Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory

Meng, Yazhe; Li, Hengnian; Jiang, Fanghua

doi:10.1007/s10509-020-3721-y

Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory

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Published: 10 January 2020

Volume 365, article number 8, (2020)
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Abstract

This paper presents a fail-safe polynomial-based method for determining the coast-terminating zero (CTZ) of fuel-optimal, time-fixed, low-thrust trajectories. The CTZ of the fuel-optimal bang-bang control is decided by the zeros of the switching function (SF). The state and costate differential equations along the coast arcs of the optimal trajectory have closed-form solutions; thus efficiently determining the CTZ is important. Existing methods suffer potential failures or are limited to time-free problems. The polynomial-based method consists of three steps. First, the SF is simplified by a coordinate transformation that eliminates the three orientation-related orbital elements. Then, the SF is rewritten as a combination of power and trigonometric functions of the argument eccentric anomaly, and the coefficients in the SF are simply evaluated using the states and costates at the periapsis. Finally, the SF is described as a polynomial of the eccentric anomaly by replacing the trigonometric functions with polynomials. The roots of the polynomial can provide an accurate estimation of the CTZ. Based on a Monte Carlo simulation, SF curves whose greatest possible numbers of zeros are $1,2,\ldots,7$ are listed. For each curve, the CTZ is solved by both the presented polynomial-based method and the existing sampling-searching method. The performance comparison demonstrates that the polynomial-based method provides superior reliability and efficiency.

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Change history

21 February 2020
In the original article, there is an error on page 4 regarding coordinate transformation which is corrected here. This error does not influence the results and contributions of the manuscript. The contents of page 4 except Eq. (15) and the two paragraphs immediately before it are replaced by the following:
21 February 2020
In the original article, there is an error on page 4 regarding coordinate transformation which is corrected here. This error does not influence the results and contributions of the manuscript. The contents of page 4 except Eq. (15) and the two paragraphs immediately before it are replaced by the following:

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant 11672146).

Author information

Authors and Affiliations

School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China
Yazhe Meng & Fanghua Jiang
The State Key Laboratory of Astronautic Dynamics, China Xi’an Satellite Control Center, Xi’an, 710043, China
Hengnian Li

Authors

Yazhe Meng
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Hengnian Li
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Fanghua Jiang
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Correspondence to Fanghua Jiang.

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Appendices

Appendix A: $a_{i}$ and $b_{j}$ in Eq. (23) and Eq. (24)

For Eq. (23), the coefficients of $\sin E$ and $\cos E$ are given in Eq. (A.1) and Eq. (A.3) if $E \in [-\pi ,-\pi /2 ]$, Eq. (A.1) and Eq. (A.4) and if $E \in [-\pi /2,0 ]$, Eq. (A.2) and Eq. (A.4) if $E \in [0, \pi /2 ]$, and Eq. (A.2) and Eq. (A.5) if $E \in [\pi /2,\pi ]$.

$$ \left [ { \textstyle\begin{array}{c} {{a_{0}}} \\ {{a_{1}}} \\ {{a_{2}}} \\ {{a_{3}}} \\ {{a_{4}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} 0 \\ {{S_{12}}\, {S_{13}}} \\ {{S_{12}}\, {S_{14}} - {S_{11}}\, {S^{2}_{13}}} \\ { - 2\, {S_{11}}\, {S_{13}}\, {S_{14}}} \\ { - {S_{11}}\, {S^{2}_{14}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} 0 \\ {0.9868} \\ { - 0.0507} \\ { - 0.2322} \\ { - 0.0370} \end{array}\displaystyle } \right ] $$

(A.1)

$$ \left [ { \textstyle\begin{array}{c} {{a_{0}}} \\ {{a_{1}}} \\ {{a_{2}}} \\ {{a_{3}}} \\ {{a_{4}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} {\mathrm{{0}}} \\ {{S_{12}}\, {S_{13}}} \\ {{S_{11}}\, {S^{2}_{13}} - {S_{12}}\, {S_{14}}} \\ { - 2\, {S_{11}}\, {S_{13}}\, {S_{14}}} \\ {{S_{11}}\, {S^{2}_{14}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} {\mathrm{{0}}} \\ { - \mathrm{{0}}{\mathrm{{.9868}}}} \\ {\mathrm{{0}}{\mathrm{{.0507}}}} \\ { - \mathrm{{0}}{\mathrm{{.2322}}}} \\ {\mathrm{{0}}{\mathrm{{.0370}}}} \end{array}\displaystyle } \right ] $$

