• Acta Photonica Sinica
  • Vol. 49, Issue 7, 712003 (2020)
Ji-han HU, Fen GAO, and Jin-ping NI
Author Affiliations
  • School of Optoelectronic Engineering, Xi''an Technological University, Xi''an 710032, China
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    DOI: 10.3788/gzxb20204907.0712003 Cite this Article
    Ji-han HU, Fen GAO, Jin-ping NI. Modeling and Measurement Accuracy Analysis of Six-light-screen Optical Target Based on Plane Equation[J]. Acta Photonica Sinica, 2020, 49(7): 712003 Copy Citation Text show less

    Abstract

    As a series of structural approximation need to be used in traditional geometry method, the model solving and accuracy analyzing of six-light-screen optical target based on this method are inaccurate. Here, a more accurate modeling and accuracy analyzing method based on plane equation was developed and a highly versatile engineering model and error transfer formula for six-light-screen optical target were deduced. Measurement accuracy decline caused by target distance and its error, light-screen tilt angle and its error, alignment error of light source and receiver, etc. were analyzed and compared under two kind of classical six-screen structural model. Sets of position and velocity measurement error distribution data and graphics in effective sensor area were acquired. A practical engineering layout of six-light-screen optical target with position measurement error less than 3 mm, and relative velocity measurement error less than 0.3% was proposed. This research can provide useful theoretical basis and data reference for practical design and accuracy estimation of six-light-screen optical target.
    $ x = \frac{S}{4}{\rm{cot}}\alpha - \frac{{S \times \left( {{t_2} - {t_1}} \right) \times {\rm{cos}}\left( {\alpha - \gamma } \right)}}{{\left( {{t_6} - {t_1}} \right) \times {\rm{cos}}\gamma \times {\rm{sin}}\alpha }} $ (1)

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    $ y = \frac{S}{4}{\rm{cot}}\beta - \frac{{S \times \left( {{t_3} - {t_1}} \right) \times {\rm{cos}}\left( {\beta - \theta } \right)}}{{\left( {{t_6} - {t_1}} \right) \times {\rm{cos}}\theta \times {\rm{sin}}\beta }} $ (2)

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    $ v = \frac{{\rm{S}}}{{({t_6} - t{}_1^{}) \times {\rm{cos}}\gamma \times {\rm{cos}}\theta }} $ (3)

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    $ \theta = {\rm{arctan}}\left[ {\frac{{{t_5} + {t_3} - {t_6}}}{{\left( {{t_5} - {t_3}} \right){\rm{tan}}\beta }}{\rm{cos}}\gamma } \right] $ (4)

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    $ \gamma = {\rm{arctan}}\left[ {\frac{{{t_4} + {t_2} - {t_6}}}{{\left( {{t_4} - {t_2}} \right){\rm{tan}}\alpha }}} \right] $ (5)

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    $ \left\{ {\begin{array}{*{20}{l}} {z = 0}\\ {{A_2}x + {B_2}y + {C_2}z + {D_2} = 0}\\ {{A_3}x + {B_3}y + {C_3}z + {D_3} = 0}\\ {{A_4}x + {B_4}y + {C_4}z + {D_4} = 0}\\ {{A_5}x + {B_5}y + {C_5}z + {D_5} = 0}\\ {{A_6}x + {B_6}y + {C_6}z + {D_6} = 0} \end{array}} \right. $ (6)

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    $ \left\{ {\begin{array}{*{20}{l}} {{x_n} = {x_0} + {\nu _x}{t_n}}\\ {{y_n} = {y_0} + {\nu _y}{t_n}}\\ {{z_n} = {z_0} + {\nu _z}{t_n}} \end{array}} \right. $ (7)

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    $ \left\{ \begin{array}{l} z = 0\\ {A_2}{x_0} + {A_2}{v_x}{t_2} + {B_2}{y_0} + {B_2}{v_y}{t_2} + {C_2}{v_z}{t_2} + {D_2} = 0\\ {A_3}{x_0} + {A_3}{v_x}{t_3} + {B_3}{y_0} + {B_3}{v_y}{t_3} + {C_3}{v_z}{t_3} + {D_3} = 0\\ {A_4}{x_0} + {A_4}{v_x}{t_4} + {B_4}{y_0} + {B_4}{v_y}{t_4} + {C_4}{v_z}{t_4} + {D_4} = 0\\ {A_5}{x_0} + {A_5}{v_x}{t_5} + {B_5}{y_0} + {B_5}{v_y}{t_5} + {C_5}{v_z}{t_5} + {D_5} = 0\\ {A_6}{x_0} + {A_6}{v_x}{t_6} + {B_6}{y_0} + {B_6}{v_y}{t_6} + {C_6}{v_z}{t_6} + {D_6} = 0 \end{array} \right. $ (8)

