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Software-in-the-Loop Validation of a Novel Two-Point Optimal Guidance for Perturbed Spacecraft Rendezvous and Formations

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Abstract

A novel guidance law that utilizes optimal control theory in combination with linear time-varying state-transition matrices to calculate the optimal path for a chaser spacecraft from its initial point to a desired final point and a desired amount of time is proposed in this paper. The optimal guidance law is formulated using more accurate second-order integral form of Gauss Variational Equations to map control accelerations to the dynamics in terms of relative orbital elements. When comparing the second-order form against the traditional Gauss Variational Equations, the guidance law results in a 15% reduction in control demand. In addition, this paper presents a new method that allows for the closed-loop testing of guidance, navigation and control systems with realistic software models of GPS receivers, RADAR systems for on-board relative navigation, and ground-station up-links of the target states. Specifically, utilizing the signal-to-noise ratio of various sensors, the errors that arise due to hardware noise and/or environmental factors can be modeled accurately based on the communication link. The guidance law is tested using the newly developed software-in-the-loop test-bed for a far-range formation flight scenario around the non-cooperative Alouette-2 spacecraft as well as on an arbitrary defined highly elliptical orbit.

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Notes

  1. https://news.northropgrumman.com/news/releases.

  2. https://carleton.ca/spacecraft/about/.

  3. http://ssl.mit.edu/.

  4. https://nps.edu/web/srl/facilities.

  5. https://www.asc-csa.gc.ca/eng/sciences/parabolic.asp.

  6. https://www.dlr.de/rb/en/desktopdefault.aspx/tabid-2711/.

  7. https://celestrak.com/columns/v04n05/#FAQ07.

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  1. Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Yazan Chihabi & Steve Ulrich

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  1. Yazan Chihabi
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Appendix