(A.2)

$$\begin{aligned} &\left [ { \textstyle\begin{array}{c} {{b_{0}}} \\ {{b_{1}}} \\ {{b_{2}}} \\ {{b_{3}}} \\ {{b_{4}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} {\frac{{{\pi ^{2}} ( {{S_{12}}\, {S_{14}} - {S_{11}}\, {S^{2}_{13}}} )}}{4} - \frac{{{S_{11}}\, {S^{2}_{14}}\, {\pi ^{4}}}}{{16}} + \frac{{\pi \, {S_{12}}\, {S_{13}}}}{2} - \frac{{{S_{11}}\, {S_{13}}\, {S_{14}}\, {\pi ^{3}}}}{4}} \\ {{S_{12}}\, {S_{13}} + \pi ( {{S_{12}} \, {S_{14}} - {S_{11}}\, {S^{2}_{13}}} ) - \frac{{{S_{11}}\, {S^{2}_{14}}\, {\pi ^{3}}}}{2} - \frac{{3\, {S_{11}}\, {S_{13}}\, {S_{14}}\, {\pi ^{2}}}}{2}} \\ { - {S_{11}}\, {S^{2}_{13}} - 3\, {S_{11}}\, \pi \, {S_{13}}\, {S_{14}} - \frac{{3\, {S_{11}}\, {\pi ^{2}}\, {S^{2}_{14}}}}{2} + {S_{12}}\, {S_{14}}} \\ { - 2\, {S_{11}}\, \pi \, {S^{2}_{14}} - 2 \, {S_{11}}\, {S_{13}}\, {S_{14}}} \\ { - {S_{11}}\, {S^{2}_{14}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} {0.3000} \\ { - 1.4642} \\ { - 1.6921} \\ { - 0.4644} \\ { - 0.0370} \end{array}\displaystyle } \right ] \end{aligned}$$

(A.3)

$$\begin{aligned} &\left [ { \textstyle\begin{array}{c} {{b_{0}}} \\ {{b_{1}}} \\ {{b_{2}}} \\ {{b_{3}}} \\ {{b_{4}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} {\frac{{{S_{11}}\, {S^{2}_{14}}\, {\pi ^{4}}}}{{16}} - \frac{{{\pi ^{2}} ( {{S_{12}}\, {S_{14}} - {S_{11}}\, {S^{2}_{13}}} )}}{4} + \frac{{\pi \, {S_{12}}\, {S_{13}}}}{2} - \frac{{{S_{11}}\, {S_{13}}\, {S_{14}}\, {\pi ^{3}}}}{4}} \\ {{S_{12}}\, {S_{13}} - \pi ( {{S_{12}} \, {S_{14}} - {S_{11}}\, {S^{2}_{13}}} ) + \frac{{{S_{11}}\, {S^{2}_{14}}\, {\pi ^{3}}}}{2} - \frac{{3\, {S_{11}}\, {S_{13}}\, {S_{14}}\, {\pi ^{2}}}}{2}} \\ {{S_{11}}\, {S^{2}_{13}} - 3\, {S_{11}}\, \pi \, {S_{13}}\, {S_{14}} + \frac{{3\, {S_{11}}\, {\pi ^{2}}\, {S^{2}_{14}}}}{2} - {S_{12}}\, {S_{14}}} \\ {2\, \pi \, {S_{11}}\, {S^{2}_{14}} - 2 \, {S_{11}}\, {S_{13}}\, {S_{14}}} \\ {{S_{11}}\, {S^{2}_{14}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} {\mathrm{{1}}.\mathrm{{0}}000} \\ \mathrm{{0}} \\ { - \mathrm{{0}}{\mathrm{{.4965}}}} \\ \mathrm{{0}} \\ { - 0.0370} \end{array}\displaystyle } \right ] \end{aligned}$$

