllers electric
    • G05B11/36—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential
    • G05B11/42—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential for obtaining a characteristic which is both proportional and time-dependent, e.g. P. I., P. I. D.
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/33—Director till display
    • G05B2219/33099—Computer numerical control [CNC]; Software control [SWC]

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  • the invention belongs to the technical field of precise and efficient intelligent multi-axis numerical control machining, and relates to an equivalent plane cross-coupling control method for improving the contour tracking precision of a three-axis numerical control system.
  • Multi-axis linkage contour tracking is the main function that the CNC system needs to complete during the machining of curved parts.
  • the multi-axis linkage contour tracking movement due to factors such as the following error of the single-axis servo control system, dynamic mismatch of the multi-axis control system and external disturbances, the deviation between the actual motion trajectory and the ideal motion trajectory, that is, the contour error . Due to the existence of this error, the motion accuracy of CNC machine tools is degraded, which affects the machining accuracy. Therefore, the research on the control method of contour error is of great significance for achieving precise and efficient machining.
  • cross-coupling control is the main method for contour error suppression.
  • Prior Art Document 1 "Analysis and Design of Integrated Control for Multi-Axis Motion Systems", Yeh et al, IEEE Transactions on Control Systems Technology, 2003, 11(3): 375-382, which estimates the spatial profile by a tangent approximation method.
  • the error vector uses the modulus of the contour error vector as the controlled object of the designed cross-coupling controller.
  • the controlled object is a vector of the vector, it is always positive, which results in limited design flexibility of the control algorithm.
  • the invention aims to overcome the defects of the prior art, and to invent an equivalent plane cross-coupling control method for improving the contour tracking precision of a three-axis numerical control system, which is based on the spatial curve and the actual motion position geometric information, and Newton's method, looking for the approximate ground point of the actual tool position to the ideal contour, and then establishing an equivalent plane by approximating the curve tangent to the actual tool point and the actual tool location, decoupling the 3D contour error vector into the equivalent plane Dimensional contour error scalar, and PID control of signed contour error scalar in equivalent plane, coupling two-axis control quantity in equivalent plane into real-space three-axis control quantity, reducing tri-axis contour error and improving three-axis Contour tracking accuracy.
  • the technical solution of the present invention is an equivalent plane cross-coupling control method, which is characterized in that the method finds an approximate footing point of an actual tool position to an ideal contour, and approximates a curve tangent line at an approximate foot point and an actual tool point.
  • Establish an equivalent plane decouple the 3D contour error vector into a 2D contour error scalar in the equivalent plane, and PID control the signed contour error scalar in the equivalent plane, and control the two axes in the equivalent plane.
  • the quantity coupling is the actual space three-axis control quantity, thereby improving the accuracy of the three-axis contour tracking; the specific steps of the method are as follows:
  • the first step is to establish an equivalent plane
  • u be the curve parameter
  • the curve parameter is u r
  • the tangential error d t (u) at the C(u) point on the defined curve is the vector C(u)-P
  • the projection in the tangential direction at the point C(u) is calculated as:
  • C'(u) is the first-order derivative of C(u) for the curve parameter u, and
  • represents the Euclidean norm
  • the contour error is defined as the vertical distance from the actual tool point to the ideal contour
  • the tangential error d t (u) must be zero when C(u) is exactly the closest foot point to the actual tool point P on the ideal contour.
  • the equivalent plane is established by the actual knife location P and the tangent of the curve at the approximate footing point C(u f ).
  • the equivalent plane normal vector n E is:
  • ⁇ represents the vector outer product
  • the second step is the equivalent in-plane contour error calculation and cross-coupling control
  • C x, E , C y, and E are the cross-coupling gains of the X E and Y E directions in the equivalent plane , respectively, calculated as:
  • ⁇ is the angle between C'(u f ) and X E .
  • k p , k i , and k d are proportional, integral, and differential gain, respectively;
  • the X E control amount ⁇ x, E , Y E is calculated to the control amount ⁇ y, E :
  • the third step is to calculate the volume of the three-axis control space.
