﻿ 光学非球面磁性复合流体抛光运动控制算法设计
 光学仪器  2019, Vol. 41 Issue (5): 30-37 PDF

1. 上海理工大学 机械学院，上海 200093;
2. 中国工程物理研究院 机械制造工艺研究所，四川 绵阳 621900

Design of motion control algorithm for magnetic compound fluid polishing of optical aspheric components
YAO Lei1, JIANG Chen1, SHI Peibing1, HU Jixiong1, YAN Guanghe1, ZHANG Yongbin2
1. School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China;
2. Institute of Mechanical Manufacturing Technology, China Academy of Engineering Physics, Mianyang 621900, China
Abstract: To meet the need of ultra-precision polishing of optical aspheric elements with high precision, the motion control algorithm is designed for magnetic compound fluid polishing of aspheric magnetic components. Optical aspheric magnetic compound fluid polishing principle is analyzed. The relation between the configuration of the polishing head and the aspheric surface is established. The D-H method is used to establish the kinematics model of the polishing test platform and the head posture during the polishing process is solved. Process experiments are carried out to verify the motion control algorithm. Experimental results show that the designed motion control algorithm can process optical aspherical elements reasonably.
Key words: aspheric magnetic components    magnetic compound fluid polishing    motion control algorithm    kinematic modeling    D-H method

1 试验台

 图 1 试验台 Figure 1 Test platform

 图 2 控制系统结构示意图 Figure 2 Schematic of control system structure
2 磁性复合流体抛光运动控制算法 2.1 磁性复合流体抛光加工原理

 图 3 光学非球面磁性复合流体抛光进动加工原理示意图 Figure 3 Precession machining principle of optical aspheric magnetic compound fluid polishing
2.2 抛光头位姿量求解

 ${{E}} = {{n}} = \left( {\begin{array}{*{20}{c}} { - \displaystyle\frac{{\partial z}}{{\partial x}}}&{ - \displaystyle\frac{{\partial z}}{{\partial y}}}&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{E_x}}&{{E_y}}&{{E_z}} \end{array}} \right)$ (1)

 $\left\{ {\begin{array}{*{20}{c}} {{Q_x} = {F_x} + \displaystyle\frac{{{E_x}}}{{\left| {{E}} \right|}}h} \\ {{Q_y} = {F_y} + \displaystyle\frac{{{E_y}}}{{\left| {{E}} \right|}}h} \\ {{Q_z} = {F_z} + \displaystyle\frac{{{E_z}}}{{\left| {{E}} \right|}}h} \end{array}} \right.$ (2)

2.3 轴运动量求解 2.3.1 D-H法

D-H法是由迪纳维特和哈坦伯格（Denavit和Hartenberg）于1955年提出，用D-H矩阵来表述连杆机构之间的关系，后来被广泛应用于机器人运动学建模，现已成为对机器人运动学建模的经典标准方法。D-H法的总体思想是[10-11]：将机器人看作是从基座到末端执行器由一系列的杆件通过移动副和转动副链接而成的开环尺寸链，给每个杆件的关节都固连一个坐标系，用4×4的齐次变换矩阵来描述相邻两坐标系的空间变换关系，然后依次写出从基座到末端执行器之间两相邻坐标系的齐次变换矩阵，将这些齐次变换矩阵依次连乘起来就得到机器人的总变换矩阵，从而建立机器人运动学方程，进而求解出各关节相应运动量。如果已知某机构在末端执行器坐标系下的空间位置与姿态，利用总变换矩阵求解其在基坐标系下的空间位置与姿态，称之为运动学正问题求解。在运动学正问题得以求解的基础上，如果末端执行器在基坐标系下的空间位置与姿态已知，求解各运动关节的运动量，则称之为运动学逆问题求解。

2.3.2 利用D-H法对试验台运动学建模

 图 4 试验台D-H坐标系示意图 Figure 4 D-H coordinate system of the test platform

 $\begin{split} &{ T} = {{ T}_1} {\text{·}} {{ T}_2} {\text{·}} {{ T}_3} {\text{·}} {{ T}_4} {\text{·}} {{ T}_5} {\text{·}}{{ T}_6} {\text{·}} {{ T}_7} =\\[-3pt] &\quad \left[ {\begin{array}{*{20}{c}} {\cos B}&{\cos A\sin B}&{\sin A\sin B}&{{W_1}} \\ { - \sin B}&{\cos A\cos B}&{\sin A\cos B}&{{W_2}} \\ 0&{ - \sin A}&{\cos A}&{{W_3}} \\ 0&0&0&{{W_4}} \end{array}} \right] \end{split}$ (3)