(A.4)

$$\begin{aligned} & \left [ { \textstyle\begin{array}{c} {{b_{0}}} \\ {{b_{1}}} \\ {{b_{2}}} \\ {{b_{3}}} \\ {{b_{4}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} {\frac{{{\pi ^{2}} ( {{S_{12}}\, {S_{14}} - {S_{11}}\, {S^{2}_{13}}} )}}{4} - \frac{{{S_{11}}\, {S^{2}_{14}}\, {\pi ^{4}}}}{{16}} + \frac{{\pi \, {S_{12}}\, {S_{13}}}}{2} - \frac{{{S_{11}}\, {S_{13}}\, {S_{14}}\, {\pi ^{3}}}}{4}} \\ {\frac{{{S_{11}}\, {S^{2}_{14}}\, {\pi ^{3}}}}{2} - \pi ( {{S_{12}}\, {S_{14}} - {S_{11}}\, {S^{2}_{13}}} ) - {S_{12}}\, {S_{13}} + \frac{{3\, {S_{11}}\, {S_{13}}\, {S_{14}}\, {\pi ^{2}}}}{2}} \\ { - {S_{11}}\, {S^{2}_{13}} - 3\, {S_{11}}\, \pi \, {S_{13}}\, {S_{14}} - \frac{{3\, {S_{11}}\, {\pi ^{2}}\, {S^{2}_{14}}}}{2} + {S_{12}}\, {S_{14}}} \\ {2\, {S_{11}}\, \pi \, {S^{2}_{14}} + 2 \, {S_{11}}\, {S_{13}}\, {S_{14}}} \\ { - {S_{11}}\, {S^{2}_{14}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} {0.3000} \\ {1.4642} \\ { - 1.6921} \\ {0.4644} \\ { - 0.0370} \end{array}\displaystyle } \right ] \end{aligned}$$

(A.5)

For Eq. (24), the coefficients of $\sin E$ and $\cos E$ are given in Eq. (A.6) and Eq. (A.8) if $E \in [-\pi ,-\pi /2 ]$, Eq. (A.7) and Eq. (A.9) and if $E \in [\pi /2,3\pi /2 ]$.

$$ \left [ { \textstyle\begin{array}{c} {{a_{0}}} \\ {{a_{1}}} \\ {{a_{2}}} \\ {{a_{3}}} \\ {{a_{4}}} \\ {{a_{5}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} 0 \\ {\frac{{2\, {S_{11}}}}{\pi }} \\ 0 \\ { - \frac{{8\, {S_{12}}}}{{{\pi ^{3}}}}} \\ 0 \\ {\frac{{32\, {S_{13}}}}{{{\pi ^{5}}}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} 0 \\ {0.9993} \\ 0 \\ { - 0.1650} \\ 0 \\ {7.2904 \times {{10}^{ - 3}}} \end{array}\displaystyle } \right ] $$

(A.6)

$$ \left [ { \textstyle\begin{array}{c} {{a_{0}}} \\ {{a_{1}}} \\ {{a_{2}}} \\ {{a_{3}}} \\ {{a_{4}}} \\ {{a_{5}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} {8\, {S_{12}} - 2\, {S_{11}} - 32\, {S_{13}}} \\ {\frac{{2\, {S_{11}}}}{\pi } - \frac{{24\, {S_{12}}}}{\pi } + \frac{{160\, {S_{13}}}}{\pi }} \\ {\frac{{24\, {S_{12}}}}{{{\pi ^{2}}}} - \frac{{320\, {S_{13}}}}{{{\pi ^{2}}}}} \\ {\frac{{320\, {S_{13}}}}{{{\pi ^{3}}}} - \frac{{8\, {S_{12}}}}{{{\pi ^{3}}}}} \\ { - \frac{{160\, {S_{13}}}}{{{\pi ^{4}}}}} \\ {\frac{{32\, {S_{13}}}}{{{\pi ^{5}}}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} {0.2549} \\ {0.3349} \\ {0.7055} \\ { - 0.5545} \\ {0.1145} \\ { - 0.7290 \times {{10}^{ - 3}}} \end{array}\displaystyle } \right ] $$

(A.7)

$$ \left [ { \textstyle\begin{array}{c} {{b_{0}}} \\ {{b_{1}}} \\ {{b_{2}}} \\ {{b_{3}}} \\ {{b_{4}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} {{C_{11}}} \\ 0 \\ { - \frac{{4\, {C_{12}}}}{{{\pi ^{2}}}}} \\ 0 \\ {\frac{{16\, {C_{13}}}}{{{\pi ^{4}}}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} {1.0000} \\ 0 \\ { - 0.4966} \\ 0 \\ {0.0370} \end{array}\displaystyle } \right ] $$