Appendix

$$\begin{aligned} {G}^1_{11}&= \frac{a^2}{\mu }+\frac{4\,a^3\,{\left( q_{2}\,\cos \left( u\right) -q_{1}\,\sin \left( u\right) \right) }^2}{h^2} \end{aligned}$$
(69)
$$\begin{aligned} {G}^1_{12}&= -\frac{4\,a^3\,\left( q_{2}\,\cos \left( u\right) -q_{1}\,\sin \left( u\right) \right) \,\left( q_{1}\,\cos \left( u\right) +q_{2}\,\sin \left( u\right) +1\right) }{h^2} \end{aligned}$$
(70)
$$\begin{aligned} {G}^1_{21}&= {G}^1_{12} \end{aligned}$$
(71)
$$\begin{aligned} {G}^1_{22}&= \frac{a^2}{\mu }+\frac{4\,a^3\,{\left( q_{1}\,\cos \left( u\right) +q_{2}\,\sin \left( u\right) +1\right) }^2}{h^2} \end{aligned}$$
(72)
$$\begin{aligned} {G}^1_{33}&= \frac{a^2}{\mu } \end{aligned}$$
(73)
$$\begin{aligned} {G}^2_{11}&= \frac{a\,\eta ^2\,{\cos \left( \mathrm {\theta }\right) }^2}{2\,e\,\mu } \end{aligned}$$
(74)
$$\begin{aligned} {G}^2_{12}&= \frac{r\,\left( e\,{\sin \left( \mathrm {\theta }\right) }^3-2\,\cos \left( \mathrm {\theta }\right) \,\sin \left( \mathrm {\theta }\right) \right) }{2\,e\,\mu } \end{aligned}$$
(75)
$$\begin{aligned} {G}^2_{21}&= {G}^1_{12} \end{aligned}$$
(76)
$$\begin{aligned} {G}^2_{22}&= \frac{r}{2\,e\,\mu \,\left( e\,\cos \left( \mathrm {\theta }\right) +1\right) }\,\left( {\cos \left( \mathrm {\theta }\right) }^2\,\left( 3\,e^2-4\right) -e^2\,{\cos \left( \mathrm {\theta }\right) }^4\right. \nonumber \\&\left. +6\,e\,\cos \left( \mathrm {\theta }\right) -4\,e\,{\cos \left( \mathrm {\theta }\right) }^3+4\right) \end{aligned}$$
(77)
$$\begin{aligned} {G}^2_{33}&= \frac{r\,\left( 2\,\cos \left( \mathrm {\theta }\right) +e\,\left( {\cos \left( \mathrm {\theta }\right) }^2+1\right) \right) }{2\,\mu \,\left( e\,\cos \left( \mathrm {\theta }\right) +1\right) } \end{aligned}$$
(78)
$$\begin{aligned} {G}^3_{23}&= -\frac{a\,\eta ^2\,\cos \left( u\right) }{2\,\mu \,{\left( q_{1}\,\cos \left( u\right) +q_{2}\,\sin \left( u\right) +1\right) }^2} \end{aligned}$$
(79)
$$\begin{aligned} {G}^3_{32}&= {G}^3_{23} \end{aligned}$$
(80)
$$\begin{aligned} {G}^3_{33}&= \frac{a\,\eta ^2\,{\sin \left( u\right) }^2}{2\,\mu \,\tan \left( i\right) \,{\left( q_{1}\,\cos \left( u\right) +q_{2}\,\sin \left( u\right) +1\right) }^2} \end{aligned}$$
(81)
$$\begin{aligned} {G}^4_{11}&= \frac{a\,\eta ^2\,\cos \left( \mathrm {\theta }\right) \,\sin \left( \mathrm {\theta }\right) }{e^2\,\mu } \end{aligned}$$
(82)
$$\begin{aligned} {G}^4_{22}&= \frac{a\,\eta ^2\,\sin \left( \mathrm {\theta }\right) }{e^2\,\mu }\,\left( \cos \left( \mathrm {\theta }\right) +\frac{e+\cos \left( \mathrm {\theta }\right) }{{\left( e\,\cos \left( \mathrm {\theta }\right) +1\right) }^2}+\frac{2\,\cos \left( \mathrm {\theta }\right) }{e\,\cos \left( \mathrm {\theta }\right) +1}\right) \end{aligned}$$
(83)
$$\begin{aligned} {G}^4_{33}&= \left\{ \begin{array}{ll} \frac{r}{\mu \,\left( e\,\cos \left( \mathrm {\theta }\right) +1\right) }\,\left( \frac{\sin \left( \mathrm {\theta }\right) }{e}+\frac{\cos \left( u\right) \,\sin \left( u\right) }{{\tan \left( i\right) }^2}\right) &{} \hbox { if}\ \sin (\omega ) = 0,\\ \frac{r}{2\,\mu \,\left( e\,\cos \left( \mathrm {\theta }\right) +1\right) }\,\left( \frac{2\,\sin \left( \mathrm {\theta }\right) }{e}-\frac{{\sin \left( \mathrm {\theta }\right) }^2}{\tan \left( \omega \right) }\right. &{} \\ \left. +\frac{{\sin \left( u\right) }^2}{\tan \left( \omega \right) }+\frac{2\,\cos \left( u\right) \,\sin \left( u\right) }{{\tan \left( i\right) }^2}\right) &{} \text{ if } \text{ otherwise } \end{array} \right. \end{aligned}$$
(84)
$$\begin{aligned} {G}^4_{12}&= \frac{a\,\eta ^2\,\left( 2\,{\cos \left( \theta \right) }^2-1\right) \,\left( e\,\cos \left( \theta \right) +2\right) }{2\,e^2\,\mu \,\left( e\,\cos \left( \theta \right) +1\right) } \end{aligned}$$
(85)
$$\begin{aligned} {G}^4_{21}&= {G}^4_{12} \end{aligned}$$
(86)
$$\begin{aligned} {G}^4_{23}&= \frac{r\,\sin \left( u\right) }{2\,\mu \,\tan \left( i\right) \,\left( e\,\cos \left( \theta \right) +1\right) } \end{aligned}$$
(87)
$$\begin{aligned} {G}^4_{32}&= {G}^4_{23} \end{aligned}$$
(88)
$$\begin{aligned} {G}^5_{23}&= -\frac{a\,\eta ^2\,\sin \left( u\right) }{2\,\mu \,\sin \left( \textrm{inc}\right) \,{\left( q_{1}\,\cos \left( u\right) +q_{2}\,\sin \left( u\right) +1\right) }^2} \end{aligned}$$
(89)
$$\begin{aligned} {G}^5_{32}&= {G}^3_{23} \end{aligned}$$
(90)