2.3.3 试验台逆向运动学分析与建模

 $\left[ {\begin{array}{*{20}{c}} {{Q_x}} \\ {{Q_y}} \\ {{Q_z}} \\ 1 \end{array}} \right] = {{T}} {\text{·}} \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 0 \\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{W_1}} \\ {{W_2}} \\ {{W_3}} \\ 1 \end{array}} \right]$ (4)

 $\left[ {\begin{array}{*{20}{c}} {{E_x}} \\ {{E_y}} \\ {{E_z}} \\ 0 \end{array}} \right] = {{T}} {\text{·}} \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\sin A\sin B} \\ {\cos B\sin A} \\ {\cos A} \\ 0 \end{array}} \right]$ (5)

 $\left\{ {\begin{array}{*{20}{l}} {{Q_x} = {W_1} = {P_x} - x\cos B - y\cos A\sin B + z\sin A\sin B} \\ {{Q_y} = {W_2} = {P_y} + x\sin B - y\cos A\sin B + z\sin A\cos B} \\ {{Q_z} = {W_3} = {P_z} + z\cos A + y\sin A} \end{array}} \right.$ (6)

 $\left\{ {\begin{array}{*{20}{l}} {{E_x} = \sin A\sin B} \\ {{E_y} = \cos B\sin A} \\ {E = \cos {A_{}}} \end{array}} \right.$ (7)

 $\left\{ {\begin{array}{*{20}{l}} {A = \arccos {E_z},\;0 {\text{≤}} A {\text{≤}} \displaystyle\frac{{\text{π}} }{2}} \\ {B = \arctan \displaystyle\frac{{{E_y}}}{{{E_x}}},\; - \displaystyle\frac{{\text{π}}}{2} {\text{≤}} B {\text{≤}} \displaystyle\frac{{\text{π}} }{2}} \end{array}} \right.$ (8)

 $\left\{ {\begin{array}{*{20}{l}} {x = \displaystyle\frac{{{P_x}\cos B - {Q_x}\cos B + {Q_y}\sin B - {P_y}\sin B}}{{\cos {B^2} + \sin {B^2}}}} \\ {y = \displaystyle\frac{{ - ({Q_x} - {P_x})\cos A\sin B - ({Q_y} - {P_y})\cos A\cos B + ({Q_z} - {P_z})\sin A}}{{\cos {B^2} + \sin {B^2}}}} \\ {z = \displaystyle\frac{{({Q_x} - {P_x})\sin A\sin B + ({Q_y} - {P_y})\sin A\cos B + ({Q_z} - {P_z})\cos A}}{{\cos {B^2} + \sin {B^2}}}} \end{array}} \right.$ (9)

 $\left\{ {\begin{array}{*{20}{l}} {x = \displaystyle\frac{{ - {Q_x}\cos B + {Q_y}\sin B}}{{\cos {B^2} + \sin {B^2}}}} \\ {y = \displaystyle\frac{{ - {Q_x}\sin B - {Q_y}\cos B}}{{\cos {B^2} + \sin {B^2}}}} \\ {z = \displaystyle\frac{{{Q_z} + 120}}{{\cos {B^2} + \sin {B^2}}}} \end{array}} \right.$ (10)

 图 5 算法流程图 Figure 5 Flow chart of algorithm
3 运动控制算法验证

 $\begin{split} &z(x,y) = \frac{{C\left( {{x^2} + {y^2}} \right)}}{{1 + \sqrt {1 - \left( {1 + k} \right){C^2}\left( {{x^2} + {y^2}} \right)} }} + \\ &\quad {a_1}{\left( {{x^2} + {y^2}} \right)^2} + {a_2}{\left( {{x^2} + {y^2}} \right)^3} + {a_3}{\left( {{x^2} + {y^2}} \right)^4} \end{split}$ (11)

 图 6 加工点轨迹与抛光头中心点轨迹仿真图 Figure 6 Simulation of the track of processing point and the central point of polishing head

 图 7 检测数据拟合曲线 Figure 7 Fitting curve of the test data
4 结　论

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