Absolute orbit determination using line-of-sight vector measurements between formation flying spacecraft

Abstract

The purpose of this paper is to show that absolute orbit determination can be achieved based on spacecraft formation. The relative position vectors expressed in the inertial frame are used as measurements. In this scheme, the optical camera is applied to measure the relative line-of-sight (LOS) angles, i.e., the azimuth and elevation. The LIDAR (Light radio Detecting And Ranging) or radar is used to measure the range and we assume that high-accuracy inertial attitude is available. When more deputies are included in the formation, the formation configuration is optimized from the perspective of the Fisher information theory. Considering the limitation on the field of view (FOV) of cameras, the visibility of spacecraft and the installation of cameras are investigated. In simulations, an extended Kalman filter (EKF) is used to estimate the position and velocity. The results show that the navigation accuracy can be enhanced by using more deputies and the installation of cameras significantly affects the navigation performance.

Appendices

Appendix 1

This appendix gives the calculation of gravity gradient tensors that take into account the $J_{2}$ perturbation. The tensor is defined as

$$ {{\boldsymbol{G}}_{i}} = \frac{{\partial {{{\dot{\boldsymbol{v}}}}_{i}}}}{{\partial {\boldsymbol{r_{i}}}}} = \left [ { \textstyle\begin{array}{c@{\quad }c@{\quad }c} {{g_{11}}}&{{g_{12}}}&{{g_{13}}} \\ {{g_{21}}}&{{g_{22}}}&{{g_{23}}} \\ {{g_{31}}}&{{g_{32}}}&{{g_{33}}} \end{array}\displaystyle } \right ] . $$

(1)

Each element of the tensor is given as

$$ \begin{aligned} {g_{11}} &= \frac{{\mu 3\mu ( {4{x^{4}} - {y^{4}} + 3{y^{2}}{z ^{2}} + 4{z^{2}} + 3{x^{2}} ( {{y^{2}} - 9{z^{2}}} ) } ) {J_{2}}R_{e}^{2}}}{{2{r^{9}}}} \\ &\quad {} + \frac{{\mu ( {3{x^{2}} - {r^{2}}} ) }}{ {{r^{5}}}}, \\ {g_{12}} &= \frac{{3xy\mu ( {2{r^{4}} + 5 ( {{r^{2}} - 7{z ^{2}}} ) {J_{2}}R_{e}^{2}} ) }}{{2{r^{9}}}}, \\ {g_{13}} &= \frac{{3xz\mu ( {2{r^{4}} + 5 ( {3{r^{2}} - 7 {z^{2}}} ) {J_{2}}R_{e}^{2}} ) }}{{2{r^{9}}}}, \\ {g_{22}} &= - \frac{{3\mu ( {{x^{4}} - 4{y^{4}} + 27{y^{2}}{z^{2}} - 4{z^{4}} - 3 ( {{r^{2}} - {x^{2}}} ) } ) {J_{2}}R_{e} ^{2}}}{{2{r^{9}}}} \\ &\quad {} + \frac{{\mu ( {{r^{2}} - 3{y^{2}}} ) }}{ {{r^{5}}}}, \\ {g_{23}} &= \frac{{3yz\mu ( {2{r^{4}} + 5 ( {3{r^{2}} - 7 {z^{2}}} ) {J_{2}}R_{e}^{2}} ) }}{{2{r^{9}}}}, \\ {g_{33}} &= - \frac{{15\mu ( {3{x^{4}} + 3{y^{4}} - 24{y^{2}} {z^{2}} + 8{z^{4}} + 6{x^{2}} ( {{y^{2}} - 4{z^{2}}} ) } ) {J_{2}}R_{e}^{2}}}{{2{r^{9}}}} \\ &\quad {} - \frac{{\mu ( {{r^{2}} - 3{z^{2}}} ) }}{ {{r^{5}}}}, \\ {g_{21}} &= {g_{12}}, \\ {g_{31}} &= {g_{13}}, \\ {g_{32}} & = {g_{23}} . \end{aligned} $$

(2)