$$\begin{aligned} {G}^5_{33}&= -\frac{a\,\eta ^2\,\sin \left( 2\,u\right) \,\cos \left( \textrm{inc}\right) }{2\,\mu \,{\sin \left( \textrm{inc}\right) }^2\,{\left( q_{1}\,\cos \left( u\right) +q_{2}\,\sin \left( u\right) +1\right) }^2} \end{aligned}$$
(91)
$$\begin{aligned} {G}^6_{11}&= \frac{a\,\eta \,{\sin \left( \theta \right) }^3\,\left( e\,{\cos \left( \theta \right) }^2+3\,\cos \left( \theta \right) +2\,e\right) }{\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^3} \nonumber \\&-\frac{2\,a\,\eta ^5\,\cos \left( \theta \right) \,\sin \left( \theta \right) }{e^2\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^2} \nonumber \\&-\frac{a\,\eta ^3\,{\cos \left( \theta \right) }^2\,\sin \left( \theta \right) \,\left( e\,\cos \left( \theta \right) +2\right) }{e\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^2} \nonumber \\&-\frac{2\,a\,\eta ^3\,\cos \left( \theta \right) \,\sin \left( \theta \right) \,\left( 3\,e+\cos \left( \theta \right) \,\left( e^2+2\right) \right) }{e\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^3} \end{aligned}$$
(92)
$$\begin{aligned} {G}^6_{22}&= \frac{a\,\eta \,\sin \left( \theta \right) \,{\left( \cos \left( \theta \right) +\frac{e+\cos \left( \theta \right) }{e\,\cos \left( \theta \right) +1}\right) }^2\,\left( e\,{\cos \left( \theta \right) }^2+3\,\cos \left( \theta \right) +2\,e\right) }{\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^3} \nonumber \\&+\frac{2\,a\,\eta ^5\,\sin \left( \theta \right) \,\left( \frac{4\,\cos \left( \theta \right) }{e\,\cos \left( \theta \right) +1}+\frac{e\,\left( e\,{\cos \left( \theta \right) }^3+1\right) }{{\left( e\,\cos \left( \theta \right) +1\right) }^2}\right) }{e^2\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^2} \nonumber \\&-\frac{a\,\eta ^3\,\sin \left( \theta \right) \,\left( e\,\cos \left( \theta \right) +2\right) }{e\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^4}\times \nonumber \\&\left( {\cos \left( \theta \right) }^2\,\left( 3\,e^2-4\right) -e^2\,{\cos \left( \theta \right) }^4+6\,e\,\cos \left( \theta \right) -4\,e\,{\cos \left( \theta \right) }^3+4\right) \nonumber \\&+\frac{2\,a\,\eta ^3\,\sin \left( \theta \right) \,\left( e\,\cos \left( \theta \right) +2\right) }{e\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^4}\times \nonumber \\&\left( 3\,e+\cos \left( \theta \right) \,\left( e^2+2\right) \right) \,\left( \cos \left( \theta \right) +\frac{e+\cos \left( \theta \right) }{e\,\cos \left( \theta \right) +1}\right) \end{aligned}$$
(93)
$$\begin{aligned} {G}^6_{33}&= -\frac{\eta \,r\,\sin \left( \theta \right) \,\left( 2\,\cos \left( \theta \right) +e\,\left( {\cos \left( \theta \right) }^2+1\right) \right) \,\left( e\,\cos \left( \theta \right) +2\right) }{\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^3} \nonumber \\&-\frac{\eta ^3\,r\,\sin \left( \theta \right) \,\left( e\,\cos \left( \theta \right) +2\right) }{e\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^3} \end{aligned}$$
(94)
$$\begin{aligned} {G}^6_{12}&= \frac{2\,r\,{\sin \left( \theta \right) }^2\,\left( \cos \left( \theta \right) +\frac{e+\cos \left( \theta \right) }{e\,\cos \left( \theta \right) +1}\right) \,\left( e\,{\cos \left( \theta \right) }^2+3\,\cos \left( \theta \right) +2\,e\right) }{\eta \,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^2} \nonumber \\&-\frac{2\,\eta ^3\,r\,\left( 2\,{\cos \left( \theta \right) }^2-1\right) \,\left( e\,\cos \left( \theta \right) +2\right) }{e^2\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^2} \nonumber \\&-\frac{2\,\eta \,r\,\left( 3\,e+\cos \left( \theta \right) \,\left( e^2+2\right) \right) \,\left( \cos \left( 2\,\theta \right) \,\left( e\,\cos \left( \theta \right) +2\right) +e\,\cos \left( \theta \right) \right) }{e\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^3} \nonumber \\&+\frac{2\,\eta \,r\,{\sin \left( \theta \right) }^2\,\left( 2\,\cos \left( \theta \right) -e\,{\sin \left( \theta \right) }^2\right) \,\left( e\,\cos \left( \theta \right) +2\right) }{e\,\mu \,{\left( e\,\cos \left( \theta \right) +1\right) }^2} \end{aligned}$$
(95)
$$\begin{aligned} {G}^6_{21}&= {G}^6_{12} \end{aligned}$$
(96)

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Chihabi, Y., Ulrich, S. Software-in-the-Loop Validation of a Novel Two-Point Optimal Guidance for Perturbed Spacecraft Rendezvous and Formations. J Astronaut Sci 70, 43 (2023). https://doi.org/10.1007/s40295-023-00405-8

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