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Twistor-based pose control for asteroid landing with path constraints

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Abstract

In this paper, a six-degree-of-freedom (6-DOF) control scheme for asteroid landing subject to collision avoidance and line-of-sight (LOS) constraints is developed within the newly established twistor framework. The controller guarantees a spacecraft touches down at a specified landing site in a desired attitude, meanwhile satisfies the constraints with the coupling effect between the translational and rotational motions involved. First, 6-DOF dynamics of a spacecraft relative to an asteroid is described by the twistor representation to model the translational and rotational motions in a unified way. Then, the collision avoidance and LOS constraints are formulated and expressed as functions of the twistor. Following that, the constraints are encoded into an artificial potential function (APF) and a control scheme that enforces the constraints is proposed by virtue of the APF technique with the stability of the controlled closed-loop system proved via Lyapunov analysis. Finally, numerical simulations are conducted, and the results demonstrate the effectiveness of the proposed controller.

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Funding

This study was funded by the China Postdoctoral Science Foundation (Grant No. 2018M640995).

Author information

Authors and Affiliations

  1. Unmanned System Research Institute, Northwestern Polytechnical University, Xi’an, 710072, China

    Bo Zhang

  2. Department of Automation Science and Technology, Xi’an Jiaotong University, Xi’an, 710049, China

    Yuanli Cai

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  1. Bo Zhang
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  2. Yuanli Cai
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Correspondence to Bo Zhang.

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Appendices

Appendix A: Expressions of \(\nabla f_1\), \(\nabla f_2\), \(\varPi ^{-\mathrm {T}}\), and \(\varGamma \)

  1. (1)

    The expansion of \(\nabla f_1\) is

    $$\begin{aligned} \nabla f_1 = \left( \frac{\partial f_1}{\partial {\varvec{r}} _{\mathrm {BD}}^{\mathrm {D}}} \right) ^\mathrm {T}\frac{\partial {\varvec{r}} _{\mathrm {BD}}^{\mathrm {D}}}{\partial {\varvec{B}}_{\mathrm {BD}}^\mathrm {T}} \end{aligned}$$
    (52)

