1 Correction to: Nonlinear Dyn (2021) 107: 51–76 https://doi.org/10.1007/s11071-021-06767-8

Due to author oversights during proofing, the article was published with errors found in page 56, Eq. (5c) should be left aligned instead of right aligned.

In page 58, Eq. (9c) should be left aligned instead of light aligned.

In page 63, the subsection 4.1.1 title should be in lightface as: 4.1.1 Body-(0) frame velocity: (\(\dot{\user2{e}}^{\left( 0 \right)} \left( {\varvec{t}} \right)\) \(\dot{\user2{r}}_{{\varvec{C}}}^{\left( 0 \right)} \left( {\varvec{t}} \right)\)).

In page 63, a line above Eq. (22b) the typographical errors for equation numbers ‘[7b, 7c]’ should be noted as ‘[20, 21]’.

In page 63, the subsection 4.1.2 title should be in lightface as: 4.1.2 Body-(13) frame velocity: (\(\dot{\user2{e}}^{{\left( {13} \right)}} \left( {\varvec{t}} \right)\) \(\dot{\user2{r}}_{{\varvec{C}}}^{{\left( {13} \right)}} \left( {\varvec{t}} \right)\)).

In Eq. (25) of page 63, on the right-hand side, the size of the closing parenthesis should match that of the opening parenthesis as:

$$ \left( {\begin{array}{*{20}c} {{\dot{\mathbf{e}}}^{{\left( {13} \right)}} \left( t \right)} & {{\dot{\mathbf{r}}}_{C}^{{\left( {13} \right)}} \left( t \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{\mathbf{e}}^{{\left( {13} \right)}} \left( t \right){\overleftrightarrow {\omega^{{\left( {13} \right)}} \left( t \right)}}} & {{\mathbf{e}}^{I} \dot{x}_{C}^{{\left( {13} \right)}} \left( t \right)} \\ \end{array} } \right). $$
(25)

Equation (26a) in page 64 has typographical errors in the superscripts of the second and the third lines; the corrected lines are:

$$ \begin{aligned} & = {\mathbf{e}}^{\left( 0 \right)} \left( t \right)\left( {{\overleftrightarrow {\omega^{\left( 0 \right)} \left( t \right)}}R^{(13/0)} \left( t \right) + \dot{R}^{(13/0)} \left( t \right)} \right) \\ & = {\mathbf{e}}^{{\left( {13} \right)}} \left( t \right)\left( {R^{(13/0)} \left( t \right)} \right)^{T} \left( {\overleftrightarrow {\omega^{\left( 0 \right)} \left( t \right)}R^{(13/0)} \left( t \right) + \dot{R}^{(13/0)} \left( t \right)} \right) \\ & = {\mathbf{e}}^{{\left( {13} \right)}} \left( t \right)\left\{ {\overleftrightarrow {\left( {R^{(13/0)} \left( t \right)} \right)^{T} \omega^{\left( 0 \right)} \left( t \right)} + \overleftrightarrow {\omega^{(13/0)} \left( t \right)}} \right\}. \\ \end{aligned} $$
(26a)

Equation (28a) in page 64 has typographical errors, where all ‘r’ and ‘e’ should appear in bold face as:

$$ \begin{aligned} {\dot{\mathbf{r}}}_{C}^{{\left( {13} \right)}} \left( t \right) & = {\dot{\mathbf{r}}}_{C}^{\left( 0 \right)} \left( t \right) + {\dot{\mathbf{e}}}^{\left( 0 \right)} \left( t \right)s_{C}^{{\left( {13/0} \right)}} \left( t \right) + {\mathbf{e}}^{\left( 0 \right)} \left( t \right)\dot{s}_{C}^{{\left( {13/0} \right)}} \left( t \right) \\ & = {\mathbf{e}}^{I} \left\{ {\dot{x}_{C}^{\left( 0 \right)} \left( t \right) + R^{\left( 0 \right)} \left( t \right)\left( {\overleftrightarrow {\omega^{\left( 0 \right)} \left( t \right)} s_{C}^{{\left( {13/0} \right)}} \left( t \right) + \dot{s}_{C}^{{\left( {13/0} \right)}} \left( t \right)} \right)} \right\} \\ & = {\mathbf{e}}^{I} \left\{ {\dot{x}_{C}^{\left( 0 \right)} \left( t \right) + R^{\left( 0 \right)} \left( t \right)\left( { - \overleftrightarrow {s_{C}^{{\left( {13/0} \right)}} \left( t \right)}\omega^{\left( 0 \right)} \left( t \right) + \dot{s}_{C}^{{\left( {13/0} \right)}} \left( t \right)} \right)} \right\} \\ \end{aligned} $$
(28a)

