Systematic Design of Optimal Low-Thrust Transfers for the Three-Body Problem

Abstract

We develop a computational approach for the design of continuous low thrust transfers in the planar circular restricted three-body problem. The use of low thrust propulsion allows the spacecraft to depart from the natural dynamics and enables a wider range of transfers. We generate the reachable set of the spacecraft and use this to determine transfer opportunities, analogous to the intersection of control-free invariant manifolds. The reachable set is developed on a lower dimensional Poincaré section and used to design transfer trajectories. This is solved numerically as a discrete optimal control problem using a variational integrator, which preserves the geometric structure of the motion in the three-body problem. We demonstrate our approach with two numerical simulations of transfers in the Earth-Moon three-body system.

Appendix: Costate Equations of Motion

The development of the costate equations of motions begins with determining the second order partial derivatives of the gravitational potential. Due to the symmetry of partial derivatives only three terms are required and are given by

$$\begin{array}{@{}rcl@{}} U_{xx_k} &=& \left( {1-\mu}\right) \left[{\frac{1}{r_{1_k}^{3}} - \frac{3 \left( {x_k +\mu}\right)^{2}}{r_{1_k}^{5}}}\right] + \mu \left[{\frac{1}{r_{2_{k}}^{3}} - \frac{3 \left( {x_k -1 + \mu}\right)^{2}}{r_{2_{k}}^{5}}}\right] , \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} U_{yy_{k}} &=& \left( {1-\mu}\right) \left[{\frac{1}{r_{1_k}^{3}} - \frac{3 y_{k}^{2}}{r_{1_k}^{5}}}\right] + \mu \left[{\frac{1}{r_{2_{k}}^{3}} - \frac{3 y_{k}^{2}}{r_{2_{k}}^{5}}}\right] , \end{array} $$

(21)

$$\begin{array}{@{}rcl@{}} U_{xy_{k}} &=& U_{yx_k} = \frac{-3 \left( {1-\mu}\right) \left( {x_k +\mu}\right) y_{k}}{r_{1_k}^{3}} - \frac{3\mu y_{k}\left( {x_k-1+\mu}\right)}{r_{2_{k}}^{5}} . \end{array} $$

(22)

The gradient of Eq. 10a is given as

$$\begin{array}{@{}rcl@{}} f_{1_{x}} &=& \frac{1}{1+h^{2}} \left[{h^{2} + 1 + \frac{h^{2}}{2} -\frac{h^{3}}{2} U_{yx_k} - \frac{h^{2}}{2}U_{xx_k}}\right] , \end{array} $$

(23a)

$$\begin{array}{@{}rcl@{}} f_{1_{y}} &=& \frac{1}{1+h^{2}} \left[{ \frac{h^{3}}{2} -\frac{h^{3}}{2} U_{yy_{k}} - \frac{h^{2}}{2}U_{xy_{k}}}\right] , \end{array} $$

(23b)

$$\begin{array}{@{}rcl@{}} f_{1_{\dot{x}}} &= &\frac{h}{1+h^{2}} , \end{array} $$

(23c)

$$\begin{array}{@{}rcl@{}} f_{1_{\dot{y}}} &= &\frac{h^{2}}{1+h^{2}} . \end{array} $$

(23d)

The gradient of Eq. 10b is given as

$$\begin{array}{@{}rcl@{}} f_{2_{x}} &=& h -h f_{1_{x}} - \frac{h^{2}}{2} U_{yx_k} , \end{array} $$

(24a)

$$\begin{array}{@{}rcl@{}} f_{2_{y}} &=& -h f_{1_{y}} + 1 + \frac{h^{2}}{2} - \frac{h^{2}}{2} U_{yy_{k}} , \end{array} $$

(24b)

$$\begin{array}{@{}rcl@{}} f_{2_{\dot{x}}} &=& -h f_{1_{\dot{x}}} ,\end{array} $$

(24c)

$$\begin{array}{@{}rcl@{}} f_{2_{\dot{y}}} &=& h - h f_{1_{\dot{y}}} . \end{array} $$

(24d)

The gradients of Eqs. 12c and 12d are given as

$$\begin{array}{@{}rcl@{}} \frac{\partial {r_{1_k}p}}{\partial {\bar{x}}} &= &\left( {\left( x_{k + 1} + \mu\right)^{2} + y_{k}p^{2}}\right)^{-\frac{1}{2}} \left[{\left( {x_{k + 1} + \mu}\right) f_{1_{\bar{x}}} + y_{k}p f_{2_{\bar{x}}}}\right] , \end{array} $$