  • k x,x is the X E- axis to X-axis coupling gain
  • k x,y is the X E- axis to Y-axis coupling gain
  • k y,x is the Y E- axis to X-axis coupling gain
  • k y,y is Y E- axis to Y-axis coupling gain
  • k y,z is Y E- axis to Z-axis coupling gain
  • the X-axis control amount ⁇ x the Y-axis control amount ⁇ y , and the Z-axis control amount ⁇ z are calculated:
  • the X-axis control amount ⁇ x , the Y-axis control amount ⁇ y , and the Z-axis control amount ⁇ z are respectively added to the X, Y, Z feed axis position loop control amount to realize the equivalent plane cross-coupling control, thereby reducing the three-dimensional space. Contour error.
  • the invention has the beneficial effects of inventing the equivalent plane cross-coupling control method, decoupling the three-dimensional contour error vector into the contour error scalar in the equivalent plane, which can improve the flexibility of the contour controller design; For the two-axis contour error control problem, the equivalent control of the three-dimensional contour tracking error using the two-dimensional contour controller can be realized.
  • Figure 2 Geometric model of the curved tool path in a Cartesian coordinate system
  • Figure 3 Profile error map obtained by the method of the invention and without using the method of the invention; wherein the A axis is the processing time, the unit is s, the B axis is the contour error, the unit is mm; the curve 1 indicates that the method of the invention is not obtained.
  • the contour error, curve 2 represents the contour error obtained by the method of the invention.
  • FIG. 1 is a whole process flow chart of the method
  • FIG. 2 is a geometric model diagram of a curved tool path in a Cartesian coordinate system.
  • the tool rail shown in FIG. 2 is taken as an example to describe the specific implementation process of the present invention in detail.
  • the equivalent plane cross-coupling control is performed on the three-dimensional spatial contour error in the curve-track contour tracking motion control process shown in FIG. 2, and the specific steps are as follows:
  • the first step is to establish an equivalent plane: according to the method described in the first step of the invention, based on the tangential inverse and Newton method, find the approximate foot point C of the actual tool point P to the ideal tool path contour C(u) ( u f ), and use the formula (5) to calculate the equivalent plane normal vector n E , and use the formula (6) to calculate the equivalent plane horizontal axis X E and the equivalent plane vertical axis Y E ;
  • the second step is the calculation of the equivalent in-plane contour error and the cross-coupling control: using the formula (8) to calculate the three-dimensional contour error scalar estimated value with the sign in the equivalent plane. And on the basis of PID control, using formula (11) to calculate the equivalent plane X E to control amount ⁇ x, E , Y E to control amount ⁇ y, E ;
  • the third step is to calculate the three-axis control of the space: Calculate the coupling gain of the two axes of the equivalent plane to the space axes according to formula (12), and then calculate the three-dimensional X-axis control amount ⁇ x and the Y-axis control amount ⁇ by using equation (13).
  • y , Z-axis control amount ⁇ z , the X-axis control amount ⁇ x , the Y-axis control amount ⁇ y , and the Z-axis control amount ⁇ z are respectively added to the X, Y, Z feed axis position loop control amount, in each
  • the interpolation cycle performs the above three steps to achieve equivalent plane cross-coupling control.
  • Figure 3 is a diagram showing the contour error obtained by the method of the present invention and without using the method of the present invention; wherein the A-axis is the processing time, the unit is s, the B-axis is the contour error, and the unit is mm;
  • the contour error obtained by the inventive method, curve 2 represents the contour error obtained by the method of the invention; it can be seen that the maximum contour error obtained by the method of the invention is about 0.4 mm, and the maximum contour error obtained by the method of the invention is about 0.07.
  • Mm using the equivalent plane cross-coupling control method of the invention, the maximum value of the three-dimensional space contour error is reduced by 82.5%; therefore, the method of the invention can effectively reduce the contour error during the three-axis machining process and improve the spatial contour tracking accuracy.
  • the invention faces the problem of contour error due to factors such as servo lag and external disturbance in the curve interpolation process, and the three-dimensional space contour error is difficult to control than the two-dimensional plane contour error, and an equivalent plane cross-coupling control method is invented, which can be three-dimensional
  • the decoupling of the contour error vector into the contour error scalar in the equivalent plane not only facilitates the flexible design of the contour controller, but also realizes the equivalent control of the three-dimensional contour tracking error by the two-dimensional contour controller.

