WO2021003928A1 - 工件轮廓检测的恒线速度控制方法 - Google Patents

工件轮廓检测的恒线速度控制方法 Download PDF

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WO2021003928A1
WO2021003928A1 PCT/CN2019/117478 CN2019117478W WO2021003928A1 WO 2021003928 A1 WO2021003928 A1 WO 2021003928A1 CN 2019117478 W CN2019117478 W CN 2019117478W WO 2021003928 A1 WO2021003928 A1 WO 2021003928A1
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workpiece
contour
measured
scanning
linear velocity
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PCT/CN2019/117478
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French (fr)
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周向东
唐小琦
卢少武
颜昌亚
张庆祥
曾祥兵
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东莞市三姆森光电科技有限公司
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Publication of WO2021003928A1 publication Critical patent/WO2021003928A1/zh
Priority to ZA2022/01073A priority Critical patent/ZA202201073B/en

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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B11/00—Measuring arrangements characterised by the use of optical techniques
    • G01B11/24—Measuring arrangements characterised by the use of optical techniques for measuring contours or curvatures
    • 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
    • G05B19/416—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 control of velocity, acceleration or deceleration

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  • the present invention belongs to the technical field of detection of mechanical manufacturing, and more specifically, relates to a method for controlling a constant linear velocity when detecting a workpiece contour.
  • the traditional method of detecting the contour of the workpiece is that the workpiece and the sensor are kept at a certain distance and vertical, and the workpiece is rotated at an equal angle, and the sensor signal is read point by point.
  • the sampling on the contour is unequal, which affects the contour fitting accuracy.
  • the invention discloses a constant linear speed control method during the contour detection of a workpiece.
  • the layout of the workpiece and the laser displacement sensor of the present invention is that the position of the laser displacement sensor is fixed.
  • the constant linear velocity scanning point is planned by establishing the kinematic model of the scanning point pose and axis movement , Real-time calculation of the movement of the XYC axis corresponding to the pose of each scan point, and drive and control the translation and rotation of the worktable (X/Y/C three directions) to ensure that the focus of the sensor is always on the surface of the workpiece contour, and The relative movement of the workpiece and the focus of the laser displacement sensor is kept at a constant speed.
  • the contour of the workpiece to be measured is an XY plane contour.
  • the contour is composed of two types of geometric elements: straight line and circular arc.
  • the connection relationship between adjacent elements in the contour must satisfy G1 continuous, that is, the normal vector at the connection point is the same, The tangent included angle is zero;
  • the contour of the workpiece to be measured can be generated by drawing software and saved in DXF format;
  • S4 Constant linear velocity scanning point plan, according to the laser displacement sensor sampling frequency, sampling interval and interpolation cycle, calculate the scanning speed per cycle; because the sampling frequency, sampling interval and interpolation cycle remain fixed in one measurement, the planning is The scanning speed is constant; take the geometric elements of the contour of the workpiece to be measured in turn, calculate the step length of each cycle according to the calculated scanning speed, and use the step length to discretize the geometric elements, and calculate the position and normal vector of the scanning point corresponding to each cycle;
  • the present invention adopts the laser displacement sensor to fix the position, by controlling the translation and rotation movement of the worktable (X/Y/C three directions), the workpiece and the laser displacement sensor focus will move relative to each other.
  • the contour is divided into small equidistant line segments according to the requirements of constant linear velocity, and the measurement scanning points are planned and calculated.
  • the established kinematics model is used to calculate the corresponding scanning point Axis movement displacement amount, while driving and controlling platform movement. Since the calculation process is carried out in each interpolation control cycle, a good constant linear velocity control effect can be obtained.
  • Figure 1 is a schematic diagram of the structure of a measurement and control platform device
  • Figure 2 is a flow chart of constant linear velocity control of the contour of the workpiece to be measured
  • Figure 3 is a schematic diagram of the calculation of the coordinates of the scanning point of the linear element
  • Figure 4 is a schematic diagram of the normal vector calculation of the linear contour scanning point
  • Figure 5 is a schematic diagram of the motion transformation model of the present invention.
  • Fig. 6 is a schematic diagram of the motion transformation process of Fig. 5;
