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Quantification and Mitigation of Errors in the Inertial Measurements of Distance

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Abstract

The accurate measurement of the distance travelled, velocity and acceleration at low velocities to supersonic speeds is an active area of research. The captive flight at Rail Track Rocket Sled (RTRS) facility provides a unique environment for kinematic testing at supersonic speeds. Using RTRS facility, an accurate distance measurement method is developed, tested and experimentally verified. Three accelerometers, with different noise density, identically moving, have been chosen for sensing forward motion. A number of measures such as different mountings, bias correction, capping, digital filtering and position fix have been tried in a practical implementation. To keep the measurement error within tolerable limits a novel method of obtaining position fix is proposed by using a pair of magneto-inductive sensors. The bias correction is applied in the position to derive corrected velocity and acceleration. To know the truthfulness of results and to validate the proposed methods, a system has been developed to generate reference values for computation of error. This reference system has an error of 0.046 % which is much better than reported in previous study. After mitigation of various errors using proposed methods, an error within 1.5 % was attained with one of the sensors used in trials. The proposed work identifies the elements which contribute in errors and quantify the mitigated errors in some cases and highlights the measures which bring about significant improvement in error. It also suggests how to obtain more accurate results using economical MEMS accelerometers.

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Author information

Authors and Affiliations

  1. Terminal Ballistics Research Lab, DRDO, Chandigarh, India

    P. K. Khosla

  2. Thapar University, Patiala, India

    R. Khanna

  3. CDAC, Mohali, India

    Sanjay P. Sood

Authors
  1. P. K. Khosla
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  2. R. Khanna
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  3. Sanjay P. Sood
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Corresponding author

Correspondence to P. K. Khosla.

Appendix A

Appendix A

The derivation of the equation given below is given in [3]. The details are given here so as to build on what has been done earlier, elaborate it, apply it to a new application and act as ready reference. The expectation value of the square error in position (E Ssq) at Nth sample can be determined for each sample when N = 1, 2, 3, … and so on, i.e., for calculation of 100th sample N = 100 and all the samples from 1 to 100 are considered while finding self correlation.

$$ E_{{{\text{Ssq}}}} (N) = r\left[ 0 \right]\frac{{t_{{\text{s}}}^{4} }}{{12}}\left( {N - 1} \right)\left( {N^{2} - 2N + 3} \right) + A\frac{{t_{{\text{s}}}^{4} }}{3}S_{3} - A\frac{{t_{{\text{s}}}^{4} }}{6}\left( {3N^{2} - 6N + 5} \right)S_{1} + A\frac{{t_{{\text{s}}}^{4} }}{6}\left( {N - 1} \right)\left( {N^{2} - 2N + 3} \right)S_{0} - A\frac{{t_{{\text{s}}}^{4} }}{8}\left( {N - 1} \right)^{2} e^{{ - B\left( {N - 1} \right)}} - A\frac{{t_{{\text{s}}}^{4} }}{8}\left( {N - 1} \right)^{2} $$
(12)

where sample time t s = 1/f s and r[0] represents self correlation function with zero lag

$$ r[0] = \frac{1}{N} \times \sum \limits_{{i\; = \;1}}^{N} a\left( i \right) \times a(i) $$
(13)

and the values of constants A and B are calculated as follows

$$ A = \frac{{\pi f_{\text{c}} \sigma_{\text{c}}^{2} }}{2},\quad B = \frac{{2\pi f_{\text{c}} }}{{f_{\text{s}} }}. $$

Here, f c (1 kHz) is the cut off frequency of the anti-aliasing filter and f s (2 kHz) is the rate at which the accelerometer output is sampled. The values of constants S 3, S 1, S 0 are calculated as follows

$$ S_{3} = \frac{f}{g} $$
$$ S_{1} = \frac{{{\text{e}}^{{ - B}} - {\text{e}}^{{ - B\left( {1 + M} \right)}} - M{\text{e}}^{{ - B\left( {1 + M} \right)}} + M{\text{e}}^{{ - B\left( {2 + M} \right)}} }}{{\left( {{\text{e}}^{{ - B}} - 1} \right)^{2} }} $$
(14)

where

$$ M = N - 2, \quad g = \left( {{\text{e}}^{{ - B}} - 1} \right)^{4} $$
(15)
$$ f = {\text{e}}^{ - B} + {\text{e}}^{ - 3B} {-}3M^{2} {\text{e}}^{ - B(1 + M)} + 6M^{2} {\text{e}}^{ - B(2 + M)} {-}{\text{e}}^{ - B(3 + M)} {-}3M^{2} {\text{e}}^{ - B(3 + M)} + 3M^{3} {\text{e}}^{ - B(2 + M)} {-}3M^{3} {\text{e}}^{ - B(3 + M)} + 3M{\text{e}}^{ - B(3 + M)} + M^{3} {\text{e}}^{ - B(4 + M)} {-} {\text{e}}^{ - B(1 + M)} {-}M^{3} {\text{e}}^{ - B(1 + M)} {-} 3M{\text{e}}^{ - B(1 + M)} + 4{\text{e}}^{ - 2B} {-} 4{\text{e}}^{ - B(2 + M)} $$
$$ E_{\text{rms}} \left( N \right) = {\sqrt{E_{\text{Ssq}}} (N)} $$
(16)

The RMS error at every sample can be calculated by:

$$ E_{\text{rms}} \left( N \right) = \sqrt {{{E}}_{\text{Ssq}} (N)} $$
(17)

The values at travel time of 4.2 s are given in Table 3.

Table 3 Typical values for accelerometer A2, t = 4.2 s, σ c = 0.0014 × 9.806 and N = 8400
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Khosla, P.K., Khanna, R. & Sood, S.P. Quantification and Mitigation of Errors in the Inertial Measurements of Distance. MAPAN 30, 49–57 (2015). https://doi.org/10.1007/s12647-014-0115-z

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