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Nonlinear high-fidelity modeling of spacecraft relative motion via orbit element differences

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Abstract

The ultimate need to design and develop high-fidelity dynamical models for future space missions is necessitated by the continuous and enormous interest in spacecraft formation flying, rendezvous and spacecraft proximity operations. In this paper, to obtain high-fidelity dynamics, higher-order relative motion model is developed via nonlinear mapping of orbit element differences and Hill coordinates. First, second-order variation of parameter technique of calculus of variations is applied to the direction cosine matrix (DCM), which maps vector components in inertial frame to vector components in Deputy Hill frame, and deputy spacecraft inertial position and velocity vectors in Deputy Hill frame. Second, after series of transformations and elimination of higher-order terms greater than quadratic terms, new, nonlinearly mapped radial, along-track and cross-track relative motion position and velocity equations are obtained. Using the new equation of motion, nonlinear state space model is developed. The new equations, validated via numerical simulations, are amenable for the analysis of spacecraft relative motion, formation flying, rendezvous and proximity operations in both circular and elliptical orbits.

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Author information

Authors and Affiliations

  1. Engineering and Space Systems Department, National Space Research and Development Agency, Abuja, Nigeria

    Ayansola D. Ogundele & Olufemi A. Agboola

  2. Department of Electronic and Electrical Engineering, Ladoke Akintola University of Technology (LAUTECH), Ogbomoso, Nigeria

    Olasunkanmi F. Oseni

Authors
  1. Ayansola D. Ogundele
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  2. Olufemi A. Agboola
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  3. Olasunkanmi F. Oseni
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Contributions

ADO: conceptualization, methodology, writing—original draft, writing—review and editing, literature search, software, formal analysis and validation. OAA: conceptualization, methodology, writing—review and editing, software and formal analysis. OFO: methodology, review and editing, software, formal analysis and validation.

Corresponding author

Correspondence to Ayansola D. Ogundele.

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Conflict of interest

The authors have no financial or personal relationship with other people or organizations that could inappropriately influence or bias the content of the paper. There is no conflict of interest with other people or organizations that could inappropriately influence or bias the content of the paper.

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The research meets all applicable standards with regards to the ethics of experimentation and research integrity. The following is being certified/declared true.As an expert scientist, the paper has been submitted with full responsibility following due ethical procedure and there is no duplicate publication, fraud, plagiarism, or concerns about animal or human experimentation.

Additional information

This is the revised version of the paper AAS 21-341 presented at 31st AAS/AIAA Space Flight Mechanics Meeting. The Sheraton Charlotte Hotel in Charlotte, North Carolina. February 1–4, 2021.

Appendix A: Normalized nonlinear terms of the nonlinear mapped relative motion orbit element differences

Appendix A: Normalized nonlinear terms of the nonlinear mapped relative motion orbit element differences