  • Figure 7 is a schematic diagram of the motion parameter calculation of the present invention.
  • the present invention provides a constant linear velocity control method during workpiece contour detection.
  • the specific steps are as follows:
  • the platform has an XY translation axis, a C rotation axis, and a laser displacement sensor; the workpiece to be measured is set on the platform, and the workpiece is driven by the axis for translation and rotation.
  • the normal vector of the workpiece surface and the laser sensor can be guaranteed during the detection process. In the same direction, while keeping the focus of the laser sensor always on the contour of the workpiece surface.
  • the contour of the workpiece to be measured is an XY plane contour, which is composed of two types of geometric elements: straight line and circular arc; the connection relationship between adjacent elements in the contour must satisfy G1 continuous, that is, the normal vector at the connection point is the same, and the tangent direction at the connection point The included angle is zero; the contour must be closed;
  • the contour of the workpiece to be measured can be generated by drawing software and saved in DXF format.
  • the sampling frequency, sampling interval and interpolation period of the laser displacement sensor calculate the scanning speed of each cycle; because the sampling frequency, sampling interval and interpolation period are kept fixed in one measurement, the planned scanning speed is constant; take the workpiece to be tested in turn
  • the geometric elements of the contour according to the calculated scanning speed, calculate the step length of each period of scanning, and use the step length to discretize the geometric elements, and calculate the position and normal vector of the scanning point corresponding to each period.
  • the specific calculation steps are as follows:
  • the position and normal vector of the scanning point 4 are determined by the geometric characteristics of the profile, the sampling frequency and the sampling interval of the measurement. Assuming the probe sampling frequency f (HZ) and the sampling interval l (mm), the relative velocity v (mm/s) of the laser displacement sensor 1 relative to the contour of the workpiece 2 to be measured can be calculated as:
  • the sampling interval of the sampling frequency is a fixed value, so the scanning speed v is guaranteed to be constant.
  • a fixed moving step is obtained, and the contour is divided into equidistant scanning points 4 according to the cycle, so as to realize the constant linear velocity scanning point 4 planning.
  • the position and normal vector of the scanning point 4 can be calculated. Since the contour of the workpiece 2 to be measured includes straight line and circular arc types, it needs to be calculated according to different types. In this embodiment, the linear type is taken as an example for description.
  • v is the scanning speed
  • T is the interpolation period
  • I the tangent to the contour of the workpiece 2 to be measured.
  • the starting point of the contour of the workpiece 2 to be measured during scanning Known, so the subsequent points can be calculated by recursive formula, the tangent of the straight line It can be calculated from the start and end information of the straight line.
  • the normal vector of the scanning point 4 is always perpendicular to the tangent direction of the contour of the workpiece 2 to be measured. After calculating the tangent of the contour of the workpiece 2 to be measured, implement a 90-degree transformation ROT(90) on the tangent to obtain the scanning point The law vector. For a straight profile, as shown in Figure 4. S5, constant linear velocity scanning control
  • the measuring device has a translation axis XY axis and a rotation axis C. Place the workpiece on the platform, and adjust to ensure that the coordinate system of the workpiece to be measured coincides with the coordinate system M of the machine tool (the coordinate system M in Figure 5 is the origin of the machine tool coordinate system and the center of rotation at the same time), and the laser displacement sensor 1 is installed at a fixed position.
  • the vector of focus 3 in the machine coordinate system M is The laser beam direction is
  • the scanning point can be set through the rotation of the C axis and the XY translation movement. Perform coordinate transformation and focus on laser Coincidence, while ensuring the normal vector of the scanning point And laser beam direction parallel.
  • C, X, and Y are transformation parameters, which are rotation and translation respectively.
  • Equation 3 the relationship established by Equation 3, when with it is known, the transformation parameters C and X and Y can be obtained by solving the equations formed by equations 3 and 4. Since the transformation is produced by the movement of the axis, X, Y and C are the movements of the XYC axis respectively.
  • the specific calculation first calculates the rotation angle C and then the translation amount, the specific process is as follows:

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  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Human Computer Interaction (AREA)
  • Manufacturing & Machinery (AREA)
  • Automation & Control Theory (AREA)
  • Length Measuring Devices By Optical Means (AREA)
  • Laser Beam Processing (AREA)

