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Application of Lagrangian coherent structures to Coulomb formation on elliptic orbit

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Abstract

Coulomb thrusting presents attractive prospects in many astronautical missions from the standpoint of close-proximity formation flying. In this study, a dynamical model of Coulomb formation and its Hamiltonian is derived on the basis of the Tschauner–Hempel equation. As a nonautonomous system, it is more complicated than the dynamical model based on the Clohessy–Wiltshire equation. The Lagrangian coherent structure (LCS), a useful tool for describing the dynamical behavior of nonautonomous systems, is used to study the Coulomb formation dynamics, design, and reconfiguration. Simulation results show that, in the autonomous case of Coulomb formation, the LCS coincides with the invariant manifolds in the proper Poincaré section. When it is extended to the nonautonomous case with a small-eccentricity reference orbit, the global morphology of the Coulomb formation dynamics remains almost unchanged. Based on the property that the LCS can act as the transport barrier in dynamical systems, it is used to construct homoclinic and heteroclinic orbits and search for invariant relative orbits and transfer trajectories for formation design and reconfiguration.

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Acknowledgements

The research is supported by the National Natural Science Foundation of China (11772024, 11432001), and the Fundamental Research Funds for the Central Universities (YWF-16-BJ-Y-10).

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Authors and Affiliations

  1. School of Astronautics, Beihang University, Beijing, 100191, China

    Mingpei Lin, Yaru Zheng & Ming Xu

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  1. Mingpei Lin
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Correspondence to Ming Xu.

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Appendix

Appendix

The Jacobian matrix of Eq. (11) is given by

$$ \varvec{J} = \left[ {\begin{array}{*{20}c} {\mathcal{0}_{3 \times 3} } & {\mathcal{I}_{3 \times 3} } \\ {\mathcal{J}_{21} } & {\mathcal{J}_{22} } \\ \end{array} } \right],\;\mathcal{J}_{21} = \frac{1}{{\left( {1 + e_{0} \cos f_{0} } \right)}}\left[ {\begin{array}{*{20}c} { - W_{xx} } & { - W_{xy} } & { - W_{xz} } \\ { - W_{xy} } & { - W_{yy} } & { - W_{yz} } \\ { - W_{xz} } & { - W_{yz} } & { - W_{zz} } \\ \end{array} } \right],\;\mathcal{J}_{22} = \left[ {\begin{array}{*{20}c} 0 & 2 & 0 \\ { - 2} & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right] $$