Equation (30) in page 64 has a typographical error of a missing space in a 1 × 2 matrix on the right-hand side. The corrected equation should be shown as follows:

$$ \left( {\begin{array}{*{20}c} {{\dot{\mathbf{e}}}_{{T_{k} }}^{{\left( {{\text{SJ }}13} \right)}} \left( t \right)} & {{\dot{\mathbf{r}}}_{{T_{k} }} \left( t \right)} \\ \end{array} } \right)_{path\left( i \right)} = \left( {\begin{array}{*{20}c} {{\mathbf{e}}_{{T_{k} }}^{{\left( {{\text{SJ }}13} \right)}} \left( t \right) \overleftrightarrow {\omega_{{T_{k} }}^{{\left( {{\text{SJ}} 13} \right)}} \left( t \right)}} & {{\mathbf{e}}^{I} \dot{x}_{{T_{k} }}^{{}} \left( t \right)} \\ \end{array} } \right)_{path\left( i \right)} , $$
(30)

In page 64, a line that follows Eq. (31a) has a typographical error for equation number: ‘Eq. (69)’, which should be noted as ‘Eq. (26b)’.

In page 64, a sentence above Eq. (31b) has two typographical errors for equation numbers ‘Eqs. (75) and (76)’. The corrected equation numbers are ‘Eqs. (30) and (31a)’.

In page 64, the second row of Eq. (32) should be indented.

In page 65, the first letter ‘r’ of Eq. (33a) should be in bold face as ‘r’. The corrected equation should be

$$ \begin{aligned} \dot{\user2{r}}_{{T_{k} }} \left( t \right) & = {\varvec{e}}^{I} \dot{x}_{{T_{k} }}^{{}} \left( t \right) = {\varvec{e}}^{{\left( {13} \right)}} \left( t \right)\overleftrightarrow {\omega^{{\left( {13} \right)}} \left( t \right)} \hat{s}_{{T_{k} }}^{{\left( {13} \right)}} + {\varvec{e}}^{I} \dot{x}_{C}^{{\left( {13} \right)}} \left( t \right) \\ & = - {\varvec{e}}^{{\left( {13} \right)}} \left( t \right)\overleftrightarrow {\hat{s}_{{T_{k} }}^{{\left( {13} \right)}} } \omega^{{\left( {13} \right)}} \left( t \right) + {\varvec{e}}^{I} \dot{x}_{C}^{{\left( {13} \right)}} \left( t \right) \\ & = {\varvec{e}}^{I} \left( { - R^{{\left( {13} \right)}} \left( t \right)\overleftrightarrow {\hat{s}_{{T_{k} }}^{{\left( {13} \right)}} } \omega^{{\left( {13} \right)}} \left( t \right) + \dot{x}_{C}^{{\left( {13} \right)}} \left( t \right)} \right). \\ \end{aligned} $$
(33a)

In page 65, the second line of a sentence that follows Eq. (34) has a typographical error; the corrected line should be: (13) is now expressed by Eqs. (30), (32) and (34).

In page 65, Eq. (37) has typographical errors on the left-hand side in which vectors ‘e’ and ‘r’ are incorrectly shown in lightface; the corrected Eq. (37) should be:

$$ \left( {\begin{array}{*{20}c} {{\dot{\mathbf{e}}}^{{\left( {2k - 1} \right)}} \left( t \right)} & {{\dot{\mathbf{r}}}_{C}^{{\left( {2k - 1} \right)}} \left( t \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{\mathbf{e}}^{{\left( {2k - 1} \right)}} \left( t \right)\overleftrightarrow {\omega^{{\left( {2k - 1} \right)}} \left( t \right)}} & {{\mathbf{e}}^{I} \dot{x}_{C}^{{\left( {2k - 1} \right)}} \left( t \right)} \\ \end{array} } \right), $$
(37)

In page 65, a line above Eq. (39a) has an incorrect equation number ‘Eqs. (26a) and (26b)’, which should be read as ‘Eqs. (22a) and (26b)’.

In page 66, in the second line of Eq. (41a) ‘(t)’ for ‘\(\omega^{(2k - 1/0)}\)’ must be under the left–right arrow-above symbol as:

$$ + {\mathbf{e}}^{\left( 0 \right)} \left( t \right)R^{(2k - 1/0)} \left( t \right)\overleftrightarrow {\omega^{(2k - 1/0)} \left( t \right)}\hat{s}_{{C/B_{k} }}^{{\left( {{\text{UJ }}2k - 1} \right)}} . $$
(41a)

