Correction: Tang, C. Y. and Chen, X.Y. A Class of  Coning Algorithms Based on a Half-Compressed Structure.  Sensors 2014, 14, 14289–14301

Tang, Chuanye; Chen, Xiyuan

doi:10.3390/s150204425

Open AccessCorrection

Correction: Tang, C. Y. and Chen, X.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure. Sensors 2014, 14, 14289–14301

by

Chuanye Tang

and

Xiyuan Chen

^*

Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, School of Instrument Science and Engineering, Southeast University, Sipailou 2, Nanjing 210096, China

^*

Author to whom correspondence should be addressed.

Sensors 2015, 15(2), 4425-4429; https://doi.org/10.3390/s150204425

Submission received: 23 December 2014 / Revised: 23 December 2014 / Accepted: 11 February 2015 / Published: 13 February 2015

(This article belongs to the Special Issue Optical Gyroscopes and Navigation Systems)

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1. Change in Tables/Equations

Due to an oversight by MDPI and the authors, the following numerical corrections were not made in the originally published article [1]. MDPI-Sensors and the authors would like to apologize for any inconvenience brought to the readers.

The authors wish to make the following correction to the article [1]:

The former Table 9 (labelled here as Old Table 9) and Table 10 (labelled here as Old Table 10) should be replaced by the new versions shown below (labelled here as New Table 9 and New Table 10), respectively. The z s in Tables 15 and 16 and the maneuver errors in [1] Table 17 will not be affected by the correction to Tables 9 and 10, because these z s and the maneuver errors were all calculated using the correct coefficients in New Tables 9 and 10. That means, the mistakes in Old Tables 9 and 10 are just writing errors.

Old Table 9. FTSuc algorithm coefficients.

**Old Table 9.** FTSuc algorithm coefficients.
L	N	Coefficients
3	3	ς₁₂ = ς₂₃ = 27/40, ς₁₃ = 9/20
4	4	ς₁₂ = ς₃₄ = 232/315, ς₂₃ = 178/315, ς₁₃ = ς₂₄ = 46/105, ς₁₄ = 54/105
5	5	ς₁₂ = 18575/24192, ς₁₃ = 2675/6048, ς₁₄ = 11,225/24,192, ς₁₅ = 125/252, ς₂₃ = 2575/6048, ς₂₄ = 425/672, ς₂₅ = 139,75/24,192, ς₃₄ = 1975/3024, ς₃₅ = 325/1512, ς₄₅ = 21,325/24,192

New Table 9. FTSuc algorithm coefficients.

**New Table 9.** FTSuc algorithm coefficients.
L	N	Coefficients
3	3	ς₁₂ = ς₂₃ = 27/40, ς₁₃ = 9/20
4	4	ς₁₂ = ς₃₄ = 232/315, ς₂₃ = 178/315, ς₁₃ = ς₂₄ = 46/105, ς₁₄ = 54/105
5	5	ς₁₂ = 21325/24192, ς₁₃ = 325/1512, ς₁₄ = 13975/24192, ς₁₅ = 125/252, ς₂₃ = 1975/3024, ς₂₄ = 425/672, ς₂₅ =11225/24192, ς₃₄ =2575/6048, ς₃₅ =2675/6048, ς₄₅ = 18575/24192

Old Table 10. LMSuc algorithm coefficients.

**Old Table 10.** LMSuc algorithm coefficients.
L	N	Coefficients
3	3	ς₁₂ = 0.681306, ς₁₃ = 0.444312, ς₂₃ = 0.679452
4	4	ς₁₂ = 0.739716, ς₁₃ = 0.432467, ς₁₄ = 516734, ς₂₃ = 0.571812, ς₂₄ = 0.4434453, ς₃₄ = 0.737795
5	5	ς₁₂ = 769,240, ς₁₃ = 0.438591, ς₁₄ = 0.467191, ς₁₅ = 0.495116, ς₂₃ = 0.431753, ς₂₄ = 0.625867, ς₂₅ = 0.579681, ς₃₄ = 0.656805, ς₃₅ = 0.213527, ς₄₅ = 0.881820

New Table 10. LMSuc algorithm coefficients.