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Abstract

一种等效平面交叉耦合控制方法，属于多轴数控加工技术领域，涉及一种基于等效平面的用于提高三轴数控系统轮廓跟踪控制精度的三轴交叉耦合控制方法。该方法通过切向逆推及牛顿法，寻找实际刀位点到理想轮廓的近似垂足点，进而过近似垂足点处曲线切线及实际刀位点建立等效平面，将三维轮廓误差矢量解耦成等效平面内的二维轮廓误差标量。并在等效平面内对带符号的轮廓误差标量进行PID控制，将等效平面内的两轴控制量耦合为实际空间三轴控制量，从而提高三轴轮廓跟踪精度。可提高轮廓控制器设计的灵活性，将三维轮廓误差控制等效为两轴轮廓误差控制问题，可实现利用二维轮廓控制器对三维轮廓跟踪误差的等价控制。

Description

一种等效平面交叉耦合控制方法 技术领域
本发明属于精密高效智能化多轴数控加工技术领域，涉及一种用于提高三轴数控系统空间曲线轮廓跟踪精度的等效平面交叉耦合控制方法。
背景技术
多轴联动轮廓跟踪是数控系统在进行曲面零件加工过程中需完成的主要功能。在多轴联动轮廓跟踪运动过程中，由于单轴伺服控制系统随动误差、多轴控制系统动态失匹及外部扰动等因素存在，会导致实际运动轨迹与理想运动轨迹间产生偏差，即轮廓误差。由于该误差的存在，导致数控机床运动精度下降，影响加工精度，因此，研究轮廓误差的控制方法，对实现精密高效加工具有重要意义。目前，交叉耦合控制是用于轮廓误差抑制的主要方法，然而，虽两轴交叉耦合控制方法有较多报道，但三轴交叉耦合控制器的设计方法较少，且大部分现有两轴交叉耦合控制无法推广应用到三轴轮廓跟踪中。鉴于复杂曲面零件往往需要三个进给轴进行空间联动实现加工，三轴交叉耦合控制方法需求迫切。
现有技术文献1“Analysis and Design of Integrated Control for Multi-AxisMotion Systems”，Yeh等，IEEE Transactions on Control Systems Technology，2003，11(3)：375-382，该文献通过切线近似的方法估计空间轮廓误差矢量，将轮廓误差矢量的模作为所设计的交叉耦合控制器的被控对象，然而，此时被控对象由于是矢量的模，故恒为正值，导致控制算法设计灵活性受限。文献2“A two-layeredcross coupling control scheme for a three-dimensional motion control system”，Zhang等，International Journal of Machine Tools&Manufacture，2015，98：12-20， 该文献建立一种两层交叉耦合控制结构，在底层对平面两轴轮廓误差进行控制，在顶层对第三轴的引入产生的轮廓误差进行控制，然而该方法未将实际空间轮廓误差作为被控对象，主要适用于对二维轮廓精度要求更高的场合。
发明内容
本发明旨在克服现有技术缺陷，发明一种用于提高三轴数控系统轮廓跟踪精度的等效平面交叉耦合控制方法，该方法根据空间曲线与实际运动位置几何信息，通过切向逆推及牛顿法，寻找实际刀位点到理想轮廓的近似垂足点，进而过近似垂足点处曲线切线及实际刀位点建立等效平面，将三维轮廓误差矢量解耦成等效平面内的二维轮廓误差标量，并在等效平面内对带符号的轮廓误差标量进行PID控制，将等效平面内的两轴控制量耦合为实际空间三轴控制量，降低三轴轮廓误差，提高三轴轮廓跟踪精度。
本发明的技术方案是一种等效平面交叉耦合控制方法，其特性在于，该方法通过寻找实际刀位点到理想轮廓的近似垂足点，过近似垂足点处曲线切线及实际刀位点建立等效平面，将三维轮廓误差矢量解耦成等效平面内的二维轮廓误差标量，并在等效平面内对带符号的轮廓误差标量进行PID控制，将等效平面内的两轴控制量耦合为实际空间三轴控制量，从而提高三轴轮廓跟踪精度；方法具体步骤如下：
第一步 建立等效平面
