## 4 Trajectory-Planning Method
### Calculation of Fuel Consumption during Transfer (TCFC)
\begin{array}{l}K_{\varphi 1}=\frac{V_{\text{ocf}}-F_{2}\left(t_{ \text{g}}\right)\sin\overline{\varphi}_{\text{ocf}}\cos\overline{\psi}_{\text {ocf}}-\frac{1}{2}\bar{y}_{\text{ocf}}t_{\text{g}}^{2}-V_{\text{ocf}}t_{\text{ g}}-V_{\text{ocf}}}{\left(-F_{2}\left(t_{\text{g}}\right)F_{1}\left(t_{\text{g}} \right)+F_{3}\left(t_{\text{g}}\right)F_{0}\left(t_{\text{g}}\right)\right) \cos\overline{\varphi}_{\text{ocf}}\cos\overline{\psi}_{\text{ocf}}}\\ K_{\varphi 2}=\frac{\left(V_{\text{ocf}}-F_{2}\left(t_{\text{g}}\right)\sin \overline{\varphi}_{\text{ocf}}\cos\overline{\psi}_{\text{ocf}}-\frac{1}{2} \bar{y}_{\text{ocf}}t_{\text{g}}^{2}-V_{\text{ocf}}t_{\text{g}}-V_{\text{ocf }}t_{\text{g}}\right)\cdot F_{0}\left(t_{\text{g}}\right)}{\left(-F_{2}\left(t _{\text{g}}\right)F_{1}\left(t_{\text{g}}\right)+F_{3}\left(t_{\text{g}}\right) F_{0}\left(t_{\text{g}}\right)\right)\cos\overline{\varphi}_{\text{ocf}}\cos \overline{\psi}_{\text{ocf}}}\\ K_{\psi 1}=\frac{Z_{\text{ocf}}-\frac{1}{2}g_{\text{ocf}}t_{\text{g}}^{2}-V_{ \text{ocf}}t_{\text{g}}-Z_{\text{ocf}}t_{\text{g}}+\frac{Z_{\text{ocf}}}{ \left(t_{\text{g}}\right)F_{0}\left(t_{\text{g}}\right)}\cos\overline{\psi} _{\text{ocf}}}\ K_{\psi 2}=\frac{\left(Z_{\text{ocf}}-\frac{1}{2}g_{\text{ocf}}t_{\text{g}}^{2}-V_{ \text{ocf}}t_{\text{g}}-Z_{\text{ocf}}t_{\text{g}}\right)\cdot F_{0}\left(t_{ \text{g}}\right)}{\left(F_{2}\left(t_{\text{g}}\right)F_{1}\left(t_{\text{g}} \right)-F_{3}\left(t_{\text{g}}\right)F_{0}\left(t_{\text{g}}\right)\right) \cos\overline{\psi}_{\text{ocf}}}\end{array} \right.\right. \tag{15}\]  
\[\left\{ \begin{array}{l}t_{\text{h}}=\frac{m}{m}\,F_{0}(t)=\int_{0}^{t}\frac{ l_{\text{hp}}}{t_{\text{h}}-l}\text{d}t=I_{\text{sp}}\ln\frac{t_{\text{h}}}{t_{ \text{h}}-l}\,F_{1}(t)=\int_{0}^{t}\frac{l_{\text{hp}}}{t_{\text{h}}-l}t\, \text{d}t=t_{\text{h}}F_{0}(t)-I_{\text{sp}}\cdot t\\ F_{2}(t)=\int_{0}^{t}\int_{0}^{s}\frac{l_{\text{hp}}}{t_{\text{h}}-l}\text{d}t \text{ds}=F_{0}(t)\cdot t-F_{1}(t)\,F_{3}(t)=\int_{0}^{t}\int_{0}^{s}\frac{l_{ \text{hp}}}{t_{\text{h}}-l}t\text{d}t\text{ds}=F_{2}(t)\cdot t_{\text{h}}-l^{2} I_{\text{sp}}/2\end{array} \right. \tag{16}\]  
Therefore, the predicted terminal position deviation \(DX\) can be obtained from the target point position \(X_{\text{ocf}}\) in the \(X\) direction and the current point state (\(X_{\text{ocf0}}\), \(V_{\text{xcf0}}\)). \[ \begin{array}{lcl}DX&=&X_{\text{ocf}}-X_{\text{ocf0}}-V_{\text{xcf0}}\cdot t _{\text{g}}-\frac{1}{2}\bar{y}_{\text{scf}}t_{\text{g}}^{2}-F_{2}\left(t_{\text{ g}}\right)\cos\overline{\varphi}_{\text{ocf}}\cos\overline{\psi}_{\text{ocf}}\\ &&-F_{2}\left(t_{\text{g}}\right)K_{\varphi 1}\sin\overline{\varphi}_{\text{ocf}}\cos \overline{\psi}_{\text{ocf}}+F_{3}\left(t_{\text{g}}\right)K_{\varphi 2}\sin\overline{\varphi}_{\text{ocf}} \cos\overline{\psi}_{\text{ocf}}\end{array}  \tag{17}\]  
The ATU-IGM updates the position of the injection point to eliminate the terminal position deviation. The original target point is marked as \(A\), and the updated target point is marked as \(C\). The parallel line of the \(OY\) axis is made through the point \(C\) and intersects with the semimajor axis at the point \(B\). The auxiliary line perpendicular to the \(OY\) axis is made through the point \(B\) and intersects the \(OY\) axis at the point \(D\). The initial true anomaly is \(\angle BOD=\theta_{0}\); \(\Delta\theta\) is the change amount of the true anomaly before and after updating.