Abstract

一种工件（2）轮廓检测的恒线速度控制方法，包括如下步骤：将激光位移传感器（1）位置固定，通过控制工作台的平移和旋转运动，工件（2）和激光位移传感器（1）的焦点（3）产生相对运动，在运动过程中，保证激光位移传感器（1）的焦点（3）始终在工件（2）轮廓的表面，且激光位移传感器（1）的焦点（3）相对于工件（2）运动的线速度保持恒定。该方法解决了传统轮廓检测中采用等角度旋转但轮廓上不等距的问题，具有工件（2）轮廓恒线速运动的优点，保证了工件（2）轮廓上检测点的均匀选取。

Description

工件轮廓检测的恒线速度控制方法 技术领域
本发明属于机械制造的检测技术领域，更具体地，涉及一种工件轮廓检测时的恒线速度控制方法。
背景技术
传统的工件轮廓的检测方法是工件和传感器保持一定的距离并垂直，工件做等角度转动，逐点读取传感器信号。这种检测方法，对于非圆工件，在轮廓上的取样是不等间距的，影响轮廓的拟合精度。
发明内容
本发明公开一种工件轮廓检测时的恒线速度控制方法。
本发明的工件和激光位移传感器的布局是激光位移传感器位置固定，为了满足传感器固定采样频率下等距采样的需求，通过建立扫描点位姿和轴运动的运动学模型，规划恒线速扫描点，实时计算每周期扫描点位姿对应的XYC轴的运动量，并驱动控制工作台的平移和旋转运动(X/Y/C三个方向)，保证传感器的焦点始终在工件轮廓的表面，并使工件和激光位移传感器焦点的相对运动保持恒速。
所述具体步骤如下：
S1、构建测控平台，该平台具有XY平移轴和C旋转轴，以及激光位移传感器；待测工件设在平台上，由轴驱动进行平移和旋转，通过正确的移动，可以保证工件表面和激光传感器保持垂直等距的关系；
S2、工件轮廓描述，待测工件的轮廓为XY平面轮廓，轮廓由直线和圆弧2类几何元素构成，轮廓内相邻元素的连接关系需满足G1连续，即连接点处的法矢相同、切线夹角为零；待测工件的轮廓可由绘图软件生成，并保存成DXF格式；
S3、待测工件轮廓导入和预处理，读入和解析包含轮廓的DXF文件，获取待测工件的轮廓几何信息，对待测工件的轮廓元素检查和排序，构造首尾相接的有序的轮廓；
S4:恒线速度扫描点规划，根据激光位移传感器采样频率、采样间隔以及插补周期，计算每周期的扫描速度；由于采样频率、采样间隔和插补周期在一次测量中保持固定，因此规划的扫描速度恒定；依次取待测工件轮廓的几何元素，根据计算的扫描速度，计算每周期扫描的步长，并用步长对几何元素离散，计算每周期对应的扫描点位置和法矢；
S5、恒线速度扫描控制，在每个插补周期，根据运动学逆变换公式，实时计算扫描点位姿对应的XYC轴的运动量，并驱动平台进行平移和旋转运动，保证传感器焦点相对于待测工件轮廓扫描点的运动保持恒线速度。
本发明由于采用了激光位移传感器位置固定，通过控制工作台的平移和旋转运动(X/Y/C三个方向)，工件和激光位移传感器焦点产生相对运动，在已知待测工件轮廓模型的前提下，根据采样参数将轮廓按恒线速度要求划分为等距的小线段，进行测量扫描点的规划和计算，在每个插补周期，通过建立的运动学模型计算每个扫描点对应的 轴运动位移量，同时驱动和控制平台运动。由于该计算过程在每个插补控制周期内进行，因此可以获得很好的恒线速度控制效果。
附图说明
图1是一种测控平台装置结构示意图；
图2是一种待测工件轮廓的恒线速度控制流程图；
图3是直线元素扫描点坐标计算示意图；
图4是直线轮廓扫描点法矢计算示意图；