$$\begin{aligned} {{\tilde{F}}_1}= & {} - \left( {\frac{{4{q_1}}}{p}} \right) \delta a\delta {q_1} - \left( {\frac{{4{q_2}}}{p}} \right) \delta a\delta {q_2}\nonumber \\&+ \frac{{2R}}{{{p^4}}}\left\{ {{p^3}\sin \theta - \frac{{{V_r}}}{{{V_t}}}\left( {{R^2}\left( {2a{q_1} + R\cos \theta } \right) + {p^3}\cos \theta } \right) } \right\} \nonumber \\&\times \delta \theta \delta {q_1}\nonumber \\&- \frac{{2R}}{{{p^4}}}\left\{ {{p^3}\cos \theta + \frac{{{V_r}}}{{{V_t}}}\left( {{R^2}\left( {2a{q_2} + R\sin \theta } \right) + {p^3}\sin \theta } \right) } \right\} \nonumber \\&\times \delta \theta \delta {q_2}\nonumber \\&- \left( {2\cos i} \right) \delta \theta \delta \varOmega + \left( {\sin i\sin 2\theta } \right) \delta i\delta \varOmega \nonumber \\&+ \frac{{2{R^4}}}{{{p^5}}}\left\{ {\left( {2a{q_1} + R\cos \theta } \right) \sin \theta + \left( {2a{q_2} + R\sin \theta } \right) \cos \theta } \right\} \nonumber \\&\times \delta {q_1}\delta {q_2}\nonumber \\&+ \frac{1}{p}\left\{ {p\left( {2{{\left( {\frac{{{V_r}}}{{{V_t}}}} \right) }^2} - 1} \right) + R\left( {{q_1}\cos \theta + {q_2}\sin \theta } \right) } \right\} \delta {\theta ^2}\nonumber \\&- \frac{1}{2}\left( {1 - \cos 2\theta } \right) \delta {i^2} \nonumber \\&+ \frac{2}{{{p^5}}}\left\{ { - a{p^4} + {R^4}\left( {2a{q_1} + R\cos \theta } \right) \cos \theta } \right\} \delta q_1^2\nonumber \\&+ \frac{2}{{{p^5}}}\left\{ { - a{p^4} + {R^4}\left( {2a{q_2} + R\sin \theta } \right) \sin \theta } \right\} \delta q_2^2 \nonumber \\&- \left( {1 - {{\sin }^2}i{{\sin }^2}\theta } \right) \delta {\varOmega ^2}\nonumber \\ {{{{\tilde{F}}}}_2}= & {} \left( {\frac{1}{a}} \right) \delta a\delta \theta + \left( {\frac{{\cos i}}{a}} \right) \delta a\delta \varOmega - \frac{1}{p}\left( {2a{q_1} + R\cos \theta } \right) \delta \theta \delta {q_1}\nonumber \\&- \frac{1}{p}\left( {2a{q_2} + R\sin \theta } \right) \delta \theta \delta {q_2} + \left( {\frac{{{V_r}\cos i}}{{{V_t}}}} \right) \delta \theta \delta \varOmega \nonumber \\&- \left( {2\sin i{{\sin }^2}\theta } \right) \delta i\delta \varOmega \nonumber \\&- \frac{{\cos i}}{p}\left( {2a{q_1} + R\cos \theta } \right) \delta \varOmega \delta {q_1}\nonumber \\&- \frac{{\cos i}}{p}\left( {2a{q_2} + R\sin \theta } \right) \delta \varOmega \delta {q_2} + \left( {\frac{{{V_r}}}{{{V_t}}}} \right) \delta {\theta ^2}\nonumber \\&- \frac{1}{2}\left( {\sin 2\theta } \right) \delta {i^2} + \frac{1}{2}\sin 2\theta \left( {1 - {{\cos }^2}i} \right) \delta {\varOmega ^2}\nonumber \\ {{{{\tilde{F}}}}_3}= & {} \left( {\frac{{\sin \theta }}{a}} \right) \delta a\delta i \nonumber \\&- \frac{1}{a}\left( {\sin i\cos \theta } \right) \delta a\delta \varOmega + \left\{ {\left( {\frac{{{V_r}}}{{{V_t}}}} \right) \sin \theta + 2\cos \theta } \right\} \delta \theta \delta i\nonumber \\&- \sin i\left\{ {\left( {\frac{{{V_r}}}{{{V_t}}}} \right) \cos \theta - 2\sin \theta } \right\} \delta \theta \delta \varOmega \nonumber \\&- \frac{{\sin \theta }}{p}\left( {2a{q_1} + R\cos \theta } \right) \delta i\delta {q_1}\nonumber \\&- \frac{{\sin \theta }}{p}\left( {2a{q_2} + R\sin \theta } \right) \delta i\delta {q_2} \nonumber \\&+ \frac{1}{p}\sin i\cos \theta \left( {2a{q_1} + R\cos \theta } \right) \delta \varOmega \delta {q_1}\nonumber \\&+ \frac{1}{p}\sin i\cos \theta \left( {2a{q_2} + R\sin \theta } \right) \delta \varOmega \delta {q_2} \nonumber \\&+ \frac{1}{2}\left( {\sin 2i\sin \theta } \right) \delta {\varOmega ^2} \nonumber \\ \end{aligned}$$
(40)

