Elliptical shape-based model for multi-revolution planeto-centric mission scenarios

Abstract

The paper presents a novel 3-dimensional shape-based algorithm which extends the domain of analytical solutions to planeto-centric mission scenarios, which classically entail even thousands revolutions to transfer to the final orbit. Thanks to the strong physical meaning the proposed method keeps while shaping the trajectory, the method succeeds in outputting a solution close to the real optimum. The proposed approach allows to easily formalize practical mission constraints, such as maximum thrust threshold and eclipses; free and fixed time of flight is manageable as well. The approach is almost completely analytic, which is beneficial as it significantly decreases the computational load. It is well suited for complex mission scenarios and for fast detection near optimal solutions to support the whole mission design.

A List of derivatives

Appendix reports the equations fundamental to the geometrical interpolation of the trajectory. The subscript ’1’ indicates the departure orbit, while the subscript ’2’ indicates the arrival orbit.

1.1 A.1 Departure orbit

The inclination of the initial orbit with respect to the reference plane can be computed as in Eq. 39.

$$\begin{aligned} \left\{ \begin{array}{lll} \cos {\alpha _{1}} = \hat{\mathbf{h }}_{1} \cdot \hat{\mathbf{h }}_{REF}\\ \sin {\alpha _{1}} = \xi _{1} \root \of {1-\cos {\alpha _{1}}^2}\\ \left\{ \begin{array}{ll} \xi _{1} = 1 \;\;\;\;\;\;\;\;\;\; if \;\;\;\; \mathbf{v} _{i} \cdot \hat{\mathbf{h }}_{REF} > 0 \\ \xi _{1} = -1 \;\;\;\;\;\;\;\;\;\; if \;\;\;\; \mathbf{v} _{i} \cdot \hat{\mathbf{h }}_{REF} < 0 \\ \end{array} \right. \end{array} \right. \end{aligned}$$

(39)

The declination ($\delta (x)_1$) of the initial orbit over the reference plane can be computed using Eq. 40, while its derivatives can be computed using Eqs. 41,  42 and  43:

$$\begin{aligned} \sin {\delta _{1}}= & {} \sin {\alpha _{1}}\frac{\sin (\psi x)}{\sin {\beta _{1}}} \end{aligned}$$

(40)

$$\begin{aligned} \delta _{1}'= & {} \frac{\psi \sin {\alpha _{1}} \cos {(\psi x)} - \beta _{1}'\cos {\beta _{1}}\sin {\delta _{1}}}{\cos {\delta _{1}}\sin {\beta _{1}}} \end{aligned}$$

(41)

$$\begin{aligned} \delta _{1}''= & {} \frac{-\psi ^2\sin {\alpha _{1}\sin {(\psi x)}} + \sin {\beta _{1}}\sin {\delta _{1}}\left( \delta _{1}'^{2} + \beta _{1}'^{2} \right) }{\cos {\delta _{1}}\sin {\beta _{1}}}+ \nonumber \\&- \frac{2\delta _{1}'\beta _{1}'\cos {\delta _{1}}\cos {\beta _{1}} + \beta _{1}''\sin {\delta _{1}}\cos {\beta _{1}}}{\cos {\delta _{1}}\sin {\beta _{1}}} \end{aligned}$$

(42)

$$\begin{aligned} \delta _{1}'''= & {} \frac{-\psi ^3\sin {\alpha _{1}\cos {(\psi x)}} + \sin {\beta _{1}}\sin {\delta _{1}} \left( 3\delta _{1}'\delta _{1}'' + 3\beta _{1}'\beta _{1}'' \right) }{\cos {\delta _{1}}\sin {\beta _{1}}} \nonumber \\&+\cos {\beta _{1}}\frac{ -\cos {\delta _{1}} \left( 3\delta _{1}'\delta _{1}'' + 3\beta _{1}'\beta _{1}'' \right) + \sin {\delta _{1}} \left( 3\delta _{1}'^{2}\beta _{1}' + \beta _{1}'^{3}-\beta _{1}''' \right) }{\cos {\delta _{1}}\sin {\beta _{1}}} \nonumber \\&+\frac{\sin {\beta _{1}}\cos {\delta _{1}} \left( 3\beta _{1}'^2\delta _{1}' + \delta _{1}'^3 \right) }{\cos {\delta _{1}}\sin {\beta _{1}}}. \end{aligned}$$

(43)

In the previous equations, another spherical angle ($\beta (x)_1$ in Fig. 2) is introduced together with its derivatives. They can be computed using Eq. 44:

$$\begin{aligned} \left\{ \begin{array}{llll} \beta _{1} = \arccos {\left( \sin {\alpha _{1}} \cos (\psi x)\right) }\\ \beta _{1}' = \psi \sin {\alpha _{1}}\frac{\sin {(\psi x)}}{\sin {\beta _{1}}}\\ \beta _{1}'' = \frac{\psi ^2\sin {\alpha _{1}}\cos {(\psi x)} - \cos {\beta _{1}\beta _{1}'^2}}{\sin {\beta _{1}}}\\ \beta _{1}''' = \frac{-\psi ^3\sin {\alpha _{1}}\sin {(\psi x)} - 3\beta _{1}'\beta _{1}''\cos {\beta _{1}} + \beta _{1}'^3\sin {\beta _{1}}}{\sin {\beta _{1}}}\\ \end{array} \right. \end{aligned}$$