(A.8)

$$ \left [ { \textstyle\begin{array}{c} {{b_{0}}} \\ {{b_{1}}} \\ {{b_{2}}} \\ {{b_{3}}} \\ {{b_{4}}} \end{array}\displaystyle } \right ] = \left [ { \textstyle\begin{array}{c} {{C_{11}} - 4\, {C_{12}} + 16\, {C_{13}}} \\ {\frac{{8\, {C_{12}}}}{\pi } - \frac{{64\, {C_{13}}}}{\pi }} \\ {\frac{{96\, {C_{13}}}}{{{\pi ^{2}}}} - \frac{{4\, {C_{12}}}}{{{\pi ^{2}}}}} \\ { - \frac{{64\, {C_{13}}}}{{{\pi ^{3}}}}} \\ {\frac{{16\, {C_{13}}}}{{{\pi ^{4}}}}} \end{array}\displaystyle } \right ] \approx \left [ { \textstyle\begin{array}{c} {0.2958} \\ {1.4705} \\ { - 1.6953} \\ {0.4651} \\ { - 0.0370} \end{array}\displaystyle } \right ] $$

(A.9)

Appendix B: $M_{\mathit{CE}}^{k}$ and ${C_{k}^{\mathrm{{den}}}}$ in Eq. (19)

The nonzero components of $M_{\mathit{CE}}^{k}$ and ${C_{k}^{\mathrm{ {den}}}}$ are:

$$\begin{aligned} &M_{\mathit{CE}}^{1} ( 6,6 )=36e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e+1 )}^{2}},\\ &M_{\mathit{CE}}^{2} ( 6,6 )=36{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e+1 )}^{2}},\\ &M_{\mathit{CE}}^{3} ( 6,6 )=-24{{e}^{2}} {{ \bigl( 1- {{e}^{2}} \bigr)}^{3/2}} {{ ( e+1 )}^{2}},\\ &M_{\mathit{CE}}^{4} ( 3,6 )=48{{ \bigl( 1-{{e}^{2}} \bigr)} ^{7/2}},\\ &M_{\mathit{CE}}^{4} ( 6,6 )=-24\sqrt{1-{{e}^{2}}} {{ ( e+1 )}^{2}} \bigl( - {{e}^{3}}-4{{e}^{2}}+e+4 \bigr),\\ &M_{\mathit{CE}}^{5} ( 1,6 )=-48ae{{ \bigl( {{e}^{2}}-1 \bigr)} ^{4}}, \\ &M_{\mathit{CE}}^{5} ( 2,6 )=48{{ \bigl( {{e}^{2}}-1 \bigr)} ^{4}}, \qquad M_{\mathit{CE}}^{6} ( 1,6 )=48a{{ \bigl( {{e}^{2}}-1 \bigr)} ^{4}},\\ &M_{\mathit{CE}}^{7} ( 1,3 )=-16a{{e}^{2}} {{ \bigl( {{e} ^{2}}-1 \bigr)}^{2}} {{ ( e-1 )}^{3}} ( e+1 ), \\ &M_{\mathit{CE}}^{7} ( 1,6 )=-32a{{e}^{2}} {{ \bigl( {{e} ^{2}}-1 \bigr)}^{2}} {{ ( e-1 )}^{2}} ( e+1 ), \\ &M_{\mathit{CE}}^{7} ( 2,3 )=8e{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{3}} ( e+1 ), \\ &M_{\mathit{CE}}^{7} ( 2,6 )=16e{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{2}} ( e+1 ), \\ &M_{\mathit{CE}}^{7} ( 4,5 )=-2e{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{2}}, \\ &M_{\mathit{CE}}^{8} ( 1,3 )=16ae{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{3}} \bigl( -{{e}^{3}}-{{e}^{2}}+3e+3 \bigr), \\ &M_{\mathit{CE}}^{8} ( 1,6 )=16ae{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{2}} ( e+1 ) \bigl( {{e}^{2}}+3e+6 \bigr), \\ &M_{\mathit{CE}}^{8} ( 2,3 )=-8{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{3}} \bigl( -{{e}^{3}}-{{e}^{2}}+3e+3 \bigr), \\ & \begin{aligned} M_{\mathit{CE}}^{8} ( 2,6 )&=- ( 16e-16 ) {{ ( e-1 )}^{2}} {{ ( e+1 )}^{3}} \\ &\quad {}\times \bigl( {{e} ^{3}}+{{e}^{2}}+e-3 \bigr), \end{aligned} \\ &M_{\mathit{CE}}^{8} ( 4,5 )=2{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} \bigl( {{e}^{2}}+1 \bigr){{ ( e-1 )}^{2}}, \\ &M_{\mathit{CE}}^{9} ( 1,3 )=-16a{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{3}} \bigl( -{{e}^{3}}-{{e}^{2}}+2e+2 \bigr), \\ &M_{\mathit{CE}}^{9} ( 1,6 )=-16a{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{2}} ( e+1 ) \bigl( {{e}^{2}}+3e+4 \bigr), \\ &M_{\mathit{CE}}^{9} ( 2,3 )=8e{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{3}} ( e+1 ), \\ &M_{\mathit{CE}}^{9} ( 2,6 )=-16{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{2}} ( e+1 ), \\ &M_{\mathit{CE}}^{9} ( 4,5 )=-2e{{ \bigl( {{e}^{2}}-1 \bigr)} ^{2}} {{ ( e-1 )}^{2}}, \\ & M_{\mathit{CE}}^{10} ( 1,1 )=-16{{a}^{2}} {{e}^{3}} {{ \bigl( 1-{{e}^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )}^{2}},\\ &M_{\mathit{CE}}^{10} ( 1,2 )=16a{{e}^{2}} {{ \bigl( 1- {{e}^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )} ^{2}}, \\ &M_{\mathit{CE}}^{10} ( 2,2 )=-4e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )}^{2}}, \\ &M_{\mathit{CE}}^{10} ( 3,3 )=-4e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} ( e+1 ),\\ &M_{\mathit{CE}}^{10} ( 3,6 )=-16e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{2}} ( e+1 ),\\ &M_{\mathit{CE}}^{10} ( 4,4 )=-e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{2}}, \\ &M_{\mathit{CE}}^{10} ( 5,5 )=-e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} ( e+1 ),\\ & \begin{aligned} M_{\mathit{CE}}^{10} ( 6,6 )&=4e\sqrt{1-{{e}^{2}}} ( e+1 ) \bigl( {{e}^{5}}+{{e}^{4}} \\ &\quad {}+3{{e}^{3}}-5{{e}^{2}}-4e+4 \bigr), \end{aligned} \\ & M_{\mathit{CE}}^{11} ( 1,1 )=48{{a}^{2}} {{e}^{2}} {{ \bigl( 1-{{e}^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )}^{2}}, \\ & M_{\mathit{CE}}^{11} ( 1,2 )=-48ae{{ \bigl( 1-{{e}^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )} ^{2}}, \\ & M_{\mathit{CE}}^{11} ( 2,2 )=12{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )}^{2}}, \\ & M_{\mathit{CE}}^{11} ( 3,3 )=4{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} \bigl( -2{{e}^{3}}-2{{e}^{2}}+3e+3 \bigr),\\ & \begin{aligned} M_{\mathit{CE}}^{11} ( 3,6 )&=16\sqrt{1-{{e}^{2}}} {{ ( e-1 )}^{2}} ( e+1 ) \\ &\quad {}\times\bigl( -2{{e}^{3}}-3 {{e}^{2}}+2e+3 \bigr), \end{aligned} \\ & M_{\mathit{CE}}^{11} ( 4,4 )={{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} \bigl( 2{{e}^{2}}+1 \bigr){{ ( e-1 )}^{2}},\\ & M_{\mathit{CE}}^{11} ( 5,5 )={{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} ( e+1 ), \\ & \begin{aligned} M_{\mathit{CE}}^{11} ( 6,6 )&=4\sqrt{1-{{e}^{2}}} ( e+1 ) \bigl( {{e}^{5}}+{{e}^{4}}+3{{e}^{3}} \\ &\quad {}+11{{e}^{2}}-4e-12 \bigr), \end{aligned} \\ & M_{\mathit{CE}}^{12} ( 1,1 )=-48{{a}^{2}}e{{ \bigl( 1- {{e}^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )} ^{2}},\\ & M_{\mathit{CE}}^{12} ( 1,2 )=16a{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} \bigl( {{e}^{2}}+2 \bigr){{ ( e-1 )}^{4}} {{ ( e+1 )}^{2}},\\ & M_{\mathit{CE}}^{12} ( 2,2 )=-12e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )}^{2}}, \\ & M_{\mathit{CE}}^{12} ( 3,3 )=4e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} \bigl( -{{e}^{3}}-{{e}^{2}}+2e+2 \bigr), \\ & M_{\mathit{CE}}^{12} ( 3,6 )=16{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{2}} \bigl( {{e}^{3}}+4{{e}^{2}}+4e+1 \bigr), \\ & M_{\mathit{CE}}^{12} ( 4,4 )=-e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} \bigl( {{e}^{2}}+2 \bigr){{ ( e-1 )}^{2}}, \\ & M_{\mathit{CE}}^{12} ( 5,5 )=e{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} ( e+1 ), \\ & \begin{aligned} M_{\mathit{CE}}^{12} ( 6,6 )&=-4\sqrt{1-{{e}^{2}}} ( e+1 ) \bigl( {{e}^{6}}+{{e}^{5}}-5{{e}^{4}}-21{{e}^{3}} \\ &\quad {}-4 {{e}^{2}}+20e+8 \bigr), \end{aligned} \\ & M_{\mathit{CE}}^{13} ( 1,1 )=16{{a}^{2}} {{ \bigl( 1- {{e}^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )} ^{2}}, \\ & M_{\mathit{CE}}^{13} ( 1,2 )=-16ae{{ \bigl( 1-{{e}^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )} ^{2}}, \\ & M_{\mathit{CE}}^{13} ( 2,2 )=4{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{4}} {{ ( e+1 )}^{2}}, \\ & M_{\mathit{CE}}^{13} ( 3,3 )=-4{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} \bigl( -3{{e}^{3}}-3{{e}^{2}}+4e+4 \bigr), \\ & M_{\mathit{CE}}^{13} ( 3,6 )=-16{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{2}} ( e+1 ){{ ( e+2 )} ^{2}}, \\ & M_{\mathit{CE}}^{13} ( 4,4 )={{e}^{2}} {{ \bigl( 1-{{e} ^{2}} \bigr)}^{3/2}} {{ ( e-1 )}^{2}} , \\ & M_{\mathit{CE}}^{13} ( 5,5 )=-{{ \bigl( 1-{{e}^{2}} \bigr)} ^{3/2}} {{ ( e-1 )}^{3}} ( e+1 ), \\ & \begin{aligned} M_{\mathit{CE}}^{13} ( 6,6 )&=-4\sqrt{1-{{e}^{2}}} ( e+1 ) \bigl( {{e}^{5}}+9{{e}^{4}}+19{{e}^{3}} \\ &\quad {}+11{{e} ^{2}}-20e-20 \bigr), \end{aligned} \\ &M_{\mathit{CE}}^{l}=p{{e}^{2}} {{ ( 1+e )}^{9/2}} {{ ( 1-e )}^{5/2}}M_{\mathit{CE}}^{l},l=1,2 \cdots 13,\\ &C_{8}^{\text{den}}=-eC,C_{13}^{\text{den}}=C, where C=4{{e}^{2}} {{ \bigl( 1-{{e}^{2}} \bigr)}^{8}}. \end{aligned}$$