**New Table 10.** LMSuc algorithm coefficients.
L	N	Coefficients
3	3	ς₁₂ = 0.679452, ς₁₃ = 0.444312, ς₂₃ = 0.681306
4	4	ς₁₂ = 0.737795, ς₁₃ = 0.434453, ς₁₄ = 516734, ς₂₃ = 0.571812, ς₂₄ =0.432467, ς₃₄ = 0.739716
5	5	ς₁₂ = 0.881820, ς₁₃ = 0.213527, ς₁₄ = 0.579681, ς₁₅ = 0.495116, ς₂₃ = 0.656805, ς₂₄ = 0.625867, ς₂₅ =0.467191, ς₃₄ =0.431753, ς₃₅ =0.438591, ς₄₅ = 0.769240

The former Equation (12) of [1]:

\begin{matrix} G Γ (t_{l - 1}) = \bar{G}, G \equiv {(g_{i})}_{M \times 1}, \bar{G} \equiv {({\bar{g}}_{j})}_{M \times 1}, Γ (t_{l - 1}) \equiv {(γ_{j i} (t_{l - 1}))}_{M \times M} \\ γ_{j i} (t_{l - 1}) = {\begin{matrix} {(- t_{l - 1})}^{i - j} & , & j = 1 \\ {(- t_{l - 1})}^{i - j} (i - 1)! / (j - 1)! & , & 1 < j \leq i \\ 0 & , & j > i \end{matrix} \end{matrix}

(12)

Should be replaced by the new Equation (12):

\begin{matrix} G Γ (t_{l - 1}) = \bar{G}, G \equiv {(g_{i})}_{M \times 1}, \bar{G} \equiv {({\bar{g}}_{j})}_{M \times 1}, Γ (t_{l - 1}) \equiv {(γ_{j i} (t_{l - 1}))}_{M \times M} \\ γ_{j i} (t_{l - 1}) = {\begin{matrix} {(- t_{l - 1})}^{i - j} & , & j = 1 \\ {(- t_{l - 1})}^{i - j} (i - 1)! / ((i - j)! (j - 1)!) & , & 1 < j \leq i \\ 0 & , & j > i \end{matrix} \end{matrix}

(12)

Affected by the correction to Equation (12), the former Table 17 (labelled here as Old Table 17) of [1] should be replaced by the new version (labelled here as New Table 17). The correction to Table 17 will not affect the conclusions of [1].

Old Table 17. Maximum maneuver error over 2 s maneuver.

**Old Table 17.** Maximum maneuver error over 2 s maneuver.
L	N	Maximum Maneuver Error, μ Rad

		FTSc	LMSc	FTShc	LMShc	FTSuc	LMSuc
3	3	1.00e–2	−1.88e–2	−3.34e–3	3.65e–3	2.86e–6	−2.52e–2
4	4	3.24e–2	3.25e–2	−5.51e–3	−5.54e–3	1.48e–12	9.66e–4
5	5	7.32e–2	7.33e–2	−7.50e–3	−7.52e–3	−7.23e–13	3.25e–5

New Table 17. Maximum maneuver error over 2 s maneuver.

**New Table 17.** Maximum maneuver error over 2 s maneuver.
L	N	Maximum Maneuver Error, μ rad

		FTSc	LMSc	FTShc	LMShc	FTSuc	LMSuc
3	3	−1.09e–2	−7.29e–3	3.63e–3	7.48e–3	1.39e–6	3.66e–3
4	4	−3.52e–2	−3.54e–2	5.98e–3	5.89e–3	1.67e–12	−1.40e–4
5	5	−7.97e–2	−7.97e–2	8.15e–3	8.17e–3	−3.09e–13	−4.73e–6

2. Change in Main Body Paragraphs

Due to an obscurity on how Equations (13) and (14) of [1] were built, the authors wish to insert some additional sentences to explain how Equations (13) and (14) of [1] can be converted from Equations (59) and (13) of Song (reference [9] of [1]).

Below we respectively denote the Song ς_ij and the [1] ς_ij using (ς_ij)_S and (ς_ij)_T.

