The eccentric case of a fast-rotating, gravity-gradient-perturbed satellite attitude solution

Abstract

A closed-form first-order perturbation solution for the attitude evolution of a triaxial space object in an elliptical orbit is presented. The solution, derived using the Lie–Deprit method, takes into account gravity-gradient torque and is facilitated by an assumption of fast rotation of the object. The formulation builds on the earlier implementation of Lara and Ferrer, which assumes a circular orbit. The previously presented work—which assumes spin about an object’s axis of maximum inertia—is further extended by the explicit presentation of the transformations required to apply the solution to an object spinning about its axis of minimum inertia. Additionally, several numerical analyses are presented to more completely assess the utility of the solution. These studies (1) validate the elliptical solution, (2) assess the impact of varying the small parameter of the perturbation procedure, (3) analyze the assumption of fast rotation, and (4) apply the solution to the common and important scenario of a tumbling rocket body.

Appendices

Appendix A: Spin about axis of minimum inertia

As presented in Lara and Ferrer (2013), the transformation from Andoyer variables to fully reduced action-angle variables (“solution variables”) is only applicable to an object spinning about its axis of maximum inertia. However, rotation about the axis of minimum inertia may be considered by revising the transformation procedure as follows. Transformed variables are denoted by a subscript T.

The problem is reorganized such that the axis of minimum inertia is aligned most closely with the body z-axis. Under the assumption $A \le B \le C$,

$$\begin{aligned} A_T&= C,&B_T&= B,&C_T&= A \end{aligned}$$

(26)

$$\begin{aligned} \varvec{\omega }_{x,T}&= \varvec{\omega }_z,&\varvec{\omega }_{y,T}&= -\varvec{\omega }_y,&\varvec{\omega }_{z,T}&= \varvec{\omega }_x \end{aligned}$$

(27)

$$\begin{aligned} \varvec{R}_T(1,1:3)&= \varvec{R} (3,1:3),&\varvec{R}_T (2,1:3)&= - \varvec{R}(2,1:3),&\varvec{R}_T (3,1:3)&= \varvec{R}(1,1:3). \end{aligned}$$

(28)

The transformed values are then converted to Andoyer variables (Celletti 2006), and transformation to action-angle variables proceeds identically to the case of spin about the maximum inertia axis. When evaluating the analytical solution, the transformed moments of inertia must be used. Further, it must be remembered that the resulting attitude solutions do not describe the orientation of the original body-fixed frame with respect to the inertial frame. For example, once the result is calculated in (transformed) action-angle variables and converted to a (transformed) angular velocity vector and rotation matrix, Eqs. (26)–(28) must be applied to obtain the $\varvec{\omega }$ and $\varvec{R}$ of the original body-fixed frame with respect to the original inertial frame.

Appendix B: Original Lara–Ferrer equations for a circular orbit

In Sect. 2, the equations of the perturbation solution for the elliptical orbit case that are the same or only slightly modified from the original circular case are omitted in the interest of reducing clutter in the presentation of the solution. For completeness, however, the relevant equations are given in this appendix. The corresponding equations in the text of the paper in which the solution for a circular orbit is described (Lara and Ferrer 2013) are referenced when appropriate; equations from Lara and Ferrer (2013) are denoted as “L–F Eq. (X),” where X is the equation number in Lara and Ferrer (2013). It is emphasized that these equations are the original equations, applicable to circular orbits only. The revisions described in Sect. 2 of the present paper must be performed in order to achieve validity for eccentric orbits.

To simplify the form of the solution, Lara and Ferrer (2013) define the auxiliary variables f (L–F Eq. (11)), m (L–F Eq. (12)), and $\Delta$ (L–F Eq. (5)) as

$$\begin{aligned} f&= \frac{C \left( B-A \right) }{ \left( C - B \right) A} \end{aligned}$$

(29)

$$\begin{aligned} m&= \frac{\left( C - \Delta \right) \left( B - A \right) }{ \left( C - B \right) \left( \Delta - A \right) } \end{aligned}$$

(30)

$$\begin{aligned} \Delta&= \frac{G^2}{2 \Phi }. \end{aligned}$$

(31)

L–F Eq. (18) gives the torque-free Hamiltonian in solution variables:

$$\begin{aligned} \Phi&= \frac{G^2}{2 A} \left( 1 - \frac{C - A}{C} \frac{f}{f + m} \right) . \end{aligned}$$

(32)

L–F Eqs. (27) and (30) give variable relationships necessary for performing calculations:

$$\begin{aligned} L \frac{\partial m}{\partial L}&= - \frac{L}{G} \frac{\pi }{K(m)} \frac{\left( f + m \right) ^{3/2}}{\sqrt{f \left( 1 + f \right) }} \end{aligned}$$

(33)

$$\begin{aligned} \frac{L}{G}&= \frac{2}{\pi } \sqrt{1 + f} \sqrt{\frac{f}{f+m}} \left[ \Pi \left( -f | m \right) - \frac{m}{f+m} K (m) \right] . \end{aligned}$$