图5是本发明的运动变换模型示意图；
图6是图5的运动变换过程示意图；
图7是本发明的运动参数计算原理图。
具体实施方式
为使本发明目的、技术方案及优点更加清楚明确，以下参照附图3～7及对应公式，对本发明的恒线速度控制的结构和原理进行进一步的详细说明。
本发明提供一种工件轮廓检测时的恒线速度控制方法，具体步骤如下：
S1、构建测控平台
该平台具有XY平移轴和C旋转轴，以及激光位移传感器；待测工件设在平台上，由轴驱动进行平移和旋转，通过正确的移动，可以保证检测过程中，工件表面法矢和激光传感器同向、同时保持激光传感器焦点始终在工件表面轮廓。
S2、工件轮廓描述
为了满足检测条件、待测工件轮廓需要满足：
待测工件的轮廓为XY平面轮廓，轮廓由直线和圆弧2类几何元素构成；轮廓内相邻元素的连接关系需满足G1连续，即连接点处的法矢相同，连接点处的切向夹角为零；轮廓必须是封闭的；
待测工件的轮廓可由绘图软件生成，并保存成DXF格式。
S3、待测工件轮廓导入和预处理
待测工件轮廓导入和预处理。由于在进行绘制和编辑轮廓时，几何元素的顺序不一定是有序的，因此读入轮廓元素后，需要对待测工件的轮廓元素进行有序化和封闭性检查。
S4、恒线速度扫描点规划
根据激光位移传感器采样频率、采样间隔以及插补周期，计算每周期的扫描速度；由于采样频率、采样间隔和插补周期在一次测量中保持固定，因此规划的扫描速度恒定；依次取待测工件轮廓的几何元素，根据计算的扫描速度，计算每周期扫描的步长，并用步长对几何元素离散，计算每周期对应的扫描点位置和法矢。具体计算步骤如下：
1)恒线速度计算
扫描点4的位置和法矢由轮廓的几何特性以及测量的采样频率及采样间隔等决定。假设测头采样频率f(HZ)，采样间隔l(mm)，可计算激光位移传感器1相对与待测工件2的轮廓的相对运动速度v(mm/s)为：
v＝l·f     (1)
在测量过程中，采样频率采样间隔为固定数值，因此保证扫描速 度v为恒速。以此恒速，获取固定移动步长，将轮廓按周期拆分成等距的扫描点4，实现恒线速度扫描点4规划。
2)扫描点计算
已知扫描速度v、待测工件2的几何轮廓及插补周期，可计算出扫描点4的位置和法矢。由于待测工件2的轮廓包括直线和圆弧类型，因此需要根据不同的类型进行计算。本实施方案中，以直线类型为例进行说明。
对于直线类型，其递推计算如图3所示意，为:
Figure PCTCN2019117478-appb-000001
其中v为扫描速度，T为插补周期，
Figure PCTCN2019117478-appb-000002
为待测工件2的轮廓的切线。扫描时待测工件2的轮廓的起点
Figure PCTCN2019117478-appb-000003
已知，因此后续点可通过递推公式计算出来，直线的切线
Figure PCTCN2019117478-appb-000004
可通过直线起终点的信息计算出来。
3)扫描点法矢计算
扫描点4法矢总是垂直于待测工件2的轮廓的切线方向，计算出待测工件2的轮廓切线切线后，对切线实施一个旋转90度的变换ROT(90)，就可获得扫描点的法矢。对于直线轮廓，如图4所示意。S5、恒线速度扫描控制
在每个插补周期，根据建立的变换模型计算实时计算扫描点4位姿对应的XYC轴的运动量，并驱动平台进行平移和旋转运动，保证激光位移传感器1的焦点相对于待测工件2的轮廓扫描点4的运动保持恒线速度。具体步骤如下：
1)建立变换模型