设从参数曲线插补器中获得的理想轮廓参数方程为C＝C(u)，u为曲线参数，理想刀位点为R＝[r x,r y,r z]，理想刀位点处曲线参数为u r，实际刀位点为P＝[p x,p y,p z]；定义曲线上C(u)点处的切向误差d t(u)为向量C(u)-P在C(u)点处切矢方向上的投影，计算为：
Figure PCTCN2018071688-appb-000001
其中C′(u)为C(u)对曲线参数u的一阶导矢，|| ||表示欧几里得范数；
由于轮廓误差定义为实际刀位点到理想轮廓的垂直距离，当C(u)恰好为理想轮廓上距离实际刀位点P最近的垂足点时，切向误差d t(u)必为零；通过求解方程d t(u)＝0找到P到理想轮廓近似垂足点处参数u f，以为建立等效平面奠定基础；首先，通过切向逆推计算逆推点参数u b：
Figure PCTCN2018071688-appb-000002
其次，将参数u b作为牛顿法初值，利用牛顿法求取方程d t(u)＝0的解u N：
Figure PCTCN2018071688-appb-000003
最后，判断牛顿法是否收敛，若|d t(u N)|＜|d t(u b)|，说明牛顿法收敛，令垂足点参数u f＝u N，否则，在u b处重新利用切向逆推计算垂足点参数，据此，垂足点参数为u f计算为：
Figure PCTCN2018071688-appb-000004
过实际刀位点P和近似垂足点C(u f)处曲线切线建立等效平面，等效平面法向量n E为：
Figure PCTCN2018071688-appb-000005
其中×表示向量外积；
以等效平面与原始空间直角坐标系XY平面交线作为等效平面水平轴X E，以垂直于X E的方向作为等效平面竖直轴Y E，二者计算方法为：
Figure PCTCN2018071688-appb-000006
第二步 等效平面内轮廓误差计算与交叉耦合控制
在等效平面内计算带有正负号的轮廓误差估计值，实际刀位点P到近似垂足点C(u f)的X E向随动误差e x,E及Y E向随动误差e y,E为：
Figure PCTCN2018071688-appb-000007
计算等效平面内轮廓误差估计值
Figure PCTCN2018071688-appb-000008
Figure PCTCN2018071688-appb-000009
其中C x,E、C y,E分别为等效平面内X E和Y E向交叉耦合增益，计算为：
Figure PCTCN2018071688-appb-000010
θ为C′(u f)与X E的夹角，且
Figure PCTCN2018071688-appb-000011
以带有正负号的等效平面内轮廓误差估计值
Figure PCTCN2018071688-appb-000012
为控制对象，进行PID控制，得到t时刻交叉耦合控制量U c(t)为：
Figure PCTCN2018071688-appb-000013
其中k p、k i、k d分别为比例、积分、微分增益；
根据交叉耦合控制量U c(t)，计算X E向控制量Δ x,E、Y E向控制量Δ y,E：
Figure PCTCN2018071688-appb-000014
第三步 空间三轴控制量计算
根据等效平面水平轴、竖直轴与原始空间直角坐标系X、Y、Z轴关系，计算等效平面两轴到空间各轴的耦合增益：
Figure PCTCN2018071688-appb-000015
其中，k x,x为X E轴到X轴耦合增益，k x,y为X E轴到Y轴耦合增益，k y,x为Y E轴到X轴耦合增益，k y,y为Y E轴到Y轴耦合增益，k y,z为Y E轴到Z轴耦合增益；
进而计算X轴控制量Δ x、Y轴控制量Δ y、Z轴控制量Δ z：
Figure PCTCN2018071688-appb-000016
将X轴控制量Δ x、Y轴控制量Δ y、Z轴控制量Δ z分别加入到X、Y、Z进给轴位置环控制量中，实现等效平面交叉耦合控制，从而降低三维空间轮廓误差。