Appendix 2

$$\begin{aligned} {{\boldsymbol{Q}}_{k}} = \int _{{t_{k}}}^{{t_{k + 1}}} {\boldsymbol{\varPhi }} ( {{t_{k + 1}},\tau } ) {\boldsymbol{Q}} ( \tau ) {\boldsymbol{\varPhi }} { ( {{t_{k + 1}},\tau } ) ^{T}}d\tau = \int _{{t_{k}}}^{{t_{k + 1}}} {\left [ { \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {{\boldsymbol{\varPhi }}_{1rv}^{2}{q^{2}}}&{{{\boldsymbol{\varPhi }}_{1rv}}{{\boldsymbol{\varPhi }} _{1vv}}{q^{2}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}} \\ {{{\boldsymbol{\varPhi }}_{1vv}}{{\boldsymbol{\varPhi }}_{1rv}}{q^{2}}}&{{\boldsymbol{\varPhi }}_{1vv} ^{2}{q^{2}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}} \\ {{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{\boldsymbol{\varPhi }}_{2rv} ^{2}{q^{2}}}&{{{\boldsymbol{\varPhi }}_{2rv}}{{\boldsymbol{\varPhi }}_{2vv}}{q^{2}}} \\ {{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{0}}_{3 \times 3}}}&{{{\boldsymbol{\varPhi }} _{2vv}}{{\boldsymbol{\varPhi }}_{2rv}}{q^{2}}}&{{\boldsymbol{\varPhi }}_{2vv}^{2}{q^{2}}} \end{array}\displaystyle } \right ] } d\tau . \end{aligned}$$

(3)

The integrations are computed as

$$\begin{aligned} &{ \int _{{t_{k}}}^{{t_{k + 1}}} {{\boldsymbol{\varPhi }}_{\mathit{irr}}^{2} ( {{t_{k + 1}}, \tau } ) {q^{2}} ( \tau ) } d\tau} \\ &{ \quad = {{\boldsymbol{I}}_{1 \times 3}}{{ \cdot {T^{3}}} /3} + ( {{{\boldsymbol{G}}_{i0}} + {{\boldsymbol{G}} _{i}}} ) {{{T^{5}}} /{30}}} \\ &{ \qquad {}+ { ( {{{\boldsymbol{G}}_{i0}} + {{\boldsymbol{G}}_{i}}} ) ^{2}}{{{T^{7}}} /{1008,}}} \end{aligned}$$

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$$\begin{aligned} &{ \int _{{t_{k}}}^{{t_{k + 1}}} {{{\boldsymbol{\varPhi }}_{\mathit{irv}}} ( {{t_{k + 1}}, \tau } ) {{\boldsymbol{\varPhi }}_{\mathit{ivv}}} ( {{t_{k + 1}},\tau } ) {q^{2}} ( \tau ) } d\tau} \\ &{ \quad = {{\boldsymbol{I}}_{1 \times 3}}{{ \cdot {T^{2}}} / 2} + ( {3{{\boldsymbol{G}}_{i0}} + 5{{\boldsymbol{G}} _{i}}} ) {{{T^{4}}} /2}} \\ &{ \qquad {}+ ( {{{\boldsymbol{G}}_{i0}} + {{\boldsymbol{G}}_{i}}} ) ( {{{\boldsymbol{G}} _{i0}} + 2{{\boldsymbol{G}}_{i}}} ) {{T^{6}}} /432,} \end{aligned}$$

(5)

$$\begin{aligned} &{ \int _{{t_{k}}}^{{t_{k + 1}}} {{{\boldsymbol{\varPhi }}_{\mathit{ivv}}} ( {{t_{k + 1}}, \tau } ) {{\boldsymbol{\varPhi }}_{\mathit{irv}}} ( {{t_{k + 1}},\tau } ) {q^{2}} ( \tau ) } d\tau } \\ &{ \quad = {{\boldsymbol{I}}_{1 \times 3}}{{ \cdot {T^{2}}} /2} + ( {3{{\boldsymbol{G}}_{i0}} + 5{{\boldsymbol{G}} _{i}}} ) {{{T^{4}}} /2}} \\ &{ \qquad {} + ( {{{\boldsymbol{G}}_{i0}} + 2{{\boldsymbol{G}}_{i}}} ) ( {{{\boldsymbol{G}} _{i0}} + {{\boldsymbol{G}}_{i}}} ) {{{T^{6}}} / {432}},} \end{aligned}$$

(6)

$$\begin{aligned} &{ \int _{{t_{k}}}^{{t_{k + 1}}} {{\boldsymbol{\varPhi }}_{2vv}^{2} ( {{t_{k + 1}}, \tau } ) {q^{2}} ( \tau ) } d\tau } \\ &{ \quad = {{\boldsymbol{I}}_{1 \times 3}} \cdot T + 2 ( {{{\boldsymbol{G}}_{i0}} + 2{{\boldsymbol{G}}_{i}}} ) {{{T^{3}}} /{18}}} \\ &{ \qquad {} + { ( {{{\boldsymbol{G}}_{i0}} + 2{{\boldsymbol{G}}_{i}}} ) ^{2}}{{{T^{5}}} / {180.}}} \end{aligned}$$

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