Correction to: A hybrid parallel Harris hawks optimization algorithm for reusable launch vehicle reentry trajectory optimization with no-fly zones

1 Correction to: Soft Computing https://doi.org/10.1007/s00500-021-06039-y

There are some errors in the layout of the paper, the correct ones are as follows:

1. “[1,1]” should be “[− 1,1]” (Line 353)

2. “${\varvec{X}}(t + 1) = \Delta {\varvec{X}}(t) - E\left| {J{\varvec{1}}{\varvec{X}}_{{{\text{rabbit}}}} (t) - {\varvec{X}}(t)} \right| \,$” should be “${\varvec{X}}(t + 1) = \Delta {\varvec{X}}(t) - E\left| {J{\varvec{X}}_{{{\text{rabbit}}}} (t) - {\varvec{X}}(t)} \right| \,$” (Line 376)

3. “${\varvec{V}}_{i} = {\varvec{X}}_{i} + F({\varvec{X}}_{best} - {\varvec{X}}_{r1} ) + F({\varvec{X}}_{r2} - {\varvec{X}}_{r3} )$” should be “${\varvec{V}}_{i} = {\varvec{X}}_{i} + F({\varvec{X}}_{best} - {\varvec{X}}_{r1} ) + F({\varvec{X}}_{r2} - {\varvec{X}}_{r3} )$” (Line 450)

4. “$\overline{u}_{k} = \left\{ {\begin{array}{*{20}l} {\lambda u_{k - 1} + (1 - \lambda ),} \hfill & {{\text{if}}\quad {\text{sign}}(u_{k} - u_{k - 1} ){\text{sign}}(u_{k + 1} - u_{k} ) < 0} \hfill \\ {u_{k} ,} \hfill & {{\text{else}}} \hfill \\ \end{array} } \right.$” should be “$\overline{u}_{k} = \left\{ {\begin{array}{*{20}l} {\lambda u_{k - 1} + (1 - \lambda )u_{k + 1} ,} \hfill & {{\text{if}}\quad {\text{sign}}(u_{k} - u_{k - 1} ){\text{sign}}(u_{k + 1} - u_{k} ) < 0} \hfill \\ {u_{k} ,} \hfill & {{\text{else}}} \hfill \\ \end{array} } \right.$” (Line 541)

5. “${\varvec{X}}_{i} (t + 1) = \left\{ {\begin{array}{*{20}l} {{\varvec{X}}_{i} (t) + F({\mathbf{X}}_{{{\text{rabbit}}}} (t) - {\varvec{X}}_{r1} (t)) + F({\varvec{X}}_{r2} (t) - {\varvec{X}}_{r3} (t))} \hfill & {q \ge 0.5} \hfill \\ {({\varvec{X}}_{{{\text{rabbit}}}} (t) - {\varvec{X}}_{m} (t)) - r_{4} (\varvec{LB} + r_{5} (\varvec{UB} - \varvec{LB}))} \hfill & {q < 0.5} \hfill \\ \end{array} } \right.$” should be “${\varvec{X}}_{i} (t + 1) = \left\{ {\begin{array}{*{20}l} {{\varvec{X}}_{i} (t) + F({\mathbf{X}}_{{{\text{rabbit}}}} (t) - {\varvec{X}}_{r1} (t)) + F({\varvec{X}}_{r2} (t) - {\varvec{X}}_{r3} (t))} \hfill & {q \ge 0.5} \hfill \\ {({\varvec{X}}_{{{\text{rabbit}}}} (t) - {\varvec{X}}_{{\text{m}}} (t)) - r_{4} (\varvec{LB} + r_{5} (\varvec{UB} - \varvec{LB}))} \hfill & {q < 0.5} \hfill \\ \end{array} } \right.$” (Line 551)

6. “u=--5” should be “u= − 5” (Line 661)

7. The last three rows of Tables 3 and 4 should be right aligned.

Table 3 Comparison of the results of HHO variants

The correct format is below (Table 4 is the same):

8. “Figs. 8a–c” should be “Figs. 8 (a)–(c)”; “Figs. 8 c” should be “Figs. 8 (c)” (Line 799 and Line 803)

9. “φ” should be “φ” (Line 823, Line 829, Line 895 and Table 5)

10. “$[\dot{Q}_{\max } ,q_{\max } ,n_{z\max } ] = [200 \, Btu/ft^{2} /s, \, 13406 \, Pa, \, 2.5 \, g_{0} ]$” should be “$[\dot{Q}_{\max } ,q_{\max } ,n_{z\max } ] = [200{\text{ Btu/ft}}^{{2}} {\text{/s}}, \, 13406{\text{ Pa}}, \, 2.5{\text{ g}}_{0} ]$” (Lines 869-870)

11. “0.59” should be “-0.59” (Line 922)

12. The figure notes for Figures 10 and 11 should be in black font.

13. “dos Coelho LS” should be “Coelho L dos S” (Line 1075)

14. “Grey Wolf Optimizer” should be “Grey wolf optimizer” (Line 1151)

15. The order of the two references corresponding to lines 1151–1157 is wrong, and the correct format is

Mirjalili S, Mirjalili SM, Hatamlou A (2016) Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl 27:495–513. https://doi.org/10.1007/s00521-015-1870-7

Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007

16. The order of the two references corresponding to lines 1207–1214 is wrong, and the correct format is

Zhang DN, Liu Y (2011) RLV reentry trajectory optimization through hybridization of an improved GA and a SQP algorithm. In: AIAA Guidance, Navigation, and Control Conference. https://doi.org/10.2514/6.2011-6658

Zhang HP, Wang HL, Li N et al (2020) Time-optimal memetic whale optimization algorithm for hypersonic vehicle reentry trajectory optimization with no-fly zones. Neural Comput Appl 32:2735–2749. https://doi.org/10.1007/s00521-018-3764-y

The original article has been corrected.