where

$$\begin{aligned} \frac{\partial f_1}{\partial {\varvec{r}}_{\mathrm {BD}}^{\mathrm {D}}}=\left[ \begin{array}{c} 2 k_{f_1} \alpha \gamma x_{\mathrm {BD}}^{\mathrm {D}} \left[ \alpha \left( (x_{\mathrm {BD}}^{\mathrm {D}})^2 + \beta (y_{\mathrm {BD}}^{\mathrm {D}})^2 \right) \right] ^{\gamma - 1}\\ 2 k_{f_1} \alpha \beta \gamma y_{\mathrm {D}} \left[ \alpha \left( x_{\mathrm {D}}^2 + \beta y_{\mathrm {D}}^2 \right) \right] ^{\gamma - 1}\\ -k_{f_1} \end{array} \right] \end{aligned}$$
(53)
$$\begin{aligned}&\left( \frac{\partial {\varvec{r}}_{\mathrm {BD}}^{\mathrm {D}}}{\partial {\varvec{B}}_{\mathrm {BD}}^\mathrm {T}} \right) ^\mathrm {T}= \\&\left[ \begin{array}{ccc} \frac{A_{11}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} &{}\quad \frac{A_{12}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} &{}\quad \frac{A_{13}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} \\ \frac{A_{21}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} &{}\quad \frac{A_{22}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} &{}\quad \frac{A_{23}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} \\ \frac{A_{31}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} &{}\quad \frac{A_{32}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} &{}\quad \frac{A_{33}}{\left( b_1^2+b_2^2+b_3^2+1\right) ^3} \\ 0 &{}\quad 0 &{}\quad 0 \\ \frac{4 \left( b_1^2-b_2^2-b_3^2+1\right) }{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} &{}\quad \frac{8 (b_1 b_2+b_3)}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} &{}\quad -\frac{8 (b_2-b_1 b_3)}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} \\ \frac{8 (b_1 b_2-b_3)}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} &{} -\frac{4 \left( b_1^2-b_2^2+b_3^2-1\right) }{\left( b_1^2+b_2^2+b_3^2 +1\right) ^2} &{}\quad \frac{8 (b_1+b_2 b_3)}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} \\ \frac{8 (b_2+b_1 b_3)}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} &{}\quad -\frac{8 (b_1-b_2 b_3)}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} &{}\quad -\frac{4 \left( b_1^2+b_2^2-b_3^2-1\right) }{\left( b_1^2+b_2^2+b_3^2 +1\right) ^2} \\ 0 &{}\quad 0 &{}\quad 0 \\ \end{array} \right] \\ A_{11}&= -8 \left[ b_5 b_1^3+3 (b_2 b_6+b_3 b_7) b_1^2\right. \\&\quad \left. +\,\left( -3 b_5 b_2^2 +4 b_7 b_2-3 b_3^2 b_5+b_5-4 b_3 b_6\right) b_1 \right. \\&\quad \left. -\,\left( b_2^2+b_3^2+1\right) (b_2 b_6+b_3 b_7)\right] \\ A_{12}&= 4 \left[ -4 b_1 \left( -b_6 b_1^2+2 b_2 b_5 b_1-2 b_7 b_1+2 b_3 b_5\right. \right. \\&\quad \left. \left. +\, b_2^2 b_6-b_3^2 b_6+b_6+2 b_2 b_3 b_7\right) \right. \\&\quad \left. -\,2 \left( b_1^2+b_2^2+b_3^2+1\right) (-b_2 b_5+b_1 b_6+b_7) \right] \\ A_{13}&= 8 \left[ b_7 b_1^3-3 (b_3 b_5+b_6) b_1^2\right. \\&\quad \left. +\,\left( b_7 b_2^2 +4 (b_5-b_3 b_6) b_2-3 \left( b_3^2+1\right) b_7\right) b_1 \right. \\&\quad \left. +\,\left( b_2^2+b_3^2+1\right) (b_3 b_5+b_6)\right] \\ A_{21}&= 8 \left[ b_5 b_2^3-3 (b_1 b_6+b_7) b_2^2\right. \\&\quad \left. +\,\left( \left( -3 b_1^2 +b_3^2-3\right) b_5+4 b_3 (b_6-b_1 b_7)\right) b_2 \right. \\&\quad \left. +\,\left( b_1^2+b_3^2+1\right) (b_1 b_6+b_7)\right] \\ A_{22}&= 4 \left[ -4 b_2 \left( -b_6 b_1^2+2 b_2 b_5 b_1-2 b_7 b_1\right. \right. \\&\quad \left. \left. +\,2 b_3 b_5 +b_2^2 b_6-b_3^2 b_6+b_6+2 b_2 b_3 b_7\right) \right. \\&\quad +\, \left. 2 \left( b_1^2+b_2^2+b_3^2+1\right) (b_1 b_5+b_2 b_6+b_3 b_7)\right] \\ A_{23}&= 4 \left[ -4 b_2 \left( -b_7 b_1^2+2 b_3 b_5 b_1+2 b_6 b_1\right. \right. \\&\quad \left. \left. -\,\,2 b_2 b_5 +2 b_2 b_3 b_6-b_2^2 b_7+b_3^2 b_7+b_7\right) \right. \\&\quad \left. -\,2 \left( b_1^2+b_2^2+b_3^2+1\right) (b_5-b_3 b_6+b_2 b_7)\right] \\ A_{31}&= 8 \left[ b_5 b_3^3+3 (b_6-b_1 b_7) b_3^2-\left( \left( 3 b_1^2 -b_2^2+3\right) b_5\right. \right. \\&\quad \left. \left. +\,4 b_2 (b_1 b_6+b_7)\right) b_3 \right. \\&\quad \left. +\,\left( b_1^2+b_2^2+1\right) (b_1 b_7-b_6)\right] \\ A_{32}&= 8 \left[ b_6 b_3^3-3 b_2 b_7 b_3^2\right. \\&\quad \left. +\,\left( \left( b_1^2-3 b_2^2 -3\right) b_6+4 b_1 b_7\right) b_3 + \left( b_1^2+b_2^2+1\right) b_2 b_7\right. \\&\quad \left. +\left( b_1^2-4 b_2 b_3 b_1+b_2^2-3 b_3^2+1\right) b_5 \right] \\ A_{33}&= 4 \left[ -4 b_3 \left( -b_7 b_1^2+2 b_3 b_5 b_1+2 b_6 b_1\right. \right. \\&\quad \left. \left. -\,2 b_2 b_5 +2 b_2 b_3 b_6-b_2^2 b_7+b_3^2 b_7+b_7\right) \right. \\&\quad \left. +\, 2 \left( b_1^2+b_2^2+b_3^2+1\right) (b_1 b_5+b_2 b_6+b_3 b_7)\right] \end{aligned}$$
  1. (2)

