Investigation and modeling of the uncertainty in laser-Doppler-velocimeter-based radius measurement

Abstract Laser Doppler velocimeter (LDV) allows for measurement of the lateral velocity of rotating cylindrical workpiece, thus, enabling the absolute radius measurement, showing great application prospects. However, there exist adverse factors that decrease the measurement accuracy, for example, variation of fringe spacing, fluctuation of rotating speed, spindle error motions, and Doppler frequency estimation. In this context, this paper comprehensively investigates the uncertainty in the LDV-based radius measurement. First, an LDV sensor was constructed and calibrated. Then, a primary model of the LDV-based radius measurement is presented: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mtext>d</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>⋅</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mo>⋅</mml:mo> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> , which suggests that the measurand is indirectly calculated from three input quantities: the Doppler frequency <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mtext>d</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> , the fringe spacing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> , and the rotation frequency <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> , each of which owns an estimation uncertainty, propagating into the measurement result. Based on this, uncertainty propagation in LDV-based radius measurement is investigated: (1) By calculating the partial derivative of the measurement model, uncertainty propagation of the rotational frequency fluctuation is derived and the resulting uncertainty is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>2.2</mml:mn> <mml:mrow> <mml:mrow> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mtext>μ</mml:mtext> <mml:mi>m</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> . (2) Random uncertainty in Doppler frequency estimation leads to stochastic dispersion in radius estimates. The resulting uncertainty is statistically derived to be <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>11.4</mml:mn> <mml:mrow> <mml:mrow> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mtext>μ</mml:mtext> <mml:mi>m</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> . (3) Measurement procedure is numerically simulated to verify that a sinusoidal systematic uncertainty can be caused by the fringe spacing variation and the workpiece runout. The resulting uncertainty is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mtext>10</mml:mtext> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mn>.6</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mstyle scriptlevel="0"/> <mml:mtext>μ</mml:mtext> <mml:mi>m</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> . (4) Spectral analysis reveals that spindle error motions introduce a significant uncertainty, accounting for 74% of the total uncertainty. For a comprehensive modeling of uncertainty, the spindle error motions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is incorporated into the measurement model: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mrow> <mml:m