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

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.

The former Equation (12) of [1]: Should be replaced by the new Equation (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].

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:

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:

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:

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

Set: where i is a positive integer, and $\frac{d^0}{d{t}^0}\left({\omega (t)|}_{t=t_{m-1}}\right)$ 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.

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.