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    $ {\rm{\Delta }}\beta = {\rm{arctan}}\left( {\frac{{{\rm{\Delta }}{z_{{\rm{u}}5}} - {\rm{\Delta }}{z_{{\rm{d}}5}} + H{\rm{tan}}\beta }}{{{\rm{\Delta }}{y_{{\rm{u}}5}} - {\rm{\Delta }}{y_{{\rm{d}}5}} + H}}} \right) - \beta $ (9)

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    $ {\rm{\Delta }}{B_5} = \frac{{{\rm{\Delta }}{z_{{\rm{u}}5}} - {\rm{\Delta }}{z_{{\rm{d}}5}} + H{\rm{tan}}\beta }}{{{\rm{\Delta }}{y_{{\rm{u}}5}} - {\rm{\Delta }}{y_{{\rm{d}}5}} + H}} - {B_5} $ (10)

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    $ {\rm{\Delta }}{\sigma _5} = {\rm{arctan}}\left( {\frac{{{\rm{\Delta }}{m_5}}}{W}} \right) $ (11)

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    $ {\rm{\Delta }}{A_5} = \frac{{{\rm{\Delta }}m}}{W} $ (12)

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    $ {\rm{\Delta }}{D_5} = \frac{{{\rm{\Delta }}m}}{2} $ (13)

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    $ \left( {{A_5} + \frac{{{\rm{\Delta }}m}}{W}} \right)\left( {{x_0} + {v_x}{t_5}} \right) + \left( {\frac{{{\rm{\Delta }}{z_{{\rm{u}}5}} - {\rm{\Delta }}{z_{{\rm{d}}5}} + H{\rm{tan}}\beta }}{{{\rm{\Delta }}{y_{{\rm{u}}5}} - {\rm{\Delta }}{y_{{\rm{d}}5}} + H}}} \right)\left( {{y_0} + {v_y}{t_5}} \right) + {v_z}{t_5} + \left( {{D_5} + \frac{{{\rm{\Delta }}m}}{2}} \right) = 0 $ (14)

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    $ \left( {{A_n} + \frac{{{\rm{\Delta }}m}}{W}} \right)\left( {{x_0} + {v_x}{t_n}} \right) + \left( {\frac{{{\rm{\Delta }}{z_{{\rm{u}}n}} - {\rm{\Delta }}{z_{{\rm{d}}n}} + H{\rm{tan}}\beta }}{{{\rm{\Delta }}{y_{{\rm{u}}n}} - {\rm{\Delta }}{y_{{\rm{d}}n}} + H}}} \right)\left( {{y_0} + {v_y}{t_n}} \right) + {v_z}{t_n} + \left( {{D_n} + \frac{{{\rm{\Delta }}m}}{2}} \right) = 0 $ (15)

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    $ \left( {\frac{{{\rm{\Delta }}{z_{{\rm{u}}n}} - {\rm{\Delta }}{z_{{\rm{d}}n}} + W{\rm{tan}}\alpha }}{{{\rm{\Delta }}{x_{{\rm{u}}n}} - {\rm{\Delta }}{x_{{\rm{d}}n}} + W}}} \right)\left( {{x_0} + {v_x}{t_n}} \right) + \left( {{B_5} + \frac{{{\rm{\Delta }}m}}{H}} \right)\left( {{y_0} + {v_y}{t_n}} \right) + {v_z}{t_n} + \left( {{D_n} + \frac{{{\rm{\Delta }}m}}{2}} \right) = 0 $ (16)

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    $ \left( {\frac{{{\rm{\Delta }}m}}{W}} \right)\left( {{x_0} + {v_n}{t_n}} \right) + \left( {\frac{{{\rm{\Delta }}{z_{{\rm{u}}n}} - {\rm{\Delta }}{z_{{\rm{d}}n}}}}{{{\rm{\Delta }}{y_{{\rm{u}}n}} - {\rm{\Delta }}{y_{{\rm{d}}n}} + H}}} \right)\left( {{y_0} + {v_y}{t_n}} \right) + {v_z}{t_n} + \left( {{D_n} + \frac{{{\rm{\Delta }}m}}{2} + {\rm{\Delta }}S} \right) = 0 $ (17)

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    Ji-han HU, Fen GAO, Jin-ping NI. Modeling and Measurement Accuracy Analysis of Six-light-screen Optical Target Based on Plane Equation[J]. Acta Photonica Sinica, 2020, 49(7): 712003
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