(34)

L–F Eq. (49) gives the Hamiltonian for gravity-gradient-perturbed rotation, simplified by the assumption of fast rotation, in Andoyer variables:

$$\begin{aligned} \mathcal {H}&= \frac{M^2}{2 C} \left\{ \left( \frac{ \sin ^2 \nu }{A / C} + \frac{\cos ^2 \nu }{B / C} \right) s_J^2 + c_J^2 + \frac{1}{4} \left( \frac{n}{M/C} \right) ^2 \right. \nonumber \\&\quad \cdot \left. \left( 1 - 3 s_I^2 \sin ^2 \phi \right) \left[ \left( 2 - \frac{B}{C} - \frac{A}{C} \right) \left( 1 - 3 c_J^2 \right) \right. \right. \nonumber \\&\left. \left. -\, \left( \frac{B}{C} - \frac{A}{C} \right) \left( 3 -3 c_J^2 \right) \cos \left( 2 \nu \right) \right] \right\} \end{aligned}$$

(35)

$$\begin{aligned} \phi&\triangleq h - \theta . \end{aligned}$$

(36)

Meanwhile, the disturbing potential in solution variables is given by L–F Eq. (59), with u defined in L–F Eq. (76):

$$\begin{aligned} U&= \frac{n^2}{8} \left( 1 - 3 s_I^2 \sin ^2 \phi \right) \left\{ \left( 2 C - B - A \right) \right. \nonumber \\&\cdot \left. \left[ 1 - 3 \frac{f}{f+m} \mathrm {dn}^2 \left( u | m \right) \right] + 3 \left( B - A \right) \right. \nonumber \\&\cdot \left. \left[ 1 - \frac{f}{f+m} \mathrm {dn}^2 \left( u | m \right) \right] \left[ 1 - 2 \frac{\left( 1+f \right) \mathrm {sn}^2 \left( u | m \right) }{1 + f \mathrm {sn}^2 \left( u | m \right) } \right] \right\} \end{aligned}$$

(37)

$$\begin{aligned} u&= \frac{2 K (m)}{\pi } l. \end{aligned}$$

(38)

The full fast-rotating Hamiltonian in solution variables is given by L–F Eq. (60):

$$\begin{aligned} \mathcal {H}&= \Phi - n H + U, \end{aligned}$$

(39)

where $\Phi$ is given by Eq. (32), n is the orbital mean motion, H is the conjugate momentum of h, and U is given by Eq. (37).

The $H_{0,2}$ contribution to the averaged Hamiltonian of the first Lie–Deprit transformation is given by L–F Eq. (65), with the auxiliary variable $\kappa$ defined in L–F Eq. (66):

$$\begin{aligned} \mathcal {H}_{0,2}&= \frac{n^2}{4} \kappa \left( 1 - 3 s_I^2 \sin ^2 \phi \right) \end{aligned}$$

(40)

$$\begin{aligned} \kappa&= \left( B - A \right) \left\{ \frac{C-A}{B-A} + 1 - 3 \frac{1+f}{m+f} \left[ 1 + \frac{C-B}{B} \frac{E(m)}{K(m)} \right] \right\} . \end{aligned}$$

(41)

The $W_2$ term of the generating function of the first Lie–Deprit transformation is given by L–F Eq. (67). Calculations are facilitated by L–F Eqs. (68) and (35), which give $Z \left( \Psi | m \right)$ and $\Psi$, respectively:

$$\begin{aligned} W_2&= -\frac{3}{2} \frac{n^2}{G} \left( C - B \right) A \sqrt{ f \frac{1+f}{f+m}} Z \left( \Psi | m \right) \left( 1 - 3 s_I^2 \sin ^2 \phi \right) \end{aligned}$$

(42)

$$\begin{aligned} Z \left( \Psi | m \right)&= E \left( \Psi | m \right) - \frac{E(m)}{K(m)} F \left( \Psi | m \right) \end{aligned}$$

(43)

$$\begin{aligned} \Psi&= \mathrm {am} \left( -\frac{2 K (m)}{\pi } l | m \right) . \end{aligned}$$

(44)

L–F Eqs. (77)–(82) give the variable transformations for the first Lie–Deprit transformation, which averages over the angle l:

$$\begin{aligned} \Delta l&=\frac{3}{4} \frac{n^2}{G^2} \left( C - B \right) A \frac{\pi }{2 K(m)} \nonumber \\&\quad \cdot \left[ Z \left( \Psi | m \right) - 2 \left( f+m \right) \frac{\mathrm {d} Z \left( \Psi | m \right) }{\mathrm {d} m} \right] \left( 1 - 3 s_I^2 \sin ^2 \phi \right) \end{aligned}$$

