Statistical Analysis of Optimal Stationkeeping Location and Coast Duration Using Stretching Directions

Abstract

The methodologies for orbit maintenance continue to develop as the number of cislunar space missions grows. Many of the studies on stationkeeping within cislunar space assume a fixed number of evenly spaced correction maneuvers or plan maneuvers based on the control methodology and mission constraints. From these studies, the maneuver location and coast duration have been identified as essential design parameters that affect the cost of fuel consumption and aid in understanding when and where maneuvers should be placed. The Cauchy–Green tensor allows the analysis of combinations of these design parameters in the form of stability maps. This manuscript derives a modified Cauchy–Green tensor for one and two impulsive stationkeeping maneuvers. The stability maps obtained from the modified Cauchy–Green tensors are used to determine combinations of maneuver location and coast duration that minimize fuel consumption and the final position and velocity errors. In addition, a guidance policy, dependent on uncertainties and an initial state perturbation, is derived from the two-impulse stationkeeping analysis to determine whether performing a corrective maneuver is more beneficial to the overall system’s performance. The solutions obtained from this analysis yield a tool for mission design that is dependent mainly on the problem dynamics, navigation, and thrust uncertainties.

Appendices

Appendix A: Jacobian Matrix for the CR3BP

$$\begin{aligned} F({\textbf{X}}) = \begin{bmatrix} {\dot{x}} \\ \\ {\dot{y}}\\ \\ {\dot{z}} \\ \\ x - (1-\mu )\frac{(x-x_{1})}{\rho _{1}^{3}} - \mu \frac{(x-x_{2})}{\rho _{2}^{3}} + 2{\dot{y}} \\ \\ (1 - \frac{1-\mu }{\rho _{1}^{3}} - \frac{\mu }{\rho _{2}^{3}})y - 2{\dot{x}} \\ \\ -(\frac{1-\mu }{\rho _{1}^{3}} + \frac{\mu }{\rho _{2}^{3}})z \end{bmatrix} \end{aligned}$$

(A1)

$$\begin{aligned} {\textbf{A}}(t) = \frac{\partial F({\textbf{X}})}{\partial {\textbf{X}}} = \begin{bmatrix} {0}_{3x3} &{} {I}_{3x3} \\ {A}_{x_{21}} &{} {A}_{x_{22}} \end{bmatrix} \end{aligned}$$

(A2)

$$\begin{aligned} {A}_{x_{22}} = \begin{bmatrix} 0 &{} 2 &{} 0 \\ -2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{bmatrix} \end{aligned}$$

(A3)

$$\begin{aligned} {\textbf {A}}_{x_{21}} = \begin{bmatrix} A_{4x}&{} A_{4y} &{}A_{4z}\\ A_{5x}&{} A_{5y} &{}A_{5z}\\ A_{6x}&{} A_{6y} &{}A_{6z} \end{bmatrix} \end{aligned}$$

(A4)

$$\begin{aligned} \begin{bmatrix} A_{4x} \\ \\ A_{4y} \\ \\ A_{4z}\\ \\ A_{5x} \\ \\ A_{5y}\\ \\ A_{5z}\\ \\ A_{6x} \\ \\ A_{6y} \\ \\ A_{6z} \end{bmatrix} =\begin{bmatrix} \frac{3(1-{\mu })(x-{x_{1}})^2}{\rho _{1}^{5}}-\frac{1-{\mu }}{\rho _{1}^{3}}-\frac{{\mu }}{\rho _{2}^{3}}+\frac{3 {\mu }(x-{x_{2}})^2}{\rho _{2}^{5}}+1\\ \\ \frac{3 (1-{\mu }) y (x-{x_{1}})}{\rho _{1}^{5}}+\frac{3 {\mu } y (x-{x_{2}})}{\rho _{2}^{5}}\\ \\ \frac{3 (1-{\mu }) z (x-{x_{1}})}{\rho _{1}^{5}}+\frac{3 {\mu } z (x-{x_{2}})}{\rho _{2}^{5}} \\ \\ A_{4y}\\ \\ y \left( \frac{3 (1-{\mu }) y}{\rho _{1}^{5}}+\frac{3 {\mu } y}{\rho _{2}^{5}}\right) -\frac{1-{\mu }}{\rho _{1}^{3}}-\frac{{\mu }}{\rho _{2}^{3}}+1 \\ \\ y \left( \frac{3 (1-{\mu }) z}{\rho _{1}^{5}}+\frac{3 {\mu } z}{\rho _{2}^{5}}\right) \\ \\ A_{4z}\\ \\ A_{5z}\\ \\ z \left( \frac{3 (1-{\mu }) z}{\rho _{1}^5}+\frac{3 {\mu } z}{\rho _{2}^5}\right) -\frac{1-{\mu }}{\rho _{1}^3}-\frac{{\mu }}{\rho _{2}^3} \end{bmatrix} \end{aligned}$$

(A5)

Appendix B: Stability Maps for Three Orbit Periods

From an automation perspective, maintaining consistent maneuver locations across multiple orbit revolutions is desirable. Given the periodic nature of orbits in the CR3BP, it is expected that the stability map patterns repeat. This is demonstrated by regenerating the stability maps in Figs. 5 and 6 for three orbit revolutions, as illustrated in Figs. 11 and 12 for the NRHO and $L_2$ Lyapunov, respectively. Similarly, Figs. 7 and 8 exhibit the same behavior, as evident in Figs. 13 and 14 for the NRHO and $L_2$ Lyapunov, respectively.

Fig. 11

Maximum deformation using ${\tilde{{\textbf{C}}}}$ for NRHO (3 revolutions)

Fig. 12

Maximum deformation using ${\tilde{{\textbf{C}}}}$ for Lyapunov (3 revolutions)

Fig. 13

Maximum deformation using ${\tilde{{\textbf{C}}}}_{x0}$ for NRHO (3 revolutions)

Fig. 14

Maximum deformation using ${\tilde{{\textbf{C}}}}_{x0}$ for Lyapunov (3 revolutions)