(44)

$\varDelta L_1(x)$ is fundamental to compute the Longitude (6th MEE) on the initial orbit at each x and, as a consequence, the attractor distance from the attractor on the departure orbit as function of x; it can be computed with Eqs. 45 and  46:

$$\begin{aligned}&\left\{ \begin{array}{lll} \sin {\left( \varDelta L_{1}\right) } = \frac{1}{\sin {\alpha _{1}}}\sin {\delta _{1}} \\ \cos {\left( \varDelta L_{1}\right) } = \cos {(\psi x)}\cos {\delta _{1}} \\ \end{array} \right. \end{aligned}$$

(45)

$$\begin{aligned}&\left\{ \begin{array}{lll} \varDelta L_{1}' = \frac{\delta _{1}'}{\sin {\alpha _{1}}\cos {(\psi x)}}\\ \varDelta L_{1}'' = \frac{\delta _{1}'' + \psi \sin {\alpha _{1}}\sin {(\psi x)}\varDelta L_{1}'}{\sin {\alpha _{1}}\cos {(\psi x)}}\\ \varDelta L_{1}''' = \frac{\delta _{1}''' + 2\psi \sin {\alpha _{1}}\sin {(\psi x)}\varDelta L_{1}'' + \psi ^2\sin {\alpha _{1}}\cos {(\psi x)}\varDelta L_{1}'}{\sin {\alpha _{1}}\cos {(\psi x)}}\\ \end{array} \right. \end{aligned}$$

(46)

1.2 A.2 Target orbit

The inclination of the arrival orbit with respect to the reference plane can be computed as in Eq. 47:

$$\begin{aligned} \left\{ \begin{array}{lll} \cos {\alpha _{2}} = \hat{\mathbf{h }}_{2} \cdot \hat{\mathbf{h }}_{REF}\\ \sin {\alpha _{2}} = \xi _{2} \root \of {1-\cos {\alpha _{2}}^2}\\ \begin{array}{ll} \xi _{2} = 1 \;\;\;\;\;\;\;\;\;\; if \;\;\;\; \mathbf{v} _{f} \cdot \hat{\mathbf{h }}_{REF} < 0 \\ \xi _{2} = -1 \;\;\;\;\;\;\;\;\;\; if \;\;\;\; \mathbf{v} _{f} \cdot \hat{\mathbf{h }}_{REF} > 0 \\ \end{array} \end{array} \right. \end{aligned}$$

(47)

The declination ($\delta (x)_2$) of the arrival orbit over the reference plane can be computed using Eq. 48, while its derivatives can be computed using Eqs. 49,  50 and  51:

$$\begin{aligned} \sin {\delta _{2}}= & {} \sin {\alpha _{2}}\frac{\sin \left( \psi \left( 1- x\right) \right) }{\sin {\beta _{2}}} \end{aligned}$$

(48)

$$\begin{aligned} \delta _{2}'= & {} \frac{-\psi \sin {\alpha _{2}} \cos {(\psi (1- x))} - \beta _{2}'\cos {\beta _{2}}\sin {\delta _{2}}}{\cos {\delta _{2}}\sin {\beta _{2}}} \end{aligned}$$

(49)

$$\begin{aligned} \delta _{2}''= & {} \frac{-\psi ^2\sin {\alpha _{2}\sin {(\psi (1- x))}} + \sin {\beta _{2}}\sin {\delta _{2}}\left( \delta _{2}'^{2} + \beta _{2}'^{2} \right) }{\cos {\delta _{2}}\sin {\beta _{2}}} +\nonumber \\&- \frac{2\delta _{2}'\beta _{2}'\cos {\delta _{2}}\cos {\beta _{2}} + \beta _{2}''\sin {\delta _{2}}\cos {\beta _{2}}}{\cos {\delta _{2}}\sin {\beta _{2}}} \end{aligned}$$

(50)

$$\begin{aligned} \delta _{2}'''= & {} \frac{\psi ^3\sin {\alpha _{2}\cos {(\psi (1- x)}} + \sin {\beta _{2}}\sin {\delta _{2}} \left( 3\delta _{2}'\delta _{2}'' + 3\beta _{2}'\beta _{2}'' \right) }{\cos {\delta _{2}}\sin {\beta _{2}}}\nonumber \\&+\cos {\beta _{2}}\frac{\cos {\delta _{2}} \left( 3\delta _{2}'\delta _{2}'' + 3\beta _{2}'\beta _{2}'' \right) +\sin {\delta _{2}} \left( 3\delta _{2}'^{2}\beta _{2}' + \beta _{2}'^{3}-\beta _{2}''' \right) }{\cos {\delta _{2}}\sin {\beta _{2}}}\nonumber \\&+\frac{\sin {\beta _{2}}\cos {\delta _{2}} \left( 3\beta _{2}'^2\delta _{2}' + \delta _{2}'^3 \right) }{\cos {\delta _{2}}\sin {\beta _{2}}} \end{aligned}$$