(25a)

$$\begin{array}{@{}rcl@{}} \frac{\partial {r_{2_{k}}p}}{\partial {\bar{x}}} & = & \left( {\left( x_{k + 1} - 1 + \mu\right)^{2} + y_{k}p^{2}}\right)^{-\frac{1}{2}} \left[{\left( {x_{k + 1} - 1 + \mu}\right) f_{1_{\bar{x}}} + y_{k}p f_{2_{\bar{x}}}}\right] . \end{array} $$

(25b)

The second order partial derivatives of the gravitational potential at k + 1 are given as

$$\begin{array}{@{}rcl@{}} \frac{\partial {U_{xx_{k + 1}}}}{\partial {\bar{x}}} &=& \left( {1-\mu}\right)\left[{\frac{1}{r_{1_k}p^{3}} f_{1_{\bar{x}}} - \frac{3 \left( {x_{k + 1} +mu}\right)}{r_{1_k}p^{4}} \frac{\partial {r_{1_k}p}}{\partial {\bar{x}}}}\right] \\&&+ \mu \left[{\frac{1}{r_{2_{k}}p^{3}} f_{1_{\bar{x}}} - \frac{-3 \left( {x_{k + 1} -1 + \mu}\right)}{r_{2_{k}}p^{4}} \frac{\partial {r_{2_{k}}p}}{\partial {\bar{x}}}}\right] , \end{array} $$

(26a)

$$\begin{array}{@{}rcl@{}} \frac{\partial {U_{yx_{k + 1}}}}{\partial {\bar{x}}} &=& \left( {1-\mu}\right)\left[{\frac{1}{r_{1_k}p^{3}} f_{2_{\bar{x}}} - \frac{3 y_{k}p}{r_{1_k}p^{4}} \frac{\partial {r_{1_k}p}}{\partial {\bar{x}}}}\right] \\&&+ \mu \left[{\frac{1}{r_{2_{k}}p^{3}} f_{2_{\bar{x}}} - \frac{-3 y_{k}p}{r_{2_{k}}p^{4}} \frac{\partial {r_{2_{k}}p}}{\partial {\bar{x}}}}\right] . \end{array} $$

(26b)

The gradient of Eqs. 10c and 10d are given as

$$\begin{array}{@{}rcl@{}} f_{3_{x}} &= &2 f_{2_{x}} + \frac{h}{2} \left( {f_{1_{x}} + 1}\right) - \frac{h}{2} U_{xx_{k + 1}} - \frac{h}{2} U_{xx_k} , \end{array} $$

(27a)

$$\begin{array}{@{}rcl@{}} f_{3_{y}} &= &-2 + 2 f_{2_{y}} + \frac{h}{2} f_{1_{y}} - \frac{h}{2} U_{xy_{k}p} -\frac{h}{2} U_{xy_{k}} , \end{array} $$

(27b)

$$\begin{array}{@{}rcl@{}} f_{3_{\dot{x}}} &=& 1 + 2 f_{2_{\dot{x}}} + \frac{h}{2} f_{1_{\dot{x}}} - \frac{h}{2} U_{x\dot{x}_{k + 1}} , \end{array} $$

(27c)

$$\begin{array}{@{}rcl@{}} f_{3_{\dot{y}}} &=& 2 f_{2_{\dot{y}}} , \end{array} $$

(27d)

$$\begin{array}{@{}rcl@{}} f_{4_{x}} &=& 2 - 2 f_{1_{x}} + \frac{h}{2} f_{2_{x}} - \frac{h}{2} U_{yx_{k + 1}} - \frac{h}{2} U_{yx_k} , \end{array} $$

(28a)

$$\begin{array}{@{}rcl@{}} f_{4_{y}} &=& -2f_{1_{y}} - \frac{h}{2} \left( {f_{2_{y}} + 1}\right) - \frac{h}{2} U_{yy_{k}p} -\frac{h}{2} U_{yy_{k}} , \end{array} $$

(28b)

$$\begin{array}{@{}rcl@{}} f_{4_{\dot{x}}} &=& - 2 f_{1_{\dot{x}}} + \frac{h}{2} f_{2_{\dot{x}}} - \frac{h}{2} U_{y\dot{x}_{k + 1}} , \end{array} $$

(28c)

$$\begin{array}{@{}rcl@{}} f_{4_{\dot{y}}} &=& 1 - 2 f_{1_{\dot{y}}} + \frac{h}{2} f_{2_{\dot{y}}} - \frac{h}{2} U_{y\dot{y}_{k + 1}} . \end{array} $$

(28d)

These gradient equations are in a cascade type structure. Equations 27a–d and 28a–d are functions of Eqs. 10c and 10d. As a result, the accuracy of the Jacobian will tend to decrease as the first order approximation errors accumulate.