本发明的有益效果是发明了等效平面交叉耦合控制方法，将三维轮廓误差矢量解耦成等效平面内的轮廓误差标量，可提高轮廓控制器设计的灵活性；将三维轮廓误差控制等效为两轴轮廓误差控制问题，可实现利用二维轮廓控制器 对三维轮廓跟踪误差的等价控制。
附图说明
图1—方法整体流程图；
图2—直角坐标系中曲线刀轨几何模型图；
图3—利用本发明方法和不利用本发明方法得到的轮廓误差图；其中，A轴为加工时间，单位为s，B轴为轮廓误差，单位为mm；曲线1表示不利用本发明方法得到的轮廓误差，曲线2表示利用本发明方法得到的轮廓误差。
具体实施方式
结合技术方案与附图详细说明本发明的具体实施方式。
在曲线插补加工过程中，由于单轴随动误差及多轴动态失匹等因素存在，导致实际运动轨迹与理想轨迹产生偏差，即轮廓误差。为降低三轴轮廓误差，提高加工精度，发明一种等效平面交叉耦合控制方法。
附图1为方法整体流程图，附图2为直角坐标系中曲线刀轨几何模型图，以附图2所示刀轨为例，详细说明本发明具体实施过程。
根据附图1所示方法整体流程，对附图2所示曲线刀轨轮廓跟踪运动控制过程中的三维空间轮廓误差进行等效平面交叉耦合控制，具体步骤为：
第一步 建立等效平面：根据发明内容中第一步所述方法，基于切向逆推及牛顿法，寻找实际刀位点P到理想刀轨轮廓C(u)的近似垂足点C(u f)，并利用公式(5)计算等效平面法向量n E，利用公式(6)计算等效平面水平轴X E及等效平面竖直轴Y E；
第二步 等效平面内轮廓误差计算与交叉耦合控制：利用公式(8)计算等效平面内带有正负号的三维轮廓误差标量估计值
Figure PCTCN2018071688-appb-000017
并在对其进行PID控制的基础 上，利用公式(11)计算等效平面内X E向控制量Δ x,E、Y E向控制量Δ y,E；
第三步 空间三轴控制量计算：根据公式(12)计算等效平面两轴到空间各轴的耦合增益，进而利用公式(13)计算三维空间X轴控制量Δ x、Y轴控制量Δ y、Z轴控制量Δ z，将X轴控制量Δ x、Y轴控制量Δ y、Z轴控制量Δ z分别加入到X、Y、Z进给轴位置环控制量中，在每一个插补周期执行上述三个步骤，实现等效平面交叉耦合控制。
附图3所示为利用本发明方法和不利用本发明方法得到的轮廓误差图；其中，A轴为加工时间，单位为s，B轴为轮廓误差，单位为mm；曲线1表示不利用本发明方法得到的轮廓误差，曲线2表示利用本发明方法得到的轮廓误差；可见，不利用本发明方法得到的轮廓误差最大值约为0.4mm，利用本发明方法得到的轮廓误差最大值约为0.07mm，利用本发明等效平面交叉耦合控制方法将三维空间轮廓误差最大值降低了82.5％；因此，本发明方法可有效降低三轴加工过程中的轮廓误差，提高空间轮廓跟踪精度。
本发明面向曲线插补过程中由于伺服滞后及外界扰动等因素存在轮廓误差，且三维空间轮廓误差较二维平面轮廓误差控制困难的问题，发明一种等效平面交叉耦合控制方法，可将三维轮廓误差矢量解耦成等效平面内的轮廓误差标量，不仅有利于轮廓控制器的灵活设计，更可实现利用二维轮廓控制器对三维轮廓跟踪误差的等价控制。

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  1. 一种等效平面交叉耦合控制方法，其特性在于，该方法通过寻找实际刀位点到理想轮廓的近似垂足点，过近似垂足点处曲线切线及实际刀位点建立等效平面，将三维轮廓误差矢量解耦成等效平面内的二维轮廓误差标量，并在等效平面内对带符号的轮廓误差标量进行PID控制，将等效平面内的两轴控制量耦合为实际空间三轴控制量，从而提高三轴轮廓跟踪精度；方法具体步骤如下：
    第一步建立等效平面
    设从参数曲线插补器中获得的理想轮廓参数方程为C＝C(u)，u为曲线参数，理想刀位点为R＝[r x,r y,r z]，理想刀位点处曲线参数为u r，实际刀位点为P＝[p x,p y,p z]；定义曲线上C(u)点处的切向误差d t(u)为向量C(u)-P在C(u)点处切矢方向上的投影，计算为：
    Figure PCTCN2018071688-appb-100001