and

$$\begin{aligned} {{{{\tilde{F}}}}_4}= & {} \frac{1}{{2ap}}\left( {p + 2R} \right) \delta a\delta \theta + \frac{1}{{ap{V_t}}}\left( {3{V_r}a{q_1} - h\sin \theta } \right) \delta a\delta {q_1}\nonumber \\&+ \frac{1}{{ap{V_t}}}\left( {3{V_r}a{q_2} + h\cos \theta } \right) \delta a\delta {q_2} + \left( {\frac{{3\cos i}}{{2a}}} \right) \delta a\delta \varOmega \nonumber \\&- \frac{1}{{{p^4}V_t^2}}\left\{ {{p^2}V_t^2a{q_1}\left( {p + 2R} \right) } \right. \nonumber \\&\left. { + 2R{V_r}\left( {{p^3}\left( {{V_t}\sin \theta - {V_r}\cos \theta } \right) - {R^2}{V_r}\left( {2a{q_1} + R\cos \theta } \right) } \right) } \right\} \nonumber \\&\times \delta \theta \delta {q_1}\nonumber \\&- \frac{1}{{{p^4}V_t^2}}\left\{ {{p^2}V_t^2a{q_2}\left( {p + 2R} \right) } \right. \nonumber \\&\left. { - 2R{V_r}\left( {{p^3}\left( {{V_t}\cos \theta + {V_r}\sin \theta } \right) + {R^2}{V_r}\left( {2a{q_2} + R\sin \theta } \right) } \right) } \right\} \nonumber \\&\times \delta \theta \delta {q_2}\nonumber \\&+ \frac{{{V_r}}}{{{V_t}}}\left( {2\cos i} \right) \delta \theta \delta \varOmega + \frac{1}{{{V_t}}}\left( {2{V_t}\sin i\cos 2\theta } \right) \delta i\delta \varOmega \nonumber \\&+ \frac{1}{{{V_t}{p^5}}}\left\{ {2a{p^3}\left( {3{V_r}a{q_1}{q_2} - h\left( {{q_1}\cos \theta - {q_2}\sin \theta } \right) } \right) } \right. \nonumber \\&\left. { - 2{V_r}{R^4}\left( {\left( {2a{q_1} + R\cos \theta } \right) \sin \theta + \left( {2a{q_2} + R\sin \theta } \right) \cos \theta } \right) } \right\} \nonumber \\&\quad \delta {q_1}\delta {q_2}\nonumber \\&- \frac{1}{p}\cos i\left( {3a{q_1} + 2R\cos \theta } \right) \delta \varOmega \delta {q_1} \nonumber \\&- \frac{1}{p}\cos i\left( {3a{q_2} + 2R\sin \theta } \right) \nonumber \\&\times \delta \varOmega \delta {q_2}\nonumber \\&+ \left( {\frac{{3{V_r}}}{{4{V_t}{a^2}}}} \right) \delta {a^2} \nonumber \\&- \frac{{{V_r}}}{{p{V_t}}}\left\{ {2p{{\left( {\frac{{{V_r}}}{{{V_t}}}} \right) }^2} + R\left( {{q_1}\cos \theta + {q_2}\sin \theta } \right) - p} \right\} \delta {\theta ^2}\nonumber \\&+ \frac{1}{{2{V_t}}}\left\{ { - 2\left( {{V_r}{{\sin }^2}\theta + {V_t}\sin 2\theta } \right) + {V_r}\left( {1 - \cos 2\theta } \right) } \right\} \delta {i^2}\nonumber \\&+ \frac{1}{{{V_t}{p^5}}}\left\{ {a{p^3}\left( {{V_r}\left( {p + 3aq_1^2} \right) + 2h{q_1}\sin \theta } \right) } \right. \nonumber \\&\left. { - 2{V_r}\left( { - a{p^4} + {R^4}\left( {2a{q_1} + R\cos \theta } \right) \cos \theta } \right) } \right\} \delta q_{1}^{2}\nonumber \\&+ \frac{1}{{{V_t}{p^5}}}\left\{ {a{p^3}\left( {{V_r}\left( {p + 3aq_2^2} \right) - 2h{q_2}\cos \theta } \right) } \right. \nonumber \end{aligned}$$