如图5，该测量设备具有平移轴XY轴和旋转轴C。将工件安放于平台，通过调整保证待测工件坐标系和机床坐标系M重合(图5中坐标系M为机床坐标系原点，同时也为旋转中心)，激光位移传感器1安装到固定位置，激光焦点3在机床坐标系M下的矢量为
Figure PCTCN2019117478-appb-000005
激光光束方向为
Figure PCTCN2019117478-appb-000006
在进行待测工件2的轮廓上任意一点位置的扫描测量时，如图6所示，可通过C轴的旋转和XY平移运动，将扫描点
Figure PCTCN2019117478-appb-000007
进行坐标变换，并和激光焦点
Figure PCTCN2019117478-appb-000008
重合，同时保证扫描点法矢
Figure PCTCN2019117478-appb-000009
和激光光束方向
Figure PCTCN2019117478-appb-000010
平行。
由此建立的变换模型如下：
如果扫描点4位置为
Figure PCTCN2019117478-appb-000011
法矢为
Figure PCTCN2019117478-appb-000012
激光焦点位置为
Figure PCTCN2019117478-appb-000013
方向为
Figure PCTCN2019117478-appb-000014
通过M r旋转变换矩阵和M t平移变换矩阵分别对
Figure PCTCN2019117478-appb-000015
和
Figure PCTCN2019117478-appb-000016
施加变换，使变换后的
Figure PCTCN2019117478-appb-000017
和
Figure PCTCN2019117478-appb-000018
同向，
Figure PCTCN2019117478-appb-000019
和
Figure PCTCN2019117478-appb-000020
重合，则变换公式如下：
Figure PCTCN2019117478-appb-000021
Figure PCTCN2019117478-appb-000022
其中
Figure PCTCN2019117478-appb-000023
Figure PCTCN2019117478-appb-000024
矩阵中C,X,Y为变换参数，分别为旋转和平移量。
2)XYC运动量计算
如图7所示，由式3建立的关系，当
Figure PCTCN2019117478-appb-000025
和
Figure PCTCN2019117478-appb-000026
已知时，通过解 式3、4构成的方程组，便可获得变换参数C和X、Y。由于变换是由轴的运动产生的，因此X、Y和C分别为XYC三轴的运动量。具体计算先计算旋转角度C再计算平移量，具体过程如下：
由式3得：
Figure PCTCN2019117478-appb-000027
由于
Figure PCTCN2019117478-appb-000028
固定，总是为[1 0 0]，因此解得旋转量C为：
C＝acos(n x)(6)
由式4：
Figure PCTCN2019117478-appb-000029
解得平移量XY为：
X＝q x-cos(C)·p x-sin(C)·p y     (8)
Y＝q y+sin(C)·p x-cos(C)·p y     (9)