    其中C′(u)为C(u)对曲线参数u的一阶导矢，|| ||表示欧几里得范数；
    由于轮廓误差定义为实际刀位点到理想轮廓的垂直距离，当C(u)恰好为理想轮廓上距离实际刀位点P最近的垂足点时，切向误差d t(u)必为零；通过求解方程d t(u)＝0找到P到理想轮廓近似垂足点处参数u f，以为建立等效平面奠定基础；首先，通过切向逆推计算逆推点参数u b：
    Figure PCTCN2018071688-appb-100002
    其次，将参数u b作为牛顿法初值，利用牛顿法求取方程d t(u)＝0的解u N：
    Figure PCTCN2018071688-appb-100003
    最后，判断牛顿法是否收敛，若|d t(uN)|＜|d t(u b)|，说明牛顿法收敛，令垂足点参数u f＝u N，否则，在u b处重新利用切向逆推计算垂足点参数，据此，垂足点参数为u f计算为：
    Figure PCTCN2018071688-appb-100004
    过实际刀位点P和近似垂足点C(u f)处曲线切线建立等效平面，等效平面法向量n E为：
    Figure PCTCN2018071688-appb-100005
    其中×表示向量外积；
    以等效平面与原始空间直角坐标系XY平面交线作为等效平面水平轴X E，以垂直于X E的方向作为等效平面竖直轴Y E，二者计算方法为：
    Figure PCTCN2018071688-appb-100006
    第二步等效平面内轮廓误差计算与交叉耦合控制
    在等效平面内计算带有正负号的轮廓误差估计值，实际刀位点P到近似垂足点C(u f)的X E向随动误差e x,E及Y E向随动误差e y,E为：
    Figure PCTCN2018071688-appb-100007
    计算等效平面内轮廓误差估计值
    Figure PCTCN2018071688-appb-100008
    Figure PCTCN2018071688-appb-100009
    其中C x,E、C y,E分别为等效平面内X E和Y E向交叉耦合增益，计算为：
    Figure PCTCN2018071688-appb-100010
    θ为C′(u f)与X E的夹角，且
    Figure PCTCN2018071688-appb-100011
    以带有正负号的等效平面内轮廓误差估计值
    Figure PCTCN2018071688-appb-100012
    为控制对象，进行PID控制，得到t时刻交叉耦合控制量U c(t)为：
    Figure PCTCN2018071688-appb-100013
    其中k p、k i、k d分别为比例、积分、微分增益；
    根据交叉耦合控制量U c(t)，计算X E向控制量Δ x,E、Y E向控制量Δ y,E：
    Figure PCTCN2018071688-appb-100014
    第三步空间三轴控制量计算
    根据等效平面水平轴、竖直轴与原始空间直角坐标系X、Y、Z轴关系，计算等效平面两轴到空间各轴的耦合增益：
    Figure PCTCN2018071688-appb-100015
    其中，k x,x为X E轴到X轴耦合增益，k x,y为X E轴到Y轴耦合增益，k y,x为Y E轴到X轴耦合增益，k y,y为Y E轴到Y轴耦合增益，k y,z为Y E轴到Z轴耦合增益；
    进而计算X轴控制量Δ x、Y轴控制量Δ y、Z轴控制量Δ z：
    Figure PCTCN2018071688-appb-100016
    将X轴控制量Δ x、Y轴控制量Δ y、Z轴控制量Δ z分别加入到X、Y、Z进给轴位置环控制量中，实现等效平面交叉耦合控制，从而降低三维空间轮廓误差。
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