    The expression of \(\nabla f_2\) is

    $$\begin{aligned} \nabla f_2 = \left[ \begin{array}{c} -\frac{2 \left( (b_3 b_5-b_6) b_1^2+2 \left( b_7 b_3^2+b_2 b_6 b_3+b_2 b_5+b_7\right) b_1-\left( b_2^2+b_3^2+1\right) (b_3 b_5-b_6)\right) }{\left( b_1^2+b_2^2+b_3^2+1\right) ^3 \sqrt{\frac{b_5^2+b_6^2+b_7^2}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2}}} \\ -\frac{2 \left( (b_5+b_3 b_6) b_2^2+2 \left( b_7 b_3^2+b_1 b_5 b_3-b_1 b_6+b_7\right) b_2-\left( b_1^2+b_3^2+1\right) (b_5+b_3 b_6)\right) }{\left( b_1^2+b_2^2+b_3^2+1\right) ^3 \sqrt{\frac{b_5^2+b_6^2+b_7^2}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2}}} \\ -\frac{2 \left( (b_1 b_5+b_2 b_6) b_3^2-2 \left( b_7 b_2^2-b_2 b_5+b_1 (b_6+b_1 b_7)\right) b_3-\left( b_1^2+b_2^2+1\right) (b_1 b_5+b_2 b_6)\right) }{\left( b_1^2+b_2^2+b_3^2+1\right) ^3 \sqrt{\frac{b_5^2+b_6^2+b_7^2}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2}}}\\ 0 \\ \frac{\sqrt{\frac{b_5^2+b_6^2+b_7^2}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2}} \left( b_5 b_7 b_1^2+2 \left( b_5 b_6+b_3 \left( b_6^2+b_7^2\right) \right) b_1+b_2^2 b_5 b_7-\left( b_3^2+1\right) b_5 b_7+2 b_2 \left( b_6^2-b_3 b_5 b_6+b_7^2\right) \right) }{\left( b_5^2+b_6^2+b_7^2\right) ^2}\\ \frac{\sqrt{\frac{b_5^2+b_6^2+b_7^2}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2}} \left( b_6 b_7 b_1^2-2 \left( b_5^2+b_3 b_6 b_5+b_7^2\right) b_1+b_2^2 b_6 b_7-\left( b_3^2+1\right) b_6 b_7+2 b_2 \left( b_3 \left( b_5^2+b_7^2\right) -b_5 b_6\right) \right) }{\left( b_5^2+b_6^2+b_7^2\right) ^2}\\ -\frac{\sqrt{\frac{b_5^2+b_6^2+b_7^2}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2}} \left( \left( b_1^2+b_2^2-b_3^2-1\right) b_5^2+2 (b_2+b_1 b_3) b_7 b_5+b_6 \left( \left( b_1^2+b_2^2-b_3^2-1\right) b_6-2 (b_1-b_2 b_3) b_7\right) \right) }{\left( b_5^2+b_6^2+b_7^2\right) ^2} \\ 0 \end{array} \right] \end{aligned}$$
  2. (3)

    \({\varvec{\varPi }}_{\mathrm {BD}}^{-\mathrm {T}}\) is expanded as

    $$\begin{aligned} {\varvec{\varPi }}_{\mathrm {BD}}^{-\mathrm {T}}&= \frac{1}{\left( b_1^2+b_2^2+b_3^2+1\right) ^2} \left[ \begin{array}{ccc} 4 \left( b_1^2-b_2^2-b_3^2+1\right) &{} 8 (b_1 b_2 + b_3) &{} -8 (b_2 - b_1 b_3) \\ 8 (b_1 b_2 - b_3) &{} -4 \left( b_1^2-b_2^2+b_3^2-1\right) &{} 8 (b_1 + b_2 b_3) \\ 8 (b_2 + b_1 b_3) &{} -8 (b_1 - b_2 b_3) &{} -4 \left( b_1^2+b_2^2-b_3^2-1\right) \\ \end{array} \right] \end{aligned}$$
  3. (4)