After setting p = N + 1− i and q = N + 1− j, we can rewrite Equation (5) of Song [9] as:

δ {\hat{ϕ}}_{unc} (t) = \sum_{p = 1}^{N - 1} \sum_{q = p + 1}^{N} {(ς_{N + 1 - p, N + 1 - q})}_{S} Δ α_{p} \times Δ α_{q}

(a1)

If δϕ̂_unc(t) and (ς_N₊₁₋_p,N₊₁₋_q)_S are respectively denoted by δϕ̂_l and ξ_pq, Equation (a1) can be rewritten as:

δ {\hat{ϕ}}_{l} = \sum_{p = 1}^{N - 1} \sum_{q = p + 1}^{N} ξ_{p q} Δ α_{p} \times Δ α_{q}

(a1)

Comparing Equation (a2) with the [1] Equation (3), we will find that both equations are the same expression under ξ_pq = (ς_ij)_T with p = i and q = j.

Thus, to make Song [9] Equation (5) of and [1] Equation (3) equivalent will achieve (ς_ij)_T = (ς_N₊₁₋_i_,_N₊₁₋_j)_S. Using this relationship, we have respectively converted ς_ij s in Tables 1 and 2 of Song [9] to ς_ij s in New Tables 9 and 10, also we can convert Song [9] Equation (13) to [1] Equation (14), when Song [9] n is replaced by L.

Now we rewrite Song [9] Equation (59) as:

\begin{matrix} \begin{array}{l} δ {\hat{ϕ}}_{unc} - δ ϕ_{c} = z_{3}^{'} ω (t_{m - 1}) \times \dot{ω} (t_{m - 1}) {(t - t_{m - 1})}^{3} + z_{4}^{'} ω (t_{m - 1}) \times \ddot{ω} (t_{m - 1}) {(t - t_{m - 1})}^{4} \\ + (z_{51}^{'} ω (t_{m - 1}) \times \overset{⃛}{ω} (t_{m - 1}) + z_{52}^{'} \dot{ω} (t_{m - 1}) \times \ddot{ω} (t_{m - 1})) {(t - t_{m - 1})}^{5} \\ + (z_{61}^{'} ω (t_{m - 1}) \times \dot{\overset{⃛}{ω}} (t_{m - 1}) + z_{62}^{'} \dot{ω} (t_{m - 1}) \times \overset{⃛}{ω} (t_{m - 1})) {(t - t_{m - 1})}^{6} \\ + (z_{71}^{'} ω (t_{m - 1}) \times \ddot{\overset{⃛}{ω}} (t_{m - 1}) + z_{72}^{'} \dot{ω} (t_{m - 1}) \times \dot{\overset{⃛}{ω}} (t_{m - 1}) + z_{73}^{'} \ddot{ω} (t_{m - 1}) \times \overset{⃛}{ω} (t_{m - 1})) {(t - t_{m - 1})}^{7} \\ + o ({(t - t_{m - 1})}^{9}) \end{array} \\ z_{3}^{'} = \frac{1}{6} (f_{3} - \frac{1}{2}), z_{4}^{'} = \frac{1}{24} (f_{4} - 1), z_{51}^{'} = \frac{1}{120} (f_{51} - \frac{3}{2}), z_{52}^{'} = \frac{1}{120} (f_{52} - 1), z_{61}^{'} = \frac{1}{720} (f_{61} - 2) \\ z_{62}^{'} = \frac{1}{720} (f_{62} - \frac{5}{2}), z_{71}^{'} = \frac{1}{5040} (f_{71} - \frac{5}{2}), z_{72}^{'} = \frac{1}{5040} (f_{72} - \frac{9}{2}), z_{73}^{'} = \frac{1}{5040} (f_{73} - \frac{5}{2}) \end{matrix}

(a3)

where f s are of Song [9], rather than of [1].

Set:

g_{i} = \frac{d^{i - 1}}{d t^{i - 1}} ({ω (t) |}_{t = t_{m - 1}}) / (i - 1)!, i = 1, 2, \dots

(a4)

where i is a positive integer, and

\frac{d^{0}}{d t^{0}} ({ω (t) |}_{t = t_{m - 1}})

denotes ω(t_m₋₁).

Then Equation (a3) can be converted into [1] Equation (13), when δϕ̂_unc(t) − δϕ_c(t), t_m₋₁ and n are respectively replaced by δϕ̂_l − δϕ_l, t_l₋₁ and L.

