Correction: Tang, C. Y. and Chen, X.Y. A Class of Coning Algorithms Based on a Half-Compressed Structure.

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.

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

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 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

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

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

The former Equation (12) of [1]:

$$\begin{array}{c}G\Gamma (t_{l-1})=\overline{G},G\equiv {\left(g_i\right)}_{M\times 1},\overline{G}\equiv {\left({\overline{g}}_j\right)}_{M\times 1},\Gamma (t_{l-1})\equiv {\left({\gamma }_{ji}(t_{l-1})\right)}_{M\times M}\\ {\gamma }_{ji}\left(t_{l-1}\right)=\{\begin{array}{ccc}{\left(-t_{l-1}\right)}^{i-j}& ,& j=1\\ {\left(-t_{l-1}\right)}^{i-j}(i-1)!/(j-1)!& ,& 1<j\le i\\ 0& ,& j>i\end{array}\end{array}$$ (12)

Should be replaced by the new Equation (12):

$$\begin{array}{c}G\Gamma (t_{l-1})=\overline{G},G\equiv {\left(g_i\right)}_{M\times 1},\overline{G}\equiv {\left({\overline{g}}_j\right)}_{M\times 1},\Gamma (t_{l-1})\equiv {\left({\gamma }_{ji}(t_{l-1})\right)}_{M\times M}\\ {\gamma }_{ji}\left(t_{l-1}\right)=\{\begin{array}{ccc}{\left(-t_{l-1}\right)}^{i-j}& ,& j=1\\ {\left(-t_{l-1}\right)}^{i-j}(i-1)!/((i-j)!(j-1)!)& ,& 1<j\le i\\ 0& ,& j>i\end{array}\end{array}$$ (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].

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:

$$\delta {\widehat{\varphi }}_{\mathit{\text{unc}}}(t)=\sum _{p=1}^{N-1}\sum _{q=p+1}^N{\left({\varsigma }_{N+1-p,N+1-q}\right)}_S\Delta {\alpha }_p\times \Delta {\alpha }_q$$ (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:

$$\delta {\widehat{\varphi }}_l=\sum _{p=1}^{N-1}\sum _{q=p+1}^N{\xi }_{pq}\Delta {\alpha }_p\times \Delta {\alpha }_q$$ (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:

$$\begin{array}{c}\begin{array}{l}\delta {\widehat{\varphi }}_{\mathit{\text{unc}}}-\delta {\varphi }_c=z_3^{\prime }\omega (t_{m-1})\times \dot{\omega }(t_{m-1}){\left(t-t_{m-1}\right)}^3+z_4^{\prime }\omega (t_{m-1})\times \ddot{\omega }(t_{m-1}){\left(t-t_{m-1}\right)}^4\\ +\left(z_{51}^{\prime }\omega (t_{m-1})\times \stackrel{⃛}{\omega }(t_{m-1})+z_{52}^{\prime }\dot{\omega }(t_{m-1})\times \ddot{\omega }(t_{m-1})\right){\left(t-t_{m-1}\right)}^5\\ +\left(z_{61}^{\prime }\omega (t_{m-1})\times \dot{\stackrel{⃛}{\omega }}(t_{m-1})+z_{62}^{\text{'}}\dot{\omega }(t_{m-1})\times \stackrel{⃛}{\omega }(t_{m-1})\right){\left(t-t_{m-1}\right)}^6\\ +\left(z_{71}^{\text{'}}\omega (t_{m-1})\times \ddot{\stackrel{⃛}{\omega }}(t_{m-1})+z_{72}^{\text{'}}\dot{\omega }(t_{m-1})\times \dot{\stackrel{⃛}{\omega }}(t_{m-1})+z_{73}^{\text{'}}\ddot{\omega }(t_{m-1})\times \stackrel{⃛}{\omega }(t_{m-1})\right){\left(t-t_{m-1}\right)}^7\\ +\mathrm{o}\left({\left(t-t_{m-1}\right)}^9\right)\end{array}\\ z_3^{\prime }=\frac{1}{6}\left(f_3-\frac{1}{2}\right),z_4^{\prime }=\frac{1}{24}(f_4-1),z_{51}^{\prime }=\frac{1}{120}\left(f_{51}-\frac{3}{2}\right),z_{52}^{\prime }=\frac{1}{120}(f_{52}-1),z_{61}^{\prime }=\frac{1}{720}(f_{61}-2)\\ z_{62}^{\prime }=\frac{1}{720}\left(f_{62}-\frac{5}{2}\right),z_{71}^{\prime }=\frac{1}{5040}\left(f_{71}-\frac{5}{2}\right),z_{72}^{\prime }=\frac{1}{5040}\left(f_{72}-\frac{9}{2}\right),z_{73}^{\prime }=\frac{1}{5040}\left(f_{73}-\frac{5}{2}\right)\end{array}$$ (a3)

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

Set:

$$g_i=\frac{d^{i-1}}{d{t}^{i-1}}\left({\omega (t)|}_{t=t_{m-1}}\right)/(i-1)!,i=1,2,\dots$$ (a4)

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.

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.

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.