    The expansion of \({\varvec{\varGamma }}\) is

    $$\begin{aligned} {\varvec{\varGamma }} = \left[ \begin{array}{cc} {\varvec{\varGamma }}_1 &{} {\varvec{\varGamma }}_2\\ {\varvec{0}}_{4\times 4} &{} {\varvec{\varGamma }}_3 \end{array} \right] \end{aligned}$$

    where

    $$\begin{aligned} {\varvec{\varGamma _{1}}}&= \left[ \begin{array}{cccc} b_1^2-b_2^2-b_3^2+1 &{} 2 b_1 b_2+2 b_3 &{} 2 b_1 b_3-2 b_2 &{} 0\\ 2 b_1 b_2-2 b_3 &{} -b_1^2+b_2^2-b_3^2+1 &{} 2 b_1+2 b_2 b_3 &{} 0\\ 2 b_2+2 b_1 b_3 &{} 2 b_2 b_3-2 b_1 &{} -b_1^2-b_2^2+b_3^2+1 &{} 0\\ 0 &{} 0 &{} 0 &{} b_1^2+b_2^2+b_3^2+1 \end{array} \right] \\ {\varvec{\varGamma _{2}}}&= \left[ \begin{array}{cccc} 2 b_1 b_5-2 b_2 b_6-2 b_3 b_7 &{} 2 b_2 b_5+2 b_1 b_6+2 b_7 &{} 2 b_3 b_5-2 b_6+2 b_1 b_7 &{} 0\\ 2 b_2 b_5+2 b_1 b_6-2 b_7 &{} -2 b_1 b_5+2 b_2 b_6-2 b_3 b_7 &{} 2 b_5+2 b_3 b_6+2 b_2 b_7 &{} 0\\ 2 b_3 b_5+2 b_6+2 b_1 b_7 &{} -2 b_5+2 b_3 b_6+2 b_2 b_7 &{} -2 b_1 b_5-2 b_2 b_6+2 b_3 b_7 &{} 0\\ 0 &{} 0 &{} 0 &{} 2 b_1 b_5+2 b_2 b_6+2 b_3 b_7 \end{array} \right] \\ {\varvec{\varGamma _{3}}}&= \left[ \begin{array}{cccc} b_1^2-b_2^2-b_3^2+1 &{} 2 b_1 b_2+2 b_3 &{} 2 b_1 b_3-2 b_2 &{} 0\\ 2 b_1 b_2-2 b_3 &{} -b_1^2+b_2^2-b_3^2+1 &{} 2 b_1+2 b_2 b_3 &{} 0\\ 2 b_2+2 b_1 b_3 &{} 2 b_2 b_3-2 b_1 &{} -b_1^2-b_2^2+b_3^2+1 &{} 0\\ 0 &{} 0 &{} 0 &{} b_1^2+b_2^2+b_3^2+1 \end{array} \right] \end{aligned}$$

Appendix B: Reduction of \({\dot{V}}_1^\prime \) and \({\dot{V}}_2^\prime \)

Substituting the definitions of \({\varvec{P}}\) and the velocity motor into (31) and (32), one can write \({\dot{V}}_1^\prime \) and \({\dot{V}}_2^\prime \) as:

$$\begin{aligned} {\dot{V}}_1^\prime&=\, {\varvec{Q}}^\mathrm {T}\left[ p {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}+ \varepsilon p_{\varepsilon } \left( \dot{{\varvec{r}}}_\mathrm {BD}^\mathrm {B}+ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{r}}_\mathrm {BD}^\mathrm {B}\right) \right] ^s \\&\quad \circ \left\{ \left[ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}+ \varepsilon \left( \dot{{\varvec{r}}}_\mathrm {BD}^\mathrm {B}+ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{r}}_\mathrm {BD}^\mathrm {B}\right) \right] \right. \\&\quad \, \left. \times {\varvec{M}} \left[ {\varvec{\omega }}_{\mathrm {BI}}^{\mathrm {B}} + \varepsilon \left( \dot{{\varvec{r}}}_{\mathrm {BI}}^{\mathrm {B}} + {\varvec{\omega }}_{\mathrm {BI}}^{\mathrm {B}} \times {\varvec{r}}_{\mathrm {BI}}^{\mathrm {B}} \right) \right] \right\} \\ {\dot{V}}_2^\prime&=\, {\varvec{Q}}^\mathrm {T}\left[ p {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}+ \varepsilon p_{\varepsilon } \left( \dot{{\varvec{r}}}_\mathrm {BD}^\mathrm {B}+ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{r}}_\mathrm {BD}^\mathrm {B}\right) \right] ^s \\&\quad \circ \left\{ {\varvec{V}}_{\mathrm {DI}}^{\mathrm {B}} \times \left[ m \left( \dot{{\varvec{r}}}_\mathrm {BD}^\mathrm {B}+ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{r}}_\mathrm {BD}^\mathrm {B}\right) \right. \right. \\&\quad \,\, \left. \left. + \varepsilon J {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\right] \right. \\&\quad \, \left. - {\varvec{M}} \left\{ \left[ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}+ \varepsilon \left( \dot{{\varvec{r}}}_\mathrm {BD}^\mathrm {B}+ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{r}}_\mathrm {BD}^\mathrm {B}\right) \right] \times {\varvec{V}}_{\mathrm {DI}}^{\mathrm {B}} \right\} \right\} \end{aligned}$$