Claims (5)

  1. 一种工件轮廓检测时的恒线速度控制方法，包括如下步骤：
    S1、构建测控平台，该平台具有XY平移轴和C旋转轴，以及激光位移传感器；待测工件设在平台上，由轴驱动进行平移和旋转，保证工件表面和激光传感器保持垂直等距的关系；
    S2、工件轮廓描述，待测工件的轮廓为XY平面轮廓，轮廓由直线和圆弧两类几何元素构成，轮廓内相邻元素的连接关系需满足G1连续，待测工件的轮廓由绘图软件生成，并保存成DXF格式；
    S3、待测工件轮廓导入和预处理，读入和解析包含轮廓的DXF文件，获取待测工件的轮廓几何信息，对待测工件的轮廓元素检查和排序，构造首尾相接的有序的轮廓；
    S4:恒线速度扫描点规划，根据激光位移传感器采样频率、采样间隔以及插补周期，计算每周期的扫描速度；依次取待测工件轮廓的几何元素，根据计算的扫描速度，计算每周期扫描的步长，并用步长对几何元素离散，计算每周期对应的扫描点位置和法矢；
    S5、恒线速度扫描控制，在每个插补周期，根据运动学逆变换公式，实时计算扫描点位姿对应的XYC轴的运动量，并驱动平台进行平移和旋转运动，保证传感器焦点相对于待测工件轮廓扫描点的运动保持恒线速度。
  2. 根据权利要求1所述的工件轮廓检测时的恒线速度控制方法，其特征在于：所述恒线速度扫描点规划的具体计算步骤如下：
    1)恒线速度计算，
    扫描点的位置和法矢由轮廓的几何特性以及测量的采样频率及采样间隔等决定，假设测头采样频率f(HZ)，采样间隔l(mm)，可计算激光位移传感器1相对与待测工件的轮廓的相对运动速度v(mm/s)为：
    v＝l·f    式1
    在测量过程中，采样频率采样间隔为固定数值，因此保证扫描速度v为恒速；以此恒速，获取固定移动步长，将轮廓按周期拆分成等距的扫描点，实现恒线速度扫描点规划；
    2)扫描点计算，
    已知扫描速度v、待测工件的几何轮廓及插补周期，可计算出扫描点4的位置和法矢。由于待测工件的轮廓包括直线和圆弧类型，因此需要根据不同的类型进行计算；
    对于直线类型，其递推计算为:
    Figure PCTCN2019117478-appb-100001
    其中v为扫描速度，T为插补周期，
    Figure PCTCN2019117478-appb-100002
    为待测工件2的轮廓的切线；扫描时待测工件的轮廓的起点
    Figure PCTCN2019117478-appb-100003
    已知，因此后续点可通过递推公式计算出来，直线的切线
    Figure PCTCN2019117478-appb-100004
    可通过直线起终点的信息计算出来；
    3)扫描点法矢计算，
    扫描点法矢总是垂直于待测工件的轮廓的切线方向，计算出待测工件的轮廓切线切线后，对切线实施一个旋转90度的变换 ROT(90)，就可获得扫描点的法矢。
  3. 根据权利要求1或2所述的工件轮廓检测时的恒线速度控制方法，其特征在于：所述构建测控平台包括，将工件安放于平台，通过调整保证待测工件坐标系和机床坐标系M重合，激光位移传感器安装到固定位置，激光焦点在机床坐标系M下的矢量为
    Figure PCTCN2019117478-appb-100005
    激光光束方向为
    Figure PCTCN2019117478-appb-100006
    在进行待测工件的轮廓上任意一点位置的扫描测量时，通过C轴的旋转和XY平移运动，将扫描点
    Figure PCTCN2019117478-appb-100007
    进行坐标变换，并和激光焦点
    Figure PCTCN2019117478-appb-100008
    重合，同时保证扫描点法矢
    Figure PCTCN2019117478-appb-100009
    和激光光束方向
    Figure PCTCN2019117478-appb-100010
    平行。
  4. 根据权利要求3所述的工件轮廓检测时的恒线速度控制方法，其特征在于：由此建立的变换模型如下：
    如果扫描点位置为
    Figure PCTCN2019117478-appb-100011
    法矢为
    Figure PCTCN2019117478-appb-100012
    激光焦点位置为
    Figure PCTCN2019117478-appb-100013
    方向为
    Figure PCTCN2019117478-appb-100014
    通过M r旋转变换矩阵和M t平移变换矩阵分别对
    Figure PCTCN2019117478-appb-100015
    和
    Figure PCTCN2019117478-appb-100016
    施加变换，使变换后的
    Figure PCTCN2019117478-appb-100017
    和
    Figure PCTCN2019117478-appb-100018
    同向，
    Figure PCTCN2019117478-appb-100019
    和
    Figure PCTCN2019117478-appb-100020
    重合，则变换公式如下：
    Figure PCTCN2019117478-appb-100021
    Figure PCTCN2019117478-appb-100022
    其中
    Figure PCTCN2019117478-appb-100023
    Figure PCTCN2019117478-appb-100024
    矩阵中C,X,Y为变换参数，分别为旋转和平移量。
  5. 根据权利要求4所述的工件轮廓检测时的恒线速度控制方法，其特征在于：所述实时计算扫描点位姿对应的XYC轴的运动量包括，
    由式3建立的关系，当
    Figure PCTCN2019117478-appb-100025
    和
    Figure PCTCN2019117478-appb-100026
    已知时，通过解式3、4构成的方程组，便可获得变换参数C和X、Y；由于变换是由轴的运动产生的，因此X、Y和C分别为XYC三轴的运动量；具体计算先计算旋转角度C再计算平移量，具体过程如下：
    由式3得：
    Figure PCTCN2019117478-appb-100027
    由于
    Figure PCTCN2019117478-appb-100028
    固定，总是为[1 0 0]，因此解得旋转量C为：
    C＝a cos(n x)    (6)
    由式4：
    Figure PCTCN2019117478-appb-100029
    解得平移量XY为：
    X＝q x-cos(C)·p x-sin(C)·p y    (8)
    Y＝q y+sin(C)·p x-cos(C)·p y    (9)
    。
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