Abstract
We derive the dynamic equations of attitude, orbit and flexible modes of a generic satellite. We will follow the Lagrangian approach, and the equations will be set in a form specifically suitable for attitude and orbit control. After the derivation of the flexible body Lagrangian equations we treat some specific complements that are useful for space applications: momentum management systems , generalized forces due a central gravitational field , first invariants of motion in a central gravitational field, transformations to uncouple orbital and attitude equations and methods to linearize the equations around a reference orbital and attitude trajectory. We close the chapter by presenting a typical linear model of a flexible satellite as used in practical linear control engineering design.
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Notes
- 1.
The velocity of the origin in \(\mathbf {BRF}\) components has been designated with v, in order to get the inertial acceleration in \(\mathbf {BRF}\): \( a=M^T (\frac{d}{dt} M v) \), we need to bring v in \(\mathbf {ECI}\) components, derive each component and then bring them again in \(\mathbf {BRF}\). This sequence of operations brings to apply the operator \( \dot{v}+[\omega , v]\) to the initial vector v, this can be easily derived from Eq. 2.3 given in Sect. 2.3. The same rule applies to any vector in \(\mathbf {BRF}\) when we have to derive its inertial components.
- 2.
We define first integrals of motion or invariants of a system of differential equations functions of the state variables that are constant on any trajectory of motion. These first integrals can help to find closed form solutions. The first integrals are also called invariants.
- 3.
We do not treat here the effects due to the elasticity of the rotor and of the satellite.
- 4.
The expression of the generalized Lagrangian forces \(Q^\varepsilon _{k}\) acting on the satellite are null because we have assumed that the momentum management system is mounted on a hard point. Actually at frequencies \(\gg \) \(10\) Hz where the disturbances due to the rotor unbalances are generated, the satellite structure cannot be considered rigid and their effects are normally evaluated in the frame of the satellite’s structural dynamics.
- 5.
We remember that \(C_{I0}\) from Eq. 3.70 is the part of the interface torque necessary to transmit the satellite motion to the rotor.
- 6.
We use here the product rule: \(a^T [b,c]=b^T [c, a]\).
- 7.
ECEF is similar to \(\mathbf {ECI}\) presented in Chap. 2, but it is fixed with the Earth. The Z axis is aligned with the \(\mathbf {ECI}\) Z axis, orthogonal to the equatorial plane, the X axis points to the Greenwich meridian, the Y axis completes the triad. Therefore, omitting Earth precession and nutation corrections, the transformation from \(\mathbf {ECI}\) to \(\mathbf {ECEF}\) is accomplished with a simple rotation in the X/Y plane.
- 8.
In Eq. A2.1 the Levi-Civita operator is here replaced by the skew matrix. In fact the application of the Levi-Civita operator on a vector A produces a matrix whose ij-th element is \( skew(A)_{ij} = e_{ikj} A_{k}\).
- 9.
We use here the double vector product rule: \([a,[b,c]]=b (c^T a) - c (a^T b)\).
- 10.
From a mathematical point of view a generic damping matrix precludes the existence of a real modal matrix. A sufficient condition to have a real modal matrix requires a damping matrix proportional to the stiffness matrix . We will assume this hypothesis from now in order not to use complex modes. It is usual engineering practice to specify directly the damping factor for each free mode and to create the modal matrix from the cantilevered mode description without damping in order to avoid complex modes.
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Mazzini, L. (2016). The Dynamics of the Flexible Satellite. In: Flexible Spacecraft Dynamics, Control and Guidance. Springer Aerospace Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25540-8_3
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DOI: https://doi.org/10.1007/978-3-319-25540-8_3
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