To confirm the correctness of [1] Equations (13) and (14), the z s in [1] Equation (13) are calculated for LMSuc using the [1] f s (see [1] Equation (14)) and ς s in New Table 10. Also the z^′ s in Equation (a3) are calculated for UncExp using the Song [9] f s (see Song [9] Equations (13)) and ς s in Song [9] Table 1. The z s for LMSuc and the z^′ s for UncExp are listed in Tables a1 (the copy of [1] Table 16) and a2, respectively.

Table a1. The z s for [1] LMSuc.

**Table a1.** The z s for [1] LMSuc.
L	N	z₃	z₄	z₅₁	z₅₂	z₆₁	z₆₂	z₇₁	z₇₂	z₇₃
3	3	−2.29e–5	0	−9.12e–4	−2.56e–4	−1.83e–3	−8.46e–4	−2.57e–3	−1.48e–3	−5.53e–4
4	4	4.95e–7	−1.30e–8	−2.00e–8	−1.04e–6	1.32e–7	−1.02e–8	−6.17e–5	−7.30e–5	−2.84e–5
5	5	1.07e–8	1.07e–9	2.24e–9	2.21e–8	3.03e–9	1.55e–9	−3.49e–5	2.08e–9	3.45e–6

Table a2. The z^′ s for Song [9] UncExp.

**Table a2.** The z^′ s for Song [9] UncExp.
n	N	z₃^′	z₄^′	z₅₁^′	z₅₂^′	z₆₁^′	z₆₂^′	z₇₁^′	z₇₂^′	z₇₃^′
3	3	−2.29e–5	0	−1.52e–4	−1.28e–4	−7.63e–5	−1.41e–4	−2.15e–5	−6.18e–5	−4.61e–5
4	4	4.95e–7	−6.51e–9	−3.34e–9	−5.21e–7	5.49e–9	−1.70e–9	−5.14e–7	−3.04e–6	−2.37e–6
5	5	1.07e–8	5.33e–10	3.73e–10	1.10e–8	1.26e–10	2.58e–10	−2.91e–7	8.67e–11	2.87e–7

Comparing the z₃^′,z₄^′, z₅₁^′ and z₅₂^′ in Table a2 with those in Equations (65)–(67) of Song [9], we can find that the former is consistent with the later except for z₄^′ and z₅₁^′. (The z₄^′ and z₅₁^′ in Equations (66) and (67) of Song [9] are zero, while the z₄^′ and z₅₁^′ in Table a2 for UncExp4 and UncExp5 are near zero. The difference between z₄^′ and z₅₁^′ of Table a2 and those of Song [9] is due to round-off (to six places) in the Song [9] ς s used in [1].) This has been confirmed independently by a Reviewer of [1] that found identical results when using Song [9] equations and Song rounded ς s.

The authors wish to express their appreciation to a reviewer of [1] for his insightful comments and constructive suggestions used in the original article, also for his valuable suggestions used in this correction.

References

Tang, C.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure. Sensors 2014, 14, 14289–14301. [Google Scholar]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

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Tang, C.; Chen, X. Correction: Tang, C. Y. and Chen, X.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure. Sensors 2014, 14, 14289–14301. Sensors 2015, 15, 4425-4429. https://doi.org/10.3390/s150204425

AMA Style

Tang C, Chen X. Correction: Tang, C. Y. and Chen, X.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure. Sensors 2014, 14, 14289–14301. Sensors. 2015; 15(2):4425-4429. https://doi.org/10.3390/s150204425

Chicago/Turabian Style

Tang, Chuanye, and Xiyuan Chen. 2015. "Correction: Tang, C. Y. and Chen, X.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure. Sensors 2014, 14, 14289–14301" Sensors 15, no. 2: 4425-4429. https://doi.org/10.3390/s150204425

APA Style

Tang, C., & Chen, X. (2015). Correction: Tang, C. Y. and Chen, X.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure. Sensors 2014, 14, 14289–14301. Sensors, 15(2), 4425-4429. https://doi.org/10.3390/s150204425

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Correction: Tang, C. Y. and Chen, X.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure. Sensors 2014, 14, 14289–14301

1. Change in Tables/Equations

2. Change in Main Body Paragraphs

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