(45)

$$\begin{aligned} \Delta h&= \frac{3}{4} \frac{n^2}{G^2} \left( C - B \right) A \sqrt{f \frac{1+f}{m+f}} Z \left( \Psi | m \right) 6 c_I \sin ^2 \phi \end{aligned}$$

(46)

$$\begin{aligned} \Delta g&= - \frac{3}{4} \frac{n^2}{G^2} \left( C - B \right) A \sqrt{f \frac{1+f}{m+f}} Z \left( \Psi | m \right) \nonumber \\&\quad \cdot \left( 1 - 3 s_I^2 \sin ^2 \phi \right) - \frac{L}{G} \Delta l - \frac{H}{G} \Delta h \end{aligned}$$

(47)

$$\begin{aligned} \Delta L&= \frac{3}{4} \frac{n^2}{G^2} \left( C - B \right) \cdot \nonumber \\&\quad A \sqrt{f \frac{1+f}{m+f}} \frac{2 K(m)}{\pi } \left[ \frac{E(m)}{K(m)} - \mathrm {dn}^2 \left( u | m \right) \right] \left( 1 - 3 s_I^2 \sin ^2 \phi \right) \end{aligned}$$

(48)

$$\begin{aligned} \Delta G&= 0 \end{aligned}$$

(49)

$$\begin{aligned} \Delta H&= \frac{3}{4} \frac{n^2}{G^2} \left( C - B \right) A \sqrt{f \frac{1+f}{m+f}} Z \left( \Psi | m \right) 3 s_I^2 \sin \left( 2 \phi \right) , \end{aligned}$$

(50)

where (L–F Eq. (75))

$$\begin{aligned} \frac{\mathrm {d} Z \left( \Psi | m \right) }{\mathrm {d} m}&= \frac{\mathrm {cn} \left( u | m \right) }{2 \left( 1 - m \right) }. \end{aligned}$$

(51)

Appendix C: Transformations between Andoyer and action-angle variables

In this appendix, the transformations between Andoyer variables ($\lambda$, $\mu$, $\nu$, $\Lambda$, M, N) and the action-angle variables or Lara–Ferrer solution variables (l, g, h, $\tau$, L, G, H, T) are described (Ferrer and Lara 2010; Lara and Ferrer 2013). The transformation from Andoyer variables to solution variables is accomplished via

$$\begin{aligned} l&= -\frac{\pi }{2 \mathrm {K} (m)} \mathrm {F} \left( \Psi | m \right) \end{aligned}$$

(52)

$$\begin{aligned} g&= \mu + \sqrt{1 + f} \sqrt{\frac{f + m}{f}} \left[ \frac{\Pi (-f | m)}{\mathrm {K} (m)} \mathrm {F} (\Psi | m) - \Pi (-f, \Psi | m) \right] \end{aligned}$$

(53)

$$\begin{aligned} h&= \lambda \end{aligned}$$

(54)

$$\begin{aligned} L&= \frac{2 M}{\pi } \sqrt{1 + f} \sqrt{\frac{f + m}{f}} \left[ \Pi (-f | m) - \frac{m}{f+m} \mathrm {K} (m) \right] \end{aligned}$$

(55)

$$\begin{aligned} G&= M \end{aligned}$$

(56)

$$\begin{aligned} H&= \Lambda , \end{aligned}$$

(57)

where $\Psi$ is calculated using

$$\begin{aligned} \cos \Psi&= \frac{\sqrt{1 + f} \sin \nu }{\sqrt{1 + f \sin ^2 \nu }} \end{aligned}$$

(58)

$$\begin{aligned} \sin \Psi&= \frac{\cos \nu }{\sqrt{1 + f \sin ^2 \nu }}. \end{aligned}$$

(59)

The transformation from solution variables to Andoyer variables is performed using

$$\begin{aligned} \lambda&= h \end{aligned}$$

(60)

$$\begin{aligned} \mu&= g - \sqrt{1+f} \sqrt{\frac{f+m}{f}} \left[ \frac{\Pi (-f | m)}{\mathrm {K} (m)} \mathrm {F} (\Psi | m) - \Pi (-f, \Psi | m) \right] \end{aligned}$$

(61)

$$\begin{aligned} \cos \nu&= \frac{\sqrt{1+f} \sin \Psi }{\sqrt{1+f\sin ^2 \Psi }} \end{aligned}$$

(62)

$$\begin{aligned} \sin \nu&= \frac{\cos \Psi }{\sqrt{1+f\sin ^2 \Psi }} \end{aligned}$$

(63)

$$\begin{aligned} \Lambda&= H \end{aligned}$$

(64)

$$\begin{aligned} M&= G \end{aligned}$$

(65)

$$\begin{aligned} N&= G \sqrt{\frac{f}{f+m}} \sqrt{1 - m \sin ^2 \Psi }, \end{aligned}$$

(66)

where $\Psi$ is given by Eq. (44).

It is noted that no transformations are required for $\tau$ and T because these variables are associated with the use of the extended phase space rather than the rotational state of the body.