(51)

In the previous equations, another spherical angle ($\beta (x)_2$ in Fig. 2) is introduced together with its derivatives. They can be computed using Eq. 52:

$$\begin{aligned} \left\{ \begin{array}{llll} \beta _{2} = \arccos {\left( \sin {\alpha _{2}} \cos (\psi (1- x))\right) }\\ \beta _{2}' = -\psi \sin {\alpha _{2}}\frac{\sin {(\psi (1- x))}}{\sin {\beta _{2}}}\\ \beta _{2}'' = \frac{-\psi ^2\sin {\alpha _{2}}\cos {(\psi (1- x))} - \cos {\beta _{2}\beta _{2}'^2}}{\sin {\beta _{2}}}\\ \beta _{2}''' = \frac{\psi ^3\sin {\alpha _{2}}\sin {(\psi (1- x))} - 3\beta _{2}'\beta _{2}''\cos {\beta _{2}} + \beta _{2}'^3\sin {\beta _{2}}}{\sin {\beta _{2}}} \end{array} \right. \end{aligned}$$

(52)

The angle $\varDelta L_2(x)$ is fundamental to compute the Longitude (6th MEE) on the arrival orbit at each x and so the attractor distance of the arrival orbit as function of x; it can be computed with Eqs. 53 and  54:

$$\begin{aligned}&\left\{ \begin{array}{ll} \sin {\left( \varDelta L_{2}\right) } = \frac{1}{\sin {\alpha _{2}}}\sin {\delta _{2}} \\ \cos {\left( \varDelta L_{2}\right) } = \cos {\left( \psi (1- x)\right) }\cos {\delta _{2}} \\ \end{array} \right. \end{aligned}$$

(53)

$$\begin{aligned}&\left\{ \begin{array}{lll} \varDelta L_{2}' = \frac{\delta _{2}'}{\sin {\alpha _{2}}\cos {(\psi (1- x))}}\\ \varDelta L_{2}'' = \frac{\delta _{2}'' - \psi \sin {\alpha _{2}}\sin {(\psi (1- x))}\varDelta L_{2}'}{\sin {\alpha _{2}}\cos {(\psi (1- x))}}\\ \varDelta L_{2}''' = \frac{\delta _{2}''' + 2\psi \sin {\alpha _{2}}\sin {(\psi (1- x))}\varDelta L_{2}'' + \psi ^2\sin {\alpha _{2}}\cos {(\psi (1- x))}\varDelta L_{2}'}{\sin {\alpha _{2}}\cos {(\psi (1- x))}}\\ \end{array} \right. \end{aligned}$$

(54)

1.3 A.3 Attractor distances

To compute the attractor distance, the longitude at each position x on the initial and final orbits using Eqs. 55 and  56 has to be computed first:

$$\begin{aligned}&\left\{ \begin{array}{ll} l_{1}(x) = L_{1} + \varDelta L_{1}(x)\\ l_{1}'(x) = \varDelta L_{1}'(x)\\ \end{array} \right. \end{aligned}$$

(55)

$$\begin{aligned}&\left\{ \begin{array}{ll} l_{2}(x) = L_{2} - \varDelta L_{2}(x)\\ l_{2}'(x) = -\varDelta L_{2}'(x)\\ \end{array} \right. \end{aligned}$$

(56)

The attractor distance on the initial and final orbits can be computed using Eq. 57:

$$\begin{aligned} \left\{ \begin{array}{llll} s_{i}(x) = \frac{p_i}{q_{i}(x)}\\ s_{i}(x)' = -\frac{p_{i}q_{i}'}{q_{i}^2}\\ s_{i}(x)'' = 2\frac{p_{i}q_{i}'^2}{q_{i}^3} - \frac{p_{i}q_{i}''}{q_{i}^2}\\ s_{i}(x)''' = -6\frac{p_{i}q_{i}'^3}{q_{i}^4} +6 \frac{p_{i}q_{i}'q_{i}''}{q_{i}^3} -\frac{p_{i}q_{i}'''}{q_{i}^2}\\ \end{array} \right. \end{aligned}$$

(57)

The q term appears with its derivatives; it can be computed using Eq. 58:

$$\begin{aligned} \left\{ \begin{array}{llll} q_{i}(x) = 1+f_{i}\cos {l_{i}(x)} + g_{i}\sin {l_{i}(x)}\\ q_{i}(x)' = \left( -f_{i}\sin {l_{i}} + g_{i}\cos {l_{i}} \right) \varDelta L_{i}'\\ q_{i}(x)'' = \left( 1-q_{i}\right) \varDelta L_{i}'^2 + q_{i}'\frac{\varDelta L_{i}''}{\varDelta L_{i}'^2}\\ q_{i}(x)''' = -q_{i}'\varDelta L_{i}'^2 + 3\left( 1-q_{i}\right) \varDelta L_{i}'\varDelta L_{i}'' + q_{i}'\frac{\varDelta L_{i}'''}{\varDelta L_{i}'}\\ \end{array} \right. \end{aligned}$$

(58)