Appendix C: $\dfrac{{d S}}{{d E}}$

$$\begin{aligned} \dfrac{{dS}}{{dE}} &= - \dfrac{1}{m} \dfrac{{d \Vert {{\mathbf{{M}}^{T}}{\boldsymbol{\lambda }}} \Vert }}{{dE}} \\ & = - \dfrac{1}{m} \dfrac{{{{\boldsymbol{\lambda }}^{T}}{\mathbf{{M}}}}}{{ \Vert {{\mathbf{{M}}^{T}}{\boldsymbol{\lambda }}} \Vert }} \dfrac{{d ( {{\mathbf{{M}}^{T}}{\boldsymbol{\lambda }}} )}}{{dE}} \\ & = - \dfrac{1}{m} \dfrac{{{{\boldsymbol{\lambda }}^{T}}{\mathbf{{M}}}}}{{ \Vert {{\mathbf{{M}}^{T}}{\boldsymbol{\lambda }}} \Vert }} \biggl[ {{ \dfrac{{d ( \mathbf{{M}} )}}{{dE}}}^{T}} {\boldsymbol{\lambda }} + { \mathbf{{M}}^{T}}\dfrac{{d{\boldsymbol{\lambda }}}}{{dE}} \biggr]. \end{aligned}$$

(C.1)