Let \({\varvec{\alpha }}=\dot{{\varvec{r}}}_\mathrm {BD}^\mathrm {B}+ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{r}}_\mathrm {BD}^\mathrm {B}\), \({\varvec{\beta }} = \dot{{\varvec{r}}}_{\mathrm {BI}}^{\mathrm {B}} + {\varvec{\omega }}_{\mathrm {BI}}^{\mathrm {B}} \times {\varvec{r}}_{\mathrm {BI}}^{\mathrm {B}}\), \({\varvec{V}}_{\mathrm {DI}}^{\mathrm {B}} = {\varvec{\zeta }} + \varepsilon {\varvec{\eta }}\), and then \({\dot{V}}_1^\prime \) and \({\dot{V}}_2^\prime \) can be reduced to

$$\begin{aligned} {\dot{V}}_1^\prime&=\, \left( p_{\varepsilon } q {\varvec{\alpha }} + \varepsilon p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\right) \\&\quad \circ \left[ \left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}+ \varepsilon {\varvec{\alpha }} \right) \times \left( m {\varvec{\beta }} + \varepsilon J {\varvec{\omega }}_{\mathrm {BI}}^{\mathrm {B}} \right) \right] \\&=\, \left( p_{\varepsilon } q {\varvec{\alpha }} + \varepsilon p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\right) \\&\quad \circ \left\{ m {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\beta }} + \varepsilon \left[ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times \left( J {\varvec{\omega }}_{\mathrm {BI}}^{\mathrm {B}} \right) + m {\varvec{\alpha }} \times {\varvec{\beta }} \right] \right\} \\&=\, m p_{\varepsilon } q {\varvec{\alpha }} \cdot \left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\beta }} \right) + p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\\&\quad \cdot \left[ {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times \left( J {\varvec{\omega }}_{\mathrm {BI}}^{\mathrm {B}} \right) + m {\varvec{\alpha }} \times {\varvec{\beta }} \right] \\&=\, m p_{\varepsilon } q {\varvec{\alpha }} \cdot \left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\beta }} \right) \\&\quad \,+ m p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\cdot \left( {\varvec{\alpha }} \times {\varvec{\beta }} \right) \\ {\dot{V}}_2^\prime&=\, \left( p_{\varepsilon } q {\varvec{\alpha }} + \varepsilon p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\right) \\&\quad \circ \left\{ m {\varvec{\zeta }} \times {\varvec{\alpha }} + \varepsilon \left[ {\varvec{\zeta }} \times \left( {\varvec{J}} {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\right) + m {\varvec{\eta }} \times {\varvec{\alpha }} \right] \right. \\&\quad \left. - \left[ m \left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\eta }} + {\varvec{\alpha }} \times {\varvec{\zeta }} \right) + \varepsilon {\varvec{J}}\left( {\varvec{\omega }} _\mathrm {BD}^\mathrm {B}\times {\varvec{\zeta }} \right) \right] \right\} \\&=\, m p_{\varepsilon } q {\varvec{\alpha }} \cdot \left( {\varvec{\zeta }} \times {\varvec{\alpha }} \right) + p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\\&\quad \cdot \left[ {\varvec{\zeta }} \times \left( {\varvec{J}} {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\right) + m {\varvec{\eta }} \times {\varvec{\alpha }} \right] \\&\quad - m p_{\varepsilon } q {\varvec{\alpha }} \cdot \left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\eta }} + {\varvec{\alpha }} \times {\varvec{\zeta }} \right) \\&\quad - p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\cdot {\varvec{J}}\left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\zeta }} \right) \\&=\, p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\cdot \left[ {\varvec{\zeta }} \times \left( {\varvec{J}} {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\right) \right] - m p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\cdot \left( {\varvec{\alpha }} \times {\varvec{\eta }} \right) \\&\quad - m p_{\varepsilon } q {\varvec{\alpha }} \cdot \left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\eta }} \right) - p q_{\varepsilon } {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\cdot {\varvec{J}} \left( {\varvec{\omega }}_\mathrm {BD}^\mathrm {B}\times {\varvec{\zeta }} \right) \end{aligned}$$

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Zhang, B., Cai, Y. Twistor-based pose control for asteroid landing with path constraints. Nonlinear Dyn 100, 2427–2448 (2020). https://doi.org/10.1007/s11071-020-05610-w

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