| Summary: | The elliptical paraboloid array plays an important role in precision measurement, astronomical telescopes, and communication systems. The calibration of the vertex distance of elliptical paraboloids is of great significance to precise 2D displacement measurement. However, there are some difficulties in determining the vertex position with contact measurement. In this study, an elliptical paraboloid array and an optical slope sensor for displacement measurement were designed and analyzed. Meanwhile, considering the geometrical relationship and relative angle between elliptical paraboloids, a non-contact self-calibration method for the vertex distance of the elliptical paraboloid array was proposed. The proposed self-calibration method was verified by a series of experiments with a high repeatability, within <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mtext> </mtext> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics> </math> </inline-formula> in the X direction and within <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mtext> </mtext> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics> </math> </inline-formula> in the Y direction. Through calibration, the displacement measurement system error was reduced from <inline-formula> <math display="inline"> <semantics> <mrow> <mn>100</mn> <mtext> </mtext> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mtext> </mtext> <mi mathvariant="sans-serif">μ</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics> </math> </inline-formula>. The self-calibration method of the elliptical paraboloid array has great potential in the displacement measurement field, with a simple principle and high precision.
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