The non-zero terms in ${\dfrac{{\partial {\mathbf{{M}}}}}{{\partial E}}}$ are:

$$\begin{aligned} &\dfrac{{\partial {\mathbf{{M}}} ( {1,2} )}}{{\partial E}} = - \dfrac{{2e\sqrt{{p^{3}}} \sin L}}{{\sqrt{1 - {e^{2}}} w}}, \qquad \dfrac{{\partial {\mathbf{{M}}} ( {2,1} )}}{{\partial E}} = \dfrac{{\sqrt{p} \cos L \, w}}{{\sqrt{1 - {e^{2}}} }}, \\ &\dfrac{{\partial {\mathbf{{M}}} ( {2,2} )}}{{\partial E}} = -\dfrac{{\sqrt{p} \sin L ( {{e^{2}} \cos ^{2}{{ L }} - {e^{2}} + 2\, e \cos L + 2} )}}{{\sqrt{1 - {e^{2}}} \, w}}, \\ &\dfrac{{\partial {\mathbf{{M}}} ( {3,1} )}}{{\partial E}} =\dfrac{{\sqrt{p} \sin L \, w}}{{\sqrt{1 - {e^{2}}} }}, \\ &\dfrac{{\partial {\mathbf{{M}}} ( {3,2} )}}{{\partial E}} = \dfrac{{\sqrt{p} ( {{e^{2}} \cos ^{3} {{L}} + 2\, e \cos ^{2} {{ L }} + e + 2 \cos L} )}}{{\sqrt{1 - {e^{2}}} \, w}}, \\ &\dfrac{{\partial {\mathbf{{M}}} ( {4,3} )}}{{\partial E}} = - \dfrac{{\sqrt{p} \sin L }}{{2\, \sqrt{1 - {e^{2}}} \, w}}, \\ &\dfrac{{\partial {\mathbf{{M}}} ( {5,3} )}}{{\partial E}} = \dfrac{{\sqrt{p} ( {e + \cos L } ) }}{{2\, \sqrt{1 - {e^{2}}} \, w}}, \\ &\dfrac{{d{\boldsymbol{\lambda }}}}{{dE}} = \dfrac{1}{{\sqrt{1 - {e^{2}}} }} \left ( \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&0&0&0&0&{ - \frac{{3 ( {e \cos L + 1} )}}{{2\, p}}} \\ 0&0&0&0&0&{2 \cos L} \\ 0&0&0&0&0&{2 \sin L} \\ 0&0&0&0&0&0 \\ 0&0&0&0&0&0 \\ 0&0&0&0&0&{ - 2\, e \sin L } \end{array}\displaystyle \right )^{T}{\boldsymbol{\lambda }}, \end{aligned}$$

where $w = 1+e\cos L$.

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Meng, Y., Li, H. & Jiang, F. Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory. Astrophys Space Sci 365, 8 (2020). https://doi.org/10.1007/s10509-020-3721-y

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Received: 16 August 2019
Accepted: 02 January 2020
Published: 10 January 2020
DOI: https://doi.org/10.1007/s10509-020-3721-y

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Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory

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Deadzone-Quadratic Penalty Function for Predictive Extended Cruise Control with Experimental Validation

Efficient optimization method for variable-specific-impulse low-thrust trajectories with shutdown constraint

Determination of Periodic Trajectories of Dynamic Systems Subject to Switching Input Constraints

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21 February 2020

21 February 2020

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Appendices

Appendix A: \(a_{i}\) and \(b_{j}\) in Eq. (23) and Eq. (24)

Appendix B: \(M_{\mathit{CE}}^{k}\) and \({C_{k}^{\mathrm{{den}}}}\) in Eq. (19)

Appendix C: \(\dfrac{{d S}}{{d E}}\)

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Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory

Abstract

Access this article

Similar content being viewed by others

Deadzone-Quadratic Penalty Function for Predictive Extended Cruise Control with Experimental Validation

Efficient optimization method for variable-specific-impulse low-thrust trajectories with shutdown constraint

Determination of Periodic Trajectories of Dynamic Systems Subject to Switching Input Constraints

Change history

21 February 2020

21 February 2020

References

Acknowledgements

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Authors and Affiliations

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Publisher’s Note

Appendices

Appendix A: \(a_{i}\) and \(b_{j}\) in Eq. (23) and Eq. (24)

Appendix B: \(M_{\mathit{CE}}^{k}\) and \({C_{k}^{\mathrm{{den}}}}\) in Eq. (19)

Appendix C: \(\dfrac{{d S}}{{d E}}\)

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