In page 66, the left-hand side ‘e’ of Eq. (42a) should appear in bold face as:

$$ \left( {\begin{array}{*{20}c} {{\dot{\mathbf{e}}}^{{\left( {2k} \right)}} \left( t \right)} & {{\dot{\mathbf{r}}}_{C}^{{\left( {2k} \right)}} \left( t \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{\mathbf{e}}^{{\left( {2k} \right)}} \left( t \right)\overleftrightarrow {\omega^{{\left( {2k} \right)}} \left( t \right)}} & {{\mathbf{e}}^{I} \dot{x}_{C}^{{\left( {2k} \right)}} \left( t \right)} \\ \end{array} } \right). $$
(42a)

In page 66, on the second line of Eq. (42c), the first ‘(t)’ should appear under the left–right arrow-above symbol as:

$$ \left( { - \overleftrightarrow {s_{C}^{{\left( {2k/2k - 1} \right)}} \left( t \right)}\omega^{{\left( {2k - 1} \right)}} \left( t \right) + e_{3} \dot{d}^{\left( k \right)} \left( t \right)} \right). $$
(42c)

In page 66, the first ‘e’ of Eq. (43a) should be in bold face. In addition, on the right-hand side, the size of a pair of parentheses should match as shown in the following:

$$ \left( {\begin{array}{*{20}c} {{\dot{\mathbf{e}}}_{{T_{k} }}^{{\left( {{\text{SJ }}13} \right)}} \left( t \right)} & {{\dot{\mathbf{r}}}_{{T_{k} }} \left( t \right)} \\ \end{array} } \right)_{{path\left( {ii} \right)}} = \left( {\begin{array}{*{20}c} {{\mathbf{e}}_{{T_{k} }}^{{\left( {{\text{SJ }}13} \right)}} \left( t \right) \overleftrightarrow {\omega_{{T_{k} }}^{{\left( {{\text{SJ}} 13} \right)}} \left( t \right)}} & {{\mathbf{e}}^{I} \dot{x}_{{T_{k} }}^{{\left( {{\text{SJ }}13} \right)}} \left( t \right)} \\ \end{array} } \right)_{{path\left( {ii} \right)}} , $$
(43a)

In page 68, the sentence that follows Eq. (48c) has a typographical error ‘\(\phi_{2}^{\left( k \right)} \left( t \right)/ = \pm \pi /2\)’, which should appear as ‘\(\phi_{2}^{\left( k \right)} \left( t \right) \ne \pm \pi /2\)’.

In pate 71, a line that follows Eq. (Aa) should not be indented.

In page 72, Eq. (A3d) has a typographical error ‘\({\Delta }\)’ for a dot product sign ‘\(\cdot\)’ of vectors. The corrected equation is as follows:

$$ {\varvec{e}}_{3}^{\left( 0 \right)} \left( t \right): \quad w_{3}^{*\left( k \right)} = {\varvec{e}}_{3}^{\left( 0 \right)} \left( t \right) \cdot {\varvec{e}}^{{*\left( {13} \right)}} \left( t \right)\hat{s}_{{T_{k} }}^{{\left( {13} \right)}} + s_{3C}^{{*\left( {13/0} \right)}} - \hat{h}_{UJ}^{\left( 0 \right)} . $$
(A3d)

In page 72, the second line of Eq. (A5) should be indented.

In page 73, the second line of Eq. (A9a) should be indented.

In page 75, all membership signs ‘\(\in\)’ in Eq. (B3) should be replaced by epsilons ‘\(\epsilon\)’ as:

$$ \epsilon_{mnq} R_{mi} R_{nj} R_{qk} = \epsilon_{ijk} \det R = \epsilon_{ijk} . $$
(B3)

In page 75, Eq. (B4) has missing terms on the first line. In addition, all membership signs ‘\(\in\)’ should be replaced by epsilons ‘\(\epsilon\)’. The corrected equation is

$$ \begin{aligned} \left( {R^{T} \mathop{\omega }\limits^{\leftrightarrow} R} \right)_{ij} & = R_{mi} \left( {\mathop{\omega }\limits^{\leftrightarrow} } \right)_{mn} R_{nj} = - R_{mi} \epsilon_{mnk} \omega_{k} R_{nj} \\ & = - \epsilon_{mnk} R_{mi} R_{nj} \omega_{k} = - \epsilon_{mnk} R_{mi} R_{nj} \delta_{kp} \omega_{p} \\ & = - \left( {\epsilon_{mnk} R_{mi} R_{nj} R_{kq} } \right)R_{pq} \omega_{p} = - \epsilon_{ijq} R_{pq} \omega_{p} \\ & = - \epsilon_{ijq} \left( {R^{T} \omega } \right)_{q} = \left( {\overleftrightarrow {R^{T} \omega }} \right)_{ij} . \\ \end{aligned} $$
(B4)

In the second column of page 75, on the second line from the bottom, the reference number has an error ‘by Holm [7a]’, which should be read as: ‘by Holm [19]’.

The author group and publisher regret the mistake and ask readers to consider the noted corrections, which stand to correct the original article.