where

$$ \begin{aligned} W_{xx} & = - 1 + \frac{1}{{\bar{r}_{i}^{3} }} - \frac{{3\left( {1 + \bar{x}} \right)^{2} }}{{\bar{r}_{i}^{5} }} + \sum\limits_{i = 1}^{n} {\frac{1}{{\bar{r}_{si}^{3} {\text{e}}^{{\bar{r}_{si} /\bar{\lambda }_{\text{d}} }} }}\left( {1 + \frac{{\bar{r}_{i} }}{{\bar{\lambda }_{\text{d}} }} + \frac{{\left( {\bar{x} - \bar{x}_{i} } \right)^{2} }}{{\bar{\lambda }_{\text{d}} \bar{r}_{si} }} - \frac{{\left( {1 + {{\bar{r}_{si} } \mathord{\left/ {\vphantom {{\bar{r}_{si} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)\left( {3 + {{\bar{r}_{i} } \mathord{\left/ {\vphantom {{\bar{r}_{i} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)}}{{\bar{r}_{si}^{2} }}\left( {\bar{x} - \bar{x}_{i} } \right)^{2} } \right)} \\ W_{yy} & = - 1 + \frac{1}{{\bar{r}_{i}^{3} }} - \frac{{3\bar{y}^{2} }}{{\bar{r}_{i}^{5} }} + \sum\limits_{i = 1}^{n} {\frac{1}{{\bar{r}_{si}^{3} {\text{e}}^{{r_{si} /\lambda_{\text{d}} }} }}\left( {1 + \frac{{\bar{r}_{i} }}{{\bar{\lambda }_{\text{d}} }} + \frac{{\left( {\bar{y} - \bar{y}_{i} } \right)^{2} }}{{\bar{\lambda }_{\text{d}} r_{si} }} - \frac{{\left( {1 + {{\bar{r}_{si} } \mathord{\left/ {\vphantom {{\bar{r}_{si} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)\left( {3 + {{\bar{r}_{i} } \mathord{\left/ {\vphantom {{\bar{r}_{i} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)}}{{r_{si}^{2} }}\left( {\bar{y} - \bar{y}_{i} } \right)^{2} } \right)} \\ W_{zz} & = {\text{e}}_{0} \cos f_{0} + \frac{1}{{\bar{r}_{i}^{3} }} - \frac{{3\bar{z}^{2} }}{{\bar{r}_{i}^{5} }} + \sum\limits_{i = 1}^{n} {\frac{1}{{\bar{r}_{si}^{3} {\text{e}}^{{r_{si} /\lambda_{\text{d}} }} }}\left( {1 + \frac{{\bar{r}_{i} }}{{\bar{\lambda }_{\text{d}} }} + \frac{{\left( {\bar{z} - \bar{z}_{i} } \right)^{2} }}{{\bar{\lambda }_{\text{d}} \bar{r}_{si} }} - \frac{{\left( {1 + {{\bar{r}_{si} } \mathord{\left/ {\vphantom {{\bar{r}_{si} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)\left( {3 + {{\bar{r}_{i} } \mathord{\left/ {\vphantom {{\bar{r}_{i} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)}}{{r_{si}^{2} }}\left( {\bar{z} - \bar{z}_{i} } \right)^{2} } \right)} \\ W_{xy} & = - \frac{{3\left( {1 + \bar{x}} \right)\bar{y}}}{{\bar{r}_{i}^{5} }} + \sum\limits_{i = 1}^{n} {\frac{1}{{\bar{r}_{si}^{3} {\text{e}}^{{\bar{r}_{si} /\bar{\lambda }_{d} }} }}\left( {\frac{{\left( {\bar{x} - \bar{x}_{i} } \right)\left( {\bar{y} - \bar{y}_{i} } \right)}}{{\bar{\lambda }_{\text{d}} \bar{r}_{si} }} - \frac{{\left( {1 + {{\bar{r}_{si} } \mathord{\left/ {\vphantom {{\bar{r}_{si} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)\left( {3 + {{\bar{r}_{i} } \mathord{\left/ {\vphantom {{\bar{r}_{i} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)}}{{\bar{r}_{si}^{2} }}\left( {\bar{x} - \bar{x}_{i} } \right)\left( {\bar{y} - \bar{y}_{i} } \right)} \right)} \\ W_{xz} & = - \frac{{3\left( {1 + \bar{x}} \right)\bar{z}}}{{\bar{r}_{i}^{5} }} + \sum\limits_{i = 1}^{n} {\frac{1}{{\bar{r}_{si}^{3} {\text{e}}^{{\bar{r}_{si} /\bar{\lambda }_{d} }} }}\left( {\frac{{\left( {\bar{x} - \bar{x}_{i} } \right)\left( {\bar{z} - \bar{z}_{i} } \right)}}{{\bar{\lambda }_{\text{d}} \bar{r}_{si} }} - \frac{{\left( {1 + {{\bar{r}_{si} } \mathord{\left/ {\vphantom {{\bar{r}_{si} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)\left( {3 + {{\bar{r}_{i} } \mathord{\left/ {\vphantom {{\bar{r}_{i} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)}}{{\bar{r}_{si}^{2} }}\left( {\bar{x} - \bar{x}_{i} } \right)\left( {\bar{z} - \bar{z}_{i} } \right)} \right)} \\ W_{yz} & = - \frac{{3\bar{y}\bar{z}}}{{\bar{r}_{i}^{5} }} + \sum\limits_{i = 1}^{n} {\frac{1}{{\bar{r}_{si}^{3} {\text{e}}^{{\bar{r}_{si} /\bar{\lambda }_{d} }} }}\left( {\frac{{\left( {\bar{y} - \bar{y}_{i} } \right)\left( {\bar{z} - \bar{z}_{i} } \right)}}{{\bar{\lambda }_{\text{d}} \bar{r}_{si} }} - \frac{{\left( {1 + {{\bar{r}_{si} } \mathord{\left/ {\vphantom {{\bar{r}_{si} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)\left( {3 + {{\bar{r}_{i} } \mathord{\left/ {\vphantom {{\bar{r}_{i} } {\bar{\lambda }_{\text{d}} }}} \right. \kern-0pt} {\bar{\lambda }_{\text{d}} }}} \right)}}{{\bar{r}_{si}^{2} }}\left( {\bar{y} - \bar{y}_{i} } \right)\left( {\bar{z} - \bar{z}_{i} } \right)} \right)} \\ \end{aligned} $$

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Lin, M., Zheng, Y. & Xu, M. Application of Lagrangian coherent structures to Coulomb formation on elliptic orbit. Nonlinear Dyn 102, 2649–2668 (2020). https://doi.org/10.1007/s11071-020-05968-x

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