Decoupling and quasi-linearization methods for boundary value problems in relative orbital mechanics

Abstract

Accurate and real-time orbit computational methods are highly required in space tasks of the space vehicles. A decoupling method and a quasi-linearization method are proposed to efficiently and accurately solve relative orbit transfer problems. The orbit transfer problem is transformed into a group of orbit propagation problems that are much easier to solve. A novel local variational iteration method is provided for orbit propagation. Relative orbit transfer problems based on the Tschauner–Hempel model are solved by using the decoupling method without initial guess and iteration. The proposed quasi-linear local variational iteration method, which combines the quadratic convergence of quasi-linearization and the high computational performance of the local variational iteration method, is successfully applied to solve two types of strongly nonlinear relative orbit transfer problems. Comparisons are made with the Newton’s shooting and finite-difference-based solver. The QLVIM can derive far more accurate solutions while consuming much less computing time. Simulations also manifest that the proposed method covers longer solvable time span; i.e., it shows better convergency. The initial guess of the QLVIM is easier to obtain. Rather than a refined initial guess of the initial velocity, the proposed method only needs a rough trajectory connecting the boundary positions. It is also very convenient to extend the proposed methods to other engineering boundary value problems.

Appendices

Appendix A

See Table

Table 9 Expressions of basis functions

9.

In Sect. 2.3, ${\tilde{\varvec{R}}}$ is a row rearranging matrix that moves the ($dM + m$)th row of unit matrix to the($mD + d$)th row, where $d = 1,2,...,D,{\kern 1pt}$$m = 1,2,...,M$, D is the dimension of governing system, M is the number of collocation nodes. The details of ${{\varvec{\Phi}}}(t)$, ${\dot{\varvec{\Phi }}}(t)$, $\int_{{t_{0} }}^{t} {{{\varvec{\Phi}}}(\tau )} d\tau$ are listed in Table 9.

Appendix B

The motion of the chief satellite $S_{0}$ can be described by a set of equations as

$$ \left\{ \begin{gathered} \dot{r} = v_{x} \hfill \\ \dot{v}_{x} = - \frac{\mu }{{r^{2} }} + \frac{{h^{2} }}{{r^{3} }} - \frac{{k_{J2} }}{{r^{4} }}(1 - 3s_{i}^{2} s_{\theta }^{2} ) \hfill \\ \dot{h} = - \frac{{k_{J2} s_{i}^{2} s_{2\theta } }}{{r^{3} }} \hfill \\ \dot{\theta } = \frac{h}{{r^{2} }} + \frac{{2k_{J2} c_{i}^{2} s_{\theta }^{2} }}{{hr^{3} }} \hfill \\ \dot{i} = - \frac{{k_{J2} s_{2i} s_{2\theta } }}{{2hr^{3} }} \hfill \\ \end{gathered} \right. $$

Appendix C

The variables $r,\dot{r},h,\omega_{x} ,\omega_{z} ,i,\theta$ are related to the reference orbit and described by:

$$ \left\{ {\begin{array}{*{20}l} {\ddot{r} = - \frac{\mu }{{r^{2} }} + \omega_{z}^{2} r + d_{c,x} } \hfill \\ {\dot{h} = rd_{c,y} } \hfill \\ {\dot{\omega }_{x} = \frac{1}{h}(\dot{r}d_{c,x} + r\dot{d}_{c,z} - \omega_{x} rd_{c,y} )} \hfill \\ {\dot{\omega }_{z} = \frac{1}{r}(d_{c,y} - 2\omega_{z} \dot{r})} \hfill \\ {\dot{i} = \frac{{rc_{\theta } }}{h}d_{c,z} } \hfill \\ {\dot{\theta } = \frac{h}{{r^{2} }} - \frac{{rs_{\theta } c_{i} }}{{hs_{i} }}d_{c,z} } \hfill \\ \end{array} ,\quad \begin{array}{*{20}l} {d_{c,x} = d_{{J_{2} ,cx}} + d_{a,cx}} \hfill \\ {d_{c,y} = d_{{J_{2} ,cy}} + d_{a,cy} } \hfill \\ {d_{c,z} = d_{{J_{2} ,cz}} + d_{a,cz} } \hfill \\ {\dot{d}_{c,z} = \dot{d}_{{J_{2} ,cz}} + \dot{d}_{a,cz} } \hfill \\ {\Delta d_{x} = \Delta d_{{J_{2} ,x}} + \Delta d_{a,x} } \hfill \\ \begin{gathered} \Delta d_{y} = \Delta d_{{J_{2} ,y}} + \Delta d_{a,y} \hfill \\ \Delta d_{z} = \Delta d_{{J_{2} ,z}} + \Delta d_{a,z} \hfill \\ \end{gathered} \hfill \\ \end{array} } \right. $$

The variables with subscripts $J_{2}$ and $a$ are related to the zonal harmonic perturbations and atmospheric drag, respectively. Due to limited space, more details are not shown here. Readers interested may consult the reference [29].