$$\begin{aligned}&\left. { - 2{V_r}\left( { - a{p^4} + {R^4}\left( {2a{q_2} + R\sin \theta } \right) \sin \theta } \right) } \right\} \delta q_2^2 \nonumber \\&+ \left( {{{\sin }^2}i\sin 2\theta } \right) \delta {\varOmega ^2}\nonumber \\ {{{{\tilde{F}}}}_5}= & {} - \left( {\frac{{{V_r}}}{{2a{V_t}}}} \right) \delta a\delta \theta + \frac{1}{{ap}}\left( {3a{q_1} - R\cos \theta } \right) \delta a\delta {q_1} \nonumber \\&+ \frac{1}{{ap}}\left( {3a{q_2} - R\sin \theta } \right) \delta a\delta {q_2}\nonumber \\&- \left( {\frac{{3{V_r}\cos i}}{{2a{V_t}}}} \right) \delta a\delta \varOmega \nonumber \\&+ \frac{1}{{{V_t}{p^4}}}\left\{ {{p^3}\left( {{V_r}\left( {2R\cos \theta - a{q_1}} \right) - 3h\sin \theta } \right) } \right. \nonumber \\&\left. { + {V_r}\left( {2a{q_1} + R\cos \theta } \right) \left( {2{R^3} + {p^3}} \right) } \right\} \delta \theta \delta {q_1}\nonumber \\&+ \frac{1}{{{V_t}{p^4}}}\left\{ {{p^3}\left( {{V_r}\left( {2R\sin \theta - a{q_2}} \right) + 3h\cos \theta } \right) } \right. \nonumber \\&\left. { + {V_r}\left( {2a{q_2} + R\sin \theta } \right) \left( {2{R^3} + {p^3}} \right) } \right\} \delta \theta \delta {q_2}\nonumber \\&- \frac{{\cos i}}{p}\left\{ {p{{\left( {\frac{{{V_r}}}{{{V_t}}}} \right) }^2} + R - p} \right\} \delta \theta \delta \varOmega \nonumber \\&- \left( {2\sin i\sin 2\theta } \right) \delta i\delta \varOmega + \frac{1}{{{p^5}}}\left\{ {6a{p^4}{q_1}{q_2} - 2{R^5}\sin 2\theta } \right. \nonumber \\&\left. { + 2Ra\left( {{p^3} - 2{R^3}} \right) \left( {{q_2}\cos \theta + {q_1}\sin \theta } \right) } \right\} \delta {q_1}\delta {q_2}\nonumber \\&+ \frac{{\cos i}}{{p{V_t}}}\left\{ {3{V_r}a{q_1} + R\left( {{V_t}\sin \theta + {V_r}\cos \theta } \right) } \right\} \delta \varOmega \delta {q_1}\nonumber \\&+ \frac{{\cos i}}{{p{V_t}}}\left\{ {3{V_r}a{q_2} + R\left( {{V_r}\sin \theta - {V_t}\cos \theta } \right) } \right\} \delta \varOmega \delta {q_2}\nonumber \\&+ \left( {\frac{3}{{4{a^2}}}} \right) \delta {a^2} - \frac{1}{p}\left\{ {3p{{\left( {\frac{{{V_r}}}{{{V_t}}}} \right) }^2} + R\left( {{q_1}\cos \theta + {q_2}\sin \theta } \right) } \right\} \nonumber \\&\times \delta {\theta ^2}\nonumber \\&- \left( {\cos 2\theta } \right) \delta {i^2}\nonumber \\&+ \frac{1}{{{p^5}}}\left\{ 3a{p^4}\left( {1 + q_1^2} \right) + 2R\cos \theta \left( a{q_1}\left( {{p^3} - 2{R^3}} \right) \right. \right. \nonumber \\&\quad \left. \left. - {R^4}\cos \theta \right) \right\} \delta q_1^2\nonumber \\&+ \frac{1}{{{p^5}}}\left\{ 3a{p^4}\left( {1 + q_2^2} \right) + 2R\sin \theta \left( a{q_2}\left( {{p^3} - 2{R^3}} \right) \right. \right. \nonumber \\&\left. \left. - {R^4}\sin \theta \right) \right\} \delta q_2^2\nonumber \\&+ \left( {{{\sin }^2}i\cos 2\theta } \right) \delta {\varOmega ^2}\nonumber \\ {{{{\tilde{F}}}}_6}= & {} - \frac{1}{{2a{V_t}}}\left( {{V_t}\cos \theta + 3{V_r}\sin \theta } \right) \delta a\delta i \nonumber \\&+ \frac{{\sin i}}{{2a{V_t}}}\left( {3{V_r}\cos \theta - {V_t}\sin \theta } \right) \delta a\delta \varOmega \nonumber \\&- \frac{1}{{2pV_t^2}}\left\{ {2p\sin \theta \left( {2V_t^2 + V_r^2} \right) + 2p{V_r}{V_t}\cos \theta } \right. \nonumber \\&\left. { - h{V_t}\left( {{q_1}\sin 2\theta + 2{q_2}{{\sin }^2}\theta } \right) } \right\} \delta \theta \delta i \nonumber \\&+ \frac{{\sin i}}{{2pV_t^2}}\left\{ {2p\cos \theta \left( {2V_t^2 + V_r^2} \right) } \right. \nonumber \\&\left. { - 2p{V_r}{V_t}\sin \theta - h{V_t}\left( {2{q_1}{{\cos }^2}\theta + {q_2}\sin 2\theta } \right) } \right\} \delta \theta \delta \varOmega \nonumber \\&+ \frac{1}{{2p{V_t}}}\left\{ {{V_r}\left( {R\sin 2\theta + 6a{q_1}\sin \theta } \right) + 2{V_t}\left( {R + a{q_1}\cos \theta } \right) } \right\} \delta i\delta {q_1}\nonumber \\&+ \frac{1}{{p{V_t}}}\left\{ {R{V_r}{{\sin }^2}\theta + a{q_2}\left( {3{V_r}\sin \theta + {V_t}\cos \theta } \right) } \right\} \delta i\delta {q_2}\nonumber \\&- \frac{{\sin i}}{{p{V_t}}}\left\{ {R{V_r}{{\cos }^2}\theta + a{q_1}\left( {3{V_r}\cos \theta - {V_t}\sin \theta } \right) } \right\} \delta \varOmega \delta {q_1}\nonumber \\&- \frac{{\sin i}}{{2p{V_t}}}\left\{ {{V_r}\left( {R\sin 2\theta + 6a{q_2}\cos \theta } \right) - 2{V_t}\left( {R + a{q_2}\sin \theta } \right) } \right\} \delta \varOmega \delta {q_2}\nonumber \\&+ \frac{1}{2}\left( {\sin 2i\cos \theta } \right) \delta {\varOmega ^2} \end{aligned}$$
(41)

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Ogundele, A.D., Agboola, O.A. & Oseni, O.F. Nonlinear high-fidelity modeling of spacecraft relative motion via orbit element differences. AS 5, 591–605 (2022). https://doi.org/10.1007/s42401-022-00155-9

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