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Journals >Optical Instruments >Volume 41 >Issue 5 >Page 30 > Article

Optical Instruments
Vol. 41, Issue 5, 30 (2019)

Design of motion control algorithm for magnetic compound fluid polishing of optical aspheric components

Lei YAO¹, Chen JIANG^1,*, Peibing SHI¹, Jixiong HU¹..., Guanghe YAN¹ and Yongbin ZHANG²|Show fewer author(s)

Author Affiliations

¹School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

²Institute of Mechanical Manufacturing Technology, China Academy of Engineering Physics, Mianyang 621900, China

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DOI: 10.3969/j.issn.1005-5630.2019.05.005 Cite this Article

Lei YAO, Chen JIANG, Peibing SHI, Jixiong HU, Guanghe YAN, Yongbin ZHANG. Design of motion control algorithm for magnetic compound fluid polishing of optical aspheric components[J]. Optical Instruments, 2019, 41(5): 30 Copy Citation Text

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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.

Keywords

aspheric magnetic components D-H method kinematic modeling magnetic compound fluid polishing motion control algorithm

${{E}} = {{n}} = \left( {

\begin{matrix} - \frac{\partial z}{\partial x} & - \frac{\partial z}{\partial y} & 1 \end{matrix}

} \right) = \left( {

\begin{matrix} E_{x} & E_{y} & E_{z} \end{matrix}

} \right)$

(1)

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$\left\{ {

\begin{matrix} Q_{x} = F_{x} + \frac{E_{x}}{| E |} h \\ Q_{y} = F_{y} + \frac{E_{y}}{| E |} h \\ Q_{z} = F_{z} + \frac{E_{z}}{| E |} h \end{matrix}

} \right.$

(2)

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$

\begin{aligned} T = T_{1} · T_{2} · T_{3} · T_{4} · T_{5} · T_{6} · T_{7} = \\ [\begin{array}{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}] \end{aligned}

$

(3)

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$\left[ {

\begin{matrix} Q_{x} \\ Q_{y} \\ Q_{z} \\ 1 \end{matrix}

} \right] = {{T}} {\text{·}} \left[ {

\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}

} \right] = \left[ {

\begin{matrix} W_{1} \\ W_{2} \\ W_{3} \\ 1 \end{matrix}

} \right]$

(4)

View in Article

$\left[ {

\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \\ 0 \end{matrix}

} \right] = {{T}} {\text{·}} \left[ {

\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}

} \right] = \left[ {

\begin{matrix} \sin A \sin B \\ \cos B \sin A \\ \cos A \\ 0 \end{matrix}

} \right]$

(5)

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$\left\{ {

\begin{array}{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)

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$\left\{ {

\begin{array}{l} E_{x} = \sin A \sin B \\ E_{y} = \cos B \sin A \\ E = \cos A_{} \end{array}

} \right.$

(7)

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$\left\{ {

\begin{array}{l} A = \arccos E_{z}, 0 ≤ A ≤ \frac{π}{2} \\ B = \arctan \frac{E_{y}}{E_{x}}, - \frac{π}{2} ≤ B ≤ \frac{π}{2} \end{array}

} \right.$

(8)

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$\left\{ {

\begin{array}{l} x = \frac{P_{x} \cos B - Q_{x} \cos B + Q_{y} \sin B - P_{y} \sin B}{\cos B^{2} + \sin B^{2}} \\ y = \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 = \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)

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$\left\{ {

\begin{array}{l} x = \frac{- Q_{x} \cos B + Q_{y} \sin B}{\cos B^{2} + \sin B^{2}} \\ y = \frac{- Q_{x} \sin B - Q_{y} \cos B}{\cos B^{2} + \sin B^{2}} \\ z = \frac{Q_{z} + 120}{\cos B^{2} + \sin B^{2}} \end{array}

} \right.$

(10)

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$

\begin{aligned} z (x, y) = \frac{C (x^{2} + y^{2})}{1 + \sqrt{1 - (1 + k) C^{2} (x^{2} + y^{2})}} + \\ a_{1} {(x^{2} + y^{2})}^{2} + a_{2} {(x^{2} + y^{2})}^{3} + a_{3} {(x^{2} + y^{2})}^{4} \end{aligned}

$

(11)

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Lei YAO, Chen JIANG, Peibing SHI, Jixiong HU, Guanghe YAN, Yongbin ZHANG. Design of motion control algorithm for magnetic compound fluid polishing of optical aspheric components[J]. Optical Instruments, 2019, 41(5): 30

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